Contents
1
Introduction
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1
Structural Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geometrical Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . .
Material Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contact (Boundary) Nonlinearities . . . . . . . . . . . . . . . . . . . . .
1-2
1-2
1-2
1-4
Solution Procedures of Nonlinear Problems . . . . . . . . . . . . . . . . 1-4
Solution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4
Concept of Time Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6
NSTAR: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6
2
Geometrically Nonlinear Analysis
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1
Large Displacement Nonlinear Analyses . . . . . . . . . . . . . . . . . . 2-2
Finite Strain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2
Large Deflection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3
3
Material Models and Constitutive Relations
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1
Linear Elastic Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1
Isotropic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1
Orthotropic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2
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Contents
Laminated Composite and Failure Criterion for Laminated Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3
Laminated Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3
Failure Criterion for Laminated Composite Materials . . . . . . . 3-3
Nonlinear Elastic Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5
Hyperelastic Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8
Mooney-Rivlin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
Ogden Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12
Blatz-Ko Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15
Plasticity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Huber-von Mises Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Drucker-Prager Elastic-Perfectly Plastic Model . . . . . . . . . .
Tresca-Saint Venant Yield Criterion
(or the constant maximum shearing stress condition) . . . . . .
Comparison of Tresca and von Mises Criteria for Plasticity .
3-17
3-17
3-20
Superelastic Models: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nitinol Model (Shape-Memory-Alloy) . . . . . . . . . . . . . . . . .
The Nitinol Model Formulation: . . . . . . . . . . . . . . . . . . . . . .
The Yield Criterion: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Flow Rule: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-24
3-24
3-25
3-27
3-28
3-21
3-23
Creep and Viscoelastic Models . . . . . . . . . . . . . . . . . . . . . . . . 3-30
Creep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-30
Linear Isotropic Viscoelastic Model . . . . . . . . . . . . . . . . . . . 3-32
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Wrinkling Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-35
2
A Bounding Surface Model for Concrete . . . . . . . . . . . . . . . . .
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Damage Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Model Parameters and Feature . . . . . . . . . . . . . . . . . . . .
3-37
3-37
3-38
3-39
User-defined Material Models . . . . . . . . . . . . . . . . . . . . . . . . .
Preparing the NSTAR Executable File . . . . . . . . . . . . . . . . .
Requirements for Windows NT/2000 . . . . . . . . . . . . . . . . . .
Model Definition Procedure . . . . . . . . . . . . . . . . . . . . . . . . .
3-42
3-43
3-43
3-44
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Useful FUNCTION Statements to Access Information from Data Base
3-48
Useful COMMON Statements to Access Information From Data Base
3-50
Element Nodal connectivity Common . . . . . . . . . . . . . . . . . 3-54
Useful Subroutines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-59
User-Defined Creep Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-60
Model Definition Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 3-61
Modifying the CREPUM Subroutine . . . . . . . . . . . . . . . . . . . 3-61
Strain Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-64
Automatic Determination of Material Properties from Test Data3-67
MPCTYPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-67
MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-69
Evaluation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-69
Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-70
Birth and Death of Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 3-82
4
Gap/Contact Problems
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1
Hybrid Technique for Gap/Contact Problems: General Description4-2
Hybrid Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2
Gap Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3
Contact Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5
GAP Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two-Node Gap Element (Node-to-Node Gap). . . . . . . . . . . . .
One-Node Gap Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-6
4-6
4-8
4-9
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Automatic Generation of Gap Elements . . . . . . . . . . . . . . . . . . 4-12
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12
Contact/Gaps Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15
Triangular Sub-Surfaces for Target Surface . . . . . . . . . . . . . 4-15
Automatic Soft Springs for Contact Source or Target . . . . . . 4-15
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Contents
A New Solution Strategy for Initial Interference . . . . . . . . . . 4-15
Troubleshooting for Gap/Contact Problems . . . . . . . . . . . . . . . 4-16
5
Numerical Procedures
Static Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Incremental Control Techniques . . . . . . . . . . . . . . . . . . . . . . .
Thermal Loading for Displacement/Arc Length Controls . . . .
Iterative Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
Line Search Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Termination Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-1
5-1
5-4
5-4
5-8
5-9
Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rayleigh Damping Effects . . . . . . . . . . . . . . . . . . . . . . . . . .
Concentrated Dampers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Base Motion Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inclusion of Dead Loads in Dynamic Analysis . . . . . . . . . . .
5-11
5-13
5-13
5-13
5-14
Adaptive Automatic Stepping Technique . . . . . . . . . . . . . . . . .
Step Size Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Safe-guard Against Equilibrium Iteration Failures . . . . . . . .
Safe-guard Against Converging to Incorrect Solutions . . . . .
5-15
5-15
5-15
5-16
J-Integral Evaluation for Nonlinear Fracture Mechanics NLFM 5-17
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-18
Modification for Temperature . . . . . . . . . . . . . . . . . . . . . . . . 5-19
Axisymmetric Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19
The Requirements in Selection of the Path . . . . . . . . . . . . . . 5-20
Requirements for JI and JII Evaluation . . . . . . . . . . . . . . . . . 5-20
Symmetric Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20
Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21
J-Integral Path Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21
Frequencies and Mode Shapes in a Nonlinear Environment . . . 5-22
Buckling Analysis in a Nonlinear Environment . . . . . . . . . . . . 5-23
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Release of Global Prescribed Displacements . . . . . . . . . . . . . . 5-24
4
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Defining Temperatures Versus Time Relative to a Reference Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-25
Modified Central Difference Technique for Dynamic Time Integration
5-26
Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28
Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28
Combination of Force Control and Displacement/Arc-Length Control
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-30
Artificial Bulk Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-31
6
Element Library
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1
7
Commands and Examples
Command Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1
Elastoplastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2
Geometrically Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . . 7-5
Elastoplastic Large Displacement Analysis . . . . . . . . . . . . . . . . 7-6
Nonlinear Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7
Linear Dynamic Analysis (Time-History) . . . . . . . . . . . . . . . . . 7-9
Analysis Including Temperature Loading . . . . . . . . . . . . . . . . . 7-10
Structural Analysis with Temperature-Dependent Material Properties711
Elastic Creep Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-13
Static Analysis Using Displacement Control Technique . . . . . . 7-14
Static Analysis Using Arc-Length Control Technique . . . . . . . 7-15
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Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16
Elastoplastic Nonlinear Analysis Example . . . . . . . . . . . . . . 7-16
Large Displacement Nonlinear Analysis Example . . . . . . . . . 7-24
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Contents
8
Verification Problems
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1
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COSMOSM Advanced Modules
1
Introduction
Introduction
Developing a reliable model capable of predicting the behavior of structural
systems represents one of the most difficult tasks to face the analyst. The finite
element method provides a convenient vehicle for performing this task due to its
versatility and the great advancement in its adaptation to computer use.
However, the success of a finite element analysis depends largely on how
accurately the geometry, the material behavior, and the boundary conditions of the
actual problem are idealized.
While elements with their geometric characteristics and boundary conditions are
used to describe the geometric domain of the problem, material models
(constitutive relations) are introduced to capture the material behavior.
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All real structures behave nonlinearly in one way or another. Still, in some cases,
due to the particular nature of the problem a linear analysis may be adequate.
However, in many other situations a linear solution has proven to be catastrophic
and a nonlinear analysis becomes a must.
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Chapter 1 Introduction
Structural Nonlinearities
In this section, major sources of structural nonlinearities encountered in practical
applications will be presented.
Geometrical Nonlinearities
In nonlinear finite element analysis, a major source of nonlinearities is due to the
effect of large displacements on the overall geometric configuration of structures.
Structures undergoing large displacements can have significant changes in their
geometry due to load-induced deformations which can cause the structure to
respond nonlinearly in a stiffening and/or a softening manner. For example, cablelike structures (Figure 1-1a) generally display a stiffening behavior on increasing
the applied loads while arches may first experience softening followed by stiffening, a behavior widely-known as the snap-through buckling (Figure 1-1b).
Material Nonlinearities
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Another important source of nonlinearities stems from the nonlinear relationship
between the stress and strain which has been recognized in several structural
behaviors. Several factors can cause the material behavior to be nonlinear. The
dependency of the material stress-strain relation on the load history (as in plasticity
problems), load duration (as in creep analysis), and temperature (as in thermoplasticity) are some of these factors. This class of nonlinearities, known as material
nonlinearities, can be idealized to simulate such effects which are pertinent to
different applications through the use of constitutive relations. Yielding of beamcolumn connections during earthquakes (Figure 1-2) is one of the applications in
which material nonlinearities are plausible.
1-2
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Generalized Force
Figure 1-1a. Cable-like Structure
Qualitative ForceDisplacement Curve for
Nonlinear
Suspension Bridge [10]
Linear
Generalized Displacement
BRIDGE
MO DE LING
Cable Nodes
Suspension Bridge
Figure 1-1b. Pressure (p) Versus Center Deflection (D) for Shallow Spherical Cap
P
P
Linear
Nonlinear
D
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D
COSMOSM Advanced Modules
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Chapter 1 Introduction
Figure 1-2. Loading and Unloading of Beam-Column Connection
Under Dynamic Loading
σ
Force
Time
Be a m-C olumn
C onne ction
Applie d
Loa ding
S tre ss-S tra in C urve
ε
Contact (Boundary) Nonlinearities
A special class of nonlinear problems is
concerned with the changing nature of the
boundary conditions of the structures
involved in the analysis during motion.
This situation is encountered in the
analysis of contact problems. Pounding of
structures, gear-tooth contacts, fitting
problems, threaded connections, and
impact bodies are several examples
requiring the evaluation of the contact
boundaries. The evaluation of contact
boundaries (nodes, lines, or surfaces) can
be achieved by using gap (contact)
elements between nodes on the adjacent
boundaries.
Figure 1-3. Pounding of Structures
Due to Seismic Motion
Solution Procedures of Nonlinear Problems
Solution Strategies
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For nonlinear problems, the stiffness of the structure, the applied loads, and/or
boundary conditions can be affected by the induced displacements. The equilibrium
of the structure must be established in the current configuration which is unknown
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a priori. At each equilibrium state along the equilibrium path, the resulting set of
simultaneous equations will be nonlinear. Therefore, a direct solution will not be
possible and an iterative method will be required.
Several strategies have been devised to perform nonlinear analysis. As opposed to
linear problems, it is extremely difficult, if not impossible, to implement one single
strategy of general validity for all problems. Very often, the particular problem at
hand will force the analyst to try different solutions procedures or to select a certain
procedure to succeed in obtaining the correct solution (for example, “Snapthrough” buckling problems of frames and shells (Figure 1-4) require deformationcontrolled loading strategies such as Displacement and Arc-length based controls
rather than Force-controlled loading).
Figure 1-4a. Load-Deflection Curve of William-Toggle Frame
where Force Control Strategy Fails
P
P
D
H
L
L
D
Figure 1-4b. Load-Deflection Curve for Hinged Cylindrical Shell
where Only Arc-Length Control Strategy Succeeds
P
h
P
L
w
θ
θ
L
R
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Chapter 1 Introduction
For these reasons, it is imperative that a computer program used for nonlinear
analyses should possess several alternative algorithms for tackling wide spectrum
of nonlinear applications. Such techniques would lead to increased flexibility and
the analyst would have the ability to obtain improved reliability and efficiency for
the solution of a particular problem.
Concept of Time Curve
For nonlinear static analysis, the loads are applied in incremental steps through the
use of “time” curves. The “time” value represents a pseudo-variable which denotes
the intensity of the applied loads at a certain step. While, for nonlinear dynamic
analysis and nonlinear static analysis with time-dependent material properties (e.g.,
creep), “time” represents the real time associated with the loads' application.
The choice of “time” step size depends on several factors such as the level of
nonlinearities of the problems and the solution procedure. A computer program
should be equipped with an adaptive automatic stepping algorithm to facilitate the
analysis and to reduce the solution cost.
NSTAR: An Overview
This brief section is intended to introduce the COSMOSM nonlinear structural
analysis module NSTAR and outline its major features for performing nonlinear
structural static and dynamic analyses.
The following present some of NSTAR capabilities:
Extensive 1D, 2D and 3D Element Library (Chapter 6)
Geometric Nonlinearities
Large displacements (total and updated Lagrangian formulations)
Large strain formulation for rubber-like materials (total Lagrangian formulation)
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Large strain formulation for von Mises elastoplastic materials (updated
Lagrangian formulation)
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Material Models and Constitutive Relations
Linear elasticity
Nonlinear elasticity
• Arbitrarily user-defined stress-strain curve
Hyperelasticity
• Mooney-Rivlin model
• Ogden model
• Blatz-Ko
Plasticity
• Huber-von Mises yield criterion with isotropic or kinematic hardening rules
• Tresca-Saint Venant yield criterion with isotropic or kinematic hardening
rules
• Drucker-Prager elastic-perfectly plastic model
• Concrete model
Creep and viscoelasticity
• Classical power law for creep
• Exponential creep law
• Linear isotropic viscoelastic model
Failure criterion for laminated composite materials
Wrinkling membrane for fabric materials
Temperature-dependent material properties for thermo-elastic-plastic analysis
User-defined material models
Contact Problems
Gaps, contact lines, and contact surfaces with generalized friction
Numerical Procedures
Solution control techniques
• Force-controlled loading strategy
• Deformation-controlled loading strategies
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- Displacement control technique
- Riks arc-length control technique
Equilibrium iterations schemes
COSMOSM Advanced Modules
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Chapter 1 Introduction
• Regular Newton-Raphson (tangent method)
• Modified Newton-Raphson
• Quasi-Newton BFGS (Broyden-Fletcher-Goldfarb-Shanno) (secant method)
Termination schemes
• Convergence criteria
• Divergence criteria
Line search option to improve convergence
User-controlled solution tolerances and equilibrium iterations interval
Direct time implicit integration techniques
• Newmark-Beta method
• Wilson-Theta method
Damping effects
• Rayleigh damping
• Concentrated dampers
Base motion effects
Restart option
Adaptive automatic stepping algorithm
Loadings
Concentrated loads (forces and moments)
Pressure (with displacement-dependency option)
Thermal
Centrifugal
Gravity
Time curves to scale loading and control the rate of application
Other Features
Buckling analysis
• Limit load analysis
• Post-buckling analysis (snap-through, snap-through/snap-back, and multiple
snap-through/snap back problems)
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Coupled degrees of freedom
Prescribed non-zero displacements associated with time curves.
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Initial conditions for dynamic analysis
Reaction force calculations
Postprocessing
Listing of displacements, strains, stresses, and gap forces.
Listing of extreme values of displacement, strain and stress components
Deformed shape plots at the user-specified steps
Color-filled, colored lines, colored vectorial contour plots on undeformed and
deformed shapes for:
• Displacements
• Strains
• Stresses
Animation for:
• Displacements
• Strains
• Stresses
X-Y plots for the response of user-specified nodes during the analysis as a
function of time. Responses include:
•
•
•
•
•
Acceleration
Velocity
Displacement
Reaction force
Stress
X-Y plots for the displacement response of user-specified nodes versus load
factor multiplier for post-buckling analysis.
References
Bathe, K. J., Finite Element Procedures in Engineering Analysis, Prentice-Hall,
1982.
2.
Belytschko, T., and Hughes, T. (eds.), Computational Methods for Transient
Analysis, North-Holland, Amsterdam, 1983.
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1.
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Part 1 NSTAR / Nonlinear Analysis
Bergan, P. G., “Solution Algorithms for Nonlinear Structural Problems,”
Comput. Struct., Vol. 12, pp. 497-509, 1980.
4.
Chen, W. F., and Saleeb, A. F., Constitutive Equations for Engineering
Materials, Vol. 1, Elasticity and Modeling, John Wiley, 1981.
5.
Cook, D. R., Malkus, D. S., and Plesha, M. E., “Concept sand Applications of
Finite Element Analysis, “Third edition, Wiley, 1989.
6.
Grisfield, M. A., Finite Elements and Solution Procedures for Structural
Analysis, Vol. I: Linear Analysis, Pineridge Press Limited, U.K., 1986.
7.
Hill, R., The Mathematical Theory of Plasticity, Oxford University Press,
London, 1950.
8.
Kulak, R. F., “Adaptive Contact Elements for Three-Dimensional Explicit
Transient Analysis,” Comp. Meth. Appl. Mech. Eng., 72, pp. 125-151, 1989.
9.
Kardestuncer, H., “Finite Element Handbook,” McGraw-Hill, 1987.
10.
Niazy, A-S.M., “Seismic Performance Evaluation of Suspension Bridges,”
Ph.D. dissertation, Civil Eng. Dept., USC, 1991.
11.
Owen, D. R. J., and Hinton, E., Finite Elements in Plasticity: Theory and
Practice, Pineridge Press, Swansea, U.K., 1980.
12.
Parisch, H., “A Consistent Tangent Stiffness Matrix for Three-Dimensional
Non-linear Contact Analysis,” Int. J. Num. Meth. Eng., Vol. 28, pp. 1803-1812,
1989.
13.
Ramm, E., “Strategies for Tracing the Nonlinear Response near Limit Points,”
in Nonlinear Finite Element Analysis in Structural Mechanics, edited by W.
Wunderlich, E. Stein, and K. Bathe, Springer-Verlag, Berlin, 1981.
14.
Riks, E., “An Incremental Approach to the Solution of Snapping and Buckling
Problems,” Int. J. Numer. Meth. Eng., 15:529-551 (1979).
15.
Zienkiewicz, O. C., and, Taylor, R. L., The Finite Element Method, Fourth
edition, Vol. 2, 1991.
In
de
x
3.
COSMOSM Advanced Modules
1-10
2
Geometrically
Nonlinear Analysis
Introduction
In finite element analysis, the overall stiffness of a structure depends on the
stiffness contribution of each of its finite elements. Throughout the history of the
load application, the structure is displaced from its original position and the nodal
coordinates will change causing the elements to deform and to change their spatial
orientations.
In
de
x
In small displacement analysis, the displacement-induced deformations of the
element and the change in its spatial orientation (rotation) are assumed small
enough that the change in its stiffness contribution to the overall structural stiffness
can be ignored. On the other hand, in large displacement analysis the displacementinduced deformations can be finite and the local element stiffness will change due
to the change in the element geometrical shape (length, area, thickness, or volume).
Also, the spatial-orientation change can no longer be infinitesimal so that the
transformation of the element local stiffness into global-stiffness contribution will
also change. In order to consider the finite change in the geometry of the structures
in the analysis, auxiliary strain measures (e.g., Green-Lagrange and Almansi strain
tensors) with their conjugate stress tensors (e.g., Second Piola-Kirchhoff and
Cauchy stress tensors) must be introduced.
COSMOSM Advanced Modules
2-1
Chapter 2 Geometrically Nonlinear Analysis
Large Displacement Nonlinear Analyses
The current literature on geometrically nonlinear analysis contain several large
displacement formulations for finite element applications. The main differences
arise from the simplifications and assumptions imposed on the kinematic relations
and the form of stress rate.
The use of the most general large displacement formulation will render “correct”
solutions, however, in many cases the use of a more restrictive formulation could be
attractive because of its computational efficiency.
Two main approaches have been introduced, namely; the Total Lagrangian (T.L.)
formulation and the Updated Lagrangian (U.L.) formulation. These two approaches
are generally used with continuum elements (PLANE2D, TRIANG, SOLID, and
TETRA4/10). Another approach, used with skeletal elements (TRUSS2D/3D,
BEAM2D/3D, and IMPIPE) and shell elements (SHELL3/4, SHELL3T/4T, and
SHELL3L/4L), incorporates a co-rotational system of axes attached to the element
during motion and use strain measures of infinitesimal displacement analysis.
Finite Strain Analysis
In this analysis, as
Figure 2-1. Large Strain Analysis
the structure
deflects, the localDeformed
Symmetry
Element
ized deformations
y'
are large such that
Deformed
the strains are no
x'
longer infinitesiy
ψ
Undeformed
mal. The local element stiffness will
L ∆
change as a result of
θ
the element shape
p
change. Large strain
x
Circular Rubber Plate Under
effect is considered
Displacement- Dependent
by adjusting the elePressure Loading
ment shape. No
assumptions are made on the magnitude of the strains.
In
de
x
The deformation of rubber-like materials is a typical analysis in which finite strains
are experienced.
2-2
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Large Deflection Analysis
In this category, the change in the spatial orientation (rotation) of the elements can
be finite but the induced strains must remain small. The overall stiffness of the
structure will change as a result of the change of the global stiffness contribution of
the element due to the change in its spatial orientation. Large deflection effect is
taken into consideration by updating the element orientation during the analysis.
Figure 2-2. Large Deflection (Finite Rotation but Small Strains) Analysis
Deformed
Element
y'
y
x'
ψ
L
∆
∆
< 0.4
L
θ
θ
M
Cantilever Beam with
End Moment
x
In
de
x
Snap-through buckling
Figure 2-3. Multiple Snap-Through/Snap-Back
Buckling of Cylindrical Shell
analysis is an example where
large deflection analysis is
required. This type of
P
h
P
buckling is characterized by
L
w
a loss of the stiffness of the
L
structure at a certain loading
θ
θ
R
condition, known as the limit
load, at which the structure
w
becomes unstable. To trace
the structural behavior
beyond the limit load in
the postbuckling range, a
deformation-controlled
loading strategy (Arc-length or Displacement controls) is used (refer to
NL_CONTROL (Analysis > NONLINEAR > Solution Control) command). It has
to be noted that if a snap-back behavior (see Figure 2-3), in the postbuckling range
is experienced, the Arc-length must be used.
COSMOSM Advanced Modules
2-3
Chapter 2 Geometrically Nonlinear Analysis
In
de
x
References
2-4
1.
Allen, H. G., and Al-Qarra, H. H., “Geometrically Nonlinear Analysis of
Structural Membranes,” Comput. Struct., Vol. 25, No. 6, pp. 871-876, 1987.
2.
Bathe, K. J., Finite Element Procedures in Engineering Analysis, Prentice-Hall,
1982.
3.
Biot, M. A., Mechanics of Incremental Deformations, Wiley, N.Y., 1965.
4.
Belytschko, T., and Hughes, T. (eds.), Computational Methods for Transient
Analysis, North-Holland, Amsterdam, 1983.
5.
Chen, W. F., and, Mizunu, E., Nonlinear Analysis in Soil Mechanics, Elsevier,
1990.
6.
Cook, D. R., Malkus, D. S., and Plesha, M. E., “Concepts and Applications of
Finite Element Analysis,” Third edition, Wiley, 1989.
7.
Fujikake, M., Kojima, O., and Fukushima, S., “Analysis of Fabric Tension
Structures,” Comput. Struct., Vol. 32, No. 3/4, pp. 537-547, 1989.
8.
Nughes, T. J. R., “Nonlinear Finite Element Shell Formulation Accounting for
Large Membrane Strains,” Computer Methods in Applied Mechanics and
Engineering, Vol. 39, 1983.
9.
Kardestuncer, H., “Finite Element Handbook,'' McGraw-Hill, 1987.
10.
Malvern, L. E., “Introduction to the Mechanics of a Continuous Medium,”
Prentice-Hall, 1969.
11.
Oden, J. T., “Finite Element of Nonlinear Continua,” McGraw-Hill, 1972.
12.
Timoshenko, S., and Woinowskey, S., “Theory of Plates and Shells,” McGrawHill, 1959.
13.
Weaver, W., and Gere, J. M., “Matrix Analysis of Framed Structures,” third
edition, Van Nostrand Reinhold, 1990.
14.
Wempner, G. A., “Mechanics of Solids with Applications to Thin Bodies,”
McGraw-Hill, 1973.
15.
Vunderlich, W., “Incremental Formulations for Geometrically Nonlinear
Problems,” in Formulations and Computational Algorithms in Finite Element
Analysis: U.S.- Germany Symposium on Finite Element Method, edited by
Bathe, K., Oden, J., and Wunderlich, W., MIT, 1976.
16.
Zienkiewicz, O. C., and, Taylor, R. L., The Finite Element Method, Fourth
edition, Vol. 2, 1991.
COSMOSM Advanced Modules
3
Material Models and
Constitutive Relations
Introduction
In this chapter, a brief discussion will be presented on the different material models
and constitutive relations implemented in NSTAR to simulate the complex
behavior of engineering materials. The description is not intended to be detailed but
rather illustrative to give the user some guide lines to select the material model best
suited for the analysis. For the users interested in further theoretical details, a list of
references is provided at the end of the chapter.
Linear Elastic Models
Isotropic
For definitions and usage, refer to Basic System User Guide and COSMOSM
Command Reference Manuals.
In
de
x
The linear elastic isotropic material model can be used with the following element
groups:
• TRUSS2D & TRUSS3D
• BEAM2D & BEAM3D
COSMOSM Advanced Modules
3-1
Chapter 3 Material Models and Constitutive Relations
•
•
•
•
•
•
•
•
•
PLANE2D 4/8 nodes
(plane stress, plane strain, and axisymmetric)
TRIANG 3/6 nodes
(plane stress, plane strain, and axisymmetric)
SOLID 8/20 nodes
TETRA4 and TETRA10
SHELL3, SHELL3T, and SHELL3L
SHELL4, SHELL4T, and SHELL4L
SHELL6 and SHELL6T
SPRING
IMPIPE
The parameters required to describe this model can be associated with temperature
curves to perform thermo-elastic analysis.
Orthotropic
For definitions and usage, refer to the Basic System User Guide and COSMOSM
Command Reference Manuals.
The linear elastic orthotropic material model can be used with the following
element groups:
•
•
•
•
•
•
•
PLANE2D 4/8 nodes
(plane stress, plane strain, and axisymmetric)
TRIANG 3/6 nodes
(plane stress, plane strain, and axisymmetric)
SOLID 8/20 nodes
TETRA4 & TETRA10
SHELL3, SHELL3T, and SHELL3L
SHELL4, SHELL4T, and SHELL4L
SHELL6 and SHELL6T
In
de
x
The parameters required to describe this model can be associated with temperature
curves to perform thermo-elastic analysis.
3-2
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Laminated Composite and Failure Criterion for
Laminated Composites
Laminated Composite
For definitions and usage, refer to Basic System User Guide and COSMOSM
Command Reference Manuals.
The linear elastic laminated composite material model can be used with SHELL3L
and SHELL4L elements.
The parameters required to describe this model can be associated with temperature
curves to perform thermo-elastic analysis.
Failure Criterion for Laminated Composite Materials
A failure criterion provides the mathematical relation for strength under combined
stresses. Unlike conventional isotropic materials where one constant will suffice for
failure stress level and location, laminated composite materials require more
elaborate methods to establish failure stresses. The strength of a laminated
composite can be based on the strength of individual plies within the laminate. In
addition, the failure of plies can be successive as the applied load increases. There
may be a first ply failure followed by other ply failures until the last ply fails,
denoting the ultimate failure of the laminate. Progressive failure description is
therefore quite complex for laminated composite structures.
A simpler approach for establishing failure consists of determining the structural
integrity which depends on the definition of an allowable stress field. This stress
field is usually characterized by a set of allowable stresses in the material principal
directions. The table below lists the components of allowable stresses referenced in
the failure theories and their material names in COSMOSM.
In
de
x
A failure criterion, the Tsai-Wu failure criteria, which makes use of the allowable
stresses input by the user is currently available. This failure criterion is used to
calculate a failure index (F.I.) from the computed stresses and user-supplied
material strengths. A failure index of 1 denotes the onset of failure, and a value less
than 1 denotes no failure.
COSMOSM Advanced Modules
3-3
Chapter 3 Material Models and Constitutive Relations
Table 3-1.
Required Material Strength Components for
Failure Criteria of Composites
Symbol
Description
Material
Property Name
X1 T
Tensile strength in the material
longitudinal direction
SIGXT
X1C
Compressive strength in the
material longitudinal direction
SIGXC
X2 T
Tensile strength in the material
transverse direction
SIGYT
X2C
Compressive strength in the
material transverse direction
SIGYC
S12
In-plane shear strength in the
material x-y plane
SIGXY
The failure criterion of Tsai-Wu is applicable to SHELL3L and SHELL4L elements
only. All components of material strengths for all layers must be input in order to
compute the failure indices.
The failure indices are computed for all layers in each element of the model.
The Tsai-Wu failure criterion (also known as the Tsai-Wu tensor polynomial
theory) is commonly used for orthotropic materials with unequal tensile and
compressive strengths. The failure index using this theory is computed using the
following equation:
where,
In
de
x
The application of Tsai-Wu failure criterion is requested by specifying a value of 7
in option 5 of the EGROUP (Propsets > Element Group) command for SHELL3L
and SHELL4L elements.
3-4
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
The program computes the failure indices for all layers in each element and prints
them along with the element stresses in the output file.
Three types of failure status are shown:
1.
OK
failure index < 1
2.
FAIL1
failure index ≥ 1
and
σ1 < X1T
if σ1 > 0
or
σ1 > – X1C
if σ1 < 0
FAIL2
failure index ≥ 1
and
σ1 ≥ X1T
if σ1 > 0
or
σ1 ≤ – X1C
if σ1 < 0
3.
where σ1 is the normal stress in the 1st material direction.
Nonlinear Elastic Model
For the particular case of stress history as related to proportional loading, where
components of stress tensor vary monotonically in constant ratio to each other, the
strains can be expressed in terms of the final state of stress in the following form:
where,
is the secant material matrix, Es is the secant modulus, and ν is the
In
de
x
Poisson's ratio.
To incorporate this model in the analysis, Poisson's ratio NUXY (not needed for
TRUSS2D/3D or BEAM2D/3D) should be defined, the MPCTYPE (LoadsBC >
FUNCTION CURVE > Material Curve Type) command with the elastic option
should be activated, and the stress-strain curve should be defined using the MPC
(LoadsBC > FUNCTION CURVE > Material Curve) command.
COSMOSM Advanced Modules
3-5
Chapter 3 Material Models and Constitutive Relations
The total strain vector is used to compute the effective strain
to get the secant
modulus from the user-defined stress-strain curve (using the MPC (LoadsBC >
FUNCTION CURVE > Material Curve) command). For the three dimensional
case,
The stress-strain curve from the third (compressive) to the first (tensile) quadrants
are applicable to this model for two and three dimensional elements with some
modifications. A method of interpolation is used to get the secant and tangent
material moduli. Defining a ratio R which is a function of the volumetric strain
Φ, effective strain, and the Poisson's ratio, R has the following expression:
Es and Et are then computed by using the equation
It is noted that R = 1 represents the uniaxial tensile case and R = -1 is for the
compressive case. These two cases are set to be the upper and lower bound such
that when R exceeds these two values, the program will push it back to the limit.
✍ Cable (no-compression) type behavior. For the element to behave as a CABLE
In
de
x
(non-compression) element, the element has to be associated with a stress-strain
curve as shown in Figure 3-2. If the stress-strain curve in the third quadrant is
not input, the program assumes that a symmetric behavior in tension and
compression exists. It should be mentioned that users may have to specify initial
force (r3) and/or initial strain (r4) for the cable-type behavior of the TRUSS2D
3D elements.
3-6
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
✍ Nonlinear SPRING. For the nonlinear SPRING element (Element Group
Op. 5 = 1), the MPC represents force-displacement curve for the axial spring
(Element Group Op. 1 = 0) and torque-rotation curve for the torsional spring
(Element Group OP. 1 = 1).
Figure 3-1. Curved Description Nonlinear Elastic Model
σ
(R=-1)
Compression
σ9
σ8
σ7
σ6
Tension
ε1
ε2
ε3
ε 4 ε5
Es(0)
s
E (t)
σ5
ε6
(R=1)
ε7 ε ε8
ε9
ε
σ4
σ3
σ2
σ1
Input Stress-Strain Curve
The nonlinear elastic material model can be used with the following element
groups:
Figure 3-2. Stress-Strain Curve for Cable-Type Behavior
Stress
Tension
Compression
Strain
In
de
x
(ε 1 , σ1)
(ε2 , σ 2 )
= (0, 0)
• TRUSS2D & TRUSS3D
COSMOSM Advanced Modules
3-7
Chapter 3 Material Models and Constitutive Relations
•
•
•
•
•
•
•
•
BEAM2D & BEAM3D
PLANE2D 4/8 nodes
(plane stress, plane strain, and axisymmetric)
TRIANG 3/6 nodes
(plane stress, plane strain, and axisymmetric)
SOLID 8/20 nodes
TETRA4 and TETRA10
SHELL3T and SHELL4T
SHELL6 and SHELL6T
SPRING
Hyperelastic Models
Incompressible Rubber-Like Materials
Hyperelastic material models can be used for modeling rubber-like materials where
solutions involve large deformations. The material is assumed nonlinear elastic,
isotropic and incompressible.
The finite element formulation for such materials has numerical difficulties due to
incompressibility. Two approaches have been devised to treat the incompressible
constraint, namely; the mixed finite element formulation and the penalty finite
element formulation. Mixed formulation uses a separate interpolation of a stress
variable that is related to the hydrostatic pressure. The penalty function method
assembles the additional degrees of freedom into the global stiffness matrix. This
method introduces an artificial compressibility and has a formulation in which the
displacement degrees of freedom are the only unknowns. Compared to the mixed
approach, the penalty finite element has fewer independent variables.
In COSMOSM, the displacement formulation (full or reduced integration) is based
on the introduction of compressibility to the strain energy density function. This
treatment is identical to the finite element penalty approach in principle. The
introduction of the penalty function modifies the strain energy function from
incompressible type to the nearly incompressible one. The stability, the
convergence, and the numerical results of nonlinear analysis depend on the type of
penalty function employed.
In
de
x
A mixed or displacement-pressure (u/p) formulation is also available in
COSMOSM. This technique explicitly replaces the pressure computed from the
displacement field by a separately interpolated pressure using a general procedure.
3-8
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
A consistent penalty method is used for eliminating the separate pressure degrees of
freedom by introducing an artificial compressibility and then for each element,
using full integration to form the element stiffness matrix. Static condensation is
used to eliminate the pressure degrees of freedom at the element level and therefore
the global stiffness matrix contains only displacement degrees of freedom.
All the skills needed for nonlinear analysis apply to the hyperelastic models. The
load step, the mesh size and its distribution, the integration rule (e.g., reduced or
full),.. etc. require careful considerations. However, in some cases especially when
lack of good understanding of the problem exists, nothing but trial and error will
prove successful. Higher order elements provide more numerical stability than
lower order elements.
Mooney-Rivlin Model
The Mooney-Rivlin strain energy density function is expressed as:
1.
Displacement formulation:
(3-1)
(3-2)
(3-3)
where I, II, and III are invariants of the right Cauchy-Green deformation tensor
and can be expressed in terms of principal stretch ratios; A, B, C, D, E, and F are
Mooney material constants, and
(3-4)
(3-5)
In
de
x
2.
Displacement - pressure formulation:
w=
+Q
COSMOSM Advanced Modules
3-9
Chapter 3 Material Models and Constitutive Relations
where
= Strain energy density function due to displacement field
=
+
= A(J1 - 3) + B(J2 - 3) + 1/2 K(J3 - 1)2
= C(J1 - 3)(J2 - 3) + D(J1 - 3)2 + E(J2 - 3)2 + F(J1 - 3)3
J1,J2,J3
= Reduced invariants
J1
= I1 I3-1/3
J2
= I2 I3-2/3
J3
= J = I31/2
I1,I2,I3
= Invariants of the right Cauchy-Green deformation tensor.
Q
= The additional strain energy density function due to
displacement and pressure fields.
=
= The hydrostatic pressure as computed directly from the
displacement.
= The hydrostatic pressure as computed from the separately
interpolated pressure variables
=
For constant field,
For linear field,
K = bulk modulus = E / [3(1 - 2ν)],
E = 6(A + B)
In
de
x
It has to be noted that as the material approach incompressibility, the third
invariant, III, approaches unity, while constants Y and K approach infinity. Thus,
for values of Poisson's ratio close to 0.5, the last term in w1 remains bounded, and a
solution can be obtained. For problems with significant localized hydrostatic
stresses, the displacement-pressure formulation is recommended so that “locking”
of displacements does not occur.
3-10
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
The material properties for Mooney-Rivlin model are input through the use of the
MPROP (Propsets > Material Property) command. Up to six Mooney-Rivlin
constants can be input. The input quantities can be:
MOONEY_A, MOONEY_B, MOONEY_C, MOONEY_D, MOONEY_E, and
MOONEY_F.
The Mooney-Rivlin material model can be used with the following element groups:
•
•
•
•
•
•
PLANE2D 4/8 nodes
(plane stress, plane strain, and axisymmetric)
TRIANG 3/6 nodes
(plane stress, plane strain, and axisymmetric)
SOLID 8/20 nodes
TETRA4 and TETRA10
SHELL3T and SHELL4T
SHELL6 and SHELL6T
The displacement-pressure formulation is available for element groups: PLANE2D
4- to 8-node (plane strain and axisymmetric) and SOLID 8-node.
Element Type
No. of Pressure DOF
4-node PLANE2D
1
5- to 8-node PLANE2D
3
8-node SOLID
1
✍ Notes:
1. Always use the Total Lagrangian Formulation for element groups
PLANE2D, TRIANG, SOLID, and TETRA4/10. For SHELL3T/4T
elements the formulation is automatically adjusted inside the program when
the large displacement option is used.
2. Use the NR (Newton-Raphson) iterative method.
3. If the structure is subjected to pressure loading, use the displacementdependent loading option.
In
de
x
4. For PLANE2D and SOLID elements, if significant localized hydro-static
stresses develop in the structure, use the option for displacement-pressure
formulation.
5. Values of Poisson's ratio greater or equal to 0.48 but less than 0.5 are
acceptable. When the displacement-pressure formulation is used, Poisson's
ratio is recommended in the range from 0.499 to 0.4999.
COSMOSM Advanced Modules
3-11
Chapter 3 Material Models and Constitutive Relations
6. Rubber-like materials usually deform rapidly at low magnitudes of loads
thus requiring a slow initial loading.
7. When dealing with rubber-like materials, due to the highly nonlinear
behavior of the problem, rapid increase in loading will often result in either
negative diagonal terms in the stiffness or divergence during equilibrium
iterations. In either case, the restart option can be used to restart the solution
from the last converged step after the loading is modified. A more
convenient way is to use the option for the automatic-adaptive stepping
algorithm.
8. The displacement or the arc-length control may prove to be more effective
than force control when negative diagonal terms repeatedly occur under
various loading rates.
9. For cases of PLANE2D plane stress option and SHELL3T/4T elements, the
analysis is simplified since incompressibility does not result in unbounded
terms. The formulation is derived assuming perfect incompressibility
(Poisson's ratio of 0.5), and therefore NUXY is neglected. Displacementpressure formulation is not considered.
10. Constants A and B must be defined such that (A+B) > 0. For more
information about how to determine the values of the A and B constants,
refer to the work by Kao and Razgunas cited at the end of this chapter.
Ogden Model
The Ogden strain energy density function, defined as,
(3-6)
where
λi
= Principal stretches
α i, µ i
= Material constants
N
= Number of terms in the function
is considered one of the most successful functions in describing the large
deformation range of rubber-like materials.
In
de
x
The penalty function used in COSMOSM Ogden model take the form of the one
used in Mooney-Rivlin model. The strain energy function actually used is a
modified type of the Ogden function:
3-12
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
1.
Displacement formulation:
(3-7)
where
2.
J
= Ratio of the deformed volume to the undeformed volume
N
= Number of terms in the function
G(J)
= J2 - 1
4/α
=
,
ν = Poisson ratio
Displacement-pressure formulation:
w=
+Q
where
L1,L2,L3 = Reduced principal values of right Cauchy-Green deformation
tensor.
LI
= λi2 I3-1/3
K
=
In
de
x
The 3-term (modified Ogden) models are widely used. Up to 4-term models (N = 4)
are available in the COSMOSM nonlinear module.
Besides the material constants mentioned above, Poisson ratio is another input to be
defined by the user. For most cases, satisfactory results can be obtained by assigning Poisson's ratio from 0.49 to 0.499 for displacement formulation and 0.499 to
0.4999 for u/p (displacement-pressure) formulation. Further, for displacement
COSMOSM Advanced Modules
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Chapter 3 Material Models and Constitutive Relations
formulation, increasing Poisson's ratio will not have significant effect on numerical
results unless considerable volumetric strain is involved. When Poisson's ratio is
extremely close to 0.5, it may cause solution termination due to negative diagonal
terms in the stiffness matrix or lack of convergence.
Like Mooney-Rivlin Hyperelastic model, the Total Lagrangian Formulation is used
for the modified Ogden model.
The material properties for Ogden model are input through the use of the MPROP
(Propsets > Material Property) command. The required quantities are:
• MU1, MU2, MU3, MU4
• ALPH1, ALPH2, ALPH3, ALPH4
• NUXY (not required for plane stress element)
The Ogden model can be used with the following element groups:
•
•
•
•
•
PLANE2D 4/8 nodes
(plane stress, plane strain, and axisymmetric)
TRIANG 3/6 nodes
(plane stress, plane strain, and axisymmetric)
SOLID 8/20 nodes
TETRA4 & TETRA10
SHELL3T & SHELL4T
The displacement-pressure formulation is available for element groups: PLANE2D
4-to 8-node (plane strain and axisymmetric) and SOLID 8-node.
Element Type
No. of Pressure DOF
4-node PLANE2D
1
5- to 8-node PLANE2D
3
8-node SOLID
1
At present, COSMOSM provides two different incompressible hyperelastic models
namely; the Mooney-Rivlin (M-R) model and the Ogden model. The following
considerations may be of use to predict which model is more appropriate to
incorporate for a particular problem.
• A 2-term M-R model is a special case of the Ogden model. Two-term Ogden
function can be transformed to M-R strain function by taking:
In
de
x
α1 = 2
α2 = -2
3-14
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
µ1 = 2 *MOONEY_A
µ2 = -2 *MOONEY_B
• M-R model compares favorably with experimental data of Rubber-like materials
undergoing large deformations. As in the cases of simple tension, pure shear,
and equibiaxial extension, the two-term M-R model can fit the experimental
data for a finite strain range up to 150% (principal stretch ratio 2.5). Ogden
model, however, can describe these deformations for very large range of strain
up to 500% or 600% (principal stretches 6 or 7).
• M-R model constants are easier to obtain from experimental tests than Ogden's
constants. M-R strain energy function is considered the most widely used
constitutive law in the stress analysis of elastomers.
• M-R model may have higher computational efficiency than the Ogden model
does since for Ogden model extra calculations are needed for transformation
between the global or the user-defined coordinates and the principal directions.
✍ Notes (1) through (9) under the Mooney-Rivlin model are applicable.
Blatz-Ko Model
Compressible Foam-Like Materials
The Blatz-Ko strain energy density function is useful for modeling compressible
polyurethane foam type rubbers and can be expressed as:
(3-8)
where
G
= Shear modulus under infinitesimal deformations = E/2(1+ν)
E
= Young's modulus of elasticity
ν
= Poisson's ratio = 0.25
Ik
= Invariants of
In
de
x
= Cauchy-Green deformation tensor =
= Lagrangian strain tensor
COSMOSM Advanced Modules
3-15
Chapter 3 Material Models and Constitutive Relations
= Identify matrix
The above expression, contains only one material constant G. Since ν = 0.25 for the
Blatz-Ko model, the only material property which is considered is the Young's
modulus. thus,
Similar to other hyperelastic models, this model works with total Lagrangian
formulation only.
The Blatz-Ko model is currently supported by the following element groups:
•
•
•
•
•
PLANE2D
SOLID
TRIANG
TETRA4
TETRA10
✍ The selected Blatz-Ko model is a simplified form of the expression obtained by
Blatz and Ko (1962) to model the deformation of a highly compressible
polyurethane foam rubber. The strain energy was approximated by the
following expression:
(3-9)
where
A specific form of this three-parameter family of elastic potential was later
proposed in which the following values of the constants α, β, and ν were assumed:
In
de
x
β = 0, ν = 0.25, and α = 0.5
3-16
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Plasticity Models
Huber-von Mises Model
The yield criterion can be written in the form:
(3-10)
where
is the effective stress and σY is the yield stress from uniaxial tests. Von
Mises model can be used to describe the behavior of metals.
In using this material model, the following considerations should be noted:
• Small strain plasticity is
assumed when small
displacement or large
displacement with total
Lagrangian formulation is
used.
• Large strain plasticity is
assumed when large
displacement with updated
Lagrangian formulation is
used for certain element
groups.
Figure 3-3. Bilinear Stress-Strain Curve
Stress, σ
1
σy
Et
E
1
Strain, ε
• An associated flow rule
assumption is made.
• A linear combination of isotropic and kinematic hardening is assumed, where
both the radius and the center of yield surface in deviatoric space can vary with
respect to the loading history.
• Pure isotropic hardening. The radius of the yield surface expnads but its
center remains fixed in deviatoric space.
• Pure kinematic hardening. The radius of the yield surface remains constant
In
de
x
while its center can move in deviatoric space.
The parameter RK defines the proportion of kinematic and isotropic hardenings.
The default value of RK is 0.85 (mixed hardening). For pure isotropic and pure
kinematic hardening, RK takes the values 0 and 1 respectively.
COSMOSM Advanced Modules
3-17
Chapter 3 Material Models and Constitutive Relations
Figure 3-4. Curved Description Plastic Model
Stress, σ
σy
ε1 ε3
ε
ε 2 ε4 5
ε6
Strain, ε
• A bilinear or multi-linear uniaxial stress-strain curve for plasticity can be input.
For bilinear stress-strain curve definition, material parameters SIGYLD and
ETAN are input through the use of MPROP (Propsets > Material Property)
command. For multi-linear stress-strain curve definition, the MPCTYP
(LoadsBC > FUNCTION CURVE > Material Curve Type) (with plasticity
option) and MPC (LoadsBC > FUNCTION CURVE > Material Curve)
commands are used.
• The SIGYLD and ETAN parameters for bilinear stress-strain curve description
can be associated with temperature curves to perform thermoplastic analysis.
• The use of NR (Newton-Raphson) iterative method is recommended.
The Huber-von Mises model can be used with the following element groups:
• TRUSS2D & TRUSS3D
• BEAM2D & BEAM3D
• PLANE2D 4/8 nodes
• TRIANG 3/6 nodes
In
de
x
• SOLID8/20 nodes
• TETRA4 & TETRA10
• SHELL3T & SHELL4T
*Large Strain Plasticity
3-18
COSMOSM Advanced Modules
(thermo-plasticity not available)
(plane stress, plane strain, and
axisymmetric)*
(plane stress, plane strain, and
axisymmetric)*
*
*
(thermo-plasticity not available)*
Part 1 NSTAR / Nonlinear Analysis
U/P Formulation:
The displacement-pressure formulation is available for element groups: PLANE2D
4- to 8-node (plane strain and axisymmetric) and SOLID 8-node, for large strain
plasticity.
Element Type
No. of Pressure DOF
4-node PLANE2D
1
5- to 8-node PLANE2D
3
8-node SOLID
1
Large Strain Analysis:
In the theory of large strain plasticity, a logarithmic strain measure is defined as:
where
is the right stretch tensor usually obtained from the right polar
decomposition of the deformation gradient
The incremental logarithmic strain is estimated as:
is the rotation tensor).
where
is the strain-displacement matrix estimated at n+1/2 and is the
incremental displacements vector. It is noted that the above form is a second-order
approximation to the exact formula.
The stress rate is taken as the Green-Naghdi rate so as to make the constitutive
model properly frame-invariant or objective. By transforming the stress rate from
the global system to the R-system
In
de
x
the entire constitutive model will be form-identical to the small strain theory.
The large strain plasticity theory in COSMOSM is applied to the von Mises yield
criterion, associative flow rule and isotropic or kinematic hardening (bilinear or
multilinear). Temperature-dependency of material property is supported by bilinear
COSMOSM Advanced Modules
3-19
Chapter 3 Material Models and Constitutive Relations
hardening. The radial-return algorithm is used in the current case. The basic idea is
to approximate the normal vector
by:
The illustration of the process is
shown in Figure 3-5.
Figure 3-5
The element force vector and stiffness
matrices are computed based on the
updated Lagrangian formulation. The
Cauchy stresses, logarithmic strains
and current thickness (shell elements
only) are recorded in the output file.
σ3
σn
~
σ trn+1
~
σ n+1
~
The elasticity in the current case is
modeled in hyperelastic form that
assumes small elastic strains but
σ1
allows for arbitrarily large plastic
strains. For large strain elasticity
problems (rubber-like), you can use
hyperelastic material models such as Mooney-Rivlin.
σ2
✍ Cauchy (true) stress and logarithmic strain should be used in defining the multilinear stress-strain curve.
Drucker-Prager Elastic-Perfectly Plastic Model
The yield criterion can be defined as:
(3-11)
where α and k are material constants which are assumed unchanged during the
analysis, σm is the mean stress and is the effective stress. α and k are functions of
two material parameters φ and c obtained from experiments where φ is the angle of
internal friction and c is the material cohesion strength.
In
de
x
Drucker-Prager model can be used to simulate the behavior of granular soil
materials such as sand and gravel.
3-20
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
In using this material model, the following considerations should be noted:
• Small strains assumption is made.
• Problems with large displacements can be handled provided that small strains
assumption is still valid.
• The use of NR (Newton-Raphson) iterative method is recommended.
• Material parameters φ and c must be bounded in the following ranges:
90 ≥ φ ≥ 0
(in degrees)
c≥0
(in force/unit area)
• For most soil mechanics problems, gravitational acceleration can have
significant effect, therefore the ACEL (LoadsBC > STRUCTURAL >
GRAVITY > Define Acceleration) command is usually used along with the
A_NONLINEAR (Analysis > NONLINEAR > NonL Analysis Options)
command in which the loading option G is activated.
The material parameters for the Drucker-Prager model are input through the
MPROP (Propsets > Material Property) command. The required inputs include the
following:
COHESN = Material cohesion strength
FRCANG = Friction angle
The Drucker-Prager model can be used with the following element groups:
•
•
•
•
PLANE2D 4/8 nodes
(plane strain, and axisymmetric)
TRIANG 3/6 nodes
(plane strain, and axisymmetric)
SOLID 8/20 nodes
TETRA4 & TETRA10
Tresca-Saint Venant Yield Criterion
(or the constant maximum shearing stress condition)
In
de
x
This criterion is based on the assumption that in the state of yielding, the maximum
shearing stress at all points of a medium is the same, and is equal to half of the yield
stress that is obtained from a uniaxial tension test for the given material.
COSMOSM Advanced Modules
3-21
Chapter 3 Material Models and Constitutive Relations
In the three-dimensional case, this is expressed as:
2 | τ1 | = | σ2 - σ3 | ≥ σy
2 | τ2 | = | σ3 - σ1 | ≥ σy
2 | τ3 | = | σ1 - σ2 | ≥ σy
The elastic state is represented by the inequality signs. In the state of yielding there
must be equality in one or two of these conditions. In other words, yielding is based
on the maximum shearing stress which is equal to half the difference between the
maximum and minimum principal stresses. Thus, based on this criterion, the
intermediate principal stress does not influence the state of yielding.
Shearing Stress Intensity
The shearing stress intensity is defined by the square root of the second invariant of
the stress deviator and can be expressed as:
State of Pure Shear
The state of pure shear is defined as:
σ1 = τ ,
σ2 = -τ ,
σ3 = 0 .
τmax = τ
For this state, the shearing stress intensity and the maximum shearing stress are
equivalent:
Τ = τmax = τ
Using the Tresca conditions the shearing stress at the yield point is obtained to be
half of the tensile yield stress:
τy = σ / 2 = 0.5 σy
In
de
x
Based on the von Mises yield criterion the shearing yield stress is equivalent to:
3-22
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Comparison of Tresca and von Mises Criteria for Plasticity
It has been observed that for polycrystalline materials (ductile metals), von Mises
condition of constant shearing stress intensity in the state of yielding agrees
somewhat better, in general, with experimental data. There are other cases,
however, that the Tresca-Saint Venant conditions appear to be in better agreement
with experimental data. Thus, the two methods may be regarded as equally possible
formulations of the yield condition.
✍ For the states of uniaxial or equibiaxial stress, the two criteria are equivalent.
✍ At other stress states yielding occurs at lower stress values according to the
Tresca conditions; under equal loading conditions, the Tresca criterion predicts
larger plastic deformation than von Mises.
✍ Maximum deviation between the two techniques occurs for the state of pure
In
de
x
shear. At this stress state, based on the Tresca conditions, yielding occurs at
87% of von Mises stress.
COSMOSM Advanced Modules
3-23
Chapter 3 Material Models and Constitutive Relations
Superelastic Models:
The term Superelastic is used for a material with the ability to undergo large
deformations in loading-unloading cycles without showing permanent
deformations.
Nitinol Model (Shape-Memory-Alloy)
Shape-memory-alloys (SMA) such as Nitinol present the Superelastic effect. In fact,
under loading-unloading cycles, even up to 10-15% strains, the material shows a
hysteretic response, a stiff-soft-stiff path for both loading and unloading, and no
permanent deformation.
Figure A typical stress-strain response for a Nitinol bar under uniaxial loading
conditions.
✍ The material behaves differently in tension and compression)
In
de
x
As it is shown by the response curve, the shape-memory-alloys show a distinctive
macroscopic behavior, not present in most traditional materials, which finds its
justification in the underlying macro-mechanics.
3-24
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
These alloys present reversible martensitic phase transformations, that is, a solidsolid diffusion-less transformations between a crystallographically more-ordered
phase, “austenite”, and a crystallographically less-ordered phase, “martensite”.
The soft portions of the response curve represent the areas where a phase
transformation: a conversion of austenite into martensite (loading), and martensite
into austenite (unloading) occurs.
For the sake of simplicity, however, we will refer to the soft behavior on the response
curve as “plastic”, and to the stiff portions as “elastic”.
According to this definition, the material first behaves elastically until a certain
stress level is reached (the initial yield stress in loading). If the loading continues,
the material shows an elastoplastic behavior until the plastic strain reaches its
ultimate value. From this point onward, the material behaves elastically again for
any increases in loading.
For unloading, again the material always starts to unload elastically until the stress
is reduced to the initial yield stress in unloading. The material will then unload in an
elastoplastic manner until all the accumulated plastic strain (from the loading phase)
is lost. And from that point onward, the material will unload elastically until it
returns to its original shape (no permanent deformation) and zero stress under zero
loads.
The Nitinol Model Formulation:
Since Nitinol material is usually used for its ability to undergo finite strains, the large
strain theory utilizing logarithmic strains along with the updated Lagrangian
formulation is employed for this model.
In
de
x
The constitutive model is, thus, constructed to relate the logarithmic strains & the
Kirchhoff stress components. However, ultimately the constitutive matrix and the
stress vector are both transformed to present the Cauchy (true) stresses.
COSMOSM Advanced Modules
3-25
Chapter 3 Material Models and Constitutive Relations
σst1, σft1 = Initial & Final yield stress for tensile loading,
= Initial & Final yield stress for tensile unloading,
3-26
[SIGT_S1, SIGT_F1]
[SIGT_S2, SIGT_F2 ]
= Initial & Final yield stress for compressive loading, [SIGC_S1, SIGC_F1]
= Initial & Final yield stress for compressive unloading, [SIGC_S2, SIGC_F2]
In
de
x
σst2, σft2
σsc1, σfc1
σsc2, σfc2
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
The exponential flow rule, utilizes additional input constants, βt1, βt2, βc1, βc2 :
βt1 = material parameter, measuring the speed of transformation for
tensile loading, [BETAT_1]
βt2
= material parameter, measuring the speed of transformation for
tensile unloading, [BETAT_2]
βc1 = material parameter, measuring the speed of transformation for
compressive loading, [BETAC_1]
βc2
= material parameter, measuring the speed of transformation for compressive
unloading, [BETAC_2]
The Yield Criterion:
To model the possibility of pressure-dependency of the phase-transformation, a
Drucker-Prager-type loading function is used for the yield criterion:
i
In
de
x
F(τ) = σ + 3α p – Rf
=0
Where:
σ = effective Stress
COSMOSM Advanced Modules
3-27
Chapter 3 Material Models and Constitutive Relations
p = mean Stress (or hydrostatic pressure)
α = (e2/3) (σsc1 − σst1) / (σsc1 + σst1)
i
i
Rf = [σf (e2/3 + α)] : i = 1 Loading
= 2 Unloading
The Flow Rule:
Through adoption of the logarithmic strain definition, the deviatoric and volumetric
components of the strain and stress tensors and their relations can be correctly
expressed in a decoupled form.
First, we consider the total plastic & elastic strain vectors to be presented by:
εp = εul ξs (n + α m)
εe = ε − εp
As a result, the Kirchhoff stress vector can be evaluated from:
τ = pm+ t
p = K (θ − 3α εul ξs )
t
= 2G (e − εul ξs n )
In the above formulations:
εul = scalar parameter representing the maximum material deformation [EUL]
(obtainable by detwinning of the multiple-variant martensite)
ξs = parameter between zero & one, as a measure of the plastic straining
In
de
x
θ
3-28
= volumetric strain = ε11 + ε22 + ε33
e
= deviatoric strain vector
t
= deviatoric stress vector
n
= norm of the deviatoric stress: t /σ
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
m
= the identity matrix in vector form: {1,1,1,0,0,0}T
K & G = the bulk & Shear elastic moduli: { K = E/[3(1-2ν)], G =E/[2(1+ν)] }
The linear flow rule in the incremental form can be expressed, accordingly:
∆ξs = (1. − ξs) ∆F / (F – Rf1 )
: Loading
∆ξs = ξs ∆F / (F – Rf2)
: Unloading
And the exponential flow rule, used when a nonzero βι is defined:
∆ξs = β1 (1. − ξs) ∆F / (F – Rf1)
∆ξs = β2 ξs ∆F / (F – Rf2)
: Loading
: Unloading
✍ In general, shape-memory-alloys are found to be insensitive to rate-effects.
Thus, in the above formulation “time” represents a pseudo variable, and its
length does not affect the solution.
✍ All the equations are presented here for tensile loading-unloading, since similar
expressions (with compressive property parameters) can be used for the
compressive loading-unloading conditions.
✍ The incremental solution algorithm here uses a return-map procedure in
In
de
x
evaluation of stresses and the constitutive equations, for a solution step.
Accordingly, the solution consists of two parts. Initially, a trial state is
computed; then if the trial state violates the flow criterion, an adjustment is
made to return the stresses to the flow surface.
COSMOSM Advanced Modules
3-29
Chapter 3 Material Models and Constitutive Relations
Creep and Viscoelastic Models
Creep
Creep is a time dependent strain produced under a state of constant stress. Creep is
observed in most engineering materials especially metals at elevated temperatures,
high polymer plastics, concrete, and solid propellant in rocket motors. Since creep
involves larger time scale than structural dynamics, its effect can be neglected in
dynamic analysis.
Creep curve is a graph between strain versus time. Three different regimes can be
distinguished in a creep curve; primary, secondary, and tertiary (see the following
figure). Usually primary and secondary regimes are of interest.
Figure 3-6. Creep Curve
ε
x
σ and T are constants
ε0
Primary
Range
Secondary
Range
Tertiary
Range
tR
t
An elastic creep analysis is available in NSTAR. Two creep laws based on an
“Equation of State” approach are implemented. Each law defines an expression for
the uniaxial creep strain in terms of the uniaxial stress and time.
In
de
x
Classical Power Law for Creep (Bailey-Norton law)
(3-12)
3-30
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
where:
T
= Temperature (°Kelvin) (= inputting temperature
+ reference temperature + offset temperature)
CT
= A material constant defining the creep temperaturedependency [command MPROP, CREEPTC, value
(default = 0) (Propsets > Material Property)]
Exponential Creep Law
(3-13)
Both laws represent primary and secondary creep regimes in one formulae. Tertiary
creep regime is not considered. In the above formula, the constants C0 to C6 are
creep constants that must be defined by CREEPC and CREEPX through the use of
MPROP (Propsets > Material Property) command based on the material creep
properties. “t” is the current real (not pseudo) time and σ is the total uniaxial stress
at time t.
To extend these laws to multiaxial creep behavior, the following assumptions are
made:
• The uniaxial creep law remains valid if the uniaxial creep strain and the uniaxial
stress are replaced by their effective values.
• Material is isotropic
• The creep strains are incompressible
For a numerical creep analysis, where cyclic loading may be applied, based on the
strain hardening rule, the current creep strain rates are expressed as a function of
the current stress and the total creep strain:
(3-14)
where:
In
de
x
= Effective stress at time t
= Total effective creep strain at time t
COSMOSM Advanced Modules
3-31
Chapter 3 Material Models and Constitutive Relations
= Components of the deviatoric stress tensor at time t
✍ Automatic Selection of step-size based on solution accuracy when AutoStepping is used.
Non proportional loadings, in particular, lead to considerable stress variations
from one step to next. The check on the creep strain increment is not enough to
ensure accuracy of creep strains. Additional Checks are added to limit the creep
strain rates using the total effective creep strain and ratio of the effective creep
strain increment to the total effective creep strain, based on the creep tolerance
that is input.
ORNL (Oak Ridge National Laboratory) auxiliary strain hardening rules are used
to extend creep behavior to cyclic loading conditions.
The creep models can be used with the following element groups:
•
•
•
•
•
TRUSS2D & TRUSS3D
PLANE2D 4/8 nodes
(plane stress, plane strain, and axisymmetric)
TRIANG 3/6 nodes
(plane stress, plane strain, and axisymmetric)
SOLID 8/20 nodes
TETRA4 & TETRA10
✍ Modification for Kelvin temperatures.
The absolute temperature is evaluated from:
T(absolute) = T + Tref + T(offset)
✍ Creep effects are included in evaluation of J-integral.
Linear Isotropic Viscoelastic Model
In
de
x
Elastic materials having the capacity to dissipate the mechanical energy due to
viscous effects are characterized as viscoelastic materials. In COSMOSM, a linear
isotropic viscoelastic material model is available in the time domain analysis. For
multiaxial stress state, the constitutive relation may be defined as:
3-32
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
(3-15)
where and φ are the deviatoric and volumetric strains; G(t-τ) and K(t-τ) are shear
and bulk relaxation functions. The relaxation functions can then be represented by
the mechanical model, (shown in Figure 3-7) which is usually referred to as a
Generalized Maxwell Model having the expressions as the following:
Figure 3-7. Generalized Maxwell Model
G0
η1
η2
η3
ηi
ηN
G1
G2
G3
Gi
GN
ε
Gi = G0 g i
ηi = Gi τGi
(3-16)
(3-17)
where G0 and K0 are instantaneous shear and bulk moduli; gi, ki, τiG, and τiK are the
i-th shear and bulk moduli and corresponding times.
The effect of temperature on the material behavior is introduced through the timetemperature correspondence principle. The mathematical form of the principle is:
In
de
x
(3-18)
COSMOSM Advanced Modules
3-33
Chapter 3 Material Models and Constitutive Relations
where γ t is the reduced time and γ is the shift function. In COSMOSM, the WLF
(Williams-Landel-Ferry) equation is used to approximate the function:
(3-19)
where T0 is the reference temperature which is usually picked as the Glass
transition temperature; C1 and C2 are material dependent constants.
The material properties for the viscoelastic model are input through the MPROP
(Propsets > Material Property) command. The required parameters include the
following:
Parameter
Linear elastic
parameters
Relaxation function
parameters
WLF equation
parameters
GEOSTAR Symbol
Description
EX
Elastic modulus
NUXY
Poisson’s ratio
GXY (optional)
Shear modulus
G1, G2, G3, ......, G8
represent g1, g2, ..., g8 in EQ. (3-16)
TAUG1, TAUG2, TAUG3, ......,
represent τ1G, τ2G, ..., τ8G in EQ. (3-16)
TAUG8
K1, K2, K3, ......, K8
represent k1, k2, ..., k8 in EQ. (3-17)
TAUK1, TAUK2, TAUK3, ......,
TAUK8
K
K
K
represent τ1 , τ2 , ..., τ8 in EQ. (3-17)
REFTEMP
represents T0 in EQ. (3-19)
VC1
represents C1 in EQ. (3-19)
VC2
represents C2 in EQ. (3-19)
The viscoelastic model can be used with the following element groups:
In
de
x
•
•
•
•
•
•
•
3-34
TRUSS2D & TRUSS3D
BEAM2D & BEAM3D
PLANE2D 4/8 nodes
(plane stress, plane strain, and axisymmetric)
TRIANG 3/6 nodes
(plane stress, plane strain, and axisymmetric)
SOLID 8/20 nodes
TETRA4 & TETRA10
SHELL3T & SHELL4T
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
✍ For TRUSS2D/3D elements, only uniaxial stress state is considered. The
relaxation functions are reduced to the extension part. The extension relaxation
moduli are input through K1, TAUK1,.... . The parameters G1, TAUG2, are not
required.
✍ For BEAM2D/3D elements, (1) is applied to the extension direction; G1,
TAUG1, ..... are needed only when the torsional degree of freedom is
considered.
✍ Creep strain is printed if requested (see the STRAIN_OUT (Analysis >
OUTPUT OPTIONS > Set Strain Output) command) which is defined as the
difference between the total mechanical strain and the linear strain.
Wrinkling Membrane
The Wrinkling Membrane material is usually used to model fabric tension
structures such as covers of indoor tennis courts and swimming pools. The
wrinkling membrane used in such structures is very thin and flexible. It has inplane stiffness but does not have any flexural stiffness. For most of the cases,
membrane prestresses are mechanically introduced in the fabric tension structures.
The prestress and the geometry give the membrane out-of-plane stiffness.
The shape of the structure is known in most structural analysis applications.
However, in the case of fabric tension structures, we first have to determine the
shape of the membrane surface. The equilibrium shape depends on the boundary
conditions and the prestress in the membrane. The above process is called shapefinding analysis and is usually an iterative process in which the user should try
several shapes.
Significant changes in the geometries and stresses might occur when membrane
structures are subjected to loads (wind or snow). Moreover, the membrane cannot
resist any compressive stresses, therefore, wrinkling occurs.
During the analysis, the strains and stresses in the principal directions are
calculated.
In
de
x
1.
If
ε1 > 0 and
ε1 > 0 and
ε2 > 0 or
ε2 ≤ 0 and
COSMOSM Advanced Modules
-ε2 / ε1 ≤ 0
3-35
Chapter 3 Material Models and Constitutive Relations
where ε1, ε2, and σ1, σ2 are strains and stresses in the principal
directions 1 and 2 and ν is the Poisson's ratio, then the wrinkling does
not occur. A linear elastic isotropic material matrix is used for the
membrane.
2.
If
ε1
> 0 and
ε2 ≤ 0 and
-ε2 / ε1 > ν
then wrinkling can occur. The membrane yields the stress state
ε1 = E ε1; and ε2 = 0.
3.
If
ε1 ≤ 0 and
ε2 ≤ 0
then biaxial wrinkling occurs and the element is inactive.
The material properties required for this material model are the same as in the
Linear Elastic-Isotropic material model.
It is noted that the Wrinkling Membrane model is supported by the plane stress
option of the PLANE2D and TRIANG elements and the membrane option of the
SHELL3/4T elements only.
Figure 3-8a. Fabric Tension Structure
Figure 3-8b. Principal Stresses in Membrane
Y
Z
σ2
σ1
In
de
x
X
3-36
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
A Bounding Surface Model for Concrete
Introduction
The concrete model is a three-dimensional, rate-independent model with a
bounding surface. The model adopts a scalar representation of the damage related
to the strain and stress states of the material. The bounding surface in the stress
space shrinks uniformly as the damage due to strain softening and/or tension cracks
accumulates. The material parameters depend on the damage level, the hydrostatic
pressure, and the distance between the current stress point and the bounding
surface.
The bounding surface function is:
where
In
de
x
σij is the normalized stress tensor (with respect to the ultimate compression strength
fc'), I1 and j2 are the first stress and the second deviatoric normalized stress
invariants respectively, θ is the loading angle, and kmax is the maximum damage
coefficient.
COSMOSM Advanced Modules
3-37
Chapter 3 Material Models and Constitutive Relations
Figure 3-9. Bounding Surface
σ ΙΙ
Stress Point
Hydroaxis
R
r
θ
Projection of
Tensile Semiaxis
Bounding Surface of
Lowe r Damage Level
Bounding Surface of
Highe r Damage Level
σΙ
σ ΙΙΙ
Damage Coefficient
The damage coefficient represents the damage due to strain hardening or softening.
The damage coefficient value is always positive and its magnitude in conjunction
with the hydrostatic pressure represents the damage due to compression and tension
cracking. For instance, the damage in a uniaxial compression test at the ultimate
strength is normalized to be 1.0 and its value is approximately 0.20 for uniaxial
tension test. The damage is obtained by integrating the incremental damage
coefficient that depends on the plastic strain and the distance from the current stress
state and the bounding surface.
In
de
x
where
3-38
HP
= Plastic shear modulus
FI(I1,θ)
is function of I1 and θ and loading conditions
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
D
= Normalized distance r/R
r
= Distance from the projection of the current stress point on the
deviatoric plane to the hydroaxis
R
= Distance of the bounding surface from the hydroaxis along
the deviatoric stress direction
The Model Parameters and Feature
The model is defined by two material parameters which are:
1.
FPC = the concrete ultimate strength (fc')
2.
EPSU = the ultimate strain (the strain at stress of fc' in the uniaxial compression
test, εo)
The low strain elasticity modulus (E), bulk modulus (Kt), and shear modulus
In
de
x
(HP) are set.
COSMOSM Advanced Modules
3-39
Chapter 3 Material Models and Constitutive Relations
The parameters are temperature independent. Moreover, the model should be
used in conjunction with small strain formulation.
The model can be used with the following element groups.
• TRUSS2D & TRUSS3D
• PLANE2D & TRIANG
• SOLID (8/20 nodes),
(plane stress, plane strain, and axisymmetric)
TETRA4 & TETRA10
Due to the strain softening, it is preferable to use the Displacement Control or the
Arc-Length Control technique with Newton-Raphson or Modified NewtonRaphson in the solution.
Figure 3-10. Uniaxial Compression Test
Tension Behavior
In
de
x
Under tension stresses, the model behaves as a nonlinear strain hardening material
until it reaches the tension strength and starts to behave as a perfectly plastic
material. The maximum tensile strength for uniaxial test is considered as:
3-40
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Strain Softening
In
de
x
In concrete structure under loads, some zones may reach failure before the overall
failure of the structure. Therefore, a realistic modeling of the post-failure behavior
is of primary importance.
COSMOSM Advanced Modules
3-41
Chapter 3 Material Models and Constitutive Relations
Stress-Strain Relationship
References
1.
Moussa, R. A. and Buykozturk, O., “A Bounding Surface Model for Concrete,”
Nuclear Engineering and Design, Vol. 121, pp 113-125, 1990.
2.
Chen, E. S., and Buyukozturk, O., “Damage Model for Concrete in Multiaxial
Cyclic Stress,” J. Engineering Mechanics, ASCE, 111(6) 1985.
User-defined Material Models
Material behavior is often complex, and the analysts are sometimes faced with
problems that require more sophisticated material models than those commonly
used. COSMOSM allows the users to define their own material models.
This option can be used with the following element groups:
In
de
x
•
•
•
•
•
•
•
3-42
TRUSS2D & TRUSS3D
PLANE2D 4/8 nodes
(plane stress, plane strain, and axisymmetric)
TRIANG 3/6 nodes
(plane stress, plane strain, and axisymmetric)
SOLID 8/20 nodes
TETRA4 & TETRA10
SHELL 3/4/6/3T/4T/6T
BEAM2D & BEAM3D
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
The user will be required to modify a FORTRAN subroutine named UMODEL in a
FORTRAN file named “NSUM.F” to be able to incorporate the new model(s). The
following sections describe the procedure for defining these models and linking
them to NSTAR. In addition, some useful Function statements, COMMON
statements, and subroutines are presented to provide the user with the information
needed to define a user's model.
Preparing the NSTAR Executable File
For Windows user’s the required files are stored in a subdirectory called NL User
Material Model off the COSMOSM installation directory. These files must be used
as explained in the following sections.
Requirements for Windows NT/2000
Microsoft Visual Studio Basic 6.0
Contents of the Program
The NSTAR material model utility files stored in the NL User Material Model
subdirectory consist of two sample files, the object library, and the make file which
creates the special NSTAR module. These files are as noted below:
FORTRAN sample files “Nsum.f” and Nsumc.F” for the user material model
and the user creep law, respectively.
2.
The Nstar.lib object library for Windows NT/2000 is required to create the
special NSTAR executable file.
3.
The make file “makex” which should be used to compile, link and generate the
NSTAR executable file for Windows NT/2000 platforms.
In
de
x
1.
COSMOSM Advanced Modules
3-43
Chapter 3 Material Models and Constitutive Relations
Procedures
1.
Modify subroutine UMODEL in the FORTRAN file named “Nsum.f” (and / or
subroutine CREPUM in file “nsumc.f” for the user creep laws).
2.
Modify the file “makex” to specify the proper COSMOSM directory where
Nstar.exe is to be generated (rename the current Nstar.exe to a different name to
save).
3.
For compilation and linking type the command: “nmake makex” in a DOS
window (nmake.exe nmakx.err are properties of Microsoft Linker).
The Nstar.exe file should then be copied into the COSMOSM directory. It is
recommended that you save (or rename) the original execution file for future use.
Model Definition Procedure
1.
Prepare the modified Nstar.exe file after modifying the UMODEL sub-routine
and appending user subroutines to the “NSUM.F” file (an example to show the
procedure for modifying the UMODEL will be presented in the next section).
2.
When using the EGROUP (Propsets > Element Group) command to define an
element group, assign a negative integer number in the range [-20, -1] to define
the type of material model (i.e., MODEL = [-20, -1].
3.
When using MPROP (Propsets > Material Property) command to define a
material property set, the user can input up to 20 additional properties (MCij,
i = 1, 6 and j = i, 6 and MC66 not included) to be used in the definition of the
user model.
Modifying the UMODEL Subroutine
Subroutine UMODEL calls the proper user subroutine(s) based on the material
model (MODEL) and the element type (IGTYP). Upon entrance, the STRAIN
vector is known. The user must define the STRESS vector and the stress-strain
matrix [SS], before returning to the main routine.
In
de
x
In addition, if there are state variables to be saved, they must be placed into
common VSTORE, as explained in the Useful COMMON statements section.
In the following, the UMODEL subroutine and its arguments are explained.
3-44
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
SUBROUTINE UMODEL
(STRAIN, STRESS, SS, NDIMS, IGTYP, NEL, IPT, NPT, IOUT, MODEL)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION STRAIN(NDIMS), STRESS(NDIMS), SS(NDIMS,NDIMS)
Variable
Description
STRAIN
Mechanical strain vector in global Cartesian coordinates (thermal
and creep strains are removed)
Green-Lagrange strains (for Total Lagrangian) or Almansi strains
(for Updated Lagrangian)
TRUSS2D, TRUSS3D,
BEAM2D or BEAM3D
STRESS
{E11} (Updated Lagrangian only)
PLANE2D or TRIANG
{E11,E22,E12,E33}
SOLID, or TETRA4/10
{E11,E22,E33,E12,E23,E13}
SHELL 3/4/6/3T/4T/6T*
{E11,E22,E123}
2nd Piola-Kirchhoff stresses (for Total Lagrangian) or Cauchy
stresses (for Updated Lagrangian) in the global Cartesian
directions
TRUSS2D/3D
{S11}(Updated Lagrangian only)
PLANE2D or TRIANG
{S11,S22,S12,S33}
SHELL 3/4/6/3T/4T/6T
{E11,E22,E12}
SOLID, or TETRA4/10
{S11,S22,S33,S12,S23,S13}
SHELL 3/4/6/3T/4T/6T*
{S11,S22,S12}
SS
Strain-Stress Matrix:
{STRESS} = [SS] {STRAIN}
NDIMS
Number of Strain or Stress Components
TRUSS2D, TRUSS3D
BEAM2D or BEAM3D = 1
PLANE2D or TRIANG = 4
SOLID, TETRA4, or TETRA10 = 6
SHELL 3/4/6/3T/4T/6T = 3
IGTYP
Element Group Type
PLANE2D or TRIANG (Axisymmetric option) = 0
PLANE2D or TRIANG (Plane-Strain option) = 1
PLANE2D or TRIANG (Plane-Stress option) = 2
In
de
x
SOLID, TETRA4, or TETRA10 = 3
TRUSS2D or TRUSS3D = 4
COSMOSM Advanced Modules
3-45
Chapter 3 Material Models and Constitutive Relations
Variable
Description
SHELL 3/4/6/3T/4T/6T = 5
BEAM2D or BEAM3D = 6
NEL
Element number
IPT
Location (Gauss point) number in the element
NPT
Total number of Locations (Gauss points) for this element
IOUT
Output unit number for printing
MODEL
User material model number + 100 [101-120]
*In the case of shells, the above stresses and strains refer to in-plane terms
only. To define out-of-plane (transverse) effects, common SHLCUR need to
be used (refer to the Useful Common statements section).
Example
SUBROUTINE UMODEL (STRAIN,STRESS,SS,NDIMS,
IGTYP,NEL,IPT,NPT,IOUT,MODEL)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION
STRAIN(NDIMS), STRESS(NDIMS), SS(NDIMS,NDIMS)
MODL = MODEL - 100
IGTP = IGTYP + 1
GO TO (10,20), MODL
10
GO TO (11,11,11,13,14,15,14), IGTP
C--PLANE2D /TRIANG
11 CALL MOD2D1
(STRAIN,STRESS,SS,NDIMS,
IGTYP,NEL,IPT,NPT,IOUT)
RETURN
C--SOLID /TETRA4 /TETRA10
13 CALL MOD3D1
(STRAIN,STRESS,SS,NDIMS,
IGTYP,NEL,IPT,NPT,IOUT)
RETURN
C--TRUSS2D / TRUSS3D / BEAM2D / BEAM3D
14 CALL MODTR1
(STRAIN,STRESS,SS,NDIMS,
IGTYP,NEL,IPT,NPT,IOUT)
C--SHELL3 / SHELL4
15 CALL MODSH1
(STRAIN,STRESS,SS,NDIMS,
IGTYP,NEL,IPT,NPT,IOUT)
In
de
x
RETURN
20 GO TO (21,21,21,23,24,25,24), IGTP
3-46
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
C--PLANE2D /TRIANG
21 CALL MOD2D2
(STRAIN,STRESS,SS,NDIMS,
IGTYP,NEL,IPT,NPT,IOUT)
RETURN
C--SOLID /TETRA4 /TETRA10
23 CALL MOD3D2
(STRAIN,STRESS,SS,NDIMS,
IGTYP,NEL,IPT,NPT,IOUT)
RETURN
C--TRUSS2D / TRUSS3D
24 CALL MODTR2
(STRAIN,STRESS,SS,NDIMS,
IGTYP,NEL,IPT,NPT,IOUT)
RETURN
END
SUBROUTINE UMINIT (IGTYP,MODEL)
Subroutine to request special treatments and/or additional information for the
user model formulation.
This subroutine is provided in the Fortran file NSUM.f, and is called internally
by the program per element group.
Based on the element type (IGTYP) and model number (MODEL>101), this
subroutine can be altered to activate (or deactivate) flags, to request:
1. Additional storage for the state variables
2. Nodal Coordinate per element
3. Evaluation (Gaussian Point) Location & Jacobian
4. SHELL formulation based on initial-configuration (Lagrangian Strains,
2nd Piola-Kirchhoff stresses)
5. Orthotropic transformation matrix
6. Deformation Gradient Tensor
The activation of each feature, allows for the common statement that is
associated with it to become available, during the model formulation.
Optional Common Statements:
In
de
x
IMPLICIT REAL*8 (A-H,O-Z)
COMMON /VSMORE/ MDROW, MOREX, VSMOR(63*10)
COMMON /ELXYZS/ EXYZ0(3,21), EXYZN(3,21), IELXYZ
COSMOSM Advanced Modules
3-47
Chapter 3 Material Models and Constitutive Relations
COMMON RRR,SSS,TTT, WEIGHT, JACOB, XXX,YYY,ZZZ,IPOSIT
COMMON /FMUTYP/ ICONF
COMMON /DIRECS/ ORTC(3,3), T(6,6), NORTH
COMMON /DGRADS/ FMTX(3,3), IDFGRD
Default Setting:
MOREX = 0 -> Additional storage is not required.
IELXYZ = 0 -> Nodal coordinates are not required.
IPOSIT = 0 -> Location information is not required.
ICONF
= 1 -> Shell formulation is based on the deformed geometry.
NORTH = 1 -> Orthotropic formulation is activated.
IDFGRD = 1 -> Deformation Gradient Tensor is required.
✍ For more information regarding the above common statements, see the Useful
COMMON Statements section.
Useful FUNCTION Statements to Access Information from Data Base
To Get Any Real Constant, Use Function
REAL*8
FUNCTION RCNST (i)
Variable
Description
i
real constant number)
Example
To assign the 5th real constant in the set for the current element to RX, set:
In
de
x
REAL*8 RX, RCNST
RX = RCNST (5)
3-48
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
To Get Any Material Property, Use Function
REAL*8
FUNCTION PROPRT (id, temp)
Variable
Description
id
property id number (see the table below)
temp
temperature (used only if property is temperature-dependent)
Example
To get modulus of elasticity, Poisson's ratio, and density for the current element,
set:
REAL*8 EX,NUXY,DENSITY, PROPRT,ETEMP, US1
EX = PROPRT (1,ETEMP)
NUXY = PROPRT (3,ETEMP)
DENSITY= PROPRT (16,ETEMP)
US1 = PROPRT (-36,ETEMP)
Table 3.1. Property ID Table
id =
1-EX
2-ALPX
3-NUXY
4-SIGYLD
5-ETAN
6-EY
id =
19-26 -> Hyperelastic Constants
-ij
13-GXZ
14-GYZ
15-VISCM
16-DENS
17-FRCANG
18-COHESN
-> User Defined Properties
where i and j are the indices associated with the user's
material properties constants MCij (i=1, 6 and j=i, 6 & MC66
not included). For example, id=-36 refer to MC36.
In
de
x
id =
7-EZ
8-NUXZ
9-NUYZ
10-ALPY
11-ALPZ
12-GXY
COSMOSM Advanced Modules
3-49
Chapter 3 Material Models and Constitutive Relations
Useful COMMON Statements to Access Information From Data Base
General Information Common
REAL*8 UTEMPO, UTEMPN, TIMCUR, TIMINC
COMMON /UVARBL/ UTEMPO, UTEMPN, TIMCUR, TIMINC, ISTEP,
IEQT, IREF, ISPR, LGDFM
Variable
Description
UTEMPN
Average element temperature at the current step
UTEMPO
Average element temperature at the last step
TIMCUR
Time value at the current step (t + ∆t)
TIMINC
Time increment for the current step (∆t)
ISTEP
Current step number
IEQT
Equilibrium iteration number
IREF
= 0 -> Stiffness is to be reformed (otherwise IREF>0)
ISPR
> 0 -> During stress printings (otherwise ISPR = 0)
LGDFM
= 0 -> Small deflection theory
= 1 -> Large displacement analysis
(Updated Lagrangian formulation, U.L.)
= 2 -> Large displacement analysis
(Total Lagrangian formulation, T.L.)
State Variables Common
Variables to be saved and recovered from one previous step, if any.
REAL*4 VSTOR
COMMON /VSTORE/ NEXIS, NDMV, VSTOR (126)
Variable
Description
NEXIS
> 0 -> previous values are restored.
In
de
x
= 0 -> parameters are initialized to zero (occurs only at the start of
solution) (not to be changed).
3-50
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Variable
Description
NDMV
Length of the variables to be stored (126 Real*4 or 63 Real*8) (not
to be changed).
VSTOR
Vector of state variables to be saved or recovered (to be changed).
Example
1.
Bring in A1(n1), A2(n2), A3(n3) from last step.
2.
Save these variables (updated to the current step by the user) to be used in the
next step.
Define the common as:
REAL*8 A1
REAL*4 A2
INTEGER A3
COMMON /VSTORE/ NEXIS,NDMV, A1(n1),A2(n2),A3(n3)
NDMV = (n1*2 + n2 + n3) < or = 126
Upon entrance, A1, A2, A3 represent values from last step. Therefore, They can be
used to define material law and stresses for the current step. Before leaving, these
parameters must be updated to represent values at the current step and will be saved
for use in the next step.
Deformation Gradient Tensor Common
REAL*8 FMTX
COMMON /DGRADS/ FMTX(3,3), IDFGRD
Variable
Description
IDFGRD
> 0 -> Deformation gradient matrix is required
= 0 -> Deformation gradient matrix is not used (default = 1)
FMTX
Deformation Gradient Tensor (only when IDFGRD>0)
T.L. -> Right deformation tensor
In
de
x
U.L. -> Left deformation tensor
COSMOSM Advanced Modules
3-51
Chapter 3 Material Models and Constitutive Relations
✍ If the deformation gradient tensor is not required, the user can set IDFGRD = 0
to save time.
Orthotropic Directions Transfo rmation Matrices
REAL*8 ORTC,ORTT
COMMON /DIRECS/ ORTC(3,3), ORTT(6,6), NORTH
Variable
Description
NORTH
> 0 -> Calculate the orthotropic transformation matrices
= 0 -> do not calculate orthotropic matrices (default =1)
ORTC
Matrix of the orthotropic direction cosines with respect to the global
coordinate system
T.L. -> Global undeformed coordinates
U.L. -> Global deformed coordinates
[Global Stress Tensor] =
[ORTC] * [Local Stress Tensor] * [ORTC(transpose)]
[ORTC] = [Tij], i=1,3 & j=1,3
Tij = Cosine angle between jth orthotropic direction and
the ith global coordinate
ORTT
Transformation matrix between the orthotropic directions and the
global coordinate system (when stress or strain are represented by
vectors)
{Global Stress Vector} =
[ORTT(transpose)] * {Local Stress Vector}
[SS(global)] = Constitutive law in global system =
[ORTT(transpose)] * [SS(local)] * [ORTT]
[ORTT] = [TTij], i=1,nc & j=1,nc
nc = number of stress (or strain) components
✍ If the user model is not orthotropic, user can set NORTH = 0 to avoid
In
de
x
unnecessary calculations.
3-52
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Additional State Variables Common
Provides additional storage for the state variables in excess of the storage capacity
of common VSTORE.
REAL*4
VSMOR
COMMON /VSMORE/ MDROW, MOREX, VSMOR(126*10)
Variable
Description
MDROW
> 0 -> Previous values are restored.
= 0 -> Variables are initialized to zero. (not to be altered)
MOREX
= 0 -> Additional storage is not required.
= m: 1<m<10 -> m*126 additional variables are needed to
be saved &
VSMOR
Vector of additional state variables to be saved and recovered.
✍ Parameter MOREX must be reset (if non-zero) in subroutine UMINIT,
according to the element type and model.
✍ Select m such that: (m-1)*126 < no. of additional variables < m*126
Element Nodal Coordinates Common
Provides information regarding the initial & current positions of the nodes
associated with the element
REAL*8
EXYZ0,EXYZN
COMMON /ELXYZS/ EXYZ0(3,21), EXYZN(3,21),IELXYZ
Variable
Description
IELXYZ
=1 -> Provide nodal coordinates
=0 -> Do not provide nodal coordinates
In
de
x
EXYZ0(i,N) Coordinate i of the Nth node of the element (Initial
position)
EXYZN(i,N) Coordinate i of the Nth node of the element (Current
position)
COSMOSM Advanced Modules
3-53
Chapter 3 Material Models and Constitutive Relations
✍ 1< i < 3, represents one of the 3 global directions.
✍ 1< N < Number of nodes per element
✍ Parameter IELXYZ must be reset in subroutine UMINIT, according to the
element type and model.
Element Nodal connectivity Common
COMMON /ELNODS/ NENOD(21), NNODMX, NELCUR
Variable
Description
NELCUR
Element id number
NNODMX
Number of nodes for this element
NENOD
Vector of Node numbers
Evaluation Point Information Common
Provides information regarding the location of the point, inside the element, for
which the U-Model subroutine is called.
REAL*8 RRR,SSS,TTT,WEIGHT,JACOB,XXX,YYY,ZZZ
COMMON /POSITN/ RRR, SSS, TTT, WEIGHT, JACOB, XXX, YYY, ZZZ,
IPOSIT
Variable
Description
IPOSIT
=1 -> Provide information about location
=0 -> Do not provide location information
RRR,SSS,TTT
Gaussian coordinates for this point [-1 to 1]
WEIGHT
Multiplier for integration
JACOB
Jacobian evaluated at this point
XXX,YYY,ZZZ Global coordinates for this point
In
de
x
Element Type Exceptions
3-54
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
PLANE2D and TRIANG
TTT:
not used.
ZZZ:
Thickness at this point
Associated Volume = Weight *|JACOB| * Thickness
SOLID and Tetrahedral elements
Associated Volume = Weight *|JACOB|
All Shell elements
RRR,SSS:
not used.
TTT:
Gaussian coordinate in the thickness direction [-.5 to .5]
JACOB:
Initial AREA
Associated Volume = Weight * Thickness *AREA
Beam elements
RRR:
Gaussian coordinate in the Length direction [0 to 1]
SSS,TTT:
Gaussian coordinates on the Cross Section [-.5 to .5]
JACOB:
AREA of the Cross Section
XXX,YYY, ZZZ:
Coordinates with respect to the element system
Associated Volume = Weight * Length * AREA
Solution Termination Common
To Stop/Restart solution when errors are detected.
In
de
x
COMMON /AUTOFLG/ IAUTSTP, JCHECK, KSTOP
Variable
Description
JCHECK
= 9 -> Set this parameter =9 when errors are detected. It
prompts to Stop the solution, or restart the step with a
smaller time increment in case of auto-stepping.
Previous-Step Information Common for Beam Elements
COSMOSM Advanced Modules
3-55
Chapter 3 Material Models and Constitutive Relations
Additional information that is provided in regards to the previous solution step:
Time=t, and can be used in the current solution step: Time=t+dt)
REAL*8
EPST,SIGT,GTAN
COMMON/ BMLAST/ EPST, SIGT, GTAN
Variable
Description
EPST
Axial Strain at time=t
SIGT
Axial Stress at time=t
GTAN
Current Shear Strength for Beam with torsion
= Gxy (GTAN can be changed in the U-MODEL routines
for BEAM3D with torsion)
Formulation Type common for Shell Elements
Defines whether the material model formulation is based on the element’s initial
state, or the deformed state.
COMMON/FMUTYP/ ICONF
Variable
Description
ICONF
= 1 -> Formulation is based on the current state of deformation
[Eulerian Strains, Cauchy Stresses]
= 0 -> Formulation is based on initial configuration at time 0
[Green-Lagrange Strains, 2nd PK stresses]
✍ Parameter ICONF must be set in subroutine UMINIT, based on the model
formulation for shell elements (IGTYP=5).
✍ Among the available material models for SHELL in NSTAR, only the Hyper-
In
de
x
elastic models are formulated based on the initial configuration of the element.
3-56
COSMOSM Advanced Modules
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Right Cauchy-Green Strains Common for Shell Elements
This strain tensor is evaluated from Lagrangian strains, based on the element’s
initial configuration.
REAL*8
C11,C22,C33,C12,C23,C13
COMMON /CGDEFO/ C11, C22, C33, C12, C23, C13
Variable
Description
C11,…,...
C13
Components of the right Cauchy-Green Tensor
Previous-step Information Common for Shell Elements
Additional information that is provided in regards to the previous solution step:
(Time=t), which can be used in the solution of the current step (Time=t+dt).
REAL*8
STRNT,STNTH,THICKT,AREAT,AREAN
COMMON /SHLAST/ STRNT(6), STNTH(4), THICKT, AREAT, AREAN
Description
STRNT
Mechanical Strains at time t:
{E11t, E22t, E12t, E33t, E13t, E23t}
STNTH
Thermal Strains at time t:
{ETh11, Eth22, Eth12, Eth33}
THICKT
Thickness at time t
AREAT
Area at time t
AREAN
Area at time t+dt
In
de
x
Variable
COSMOSM Advanced Modules
3-57
Chapter 3 Material Models and Constitutive Relations
Transverse (Out-of-Plane) Strains/Stresses Common for Shell Elements
Additional information that is required to be defined in the U-MODEL routine for
the current solution step: time=t+dt.
REAL*8
GAMA,TAU,TAG,EPSZ,ENRGY
COMMON/SHLCUR/ GAMA(2), TAU(2), TAG(2,2), EPSZ, ENRGY
Variable
Description
GAMA
Transverse Shearing Strains at time t+dt (known)
{E13, E23}
TAU
Transverse Shearing Stresses at time t+dt (to be defined)
{S13, S23}
TAG
Current Transverse Shearing strain-stress matrix.
d{TAU}=[TAG].d{GAMA} (to be defined)
EPSZ
Strain normal to the plane at time t+dt (optional).
{E33}
ENRGY
Energy at time t+dt (optional).
{E33}
✍ Here the shearing strains, {GAMA}, are provided by the program. The
transverse shear stresses, {TAU}, and the GAMA-TAU matrix, [TAG], must be
defined by the user.
✍ d{TAU} and d{GAMA} are used to represent variations of the shearing
stresses & strains, at time t+dt.
✍ The normal-to-plane strain component, EPSZ, is used in evaluation of the
current thickness. It is initially evaluated based on the assumption that the
material is incompressible. However, it can be corrected in the U-Model
routine, for compressible materials.
In
de
x
✍ The last parameter, ENRGY, is used only for printout.
3-58
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Useful Subroutines
Subroutine to Form Transformation Matrix Between Vectors
of Stress or Strain in Different Coordinate Systems
REAL*8 T(3,3), TT(LNG,LNG)
CALL SYSTRF (TT,T,IGTYP,LNG)
If [T] relates stress (or strain) tensors in two systems a and b:
[Tensor(a)] = [T] * [Tensor(b)] * [T(transpose)]
Then, [TT] is defined such that
{Vector(a)} = [TT(transpose)] *{Vector(b)}
[TT] is evaluated according to:
ij
= refer to stress components in system (a)
mn = refer to stress components in system (b)
✍ Regardless of the size (LNG), the rows and columns of [TT] to be defined
In
de
x
depend on the size of the stress vector and the order in which it is defined
according to the element type. For example, in the case of PLANE2D elements,
only the first four rows and columns of [TT] are evaluated. Rows 1,2, and 4
correspond to the normal components in the x, y, and z directions, and row 3
represents the shear term in the xy-plane.
COSMOSM Advanced Modules
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Chapter 3 Material Models and Constitutive Relations
Subroutine to Get [TC] = [TA] (transpose) * [TB]
REAL*8 TA(NA1,NA2), TB(NB1,NB2), TC(NC1,NC2)
CALL MULATB (TA, TB, TC, NA1, NA2, NB1, NB2, NC1, NC2)
NA1, NA2 = number of rows, columns in [TA]
NB1, NB2 = number of rows, columns in [TB]
NC1, NC2 = number of rows, columns in [TC]
✍ Since the rows and columns may not be equal for the three matrices, minimums
of (NA2, NC1), (NA1, NB1), and (NB2,NC2) are used in matrix multiplication.
User-Defined Creep Laws
This feature allows the user to create his/her own creep law and implement it into
NSTAR in a procedure similar to the one used for user material models.
Accordingly, a library of NSTAR object files will be provided to the customer plus
directions as to how to code his/her own creep law and link with the NSTAR
library. The user creep law(s) can be used to model creep behavior in conjunction
with most available or user defined material models in NSTAR.
A creep law gives an expression for the creep behavior in an uniaxial environment;
In the multi-axial case, the uniaxial stresses or strains are replaced by their effective
(or equivalent) values in the creep formulation. Here, the user is only required to
provide the uniaxial expression that yields the uniaxial creep strain rate; the
extension of the model to the multi-axial case (based on the element type), the
considerations for stress reversals, and the accumulation and storage of pertinent
data is done internally by NSTAR.
User creep laws can be used with the following element groups:
In
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x
•
•
•
•
•
3-60
TRUSS2D & TRUSS3D
PLANE2D 4/8 nodes (plane-stress, plane-strain, axisymmetric)
TRIANG 3/6 nodes (plane-stress, plane-strain, axisymmetric)
SOLID 8/20 nodes
TETRA4 & TETRA10
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
The user will be required to modify a FORTRAN subroutine named CREPUM in a
FORTRAN file named 'NSUMC.F' to be able to incorporate the new creep law(s).
The linking procedure is already explained in the section regarding the user-defined
material models (Replace 'NSUM.F' by 'NSUMC.F' or just add 'NSUMC.F' to the
list of files to be used). The following sections define the procedure for defining
user creep laws.
Model Definition Procedure
1.
Prepare the modified NSTAR.EXE file after the CREPUM subroutine and
appending user subroutines to the 'NSUMC.F' file are completed (an example to
show the procedure for modifying the CREPUM routine will be presented in the
next section).
2.
When using the EGROUP command to define an element group, assign a
negative integer number (-1,-2,...,-n) for the creep option to define the type of
user creep law.
3.
When using MPROP command to define a material property set, the user can
input up to 21 creep properties (Mcij, i=1,6,j=i,6) to be used in the definition of
the user creep law. If the user creep law is defined in conjunction with a user
material model, the above constants can be used to specify properties for both
the material model and the creep law (for example, the user can assign MC11,
MC12, and MC13 to represent properties for the user model, while MC22,
MC23, MC24, MC25 are to serve as creep properties.
Modifying the CREPUM Subroutine
Subroutine CREPUM calls the proper user subroutine(s) based on the creep law
(independent of element type). Upon entrance, current effective creep strain,
current effective stress, and current temperature is known. The user must define the
current effective creep strain rate before returning to the main routine.
✍ Creep behavior under stress reversals is based on the ORNL auxiliary hardening
rules for all creep laws.
In
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x
✍ The user does not need to be concerned with storage of data, since at each time
step, the current creep strains and the current creep origins (for stress reversals)
are stored and recovered internally by NSTAR.
COSMOSM Advanced Modules
3-61
Chapter 3 Material Models and Constitutive Relations
✍ Function statements RCNST & PROPRT, as explained in the user material
model section, can also be used for the user creep model to access information
from data base.
✍ Among the common blocks defined for user material model, only common
block UVARBL can be used in the user creep subroutine to access information
such as current time, current time increment, element temperature, and etc.
✍ The customer is responsible to have the proper Compiler and linker (See section
on user-defined material models).
In the following, the CREPUM subroutine and it's arguments are explained. In
addition, an ex-ample of a power creep law is given.
SUBROUTINE CREPUM
(EDOT,EHBAR,SIGBAR,TEMP,TREF,TOFSET,NCTYPU,NEL,IPT,IOUT)
IMPLICIT REAL*8 (A-H,O-Z)
Variable
Description
EDOT
Effective creep strain rate (to be defined)
EHBAR
Effective Creep Strain
SIGBAR
Effective Stress
TEMP
Current Temperature = Temp(t+dt/2)
Reference temperature
TOFSET
Offset units for evaluation of absolute temperature
NCTYPU
id number for the user creep law = -1,-2,...
NEL
Element number
IPT
Location (Gauss point) number in the element
IOUT
Output unit number for printing
In
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x
TREF
3-62
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Example
NCTYPE=-NCTYPU
GO TO (100,200,900,900,900) NCTYPE
C
C** NCTYPE=1 -> User creep law type 1
C
100 CONTINUE
CALL CREEP1 (EDOT,EHBAR,SIGBAR,TEMP,TREF,TOFSET,NEL,IPT,IOUT)
RETURN
C
C
C** NCTYPE=2 -> User creep law type 2
C
200 CONTINUE
CALL CREEP2 (EDOT,EHBAR,SIGBAR,TEMP,TREF,TOFSET,NEL,IPT,IOUT)
RETURN
C
C
C** Non-existent Creep Models:
C
900 CONTINUE
RETURN
C
END
Example
Define Effective Creep Strain rate for a power law given by:
(1/sec)
Where effective stress (SIGBAR) is in ksi, and T is Temperature in Fahrenheit. Lets
assume that the creep constants are input by User constants:
In
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x
MPROP, , MC11, 4.64E-8
MPROP, , MC12, 12.5
MPROP, , MC13, 53712
And,
COSMOSM Advanced Modules
3-63
Chapter 3 Material Models and Constitutive Relations
TOFSET, 460.
TREF, 70.
(Note that the above law is equivalent to the NSTAR classical power law with
C2 = 1.0)
Coding
SUBROUTINE CREEP1
(EDOT,EHBAR,SIGBAR,TEMP,TREF,TOFSET,NEL,IPT,IOUT)
IMPLICIT REAL*8 (A-H,O-Z)
C0 = PROPRT(-11,TEMP)
C1 = PROPRT(-12,TEMP)
CT = PROPRT(-13,TEMP)
TKLVN = TEMP + TREF + TOFSET
EDOT = C0 * (SIGBAR)**C1 * DEXP(-CT/TKLVN)
RETURN
END
Strain Output
Five types of strain output are available in NSTAR namely,
1. Total strain
2. Thermal strain
3. Creep strain
4. Plastic strain
In
de
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5. Principal strain
3-64
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
For Small Displacement formulation
(Element Group Op. 6 = 0) for all structural
elements (trusses, beams, pipes, and shells)
which are not associated with hyperelastic
or large strain plastic material models, the
infinitesimal strains are output. For large
strain plasticity model, the logarithmic strains
are output. Otherwise, the Unit Extensions are
adapted, which are defined as follows:
1.
Figure 3-11
X2
dl1
eR: Extension ratio with respect to the
original length in the direction XR (R = 1,
2, or 3 to denote the direction).
dL1
X1
where
λR
= Stretch ratio in the direction XR
= Lagrangian strain tensor
2.
g12, g23, g13: Change of angle
between two axes originally in
the XR and XS directions
θRs
= θR + θs
Figure 3-12
X2
= π/2 - ϑRs
dl 2
= sin-1 (cos ϑRs)
θ2
ϑ1 2
dl1
θ1
dL 2
X1
In
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dL 1
COSMOSM Advanced Modules
3-65
Chapter 3 Material Models and Constitutive Relations
3.
Logarithmic strain :
Principal strains are defined as follows:
1.
Small displacement or large strain plasticity formulation:
where
= infinitesimal strain tensor or logarithmic strain tensor
(large strain plasticity)
2.
e
= principal strain
δij
= 1 if i = j and δij = 0 otherwise
Large displacement formulation (excluding large strain plasticity):
e = λ -1
where
= Left and Right Cauchy-Green strain tensors
= Principal stretch ratio
In
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λ
3-66
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Automatic Determination of Material Properties
from Test Data
NSTAR offers automated facilities (curve-fitting) for the analyst to determine
material constants for various models, namely, Mooney-Rivlin, Ogden, and
viscoelasticity. Two commands, MPCTYPE (LoadsBC > FUNCTION CURVE >
Material Curve Type) and MPC (LoadsBC > FUNCTION CURVE > Material
Curve), are required in the data entry level. The commands and the corresponding
options are explained below:
MPCTYPE
Geo Panel: LoadsBC > FUNCTION CURVE > Material Curve Type
For Mooney-Rivlin and Ogden Models
Variable
Description
type
type of material property curve
= 2 Mooney-Rivlin
= 3 Ogden
icode
Decimal code for experiment type, IJK
I = 1 Uniaxial test is present
J = 1 Plane strain test is present
K = 1 Equibiaxial test is present
number1
Number of terms for approximation
= 2(linear), or 5(quadratic), or 6(cubic) for Mooney-Rivlin model
= 1 to 4 for Ogden model
Minimum and maximum values of ALPH(1),...
In
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value1,
value2,
etc.:
COSMOSM Advanced Modules
3-67
Chapter 3 Material Models and Constitutive Relations
✍ Notes:
1. An experiment can be used alone or combined with the other experimnts.
For example, IJK = 101 means both uniaxial and equibiaxial test data are
present.
2. To obtain a set of accurate material constants for Mooney-Rivlin model, both
uniaxial and plane strain tests should be employed. Equibiaxial test data can
also be collected.
3. For Mooney-Rivlin model, the singular use of plane strain data is not
acceptable because of the linear dependency of the material constants in the
plane strain solution.
4. Number of terms might be increased inside the program. The preset criteria
are listed below for Ogden model:
with uniaxial and/or plane strain only:
- strain > 125% ---> 2-term approximation
with equibiaxial added:
- strain > 125% ---> 3-term approximation
- strain > 700% ---> 4-term approximation
5. It is suggested that for the Mooney-Rivlin model the analyst use 5-term
approximation when strain > 125% and 6-term approximation when strain >
600%.
6. Value1 and the rest are only for Ogden model. For 1-term approximation,
value1 and value2 are required. Likewise are 2-term, etc. Default values will
be used if they are not input.
For Viscoelastic Model
Variable
Description
type
type of material property curve
=4
icode
Decimal code for relaxation type, IJK
I = 1 Shear relaxation function is present
J = 1 Bulk relaxation function is present
K is not used (= 0)
number1
Number of terms for shear relaxation approximation
number2
Number of terms for bulk relaxation approximation
In
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x
= 1 to 8
= 1 to 8
3-68
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
✍ Nonzero value of number1 is input only when shear relaxation is present.
Likewise is for number2.
MPC
Geo Panel:
LoadsBC > FUNCTION CURVE > Material Curve
For Mooney-Rivlin and Ogden models, stress is defined as the nominal stress, i.e.,
force/original area (> 0) and strain is defined as the stretch ratio, i.e., final length/
original length (> 0). For viscoelastic model, the definition of stress versus strain is
replaced by relaxation function versus time.
✍ Multiple stress-strain curves can be input under the same material set number.
The input sequence is: uniaxial, plane strain, and equibiaxial.
✍ The relaxation function must be in a descending order starting with a non-zero
time.
✍ The instantaneous shear and bulk moduli are calculated in the program by using
properties EX and NUXY.
Evaluation Process
After the experimental data are input, the nonlinear program mathematically fits the
data. There are a few criteria preset inside the program for the evaluation process.
They are listed below:
Mooney-Rivlin constants A+B must be greater than zero.
2.
Summation of Ogden constants ALPH(I) * MU(I) must be greater than zero.
3.
Within certain range of ALPH(I), the program should be able to find a set of
ALPH(I) and MU(I) which minimize the error of stresses between the
experimental and estimated values.
In
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1.
COSMOSM Advanced Modules
3-69
Chapter 3 Material Models and Constitutive Relations
4.
The ith relaxation modulus G(I) or K(I) should contain a positive sign.
5.
Summation of shear relaxation modulus G(I) and summation of bulk relaxation
modulus K(I) should be less than the instantaneous moduli.
In the output file the analyst will see the comparison of the experimental data
versus estimated ones as well as the error of stresses or moduli. The error is defined
as below:
where S(i) is the measured stress or modulus at the ith given stress ratio or time and
Se(i) is the estimated one from the formula. The program also creates an ASCII file
named problem-name.PLT with the same format as the user-created file. The
analyst can use commands ACTXYUSR (Display > XY PLOTS > Activate User
Plot), etc. to plot the input data versus the estimated one for verification. Finally the
program creates a set of necessary material constants for the material model chosen
by the analyst.
It is important that the estimated material parameters fit the test data within the
range of strain appearing in the actual nonlinear analysis. Therefore, it is suggested
that the analyst always review the comparison either from the output file or the
user-created file before the actual nonlinear analysis is performed.
Examples
Some examples are shown below to illustrate the curve-fitting procedure:
Mooney-Rivlin Model
1.
Input commands:
In
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x
Geo Panel: Propsets > Element Group (EGROUP)
EGROUP,1,PLANE2D,0,2,0,0,3,2,0,
C* uniaxial, plane strain, and equibiaxial tests present
C* 6-term approximation
3-70
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Geo Panel: LoadsBC > FUNCTION CURVE > Material Curve Type (MPCTYPE)
MPCTYPE,1,2,111,6,
C* uniaxial test data
Geo Panel: LoadsBC > FUNCTION CURVE > Material Curve (MPC)
MPC,1,0,1,....
C* plane strain test data
MPC,1,0,23,....
C* equibiaxial test data
MPC,1,0,36,....
2.
Results in output file:
Curve Fitting for Mooney-Rivlin Material Constants:
Material Property Set 1
6-term approximation
Term Number
Mooney-Rivlin Constant
1
0.168951
2
0.007546
3
-0.001034
4
-0.000094
5
-0.000001
6
0.000038
Principal
Stretch Ratio
Stress (Data in
Uniaxial)
Stress
(Theory)
0.1031E+01
0.1640E-01
0.3181E-01
.....
.....
0.5303E+01
0.5235E+01
In
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.....
0.7375E+01
COSMOSM Advanced Modules
3-71
Chapter 3 Material Models and Constitutive Relations
Principal
Stretch Ratio
Stress (Data in
Plane Strain)
Stress
(Theory)
0.1063E+01
0.4760E-01
0.8135E-01
.....
.....
.....
0.4952E+01
0.1856E+01
0.1793E+01
Principal
Stretch Ratio
Stress (Data in
Equibiaxial)
Stress
(Theory)
0.1033E+01
0.8060E-01
0.6473E-01
.....
.....
.....
0.4429E+01
0.2465E+01
0.2465E+01
Stress Error = 0.2949E+00
3.
Graphical display from user file:
In
de
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Figure 3-13. Comparison of Results for Uniaxial Test
3-72
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure 3-14. Comparison of Results for Plane Strain Test
In
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Figure 3-15. Comparison of Results for Equibiaxial Test
COSMOSM Advanced Modules
3-73
Chapter 3 Material Models and Constitutive Relations
Ogden Model
1.
Input commands:
Geo Panel: Propsets > Element Group (EGROUP)
EGROUP,1,PLANE2D,0,2,0,0,6,2,0,
C* uniaxial, plane strain, and equibiaxial tests present
C* 3-term approximation
C* Initial trial range: ALPH(1): (1.,2.), ALPH(2): (4.5,5.5),
C* ALPH(3): (-2.5,-1.5)
Geo Panel: LoadsBC > FUNCTION CURVE > Material Curve Type
(MPCTYPE)
MPCTYPE,1,3,111,3,1.,2.,4.5,5.5,-2.5,-1.5,
C* Using the same stress-strain curves as Mooney-Rivlin model......
2.
Results in output file:
Curve Fitting for Ogden Material Constants:
Material Property Set 1
3-term approximation
ALPHI
MUI
1
1.100000
0.738110
2
4.800000
0.002134
3
-2.100000
-0.008790
Principal
Stretch Ratio
Stress (Data in
Uniaxial)
Stress
(Theory)
0.1031E+01
0.1640E-01
0.3765E-01
.....
.....
.....
0.7375E+01
0.5303E+01
0.5111E+01
In
de
x
Term Number
3-74
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Principal
Stretch Ratio
Stress (Data in
Plane Strain)
Stress
(Theory)
0.1063E+01
0.4760E-01
0.9672E-01
.....
.....
.....
0.4952E+01
0.1856E+01
0.1823E+01
Principal
Stretch Ratio
Stress (Data in
Equibiaxial)
Stress
(Theory)
0.1033E+01
0.8060E-01
0.7794E-01
.....
.....
.....
0.4429E+01
0.2465E+01
0.2488E+01
Stress Error = 0.4977E+00
3.
Graphical display from user file:
In
de
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Figure 3-16. Comparison of Results for Uniaxial Test
COSMOSM Advanced Modules
3-75
Chapter 3 Material Models and Constitutive Relations
Figure 3-17. Comparison of Results for Plane Strain Test
In
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Figure 3-18. Comparison of Results for Equibiaxial Test
3-76
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Viscoelastic Model
1.
Input commands:
Geo Panel: Propsets > Element Group (EGROUP)
EGROUP,1,PLANE2D,0,2,1,0,8,0,0,
C* Instantaneous elastic moduli
Geo Panel: Propsets > Material Property (MPROP)
MPROP,1,EX,9152,
MPROP,1,NUXY,0.3,
C* shear and bulk relaxation functions present
C* 8- and 2-term approximation for shear and bulk relaxation
C* functions respectively
Geo Panel: LoadsBC > FUNCTION CURVE > Material Curve Type
(MPCTYPE)
MPCTYP,1,4,110,8,2,
C* shear relaxation test data
Geo Panel: LoadsBC > FUNCTION CURVE > Material Curve
(MPC)
MPC,1,0,1,....
C* bulk relaxation test data
MPC,1,0,81,....
2.
Results in output file:
Curve Fitting for Viscoelastic Material Constants
Material Property Set 1
In
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8-term approximation
COSMOSM Advanced Modules
3-77
Chapter 3 Material Models and Constitutive Relations
Term Number
GI
TAUGI
1
0.34245
0.00017
2
0.12260
0.00500
3
0.58632
0.0500
4
0.44963
0.39043
5
0.29849
0
6
0.53631
2.15719
7
0.019014
50.000000
8
0.002623
95.000000
Shear Relaxation Function
Time
Data
Theory
0.5000E-05
0.3221E+04
0.3217E+04
.....
.....
.....
0.5000E+03
0.5112E+03
0.5097E+03
Relaxation Function Error = 0.2829E+03
2-term approximation
Term Number
KI
TAUKI
1
0.508247
0.050720
2
0.252678
5.759163
Bulk Relaxation Function
Time
Data
Theory
0.1500E-02
0.7513E+04
0.7513E+04
.....
.....
.....
0.1500E+02
0.2002E+04
0.1966E+04
In
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Relaxation Function Error = 0.1545E+03
3-78
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
3.
Graphical display from user file:
Figure 3-19. Comparison of Results for Shear Relaxation Function
TIM E
Figure 3-20. Comparison of Results for Bulk Relaxation Function
In
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TIM E
COSMOSM Advanced Modules
3-79
Chapter 3 Material Models and Constitutive Relations
References
Bathe, K. J., Dvorkin, E., and Ho, L. W., “Our Discrete-Kirchhoff and Isoparametric Shell Elements for Nonlinear Analysis- An Assessment,” Computers
& Structures, Vol. 16, pp. 89-98, 1983.
2.
Bathe, K. J., Finite Element Procedures in Engineering Analysis, Prentice-Hall,
1982.
3.
Blatz, P. J. and Ko, W. L., “Application of Finite Elastic Theory to the
Deformation of Rubbery Materials,” Transactions of the Society of Rheology,
Vol. 6, 1962, pp. 223-251.
4.
Chen, W. F., and, Mizunu, E., Nonlinear Analysis in Soil Mechanics, Elsevier,
1990.
5.
Chen, W. F., Plasticity for Structural Engineers, Springer-Verlag, 1988.
6.
Chen, W. F., Plasticity in Reinforced Concrete, McGraw-Hill, 1982.
7.
Chen, W. F., and Saleeb, A. F., Constitutive Equations for Engineering
Materials, Vol. 1, Elasticity and Modeling, John Wiley, 1981.
8.
Christensen, R. M., Theory of Viscoelasticity, Second edition, 1982.
9.
Drucker, D. C., and Prager, W., “Soil Mechanics and Plastic Analysis or Limit
Design,” Quarterly of Applied Mathematics, Vol. 10, pp. 157 165, 1952.
10.
Hill, R., The Mathematical Theory of Plasticity, Oxford University Press,
London, 1950.
11.
T. J. R. Hughes, “Numerical Implementation of Constitutive Models: Rate
Independent Deviatoric Plasticity,” Theoretical Foundation for Large-Scale
Computations for Non-linear Material Behavior (eds. S. Nemat-Nasser, etc.),
Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1984.
12.
Kao, B. G., and, Razgunas, L., “On the Determination of Strain Energy
Functions of Rubbers,” Proc. VI International Conference on Vehicle Structural
Mechanics, Detroit, pp. 124-154, 1986.
In
de
x
1.
3-80
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Knowles, J. K., and Eli Sternberg, “On the Ellipticity of the Equations of
Nonlinear Elastostatics for a Special Material,” Journal of Elasticity, Vol. 5,
Nos. 3-4, 1975, pp. 341-361.
14.
Kraus, H., Creep Analysis, Wiley-Interscience, New York, 1980.
15.
Ogden, R. W., “Large Deformation Isotropic Elasticity - on the Correlation of
Theory and Experiment for Incompressible Rubberlike Solids,” Proc. R. Soc.
Lond. A. 326, 565-584 (1972).
16.
Owen, D. R. J., and Hinton, E., Finite Elements in Plasticity: Theory and
Practice, Pineridge Press, Swansea, U.K., 1980.
17.
Peeken, H., Dopper, R., and Orschall, B., “A 3D Rubber Material Model
Verified in a User-Supplied Subroutine,” Computers & Structures, Vol. 26, pp.
181-189, 1987.
18.
Snyder, M. D., and, Bathe, K. J., “Formulation and Numerical Solution of
Thermo-Elastic-Plastic and Creep Problems,'' Report 82448-3, Department of
Mechanical Eng., MIT, 1977.
19.
T. Sussman and K. J, Bathe, “A Finite Element Formulation for Nonlinear
Incompressible Elastic and Inelastic Analysis”, Computers & Structures, Vol.
26, No. 1/2, pp. 357-409, 1987.
20.
G. G. Weber, A. M. Lush, A. Zavaliangos, and L. Anand, “An Objective TimeIntegration Procedure for Isotropic Rate-Independent and Rate-Dependent
Elastic-Plastic Constitutive Equations”, International Journal of Plasticity, Vol.
6, pp. 701-744, 1990.
21.
Zienkiewicz, O. C., and, Taylor, R. L., The Finite Element Method, Fourth
edition, Vol. 2, 1991.
22.
S. W. Key, C. M. Stone, and R. D. Krieg (1981),” Dynamic Relaxation Applied
to the Quasi-Static Large Deformation, Inelastic Response of Axisymmetric
Solids,” pp. 585-620 in Nonlinear Finite Element Analysis in Structural
Mechanics, W. Wunderlich et al. (eds.), Springer-Verlag, Berlin.
In
de
x
13.
COSMOSM Advanced Modules
3-81
Chapter 3 Material Models and Constitutive Relations
Birth and Death of Elements
The element birth and death feature allows you to include elements in
some stages of the solution and exclude them in other stages. Elements are excluded from participating in the solution using the EKILL
(Analysis, Nonlinear, Element_Birth/Death, Kill elements) command. Killed
elements can be brought to life to participate in the solution in subsequent runs using the ELIVE (Analysis, Nonlinear, Element_Birth/Death,
Bring elements back to life) command. Killed elements can be listed
using the EKILLLIST (Analysis, Nonlinear, Element_Birth/Death, List killed
elements) command.
All elements are considered alive by default, unless declared dead by
the EKILL command. All elements, dead or alive, must be defined
prior to running. Each solution stage requires a separate run. The
restart flag should be active for all stages other than the first stage.
The time for a solution stage starts from the end-time of the previous
stage and ends at the end-time for the current stage. The EKILL and/
or ELIVE commands can be used prior to running each solution
stage.
✍ Elements are assumed stress free at the time of their first birth. The undeformed, un-stressed shapes of elements brought to life using the ELIVE
command are defined by the deformed nodal positions obtained in the last
solution stage.
In
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Various stages of the solution must be properly associated with the time variable
and time curves.
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COSMOSM Advanced Modules
4
Gap/Contact Problems
Introduction
One of the major areas of nonlinear analysis is the solution of problems in which
separate bodies or structures may come in contact with each others. Several
methods have been developed to handle such problems. One of the techniques that
is extensively used to solve contact problems is the penalty method. In this method,
large numerical values are introduced into the stiffness matrix of the system to
simulate the rigidity between two nodes such that the two have approximately,
since the constraint exactly is not satisfied, same displacements. The penalty
method seems attractive since it preserves the size and the bandwidth of the
stiffness matrix. However, a major difficulty, typically associated with this
approach, arises in the selection of the proper penalty values. Very large penalty
values cause numerical difficulties, while small penalty values produce inaccuracy.
A compromise between the numerical performance and the accuracy of the results
is often made. Some researchers try to tackle this difficulty by implementing
algorithms that adaptively update the penalty values based on the stiffness changes
of the structure throughout the incremental solution.
In
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Another method, commonly used by researchers in contact problems, incorporates
the Lagrange multiplier method. Inherently to this approach is the introduction of
new variables (Lagrange multipliers) to the problem which in turn increase the size
and bandwidth of the matrices involved in the analysis. In addition, special care
should be devoted to avoid zero pivots in solving the equations of the system.
COSMOSM Advanced Modules
4-1
Chapter 4 Gap/Contact Problems
A third approach, used in COSMOSM NSTAR Module, uses a hybrid technique to
solve contact problems. This technique does not require assigning penalty values
and keeps the matrices size and bandwidth unchanged.
Hybrid Technique for Gap/Contact Problems:
General Description
A brief description on the hybrid technique used in solving nonlinear problems
involving contact is presented. For the simplicity of the presentation, the contact
will be assumed frictionless, however, the technique is general and is applicable to
contact problems with friction.
Hybrid Techniques
Matrix methods of structural analysis can be categorized as:
The Displacement Method
In this method, the matrix equation to be solved can be expressed as:
[K] {U} = {R}
(4-1)
where
[K]
= the stiffness matrix of the structure
{U}
= the vector of nodal displacements
{R}
= the vector of nodal forces
The unknown quantities in this matrix equation are the nodal displacements while
the prescribed quantities are the nodal forces.
The Force Method
In this method, the matrix equation to be solved can be expressed as:
In
de
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[F] {R} = {U}
4-2
COSMOSM Advanced Modules
(4-2)
Part 1 NSTAR / Nonlinear Analysis
where
[F]
= the flexibility matrix of the structure
{R}
= the vector of nodal forces
{U}
= the vector of nodal displacements
The unknown quantities in this matrix equation are the nodal forces while the
prescribed quantities are the nodal displacements.
The Hybrid Method
In this method, the displacement and the force methods are combined to solve the
matrix equation. The displacement method is used where external forces are
prescribed while the force method is utilized where the displacements are
prescribed.
In general purpose finite element programs, a displacement-based method is used.
However, in dealing with nonlinearities, such as contact, a hybrid method can be
efficient.
Gap Definition
In
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A gap is defined by two
nodes, for example, i and j.
The direction of the gap is
defined as the line connecting
node i to node j. The gap
distance is defined as the
maximum allowable relative
displacement between the two
nodes along the gap direction
(see Figure 4-1). An open gap
has no effect on the response
of the structure while a closed
gap, if rigid, limits the relative
displacements of its two
nodes along the gap direction
not to exceed the gap
distance.
COSMOSM Advanced Modules
Figure 4-1. Gap Direction
Gap
Direction
g1
gn
jn
i1
j1
in
Gap Direction
j2
g2
i3
g3
j3
i2
4-3
Chapter 4 Gap/Contact Problems
In order to analyze gaps, the force method can be used to calculate the forces at the
gap locations. Thus, each gap is replaced by two forces, equal in magnitude but
opposite in direction, which are applied to the two nodes connected by a gap.
Rewriting equation 4-2 for the gaps:
[Fg] {Rg} = {Xg}
(4-3)
where
{Rg}
= Vectors of gap forces
{Xg}
= Vectors of relative gap displacements
In order to define [Fg], a unit force is applied in the gap direction and the relative
displacements induced in all gaps are determined. This process is repeated for all
other gaps in order to obtain [Fg].
Now, consider a configuration where the effect of the gaps is neglected. The
following inequity implies that the ith gap is closed:
Uig = Ui2 - Ui1 > gi
(closed gap)
(4-4)
where
gi
= Gap distance
U i1
= Displacement induced by the external force vector {R}
Defining,
xgi = – (Uig - gi)
(4-5)
By solving equation 4-4, the gap force vector, {Rg}, is obtained. Applying these
forces to the structure, the relative gap displacement of the ith gap will equal xig.
Since, the external force vector {R} will produce {Ug} displacement vector, and the
gap forces {Rg} will produce - ({Ug} - gi) then, ({R} + {Rg}) will produce {g}.
In
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Therefore, the displaced shape of the structure will resume a position where the
relative displacement for each closed gap remains equal to its prescribed allowable
distance.
4-4
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
It should be noted that this method of solution uses no approximation and requires
no iterations. However, iterations are used to determine which gaps are closed at a
particular time, hence, forming equation 4-3 for those gaps only.
Contact Definition
A contact problem can be considered as a general case of a gap problem for the
following:
1.
The direction of the normal gap force is not fixed.
2.
The point of contact also may change, for example, if the gap is originally
between nodes i and j, the structure may displace such that point i comes in
contact with another point (see Figure 4-2).
Figure 4-2. Contact Problem
Location
at Contact
Original
Location
i'
i
j
k
Contacting Surfaces
Due to these factors, unlike simple gaps, the convergence and the accuracy of the
contact problem will depend on the incremental solution where the forces are
applied gradually to enable a node to move slowly on the surface.
In order to consider the contact between two bodies, one body is arbitrarily
declared “contactor” (source of contact), while the other is designated as the
“target”. The region of contact between the two bodies is governed by the overall
problem geometry, applied loads, material properties, and other relevant conditions.
In
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In COSMOSM, the contact problem is defined in accordance with the following
procedure:
1.
The region of contact of the “contactor” body is established by a series of nodal
points to which one-node gap elements should be assigned.
COSMOSM Advanced Modules
4-5
Chapter 4 Gap/Contact Problems
2.
The region of contact on the “target” body is defined by a series of contact lines
(in 2D problems) or surfaces (in 3D problems).
3.
The extent of contact between
the two bodies is limited to
areas defined by the one-node
gap elements. With the smalldisplacement restriction
removed, each gap can come
in contact with any of the
surface segments in that same
group.
4.
5.
Each surface (line) of the
“target” body is assigned a
positive and a negative side
based on its node connectivity
as shown in Figure 4-3. The
negative side is where the gap
elements are forbidden to
enter.
Figure 4-3. Contact Definition
Contact
Elements
Contactor
Body
Contact
Sub-Surface
+
Contact
Surface
Target Body
Surfaces defined in one group must form one continuous overall surface.
GAP Elements
Two-Node Gap Element (Node-to-Node Gap)
Two-node gap elements are used in 2D and 3D contact problems where bodies are
coming in contact with each other due to the application of external forces. The
main assumption for the this type of element is that the direction of the normal
contact force(s) and the contact points are known in advance and remain unchanged
In
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throughout the analysis. The two-node gap elements are placed between two nodes
of the contacting bodies (one node on each body) such that the direction of the gap
element, represented by the straight line joining the initial locations of its two nodes
(before deformation) (see Figure 4-4a), coincides with the normal contact force
(which is normal to the tangent line/plane at the point of contact of the two bodies).
Depending on the type of contact problems, the gap element can be specified to be
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COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
either a compressive gap (to limit the relative contraction between two nodes) or a
tensile gap (to limit the relative expansion between two nodes) (see Figure 4-4b).
Friction can be considered (only for compressive gaps) for both static and dynamic
analyses. The friction force associated with a gap element will lie in the tangent
plane.
Figure 4-4a. Two-Node Gap Element Definition
n
Before
Deformation
u2
2
Fs
1
After
Deformation
u1
Fn
1'
2'
Fn
Fs
vrel
Figure 4-4b. Two-Node Gap Elements
n
n
2
2
d ≥ g dist
g dist
d ≥ g dist
g dist
1
1
Compression Gap
Tension Gap
d = Relative position change along direction
n = ( u2- u1) • n
For more information about the element definition, commands, and examples refer
to the following:
COSMOSM User Guide Manual (for Element Definition)
COSMOSM Command Reference Manual (for Commands)
In
de
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Problems NS13-NS18 in this Manual (for Verification Problems)
COSMOSM Advanced Modules
4-7
Chapter 4 Gap/Contact Problems
One-Node Gap Elements
The one-node gap elements are used to establish the motion of a certain node on a
“contactor” entity (line or surface represented by those nodes) with respect to a
“target” entity (line or surface defined by a number of sub-surfaces).
The main advantage of one-node gap elements over the two-node gap elements are:
1.
The user does not need to know the exact location of the point of contact a
priori. The program internally will determine that location and apply the contact
forces accordingly.
2.
The direction of the contact forces is determined by the program based on the
deformed shape of the entities in contact.
3.
The nodal points on the contacting entities do not need to match each other.
4.
“Shrink fit” problems, where a portion of the model is forced to assume a new
position, can be handled through this type of gap elements.
In
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The element group definition command for a one-node gap element is the same as a
regular two-node gap element with the exception of the changes in options 4 and 5.
Option 4 is used to define the dimensionality of the contact problem (2D or 3D)
while option 5 is used to define the number of nodes used to construct the “target”
entity (2 or 3 nodes for line and 4 or 9 nodes for surfaces) (see Figure 4-5). The
“target” contact entity (line or surface), defined as an assembly of sub-entities (sublines or sub-surface) through the use of the NL_GS (Analysis > NONLINEAR >
CONTACT > Contact Surface) command, is associated with active gap element
group and should follow the gap element definition commands. If the problems
under consideration has more than one set of contact bodies, then a separate gap
element group, “contactor” and “target” must be defined for each potential contact
bodies.
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Part 1 NSTAR / Nonlinear Analysis
Figure 4-5. One-Node Contact Element
2D Contact
"Contactor" Area
of Contact
Target Node
Contactor
Contact Elements
Target Sub-line
+
r
2
1
y
7
6
3
5
4
"Target" Area of Contact
x
3D Contact
Contact Surface
Contact Element
+
6
y
5
2
Target Surface
s
1
9
3
8
7
r
4
x
Target Sub-surface
z
In
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Example
Consider a contact “target”
surface (Q) which is formed
by faces of solid or shell
elements (or simply defined
by a set of fixed nodes in
space) to belong to the
“target” body. Also consider
nodes i, j, and k (belonging to
the “contactor” body) to
represent the area that may
come in contact with surface
(Q) (see Figure 4-6).
COSMOSM Advanced Modules
Target Node
Figure 4-6. Contact Definition
Contact
Elements
j
i
Target
Surface
k
2
1
3
4
5
Q
6
7
11
12
9
8
13
14
10
15
4-9
Chapter 4 Gap/Contact Problems
The command sequence to specify this case is based on the following points:
1.
Define a GAP element group for node-to-surface contact (using the proper
option). For this case, each sub-surface is defined by 4 nodes. Activate this
element group and the corresponding real constant set.
2.
Define 3 gap elements at each of the three nodes i, j, and k.
3.
Define 8 sub-surfaces (for example sub-surface 1 is defined by nodes 1, 6, 7,
and 2) to form the “target” surface (Q).
The input commands related to this portion of contact definition are shown below:
Geo Panel: Geo Panel: Propsets > Element Group (EGROUP)
EGROUP,3,GAP,1,1,,2,4,,,
Geo Panel: Propsets > Real Constant (RCONST)
RCONST,3,3,,2,1,0.3
Geo Panel: Control > ACTIVATE > Set Entity (ACTSET)
ACTSET,EG,3
ACTSET,RC,3
Geo Panel: Meshing > ELEMENTS > Define Element (EL)
EL,101,PT,0,1,i
EL,102,PT,0,1,j
EL,103,PT,0,1,k
Geo Panel: Analysis > NONLINEAR > CONTACT > Contact Surface (NL_GS)
NL_GS,1,1,6,7,2
NL_GS,2,2,7,8,3
NL_GS,3,3,8,9,4
NL_GS,4,4,9,10,5
NL_GS,5,6,11,12,7
NL_GS,6,7,12,13,8
NL_GS,7,8,13,14,9
NL_GS,8,9,14,15,10
In
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In the above commands, the EGROUP (Propsets > Element Group) command
specifies gap elements for node-to-surface contact. The “target” sub-surfaces are
made of 4-node areas which can displace in space. The RCONST (Propsets > Real
Constant) command specifies a coefficient of friction of 0.3 between the contacting
surfaces (represented by gap elements on the “contactor” and the “target” surface
Q). The EL (Meshing > ELEMENTS > Define Element) command defines the gap
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COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
elements and the NL_GS (Analysis > NONLINEAR > CONTACT > Contact
Surface) command specifies the associated contact sub-surfaces that form the
“target” surface.
In defining “contactors” and “targets” the following should be observed:
1.
Quadratic Sub-surfaces
Figure 4-7. Sub-Surfaces Formations
An interior node of a
defined contact surface
must be surrounded by
four contact subsurfaces. Figure 4-7
shows examples of valid
and invalid definition of
contact sub-surfaces.
2.
Triangular sub-surfaces
The only restriction is
that all sub-surfaces in
one group should be
triangular defined by
same number of nodes.
3.
Invalid
Valid
Invalid
Contact is assumed to be rigid, therefore, the only real constant needed is the
coefficient of friction.
(Gap stiffness is also considered).
4.
Each sub-surface (sub-line) used to define an overall “target” surface (line) must
be defined such that the normal to the sub-surface (sub-line) points towards the
positive side of the overall surface (line).
An easier and more efficient way to model (input) a contact element group is
described in the following section.
For more information about the element definition, commands, and examples refer
to the following:
COSMOSM User Guide Manual (for Element Definition)
In
de
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COSMOSM Command Reference Manual (for Commands)
Problems NS36-NS40, and NS-99 in this Manual (for Verification Problems)
COSMOSM Advanced Modules
4-11
Chapter 4 Gap/Contact Problems
Automatic Generation of Gap Elements
To facilitate the definition of the entities involved in contact problems, the
NL_GSAUTO (Analysis > NONLINEAR > CONTACT > Contact Surface by
Geometry) can be used to automatically generate the one-node gap elements on the
“contactor” entities and line(s)/surface(s) on the “target” entities. All the entities
(contactor or target) must be meshed before issuing this command. The one-node
gap elements are created at each node on all contactor entities and the gap line(s)/
surface(s) are generated at each edge/face on the target entities. For more details on
the NL_GSAUTO (Analysis > NONLINEAR > CONTACT > Contact Surface by
Geometry) command, refer to the COSMOSM Command Reference Manual.
Example
Consider the contact problems between the two objects depicted in Figure 4-8
where the curves, the elements, and the nodes are shown. It has to be noted that the
entities involved in the contact problem must be meshed before defining the gap
elements.
In
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Figure 4-8a
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COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure 4-8b
Figure 4-8c
In
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The contact sequence to specify this case is based on the following:
1.
There are two contactors (sources of contact), curves 1 and 2, and two targets
curves 14 and 11.
2.
Curve 1 and Curve 2 can come in contact with curve 11 and 14.
3.
Define a GAP element group (group 2), node-to-line contact, to be associated
with the contact between curves 1 and 2 (sources) and 11 and 14 (targets).
COSMOSM Advanced Modules
4-13
Chapter 4 Gap/Contact Problems
4.
Contact nodes are to be created on the contactors.
5.
Contact sub-lines are to be created on contact targets.
6.
Each sub-line is defined by three nodes.
The input commands related to this portion of contact definition are shown below:
Geo Panel: Propsets > Element Group (EGROUP)
EGROUP,2,GAP,1,0,0,1,3,0,0,
Geo Panel: Control > ACTIVATE > Set Entity (ACTSET)
ACTSET,EG,2
Geo Panel:
Analysis > NONLINEAR > CONTACT > Contact Surface by
Geometry (NL_GSAUTO)
NL_GSAUTO,0,1,0,14,14,1,1,
NL_GSAUTO,0,2,0,11,11,1,1,
Geo Panel: Meshing > ELEMENTS > Merge Element (NMERGE)
EMERGE;
Figure 4-9 shows the elements after contact definition.
Figure 4-9
32
31
331
1030
34
29
2
35
36
9
3
37
8
38
17
18
19
In
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16
4
39
4-14
COSMOSM Advanced Modules
28
27
26
7
25
24
5
6
40 41
23
21 22
20
11
12
13
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15
Part 1 NSTAR / Nonlinear Analysis
Contact/Gaps Enhancement
Triangular Sub-Surfaces for Target Surface
The 3-dimensional contact analysis is modified to allow triangular sub-surfaces to
define the target surface. Each triangular sub-surfaces is defined by 3 or 6 nodes
and results from a mesh that utilizes one of the following element types:
Tetra4, tetra10, Shell3/3T/3L, and
Solid 8- to 20-node with collapsed nodes
✍ In the case of the solid with collapsed nodes, each sub-surface is defined by 4 or
8 nodes (one node is repeated once or twice)
✍ There is no restriction to the shape or relative positioning of these sub-surfaces,
except that the sub-surfaces in one gap group, must all be triangles with the
same number of nodes.
Automatic Soft Springs for Contact Source or Target
Six additional constants are added to the gap real constant set (r8 to r13). Each
constant represents a stiffness to be used to stabilize a structural part in one of the
three global directions (3 for source and 3 for target). The specified stiffness is
equally divided among the nodes on the source or the target. This capability
eliminates the need for soft springs and provides stability for the structural parts
that are unstable if contact was to be ignored.
✍ Make sure to use a different real constant set for each gap group, so that
stiffening is not extended to undesired areas.
A New Solution Strategy for Initial Interference
In
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A new feature allows that a gap group be excluded from analysis during certain
phases of solution. Whenever this option is turned on for a gap element group, that
group will not participate in the solution. However, if the flag is turned back off
prior to a restart, same group can come back to life and participate in the analysis.
This feature is particularly useful for problems where considerable initial
interference as well as geometric and/or material nonlinearities exist.
COSMOSM Advanced Modules
4-15
Chapter 4 Gap/Contact Problems
As an example, consider the case of two thick cylinders with an initial interference
that can cause some plastic deformation in the cylinders. Different means can be
utilized to fit one cylinder inside the other. The Outer cylinder may be heated, the
inner cylinder may be cooled, or pressure may be applied to one or both cylinders.
It must also be noted that the amount of plastic deformation is greatly dependent on
the procedure that is used. To solve this problem, the two cylinders are modeled by
their unstressed geometry and contact is defined along the interference area. In The
first phase of solution, the contact group need to be ignored (op7=1), the loading or
heating is prescribed such that the interference between the cylinders is eliminated.
Next, using the restart option, include contact (op7=0), and allow the forces/
temperatures to be removed gradually.
✍ This option can be specified for each gap group, independently. [option 7 in the
EGROUP
command]
✍ This option is not effective for the Node-to-node gaps.
✍ Each gap group can be killed or brought back to life, independent of other gap
groups.
Troubleshooting for Gap/Contact Problems
Following is a list of commonly encountered errors during the execution of
nonlinear gap/contact problems.
1.
During the first step, the program stops with the error message: “Stop, the
diagonal term in equation ...., node ...., direction .. is ....." (a zero or negative).
This error usually indicates that the whole model, or portion of it, is externally
statically unstable due to improper constraints. It should always be remembered
that the gap elements do not alter the stiffness. If a portion of the model is
supported only by gap elements, then that portion can be stabilized through the
use of soft trusses (see verification problems NS17 and NS18).
2.
The program runs successfully, but the postprocessing module shows that the
one node gap elements go beyond the contact surface which should have
stopped them.
In
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Make sure that you are looking at the structure's iso-scale displaced plot (both
the dimensions of the structure and the deformations have the same scale in the
plot). The default setting of the postprocessing programs show the deformed
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COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
shape with an exaggerated deformation scale. Therefore, try to use a scale factor
of 1.0 in plotting the deformation (see the DEFPLOT (Results > PLOT >
Deformed Shape) command in COSMOSM Command Reference Manual).
If the gap elements still exceed the contact surface after issuing the above
commands, then the conclusion is that the gap elements are not properly closed.
The following two possibilities then should be considered:
a. The orientation of the gap surfaces might be wrong. The gap elements are
allowed to remain on the positive side of the surface.
b. The original displacement is too large that the displaced location of the gap
elements cannot be compared correctly with respect to the contact surfaces.
This case usually occurs when one of the bodies is an unconstrained structure
supported only by soft trusses. To investigate this possibility, the analyst
should calculate the displacements of the soft trusses under the loads. If the
resultant displacements are excessive and the gap elements are pushed far
beyond the contact surface, it is likely that the gap iterations also will not
converge. This situation can be overcome by decreasing the applied load step
through modifying the “time” curve, or by using stiffer trusses to support the
unattached portion of the model.
3.
The program stops with the error message: “Stop, wrong definition for the
contact surfaces.”
Check the target surface connectivity and direction. Make sure that each target
contact surface is represented by continuous sub-surfaces.
4.
In nonlinear dynamic problems, the program converges but the structure
behaves erratically after the gaps close.
This condition often occurs due to the assumed perfect rigidity of the closed
two-node gaps. To avoid that situation, some flexibility for the contact can be
introduced through the third real constant of the two-node gap elements.
5.
The program completes one or more steps with some gap elements closed, but
finally stops with one of the following error messages:.
a. “Stop, the diagonal term in equation..., node ..., direction ... is zero or
negative."
b. “Stop, convergence not achieved for gap elements.”
In
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c. “*** ERROR: Convergence is not achieved in 200 iterations” or
“Convergence is not achieved in 100 contact iterations.”
COSMOSM Advanced Modules
4-17
Chapter 4 Gap/Contact Problems
These errors basically imply difficulties in problem convergence due to:
– System stiffness has deteriorated and become singular or close to singular
due to other nonlinearities (geometric or material).
– The load increment is too large.
In either case, reducing the load increment is most likely to solve the problem.
However, in case I, if the stiffness has extremely deteriorated, a solution
continuation may not be possible.
✍ If friction forces are present, the analysis is nonconservative (dependent on the
load application sequence). Therefore, the loads must be applied in increments
which resemble the actual load history.
✍ Contact problems which involve large-deflection analysis are likely to require
mesh refinement in the regions where contact is expected.
In
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References
4-18
1.
Bathe, K. J., Finite Element Procedures in Engineering Analysis, Prentice Hall,
1982.
2.
Belytschko, T., and Hughes, T., (eds.) Computational Methods for Transient
Analysis, North-Holland, Amsterdam, 1983.
3.
Cook, D. R., Malkus, D. S., and Plesha, M. E., “Concepts and Applications of
Finite Element Analysis,” Third edition, Wiley, 1989. Kardestuncer, H., “Finite
Element Handbook,” McGraw-Hill, 1987.
4.
Kulak, R. F., “Adaptive Contact Elements for Three-Dimensional Explicit
Transient Analysis,” Comp. Meth. Appl. Mech. Eng., 72, pp. 125-151, 1989.
5.
Mazurkiewicz, M., and Ostachowicz, W., “Theory of Finite Element Method for
Elastic Contact Problems of Solid Bodies,” Comput. Struct., Vol. 17, pp. 51-59,
1983.
6.
Parisch, H., “A Consistent Tangent Stiffness Matrix for Three-Dimensional
Non-linear Contact Analysis,” Int. J. Num. Meth. Eng.. Vol. 28, pp. 1803-1812,
1989.
7.
Zienkiewicz, O. C., and, Taylor, R. L., The Finite Element Method, Fourth
edition, Vol. 2, 1991.
COSMOSM Advanced Modules
5
Numerical Procedures
Static Analysis
There are different numerical procedures that can be incorporated in the solution of
nonlinear problems using the finite element method. A successful procedure must
include the following:
• A control technique capable of controlling the progress of the computations
along the equilibrium path(s) of the system.
• An iterative method to solve a set of simultaneous nonlinear equations
governing the equilibrium state along the path(s).
• Termination schemes to end the solution process.
Additional schemes such as line search, acceleration, and/or preconditioning may
be augmented to enhance the solution procedure.
Incremental Control Techniques
In
de
x
Different control techniques have been devised to perform nonlinear analysis.
These techniques can be classified as:
COSMOSM Advanced Modules
5-1
Chapter 5 Numerical Procedures
Force Control
Figure 5-1a. Force Control
In this strategy, the loads applied to the system
are used as the prescribed variables. Each
state (point) on the equilibrium path is
determined by the intersection of a surface
(F = constant) with the path to determine the
deformation parameters (Figure 5-1a).
Fk
In adapting this technique for finite element
analysis, the loads (base motions, prescribed
displacements, thermal, gravity, ...) are
incrementally applied according to their
associated “time” curves.
f*
Displacement Control
Figure 5-1b. Displacement
Control
In this technique, a point on the equilibrium
path is determined by the intersection of a
surface defined by a constant deformation
parameter (U = constant) with the solution
curve (Figure 5-1b).
To incorporate this technique in finite element
analysis, the pattern of the applied loads is
proportionally incremented (using a single
load multiplier) to achieve equilibrium under
the control of a specified degree of freedom.
The controlled DOF is incremented through
the use of a “time” curve.
uk
Fk
uk
u*
Arc-Length Control
In this strategy, a special parameter is prescribed by means of a constraint
(auxiliary) equation which is added to the set of equations governing the
equilibrium of the system. In the geometric sense, the control parameter can be
viewed as an “arc length” of the equilibrium path (Figure 5-1c).
In
de
x
To use this technique in finite element analysis, the pattern of the applied loads is
proportionally incremented (using a single load multiplier) to achieve equilibrium
5-2
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
under the control of a specified length (arclength) of the equilibrium path. The arclength will be automatically calculated by
the program. No “time” curve is required.
Figure 5-1c. Arc Length Control
Fk
Both Force control and Displacement
control will breakdown in the neighborhood
of turning points (known as snap-through
for force control and snap-back for
displacement control) as in Figure 5-2.
These difficulties usually are encountered
in buckling analysis of frames, rings, and
shells. Arc-Length control will successfully
overcome these difficulties.
S*
uk
Figure 5-2. Failures of Control Techniques
F
F
f
u
u
u
a. F ailure of F orce Control
b. F ailure of Displacement
Control
Load
b, c, e - limit points under force control
f, g - limit points under displacement control
e
f
b
'snap-through'
under force
control
d
'snap-back'
under
displacement
control
g
h
c
a
Displacement
In
de
x
c. F ailure of F orce and Displacement Controls
COSMOSM Advanced Modules
5-3
Chapter 5 Numerical Procedures
Thermal Loading for Displacement/Arc Length Controls
Here, the input temperatures are used as the loading pattern. The output load factor
defines the temperature factor for the state of deformation. The input temperature
pattern is assumed to be relative to reference:
{T(u)} = LF(u) *{To} + Tref
Where
{T(u)} - Temperature vector associated with displacement {u}
{To}
- vector of nodal Temperature pattern
LF(u) - Load Factor obtained for displacement {u}
Tref
- Reference Temperature
Iterative Solution Methods
In nonlinear static analysis, the basic set of equations to be solved at any “time”
step, t+∆t, is:
t + ∆t
{R} - t + ∆t{F} = 0
(5-1)
where
t + ∆t
{R}
= Vector of externally applied nodal loads
t + ∆t
{F}
= Vector of internally generated nodal forces
Since the internal nodal forces t+∆t{F} depend on nodal displacements at time t+∆t,
t+∆t
{U}, an iterative method must be used.
The following equations represent the basic outline of an iterative scheme to solve
the equilibrium equations at a certain time step, t+∆t,
In
de
x
(5-2)
5-4
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
(5-3)
(5-4)
(5-5)
where
t+∆t{R}
= Vector of externally applied nodal loads
t+∆t{F}(i-1) =
{∆R}
(i-1)
{∆U}(i)
Vector of internally generated nodal forces at iteration (i)
= The out-of-balance load vector at iteration (i)
= Vector of incremental nodal displacements at iteration (i)
t + ∆t
{U}(i) = Vector of total displacements at iteration (i)
t + ∆t
[K] (i) = The Jacobian (tangent stiffness) matrix at iteration (i)
There exists different schemes to perform the above iteration. In the following, a
brief description of three methods of the Newton type will be furnished.
Newton-Raphson (NR) Scheme
In
de
x
In this scheme, the tangential stiffness matrix is formed and decomposed at
each iteration within a particular step (Figure 5-3a). The NR method has a high
convergence rate and its rate of convergence is quadratic. However, since the
tangential stiffness is formed and decomposed at each iteration, which can be
prohibitively expensive for large systems, it may be advantageous to use another
iterative method.
COSMOSM Advanced Modules
5-5
Chapter 5 Numerical Procedures
Figure 5-3a. Newton-Raphson Iterative Method with Force Control, 1D
Load
F(U)
t + ∆t
1
R
t + ∆t
1
t + ∆t
K (1)
t + ∆t
F (1)
K (2)
1
t + ∆t
t
(0) =
K
K
t + ∆t
t + Dt
F (2)
F
R
∆U
t
t
U
(1)
∆U
(2)
∆U
(3)
F
t + ∆t
U (1)
t + ∆t
U
(2)
t + ∆t
U
(3)
t + ∆t
U
Displacement
Modified Newton-Raphson (MNR) Scheme
In
de
x
In this scheme, the tangential stiffness matrix is formed and decomposed at the
beginning of each step (or the user-specified reformation interval) and used
throughout the iterations (Figure 5-3b).
5-6
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure 5-3b. Modified Newton-Raphson iterative Method with Force Control, 1D
Load
F(U)
t + ∆t
R
1
K = tK
t + ∆t
t
t + ∆t
F (1)
t + ∆t
F (2)
F
R
∆U
t
t
U
(1)
∆U (2) ∆U (3)
F
t + ∆t
U (1)
t + ∆t
U
(2) t + ∆t
U
(3)
t + ∆t
U
Displacement
Quasi-Newton (QN) Schemes
Unlike the NR and MNR iterative schemes, the QN family of schemes employs a
lower-rank matrix to update the stiffness matrix (or its inverse) to provide secant
approximation from iteration (i-1) to iteration (i) (Figure 5-3c). Defining a
displacement increment as:
(5-6)
and an increment in the out-of-balance loads;
(5-7)
the updated iterative matrix should satisfy the QN equation:
In
de
x
(5-8)
COSMOSM Advanced Modules
5-7
Chapter 5 Numerical Procedures
Figure 5-3c. Quasi-Newton Iterative Method with Force Control, 1D
Load
F(U)
t + ∆t
R
δ (3)
δ
K(1) γ
t
(3)
γ (2)
(2)
1
γ
K
(2)
t + ∆t
(1)
1
δ
R
∆U (2)
∆U (1)
t
t
F
(1)
∆U (3)
F
U
t + ∆t
t + ∆t
U (1)
U
(2)
t + ∆t
U (3)
t + ∆t
U
Displacement
The BFGS (Broyden-Fletcher-Goldfarb-Shanno) update formula are widely used
with the QN algorithm.
A version of the above-mentioned iterative scheme is implemented in COSMOSM.
Line Search Scheme
The performance of an iterative method is very much dependent on the choice of
the “optimal” step length (ß) in the direction of increment (search direction). An
estimate of such a step can be calculated for BFGS iterative technique by requiring
that the projection of the residual load vector in that direction to vanish, i.e.;
(5-9)
and the solution is then expressed as:
In
de
x
(5-10)
5-8
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
The line search operation is expensive because one search process may involve a
large number of recalculating the residual vector. Instead of using the strict criterion
of vanishing the residual, many researchers suggested that the following criterion
will generally be adequate,
(5-11)
A value of 0.50 is recommended for the tolerance (TOL).
The line search option is supported in conjunction with the BFGS iteration method
where its effectiveness is recognized.
Termination Schemes
For any incremental procedure, based on iterative methods, to be effective,
practical termination schemes should be provided. At the end of each iteration, a
check should be made to test if the iteration converged within realistic tolerances or
it is diverging. Very loose tolerance will initiate inaccurate results, while very strict
one can needlessly make the computational cost high. On the other side, bad
divergence check can end the iterative process when the solution is not diverging or
allow the process to continue for searching unrealizable solution.
A number of procedures have been introduced as convergence criteria for
terminating an iterative process. In the following, three convergence criteria will be
discussed.
Displacement Convergence
This criterion is based on the displacement increments during iterations. It is given
by:
In
de
x
(5-12)
where |{α}| denotes the Euclidean norm of {α}, and εd is the displacement
tolerance.
COSMOSM Advanced Modules
5-9
Chapter 5 Numerical Procedures
Force Convergence
This criterion is based on the out-of-balance (residual) loads during iterations. It
requires that the norm of the residual load vector to be within a tolerance (εf) of the
applied load increment, i.e.,
(5-13)
Energy Tolerance
In this criterion, the increment in the internal energy during each iteration, which is
the work done by the residual forces through the incremental displacements, is
compared with the initial energy increment. Convergence is assumed to reach when
the following is satisfied:
(514)
where εe is the energy tolerance.
In addition, a number of schemes have been described as divergence criteria. One
of them is based on the divergence of the residual loads. Another is based on the
divergence of the incremental energy.
Procedure Activation
In
de
x
The solution techniques implemented in COSMOSM nonlinear module can be
accessed through NL_CONTROL (Analysis > NONLINEAR > Solution Control)
and A_NONLINEAR (Analysis > NONLINEAR > NonL Analysis Options)
commands. For specific input explanation, the user is referred to COSMOSM
Command Reference.
5-10
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Dynamic Analysis
The skeleton of the procedures used for nonlinear dynamic analysis follow the
same method used for nonlinear static analysis (Control + Iteration + Termination).
The discretized equilibrium equations of the dynamic system can be written in the
form:
(5-15)
where
[M]
= Mass matrix of the system
[C]
t+∆t
= Damping matrix of the system
[K]
(i)
= Stiffness matrix of the system
t+∆t
{R}
t+∆t
{F}(i-1) = Vector of internally generated nodal forces at iteration (i)
= Vector of externally applied nodal loads
t+∆t{∆U}(i) =
Vector of incremental nodal displacements at iteration (i).
= Vector of total displacements at iteration (i)
= Vector of total velocities at iteration (i)
= Vector of total accelerations at iteration (i)
Using implicit time integration schemes such as Newmark-Beta or Wilson-Theta
methods, and employing a Newton's iterative method, the above equations can be
cast in the form:
In
de
x
(5-16)
COSMOSM Advanced Modules
5-11
Chapter 5 Numerical Procedures
where
= The effective load vector
= The effective stiffness matrix
and a0, a1, a2, a3, a4, and a5 are constants of the implicit integration schemes.
Since Displacement and Arc-length controls are adapted for proportional loadings,
which is not the case in dynamic problems where loading time histories are
prescribed, then, only Force control can be incorporated for dynamic analysis.
Also, all iterative solution strategies discussed in static analysis can also be
incorporated for dynamic analysis. However, only MNR and NR methods are
available in this version for dynamic analysis. Since, the inertia of a system tends
to smoothen its dynamic response more than its static response, convergence is
generally expected to be easier than static analysis. No line search is implemented
for dynamic analysis.
In
de
x
NL_CONTROL (Analysis > NONLINEAR > Solution Control), NL_INTGR
(Analysis > NONLINEAR > Integration Options) and A_NONLINEAR (Analysis
> NONLINEAR > NonL Analysis Options) commands are used to activate the
required procedure. For specific input explanation and default values, the user is
referred to COSMOSM Command Reference.
5-12
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Rayleigh Damping Effects
To include the effects of damping forces in dynamic analysis, the proportional
Rayleigh damping matrix is introduced. The damping matrix of the system [C] is
assumed to take the form:
where
[K0]
= Initial stiffness matrix of the system
[M]
= Mass matrix of the system
α
= Rayleigh damping coefficient associated with the stiffness
matrix
β
= Rayleigh damping coefficient associated with the mass
matrix
The NL_RDAMP (Analysis > NONLINEAR > Damping Coefficient) command is
used to activate the effect of damping.
Concentrated Dampers
The effects of concentrated dampers is also included in the dynamic analysis. The
PD_CDAMP command can be used to define damping between two nodes on the
structure.
Base Motion Effects
The dynamic effects of base motion accelerations in global X, Y, and Z directions
(may resemble the effects of seismic motions on a structural system) can be utilized
in the analysis.
In
de
x
The NL_BASE (Analysis > NONLINEAR > Base Motion Parameter) command is
used to activate this procedure.
COSMOSM Advanced Modules
5-13
Chapter 5 Numerical Procedures
Inclusion of Dead Loads in Dynamic Analysis
In many cases, it is desired to start the dynamic analysis of structural systems from
the deformed configurations obtained under the effect of dead loads. Such an
analysis may be performed as follows:
1.
Suppose that the pseudo time to be used for static analysis is Ts. The true
dynamic time at a time t > Ts is equivalent to (t-Ts).
2.
Define dead loads by associating them with time curves that start from zero (at
time 0), reach their final values at the pseudo time Ts, and remain unchanged
afterwards throughout the period of the dynamic solution.
3.
Define dynamic loads by associating them with time curves that keep a value of
zero throughout the static solution pseudo time, and vary as desired during the
dynamic solution true time past Ts.
4.
Perform a static analysis for the time duration (0 to Ts).
5.
Activate the restart flag and run a dynamic analysis from Ts to Td (where Td-Ts
represents the actual true time for dynamic analysis).
Note that if Rayleigh damping is used, then the damping matrix calculations will be
based on the stiffness at the start of the dynamic run.
Figure 5-4. Time Curves for Loads
Multiplier
Multiplier
Dynamic Load
Ts
0
Static
Analysis
Td
t
Dynamic
Analysis
In
de
x
a ) Time C urve s for D e a d Loa ds
5-14
COSMOSM Advanced Modules
Ts
0
Static
Analysis
Td
t
Dynamic
Analysis
b) Time C urve s for D yna mic Loa ds
Part 1 NSTAR / Nonlinear Analysis
Adaptive Automatic Stepping Technique
In COSMOSM the user has the choice to solve nonlinear problems by directly
specifying the load and/or displacement increment to be followed or by letting the
program select its own incremental procedure based on user specified parameters.
The adaptive automatic stepping algorithm (Analysis > NONLINEAR > AutoStep
Options) provides the following:
Step Size Optimization
The algorithm automatically adjusts the incremental step so that smaller steps are
enforced in the region of the most severe nonlinearity while larger steps are allowed
when the response tends to be linear. To avoid excessive cut-backs in cases of limit
loads or buckling, a minimum step DTMIN is defined. In addition, to prevent
convergence on a higher equilibrium path during solution process, especially when
path-dependent materials or loads are present, a maximum step DTMAX is used.
The definition of DTMIN and DTMAX is optional (COSMOSM will provide
default values if not specified) but it is recommended for complex problems.
Safe-guard Against Equilibrium Iteration Failures
The scheme senses the rate of convergence and provides adaptive step adjustment
to avoid the termination of the solution process due to:
• Exceeding the number of permissible equilibrium iterations because of lack of
convergence.
• Divergence of the incremental residual load and/or energy.
• Gap/contact iterations nonconvergence.
• The presence of a negative term on the diagonal of the stiffness matrix during
iterations (due to a large load increment) under force control.
• The presence of a negative term on the diagonal of the stiffness matrix at the
In
de
x
beginning of a new step due to local singularity under force control.
COSMOSM Advanced Modules
5-15
Chapter 5 Numerical Procedures
Safe-guard Against Converging to Incorrect Solutions
• Produced by large incremental rotations:
An incremental equivalent rotation at each node is computed from:
The criterion for resetting the time increment is:
• Produced by large incremental creep strain:
For Creep analysis (Element Group Op. 7 = 1) and Viscoelastic material model
(Element Group Op. 5 = 8), the effective creep strain at each integration point is
computed:
The criterion for resetting time increment is:
Note that if CETOL is not input by the A_NONLINEAR (Analysis >
NONLINEAR > NonL Analysis Options) command, the default value is 0.01.
• Produced by large incremental plastic strain:
For Elastoplastic material models (Element Group Op. 5 = 1, 2, 5, 11) an incremental effective plastic strain at each integration point is computed:
In
de
x
The criterion for time increment is:
5-16
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
• Produced by unacceptable element status change:
When the Wrinkling Membrane model is used (Element Group Op. 5 = 10), the
element status is determined by the state of strain, namely, Taut, Wrinkled, and
Slack (details in Chapter 3: Material Models and Constitutive Relations). The
program only accepts the following element status changes:
• Encountering a state of equilibrium on a higher path that produces a negative
determinant for the deformation tensor.
• Produced by large incremental logarithmic strain in large strain plasticity model.
The criterion for time increment is:
< 40% where
is the incremental effective logarithmic strain.
The NL_AUTO (Analysis > NONLINEAR > AutoStep Options) command is used
to activate this procedure.
J-Integral Evaluation for Nonlinear Fracture
Mechanics NLFM
Evaluation of J-integral (the two-dimensional formulation) as a viable fracture
criteria for linear elastic and elastic-plastic deformations is implemented in the
NSTAR module. The two-dimensional J-integral formulation has been modified to
include axisymmetric behavior as well as thermal gradients.
The J-integral provides estimations for the stress intensity factors in a nonlinear
environment. For an elastic fracture assessment, the stress intensity factors can be
determined from the J-integral parameters:
In
de
x
where:
KI, KII
= Stress intensity factor for modes I and II
JI, JII
= J-integral values for modes I and II
J
= Total J-integral value = JI + JII
COSMOSM Advanced Modules
5-17
Chapter 5 Numerical Procedures
= plane stress
= plane strain and axisymmetric
I, II
= First and second (opening and shearing) crack modes
E, υ
= Modulus of Elasticity and Poisson's ratio
The J-integral parameters are path-independent. They can be obtained from any
arbitrary closed path which starts from one crack surface, travels around the crack
tip and ends on the other crack surface.
Since this path can be taken well away from the crack tip singularity, it requires
significantly less mesh refinement than other fracture assessment techniques.
Definition
Referring to Figure 5-5. The two-dimensional J-integral is defined by:
S
= an arbitrary path (J path) surrounding the crack tip
= Outward normal to curve S
X
= Crack axis
Y
= Normal to crack axis
w
= Strain energy density =
εij
= Infinitesimal strain tensor
σij
= Cauchy stress tensor
= The traction vector defined as: Ti = σij nj
In
de
x
= Displacement vector
5-18
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Modification for Temperature
A
= The area enclosed within Contour S
β
= Coefficient of thermal expansion
δ
= Cronecker delta
T
= Temperature
Figure 5-5. Coordinates and Curves for Definition of the J-Integral
y
x, y = Global Cartesian
coordinates
n
_
n
X, Y = Crack coordinates
n
A
= Outward normal to
curve s
Crack
Axis
Y
= Areas enclosed
within the contour s
Crack
Surface
X
x
dA
J Path (Counter
Clockwise) (Curve S)
In
de
x
Axisymmetric Formulation
COSMOSM Advanced Modules
5-19
Chapter 5 Numerical Procedures
Rc
= The crack tip radius
r
= Radius
A
= Area enclosed within the Contour S
The Requirements in Selection of the Path
1.
The J-integral path should not pass through elements at the crack tip.
2.
In case of elastic-plastic analysis, the J-integral path can pass through plastically
deformed regions.
3.
More than one path can be selected. Different paths around the same crack tip
should render similar results.
Requirements for JI and JII Evaluation
In addition to the combined mode parameter J, the J-integral values for modes I,
and II (crack opening, and shearing), can be evaluated. For this case:
1.
The entire area around the crack tip must be included in modeling.
2.
The J-path must be symmetric with respect to the crack axis. This infers that the
finite element mesh inside the J-integral path (around the crack tip) must also be
symmetric with respect to the crack axis.
3.
In the case of elastic-plastic analysis, the J-integral path can only pass through
elastically deformed regions.
4.
Axisymmetric analysis and/or temperature gradients are not available.
Symmetric Modeling
In
de
x
If due to symmetry only half of a crack is modeled, then the total J-integral equals
to JI, and JII = 0. Notice that the J value which is output in this case, is twice the
value which is obtained from the path.
5-20
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Specifications
Presently, the J-integral can be evaluated for two dimensional homogeneous
structures (with cracks) which are modeled using PLANE2D (plane strain, plane
stress, or axisymmetric) elements. Effects of thermal loading on crack parameters is
also included.
Although, the presented J-integral formulation is valid for deformation theory of
plasticity, the flow theory of plasticity (available in NSTAR) can be employed so
long as the loading is proportional and there is no unloading. Note that the
likelihood of violating these conditions is greater in presence of thermal gradients
than for purely mechanical loading.
J-Integral Path Definition
The J-integral path is defined by a number of segments each representing a side of a
finite element. The J-integral is evaluated by taking the path through the Gauss
points which are closest to the selected element side, within the element.
Each J-integral segment is defined by two nodes and a global element number. Jsegments must be defined in the proper (Counter Clockwise) order.
Figure 5-6 Part of a Finite Element Mesh Including a Crack Notch
J Segments
17
16
15
14
13
12
18
11
19
End of the J Path
Start of the J Path
20
Crack
Tip
Crack
Axis
1
2
10
X
9
i
m
8
k
j
4
5
6
7
In
de
x
3
COSMOSM Advanced Modules
5-21
Chapter 5 Numerical Procedures
In the model shown in Figure 5-6, 20 segments must be input to define the selected
J-path. Segment number 3, for example, is defined by nodes j and k and element m.
(Segment 2 is defined by nodes i and j and element m). [Command J_INTDEF
(Analysis > NONLINEAR > J INTEGRAL > Define Path)]
In case of an axisymmetric analysis and/or thermal loading, elements inside the Jintegral path must also be input. No order is required for input of these elements.
[Command J_INTELEM (Analysis > NONLINEAR > J INTEGRAL > Define
Element)]
Special care must be given to avoid merging of the nodes along the two crack free
surfaces.
Frequencies and Mode Shapes in a Nonlinear
Environment
Frequency analysis of structures which include nonlinear effects (Geometric and/or
material) is implemented in the NSTAR module.
The natural frequencies and mode shapes for a nonlinear structure are determined
by performing a frequency analysis using the nonlinear stiffness matrix which is
obtained from a nonlinear step-by-step solution.
Moreover, by performing several frequency analyses at different stages of a
nonlinear solution, the variations of the structure's natural frequencies with respect
to the level of loading (or time in the case of viscoelastic, or creep behavior) can be
detected.
The solution procedure is as follows:
The NSTAR module determines and stores the current nonlinear stiffness and
the mass matrices on files (the current stiffness matrix means the stiffness
calculated at the last solution step).
2.
The DSTAR module can then perform a frequency analysis of the structure
using the latest nonlinear stiffness matrix calculated by NSTAR.
In
de
x
1.
5-22
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
3.
The nonlinear solution can then be continued by using the RESTART option,
followed by another frequency analysis, followed by another nonlinear restart,
and so on.
✍ The preparation of files for a frequency analysis must be requested prior to
running NSTAR. [Command A_NONLINEAR (Analysis > NONLINEAR >
NonL Analysis Options)].
✍ A frequency analysis can be performed, at any time, after a nonlinear run is
successfully completed. [Command R_FREQUENCY (Analysis > FREQBUCK
> Run Frequency)]
✍ All the information pertaining to the frequency as well as the nonlinear analysis
must be input prior to the first nonlinear run (start from time zero).
✍ Prior to running for frequencies, it is necessary to request the frequency analysis
to be based on the nonlinear results. [Command A_FREQUENCY (Analysis >
FREQBUCK > Frequency Options)].
Buckling Analysis in a Nonlinear Environment
Buckling analysis of structures with inclusion of the nonlinear effects such as
material nonlinearities, and gaps is implemented in the NSTAR and DSTAR
modules. The buckling load parameter and mode shape is determined by
performing a buckling analysis using the nonlinear stiffness matrix which is
obtained from a nonlinear step-by-step solution.
Buckling analyses at different stages of a nonlinear solution, yield different
buckling loads. As long as the structure is in the pre-buckling state, the buckling
load parameter (which is obtained from a buckling analysis) must be greater than
one. As the structure approaches a buckling state (during the nonlinear analysis),
the buckling load parameter approaches one. In the post-buckling state, the
buckling load parameter is usually less than one.
In
de
x
It must be noted, however, that at any stage of a nonlinear step-by-step solution,
when the structure stiffness matrix becomes singular, the buckling analysis will
predict a buckling state, i.e., the buckling load parameter equals one.
COSMOSM Advanced Modules
5-23
Chapter 5 Numerical Procedures
✍ The preparation of files for a buckling analysis must be requested prior to
running NSTAR [Command A_NONLIN].
✍ A buckling analysis can be performed, at any time after a nonlinear run (if the
nonlinear run is not successfully completed, the results of the last successful
step will be used). [Command R_BUCK].
✍ All the information pertaining to the buckling as well as nonlinear analysis must
be input prior to the first nonlinear run (start from time zero).
✍ Prior to the first buckling run, it is necessary to request the buckling analysis to
be based on the nonlinear results. [Command A_BUCK].
Release of Global Prescribed Displacements
Prescribed displacements are the effect of unknown external forces that can be
determined from the reaction forces; Each applied force is equal in magnitude but
opposite in direction to the reaction force that is obtained for the degree of freedom
with a prescribed displacement.
Considering a node for which displacement is prescribed, it is sometimes desired to
release that node once a certain level of displacement/loading is reached. In
addition, for certain problems, it may be desired to keep the pre-release force acting
on the released node.
In another case, it may be desired to prescribe displacement(s) only after certain
level of loading or displacement is reached. For example, consider a structure that
is heated first. Next, certain nodes on it's boundary are secured at their heated
position. And finally, the structure is cooled and it may be subjected to other
loading conditions.
✍ The release periods are specified as those areas on the time curve (associated
with a prescribed displacement) where the curve value is greater than 1.E8, as
well as, the in-between areas, i.e., areas where one curve value is greater and
one smaller than 1.E8.
✍ To request to keep the pre-release forces on released degrees of freedom, a flag
In
de
x
(RFKEEP) is added to the A_NONLINEAR command.
5-24
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
✍ The release of a prescribed displacement may be sudden (dynamic) or slow
(static).
In the dynamic case, the solution can be obtained in the following steps:
a. First the static response due to prescribed displacement(s) is found.
b. Next, a release must be specified for the time duration of the dynamic
analysis.
c. The analysis type needs to be changed to dynamic.
d. Restart of the solution, yields the proper dynamic response.
In any case, displacements and forces may be prescribed for the same d.o.f.
(forces become effective whenever the node is released.
Defining Temperatures Versus Time Relative to a
Reference Temperature
If nodal temperatures are associated with a time curve that has a value of zero at
time zero, the curve is assumed to prescribe relative temperatures (relative to
reference) instead of prescribng Cotal temperatures:
Trel( n,t) = Curve_value(t) *( Tn - Tref )
Where:
Trel( n,t)
- Relative temperature at node n and time t
Tn
- Input temperature at node n
Tref
- Reference temperature specified by the TREF command
In
de
x
This assumption is made to provide ease of use for certain cases where
otherwise each node requires a separate curve to specify its total temperature.
COSMOSM Advanced Modules
5-25
Chapter 5 Numerical Procedures
Modified Central Difference Technique for
Dynamic Time Integration
A “Modified” Central Difference (explicit) technique is implemented into NSTAR
for the integration of response (in time) in a dynamic analysis. As an explicit
technique, Central Difference can be used to investigate the dynamic response of
structures that are subjected to shock loading or (high impact) collisions. Here, the
term “modified” has been used to represent the changes that are made to the
Standard Central Difference to make it more practical and suitable for use. Since
this method is only “Conditionally” Stable, it usually requires very small time
increments which yield a great number of solution steps. Furthermore, it is not easy
to guess a small-enough time increment for a general dynamic problem.
To avoid these difficulties and to speed up the solution, the following modifications
are made.
1.
Sub-Steps within Each Solution Step:
In order to reduce the impracticality and the length of time that results from too
many solution steps, a special arrangement has been made:
As usual, End_Time and Time_Inc, specified in the TIMES command, are used
to define:
NSTEP
= number of steps for output & graphs
= End_Time / Time_Inc.
The dynamic analysis, however, is performed such that:
DSTEP
= number of steps for dynamic integration
= NSTEP * isub
isub
= number of sub-steps to be specified [Default=100]
This means that isub number of (sub-) steps are performed within each solution
step. As a result, the time increment that is used for dynamic integration, will be
different from Time_Inc:
dt
= time increment for dynamic integration
= Time_Inc / isub
In
de
x
As an example, consider a certain model for which a dynamic step-by-step
solution is to be performed. Moreover, lets assume that for this analysis to be
accurate, at least a million solution steps are required. Here, by setting isub to be
1000, the number of external solution steps can be reduced to 1000.
5-26
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
2.
Evaluation of Critical Time Increment:
Since Central difference is only conditionally stable, i.e., time increment should
be smaller than a critical value, it is helpful to obtain this critical time value:
dt(critical) = Tn/π
where Tn is the system's smallest period.
To evaluate dt(critical), an iterative technique (Forward Iteration) is
implemented which evaluates the highest frequency of the finite element
system. This calculation is done once at the start of the solution (using the linear
stiffness matrix) and the critical time increment is printed in the output file.
3.
Mass and Damping:
In order to use central difference in its most efficient way, i.e., requiring no
matrix decompositions, mass and damping matrices need to be diagonal. Thus,
a Lumped mass matrix is used with a modified Rayleigh damping of the form:
[C] = β[M]
4.
Checks for Convergence of Nonlinear Solution:
While the above technique of dividing total dynamic steps into steps and substeps is 100% accurate for a linear structure (nstep=1 and isub=1000, or
nstep=100 and isub=10, or nstep=1000 and isub=1 are all equivalent), the same
may not be true for nonlinear problems where material and/or geometric
nonlinearities are to be considered. This is due to the fact that the structure
stiffness is assumed to remain constant during the sub-step calculations. In other
words, the stiffness is updated at intervals separated by isub number of (sub-)
steps. To avoid divergence due to changes of stiffness during the sub-step phases
of solution, another check is added to ensure that the out of balance forces
remain small. Using norms of vectors:
| |{F´t}| - |{Ft}| | < toln * |{Ft}|
where toln is a preset tolerance and:
{Ft} = obtained internal force vector at time t
{F´t} = expected internal force vector at time t, assuming [Kt] is constant
In
de
x
5.
Auto-Stepping for Nonlinear dynamic problems:
Auto-Stepping with considerations for the dynamic accuracy is available with
the central difference technique. The Minimum/Maximum time increments are
internally adjusted, based on the time increment that is specified by the TIMES
command:
COSMOSM Advanced Modules
5-27
Chapter 5 Numerical Procedures
Maximum Time Inc. = specified Time_Inc.
6.
Gaps and Contact:
The central difference method is extended to include Gaps and Contact
algorithm with or without Friction. For this case, however, a stiffness should be
assigned to the gap elements [default=1.5E7]. (Spring-Damper is excluded.)
7.
Other capabilities:
Other capabilities include prescribed global displacements as dynamic
excitations, local boundary conditions, and reaction force calculation. (Nonzero local prescribed displacements, and constraint equations with non-zero
right hand sides are not available.)
8.
Inclusion of dead loads in the dynamic analysis:
Similar to other dynamic methods, it is possible to perform a dynamic analysis,
using central difference, following a static analysis. This feature maybe more
useful here than it is for the unconditionally stable techniques; in this case, static
modes can not be estimated by modes with zero masses or modes with too small
periods. (See the section on guidelines.)
Advantages
The two-phase solution (steps and sub-steps), in absence of matrix decompositions,
results in a solution procedure which is faster than implicit dynamic integration
techniques such as Newmark or Wilson. The solution speed can help solve
problems that need too many solution steps such as shock or high impact.
Disadvantages
In
de
x
The technique, being conditionally stable, imposes a limit on the size of the time
increment. Since this limit depends on the smallest period of the finite element
assemblage:
5-28
1.
All degrees of freedom require to have non-zero masses; a zero mass means a
zero period which means a zero value for critical time increment.
2.
Systems having either very small masses or very large stiffnesses at some
degrees of freedom relative to others, may not be appropriate to be analyzed by
central difference.
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
✍ Specify some mass for all degrees of freedom; otherwise the program will
assign an approximate value (1% of the minimum mass used for any degree of
freedom) to all degrees of freedom with no mass. (If the reaction force flag is
on, and you get a warning message about degrees of freedom with zero masses,
you can ignore it)
✍ In order to exclude the static modes from dynamic analysis, never use small
masses or large stiffness. Instead, use one of the two following procedures:
a. First perform a static analysis raising the static forces in a very small time
duration (1.e-7 seconds), then keeping the static forces constant, restart to
perform a dynamic solution.
b. To minimize the dynamic effects due to static loads, the time curve,
associated with static loads, can be defined such that, loads are raised from
zero to their proper value in a very short time (smaller than dynamic time
inc.), and are kept constant thereafter.
✍ When nonlinearities (material and/or geometric) exist, Auto-Stepping can be
used with central difference.
✍ When there is possibility of contact/impact, it must be noted that gaps or contact
elements act as stiff springs which can result in a reduction of the critical time
increment; This effect is not included in the evaluation of the critical time
increment and must be considered by the user in selecting a proper time inc.
For best results, first perform a trial run to find the approximate time of the
impact. Next, you can divide solution time into two or more portions (using
restart), for which different time increments (and/or different number of substeps) are used (smaller time incs. and/or larger isubs may be required during
the impact phase). Make sure for every duration you re-issue the NL_AUTO
command and change the maximum time increment to the time increment you
are requesting in the TIMES command.
✍ Friction is based on the generalized model (friction may be sliding or nonsliding), regardless of the type that is specified in the input.
✍ Note that the variation of loads during the sub-step phase is assumed to be
In
de
x
linear, i.e., linear interpolation is used between the times of two consecutive
steps.
COSMOSM Advanced Modules
5-29
✍ It must also be noted that, for this technique, the accelerations and velocities are
always one step behind the displacements (and stresses). As a result, there is an
error in display of time in the xy-plots of accelerations or velocities (time
coordinates should be reduced by a time increment).
✍ If the solution requires many reductions of the time increment due to:
“Out of Balance Loads Diverging”
solution time as well as the number of steps may be reduced by selecting a
smaller time increment.
✍ If the selected time increment is larger than critical, or nonlinearities exist in the
model (impact in particular), to assure convergence, it is helpful to compare
results between two runs for which two different time increments are used.
Combination of Force Control and Displacement/
Arc-Length Control Methods
Definitions:
Parametric Loads are loads that represent a loading pattern. They are not defined
by time curves and their actual value at a state of deformation, is defined by the
load factor.
Non-Parametric Loads are loads are loads with known magnitudes defined by
multipliers and time curves.
Assumption:(for displacement or arc-length analysis)
All loads (concentrated forces, pressures, temperatures, centrifugal, gravity) that
are associated with time curve number 1 are considered to be parametric loads.
Loads that are associated with a time curve >1 are considered as non-parametric
(defined by the time curve).
✍ For displacement control, time curve 1 describes the displacement of the
In
de
x
controlled degree of freedom. For the arc length, time curve 1 is ignored.
5-30
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
A combined analysis of parametric and non-parametric loads can be performed in
two ways:
• Using displacement control, and including both types of loads in the analysis,
simultaneously.
• Starting with force control to find deformation under non-parametric loads
(curve 1 should have zero values during this phase), then restart with either
displacement or arc-length control and obtain the load factor and response due
to parametric loads while other loads are kept constant.
Artificial Bulk Viscosity
Artificial bulk viscosity can be used to provide response continuity for shock
waves. The bulk viscosity damps out the response slightly, replacing the shock
discontinuities with rapidly varying but continuous response.
The use of the bulk viscosity will not affect regions of the model away from the
shock front, while maintaining the jump conditions across the shock transition.
This technique introduces an additional viscous term, q, to relax the element
internal pressure. Two variations are implemented:
1.
Quadratic Form, used for strong shocks
q = ρ l {C0 l (δεκκ/δt)2 − C1 C2 δεκκ/δt } if δεκκ /δt < 0
q =0
2.
if δεκκ /δt >= 0
Linear Form, used for weak shocks [Strain rate << Wave speed]
q = ρ l {− C1 C2 δεκκ/δt }
Where:
εκκ = Trace of the strain tensor = ε11
+ ε22 + ε33
δ /δt = Variation with respect to time
ρ = Density
l = Equivalent Length = (Volume)(1/3)
(1/2)
In
de
x
= (Area)
for 3D
for 2D
C0
= Dimensionless Constant [default = 1.5]
C1
= Dimensionless Constant [default=0.06]
COSMOSM Advanced Modules
5-31
Chapter 5 Numerical Procedures
C2
= Material Wave Propagation Speed
[default = {(Κ+ 4G/3)/ρ }(1/2) , Κ = Bulk Modulus, G = Shear Modulus]
The above formulation can be activated, using Option 7 (Option 7 =2 or 3) in the
EGROUP command, for the SOLID, TETRA4, TETRA10, PLANE2D, &
TRIANG elements. Parameters C0, C1, & C2 can be specified using command:
MPROP, ,CREEPC,…
References
Bathe, K. J., Finite Element Procedures in Engineering Analysis, Prentice Hall,
1982.
2.
Bathe, K J., and A. Cimento, “Some Practical Procedures for the Solution of
Nonlinear Finite Elements Equations,” Comput. Meth. Appl. Mech. Eng., 22:5985, 1980.
3.
Grisfield, M. A., “A Fast Incremental/Iterative Solution Procedure That Handles
'Snap-through', “Comput. Struct., 13:55-62, 1980.
4.
Grisfield, M. A., Finite Elements and Solution Procedures for Structural
Analysis, Vol. I: Linear Analysis, Pineridge Press Limited, U.K., 1986.
5.
Geradin, M., S. Idelsohn, and M. Hogge, “Computational Strategies for the
Solution of Large Nonlinear Problems via Quasi-Newton Methods,” Comput.
Struct., 13:73-81, 1981.
6.
Geradin, M., M. Hogge, and S. Idelsohn, in T. Belytschko and T. Hughes (eds.),
“Implicit Finite Element Methods,” Computational Methods for Transient
Analysis, North-Holland, Amsterdam, 1983,chap. 4, pp. 417-471.
7.
Mathies, H., and G. Strang, “The Solution of Nonlinear Finite Element
Equations,” Int. J. Numer. Meth. Eng., 14:1613-1626, 1979.
8.
Riks, E., “An Incremental Approach to the Solution of Snapping and Buckling
Problems,” Int. J. Numer. Meth. Eng., 15:529-551, 1979.
9.
Tsai, C. T., and A. N. Palazotto, “Nonlinear and Multiple Snapping Responses
of Cylindrical Panels Comparing Displacement Control and Riks Method,”
Comput. Struct., 41:605-610, 1991.
In
de
x
1.
5-32
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Zienkiewicz, O. C., “Incremental Displacement in Nonlinear Analysis,” Int. J.
Numer. Meth. Eng., 3:587-588, 1971.
11.
Zienkiewicz, O. C., and Taylor, R. L., The Finite Element Method, Vol. II,
Fourth edition, McGraw-Hill, 1991.
In
de
x
10.
COSMOSM Advanced Modules
5-33
In
de
x
Chapter 5 Numerical Procedures
5-34
COSMOSM Advanced Modules
6
Element Library
Introduction
In
de
x
In this chapter, general information regarding the different element types, their
geometric dimensions and dimensional behavior are presented. The different
material models available for use with the different element groups are also
furnished. Tables for real constants for NSTAR elements are also provided.
COSMOSM Advanced Modules
6-1
Chapter 6 Element Library
Table 6-1. Element Library: General Information
COSMOSM
Element
Name
Description
Element
Dimensional
Behavior
TRUSS2D
Plane Truss
2
1D/ 2D
(XY-Plane)
TRUSS3D
Space Truss
3
1D/ 2D/ 3D
BEAM2D
Plane Beam
3
1D/ 2D
(XY-Plane)
BEAM3D
Space Beam
6
1D/ 2D/ 3D
IMPIPE
Immersed Pipe
6
2D/ 3D
SPRING
Axial and/or Torsional Spring
PLANE2D
4 to 8-node (Plane Stress, Strain,
Axisymmetric)
2
2D (XY-Plane)
TRIANG
3 to 6-node (Plane Stress, Strain,
Axisymmetric)
2
2D (XY-Plane)
SHELL3
3-node Triangular Thin Shell
6
2D/ 3D
SHELL4
4-node Quadrilateral Thin Shell
6
2D/ 3D
SHELL3T
3-node Triangular Thick Shell
6
2D/ 3D
SHELL6
6-node Triangular Thin Shell
6
2D/ 3D
SHELL6T
6-node Triangular Thick Shell
6
2D/ 3D
SHELL4T
4-node Quadrilateral Thick Shell
6
2D/ 3D
SHELL3L
3-node Triangular Composite Shell
6
2D/ 3D
3 to 6
1D/ 2D/ 3D
SH3LL4L
4-node Quadrilateral Composite Shell
6
2D/ 3D
SOLID
8 to 20-node Continuum Brick
3
3D
TETRA4
4-node Continuum Tetrahedron
3
3D
TETRA10
10-node Continuum Tetrahedron
3
3D
GAP
Gap/Contact with Friction
1/2/3*
1D/ 2D/ 3D
MASS
Concentrated Mass
6
1D/ 2D/ 3D
BUOY
Immersed Spherical Mass
6
1D/ 2D/ 3D
GENSTIF
General Stiffness
6
3D
In
de
x
• According to the contact nodes
6-2
Number of
DOF
/Node
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Table 6-2a. Element Library: Linear Material Model Information
COSMOSM
Element Name
Elastic
Isotropic
Orthotropic
Composite
Viscoelastic
Isotropic
TRUSS2D
•
•
TRUSS3D
•
•
BEAM2D
•
•
BEAM3D
•
•
IMPIPE
•
SPRING
•
PLANE2D
•
•
•
TRIANG
•
•
•
SHELL3
•
SHELL4
•
SHELL3T
•
SHELL6
•
SHELL6T
•
SHELL4T
•
SHELL3L
•
•
•
SHELL4L
•
•
•
SOLID
•
•
•
TETRA4
•
•
•
TETRA10
•
•
•
•
•
•
In
de
x
• Available
COSMOSM Advanced Modules
6-3
Chapter 6 Element Library
Table 6-2b. Element Library: Nonlinear Material Model Information
Elastic-Plastic
Elastic
Curved
Description
Creep
Classical
Creep
Exp
TRUSS2D
•
•
•
TRUSS3D
•
•
•
BEAM2D
•
BEAM3D
•
COSMOSM
Element Name
Hyperelastic
MooneyRivlin
Ogden
BlatzKo
IMPIPE
SPRING
•
PLANE2D
•
•
•
•
•
•
TRIANG
•
•
•
•
•
•
•
•
•
SHELL6T
•
•
•
SHELL4T
•
•
•
SHELL3
SHELL4
SHELL3T
SHELL6
SHELL3L
SH3LL4L
SOLID
•
•
•
•
•
•
TETRA4
•
•
•
•
•
•
TETRA10
•
•
•
•
•
•
In
de
x
• Available
6-4
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Table 6-2c. Element Library: Nonlinear Material Model Information
COSMOSM
Element
Name
Plastic Bilinear
+ Curve
Description
Plastic
ElasticPerfectly
Plastic
DruckerPrager
Plastic
Concrete
Fabric
Membrane
Failure
of Composites
UserDefined
von
Mises
Iso
von
Mises
Kin
TRUSS2D
•
•
•
•
TRUSS3D
•
•
•
•
BEAM2D
•
•
BEAM3D
•
•
PLANE2D
•
•
•
•
•*
•
TRIANG
•
•
•
•
•*
•
•
•
•+
SHELL6T
•
•
•+
SHELL4T
•
•
•+
IMPIPE
SPRING
SHELL3
SHELL4
SHELL3T
SHELL6
SHELL3L
•
SH3LL4L
•
SOLID
•
•
•
•
•
TETRA4
•
•
•
•
•
TETRA10
•
•
•
•
•
• Available
In
de
x
* Plane Stress
+ Membrane
COSMOSM Advanced Modules
6-5
Chapter 6 Element Library
Table 6-3. Element Library: Real Constants
COSMOSM
Element
Name
TRUSS2D
TRUSS3D
BEAM2D
No.
RC
Description
4
RC1
RC2
RC3
RC4
Cross sectional area
Cross sectional perimeter (for thermal analysis only)
Initial axial force (with large displacement options)
Initial axial strain (with large displacement options)
4
RC1
RC2
RC3
RC4
Cross sectional area
Cross sectional perimeter (for thermal analysis only)
Initial axial force (with large displacement options)
Initial axial strain (with large displacement options)
8
RC1
RC2
RC3
RC4
RC5
RC6
RC7
RC8
Cross-sectional area
Moment of inertia
Depth (diameter for circular cross section)
End release code at node 1
End release code at node 2
Shear factor in the element y-axis
Temperature difference in the element y-axis
Perimeter (for thermal analysis only)
RC1
14 to RC2
27 …
…
Cross sectional area
Moment of inertia
………
Temperature difference in the element y-axis
Perimeter (for thermal analysis only)
………
(Refer to the elements chapter in COSMOSM User Guide
for more information)
In
de
x
BEAM3D
RC Constants
6-6
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Table 6-3. Element Library: Real Constants (Continued)
COSMOSM
Element
Name
RC Constants
No.
RC
IMPIPE
17
RC1
RC2
RC3
RC4
RC5
RC6
RC7
RC8
RC9
RC10
RC11
RC12
—
RC13
RC14
RC15
RC16
RC17
Outside diameter of the pipe
Pipe thickness
Flexibility factor
End-release code
Internal pressure
Internal fluid density
Global Z-coordinate of the pipe internal fluid's free surface
Mass of internal fluid/hardware
External insulation density
Thickness of external insulation
Coefficient of buoyant force
Coefficient of axial strain correction due to external
hydrostatic and hydrodynamic pressures
Coefficient of added mass
Coefficient of fluid inertia force
Coefficient of normal drag
Coefficient of tangential drag
Prestrain
SPRING
2
RC1
RC2
Axial stiffness (for linear material only)
Rotational stiffness (for linear material only)
PLANE2D
2
RC1
RC2
Thickness (for plane stress option only)
Material angle (for linear orthotropic material only)
TRIANG
2
RC1
RC2
Thickness (for plane stress option only)
Material angle (for linear orthotropic material only)
6
RC1
RC2
RC3
RC4
RC5
RC6
Thickness
Temperature gradient
Unused constant for this element
Unused constant for this element
Normal prestress value for element x and y directions
Normal prestrain value for element x and y directions
In
de
x
SHELL3/4/6
SHELL3T/4T/
6T
Description
COSMOSM Advanced Modules
6-7
Chapter 6 Element Library
Table 6-3. Element Library: Real Constants (Concluded)
COSMOSM
Element
Name
RC Constants
No.
RC
2+3
NL
RC1
RCxx
The number of real constants to be entered depends on the
number of layers (NL). Refer to the elements chapter in
COSMOSM user guide for more information.
TETRA4R
9
RC1-9
X,Y,Z coordinates of three points (for orthotropic materials
only)
SOLID
9
RC1-9
X,Y,Z coordinates of three points (for orthotropic materials
only)
SHELL3L/4L
RC6
RC7
RC8
RC9
RC10
RC11
RC12
RC13
Allowable relative displacement between two nodes
Coefficient of friction
Spring stiffness
Preload in the gap
Maximum allowable distance beyond which gap responds
as perfectly rigid
Damper constant
Damper constant
Source Stiffness in X-direction
Source Stiffness in Y-direction
Source Stiffness in Z-direction
Target Stiffness in X-direction
Target Stiffness in Y-direction
Target Stiffness in Z-direction
7
RC1
RC2
RC3
RC4
RC5
RC6
RC7
Mass in the global Cartesian X-direction
Mass in the global Cartesian Y-direction
Mass in the global Cartesian Z-direction
Rotary inertia about the global Cartesian X-direction
Rotary inertia about the global Cartesian Y-direction
Rotary inertia about the global Cartesian Z-direction
Thermal capacity (in units of heat energy)
11
RC1
RC2
RC3
RC4
RC5
RC6
RC7
RC8
RC9
RC10
RC11
Mass in the global Cartesian X-direction
Mass in the global Cartesian Y-direction
Mass in the global Cartesian Z-direction
Rotary inertia about the global Cartesian X-direction
Rotary inertia about the global Cartesian Y-direction
Rotary inertia about the global Cartesian Z-direction
Outside diameter of the buoy
Coefficient of buoyant force
Coefficient of added mass
Coefficient of fluid inertia force
Coefficient of drag
RC1
RC2
RC3
RC4
RC5
GAP
(Node-toNode)
MASS
7
In
de
x
BUOY
Description
6-8
COSMOSM Advanced Modules
7
Commands and Examples
Command Summary
Table 7-1. Frequently Used Commands for Nonlinear Analysis
In
de
x
Command/Menu
Path
A_NONLINEAR
Analysis > NONLINEAR > NonL Analysis Options
ACTSET
Control > ACTIVATE > Set Entity
CONTACT >
Analysis > NONLINEAR >
CURDEF
LoadsBC > FUNCTION CURVE > Time/Temp Curve
INITIAL
LoadsBC > LOAD OPTIONS > Initial Cond
J INTEGRAL
Analysis > NONLINEAR >
NL_AUTO
Analysis > NONLINEAR > AutoStep Options
NL_BASE
Analysis > NONLINEAR > Base Motion Parameter
NL_CONTROL
Analysis > NONLINEAR > Solution Control
NL_INTGR
Analysis > NONLINEAR > Integration Options
NL_NRESP
Analysis > NONLINEAR > Response Options
NL_PLOT
Analysis > NONLINEAR > Plot Options
NL_PRINT
Analysis > NONLINEAR > Print Options
NL_RDAMP
Analysis > NONLINEAR > Damping Coefficient
MPC
LoadsBC > FUNCTION CURVE > Material Curve
MPCTYP
LoadsBC > FUNCTION CURVE > Material Curve Type
R_NONLINEAR
Analysis > NONLINEAR > Run NonL Analysis
RESTART
Analysis >
TIMES
LoadsBC > LOAD OPTIONS > Time Parameter
EKILL and ELIVE
Analysis > NONLINEAR >ELEMENT_BIRTH/DEATH
COSMOSM Advanced Modules
7-1
Chapter 7 Commands and Examples
The above represent some of the most frequently used commands needed to
perform nonlinear analysis using the NSTAR module. Information regarding
analysis type, direct time integration, initial conditions, damping, time and
temperature curves associated with different loading conditions and material
property sets, and numerical solution procedures are provided using these
commands. In addition, both line and surface contact problems may be considered
for analysis with the nonlinear module. Parameters required to define material
models are discussed in Chapter 3. Command descriptions are presented in
COSMOSM Command Reference Manual.
NSTAR can handle geometric, material, and contact nonlinearities. The
temperature dependency of material properties can also be handled.
As a guide to the users, a brief outline describing the application of specific
commands required to set up different categories of nonlinear problems is
presented.
Elastoplastic Analysis
In the following, only the commands essential for this nonlinear analysis case are
listed.
Command (Path)
Intended Use
EGROUP
(Propsets > Element Group)
Option 5 of this command
specifies the use of von Mises
elastoplastic model with an
isotropic or a kinematic
hardening rule for this element
group.
MPROP
In
de
x
(Propsets > Material Property)
7-2
COSMOSM Advanced Modules
Using this command the
required material properties
(EX, ETAN, NUXY, and
SIGYLD) for defining the
bilinear elastoplastic stressstrain curve are input.
Part 1 NSTAR / Nonlinear Analysis
MPCTYP and MPC
(LoadsBC>FUNCTION CURVE>
Material Curve Type, Material Curve)
These commands are used for
curve description of
elastoplastic stress-strain curve.
TIMES
(LoadsBC > LOAD OPTIONS > Time Parameter)
The command is used to define
the starting time, the ending
time and the time step
increment for all nonlinear
analysis cases. This command
is not needed if the Arc-length
Control technique is used.
ACTSET, TC, Curve_Label
(Control > ACTIVATE > Set Entity)
Commands to activate time
curves (previously defined by
the CURDEF command) and
define the accompanying forces
and/or pressure and/or
prescribed displacements. Note
that it is a good practice to
deactivate the curves
immediately after their use to
avoid errors or confusion of any
kind. This command is not
needed if the Arc-length
Control technique is used.
FND, FPT, FCR, FSF, FCT, FRG
(LoadsBC > STRUCTURAL > FORCE >)
See ACTSET, TC,
Curve_Label above.
PEL, PCR, PSF, PRG
(LoadsBC > STRUCTURAL > PRESSURE >)
See ACTSET, TC,
Curve_Label above.
DND, DPT, DCR, DSF, DCT, DRG
(LoadsBC>STRUCTURAL>DISPLACEMENT>)
See ACTSET, TC,
Curve_Label above.
ACTSET, TC, 0
See ACTSET, TC,
Curve_Label above.
In
de
x
(Control > ACTIVATE > Set Entity)
COSMOSM Advanced Modules
7-3
Chapter 7 Commands and Examples
A_NONLINEAR***
(Analysis > NONLINEAR >
NonL Analysis Options)
This command specifies the
nonlinear option (S = STATIC,
or D = DYNAMIC) and some
other relevant options and
parameters.
NL_CONTROL***
(Analysis > NONLINEAR > Solution Control)
This command specifies the
numerical procedure to be used
in nonlinear analysis. It defines
the Control technique, the
iterative method, and their
associated input.
NL_AUTO***
(Analysis > NONLINEAR > AutoStep Options)
This command is used to
activate the adaptive automatic
stepping option in nonlinear
structural analysis. Delimiters
for the step size can be
specified.
NL_NRESP
(Analysis > NONLINEAR > Response Options)
This command is used to select
nodes for which the
displacement response is to be
saved for X-Y plot purposes.
ACTSET, TC, Curve_Label NL_PLOT
(Analysis > NONLINEAR > Plot Options)
This command is used to define
sets of steps for which the
deformations and stresses are to
be plotted.
ACTSET, TC, Curve_Label NL_PRINT
(Analysis > NONLINEAR > Print Options)
This command controls the
output quantities to be written
in the output file.
ACTSET, TC, Curve_Label R_NONLINEAR
(Analysis > NONLINEAR > Run NonL Analysis)
This command runs NSTAR
which is the nonlinear module
of COSMOSM.
In
de
x
As it is evident, the commands required to completely describe the finite element
model and other properties should be defined before issuing the solution command
[R_NONLINEAR (Analysis > NONLINEAR > Run NonL Analysis)].
7-4
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Load curves essential for the incremental solution of nonlinear static
problems are represented by (pseudo) time curves. These curves are
defined using the CURDEF (LoadsBC > FUNCTION CURVE > Time/
Temp Curve) command.
*** If the default options (and/or parameters) are satisfactory, this command may be omitted.
**
Geometrically Nonlinear Analysis
The special commands required to set up the geometrically nonlinear analysis
available in NSTAR are as follows:
Command (Path)
Intended Use
EGROUP
(Propsets > Element Group)
Option 6 of this command sets
the flag on for geometrically
nonlinear analysis formulation
to be associated with this
element group.
TIMES
(LoadsBC > LOAD OPTIONS > Time Parameter)
See the Elastoplastic Analysis
section.
ACTSET, TC, Curve_Label
(Control > ACTIVATE > Set Entity)
See the Elastoplastic Analysis
section.
FND, FPT, FCR, FSF, FCT, FRG
(LoadsBC > STRUCTURAL > FORCE >)
See the Elastoplastic Analysis
section.
PEL, PCR, PSF, PRG
(LoadsBC > STRUCTURAL > PRESSURE >)
See the Elastoplastic Analysis
section.
DND, DPT, DCR, DSF, DCT, DRG
(LoadsBC > STRUCTURAL > DISPLACEMENT >) See the Elastoplastic Analysis
section.
ACTSET, TC, 0
See the Elastoplastic Analysis
section.
In
de
x
(Control > ACTIVATE > Set Entity)
COSMOSM Advanced Modules
7-5
Chapter 7 Commands and Examples
A_NONLINEAR
(Analysis > NONLINEAR > NonL Analysis Options) See the Elastoplastic Analysis
section.
NL_CONTROL
(Analysis > NONLINEAR > Solution Control)
See the Elastoplastic Analysis
section.
NL_AUTO
(Analysis > NONLINEAR > AutoStep Options)
See the Elastoplastic Analysis
section.
R_NONLINEAR
(Analysis > NONLINEAR > Run NonL Analysis)
See the Elastoplastic Analysis
section.
Again, it is assumed that the full model has been properly constructed with the help
of GEOSTAR commands.
Elastoplastic Large Displacement Analysis
Command (Path)
Intended Use
EGROUP
(Propsets > Element Group)
Options 5 and 6 are both turned
on to specify the elastoplastic
model and the geometric
nonlinear option.
MPROP
(Propsets > Material Property)
See the Elastoplastic Analysis
section.
MPCTYP and MPC
(LoadsBC>FUNCTION CURVE >
Material Curve Type, Material Curve)
See the Elastoplastic Analysis
section.
TIMES
(LoadsBC > LOAD OPTIONS > Time Parameter)
See the Elastoplastic Analysis
section.
ACTSET, TC, Curve_Label
(Control > ACTIVATE > Set Entity)
See the Elastoplastic Analysis
section.
FND, FPT, FCR, FSF, FCT, FRG
In
de
x
(LoadsBC > STRUCTURAL > FORCE >)
7-6
COSMOSM Advanced Modules
See the Elastoplastic Analysis
section.
Part 1 NSTAR / Nonlinear Analysis
PEL, PCR, PSF, PRG
(LoadsBC > STRUCTURAL > PRESSURE >)
See the Elastoplastic Analysis
section.
DND, DPT, DCR, DSF, DCT, DRG
(LoadsBC>STRUCTURAL>DISPLACEMENT>
See the Elastoplastic Analysis
section.
ACTSET, TC, 0
(Control > ACTIVATE > Set Entity)
See the Elastoplastic Analysis
section.
A_NONLINEAR
(Analysis>NONLINEAR>NonL Analysis Options)
See the Elastoplastic Analysis
section.
NL_CONTROL
(Analysis > NONLINEAR > Solution Control)
See the Elastoplastic Analysis
section.
NL_AUTO
(Analysis > NONLINEAR > AutoStep Options)
See the Elastoplastic Analysis
section.
R_NONLINEAR
(Analysis > NONLINEAR > Run NonL Analysis)
See the Elastoplastic Analysis
section.
Nonlinear Dynamic Analysis
The nonlinear dynamic analysis based on direct time integration methods requires
the following special commands:
Command (Path)
Intended Use
EGROUP
(Propsets > Element Group)
Options 5 and 6 both control the
material models and the
geometric nonlinear analysis
flags.
MPROP
(Propsets > Material Property)
See the Elastoplastic Analysis
section.
MPCTYP and MPC
See the Elastoplastic Analysis
section.
In
de
x
(LoadsBC>FUNCTION CURVE>
Material Curve Type, Material Curve)
COSMOSM Advanced Modules
7-7
Chapter 7 Commands and Examples
TIMES
(LoadsBC > LOAD OPTIONS > Time Parameter)
See the Elastoplastic Analysis
section.
ACTSET, TC, Curve_Label
(Control > ACTIVATE > Set Entity)
FND, FPT, FCR, FSF, FCT, FRG
(LoadsBC > STRUCTURAL > FORCE >)
PEL, PCR, PSF, PRG
(LoadsBC > STRUCTURAL > PRESSURE >)
ACTSET, TC, 0
(Control > ACTIVATE > Set Entity)
See the Elastoplastic Analysis
section.
A_NONLINEAR
(Analysis>NONLINEAR>NonL Analysis Options)
The analysis option should be
switched to “D” for dynamic.
NL_CONTROL
(Analysis > NONLINEAR > Solution Control)
The Force control option with
MNR or NR methods must be
used. No line search is
performed for dynamic
analysis.
NL_RDAMP
(Analysis > NONLINEAR > Damping Coefficient)
This command is used to
incorporate Rayleigh
proportional damping in the
dynamic analysis.
NL_BASE
(Analysis>NONLINEAR>Base Motion Parameter)
This command is used to
incorporate the effects of base
motion accelerations in the
dynamic analysis.
NL_INTGR
In
de
x
(Analysis > NONLINEAR > Integration Options)
7-8
COSMOSM Advanced Modules
This command can be used to
choose the direct implicit time
integration schemes. The user
can select either NewmarkBeta or Wilson-Theta methods.
If this command is not issued,
the Newmark-Beta method
with defaults values is
incorporated in the analysis.
Part 1 NSTAR / Nonlinear Analysis
NL_AUTO
(Analysis > NONLINEAR > AutoStep Options)
See the Elastoplastic Analysis
section.
NL_NRESP
(Analysis > NONLINEAR > Response Options)
This command is used to select
nodes for which the
displacement, velocity, and
acceleration responses are to be
saved for X-Y-plotting
purposes.
NL_PLOT
(Analysis > NONLINEAR > Plot Options)
See the Elastoplastic Analysis
section.
NL_PRINT
(Analysis > NONLINEAR > Print Options)
See the Elastoplastic Analysis
section.
R_NONLINEAR
(Analysis > NONLINEAR > Run NonL Analysis)
See the Elastoplastic Analysis
section.
Linear Dynamic Analysis (Time-History)
The nonlinear module NSTAR may also be used for the solution of linear dynamic
problems using time integration methods. Special commands required to set up this
case are listed below.
Command (Path)
Intended Use
TIMES
(LoadsBC > LOAD OPTIONS > Time Parameter)
See the Elastoplastic Analysis
section.
ACTSET, TC, Curve_Label
(Control > ACTIVATE > Set Entity)
See the Elastoplastic Analysis
section.
FND, FPT, FCR, FSF, FCT, FRG
(LoadsBC > STRUCTURAL > FORCE >)
See the Elastoplastic Analysis
section.
PEL, PCR, PSF, PRG
See the Elastoplastic Analysis
section.
In
de
x
(LoadsBC > STRUCTURAL > PRESSURE >)
COSMOSM Advanced Modules
7-9
Chapter 7 Commands and Examples
ACTSET, TC, 0
(Control > ACTIVATE > Set Entity)
See the Elastoplastic Analysis
section.
A_NONLINEAR
(Analysis>NONLINEAR>NonL Analysis Options)
See the Nonlinear Dynamic
Analysis section.
NL_CONTROL
(Analysis > NONLINEAR > Solution Control)
See the Nonlinear Dynamic
Analysis section.
NL_RDAMP
(Analysis > NONLINEAR > Damping Coefficient)
See the Nonlinear Dynamic
Analysis section.
NL_BASE
(Analysis>NONLINEAR>Base Motion Parameter)
See the Nonlinear Dynamic
Analysis section.
NL_INTGR
(Analysis > NONLINEAR > Integration Options)
See the Nonlinear Dynamic
Analysis section.
NL_AUTO
(Analysis > NONLINEAR > AutoStep Options)
See the Elastoplastic Analysis
section.
R_NONLINEAR
(Analysis > NONLINEAR > Run NonL Analysis)
See the Elastoplastic Analysis
section.
Analysis Including Temperature Loading
Command (Path)
Intended Use
TIMES
(LoadsBC > LOAD OPTIONS > Time Parameter)
See the Elastoplastic Analysis
section.
ACTSET, TC, Curve_Label
In
de
x
(Control > ACTIVATE > Set Entity)
7-10
COSMOSM Advanced Modules
These commands are used to
activate time curves and define
the accompanying loading
temperature. Again, it is a good
practice to deactivate the curves
immediately after their use to
avoid errors or confusion of any
kind.
Part 1 NSTAR / Nonlinear Analysis
NTND, NTPT, NTCR, NTSF, NTCT, NTRG
(LoadsBC > THERMAL > TEMPERATURE >)
See ACTSET, TC,
Curve_Label above.
ACTSET, TC, 0
(Control > ACTIVATE > Set Entity)
See ACTSET, TC,
Curve_Label above.
A_NONLINEAR
(Analysis>NONLINEAR>NonL Analysis Options)
In this command, the special
loading flag must include the
character “T''.
NL_CONTROL
(Analysis > NONLINEAR > Solution Control)
See the Elastoplastic Analysis
section.
NL_AUTO
(Analysis > NONLINEAR > AutoStep Options)
See the Elastoplastic Analysis
section.
R_NONLINEAR
(Analysis > NONLINEAR > Run NonL Analysis)
See the Elastoplastic Analysis
section.
Structural Analysis with Temperature-Dependent
Material Properties
The relevant commands needed to set up this analysis for problems with
temperature-dependent material properties are:
Command (Path)
Intended Use
EGROUP
(Propsets > Element Group)
Options 5 and 6 both control the
material models and the
geometric nonlinear analysis
flags (refer to COSMOSM
Command Reference Manual
for a list of different material
models with temperaturedependent parameters).
ACTSET, TP, Curve_Label
In
de
x
(Control > ACTIVATE > Set Entity)
COSMOSM Advanced Modules
These commands are used to
activate temperature curves and
define the associated
temperature-dependent-
7-11
Chapter 7 Commands and Examples
material properties. Again, it is
a good practice to deactivate the
curves immediately after their
use to avoid errors or confusion
of any kind.
MPROP
(Propsets > Material Property)
See ACTSET, TP,
Curve_Label above.
EX, SIGYLD, ETAN,....., etc.
ACTSET, TP, 0
(Control > ACTIVATE > Set Entity)
See ACTSET, TP,
Curve_Label above.
TIMES
(LoadsBC > LOAD OPTIONS > Time Parameter)
See the Elastoplastic Analysis
section.
ACTSET, TC, Curve_Label
(Control > ACTIVATE > Set Entity)
See the Analysis Including
Temperature Loading section.
NTND, NTPT, NTCR, NTSF, NTCT, NTRG
(LoadsBC > THERMAL > TEMPERATURE >)
See the Analysis Including
Temperature Loading section.
ACTSET, TC, 0
(Control > ACTIVATE > Set Entity)
See the Analysis Including
Temperature Loading section.
A_NONLINEAR
(Analysis > NONLINEAR >
NonL Analysis Options)
See the Analysis Including
Temperature Loading section.
NL_CONTROL
(Analysis > NONLINEAR > Solution Control)
See the Elastoplastic Analysis
section.
NL_AUTO
(Analysis > NONLINEAR > AutoStep Options)
See the Elastoplastic Analysis
section.
R_NONLINEAR
In
de
x
(Analysis > NONLINEAR > Run NonL Analysis)
7-12
COSMOSM Advanced Modules
See the Elastoplastic Analysis
section.
Part 1 NSTAR / Nonlinear Analysis
Elastic Creep Analysis
The special commands essential for setting up the input for this type of analysis are
given below:
Command (Path)
Intended Use
EGROUP
(Propsets > Element Group)
Option 7 of this command sets
the flag for creep analysis with
this element group.
MPROP
(Propsets > Material Property)
Use CREEPC or CREEPX
according to the creep law used
in the analysis to define creep
constants.
TIMES
(LoadsBC > LOAD OPTIONS > Time Parameter)
See the Elastoplastic Analysis
section.
ACTSET, TC, Curve_Label
(Control > ACTIVATE > Set Entity)
See the Elastoplastic Analysis
section.
FND, FPT, FCR, FSF, FCT, FRG
(LoadsBC > STRUCTURAL > FORCE >)
See the Elastoplastic Analysis
section.
PEL, PCR, PSF, PRG
(LoadsBC > STRUCTURAL > PRESSURE >)
See the Elastoplastic Analysis
section.
DND, DPT, DCR, DSF, DCT, DRG
(LoadsBC > STRUCTURAL > DISPLACEMENT >) See the Elastoplastic Analysis
section.
ACTSET, TC, 0
(Control > ACTIVATE > Set Entity)
See the Elastoplastic Analysis
section.
A_NONLINEAR Command
(Analysis>NONLINEAR>NonL Analysis Options)
See the Elastoplastic Analysis
section.
NL_CONTROL
(Analysis > NONLINEAR > Solution Control)
See the Elastoplastic Analysis
section.
NL_AUTO
See the Elastoplastic Analysis
section.
In
de
x
(Analysis > NONLINEAR > AutoStep Options)
COSMOSM Advanced Modules
7-13
Chapter 7 Commands and Examples
R_NONLINEAR
(Analysis > NONLINEAR > Run NonL Analysis)
See the Elastoplastic Analysis
section.
Static Analysis Using Displacement Control Technique
Command (Path)
Intended Use
TIMES
(LoadsBC > LOAD OPTIONS > Time Parameter)
See the Elastoplastic Analysis
section.
Define a Pattern of Loads (no association with time curve).
ACTSET, TC, Curve_Label = 1
(Control > ACTIVATE > Set Entity)
This curve will be associated
with the controlled degree of
freedom used in the solution
process using the Displacement
Control technique.
FND, FPT, FCR, FSF, FCT, FRG
(LoadsBC > STRUCTURAL > FORCE >)
—
PEL, PCR, PSF, PRG
(LoadsBC > STRUCTURAL > PRESSURE >)
—
A_NONLINEAR
(Analysis>NONLINEAR>NonL Analysis Options)
See the Elastoplastic Analysis
section.
NL_CONTROL
(Analysis > NONLINEAR > Solution Control)
The Displacement Control
option with MNR or NR
iterative methods is selected.
Also, the controlled degree of
freedom is specified.
NL_AUTO
(Analysis > NONLINEAR > AutoStep Options)
See the Elastoplastic Analysis
section.
NL_NRESP
In
de
x
(Analysis > NONLINEAR > Response Options)
7-14
COSMOSM Advanced Modules
This command is used to select
nodes for which the
displacement response is to be
saved for XY-plotting
Part 1 NSTAR / Nonlinear Analysis
purposes. X-Y-plots of the load
factor multiplier (LFACT), on
the Y-axis, versus the nodal
displacement components (UX,
UY, UZ, ...), on the X-axis, of
the selected nodes can be
provided.
NL_PLOT
(Analysis > NONLINEAR > Plot Options)
See the Elastoplastic Analysis
section.
NL_PRINT
(Analysis > NONLINEAR > Print Options)
See the Elastoplastic Analysis
section.
R_NONLINEAR
(Analysis > NONLINEAR > Run NonL Analysis)
See the Elastoplastic Analysis
section.
Static Analysis Using Arc-Length Control Technique
Define a Pattern of Loads (no association with time curve).
Command (Path)
Intended Use
FND, FPT, FCR, FSF, FCT, FRG
(LoadsBC > STRUCTURAL > FORCE >)
Forces and pressures defined:
ACTSET,TC,1 will be
considered. (No time curves
need to be defined).
PEL, PCR, PSF, PRG
(LoadsBC > STRUCTURAL > PRESSURE >)
A_NONLINEAR
(Analysis > NONLINEAR >
NonL Analysis Options)
See the Elastoplastic Analysis
section.
NL_CONTROL
In
de
x
(Analysis > NONLINEAR > Solution Control)
COSMOSM Advanced Modules
The Arc-Length Control option
with MNR or NR iterative
methods is selected. Also, the
parameters required for this
control are input.
7-15
Chapter 7 Commands and Examples
NL_AUTO
(Analysis > NONLINEAR > AutoStep Options)
See the Elastoplastic Analysis
section.
NL_NRESP
(Analysis > NONLINEAR > Response Options)
See the Static Analysis Using
Displacement Control
Technique section.
NL_PLOT
(Analysis > NONLINEAR > Plot Options)
See the Elastoplastic Analysis
section.
NL_PRINT
(Analysis > NONLINEAR > Print Options)
See the Elastoplastic Analysis
section.
R_NONLINEAR
(Analysis > NONLINEAR > Run NonL Analysis)
See the Elastoplastic Analysis
section.
Examples
The following are examples of nonlinear analyses.
Elastoplastic Nonlinear Analysis Example
An example of elastoplastic, nonlinear, static analysis is described. This includes
the descriptions of the steps required to set up and solve the problem, in detail from
GEOSTAR.
Statement of the Problem
In
de
x
A cantilever metal sheet is subjected to a uniform pressure along the edge as shown
in Figure 7-1. Investigate the elastoplastic response of this metal sheet using 4node, 2D plane stress elements. The material of the sheet is assumed to obey the
von Mises yield criterion.
7-16
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure 7-1. Problem Sketch
p
13
13
14
15
16
10
9
7
9
L
10
4
5
1
8
11
5
6
6
7
2
1
9
2
12
t
1 2
Time Curve
8
σ
3
3
8
4
ET
L
Problem Sketch
σy
E
ε
Stress - Strain Curve
Given
E
= EX = 21,000 N/mm2
L
= 30 mm
σy = 10 N/mm2
ν
= 0.3
ET = 5,000 N/mm2
t
= 1 mm
GEOSTAR Input
In
de
x
1.
Define the element group. For this example, the 2D plane stress element is
selected.
Geo Panel: Propsets > Element Group (EGROUP)
Element group > 1
Element category > Area
Element type (for area) > PLANE2D
COSMOSM Advanced Modules
7-17
Chapter 7 Commands and Examples
OP1:S/F flag > Solid
OP2:Integr Type > QM6
OP3:Type > Plane Stress
OP4:Stress direction > Global Cartesian
OP5:Mat > von Mises (isotropic)
OP6:Disp. > Small
OP7:Material creep > No
OP8:Strain plasticity > Small
2.
The plasticity model used in the analysis is based on von Mises yield criterion
with bilinear isotropic hardening rule. Define the bilinear stress-strain curve
(Figure 7-1) by EX, ETAN and SIGYLD options in command MPROP
(Propsets > Material Property). EX defines Young's Modulus, ETAN defines
the Tangent Modulus (ET) and SIGYLD defines the Yield Stress σy.
Geo Panel: Propsets > Material Property (MPROP)
Material property set > 1
Material property name > EX
Property value > 21000
Material property name > ETAN
Property value > 5000
Material property name > SIGYLD
Property value > 10
The default value of NUXY is 0.3. Therefore, it is not necessary to specify
Poisson's ratio unless other than the default value is required.
3.
Define the thickness of the plane stress element.
Geo Panel: Propsets > Real Constant (RCONST)
Associated element group > 1
Real constant set > 1
Start location of the real constants > 1
No. of real constants to be entered > 2
RC1: Thickness > 1.0
RC2: Material angle (beta) > 0.0
In
de
x
4.
Define the geometry of the model. Change the view to X-Y using the Viewing
(Binocular) icon.
Geo Panel:
7-18
Geometry > SURFACES > Draw w/ 4 Coord (SF4CORD)
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Surface > 1
Keypoint 1 XYZ-coordinate value > 0,0,0
Keypoint 2 XYZ-coordinate value > 30,0,0
Keypoint 3 XYZ-coordinate value > 30,30,0
Keypoint 4 XYZ-coordinate value > 0,30,0
5.
Use the Auto Scale icon to readjust the model scale. Define the elements and
nodes through mesh generation.
Geo Panel: Geo Panel:
Meshing > PARAMETRIC MESH > Surfaces
(M_SF)
Beginning surface > 1
Ending surface > 1
Increment > 1
Number of nodes per element > 4
Number of elements on first curve > 3
Number of elements on second curve > 3
Spacing ratio for first curve > 1.0
Spacing ratio for second curve > 1.0
6.
Define displacement constraints using the DCR (LoadsBC > STRUCTURAL >
DISPLACEMENT > Define Curves) command.
Geo Panel:
LoadsBC > STRUCTURAL > DISPLACEMENT > Define by
Curves (DCR)
Beginning curve > 2
Displacement label > AL: All 6 DOF
Accept defaults ...
7.
Define the starting time, final time and time increment using the TIMES
(LoadsBC > LOAD OPTIONS > Time Parameter) command.
Geo Panel: LoadsBC > LOAD OPTIONS > Time Parameter (TIMES)
Starting time > 0.0
Final time > 8
Time increment > 1
In
de
x
8.
Now define the load versus time curve using the CURDEF (LoadsBC >
FUNCTION CURVE > Time/Temp Curve) command.
Geo Panel: LoadsBC > FUNCTION CURVE > Time/Temp Curve
(CURDEF)
COSMOSM Advanced Modules
7-19
Chapter 7 Commands and Examples
Curve type > Time
Curve number > 1
Start point > 1
Time value for point 1 > 0.0
Function value for point 1 > 0
Time value for point 2 > 1.0
Function value for point 2 > 9
Time value for point 3 > 2.0
Function value for point 3 > 10
Time value for point 4 > 8.0
Function value for point 4 > 13
Time value for point 5 > 9.0
Function value for point 5 > 13
Time value for point 6 >
9.
A number of time-load curves can be defined. However, the curve associated
with the applied load must be activated prior to the definition of the load. Since
you just defined time curve 1, it is currently active.
10.
Define pressure loading using the PCR (LoadsBC > STRUCTURAL >
PRESSURE > Define Curves) command.
Geo Panel:
LoadsBC > STRUCTURAL > PRESSURE > Define Curves
(PCR)
Beginning curve > 1
Pressure magnitude > -1
Ending curve > 1
Increment > 1
Pressure at the end of direction 1 > -1
Pressure direction > Normal direction
11.
By default, COSMOSM will write displacement output for all the nodes defined
in the problem. But here, we are only interested in the displacements of the free
edge of the cantilever plate. Therefore, define the group of nodes on the edge to
be considered for displacement output. Remember that this step only affects the
output file (.OUT) and not the files used for postprocessing.
Geo Panel:
Analysis > OUTPUT OPTIONS > Set Nodal Range
(PRINT_NDSET)
Number of groups > 1
In
de
x
Beginning node of group 1 > 1
Ending node of group 1 > 4
7-20
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
12.
Define the number of nodes for which the response curve is required to be
generated. Use the NL_NRESP (Analysis > NONLINEAR > Response
Options) command for this purpose. This command specifies a time-history
plot file to be created for the defined set of nodes.
Geo Panel: Analysis > NONLINEAR > Response Options (NL_NRESP)
Starting location [1] > 1
Node 1 > 1
Node 2 > 2
Node 3 > 3
Node 4 > 4
13.
Next, define the time step number for which the displacements and stresses are
to be saved for later graphic processing. Here, results for time step number 1
through 8 are desired.
Geo Panel: Analysis > NONLINEAR > Plot Options (NL_PLOT)
Set 1 beginning step > 1
Set 1 ending step > 8
Set 1 step increment > 1
14.
Finally, having completed the description of the model and the specification of
the desired postprocessing data, the nonlinear analysis will be performed.
Geo Panel:
Analysis > NONLINEAR > Run NonL Analysis
(R_NONLINEAR)
15.
This command runs the NSTAR module and gives displacement as well as
stress output. You can examine the output file using the EDIT (File > Edit...)
command or your favorite text editor.
Postprocessing
In
de
x
16.
Plot the deformed shape at time step 8.
Geo Panel: Results > PLOT > Deformed Shape (DEFPLOT)
Step number > 8
Beginning element > 1
Ending element > 9
Increment > 1
Deformation scale flag >
Scale factor > 97.258057
COSMOSM Advanced Modules
7-21
Chapter 7 Commands and Examples
17.
Use the sliding scale button to set the scale to 0.5. Animate the deflected shape
for time steps 1 through 8.
Geo Panel: Results > PLOT > Animate (ANIMATE)
Beginning step number > 1
Ending step number > 8
Step increment > 1
Animation type > Two Way
Delay number > 0
Scale factor > 97.258057
Number of frames > 9
Save and play as AVI > No
AVI file name > test.avi
Number of iterations > 1
18.
Plot von Mises stresses at time step 8:
Geo Panel: Results > PLOT > Stress (STRPLOT)
Time step number > 8
Component > VON
Layer number > 1
Coordinate system > 0
Stress flag > Nodal stress
Face of element > Top face
Select contour option ...
Plot type > Color filled contour
Accept defaults ...
19.
Plot the variation of von Mises stresses along a line segment in the sheet.
Geo Panel: Results > PLOT > Path Graph (LSECPLOT)
Node > 2
Node > 14
Node > 14
The resulting plot is shown in Figure 7-2. The TRANSLATE command (or the
Translate icon) is used to adjust the relative position of the plotted figures.
20.
Animate the von Mises stresses for steps 1 through 8.
In
de
x
Geo Panel: Results > PLOT > Animate (ANIMATE)
Beginning step number > 1
Ending step number > 8
7-22
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Step increment > 1
Animation type > Two way Animation
Delay number >
Scale factor > 97.1118
21.
Generate a time history plot:
Geo Panel: Display > XY PLOTS > Activate Post-Proc (ACTXYPLOT)
Graph Number >1
X_variable > Time
Y_variable > UY
Node number > 2
Graph color > 12
Graph line style > Solid
Graph symbol style > 0
Graph id > 2N
Geo Panel: Display > XY PLOTS > Plot Curves (XYPLOT)
Plot graph > Yes
The resulting plot is shown in Figure 7-3.
In
de
x
Figure 7-2. The von Mises Stress Plot at Time Step 8
COSMOSM Advanced Modules
7-23
Chapter 7 Commands and Examples
Figure 7-3. Vertical Displacement Versus Time at Node 2
Large Displacement Nonlinear Analysis Example
An example of Large Displacement Static analysis is described in this section. This
includes the description of the steps required to set up and solve the problem, in
detail. Here, the problem of a cantilever beam under uniformly distributed load is
considered.
Statement of the Problem
In
de
x
Find the static response of a cantilever beam under uniform loading that causes
large displacements. Use 2D plane stress elements. The cantilever is shown in
Figure 7-4.
7-24
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure 7-4. A Cantilever Beam Under Uniform Loading
p/2
h
b
L
Problem Sketch
p
19
18
12
28
1
2
3
1
4
5
17
5
11
2
2 D Model
100
Time Curve
t
Given
L
= 10 in
b
= 1 in
ν
= 0.2
h
= 1 in
E
= 12,000 psi
p
= 10 lb/in
GEOSTAR Input
In
de
x
1.
Define the element group. For this example, the PLANE2D with plane stress
option is selected.
Geo Panel: Propsets > Element Group (EGROUP)
Element group > 1
Element category > Area
Element type (for area) > PLANE2D
OP1:S/F flag > Solid
COSMOSM Advanced Modules
7-25
Chapter 7 Commands and Examples
OP2:Integr type > QM6
OP3:Type > Plane Stress
OP4:Stress direction > Global Cartesian
OP5:Mat > Linear Elastic
OP6:Disp. formulation > Updated Lagrangian
OP7:Material creep > No
OP8:Strain plasticity > Small
2.
Define material properties (EX, NUXY).
Geo Panel: Propsets > Material Property (MPROP)
Material property set > 1
Material property name > EX
Property value > 12000
Material property name > NUXY
Property value > 0.20
3.
Define the thickness of the plane stress elements.
Geo Panel: Propsets > Real Constant (RCONST)
Associated element group > 1
Real constant set > 1
Start location of the real constants > 1
No. of real constants to be entered > 2
RC1: Thickness > 1.0
RC2: Material angle (beta) > 0.0
4.
Define the geometry of the model. Change the view to X-Y using the Viewing
icon.
Geo Panel: Geometry > SURFACES > Draw w/ 4 Coord (SF4CORD)
Surface > 1
Keypoint 1 XYZ-coordinate value > 0,0,0,
Keypoint 2 XYZ-coordinate value > 10,0,0,
Keypoint 3 XYZ-coordinate value > 10,1,0,
Keypoint 4 XYZ-coordinate value > 0,1,0,
5.
Define the elements and nodes through mesh generation.
In
de
x
Geo Panel: Meshing > PARAMETRIC MESH > Surfaces (M_SF)
Beginning surface > 1
Ending surface > 1
7-26
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Increment > 1
Number of nodes per element > 8
Number of elements on first curve > 5
Number of elements on second curve > 1
Spacing ratio for first curve > 1.0
Spacing ratio for second curve > 1.0
6.
Define displacement constraints on the boundary using the DCR (LoadsBC >
STRUCTURAL > DISPLACEMENT > Define Curves) command.
Geo Panel:
LoadsBC > STRUCTURAL > DISPLACEMENT > Define
Curves (DCR)
Beginning curve > 3
Displacement label > AL: All 6 DOF
Accept defaults ...
7.
Define the starting time, final time and time increment for the solution using the
TIMES (LoadsBC > LOAD OPTIONS > Time Parameter) command.
Geo Panel: LoadsBC > LOAD OPTIONS > Time Parameter (TIMES)
Starting time > 0.0
Final time > 100
Time increment > 1
8.
Define the load versus time curve using the CURDEF (LoadsBC > FUNCTION
CURVE > Time/Temp Curve) command.
Geo Panel:
LoadsBC > FUNCTION CURVE > Time/Temp Curve
(CURDEF)
Curve type > Time
Curve number > 1
Start point > 1
In
de
x
Time value for point 1 > 0
Function value for point 1 > 0
Time value for point 2 > 100
Function value for point 2 > 5
Time value for point 3 > <CR>
COSMOSM Advanced Modules
7-27
Chapter 7 Commands and Examples
9.
A number of time-load curves can be defined. However, the curve associated
with an applied load or pressure must be activated prior to the definition of that
load. Note that the last defined time curve is currently active.
10.
Define pressure loading using the PCR (LoadsBC > STRUCTURAL >
PRESSURE > Define Curves) command.
Geo Panel:
LoadsBC > STRUCTURAL > PRESSURE > Define by Curves
(PCR)
Beginning curve > 1
Pressure magnitude > -1.0
Accept defaults ...
Geo Panel:
LoadsBC > STRUCTURAL > PRESSURE > Define by Curves
(PCR)
Beginning curve > 2
Pressure magnitude > 1.0
Accept defaults ...
11.
By default, COSMOSM will write displacement output for all the nodes defined
in the problem. But here, we are only interested in the displacements, of the free
edge of the cantilever beam. Therefore, define the group of nodes on the edge to
be considered for displacement output. Remember that this step only affects the
output file (.OUT) and not the files used for postprocessing.
Geo Panel:
Analysis > OUTPUT OPTIONS > Set Nodal Range
(PRINT_NDSET)
Number of groups > 1
Beginning node of group 1 > 17
Ending node of group 1 > 17
12.
Define the number of nodes for which the response curve is required to be
generated. Use the NL_NRESP (Analysis > NONLINEAR > Response
Options) command for this purpose. This command specifies a time-history
plot file to be created for the defined set of nodes.
Geo Panel: Analysis > NONLINEAR > Response Options (NL_NRESP)
Starting location > 1
In
de
x
Node 1 > 17
7-28
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
13.
Define the time step number for which displacements and stresses are to be
saved for later graphic examination. Here, results for time step number 10
through 100 with increment 10 are desired.
Geo Panel: Analysis > NONLINEAR > Plot Options (NL_PLOT)
Set 1 beginning step > 10
Set 1 ending step > 100
Set 1 step Increment > 10
14.
Define the type of analysis by the A_NONLINEAR (Analysis > NONLINEAR >
NonL Analysis Options) command. Theoretically, after each time step
increment, the structural system should be in the equilibrium state. So, generally
it is advisable to keep the equilibrium option on. When the time step size is very
small and the loading is smooth, this equilibrium option can be turned off. For
this example, the number of time steps between equilibrium is set to 100 to
check equilibrium only at time step 100.
Geo Panel:
Analysis > NONLINEAR > NonL Analysis Options
(A_NONLINEAR)
Analysis option > S
Steps between reforming stiffness > 1
Steps between eqlbm. iterations > 100
Max equilibrium iterations > 20
Accept defaults ...
15.
Finally, having completed the description of the model and the specification of
postprocessing data, the nonlinear analysis will be performed.
Geo Panel:
Analysis > NONLINEAR > Run NonL Analysis
(R_NONLINEAR)
16.
This command runs the NSTAR module and gives displacement as well as
stress output. You can examine the output file using the EDIT (File > Edit...)
command or your favorite text editor.
Postprocessing
1.
Plot the deformed shape at time step 100.
Geo Panel:
Results > PLOT > Deformed Shape (DEFPLOT)
In
de
x
Accept defaults ...
Geo Panel:
LoadsBC > STRUCTURAL > DISPLACEMENT > Plot
(DPLOT)
COSMOSM Advanced Modules
7-29
Chapter 7 Commands and Examples
Accept defaults ...
The resulting plot is shown in Figure 7-5.
Figure 7-5. Deformation at Time Step 100
2.
Generate a displacement contour plot. Scale the resulting plot by a factor of 0.7
using the sliding button.
Geo Panel: Results > PLOT > Displacement (ACTDIS, DISPLOT)
Time step number > 100
Component > URES
Select contour option ...
Plot Type > Color filled contour
Accept defaults ...
Geo Panel:
LoadsBC > STRUCTURAL > DISPLACEMENT > Plot
(DPLOT)
Accept defaults ...
Geo Panel:
LoadsBC > STRUCTURAL > PRESSURE > Plot (DPLOT)
Accept defaults ...
In
de
x
The resulting plot is shown in Figure 7-6.
7-30
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure 7-6
3.
Animate the resultant displacements for steps 10 through 100 with increment
10:
Geo Panel: Results > PLOT > Animate (ANIMATE)
Beginning step number > 10
Ending step number > 100
Step increment > 10
Animation type > Two Way
Delay number > 0
Scale factor > 0.13701
Save and play as AVI > No
AVI file name > test.avi
Number of iterations > 1
4.
Plot von Mises Stresses at time step 100:
Geo Panel: Results > PLOT > Stress (STRPLOT)
Time step number > 100
Component > VON
Accept defaults ...
In
de
x
Select contour option ...
Plot type > Color filled contour
Beginning element > 1
Ending element > 5
Increment > 1
Shape of model > Deformed shape
Scale factor > Default
COSMOSM Advanced Modules
7-31
Chapter 7 Commands and Examples
Geo Panel:
LoadsBC > STRUCTURAL > DISPLACEMENT > Plot
(DPLOT)
The resulting plot is shown in Figure 7-7.
Figure 7-7
5.
To generate a time history curve:
Geo Panel: Display > XY PLOTS > Activate Post-Proc (ACTXYPOST)
Graph number > 1
X_variable > Time
Y_variable > UY
Node number > 17
Graph color > 12
Graph line Style > Solid
Graph symbol > 0
Graph id > 17N
In
de
x
Geo Panel: Display > XY PLOTS > Plot Curves (XYPLOT)
Plot graph > Yes
7-32
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
The resulting plot is shown in Figure 7-8.
In
de
x
Figure 7-8Time history.
COSMOSM Advanced Modules
7-33
In
de
x
7-34
COSMOSM Advanced Modules
8
Verification Problems
Introduction
In the following, a comprehensive set of verification problems are provided to
illustrate the various features of the nonlinear analysis module (NSTAR). The
problems are carefully selected to cover a wide range of applications in different
fields of nonlinear analyses.
In
de
x
The input files for the verification problems are available in the “...\Vprobs\
Nonlinear” folder. Where “...” refers to the COSMOSM installation folder. For
example the input file for problem NS1 is available in the file
“...\Vprobs\Nonlinear\NS1.GEO” and the input file for problem ND1 is
“...\Vprobs\Nonlinear\ND1.GEO.” NS in the problem name refers to Nonlinear
Static, and ND refers to Nonlinear Dynamic.
COSMOSM Advanced Modules
8-1
In
de
x
Nonlinear Static Analysis
8-2
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS1: Elastoplastic Compression
of a Composite Pipe Assembly
TYPE:
Static, Plastic Analysis, Truss Elements.
REFERENCE:
Crandall, Dahl and Lardner, “An Introduction to the Mechanics of Solids,” Second
Edition, McGraw-Hill, 1972, pp. 277-280
PROBLEM:
Two coaxial pipes, the inner one of 1020 CR steel and cross-sectional area As, and
the outer one of 2024-T4 aluminum alloy and of area Aa, are compressed between
heavy, flat end plates, as shown below. Determine the load-deflection curve of the
assembly as it is compressed into the plastic region by an axial force P.
Figure NS1-1
P
Y
2
L
2
1
1
X
Z
Finite Element Model
Problem Sketch
σ
P
σys
E
Ts
1299860
Es
E
Ta
σ ya
E
a
In
de
x
ε
Stress-Strain Curve
COSMOSM Advanced Modules
t
103
8-3
Chapter 8 Verification Problems
GIVEN:
As
= 7 in2
Aa
= 12 in2
L
= 10 in
σys
= 86,000 psi
σya
= 55,000 psi
ν
= 0.3
Es
= 26.875 x 106 psi
Ea
= 11 x 106 psi
ETS
= 41,322 psi
ETa
= 52,632 psi
COMPARISON OF RESULTS:
In
de
x
Figure NS1-2
8-4
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS2: Nonlinear Analysis of a Cable Assembly
TYPE:
Static, Plastic Analysis, Truss Elements.
REFERENCE:
Crandall, Dahl and Lardner, “An Introduction to the Mechanics of Solids,” Second
Edition, McGraw-Hill, 1972 page 356.
PROBLEM:
A chain hoist is attached to the ceiling through three tie rods as shown in the sketch.
The tie rods are made of cold-rolled steel with yield strength σy, and each has an area
A.
1.
What is the load-deflection relation when the deflections are elastic in all three
rods?
2.
What is the value of the load at which the three rods become plastic?
Figure NS2-1
Y
Stress, σ
Et
1
2
σy
3
2
E
L
1
θ
θ
3
X
4
Strain, ε
P
Stress-Strain Curve
In
de
x
Problem Sketch and
Finite Element Model
COSMOSM Advanced Modules
8-5
Chapter 8 Verification Problems
GIVEN:
A
= 1 in2
L
= 100 in
θ
= 30°
COMPARISON OF RESULTS:
P (lb)
Theory
σy = 30,000 psi
E
Displacement at Node 4 (inch)
= 30 x 106 psi
Et = 0.606 x 106 psi
0.0225991
0.011599
16000
0.0231981
0.023198
24000
0.0347972
0.034797
32000
0.0463962
0.046396
40000
0.0579953
0.057995
48000
0.0695943
0.069594
56000
0.0811934
0.081193
64000
0.0927924
0.092792
Ultimate load (lb).
Pu (lb)
In
de
x
Figure NS2-2
8-6
COSMOSM Advanced Modules
COSMOSM
8000
Theory
COSMOSM
81961.52
82100.0
Part 1 NSTAR / Nonlinear Analysis
NS3: Static Collapse of a Truss Structure
TYPE:
Static, Plastic Analysis, Truss Elements, Force control, BFGS iterations, Line
search.
REFERENCE:
Bathe, Klaus-Jurgen, Ozdemir, H., and Wilson, E., “Static and Dynamic Geometric
and Material Nonlinear Analysis,” Report No. UCSESM 74-4, University of
California, Berkeley, 1974, pp. 111-112.
Figure NS3-1
P
Stress, σ
4
2
4
Et
σy
2
3
36 in
E
1
3
1
48 in
Strain, ε
Stress-Strain Curve
Problem Sketch
P
24
24
t
In
de
x
Load_Time Curve
COSMOSM Advanced Modules
8-7
Chapter 8 Verification Problems
GIVEN:
A
= 1 in2
E
= 3x104 ksi
σy = 30 ksi
Et = 300 ksi
COMPARISON OF RESULTS:
Elastic-plastic displacement of the truss at node 4 is plotted.
In
de
x
Figure NS3-2
8-8
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS4: Truss with Temperature
Dependent Material Properties
TYPE:
Nonlinear static thermal analysis using truss elements.
PROBLEM:
A one dimensional structure consists of truss elements with various material
properties. Determine deflections and Thermal stresses in the structure, due to a
uniform rise of temperature to:
a.
100° F
b.
200° F
Figure NS 4-1
Y
L
2
1
3
4
5
X
Finite Element Model
Problem Sketch
E
(ksi)
E = 30000. ksi
GIVEN:
= 1 in2
α
= 0.00001 1/°F
E
= 30,000 ksi
L
= 4 in
Element 1
8
A
Temperature
E
(ksi)
30000.
20000.
Elements 2 & 4
Temperature
150
200
E
(ksi)
30000.
27500.
Element 3
Temperature
In
de
x
125
200
Elements Elastic Modulus - Temperature Curves
COSMOSM Advanced Modules
8-9
Chapter 8 Verification Problems
ANALYTICAL SOLUTION:
1.
T = 100° F
At this temperature all the elements have the same modulus of elasticity,
therefore
u(i) = 0
stress = E α T = 30 ksi
2.
T = 200 ° F
For this case, first we assume that all nodes are constrained, and later release
these nodes. If:
e = Thermal strain
u(i) = Deflection at node i (i = 1,5)
S(j) = Stress at element j (j = 1,4)
Assuming that all nodes are fixed:
e = αT = 0.00001*200 = 0.002 in/in
S(1) = E e = 30000* 0.002 = 60 ksi
S(2) = E e = 20000* 0.002 = 40 ksi
S(3) = E e = 27500* 0.002 = 55 ksi
S(4) = E e = 20000* 0.002 = 40 ksi
Releasing the nodes:
S(1) = 60-30000 u(2) / 10
S(2) = 40-20000 [u(3) - u(2)] / 10
S(3) = 55-27500 [u(4) - u(3)] / 10
S(4) = 40+20000 u(4) / 10
However, equilibrium is only satisfied if the stresses in all elements are equal, thus
yielding a solution to the above equations.
u(2) = 0.0042857 in
u(3) = 0.0007143 in
u(4) = 0.0035714 in
stress = 47.14286 ksi
COMPARISON OF RESULTS:
In
de
x
Same results are obtained using COSMOSM nonlinear module, NSTAR.
8-10
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS5: Elastic-Plastic Static Analysis
of a Metal Sheet
TYPE:
Nonlinear Static Analysis, Plasticity, 2D Isoparametric Plane Stress Elements
(4 node).
PROBLEM:
Investigate elastoplastic response of a metal sheet using nine 4-node 2D plane stress
elements. Material of the sheet is assumed to obey the von Mises yield criterion.
Figure NS5-1
p
Y
13
13
14
15
16
10
9
7
9
10
8
11
9
12
t
L
4
5
5
6
6
1
8
7
2
8
Load - Time Curve
1
2
1
2
3
3
4
σ
X
E
L
Problem Sketch and
Finite Element Model
T
σy
E
ε
In
de
x
Stress - Strain Curve
COSMOSM Advanced Modules
8-11
Chapter 8 Verification Problems
GIVEN:
E
= 21,000 N/mm2
ET
= 5,000 N/mm2
ν
= 0.3
σy
= 10 N/mm2
L
= 30 mm
Thickness = 1 mm
COMPARISON OF RESULTS:
In
de
x
Figure NS5-2
8-12
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS6: ElastoPlastic Analysis
of a Thick Walled Tube
TYPE:
Nonlinear Static Analysis, Plasticity, 2D Axisymmetric Elements (8 node).
REFERENCE:
Hodge, P. G., and White, G. H., “A Quantitative Comparison of Flow and
Deformation Theories of Plasticity,” J. Appl. Mech., Vol. 17, pp. 180-184, 1950.
PROBLEM:
Investigate elastoplastic response of a thick-walled cylinder using sixteen 4-node 2D
axisymmetric elements. The material is assumed to obey the von Mises yield
criterion.
Figure NS6-1
σ
p
x
P
E
T
σy
R1
z
Problem Sketch
E
R2
ε
y
Stress - Strain Curve
3 5
2
p
23
1
2
3
4
H
22
13
12
11.5
7.5
1 rad
21
x
R1 1 4
t
R2
1
9
12
20
In
de
x
Load - Time Curve
Finite Element Model
COSMOSM Advanced Modules
8-13
Chapter 8 Verification Problems
GIVEN:
G
= 0.333333E5 psi
ν
= 0.3
σy = 17.32 psi
R2 = 2 in
H
= 1 in
E
= 86666 psi
ET = 0 psi
R1 = 1 in
COMPARISON OF RESULTS:
In
de
x
Figure NS6-2
8-14
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS7: Large Deflection Analysis
of an Infinitely Long Plate
TYPE:
Nonlinear Static Analysis, Large Displacement, 2D Plane Strain (8 node) elements.
REFERENCE:
Timoshenko, S., Woinowsky-Kreiger, S., “Theory of Plates and Shells,” pp.422423.
PROBLEM:
Study the large displacement response of an infinitely long rectangular plate using
2D plane strain elements.1
MODELING HINTS:
Due to symmetry, only a half of the plate is modeled.
Figure NS7-1
CL
q
1.0
h
q
2b
Problem Sketch
t
20
Load_Time Curve
CL
y
4
51
1
2
3
4
5
6
7
8
9
10
52
X
53
5
Finite Element Model
In
de
x
CL
COSMOSM Advanced Modules
8-15
Chapter 8 Verification Problems
GIVEN:
E
= 10.92 E6 N/mm2
ν
= 0.3
h
= 1 mm
b
= 100 mm
q
= 1 N/mm2
COMPARISON O F RESULTS:
Theory
COSMOSM
Maximum Bending Stress
(Node 53)
12730
12778
Midspan Deflection
(Node 1)
1.526
1.5255
In
de
x
Figure NS7-2
8-16
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS8: Static Large Displacement
Analysis of a Cantilever Beam
TYPE:
Nonlinear Analysis - Large Displacement.
NS8A)
PLANE2D elements
NS8B)
BEAM2D elements
REFERENCE:
Holden, J. T., “On the Finite Deflections of Thin Beams,” Int. J. Solid Structure,
Vol. 8, pp. 1051-1055, 1972.
PROBLEM:
Investigate large displacements of a cantilever beam using five 8-node 2D plane
stress elements.
Figure NS8-1
p
p/2
10
h
p/2
b
L
t
100
Problem Sketch
Load - Time Curve
5
3
2
28
1
2
3
4
5
1
2
1
1
3
4
5
6
27
2
3
4
5
26
4
Beam2D
Finite Element Model
In
de
x
Plane2D (8 node)
Finite Element Model
COSMOSM Advanced Modules
8-17
Chapter 8 Verification Problems
GIVEN:
L
= 10 in
E
= 12,000 psi
ν
= 0.2
p
= 10 lb/in
h
= 1 in
b
= 1 in
COMPARISON OF RESULTS
Node No.
PLANE2D
27
7.17
BEAM2D
6
7.05
In
de
x
Figure NS8-2
8-18
Maximum
Deflection
Element Type
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS9: Static Large Displacement
Analysis of a Spherical Shell
TYPE:
Nonlinear Static Analysis, Large Displacement, 2D Isoparametric Axisymmetric (8
node) Elements.
REFERENCE:
Stricklin, J. A., “Geometrically Nonlinear Static and Dynamic Analysis of Shells of
Revolution,” High Speed Computing of Elastic Structures, Proceedings of the
Symposium of IUTAM, University of Liege, pp. 383-411, August 1970.
PROBLEM:
Investigate large displacements of a spherical shell using ten 8-node 2D
axisymmetric elements.
Figure NS9-1
P
p
h
H
w
o
100
R
θ
t
200
Load - Time Curve
Problem Sketch
p/2π
53
52
10
9
51
8
7
6
5
4
3
2
5
1
3
4
1
2
In
de
x
Finite Element Model
COSMOSM Advanced Modules
8-19
Chapter 8 Verification Problems
GIVEN:
E
= 10 x 106 lb/in2
ν
= 0.3
h
= 0.01675 in
R
= 4.76 in
P
= 100 lb
H
= 0.0859 in
COMPARISON OF RESULTS:
In
de
x
Figure NS9-2
8-20
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS10: Large Displacement Analysis of a
Fixed Beam with Concentrated Load
TYPE:
Nonlinear Static Analysis, Large Displacement.
NS10A)
2D Plane Stress (8-node) Elements
NS10B)
2D Beam Elements.
REFERENCE:
Noor, A. K. and Peters, J. M., “Reduced Basis Technique for Nonlinear Analysis of
Structures,” AIAA J., Vol 18, No. 4, Apr. 1980, Article No. 79-0747R, pp. 455-462.
PROBLEM:
Investigate large displacements of a fixed-fixed beam subject to a concentrated load
at mid-span using 8-node 2D plane and BEAM2D elements.
MODELING HINTS:
Due to symmetry, only a half of the beam is modeled.
Figure NS10-1
p
p
h
3113.6
b
L
Problem Sketch
100
t
Load - Time Curve
Y
Y
p/2
p/2
5
23
3
2
1
2
3
4
21
1
4
L/2
2
3
5
4
X
X
1
2
3
4
L/2
Beam2D Finite Element Model
NS10B
In
de
x
Plane2D Finite Element Model
NS10A
1
22
COSMOSM Advanced Modules
8-21
Chapter 8 Verification Problems
GIVEN:
E
= 20.684 x 1010 N/m2
b
= 0.0254 m
h
= 3.175 x 10-3 m
L
= 0.508 m
Iz
= 0.6774E-10 in4
COMPARISON OF RESULTS:
In
de
x
Figure NS10-2
8-22
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS11: Simply Supported Rectangular
Plate, Large Deflection Analysis
TYPE:
Nonlinear Analysis, SHELL3 Elements.
REFERENCE:
Levy, S., “Bending of Rectangular Plates with Large Deflections,” Technical Note,
National Advisory Committee for Aeronautics. No. 846, 1942.
PROBLEM:
Calculate the large deflection response of a simply supported square plate subjected
to uniformly distributed pressure.
MODELING HINTS:
Due to symmetry, only a quarter of the plate is modeled.
Figure NS11-1
b
h
a
Problem Sketch
500
y
13
16
z
150
X
1
K = qa4 /Eh 4
4
4
12
22
t
Time Curve
In
de
x
Finite Element Model
30
COSMOSM Advanced Modules
8-23
Chapter 8 Verification Problems
GIVEN:
E
= 1.0 x 107 psi
ν
= 0.3162
h
= 0.12 in
a
= b = 24 in
q
= Uniform applied pressure per unit area = 0.00625 kpsi
k
= Load factor
BOUNDARY CONDITIONS:
All edges are simply supported.
COMPARISON OF RESULTS:
In
de
x
Figure NS11-2
8-24
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS12: Large Deflection Analysis of a Cantilever
TYPE:
Nonlinear Static Analysis, Large Displacement.
NS12A)
SHELL3 Elements
NS12B)
2D Beam Elements
REFERENCE:
Ramm, E., “A Plate/Shell Element for Large Deflections and Rotations,” in
Formulations and Computational Algorithm in Finite Element Analysis. [M.I.T.
Press, 1977].
PROBLEM:
A cantilever beam subjected to a concentrated end moment.
Figure NS12-1
M
u
h
v
b
L
Problem Sketch
M/2
Z
Y
7
1
8
2
9
10
3
4
6.28518
12
11
X
6
5
M
M/2
t
Shell Model
1
2
3
4
80
Time Curve
5
6
X
M
In
de
x
Beam Model
COSMOSM Advanced Modules
8-25
Chapter 8 Verification Problems
GIVEN:
L
= 100 in
I
= 0.01042 in4
A
= 0.5 in2
ν
=0
h
= 0.5 in
M = Concentrated End Moment = 2 in-lb
E
= 12,000 psi
COMPARISON OF RESULTS:
In
de
x
Figure NS12-2
8-26
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS13: Analysis of Simply Supported
Beam Using Gap Elements
TYPE:
Static, Nonlinear, Gap-friction elements.
PROBLEM:
The beam is modeled using BEAM2D elements. Material of the beam is assumed to
be elastic, and large deformations are not considered. Two gap elements are used to
model the supports. Two soft truss elements are used to avoid stiffness singularities.
Figure NS13-1
p
h
b
L
Problem Sketch
Y
p
1
p
2
1
3
X
2
3
5
4
6
5
4
t
In
de
x
Finite Element Model
Time Curve
GIVEN:
COMPARISON OF RESULTS:
h
= 10 in
b
= 1.2 in
Iz
= 100 in4
The result is in good agreement with the solution
obtained using the theory of structures.
(At time step 1)
Theory
COSMOSM
E
= 30 x 106 lb/in2
P
= 1000 lbs
COSMOSM Advanced Modules
Max. Displacement (in)
0.00694
0.006944
Reactions (lb)
500
500
8-27
Chapter 8 Verification Problems
NS14: Analysis of a Propped Cantilever
Beam Using Gap Elements
TYPE:
Static, Nonlinear, Gap-friction elements.
PROBLEM:
The beam is modeled using BEAM2D elements. Material of the beam is assumed to
be elastic, and large deflections are not considered. A gap element is used to model
the simply supported end.
Figure NS14-1
p
h
b
L
Problem Sketch
p
Y
p
1
1
2
2
3
X
3
4
In
de
x
Finite Element Model
8-28
Gap
Element
t
Time Curve
GIVEN:
COMPARISON OF RESULTS:
L
= 100 in
h
= 10 in
Iz
= 100 in4
The result is in good agreement with the solution
obtained using the theory of structures.
(At time step 1)
Theory
COSMOSM
E
= 30 x 104 lb/in
b
= 1.2 in
P
= 1,000 lbs
COSMOSM Advanced Modules
Max. Displacement (in)
0.003040
0.003038
Reactions (lb)
312.5
312.5
Part 1 NSTAR / Nonlinear Analysis
NS15: Nonlinear Analysis of a Cantiliver Beam with
Gaps Under Multiple Loading Conditions
TYPE:
Nonlinear Static Analysis, Beam and Gap-friction elements.
PROBLEM:
The beam is modeled using BEAM2D elements. Five gap elements with zero gap
distances are used. Ten different load cases were selected, and the analysis was
performed in 10 solution steps. The material of the beam was assumed to be elastic;
deformations were assumed to be small.
Figure NS15-1
F
c
F
b
F
a
p
F
e
F
d
h
b
L
L
1
L
2
L
2
L
2
2
Problem Sketch
Y
F
a
F
c
F
b
F
e
F
d
p
11
12
13
14
15
X
16
In
de
x
FInite Element Model
COSMOSM Advanced Modules
8-29
Chapter 8 Verification Problems
GIVEN:
Ebeam = 30 x 106 psi
b
= 1.2 in
IZ
= 100 in4
h
= 10 in
L2
= 50 in
L1
= 100 in
Figure NS15-2
F
F
c
F
b
a
-2000
-1000
-1000
-1000
t
t
4
5
10
7
F
d
t
7 8 10
4
F
e
6
9
p
-2000
-1000
-1000
100
t
3 5 6
8
t
1
4
6
8
t
9 11
Time Curves
COMPARISON OF RESULTS:
In
de
x
The state of gaps at any time agrees with the beam deformed shape at that time. The
results can be compared with the solution obtained from linear static analysis, where
the gaps are removed but the nodes which are connected to closed gaps are fixed.
8-30
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
OBTAINED RESULTS:
Forces in Gap
Applied
Forces
Time Step
No.
No. 1
Fa = - 1000
1
312.50
Fa = - 1000
Fe = - 1000
2
Fa = - 1000
Fe = - 2000
3
Fd = - 1000
4
Fc = - 1000
5
Fb = - 1000
6
Fd = - 1000
Fe = - 1000
7
Fa = - 1000
Fc = - 2000
8
Fa = - 1000
Fb = - 1000
9
No. 3
No. 4
No. 5
390.38
498.02
482.14
353.17
1075.4
928.57
564.45
586.54
631.25
442.31
425.00
351.81
361.84
1197.4
1206.3
275.00
10
453.12
2050.0
1338.7
318.55
842.11
6100.
1900.0
In
de
x
p = - 100
No. 2
COSMOSM Advanced Modules
8-31
Chapter 8 Verification Problems
NS16: Cantilever Beam on Elastic Foundation
TYPE:
Nonlinear Static Analysis, Beam and Gap-friction elements.
PROBLEM:
The geometry of the beam and loading conditions are similar to problem NS15.
Thirty gap elements are used to model the interface between the beam and the
ground. To account for the ground elasticity, one end of each gap is connected to a
truss with equivalent stiffness. The analysis was performed for ten steps.
Figure NS16-1
F
c
F
b
F
a
p
F
e
F
d
h
50
70
50
50
150
50
100
50
(All dimensions are in inches)
Problem Sketch
Y
F
a
F
b
F
c
p
F
d
1 2
F
e
30 31
61
32
62
91
In
de
x
Finite Element Model
8-32
COSMOSM Advanced Modules
b
Beam
Cross
Section
Part 1 NSTAR / Nonlinear Analysis
GIVEN:
Ebeam
= 30 x 106 psi
b
= 1.2 in
h
= 10 in
ktruss
= 30 x 105 lb/in
Iz
= 100 in4
Figure NS16-2
F
b
F
a
F
c
-2000
-1000
-1000
-1000
t
t
4
7
5
10
F
d
t
7 8 10
4
F
e
6
9
p
-2000
-1000
-1000
100
t
3 5 6
t
8
1
4
6
8
t
9 11
Load - Time Curves
COMPARISON OF RESULTS:
In
de
x
Again, the results may be verified if the gaps are removed and truss elements which
connect to closed gaps were directly attached to the beam. The deflection of the
beam at any step confirms the state of the gaps.
COSMOSM Advanced Modules
8-33
Chapter 8 Verification Problems
OBTAINED RESULTS:
Applied
Forces
Fa = - 1000
Time
Step
1
Forces in Gap Elements
2)20.31
3)103.97
4)256.07
5)368.06
6)248.73
3)103.04
4)253.43
5)363.72
6)246.17
25)57.377
26)257.93
27)381.11
28)257.88
29)45.84
2)20.34
3)103.85
4)255.50
5)366.79
6)247.16
26)514.84
27)759.15
28)513.76
29)91.37
7)43.96
Fa = - 1000
2
Fe = - 1000
Fa = - 1000
2)20.34
7)53.97
3
Fe = - 2000
7)44.31
25)123.92
Fd = - 1000
4
20)65.66
21)257.02
22)378.34
23)256.08
24)45.57
Fc = - 1000
5
15)71.25
16)256.41
17)376.50
18)254.86
19)45.38
Fb = - 1000
6
10)81.11
11)255.88
12)373.62
13)252.88
14)45.0
Fd = - 1000
7
20)62.93
21)238.71
22)347.15
23)237.66
24)115.48
25)116.94
26)242.32
27)354.30
28)242.14
29)44.81
2)20.39
3)103.30
4)253.42
5)362.56
6)242.65
15)152.33
16)511.01
17)749.14
18)507.36
19)90.51
2)19.87
3)100.18
4)245.31
5)350.23
6)237.94
7)84.09
9)1.236
10)89.21
11)249.6
12)366.8
13)249.47
14)45.29
14)115.77
15)496.20
16)826.44
17)991.33
18)1035.2
19)1031.2
20)1025.2
21)1032.4
22)1040.3
23)1004.3
24)846.45
25)504.95
26)60.48
Fe = - 1000
Fa = - 1000
8
Fc = - 2000
Fa = - 1000
7)47.88
9
Fb = - 1000
10
In
de
x
p = - 100
8-34
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS17: Simply Supported Beam Subjected to
Pressure from a Rigid Parabolic Shaped Piston
TYPE:
Static, Nonlinear, Gap-friction elements, BEAM2D and PLANE2D elements.
MODELING:
The shape of the piston was simulated through gap distances. In order to avoid
singularities in the structure stiffness, two soft truss elements were used to hold the
piston. The problem was analyzed in one hundred steps, gradually increasing the
pressure load.
Figure NS17-1
y = (x/100)3
p
h
b
120 in
60 in
120 in
p
Problem Sketch
110
24
25
K
K
p
Y
11
23
10
22
g1
10
t
1
101
Time Curve
g7
9
1
X
2
3
4
5
6
7
8
In
de
x
Finite Element Model
COSMOSM Advanced Modules
8-35
Chapter 8 Verification Problems
GIVEN:
Gap Distances
g1
= g7 = 0.027 in
g2
= g6 = 0.008 in
g3
= g5 = 0.001 in
g4
= 0 in
Iz
= 100 in4
h
= 10 in
b
= 1.2 in
E
= 30 x 106 psi
Episton
= 30 x 108 psi
COMPARISON OF RESULTS:
The forces of gaps at any time were in good agreement with the total force applied
to the pis-ton at that time. The deformed shape of the beam at any solution step
confirmed with the forces and location of closed gaps at that step.
OBTAINED RESULTS:
Applied
Forces
Time
Step
Closed
Gaps
Pressure
Gap
Forces
Total
Total Force
(Theory) (lb)
p < 13
4
4
p = 13
777.1
777.1
780
13 < p < 14
5
3,4,5
p = 14
423.5,
206.7,
206.9
836.9
840
14 < p < 33
24
3,5
p = 33
986.3,
986.3
1972.0
1980
33 < p < 36
27
2,3,4,5
p = 36
13.50,
13.5
36 < p < 58
27
2,6
p = 58
1062.6,
1062.6
3467.0
3480
58 < p < 63
54
1,2,6,7
p = 63
215.4,
215.4,
1668.,
1668
3766.8
3780
63 < p
100
15,21
p = 109
3258.8,
3258.8
6517.6
6540
In
de
x
8-36
Forces
COSMOSM (lb)
COSMOSM Advanced Modules
2153.1
2160
Part 1 NSTAR / Nonlinear Analysis
NS18: Static Analysis of Sheets
Connected by Friction Elements
TYPE:
Static, Nonlinear, Gap elements with friction.
MODELING:
The sheets are modeled using PLANE2D plane stress elements. In order to avoid any
stiffness singularities soft truss elements are used to hold the upper sheet. In
addition, truss elements were also used in the gap locations; these additional trusses
were found to be essential for a large deflection analysis, where the stiffness had to
be reformed at every step. The normal force was kept constant, while the horizontal
force was increased gradually, in one hundred steps.
Figure NS18-1
Fy
Fy
33
26
27
34
2 in
28
29
25
20
21
22
23
F
24 x
16
17
18
19
30
F
x
2 in
4 in
31
Problem Sketch
Fy
11
12
13
14
15
6
7
8
9
10
1
2
3
4
1000
T
Fx
5
X
500
Finite Element Model
T
In
de
x
Time Curves
COSMOSM Advanced Modules
8-37
Chapter 8 Verification Problems
GIVEN:
E
= 30 x 106 psi
ν
= 0.3
Thickness
= 1 in
Friction Coefficient = 0.5
COMPARISON OF RESULTS:
In both cases (using small or large deformation theory), the structure starts slipping
as soon as the horizontal force exceeds the normal force times coefficient of friction
(step = 101, Fx > 500.), i.e., the horizontal deflection becomes large. For this case,
the force distributions among the gap elements are the same whether small or large
deformation theory is used.
OBTAINED RESULTS:
1.
Small Displacement Analysis
Gap forces (in pounds) at step 101:
2.
Gap No.
Gap Forces
1
31.08
Friction Force
-15.5
2
160.28
-80.14
3
264.0
-132.0
4
361.9
-180.9
5
182.8
-91.4
Total
1000.96
-499.94
Large Displacement Analysis
Gap forces (in pounds) at step 101:
Gap No.
Friction Force
1
31.08
-15.5
2
160.28
-80.14
3
264.0
-132.0
4
361.9
-180.9
5
182.8
-91.4
Total
1000.96
-499.94
In
de
x
8-38
Gap Forces
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS19: Large Displacement Analysis
of a S.S. Circular Plate
TYPE:
Nonlinear Static Analysis, Large Displacement, 2D Isoparametric (8 node)
Axisymmetric Elements.
REFERENCE:
Timoshenko, S., Woinowsky-Kreiger, S., “Theory of Plates and Shells,” p. 411.
PROBLEM:
Investigate the large displacement behavior of a simply supported circular plate with
radically movable edges using 2D axisymmetric elements.
Figure NS19-1
q
q
1.0
0.5
h
0.25
0.1
2a
t
Problem Sketch
40 50
60 70
Time Curve
CL
5
3
2
53
1
2
3
4
5
6
7
1
8
9
10
52
51
X
4
a
In
de
x
Finite Element Model
COSMOSM Advanced Modules
8-39
Chapter 8 Verification Problems
GIVEN:
a
= 100 cm
h
= 1 cm
E
= 2 x 108 N/cm2
ν
= 0.25
q
= 400 N/cm2
COMPARISON OF RESULTS:
Theory
COSMOSM
Maximum Deflection (cm)
7.88
7.91
Principal Stress
(at node 53) (N/cm2)
-1.1646 x 106
-1.177 x 106
In
de
x
Figure NS19-2
8-40
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS20: Large Deflection Analysis
of a Fixed-Fixed Shallow Arch
TYPE:
Nonlinear Static Analysis, Large Displacement, 2D Beam elements.
REFERENCE:
Bathe, K. J., and Bolourchi, S., “Large Displacement Analysis of 3D Beam
Structures,” Int. J. for Numerical Methods in Engineering, Vol. 14., 1979, page 977.
PROBLEM:
Find the static response of a spherical arch under apex load due to large displacement
effect using 2D Beam elements.
Figure NS20-1
P
H
β
h
b
Beam
CrossSection
L
Problem Sketch
p
34
P/2
1
2
3
4
5
t
34
Time Curve
W
10
11
12
In
de
x
13
Finite Element Model
COSMOSM Advanced Modules
8-41
Chapter 8 Verification Problems
GIVEN:
E
= 1.0 x 107 psi
ν
= 0.2
h
= 0.1875 in
L
= 34 in
A
= 0.1875 in2
Iz
= 0.00055 in4
R
= 133.114 in
β
= 7.3397°
COMPARISON OF RESULTS:
Figure NS20-2
COSMOS/M
BATHE & BOLOURCHI
w
In
de
x
TIME STEP
8-42
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS21: Elastoplastic Small Displacement
Analysis of a Cantilever Beam with Tip Moment
TYPE:
Nonlinear Static Analysis, Plasticity, BEAM2/3D Elements, and beam-sectiondefinition.
REFERENCE:
Timoshenko, S. P, and Gere, James M., “Mechanics of Materials,” pp. 289-316.
PROBLEM:
Investigate the elastoplastic response of a cantilever beam subjected to an end
moment. Both elastic-perfectly plastic and elastic-linear strain hardening models are
studied.
Figure NS21-1
M
H
B
L
Problem Sketch
σy
σ
case-2
ET
H/2
σy
ρ
case-1
E
ε
M
Stress-Strain Curve
σy
15,000
1
2
1
3
2
4
3
5
4
6
5
7
6
8
7
10
9
10
11
M
t
30
Time Curve
In
de
x
Finite Element Model
9
8
COSMOSM Advanced Modules
8-43
Chapter 8 Verification Problems
GIVEN:
The problem sketch is shown in Figure NS21-1. Ten (10) BEAM2/3D elements are
used in the analysis.
E
= 30E6 psi
ν
=0
σy
= 5,000 psi
L
= 90 in
H
= 3 in
B
= 1 in
Case–1:ET
=0
Case–2:ET
= 3E6 psi
COMPARISON OF RESULTS:
Analytical solutions in the elastoplastic range:
Case1:
Case 2:
where:
δ = Displacement
K = Curvature
Ky = Curvature at the yield point
In
de
x
Fig NS21-2 and NS21-3 show load-deflection curves of the beam with respect to
the stress-strain curve-1 and -2. Analytical solutions are also included.
8-44
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS21-2
LIMIT LOAD = 11250 LB*IN
2
1
L
0
A
D
L
B
*
I
N
1 ANALYTICAL SOLUTION
2 COSMOS/M
0.8
0.9
1
DISPLACEMENT (INCH)
Figure NS21-3
L
0
A
D
L
B
*
I
N
In
de
x
1 ANALYTICAL SOLUTION
2 COSMOS/M
DISPLACEMENT (INCH)
COSMOSM Advanced Modules
8-45
Chapter 8 Verification Problems
NS22: Clamped Square Plate
with Pressure Loading
TYPE:
Nonlinear Static Analysis, Large Displacement, 3D Isoparametric (20 node) Solid
Elements.
REFERENCE:
Timoshenko, S., Woinowsky-Kreiger, S., “Theory of Plates and Shells,” pp. 422-423
PROBLEM:
Carry out large displacement analysis of a clamped square plate using 3D (20 node)
solid elements.
MODELING HINT:
Only a quarter of the plate is modeled due to symmetry.
Figure NS22-1
Y
q
q
h
1
2a
4
8
12
7
11 16
5
10 15
14
9
13
2a
2
3
X
In
de
x
Problem Sketch and Finite
Element Model
8-46
20,000
6
COSMOSM Advanced Modules
20
Time Curve
t
Part 1 NSTAR / Nonlinear Analysis
GIVEN:
a
=1m
h
= 0.01 m
q
= 2 x 104 N/m2
E
= 10.92 x 1010 N/m2
ν
= 0.3
COMPARISON OF RESULTS:
Central Deflection (m)
Theory
COSMOSM
0.01594
0.015938
In
de
x
Figure NS22-2
COSMOSM Advanced Modules
8-47
Chapter 8 Verification Problems
NS23: Large Displacement Static Analysis
of a Clamped Sandwich Plate
TYPE:
Nonlinear Static Analysis, Large Displacement.
NS23A)
4-node Composite Shell Elements.
NS23B)
3D Isoparametric (20 Node) Solid Elements
REFERENCE:
Schmit, L. A., Monforton, G. R., “Finite Deflection Discrete Element Analysis of
Sandwich Plates and Cylindrical Shells with Laminated Faces,” AIAA Journal, Vol.
8, No. 89., pp. 1454-1461.
PROBLEM:
Perform Large
Displacement
analysis of a
clamped
sandwich plate
subject to uniform
pressure using 3D
(20 node) solid
elements and 4node composite
shell elements.
Figure NS23-1
t
f
q
6
5
8
tc
64.615
7
1
tf
4
3
2a
MODELING
HINTS:
3
14
2a
t
Time Curve
In
de
x
Problem Sketch and Finite
Only an eighth of
Element Model (20-node solid)
the plate is
modeled due to
symmetry for 20-node composite shell elements and one quarter of the plate is
modeled for 4-node composite shell elements. The properties of the core material are
adjusted to match the linear solution given in the reference.
8-48
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
)
GIVEN:
COMPARISON OF RESULTS:
a
= 25 in
(At time step 14
q
= 64.615385 psi
Schmit and
Monforton
Properties of the Face Sheets:
Ef
= 10.5 x 106 psi
νf
= 0.3
tf
= 0.015 in
Central
Deflection
(in)
2.48
COSMOSM
SHELL4T
SOLID3D
2.6804
2.5285
Gxzf = Gyzf = 0.1 psi
Properties of the Core:
Exc
= Eyc = 1 E-12 psi; Ezc = 34,500 psi
νxy
= νxzc = νyzc = 0
Gxy
= 0 psi; Gxzc = Gyzc = 50,000 psi
t
= 1 in
In
de
x
Figure NS23-2
COSMOSM Advanced Modules
8-49
Chapter 8 Verification Problems
NS24: Plate Subjected to Triangular
Temperature Loading
TYPE:
Linear Static thermal analysis using PLANE2D 8-node planes tress elements.
REFERENCE:
Heldenfels, R. R., and Roberts, W. M., “Experimental and Theoretical
Determination of Thermal Stresses in a Flat Plate,” Technical Note 2769, NACA,
Washington, Aug. 1952.
PROBLEM:
A flat plate is subjected to a triangular
thermal loading along two parallel edges.
Determine the thermal stresses in the plate.
Figure NS24-1
T
1
MODELING HINT:
Due to symmetry, a quarter of the plate is
modeled.
a
GIVEN:
a
b
T
E
ν
α
= 18 in
= 12 in
= 150°
= 10.4E6 psi
=0
= 12.7E-6/° F
COMPARISON OF RESULTS:
a
b
b
T
1
Problem Sketch
In
de
x
The stresses along the vertical middle axis
are obtained and compared to those from the
reference in the table below. On the whole good agreement is observed, although as
seen, some discrepancy exists near the top edge, where it appears that a finer mesh
is required. The two values of the stresses come from joining elements. When the
two values differ by a large amount convergence has not been achieved.
8-50
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS24-2
Node
Ref
σxy
σx
σy
σxy
1
9403
- 6.6
0.0
10858
- 3.95
- 264.6
21
3687
- 768
0.0
41
σx
573
σy
COSMOSM
- 2352
0.0
2604.5
- 1954
+18.6
3051.3
- 2003
- 104.4
121.3
- 3966
- 16.3
241.2
- 3951
- 29.3
- 5487
- 10.6
- 952
- 4084
0.0
- 1124
- 1078
- 5479
- 19.7
81
- 1573
- 5631
0.0
- 1604
- 6680
- 7.2
- 1580
- 6676
- 10.1
101
- 1727
- 6855
0.0
- 1716
- 7570
- 5.
- 1701
- 7568
- 6.9
121
- 1688
- 7718
0.0
- 1668
- 8186
- 2.9
- 1660
- 8185
- 4.2
141
- 1611
- 8115
0.0
- 1594
- 8547
- 1.2
- 1590
- 8546
- 2.3
161
- 1577
- 8392
0.0
- 1560
- 8667
0.05
In
de
x
61
COSMOSM Advanced Modules
8-51
Chapter 8 Verification Problems
NS25: Analysis of a Hollow Thick-Walled Cylinder
Subjected to Temperature and Pressure Loading
TYPE:
Static analysis, 2D axisymmetric element.
REFERENCE:
Timoshenko, S. P. and Goodier, “Theory of Elasticity,” McGraw-Hill Book Co.,
New York, l961.
Figure NS25-1
y
Ta
Pa
51
35
x
T(r)
1.0
1 23 4 5 6
7
17
2.0
Problem Sketch
Finite Element Model
GIVEN:
ANALYTICAL SOLUTION:
The hollow cylinder in plane strain is subjected to two
independent loading conditions.
E
a
1. An Internal pressure
b
2. A steady state axisymmetric temperature distribution ν
given by the equation:
α
T(r) = Ta {[ln (b/r)] [ln (b/a)]}
Pa
Ta
Where Ta is the temperature of the inner surface.
= 30 x 106 psi
= 1 in
= 2.0 in
= 0.3
= 1.0 x10-6/degree
= 100 psi
= 100°
COMPARISON OF RESULTS:
(At node No. 8)
COSMOSM
Theory
4-Node Element
(Problem S19)
8-Node Element
Radial (psi)
- 398.34
- 396.78
- 398.45
Tangenient (psi)
- 592.47
- 597.00
- 596.03
In
de
x
Stresses
8-52
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS26: Thermal Stress Analysis of a Plate,
Temperature Dependent Material Properties
TYPE:
Nonlinear static thermal analysis using:
NS26A)
PLANE2D 4-node plane stress elements
NS26B)
Solid 8-node elements
PROBLEM:
A flat plate consists of different material properties through it’s length. Determine
deflections and thermal stresses in the plate due to uniform changes of temperature
equal to 100° F and 200 °F.
CL
Figure NS26-1
E
(ksi)
Y
30000.
Elements 1 & 2
L
8
6
1
8
7
2
1
2
9
3
3
Temperature
10
E
(ksi)
4
4
5
X
Problem Sketch
30000.
20000.
Elements 3 & 4
Finite Element Model
150
Temperature
200
Material Properties
GIVEN:
COMPARISON OF RESULTS:
t
= 0.1 in
L
= 4 in
ν
=0
Since the effect of Poisson’s ratio is neglected, problem
reduces to one dimension and the solution can be obtained
as follows:
α
= 1 x 10-5/° F
E
= 30E3 ksi
Max. Stresses in x-direction (ksi):
T = 100° F
T = 200° F
- 30
- 48
- 30
- 48
COSMOSM SOLID
- 30
- 48
In
de
x
Theory
COSMOSM PLANE2D (Node 5)
COSMOSM Advanced Modules
8-53
Chapter 8 Verification Problems
NS27: Uniaxial Creep Strain
in a Bar (Cyclic Loading)
TYPE:
Nonlinear static analysis, creep, using:
NS27A)
Truss elements
NS27B)
PLANE2D plane stress elements
NS27C)
Solid elements
REFERENCE:
Harry Kraus, “Creep Analysis,” Wiley-Interscience, New York, (1980)
PROBLEM:
A bar is subjected to a cyclic loading as shown in Figure NS27-1. Determine the
creep strain in the bar as a function of time.
MODELING HINT:
The creep material properties are also given in Figure NS27-1. A time increment of
0.1 hr. is used for the analysis. The same results were obtained for all models (using
different element types); see Figure.
P (psi)
Theoretical Loading
Figure NS27-1
900.
1000.
Applied Loading
time (hr)
2.9
0.1
3.9
4.8
-1000.
c
ε x 10
P
4
Problem Sketch
Negative Creep Origin
1.0
0.8
Theoretical
0.6
COSMOSM
0.4
0.2
-0.2
2.9
1.1
3.7
3.9
Positive Creep Origin
In
de
x
8-54
time (hr)
ε c = 6.4x10 -18 σ 4.4 t 0.75 in/in
-0.4
-0.6
4.8
2.1
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS28: Creep Analysis of a Cylinder
Subject to Cyclic Internal Pressure
TYPE:
Nonlinear static analysis, creep, PLANE2D axisymmetric elements.
REFERENCE:
Mark D. Snyder, Klaus-Jurgen Bathe, “Formulation and Numerical Solution of
Thermo-Elastic-plastic and Creep Problems,” report 8248-23 (1977)
PROBLEM:
A thick walled cylinder is subjected to a cyclic loading as shown in figure.
Determine the variation of effective (von Mises) stress in the inner and outer
boundaries of the cylinder as a function of time.
MODELING HINTS:
This problem is modeled using three 8-node two-dimensional axisymmetric
elements. Forty-eight solution steps with a time increment of 0.1 hour is used. The
finite element model plus the selected properties are shown in Figure NS28-1. A
graphic representation of the results is given.
In
de
x
Figure NS28-1
COSMOSM Advanced Modules
8-55
Chapter 8 Verification Problems
GIVEN:
E
= 20E + 6 psi
ν
= 0.3
ε
c
= 6.4E-18 σ4.4 t in/in
∆t = 0.1 hr
COMPARISON OF RESULTS:
Figure NS28-2
p (psi)
365
time (Hr)
0
2.2
2.6
-365
Load - Time Curve
Effective
Stress (psi)
1200
Reference
1000
*
Y = 0.166 in
COSMOS/M
800
600
400
Y = 0.244 in
200
0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4
In
de
x
time (Hr)
8-56
COSMOSM Advanced Modules
4.8 5.2 5.6
Part 1 NSTAR / Nonlinear Analysis
NS29: Creep Analysis of a Cantilever Beam
TYPE:
Nonlinear static analysis, Creep, PLANE2D plane stress elements.
REFERENCE:
Mark D. Synder, Klaus-Jurgenm Bathe, “Formulation and Numerical Solution of
Thermo-Elastic-Plastic and Creep problems,” Report 8448-3, (1977)
PROBLEM:
A cantilever beam is subjected to a constant end moment as shown in figure NS291. Determine the variation of the bending stress along the depth of the beam after a
steady state is reached.
MODELING HINTS:
Due to symmetry, the upper half of the beam is modeled using two rows of 4 8-node
plane stress elements. The nodes which are located on the neutral axis of the beam
are assumed to have no elongation; 100 steps of solution with a time increment of 2
hrs. are used. The finite element model plus the selected proportions are shown in
Figure NS29-1. A graphical representation of the results is given in Figure NS29-2.
GIVEN:
E
= 30E6 psi
ν
= 0.3
ε
c
= 6.4E-18 σ3.15 t in/in
∆t = 2 hr
L
= 40 in
b
= 0.3 in
h
= 4 in
COMPARISON OF RESULTS:
In
de
x
Bending stress distribution in the y-direction (ksi)
COSMOSM Advanced Modules
8-57
Chapter 8 Verification Problems
Figure NS29-1
h
M
b
L
Problem Sketch
Y
37
5
4
3
2
1
h/2
35
P = 2250 lb
X
33
Finite Element Model
M
in-lb
6000
time (hr)
2.0
Load - Time Curve
Figure NS29-2
Y (in)
(Distance from N. Axis)
Steady State
(step 100)
2.
Linear Elastic
(step 1)
1.
Theoretical
ADINA
COSMOSM
σ (ksi)
1.0
2.0
3.0
In
de
x
0.0
8-58
COSMOSM Advanced Modules
4.0
5.0
6.0
7.0
xx
Part 1 NSTAR / Nonlinear Analysis
NS30: Beam Supported by Nonlinear
Springs (Flexible Gaps)
TYPE:
Nonlinear static analysis using beam and gap elements.
PROBLEM:
A beam is supported by two nonlinear springs at its ends. The springs have different
properties in tension and compression, as shown below. Determine the deflection of
the springs due to a normal force acting on the center of the beam.
MODELING HINTS:
The nonlinear spring are modeled using 4 gaps (2 tensile and 2 compressive).
Figure NS30-1
f (lbs)
250
-4.
0.5
U
- 500
Problem Sketch
F
1
1
2
Deflection Versus Force
for Nonlinear Spring
2
3
X
F (lbs)
1000
6
5
4
3
10.
4
5
-10000
Applied Force Versus Time
In
de
x
Finite Element Model
t
20.
COSMOSM Advanced Modules
8-59
Chapter 8 Verification Problems
GAP PROPERTIES:
Based on the properties of the springs, the gaps properties are defined:
ki
= 1,000 lbs/in (i = 1,4)
fi
= 500 lbs (i = 1,2)
fi
= 250 lbs (i = 3,4)
gi
= 0 in (i = 1,2)
gi
= -1 E-8 in (i = 3,4)
COMPARISON OF RESULTS:
Since no moment is taken by the two ends of the beam, the beam is simply
supported. This means that, regardless of the beam properties, the spring force is
always half of the applied force. Thus the following represents the deflection versus
spring force.
Figure NS30-2
F
s
1000.
-4.0
0.5
Us
-1000.
-2000.
-3000.
-4000.
-5000.
Spring Force - Deflection
In
de
x
Same results are obtained by COSMOSM except that the negative spring forces are
taken by gaps No. 1 and 2, while positive forces are shared by gaps No. 3 and 4.
8-60
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS31: Crushing of a Pipe between Two Anvils
TYPE:
Nonlinear static analysis, plasticity, large displacements, gaps, 2D plane strain 8node elements.
NS31)
Force control without equilibrium iterations
NS31A)
Displacement control with Newton-Raphson iterations, adaptive
automatic stepping
NS31B)
Large strain plasticity theory, force control with Newton-Raphson
iterations, adaptive auto-stepping.
REFERENCE:
ABAQUS, Example problems manual, page 4.2.8.1, Rev. 4.5.
PROBLEM:
A long straight pipe is crushed between two flat, frictionless anvils by gradually
pushing down on the anvil. Determine the variation of the relative anvil
displacement with respect to the total force/length applied to the anvil (Shear force
on the pipe section).
Figure NS31-1
Stress
ksi
60
40
L
20
.002
.004
.006
.008
Strain
t
Problem Sketch
Stress - Strain Curve
In
de
x
GIVEN:
Yield stress
= 35,000 psi
Thickness
= t = 0.349 in
Length
= L = 1 in
COSMOSM Advanced Modules
8-61
Chapter 8 Verification Problems
Poisson ratio
= 0.3
Ext. diameter
= D = 4.5 in
Modulus of elasticity
= 30E6 psi
MODELING HINTS:
NS31 and NS31A
Due to symmetry, only a quarter of the pipe is modeled. Two rows of eight
PLANE2D plane strain 8-node elements are used to model the pipe. These elements
are allowed to undergo large deflection (total Lagrangian) and plasticity (von
Mises). Another group of eight elements of the same type are used to model the
anvil. These elements, however, are defined to be elastic and close to rigid. Gap
elements are used to model contact between the anvil and the pipe section. Two soft
truss elements are added to hold the anvil to prevent singularity of the stiffness
matrix.
NS31B
The large strain plasticity analysis is performed by using the von Mises plasticity
model and the updated Lagrangian formulation. The anvil is modeled by a very stiff
BEAM element which is used as a contact line as well. The displacement is
prescribed onto the BEAM element along the vertical direction until the designed
value. No soft spring is required. The equivalent force is obtained by coupling the
second node of the BEAM element to the first and the reaction force along the
vertical direction is recorded.
Figure NS31-2
Figure NS31-3
DEF STEP: 50 = 1
PRESCRIBE
DISPLACEMENT
UY
CONTACT
LINE
In
de
x
ORIGINAL AND FINAL DEFORMED SHAPES (NS31B)
8-62
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
LOADING:
NS31
Half of the total force/length on the section, divided by the width of the mesh of the
anvil (equal to the radius of the pipe), produces a state of constant pressure (uniform
displacement) on the anvil. Thus: total force/length = 2. * radius * pressure.
Small time increments and therefore a large number of solution steps (792 steps)
were used. Equilibrium iterations were not performed.
NS31A
The adaptive automatic stepping option is utilized along with displacement control.
NS31B
The adaptive automatic stepping option is utilized along with force control.
COMPARISON OF RESULTS:
In
de
x
Figure NS31-4
COSMOSM Advanced Modules
8-63
Chapter 8 Verification Problems
Figure NS31-5
Figure NS31-6
F
O
R
C
E
/
U
N
I
T
L
E
N
G
T
H
(LB/IN)
In
de
x
RELATIVE ANVIL DISPLACEMENT (IN)
FORCE-DISPLACEMENT RESPONSE OF THE ANVIL (NS31B)
8-64
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS32: Uniaxial Elastoplastic STRAIN
in a Bar (Cyclic Loading)
TYPE:
Nonlinear Static Analysis, plasticity based on the kinematic hardening rule for cyclic
loading conditions:
NS32A)
TRUSS elements
NS32B)
PLANE2D plane stress elements
NS32C)
SOLID elements
NS32D)
BEAM elements
NS32E)
SHELL elements
NS32F)
Combined Kinematic & Isotropic Hardening (RK=0.5)
REFERENCE:
Owen, Dr. J., Hinton, E., “Finite Elements in Plasticity: Theory and Practice,”
Pineridge Press Limited, Swansea, U. K., (1980).
PROBLEM:
A bar is subjected to a cyclic loading condition. Investigate the effects of plasticity
and stress reversals on the uniaxial stress-strain curve, based on kinematic hardening
rule; compare with the isotropic hardening rule.
Compare also with the case of combined kinematic & isotropic hardening.
Figure NS32-1
p (kips)
2.2
1.5
2.2
2.199
2.199
38 40
p
2
16 18
60 62
Problem Sketch
2.2
Step
No.
2.199
In
de
x
Cyclic Loading
COSMOSM Advanced Modules
8-65
Chapter 8 Verification Problems
GIVEN:
Modulus of elasticity = 30E6 psi
Tangential modulus
= 30E5 psi
Yield stress
= 1,599 psi
NOTES:
Equilibrium iterations are performed every other step, so that at the beginning of
each load reversal equilibrium iterations are suppressed. If we run the same problem
using the isotropic hardening approach, the same results are obtained as long as the
loading is not reversed. After a stress reversal occurs, the strain-stress relations will
remain linear elastic, unless the magnitude of the load at its peak is increased. For
this case, the load curve was selected (as shown in Figure NS32-2) such that the
magnitude of maximum and minimum strains are equal.
For the case of combined kinematic & isotropic hardening, the load curve of Figure
NS32-1 is used. Here only half of the hardening (RK=0.5) is used towards expansion
of the yield radius (the other half is used to displace the center of the yield surface
based on the kinematic hardening assumption). Thus at the end of loading:
Radius of the yield surface=1.5e3+(2.2e3-1.5e3)/2=1.85e3 psi
Start of yield in the 1st reversal= 2.2e3-2*Radius=-1.5e3 psi
And at the end of one cycle of loading, reversing, and unloading:
Radius of the yield surface= 1.85e3+(2.2e3-1.5e3)/2=2.2e3 psi
Start of yield for the 2nd cycle= -2.2e3+2*Radius=2.2e3 psi
Figure NS32-2
p (kips)
4.28
3.28
2.2
1.5
2.199
step no.
32
18
44
3.279
In
de
x
3.28
8-66
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
RESULTS:
In
de
x
Figure NS32-3
COSMOSM Advanced Modules
8-67
In
de
x
Chapter 8 Verification Problems
8-68
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS33: Elastoplastic Analysis of a Thick
Walled Tube (Cyclic Loading)
TYPE:
Nonlinear static analysis, plasticity, PLANE2D axisymmetric 8-node elements.
REFERENCE:
Mark D. Snyder, Klaus-Jurgen Bathe, “Formulation and Numerical Solution of
Thermo-Elastic-Plastic and Creep Problems,” Report 8244-3, (l977).
PROBLEM:
A thick-walled cylinder is subjected to cyclic internal pressure. Plane strain
conditions are assumed in the direction of the axis of cylinder. Investigate the radial
deflection of the outer boundary of the cylinder; compare with isotropic hardening
rule.
Figure NS33-1
3
23
4
2
p
y
1
p
P
3
1 rad
x
1
Problem
Sketch
21
Finite Element Model
p (ksi)
25
15
25.
24.9
10
1
24.9
20
22
8
step
no.
24.9
In
de
x
25.
32 34
COSMOSM Advanced Modules
8-69
Chapter 8 Verification Problems
GIVEN:
E
= 30E6 psi
ν
= 0.3
ET
= 30E5 psi
σy
= 30E3 psi
Thickness
= 1 in
Int. Diameter = 2 in
COMPARISON OF RESULTS:
In
de
x
Figure NS33-2
8-70
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
In
de
x
Figure NS33-3
COSMOSM Advanced Modules
8-71
Chapter 8 Verification Problems
NS34: 3D Extension/Compression
Tests on Mooney-Rivlin Model
TYPE:
Nonlinear static analysis, Mooney-Rivlin hyperelastic material model, large
deflections using total Lagrangian formulation, displacement control, NewtonRaphson iterations.
NS34)
One 3D SOLID element
NS34A)
One 4-node PLANE2D element
REFERENCE:
Frank J. Marx, “Hyperelastic Elements (STIF84, STIF86),” ANSYS Revision 4.3
Tutorial, 1987.
PROBLEM:
A 3D sheet of material (2x2x1 inch) is subjected to biaxial equal loadings in the X
and Y directions. Investigate the behavior of Mooney-Rivlin hyperelastic material
for various ratios of A/B.
MODELING HINTS:
Due to symmetry, a quarter of the sheet is modeled using one solid 8-node element
or one 4-node PLANE2D element. Loading is applied along the two sides such that
the sheet will expand or compress equally in the X and Y directions.
In
de
x
Test cases are performed for different values of Mooney-Rivlin constants (B varies
from 0.2A to 2A).
8-72
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS34-1. (Solid Element)
Uy
Ux
Finite Element
In
de
x
Figure NS34-2
COSMOSM Advanced Modules
8-73
Chapter 8 Verification Problems
NS35: Inflation of a Simply Supported Circular
Plate, Mooney-Rivlin Material
TYPE:
Nonlinear static analysis, hyperelastic material model, large deflections using total
Lagrangian formulation, Newton-Raphson iterations.
NS35)
PLANE2D axisymmetric 8-node elements, Mooney-Rivlin model
NS35A)
PLANE2D axisymmetric 8-node elements, Mooney-Rivlin model
with adaptive automatic stepping
NS35B)
PLANE2D axisymmetric 8-node elements, Ogden model
NS35C)
PLANE2D axisymmetric 8-node elements, Mooney-Rivlin model
with u/p formulation, constraint equations, and auto-stepping.
NS35D
TRIANG axisymmetric elements, Mooney-Rivlin model with u/p
formulation
NS35E
TETRA4 axisymmetric elements, Ogden model with u/p
formulation
REFERENCE:
Frank J. Marx, “Hyperelastic Elements (STIF84, STIF86),” ANSYS Revision, 4.3
Tutorial, 1987.
PROBLEM:
A simply supported circular flat plate is subjected to an external pressure varying
from 0. to 50 psi. The plate is made of an isotropic incompressible material of the
Mooney type.
MODELING HINTS:
NS35 through NS35B
In
de
x
One row of ten PLANE2D axisymmetric 8-node elements, in the radial direction, is
used to model the plate. A Poisson’s ratio of 0.49 is defined to approximate the
incompressibility of the material. Since the displacements increase rapidly at low
pressures (0-8 psi), the problem requires a slow initial loading. A very large load
increment leads to negative diagonal terms or divergence.
8-74
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Unloading will mirror the loading perfectly. It can be achieved by reversing the
sequence of load application. A rapid load decrease yields the same result and may
lead to termination due to negative diagonal terms or lack of convergence.
NS35C
Two rows of 20 PLANE2D axisymmetric 4-node elements in the radial direction are
used to model the disk as shown in Figure NS35-4. The u/p formulation is used for
the current case with a Poisson’s ratio equal to 0.4999, which corresponds to the ratio
of bulk modulus to shear modulus (K/G) equal to 5000. From the result, it is seen
that the error of the volume ratio (V/V0) is kept within 1%. In order to satisfy the
hinged boundary condition along the edge, two linear constraint equations are
prescribed:
UX(61) + UX(63)= 0
UY(61) + UY(63)= 0
Figure NS35-1
I
D
Section I-I
I
Problem Sketch
Y
3
53
5 8
52
2
51
1
4
center
of plate
6
9
x
Finite Element Mesh (NS35 thru NS35B)
CL
NS35D
In
de
x
One row of ten TRIANG axisymmetric 6-node elements, in the radial direction, is
used to model the plate. The Poisson’s ratio of the material is 0.4999.
COSMOSM Advanced Modules
8-75
Chapter 8 Verification Problems
NS35E
TETRA4 solid elements are used to model this problem with a Poisson’s ratio of
0.4995. The material model is Ogden hyperelastic model. Axisymmetry boundary
conditions are applied to the model.
Finite Element Mesh (NS35D)
GIVEN:
Mooney-Rivlin constants:
A
= 80
B
= 20
Poisson’s Ratio in x-y:
ν
= 0.49
(NS35 through NS35B)
= 0.4999 (NS35C)
D
= 15 in
Ogden constants:
α1 = 2,
α2 = -2,
ν
µ1 = 2A = 160
µ2 = -2B = -40
= 0.4975
In
de
x
COMPARISON OF RESULTS:
8-76
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
In
de
x
Figure NS35-2
COSMOSM Advanced Modules
8-77
Chapter 8 Verification Problems
Figure NS35-3
D EFOR M ED SH AP E P LOT AT P R ESSU R E = 3 6 P SI
Figure NS35-4
DEF STEP:1 6 = 1
H IN GED B OU N D AR Y
CON D ITION S
P R ESSU R E
In
de
x
ORIGINA L A ND FINA L DEFORM ED SHA PES (NS35C)
8-78
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS35-5
In
de
x
Figure NS35-6
COSMOSM Advanced Modules
8-79
Chapter 8 Verification Problems
NS36: Initial Interference between
Two Thick Hollow Cylinders
TYPE:
Nonlinear static analysis using PLANE2D axisymmetric and contact (node to line
gap) elements.
REFERENCE:
Timoshenko, S. P., and Goodier, J. N., “Theory of Elasticity,” McGraw-Hill, New
York (1970).
PROBLEM:
A cylinder is first compressed and then placed inside another cylinder. Evaluate the
deformed shape and the forces at the interface.
MODELING HINTS:
The two cylinders are modeled separately, each based on its unstressed geometry,
using PLANE2D axisymmetric elements. The interface is modeled with contact
(node to line) elements; nodes are selected on the outer boundary of the inner
cylinder, while a line is used to define the inner surface of the outer cylinder. A one
step solution is performed; no external forces are defined.
In
de
x
Figure NS36-1
8-80
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS36-2
GIVEN:
a1 = 20 in
b1 = 22 in
h1 = 1 in
a2 = 21.25 in
b2 = 24.25 in
h2 = 1.01 in
Modulus of elasticity = 30E6 psi
Poisson’s ratio = 0.3
COMPARISON OF RESULTS:
Theory
COSMOSM Difference
Contact Loca21.570915 in 21.57546 in
tion (radius)
Pressure at
the Interface
57.231 ksi
57.756 ksi
0.02%
0.92%
NOTES:
In order to obtain accurate results when a contact problem exists, the contacting
nodes must be located inside the contacting surface. If a node moves to a position
where a normal to the surface does not exist, the program assumes the point is not in
contact with the surface (unless a step-by-step solution is performed in which
deformations are induced gradually). Therefore, in this problem we define a height
for the outer cylinder which is slightly more than the height of the inner cylinder.
In
de
x
Figure NS36-3
COSMOSM Advanced Modules
8-81
Chapter 8 Verification Problems
NS37: Contact Between Two Solid Cubes
TYPE:
Nonlinear static analysis using 8-node solid and contact (node to surface gap)
elements.
NS37)
NS37A)
8-node solid elements
TETRA10 elements, automatic soft springs
PROBLEM:
A cube is pushed against a second larger cube which stands against a rigid wall.
Verify that the two cubes will displace together by using contact elements.
MODELING HINTS:
NS37:
Each cube is modeled by one solid 8-node element. The interface is modeled by
defining nodes on the smaller cube and one 4-noded surface on the larger cube. Soft
truss elements are used to avoid stiffness singularities.
NS37A:
Each cube is modeled using TETRA10 elements. The interface in modeled by gap
elements on the smaller cube and 6-noded sub-surfaces on the large cube. To
stabilize the smaller cube in the global x-direction, a soft stiffness of 100 lb/in is
defined for the contact source in that direction (gap real constant).
GIVEN:
E
υ
p
= 30E6 psi
=0
= 4E6 psi
ANALYTICAL SOLUTION:
U
= u1 + u2
Ui = P li / Ai Ei
U1 = (4E6 x 1) / (1 x 30E6)
U2 = (4E6 x 1.4) / [(1.4 x 1.4)
x 30E6]
= 0.22857 in
In
de
x
U
8-82
COSMOSM Advanced Modules
Figure NS37-1
Part 1 NSTAR / Nonlinear Analysis
COMPARISON OF RESULTS:
NS37:
Deflection at the free end:
δ, inch
Theory
0.22857
COSMOSM
0.22857
Figure NS37-2
NS37A
Figure NS37-3
In
de
x
Here since each
cube is defined with
several elements,
the contacting
surfaces undergo
translation and
bending. As a result,
the displacements
obtained are about
8% higher than
those predicted by
the 8-node solid
model.
COSMOSM Advanced Modules
8-83
Chapter 8 Verification Problems
NS38: Contact Between Two
Parallel S.S. Circular Plates
TYPE:
Nonlinear static analysis using 8-node PLANE2D axisymmetric and contact (node
to line gap) elements.
NS38A)
Without friction
NS38B)
With friction
PROBLEM:
Two parallel simply supported circular plates stand 0.5 inches apart. Pressure
loading is applied to the upper plate. Verify that the plates come in contact and
therefore calculate the displacement at the center of the lower plate. Compare with
regular gap elements.
MODELING HINTS:
The two plates are modeled separately, using PLANE2D axisymmetric elements.
The interface is modeled with contact (node to line gap) elements. A one step
solution is performed for each case.
Figure NS38-1
P
h
t
R
Problem Sketch
CL
P
3
1
X
h
101
103
In
de
x
CL
8-84
COSMOSM Advanced Modules
Finite Element Model
Part 1 NSTAR / Nonlinear Analysis
GIVEN:
R
= 10 in
t
= 0.1 in
h
= 0.5 in
Pressure
= 6 psi
Poisson’s ratio
= 0.3
Coefficient of friction
= 0.7
Modulus of elasticity
= 10.5E + 6 psi
COMPARISON OF RESULTS:
Vertical Disp. (inch)
(center of the lower plate)
No Friction
With Friction
Contact (Node to line)
1.7347
1.7268
Regular Gaps
1.7377
1.7308
Note that for this problem the results obtained from the two methods (contact and
regular gaps) are almost identical.
In
de
x
Figure NS38-2
COSMOSM Advanced Modules
8-85
Chapter 8 Verification Problems
NS39: Contact Between Two Parallel
Simply Supported Circular Plates
TYPE:
Nonlinear static analysis using contact (node to surface gap) elements and:
NS39A)
4-node shell elements
NS39B)
20-node solid elements
NS39C)
3-node thick shell elements, Lower plate = target
NS39D)
3-node thick shell elements, Upper plate = target
PROBLEM:
Same as problem NS38, except here shell or solid elements are used to model a
quarter of the plates. As a result, contact is modeled using node to surface gaps. The
goal is to compare the results with those obtained in problem NS38.
Figure NS39-1
MODELING HINTS:
For NS39A and NS39B:
A quarter of each plate is modeled using:
a) 80, 4-node shell elements
b) 50, 20-node solid elements
In
de
x
A hole with a small radius (0.05 inches) is assumed at the center of each plate so that
triangular elements are not needed in modeling of plates.
8-86
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Contact between the two plates is modeled using 3 different groups of gaps:
1.
Nodes nearest to the center where only vertical motion is allowed are connected
by regular (node to node) gaps.
2.
Along the two axis of symmetry (θ = 0. and θ = 90.) where motion is to remain
in planes XZ and YZ, two groups of node to line gaps are used.
3.
The last group of gaps (node to surface) considers contact between the inside
nodes of the upper plate and 4-noded (8-noded when 20-node solid elements are
used) surface segments on the lower plate.
For NS39C and NS39D:
Contact between the two plates is modeled using a node-to-surface gap element
group. In both cases, target is defined by triangular 3-noded sub-surfaces.
GIVEN:
R
t
h
Coeff. of friction
Pressure
Modulus of elasticity
Poisson’s ratio
= 10 in
= 0.1 in
= 0.5 in
= 0.7
= 6 psi
= 10.5E+6 psi
= 0.3
COMPARISON OF RESULTS:
Vertical Disp. (inch)
(center of the lower plate)
No Friction
With Friction
Axisymmetric PLANE2D
(2D Analysis)
1.7347
1.7268
20-node Solid Element
(3D Analysis) (at node No. 1001)
1.7340
1.7262*
4-node Shell Element
(3D Analysis) (at node No. 101)
1.7260
— **
3-node Shell Elements
Cased C and D)
1.733
—
In
de
x
*The results obtained from PLANE2D axisymmetric and solid 20-node elements
are almost identical.
**Shear deformation is neglected based on the Linear shell theory. Therefore, when
shell elements are used in modeling of plates, friction is not considered.
COSMOSM Advanced Modules
8-87
Chapter 8 Verification Problems
Figure NS39-2
Figure NS39-3
In
de
x
Figure NS39-4
8-88
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS39-5
Figure NS39-7
In
de
x
Figure NS39-6
COSMOSM Advanced Modules
8-89
Chapter 8 Verification Problems
NS40: Bending and Inflation of a Simply
Supported Circular Plate Split
into Halves Through Its Thickness
TYPE:
Nonlinear static analysis using 8-node PLANE2D and contact (node to line gap)
elements; Mooney-Rivlin hyperelastic material model, large deflections using total
Lagrangian formulation, Newton-Raphson iterations.
REFERENCE:
Frank J. Marx, “Hyperelastic Elements (STIF84, STIF86),” ANSYS Revision, 4.3
Tutorial, 1987.
PROBLEM:
A simply supported circular plate is split into halves through its thickness. The lower
half is then subjected to an external pressure varying from 0 to 50.0 psi. The plate is
made of an isotropic incompressible material of the Mooney type.
The behavior of this plate as a whole was studied in problem NS35. When the plate
is sliced, the two slices bend together through contact. Therefore, this problem is
useful in testing the accuracy of the contact elements when large rotations exist.
MODELING HINTS:
The two slices of the plate are modeled separately, using PLANE2D axisymmetric
elements. The interface is modeled using contact (node to line gap) elements.
Pressure is increased gradually, similar to problem NS35.
In
de
x
Figure NS40-1
8-90
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
GIVEN:
Mooney-Rivlin Constants:
A
= 80
B
= 20
ν
= 0.49
COMPARISON OF RESULTS:
While in low stresses, the split plate is weaker, in higher stresses when the two slices
undergo large deflections as well as large rotations, the response of the sliced plate
is expected to be close to that of the uncut plate.
By studying the results obtained for this problem the following can be observed:
The contact line gradually deforms from flat to nearly a quarter of a circle.
Thus, the contact force (normal to the line) near the support undergoes about 90
degrees change in direction; regular gaps can not be used for this problem.
2.
At very low pressures (0., to 0.1 psi) there is up to 32% increase in response due
to the slicing of the plate.
3.
At higher pressures (.1, to 28. psi) the response of the sliced plate is higher but
remains within 2% of that of the uncut plate.
4.
When pressure exceeds 28 psi, the sliced plate starts to weaken again. It begins
to deform at a faster rate in comparison to the uncut plate, and therefore, the two
responses start to deviate.
Figure NS40-3
In
de
x
Figure NS40-2
1.
COSMOSM Advanced Modules
8-91
Chapter 8 Verification Problems
NS41: Nonlinear Elasticity of a Cantilever Beam
TYPE:
Nonlinear static analysis, nonlinear elasticity.
NS41A)
PLANE2D elements
NS41B)
SOLID elements
NS41C)
SHELL4T elements
NS41D)
BEAM elements
NS41E)
TETRA10 elements
REFERENCE:
The theoretical solution is carried out based on thin beam theory.
PROBLEM:
Determine the deflection of a cantilever beam under an end moment as shown in
Figure NS41-1. The material is nonlinear elastic as shown in Figure NS41-2.
Figure NS41-1
M (lb.- in.)
b
7000.
M/2
M/2
h
2000.
L
Problem S ketch
1.0
Y
1.001
t
2.0
Moment - Time Curve
Y
45
5
4
3
2
1
5
43
90
41
PLANE2D
(NS41A)
Y
81
6
6
3
Z
87
5
2
25
SOLID
(NS41B)
86
X
Y
26
4
1
85
11
X
236
X
SHELL4T
(NS41C)
1
2
1
3
2
4
3
5
4
In
de
x
BEAM
(NS41D)
8-92
COSMOSM Advanced Modules
Z
Finite Element Models
TETRA10
(NS41E_
228
X
Part 1 NSTAR / Nonlinear Analysis
GIVEN:
Figure NS41-2
L
= l00 in
h
= 2 in
b
= 1 in
ν
=0
(Stress)
σ psi
60.003E6
3E3
2.001
0.001
ε
(Strain)
Strain_Stress Curve
COMPARISON OF RESULTS:
1.
Linear elasticity (M = 2000 lb-in)
Deflection at
Free End (inch)
Theory
PLANE2D
COSMOSM
2.
Rotation at
Free End
- 5.0
0.1
- 5.0001
0.1000
SOLID
- 4.9991
0.1000
SHELL4T
- 5.0000
0.1000
BEAM
- 5.0001
0.1000
TETRA10
- 4.9998
0.1000
Deflection at
Free End (inch)
Rotation at
Free End
Nonlinear elasticity (M = 7000 lb-in)
Theory
PLANE2D
0.15
0.1506
SOLID
- 7.5275
0.1506
SHELL4T
- 7.4791
0.1496
BEAM
- 7.4340
0.1487
TETRA10
- 7.5156
0.1504
In
de
x
COSMOSM
- 7.5
- 7.5279
COSMOSM Advanced Modules
8-93
Chapter 8 Verification Problems
NS42: Elastoplastic Analysis of a Simply
Supported Plate, Elastic-Perfect Plastic Case
TYPE:
Nonlinear static analysis, SHELL4T elements.
PROBLEM:
Determine the response of a simply supported plate under uniform pressure. The
pressure load (p) is varied well into the plastic range.
Figure NS42-1
Y
ρ , psi
Hinge
73
2b
1
81
Hinge
5.805
5.4
X
4.05
9
2a
t
2.
Problem Sketch and Finite
Element Model
GIVEN:
a
= b = l0 in
h
= 0.l in (thickness of plate)
E
= l0E6 psi
ν
= 0.3
ET = 0.0
In
de
x
σY = 36,000 psi
8-94
COSMOSM Advanced Modules
7.
Load - Time Curve
10.
Part 1 NSTAR / Nonlinear Analysis
COMPARISON OF RESULTS:
The variation of pressure (p) versus central deflection is shown in the next figure.
The result from COSMOS7 is also enclosed.
In
de
x
Figure NS42-2
COSMOSM Advanced Modules
8-95
Chapter 8 Verification Problems
NS43: Large Displacement Response
of a Cylindrical Shell Under a Conc. Load
TYPE:
Nonlinear static analysis, SHELL4T elements.
REFERENCE:
Horrigmoe, G., “Finite Element Instability Analysis of Free-Form Shells,” Report
No. 77-2, the Norwegian Institute of Technology, The University of Trondhem,
Norway (l977).
PROBLEM:
Determine the response of a shallow cylindrical shell under a concentrated load at
the center of the shell. The curved edges are free and the straight edges are hinged
and immovable.
Figure NS43-1
Free Edge
P
_
2b
X
h
21
Hinge
5
P , (KN)
_
25
2.2156
2.128
θ
θ
Z
In
de
x
t
Proble m S ke tch
8-96
COSMOSM Advanced Modules
Y
19
58
Loa d - Time C urve
Part 1 NSTAR / Nonlinear Analysis
GIVEN:
R
= 2,540 mm
b
= 254 mm
h
= l2.7 mm
θ
= 0.l rad
E
= 3l02.75 N/mm2
ν
= 0.3
COMPARISON OF RESULTS:
The force (p) - central deflection curve is shown in the next figure and compared
with the result from reference.
Figure NS43-2
F
O
R
C
E
(*E3),
N
Reference
COSMOS/M
In
de
x
CENTRAL DEFLECTION (mm)
COSMOSM Advanced Modules
8-97
Chapter 8 Verification Problems
NS44: Buckling and Post Buckling
of a Simply Supported Plate
TYPE:
Nonlinear static analysis, SHELL4T elements.
REFERENCE:
Timoshenko and Woinosky-Krieger, “Theory of Plates and Shells,” McGraw-Hill
Book Co., 2nd Ed., pp. 389.
PROBLEM:
Find the buckling load and post buckling behavior of a simply supported isotropic
plate subjected to inplane uniform pressure p applied at x = -a and x = a.
Figure NS44-1
Z
Y
Simply
Supported
21
25
Fe
P
2a
1
5
h
2a
Simply
Supported
In
de
x
Problem Sketch and Finite Element Model
8-98
COSMOSM Advanced Modules
X
P
Part 1 NSTAR / Nonlinear Analysis
Figure NS44-2
P lb./in.
75.
67.8
Time
15
5
30
60
65
75
Fe, lb.
1.0
0.5
5
20 25 30
Time
60
75
GIVEN:
a
= 20 in
h
= l in
E
= 3E4 psi
ν
= 0.3
NOTE:
In
de
x
The post buckling behavior is obtained by applying a transverse force (Fe) at the
center of the plate at the first stage. The inplane pressure (p) is then applied and the
transverse load is reduced at this stage. The buckling load is obtained by decreasing
the inplane pressure to zero at the final stage.
COSMOSM Advanced Modules
8-99
Chapter 8 Verification Problems
COMPARISON OF RESULTS:
The inplane pressure - central deflection curve is shown in the next figure. The
theoretical buckling load is 67.78 lb/in.
Figure NS44-3
75
I
N
P
L
A
N
E 70
P
R
E
S
S
65
U
R
E
(LB/IN)
BUCKLING LOAD = 67.78 LB/IN
60
0.0
0.1
0.2
0.3
0.4
0.5
In
de
x
CENTRAL DEFLECTION, INCH
8-100
COSMOSM Advanced Modules
0.6
0.7
0.8
Part 1 NSTAR / Nonlinear Analysis
NS45: Large Displacement Response
of a Cylindrical Shell Under a Uniform Load
TYPE:
Nonlinear static analysis, SHELL4 elements.
REFERENCE:
Sabir, A. B., and Lock, A. C., “The Application of Finite Elements to the Large
Deflection Geometrically Nonlinear Behavior of Cylindrical Shells,” Variational
Methods in Engineering, South Hampton University Press (l973).
PROBLEM:
Determine the response of a cylindrical shell under normal uniform pressure (p). All
edges are clamped and immovable.
Figure NS45-1
Clamped
2L
P
1
X
h
73
81
θ
Clamped
9
θ R
-3
2
P (10 ), N/mm
3.0
2.25
Z
1.5
t
In
de
x
Y
Problem Sketch and Finite
Element Model
COSMOSM Advanced Modules
6
18
21
Pressure - Time Curve
8-101
Chapter 8 Verification Problems
GIVEN:
R
= 2,540 mm
L
= 254 mm
h
= 3.l75 mm
θ
= 0.l rad
E
= 3l02.75 N/mm2
ν
= 0.3
COMPARISON OF RESULTS:
The pressure (p) - central deflection curve is shown in the next figure and compared
with the result from reference.
Figure NS45-2
F
O
R
C
E
(*E3),
Nmm 2
Reference
COSMOS/M
In
de
x
CENTRAL DEFLECTION (mm)
8-102
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS51: Thermo-Plasticity in a
Thick-Walled Cylinder
TYPE:
Nonlinear static analysis, thermo-plasticity, temperature-dependent yield stress,
PLANE2D axisymmetric 8-node elements
REFERENCE:
Mark D. Snyder, Klaus-Jurgen Bathe, “Formulation and Numerical Solution of
Thermo-Elastic-Plastic and Creep Problems,” Report 8244-3, (1977).
PROBLEM:
A thick-walled cylinder is subjected to varying internal pressure and temperature, as
shown in the figure. Plane strain conditions are assumed in the direction of the axis
of the cylinder. Determine:
1.
Response at the outer surface of the cylinder
2.
Residual stress distributions through the wall thickness when pressure is
reduced to zero (step = 10).
In
de
x
Figure NS51-1
COSMOSM Advanced Modules
8-103
Chapter 8 Verification Problems
RESULTS:
The response and stress plots are identical to those obtained by ADINA program.
Figure NS51-2
In
de
x
Figure NS51-3
8-104
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS51-4
In
de
x
Figure NS51-5
COSMOSM Advanced Modules
8-105
Chapter 8 Verification Problems
In
de
x
Figure NS51-6
8-106
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS52: Thermo-Plasticity in a Bar
TYPE:
Nonlinear static analysis, thermo-plasticity, temperature-dependent material
properties.
NS52)
Truss elements
NS52A)
PLANE2D plane stress elements
NS52B)
Solid elements
REFERENCE:
Mark D. Snyder, Klaus-Jurgen Bathe, “Formulation and Numerical Solution of
Thermo-Elastic-Plastic and Creep Problems,” Report 8244-3, (1977).
PROBLEM:
A bar is subjected to varying pressure and temperature, as shown in the figure.
Determine strain in the bar.
In
de
x
Figure NS52-1
COSMOSM Advanced Modules
8-107
Chapter 8 Verification Problems
Figure NS52-2
RESULTS:
In
de
x
Similar results are obtained for different element types. Since there is no load
reversal, using kinematic hardening law for plasticity yields identical results.
8-108
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS52-3
In
de
x
Figure NS52-4
COSMOSM Advanced Modules
8-109
Chapter 8 Verification Problems
NS61: Bearing Capacity for a Strip Footing
TYPE:
Nonlinear static analysis, Drucker-Prager Elastoplastic material model, small
deflection, Newton-Raphson.
NS61A)
PLANE2D (plane strain)
NS61B)
SOLID elements
REFERENCE:
Joseph E. Bowels, “Foundation Analysis and Design,” McGraw-Hill Book Co., 2nd.
Ed (1977), pp. 113-124.
PROBLEM:
Find bearing capacity for a strip footing with width B sitting at the ground surface
subjected to a uniform pressure P and soil parameters shown below.
Figure NS61-1
LINE OF SYMMETRY
B/2
p
In
de
x
FINITE ELEMENT MESH
8-110
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
GIVEN:
B
= 120 in
Soil parameters:
ρ
= 1.7961E - 4 lb - sec2/in4
φ
= 27.27°
C
= 1 psi
E
= 3E4 psi
υ
= 0.3
NOTE:
Due to symmetry, half of the soil is modeled. Because the bearing capacity of
foundations depends on the self-weight of the soil, an acceleration of gravity in the
y-direction (-386.4 in/sec2) is applied to simulate this effect. When the applied
pressure approaches the limit load (bearing capacity of foundations), the soil
“bulging” takes place adjacent to the footing. This phenomenon induces difficulties
in achieving the limit load. Reducing the load increment by taking a couple of
RESTART procedures and using a regular (not modified) Newton Raphson
iterations will help solve these problems.
COMPARISON OF RESULTS:
Ultimate Bearing
Capacity qu (psi)
Terzaghi (1943)
90.0
Hansen (1970)
93.0
COSMOSM (plane strain)
88.5
COSMOSM (SOLID)
95.0
In
de
x
The deformed shape plots for PLANE2D (plane strain) and SOLID models are
shown in Figures NS61-2 and NS61-3.
COSMOSM Advanced Modules
8-111
Chapter 8 Verification Problems
Figure NS61-2
In
de
x
Figure NS61-3
8-112
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS62: Bearing Capacity for Round Footing
TYPE:
Nonlinear static analysis, Drucker-Prager Elastoplastic material model, small
deflection, Newton-Raphson iterations, PLANE2D (Axisymmetric) elements.
REFERENCE:
Joseph E. Bowles, “Foundation Analysis and Design,” McGraw - Hill Book Co.,
2nd.Ed (1977), pp. 113-124.
PROBLEM:
Find bearing capacity for a round footing with radius R sitting at the ground surface
subjected to a uniform pressure p and soil parameters shown below.
GIVEN:
R
= 6 in
Soil parameters:
ρ
= 1.7961E-4 lb • sec2/in4
φ
= 27.27 °
c
= 1 psi
E
= 3E4 psi
υ
= 0.3
NOTES:
In
de
x
The finite element mesh is as same as that shown in Figure NS61-1 except the
replacing of B/2 by R. All the precautions mentioned in Problem NS61 are also
needed in this problem.
COSMOSM Advanced Modules
8-113
Chapter 8 Verification Problems
COMPARISON OF RESULTS:
Ultimate Bearing
Capacity qu (psi)
Terzaghi (1943)
75.0
Hansen (1970)
63.0
COSMOSM
68.5
The deformed shape plot is shown in Figure NS62-1.
Figure NS62-1
DEF STEP: 19 = 19
PLANE2D (AXISYMMETRIC)
U y = 0.8053
In
de
x
DEFORMED SHAPE PLOT (p = 68 PSI)
8-114
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS63: Bending and Inflection
of a Simply Supported Plate
TYPE:
Nonlinear static analysis, Mooney-Rivlin hyperelastic material model, large
deflection, deformation-dependent pressure, local boundary conditions, and autostepping.
NS63A)
SHELL4T with regular shell analysis
NS63B)
SHELL4T with membrane element analysis
REFERENCE:
T. J. R. Hughes and E. Carnoy, “Nonlinear Finite Element Shell Formulation
Accounting for Large Membrane Strains,” Nonlinear Finite Element Analysis of
Plates and Shells. AMD Vol. 48, pp. 193-208, ASME, New York.
PROBLEM:
A simply supported circular plate is subjected to a uniform pressure. The plate is
made of an isotropic incompressible material of the Mooney-Rivlin type (see
Problem NS35). Both regular and membrane shells are tested and results are
compared.
GIVEN:
r
= 7.5 in
Mooney-Rivlin constants:
h
= 0.5 in
A= 80 psi
B= 20 psi
NOTES:
The circular plate is modeled by SHELL4T elements (shown in Figure NS63-1).
Due to axisymmetry, only a 5-degree wedge of the plate is modeled and local
boundary conditions along the line A-C are applied to remain the axisymmetry, i.e.,
Uθ = 0
In
de
x
θr =0
For the membrane element analysis, a tiny prestress (0.01 psi) is adapted to prevent
singularity. An auto-stepping algorithm is used to control the load increment such
that the convergence and accuracy of the solution are insured especially for the
membrane element analysis.
COSMOSM Advanced Modules
8-115
Chapter 8 Verification Problems
COMPARISON OF RESULTS:
The plots of normal deflection at center versus pressure are shown in Figure NS632. The result of membrane element analysis is almost identical to the result of regular
shell because the membrane strain becomes dominant in this type of rubber analysis.
Figure NS63-1
Simply
Supported
Normal
Pressure
Y
C
5 Degrees
Z
X
A
B
Raduis
Plane View
Finite Element Mesh (10 SHELL3/4T)
Figure NS63-2
C
E
N
T
R
A
L
COSM OS/M (REGULA R M EM BRA NE)
∆
D
E
F
L
E
C
T
I
O
N
(IN)
∆∆
∆ HUGHES: SHELL
∆
∆
∆
∆
∆
∆
∆
In
de
x
PRESSURE (PSI)
8-116
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS64: Failure and Strain Response of a
[0,90,90,90,90,0] Graphite-Epoxy Laminate
TYPE:
Nonlinear static analysis, Tsai-Wu failure criterion, small deflection, NewtonRaphson iterations, SHELL4L elements.
REFERENCE:
Stephen W., Tsai and H. Thomas Hahn, “Introduction to Composite Materials,”
Technomic Publishing Co., Inc. (1980), pp. 308.
PROBLEM:
Find the FPF (First Ply Failure) and UF (Ultimate Failure) stresses for a 6-layer
T300/5208 cross-ply laminate under the uniaxial tensile load. The geometry of the
laminate is shown in Figure NS64-1.
Figure NS64-1. Geometry of a 6-Layer T300/5208 Cross-Ply Laminate
y
Nx
b
x
Nx
a
GIVEN:
In
de
x
a
= b = 10 in
6h
= 1 in
Nx
> 0 (tensile loading)
E1
= 26.27E6 psi
COSMOSM Advanced Modules
8-117
Chapter 8 Verification Problems
E2
= 1.49E6 psi
ν12
= 0.28
G12
= G13 = G23 = 1.04E6 psi
F1T
= 2.17E5 psi
F1C
= 2.17E5 psi
F2T
= 5.81E5 psi
F2C
= 3.57E4 psi
F12
= 9.87E3 psi
where F1T and F1C are tensile and compressive strengths in the 1st material
direction; F2T and F2C are tensile and compressive strengths in the 2nd material
direction; F12 is the shear strength in the material 1st-2nd plane.
NOTES:
Due to symmetry, a quarter of the laminate is modeled.
COMPARISON OF RESULTS:
Reference:
R = 38.03E3 psi for 90° ply
R = 69.09E3 psi for 0° ply
where R = failure stress
COSMOSM
FPF stress = 40E3 psi on 90° ply; along direction 2
UF stress = 75E3 psi
DISCUSSION:
In
de
x
In the COSMOSM failure analysis, a two-stage failure algorithm is used, i.e., the
failure occurs in the transverse direction first and then the fiber direction. In the
current example, the stress can still be carried by the 90° ply (2,3,4 and 5) in the fiber
direction after FPF. The laminate is then degraded to the point where the secondary
failure occurs on the 0° ply in the transverse direction. Finally UF is reached and the
largest load the laminate can carry is determined.
8-118
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS65: Failure and Strain Response of a
Symmetric Angle-Ply Laminate of Graphite-Epoxy
TYPE:
Nonlinear static analysis, Tsai-Wu failure criterion, small deflection, NewtonRaphson iterations, thermal analysis, SHELL4L elements.
REFERENCE:
Stephen W., Tsai and H. Thomas Hahn, “Introduction to Composite Materials,”
Technomic Publishing Co., Inc. (1980), pp. 362-363.
PROBLEM:
Find the FPF (First Ply Failure) and UF (Ultimate Failure) stresses for a symmetric
AS/3051 angle-ply laminate under a uniaxial tensile load.
GIVEN:
a
4h
Nx
= b = 10 m
=1m
> 0 (tensile loading)
< 0 (compressive loading)
E1
= 138 Gpa
E2
= 8.96 Gpa
ν12 = 0.3
T-T0 = 1000° K
G12
α1
α2
F1T
F1C
F2T
F2C
F12
= G13 = G23 = 7.1 Gpa
= 0.18E-6 / °K
= 22.5E-6 /°K
= 1447 Mpa
= 1447 Mpa
= 51.7 Mpa
= 206 Mpa
= 93 Mpa
where F1T and F1C are tensile and compressive strengths in the 1st material
direction; F2T and F2C are tensile and compressive strengths in the 2nd material
direction; F12 is the shear strength in the material 1st-2nd plane.
NOTES:
In
de
x
Due to symmetry, a quarter of the laminate is modeled.
COSMOSM Advanced Modules
8-119
Chapter 8 Verification Problems
COMPARISON OF RESULTS:
Nx/h at UF (Mpa) Tensile/Compressive
Theta (degrees)
Reference (from Test)
COSMOSM
0.0
15.0
30.0
45.0
90.0
1520.0/-1521.0
879.0/-648.0
482.0
193.0
54.0/-214.0
1482.0/-1482.0
1128.5/-666.0
465.5/-308
200.0/-230.0
52.0/-210.0
The comparison is also shown in Figure NS65-1.
Figure NS65-1. Comparison of Results of a Symmetric AS/3051 Angle-Ply Laminate
1500
+φ
N1
1000
b
1
N1
-φ
a
500
0
15
30
45
60
75
90
+φ
-
-500
Reference (test)
-1000
Reference (computation)
COSMOS/M
-1500
DISCUSSION:
In
de
x
The behavior of angle-ply laminates is different from that of cross-ply laminates (see
NS64). No progressive failure occurs in angle-ply laminates under the uniaxial
loading. The FPF stress is therefore equal to the UF stress.
8-120
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS66: Static Equilibrium Position and
Relaxation of a Cable-Buoy System
TYPE:
Nonlinear hydrostatic and hydrodynamic analysis, large deflection, auto-stepping
algorithm, IMPIPE, BUOY, and GAP elements.
REFERENCE:
John W. Leonard, “Tension Structures - Behavior and Analysis,” McGraw-Hill
Book Company, pp. 197-201.
PROBLEM:
NS66A)
Find the static equilibrium position of a cable-buoy system (see
Figure NS66-1). The system is completely immersed and subjected
to the gravity and buoyant forces.
NS66B)
The cable-buoy system was then released. Find the dynamic
response of the system.
Figure NS66-1. Geometry of a Cable-Buoy System
Z
SWL
X
BUOY
A
CABLE
B
H
SEA
BED
In
de
x
X
COSMOSM Advanced Modules
8-121
Chapter 8 Verification Problems
GIVEN:
Cable:
Total length of riser
= 71.56 in
A
= 51.1 in
B
= 42.1 in
EA
= 4.8 lb
Mass
= 11.443E-3 lb/ft
Outside diameter
= 0.163 in
Coefficient of normal drag
= 1.2
Coefficient of tangential drag
= 0.02
Coefficient of added mass and inertia = 1
Buoy:
Mass
= 0.025 lb
Outside diameter
= 2 in
Coefficient of drag
= 0.5
Coefficient of added mass and inertia = 0.5
Water depth
= 200 in (assumed)
Water density
= 0.9366E-4 lb • sec.2/in4
RESULTS:
The static equilibrium position of the system from the current analysis is shown
in Figure NS66-2 and compared with the reference one.
2.
For the dynamic analysis, Figures NS66-3 and NS66-4 illustrate the comparisons of the horizontal and vertical position histories of the buoy; Figure NS66-5
shows total velocity histories of the buoy, and Figure NS66-6 shows the tension
histories of the cable.
In
de
x
1.
8-122
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NOTES:
1.
Problem NS66A is a static problem, so all parameters regarding hydrodynamic
analysis are assigned as zero.
2.
To obtain the static equilibrium position of the immersed structure, it is suggested the loading start from a small value with an auto-stepping algorithm and
a regular Newton-Raphson method is used.
3.
To simulate the sea bed contact for the cable-buoy system, several gap elements
are arranged near the sea bed. The sea bed contact is detected by those ‘closed’
gaps.
4.
Problem NS66B starts from an initial configuration which is the static equilibrium position of the structure, and during the analysis acceleration of gravity is
unchanged.
5.
Tension in Figure NS66-6 is the force in the element x-direction (‘Positive’ sign
means tension) excluding the hydrostatic pressure.
Figure NS66-2. Static Equilibrium Position of the Cable-Buoy System
Z
X
REFERENCE
GAP
In
de
x
COSMO
S
COSMOSM Advanced Modules
8-123
Chapter 8 Verification Problems
Figure NS66-3. Comparison of the Horizontal Position Histories of the Buoy
REFERENCE
COSMOS/M
V
E
R
T
I
C
A
L
P
O
S
I
T
I
O
N
(IN)
TIME SEC
Figure NS66-4. Comparison of the Vertical Position Histories of the Buoy
REFERENCE
COSMOS/M
V
E
R
T
I
C
A
L
P
O
S
I
T
I
O
N
(IN)
In
de
x
TIME SEC
8-124
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS66-5. Comparison of Total Velocity Histories of the Buoy
V
E
R
T
I
C
A
L
REFERENCE
COSMOS/M
P
O
S
I
T
I
O
N
(IN/
SEC)
TIME SEC
In
de
x
Figure NS66-6. Comparison of the Tension Histories of the Cable
COSMOSM Advanced Modules
8-125
Chapter 8 Verification Problems
NS67: Static Equilibrium Position
and Dynamic Analysis of a Steep S
Model Riser with Fixed Top End
TYPE:
Nonlinear hydrostatic and hydrodynamic analysis, large deflection, IMPIPE and
BUOY elements.
REFERENCE:
Rumbod Ghadimi, “A Simple and Efficient Algorithm for the Static and Dynamic
Analysis of Flexible Marine Riser,” Computer & Structures, Vol.29, No.4, pp. 541555, 1988.
PROBLEM:
NS67A)
Find the static equilibrium position of a steep S model riser. The riser
is partially immersed and subjected to the gravity and buoyant forces
as shown in Figure NS67-1.
NS67B)
Find the dynamic response of a steep S model riser in (NS67A). The
riser is subjected to the hydrodynamic force due to the wave.
Figure NS67-1. Static Equilibrium Position of the Steep S Model Riser
Z
SWL
A
X
Y
MODEL
RISER
B
In
de
x
SEA BED
8-126
H
BUOY
COSMOSM Advanced Modules
C
Part 1 NSTAR / Nonlinear Analysis
GIVEN:
Riser:
Total length of riser
= 3.97 m
A
= 1.48 m
B
= 2.58 m
C
= 0.79 m, EA = 3.012E3 N
EI
= 9.88E-5 N • m2
Mass
= 0.0836 Kg/m
Outside diameter
= 0.006 m
Coefficient of normal drag
= 1.5
Coefficient of added mass and inertia = 2
Buoy:
Mass
= 0.1 Kg
Outside diameter
= 0.1 m
Coefficient of drag
= 1.5
Coefficient of added mass and inertia = 2
Water depth
= 2.13 m
Water density
= 1000. Kg/m3
Wave height
= 0.255 m
Wave period
= 1.69 sec
RESULTS:
1.
The static equilibrium position of the riser is shown in Figure NS67-1.
2.
For the dynamic analysis, the node near SWL (ND = 6) is selected:
The horizontal and vertical displacement histories are shown in Figures NS67-2
and NS67-3. The results from the reference are listed below for comparison:
1.69 sec
100.0 mm
20.0 mm
In
de
x
The period of dynamic response
Amplitude of horizontal displacement
Amplitude of vertical displacement
COSMOSM Advanced Modules
8-127
Chapter 8 Verification Problems
NOTES:
1.
The wave length in (NS66B) is not input but computed in the program automatically.
2.
Please refer to NS66 for more notes.
Figure NS67-2. Horizontal Displacement History of Node-6
V
E
R
T
I
C
A
L
TIME INCREMENT = 0.005 SEC
D
I
S
P
L
A
C
E
M
E
N
T
M
TIME
Figure NS67-3. Vertical Displacement History of Node-6
H
O
R
I
Z
O
N
T
A
L
TIME INCREMENT = 0.005 SEC
D
I
S
P
L
M
In
de
x
TIME
8-128
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS68: Steady-State Analysis of a Cable
Towing a Submerged Body
TYPE:
Nonlinear hydrostatic analysis, large deflection, IMPIPE elements.
REFERENCE:
Anthony R. Rizzo, “FE Analysis Simulates Undersea Structures,” Mechanical
Engineering, pp. 51-54, August 1991.
PROBLEM:
Determine the peak tension in a towed cable and the depth of the towed body as
functions of the ship’s speed (see Figure NS68-1).
Figure NS68-1
SWL
SHIP SPEED
CABLE
TOWED
BODY
GIVEN:
In
de
x
Cable:
Total length of cable
= 1000 ft
E
= 144E7 psf
ν
= 0.35
Specific gravity
= 1.5
Outside diameter
= 1 in
Coefficient of normal drag
= 2.5
Coefficient of tangential drag
= 0.02
COSMOSM Advanced Modules
8-129
Chapter 8 Verification Problems
Towed body:
Weight in water
= 200 lb
Characteristic area
= 1 ft2
Coefficient of drag
=1
Water depth
= 2000 ft
Water density
= 1.9908 lb sec2/ft4
Ship speed
= 0.5, 1, 1.5, and 2 ft/sec
MODELING HINTS:
1.
The towed body is represented as two forces acting on the cable end: a vertical
force representing the weight of the towed body in water and a horizontal force
representing the drag force on the towed body (see Figure NS68-2).
2.
The ship motion is simulated with a uniform current past the cable (see Figure
NS68-2).
3.
Consider the current velocity as a function of time, such that a static analysis is
performed instead of a dynamic analysis.
4.
In order to use cable option (Op. No. 1 = 1) in IMPIPE elements, a small prestrain (Real constant-17) is applied to eliminate the difficulty at the beginning of
the analysis.
Figure NS68-2
Z
SWL
X
CURRENT
VELOCITY
DRAG
FORCE
In
de
x
WEIGHT
IN WATER
8-130
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
RESULTS:
Figure NS68-3 shows the cable configuration corresponding to different current
velocities. Figures NS68-4 and NS68-5 show the depth of the towed body and the
peak tension of the cable as a function of the tow velocity. Results from the current
analysis and the reference one are very coincident.
Figure NS68-3
SWL
ORIGINAL
CONFIGURATION
2.0
1.5
1.0
VELOCITY = 0.5 FT/SEC
Figure NS68-4
D
E
P
T
H
F
T
In
de
x
SHIP SPEED FT/SEC
COSMOSM Advanced Modules
8-131
Chapter 8 Verification Problems
Figure NS68-5
P
E
A
K
C
A
B
L
E
T
E
N
S
I
O
N
( LB)
In
de
x
SHIP SPEED FT/SEC
8-132
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS69: 3D Large Deflection Analysis
of a 45° Circular Bend
TYPE:
Nonlinear static analysis, large deflection, 3D BEAM elements.
REFERENCE:
Klaus-Jürgen Bathe and Saïd Bolourchi, “Large Displacement Analysis of ThreeDimensional Beam Structures,” IJNME, Vol. 14, pp. 961-986, 1979.
PROBLEM:
Find the large displacement response of a cantilever 45-degree bend subjected to a
concentrated end load.
GIVEN:
The bend has an average radius R, cross-section area H*B and lies on the X-Y plane
as shown in Figure NS69-1. The concentrated tip load P is applied into the zdirection.
R
= 100 in
H
= 1 in
B
= 1 in
E
= 1E7 psi
ν
=0
Figure NS69-1
z
Fixed
End
z
B
x
1
y
2
3
4
1"
5
6
8 Equal Straight BEAM
Elements
7
1"
8
9
Beam
Cross-section
In
de
x
R=100"
A
COSMOSM Advanced Modules
8-133
Chapter 8 Verification Problems
RESULTS:
Figure NS69-2 shows the deformed configurations of the bend at various load levels.
The tip coordinates predicted by COSMOSM and ADINA are listed in Table
NS69-1 for comparison.
Table NS69-1. Tip Coordinates of a 45-Degree Circular Bend.
P (lb)
ADINA (X, Y, Z) (in)
COSMOSM (X, Y, Z) (in)
300
(22.5, 59.2, 39.5)
(21.9, 59.0, 40.2)
600
(15.9, 47.2, 53.4)
(15.2, 47.4, 53.5)
Figure NS69-2
FINAL
CONFIGURATION
Z
A
P = 600 LB
A
P =300 LB
X
Y
ORIGINAL
CONFIGURATION
A
In
de
x
P = 0 lb
8-134
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS70: Static Analysis of an Elastic Dome
TYPE:
Nonlinear static analysis, large deflection, 3D BEAM elements.
REFERENCE:
B. A. Izzuddin, A. S. Elnashai and P. J. Dowling, “Large Displacement Nonlinear
Dynamic Analysis of Space Frames,” Eurodyn 90, Bochum, W. Germany, June
1990.
PROBLEM:
Find the static response of an elastic dome subjected to a concentrated load at the
crown point.
GIVEN:
The top and side views of an elastic dome are shown in Figure NS70-1 and Figure
NS70-2. Six 3D BEAM elements per member are used to model the dome with
the beam cross-section area as H*B for all the members where H = 1.22 m and
B = 0.76 m.
E
= 20690 MN/m2
ν
= 0.3
Figure NS70-1
Y
12.57M
Z
X
10.385M
21.115
In
de
x
6.283M
COSMOSM Advanced Modules
8-135
Chapter 8 Verification Problems
Figure NS70-2
P
Z
1.55m
4.55m
Y
X
12.18m
24.38m
NOTES:
Only one quarter of the dome is modeled due to symmetry. Appropriate boundary
conditions and cross-section modulus are used along the lines of symmetry. The
displacement control algorithm is used in the iteration procedure to avoid
convergence problems.
RESULTS:
Figure NS70-3 illustrates load-displacement curves of the elastic dome where ref-1,
-2 and -3 represent the following references:
ref-1:
Izzuddin, et. al. (1990),
ref-2:
Kondoh, et. al. (1986),
ref-3:
Shi and Atluri (1988).
In
de
x
“Load Factor” (L.F.) represents the applied load (P) divided by the critical load (Pcr)
where Pcr = 123.8 MN.
8-136
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS70-3
L
O
A
D
F
A
C
T
O
R
1 REF-1 AND -2
2 REF-3
3 COSMOS/M
In
de
x
DISPLACEMENT (M)
COSMOSM Advanced Modules
8-137
Chapter 8 Verification Problems
NS71: Large Deflection Analysis of a Cantilever
Beam with Different Beam Cross-Sections
TYPE:
Nonlinear static analysis, large deflection, BEAM2/3D elements, and beam-sectiondefinition.
REFERENCE:
Ramm, E., “A Plate/Shell Element for Large Deflections and Rotations, in
Formulations and Computational Algorithm in Finite Element Analysis,” M.I.T.
Press, 1977.
PROBLEM:
A cantilever beam with different cross-sections subjected to an end moment.
GIVEN:
The problem sketch is shown in Figure NS71-1. Five (5) BEAM2/3D elements are
used in the analysis.
E
= 12,000 psi
ν
=0
L
= 100 in
A and I varied
Beam-Section-Type:
1
(Rectangular),
(NS71A)
2
Circular),
(NS71B)
3
(Pipe),
(NS71C)
4
(Box),
(NS71D)
(I-Section),
(NS71E)
6
(Trapezoidal),
(NS71F)
0
(User-defined)
(NS71G)
In
de
x
5
8-138
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS71-1
Y
Y
1
X
2
3
4
5
6
Y
X
M
L
M
Proble m S ke tch
Be a m Mode l
Y
1
Be a m S e ction
Type s
2
Y
H
z
z
R
B
H = 1.0 in.
B = 1.0 in.
3
4
Y
R = 0.5642 in.
Y
5
Y
TH
z
D
z
TB
H
H
z
TB
TH
T
B
D = 1.1416 in.
T = 0.4842 in.
H = 2.0 in.
B = 2.0 in.
6
B
TB = 0.134 in.
TH = 0.134 in.
Y
7
Y
2
B2
z
Yr
H = 5.0 in.
B = 2.25 in.
(17 pts. used)
1 17
16
3
4
H
Zr
z
R
15
14
5
13
6
B1
H = 1.3333 in. B1 = 1.0 in.
B2 = 0.5 in
TB = 0.1 in.
TH = 0.1 in.
T
12
7
11
8
9
10
R = 4.0 in.
T = 0.2 in.
RESULTS:
In
de
x
The normalized analytical solution is:
The numerical results by COSMOSM are close to the above solution.
COSMOSM Advanced Modules
8-139
Chapter 8 Verification Problems
NS72: Uniformly Loaded Elastoplastic Plate
TYPE:
Nonlinear static analysis, Plasticity, Kinematic Hardening, and SHELL4T element.
REFERENCE:
Foster Wheeler Corporation, “Intermediate Heat Exchanger for Fast Flux Test
Facility: Evaluation of the Inelastic Computer Program,” prepared for Westinghouse
ARD, Livingston, N. J., 1972.
PROBLEM:
Verify the accuracy of the result of an elastoplastic flat plate in the case of uniformly
non-proportional stressing. Kinematic hardening is used.
GIVEN:
The geometry and material model is shown in Figure NS72-1 where 2*2 SHELL4T
elements are used. Two edges are simply supported and all rotations are constrained
to be zero.
E
= 30E6 psi
ν
= 0.3
σy = 30, 000 psi
ET = 1.5E6 psi
= 1 in
H
= 0.001 in
In
de
x
L
8-140
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS72-1
H = THICKNESS
Stress, σ
Et
εx
L
1
σy
E
1
L
Geomet ry Model (2x2 SHELL4T)
Strain,
ε
St ress-St rain Curve
Loading History:
1.
The plate is loaded into the plastic range in uniaxial tension in the xdirection, unloaded slightly, and reloaded.
2.
Biaxial loading then follows, with σx and σy prescribed, as shown in Figure
NS72-2, so that the effective stress remains constant as 40,000 psi.
Figure NS72-2
In
de
x
σx
COSMOSM Advanced Modules
8-141
Chapter 8 Verification Problems
Loading History:
Loading Step
σx psi
1
30,000
2
32,500
3
35,000
4
37,500
5
40,000
6
40,500
7
40,000
7 through 84
(σx2 + σy2 – σx σy)1/2 = 40,000
RESULTS:
The σx versus ε x plot is shown in Figure NS72-3. The agreement with analytical
solution is very close.
Figure NS72-3
ANALYTICAL SOLUTION
COSMOS/M
σx
S
T
R
E
S
S
K
S
I
In
de
x
εx
8-142
COSMOSM Advanced Modules
STRAIN * E-3
Part 1 NSTAR / Nonlinear Analysis
NS73: Viscoelastic Rod/Bar Subjected
to Constant Axial Load
TYPE:
Nonlinear static analysis, Time domain linear viscoelasticity, auto-stepping.
NS73A-B) PLANE2D elements
NS73C)
SOLID elements
NS73D)
TRUSS elements
NS73E)
BEAM elements
NS73F)
SHELL4T elements
REFERENCE:
R. M. Christensen, “Theory of Viscoelasticity – an Introduction,” 2nd Ed., 1982, pp.
1-76.
PROBLEM:
A rod/bar is fixed in the axial direction on one end and a constant axial load is
suddenly applied to the other end (shown in Figure NS73-1). The rod/bar is made up
of a linear viscoelastic material.
GIVEN:
The material model is shown in Figure NS73-2 where the linear viscoelastic material
is represented by a combination of linear springs and a dashpot. The extensional
relaxation function is
where
Extensional relaxation moduli:
E∞ = 1,000 psi
In
de
x
E1 = 9,000 psi
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
Extensional relaxation time:
τ1E = 1 sec
Bulk modulus:
K
= 100,000 psi (K0 = K∞)
The time-dependent material behavior inside the code is approximated with a
generalized Maxwell model:
Shear relaxation modulai:
G0 = 3K0 E0 / (9K0 – E0)
= 3370.7865 psi
G∞ = 3K∞ E∞ / (9K∞ – E∞) = 333.7041 psi
G1 = G 0 – G∞
= 3037.0824 psi
Shear relaxation time: τ1G
The input material parameters are:
E0
= E∞ + E1
υ0
= (3K0 – E0) / 6K0 = 0.4833
g1
= Gi / G0
τ1
= 0.9010
= 0.9899 sec
= 10 in
In
de
x
L
G
= 10,000 psi
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COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
For a rod:
D
= 1 in
For a bar:
B
= 3.14159 in
H
= 0.25 in
Such that
A
= π/4 in2
A Heaviside loading function is shown in Figure NS73-2. To capture the
instantaneous material behavior, a tiny time step (0.001 sec) is used in the beginning
along with the auto-stepping algorithm during which the rod/bar is relaxed toward
its long time behavior. Tolerance for creep strain increment CETOL = 5E-4.
Figure NS73-1
P (t)
2 D-P lane S tre ss (2 x2 ) S HE LL4 T (2 x2 )
L
P (t)
Truss and Be am
H
B
In
de
x
B
P lane 2
COSMOSM Advanced Modules
8-145
Chapter 8 Verification Problems
Figure NS73-2
E1
η
1
P t = 100
F
P (t)
8
E
τ
E
1
=
η1
η = Dashpot
1
E1
E 1 = Linear Spring
0.001
60
t (se c)
Loading Function
Mate rial Mode l
RESULTS:
Analytical Solution:
For a prescribed stress problem, the strain is determined by the current value and past
history of stress:
where J(t) is termed creep function. Taking the form of the generalized Kelvin
model:
In
de
x
where:
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COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Comparison:
Instantaneous behavior:
υ0 = 0.4833
COSMOSM:
υ (t = 0.001 sec) = – ε22 / ε11 = 0.4833
Long term behavior:
υ∞ = 0.4983
COSMOSM:
υ (t = 50 sec) = – ε22 / ε11 = 0.4983
Time histories of the axial strain from COSMOSM along with the analytical solution
are plotted in Figure NS73-3 for comparison.
Figure NS73-3
A
X
I
A
L
S
T
R
A
I
N
Reference
COSMOS/M
In
de
x
TIME
COSMOSM Advanced Modules
8-147
Chapter 8 Verification Problems
NS74: Transient Thermal Loading
of a Viscoelastic Slab
TYPE:
Transient thermal analysis, Nonlinear static analysis, Thermal loading, Time domain
linear viscoelasticity, Temperature-time shift, PLANE2D elements, auto-stepping.
REFERENCES:
1.
Carslaw, H. S., and Yeager, J. C., “Conduction of Heat in Solids,” Clarendon
Press, Oxford, 1959.
2.
Williams, M. L., Landel, R. F., Ferry, J. D., J. American Chemical Society, V77,
pp. 3701, 1955.
PROBLEM:
A viscoelastic slab under plane strain restraint in all directions in its plane is
subjected to a temperature loading on its faces (see Figure NS74-1). Investigate the
response of the slab corresponding to different time values.
GIVEN:
Model:
The half-thickness (H/2 = 1 in) of the slab is modeled with a row of 8-node
PLANE2D (plane strain) elements (see Figure NS74-1) for the heat transfer
analysis. The same elements are then used for the nonlinear static analysis under the
temperature loading where plane strain condition is imposed in the y-direction and
symmetry is imposed about x = 0.
Loading:
In
de
x
The outside face of the slab is prescribed a uniform temperature. The temperaturetime curve is a Heaviside function with T0 = 100 °F (Figure NS74-1).
8-148
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Part 1 NSTAR / Nonlinear Analysis
Material:
1.
Thermal material properties:
Thermal conductivity:
k
= 1 Heat/sec • in °F
Specific heat:
c
= 1 Heat / (lb • sec2/in) °F
Density:
ρ
2.
3.
= 1 lb sec2/in4
Elastic material properties:
E
= 10,000 psi
ν
= 0.4833
α
= 1E-5/ °F
Viscoelastic material properties:
Relative shear relaxation modulus:
g1 = 0.9
Shear relaxation time:
τ1G = 0.9899 sec
4.
Temperature-time shift function:
Williams-Landell-Ferry approximation is applied:
c1 = 4.92
c2 = 215° F
In
de
x
θ0 = 70° F
COSMOSM Advanced Modules
8-149
Chapter 8 Verification Problems
Figure NS74-1
Temperature Loading on Surfaces 5 and 6
UY = 0 on Surfaces 7 & 5
UZ = 0 on Surfacss 8 & 10
H/2
9
Plain
Strain
10
Plain
Strain
6
8
Y5
Line
of
Symmetry
Temp.
Loading
y
H
7
Z
X
PROBLEM SKETCH
T
E
M
P
E
R
A
T
U
R
E
T
z
X
FINITE ELEMENT MODEL: 10+1 PLANE 2D
s
TIME
ANALYSIS AND RESULTS:
Analysis 1:
The transient heat transfer analysis is performed. To capture the rapid temperature
changes at the start of transient, a tiny time step (0.0005 sec) is used in the beginning.
The time step is then increased gradually for a time period of 6.0 sec, during which
the slab is allowed to reach its thermal equilibrium condition.
Analysis 2:
In
de
x
Reading the additional data from the second input file, the nonlinear static analysis
is performed. The temperature loading is prescribed by reading the temperature
distribution previously written in “.HTO” file. An auto stepping algorithm is used
with tolerance for creep strain increment CETOL = 5E-5. Note: During the autostepping, temperatures at any time are obtained by interpolation.
8-150
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Analytical Solution and Comparison:
1.
Temperature distribution at t = 1 sec:
x, in
Node
Analytic Solution°F
COSMOSM°F
0.9
51
98.2
98.276
0.7
31
95.0
94.994
0.5
16
92.2
92.203
0.2
5
89.5
89.506
0
4
89.2
88.963
2.
The stress and strain distributions at various times during the analysis are shown
in Figure NS74-2 and Figure NS74-3.
3.
Long term behavior:
4.
The stress and strain distributions at the end of nonlinear static analysis (t = 6.0
sec) can be compared with those of an elastic slab with material properties to be
the long-term viscoelastic material properties.
E
= E∞ = 1,000 psi
ν
= ν∞ = 0.4983
εxx = (σxx – νσyy – νσzz) / E + α T
εyy = (σyy – νσxx – νσzz) / E + α T
Symmetry:
σyy = σzz
Plane strain:
εyy = εzz = 0
Unrestrained condition in the x-direction:
σxx = 0
so:
COSMOSM
σyy = – E α T / (1 – ν) = 1.9932 psi
2.986E3
In
de
x
εxx = (1 + ν) α T / (1 – ν) = 2.9864E – 3
–1.995 psi
COSMOSM Advanced Modules
8-151
Chapter 8 Verification Problems
Figure NS74-2
4
3
σ yy
(psi)
1
2
3
4
1
T=0.009 sec
T=0.1 sec
T=1.2 sec
T=6.0 sec
2
X, INCH
Figure NS74-3
4
3
2
ε xx
(*E-3)
1
2
3
4
T=0.009 sec
T=0.1 sec
T=1.2 sec
T=6.0 sec
1
In
de
x
X, INCH
8-152
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS75: Transient Response of a Viscoelastic
Cylinder Under Torsional Oscillation
TYPE:
Nonlinear static analysis, Displacement control algorithm, Time domain linear
viscoelasticity, BEAM3D elements, auto stepping.
REFERENCES:
Christensen, R. M., “Theory of Viscoelasticity – an Introduction,” 2nd Ed., 1982, pp.
48-52.
PROBLEM:
A viscoelastic right circular cylinder has one end fixed and the other end prescribed
an harmonic twisting angle (see Figure NS75-1a and NS75-1b). Find the twisting
moment required to produce the given oscillation
Figure NS75-1a
z
e (t)
R1
Ro
5
4
4
3
y
H
2
1
1
x
θ
FINITE ELEMENT MODEL
In
de
x
PROBLEM SKETCH
(t)
COSMOSM Advanced Modules
8-153
Chapter 8 Verification Problems
Figure NS75-1b
T
θ
t
TWISTING ANGLE-TIME CURVE
GIVEN:
Four (4) BEAM3D elements are used to model the cylinder (see Figure NS75-1)
where H = 1.0 in, r0 = 0.5 in and ri = 0.125 in The prescribed twisting angle is an
harmonic function having the form:
Θ (t) = θ sin wt
w/2π = 1/T
where T = 50 sec and θ = 2.075E – 3 rad. The twisting angle-time curve is shown in
Figure NS75-1.
Material properties:
= 10,000 psi
ν
= 0.4205
G
= 3,520 psi
In
de
x
E
8-154
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Shear relaxation moduli:
i
gi
τiG, sec
1
0.2832
1.5E–5
2
0.1528
1.5E–4
3
0.1403
1.5E–3
4
0.1114
1.5E–2
5
0.0869
1.5E–1
6
0.0438
1.5
7
0.0338
1.5E1
8
0.0057
1.5E2
Tolerance for creep strain increment CETOL = 5.E-5
RESULTS:
The torque-twisting angle plot is shown in Figure NS75-2 for a total period of time
100 sec (two cycles). The analytical solution is also enclosed for comparison. It is
noted that two or three cycles of twisting are generally required for the transient part
of the response to become ignorable in comparison with the steady-state response. It
is important that this transient response does not arise from inertia effects at this low
frequency, but rather due to the fading memory nature of the material.
Figure NS75-2
T
O
R
Q
U
E
1ST CYCLE
2ND CYCLE
L
B
*
I
N
ANALYTICAL
SOLUTION
COSMOS/M
In
de
x
ANGLE OF TWISTING (RAD *E-3)
COSMOSM Advanced Modules
8-155
Chapter 8 Verification Problems
NS76: Extension of an Ogden
Hyperelastic Bar/ Sheet
TYPE:
Nonlinear static analysis, displacement control algorithm, Ogden hyperelastic
material model
NS76A)
PLANE2D–plane stress elements
NS76B)
SOLID elements
NS76C)
PLANE2D–plane strain elements
NS76D)
SHELL4T elements
PROBLEM:
NS76A-B, D)An Ogden hyperelastic bar is subjected to an axial load (see Figure
NS76-1), find the response of the nominal stress versus principal
stretch.
NS76C)
An infinitively long sheet of Ogden hyperelastic material is
subjected to a uniform extension load (see Figure NS76-1), find the
response of the nominal stress versus principal stretch.
Figure NS76-1
D
L
PROBLEM SKETCH 2
50
E-E
Ux
L
0
0
50
DISPLACEMENT - TIME CURVE
In
de
x
PROBLEM SKETCH 1
8-156
COSMOSM Advanced Modules
E-E
Part 1 NSTAR / Nonlinear Analysis
GIVEN:
Problem–1:
L
= 0.1 m
B
= 0.01 m
H
= 0.005 m
Modeled with 10*1 PLANE2D–plane stress, 20*2 SOLID and 1 x 1 SHELL4T
elements.
Problem–2:
L
= 10 cm
B
= 0.5 cm
Modeled with 10*1 PLANE2D–plane strain elements.
Ogden material model constants:
i
αi
µi, MPa
1
1.3
0.618
2
5.0
0.001245
3
–2.0
–0.00982
For PLANE2D–plane strain and SOLID elements, ν = 0.499, for PLANE2D – plane
stress and SHELL4T ν is not needed to input (ν = 0.5 is assumed internally).
A displacement control algorithm is used along the axial direction by following the
time curve as shown in Figure NS76-1 where the maximum displacement is five (5)
times the initial length (L).
RESULTS:
Analytical solution:
Problem 1:
In
de
x
2nd p. k. stress in the principal direction:
COSMOSM Advanced Modules
8-157
Chapter 8 Verification Problems
where
λ1
= L/L0
λ3
= λ1-0.5
L
= Final length
L0
= Initial length
Nominal Stress t1p:
t1p
=
F
= Force
A0
= Initial area
λ1 = F/A0 = p
Problem 2:
where
λ1
= L/L0
λ2
= λ1-1
λ3
=1
Nominal Stress t1p:
t1p
=
λ1 = F/A0 = p
In
de
x
The nominal stress-principal stretch curves for both examples are shown in Figure
NS76-2 and Figure NS76-3. The results by using the Mooney-Rivlin hyperelastic
material model are also enclosed for comparison where Mooney-Rivlin material
constants: C1 = 0.11026 MPa and C2 = 0.09708 MPa. It is noted that the results by
Ogden and Mooney–Rivlin models are close only in a small range of principal
stretch.
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COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure NS76-2
1 2 &3
N
O
M
I
N
A
L
OGDEN
1 ANALYTICAL
2 COSMOS/M PLANE STRESS
3 COSMOS/M 3D SOLID
4 MOONEY-RIVLIN
4
S
T
R
E
S
S
PRINCIPAL STRETCH
Figure NS76-3
N
O
M
I
N
A
L
OGDEN
1 ANALYTICAL
2 COSMOS/M PLANE STRESS
3 MOONEY-RIVLIN
1 &2
4
S
T
R
E
S
S
In
de
x
PRINCIPAL STRETCH
COSMOSM Advanced Modules
8-159
Chapter 8 Verification Problems
NS77: Snap-Through/Snap-Back of a Thin Hinged
Cylindrical Shell Under a Central Point Load
TYPE:
Nonlinear static analysis, SHELL4 elements, Arc-length control, automatic
stepping.
NS77A)
Ignore the effect of gravity (structure is assumed weightless)
NS77B)
Include the effect of gravity
REFERENCE:
Crisfield, M. A., “A Fast Incremental/Iterative Solution Procedure That Handles
Snap-Through,” Computers & Structures, Vol. 13, pp. 55-62, 1981.
PROBLEM:
Determine the snap-through/snap-back response of a shallow cylindrical shell under
a concentrated load (P) at the center of the shell. The curved edges are free and the
straight edges are hinged and immovable.
GIVEN:
R
= 2,540 mm
b
= 254 mm
θ
= 0.10 rad
ν
= 0.30
h (thickness)
= 6.35 mm
P (reference load) = 10 N
MODELING HINTS:
Due to symmetry, a 4 x 4 mesh is used to model a quarter of the shell.
Arc-length Control Information:
Max. load parameter (approx. value)
= 100
Max. Displacement (approx, value)
= 30
Max. number of arc steps
= 50
In
de
x
Desired average number iterations/step = 5
Initial load parameter
8-160
COSMOSM Advanced Modules
= 10
Part 1 NSTAR / Nonlinear Analysis
Unloading check flag
=0
Arc-length step adjustment coefficient = 0.5
Automatic Stepping Information:
Min. (arc) step increment
= 1.E-8
Max. (arc) step increment
= 30 for NS77A and 5 for NS77B
Additional Hints for NS77B:
The central point force is associated with time curve 1 and the acceleration of gravity
is associated with time curve 2.
First, force control is used to obtain displacements under gravity loading. During this
phase, curve 2 is raised to 1.0 while curve 1 is kept at zero.
Next, control is changed to the Arc-length algorithm with active restarting to find the
response under the effect of the central force. (Time curve definitions are ignored
during this phase).
RESULTS:
The curve of the load multiplier factor (LFACT) versus central deflection is shown
in the next two figures.
Figure NS77A-1
p
1
θ
2b
In
de
x
R
COSMOSM Advanced Modules
8-161
Chapter 8 Verification Problems
In
de
x
Figure NS77B-1
8-162
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS78: Multiple Snap-through/Snap-back
of a Thick Hinged Cylindrical Shell
Under a Central Point Load
TYPE:
Nonlinear static analysis, SHELL4T elements, Arc-length control, automatic
stepping.
REFERENCE:
Tsai, C. T., and Palazotto, A. N., “Nonlinear and Multiple Responses of Cylindrical
Panels Comparing Displacement Control and Riks Method,” Computers &
Structures, Vol. 41, pp. 605-610, 1991.
PROBLEM:
Determine the multiple snap-through/snap-back response of a shallow cylindrical
shell under a concentrated load (P) at the center of the shell. The curved edges are
free and the straight edges are hinged and immovable.
GIVEN:
R
= 2,540 mm
b
= 254 mm
θ
= 0.20 rad
ν
= 0.30
h (thickness)
= 12.70 mm
P (reference load) = 10 N
MODELING HINTS:
Due to symmetry, a 4 x 12 mesh is used to model a quarter of the shell.
Arc-length Control Information:
Max. load parameter (approx. value)
= 1,000
Max. Displacement (approx. value)
= 150 mm
Max. number of arc steps
= 140
In
de
x
Desired average number iterations/step = 5
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
Initial load parameter
= 10
Unloading check flag
=0
Arc-length step adjustment coefficient= 0.50
Automatic Stepping Information:
Min. (arc) step increment
= 1 x 10E-8
Max. (arc) step increment
= 150
RESULTS:
The curve of the load multiplier factor (LFACT) versus central deflection is shown
in the next figure.
Figure NS78-1
p
1
θ
2b
In
de
x
R
8-164
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
NS79: Large Displacement Nonlinear
Static Analysis of a Cantilever Beam
Subjected to a Prescribed End Rotation
TYPE:
Nonlinear static analysis, large displacement, BEAM2D elements, prescribed
displacement associated with time curve, Force control, automatic stepping.
REFERENCE:
Ramm, E., “A Plate/Shell Element for Large Deflection and Rotations,” in
Formulations and Computational Algorithms in Finite Element Analysis. [M.I.T.
Press, 1977].
PROBLEM:
Determine the deformed shape of the beam.
GIVEN:
L
= 100 in
I
= 0.01042 in4
A
= 0.50 in2
E
= 12,000 psi
ν
=0
h
= 0.5 in
Prescribed end rotation = 6.28 rad
Automatic Stepping Information:
Min step increment
= 1.E-8
Max. step increment
= 0.02
RESULTS:
In
de
x
The deflected shape of the beam is shown in Figure NS79-2. Also, the horizontal and
vertical displacements at the tip of the beam are shown in Figure NS79-3.
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
Figure NS79-1
p
1
θ
h
b
L
1
Problem Sketch
1
2
1
3
2
Time Curve
4
3
5
4
6
5
7
6
8
7
Finite Element Model
Figure NS79-2
In
de
x
DEFLECTED SHAPE
OF THE BEAM
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8
10
9
10
11
θ
t
Part 1 NSTAR / Nonlinear Analysis
In
de
x
Figure NS79-3
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
NS80: Rubber O-Ring Squeezed Between
Two Parallel Steel Plates
TYPE:
Nonlinear static analysis, Contact with friction, Mooney-Rivlin model, Large
Deflection formulation, Displacement Control, PLANE2D Axisymmetric elements.
PROBLEM:
A rubber O-ring is squeezed between two steel plates. The plates travel a relative
distance of 0.0347" towards each other. Determine stresses in the ring, and total
force required:
a) Coefficient of friction = 0.01
b) Coefficient of friction = 0.5
MODELING:
Due to symmetry, half of the ring cross section and one of the plates are modeled.
Two soft springs are used to hold the plate, to prevent singularity of the stiffness
matrix. Uniform pressure with a magnitude of one is applied to the top surface of the
plate.
While in part (a), the O-ring expands (slides), in part (b), friction is large enough to
prevent the ring from sliding. Therefore, in part (b), contact with generalized friction
is required. Part (a) can use either general or sliding friction options. However, the
sliding option is faster in dealing with sliding problems.
Figure NS80-1
Top Steel Plate
Bottom Steel Plate
In
de
x
/ = 1. 123"
8-168
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0.3"
0. 3"
. 139"
0. 3475
Part 1 NSTAR / Nonlinear Analysis
PROPERTIES:
1.
2.
O-Ring:
C1
= 175 psi
C2
= 10 psi
Poisson’s ratio
= 0.49
Plates:
Young’s Modulus = 30E6 psi
Poisson’s ratio
3.
= 0.3
Soft Trusses
Young’s Modulus = 1 psi
Area
= 1 in
Length
= 0.3 in
RESULTS:
In both cases, 12 solution steps are used to attain the prescribed plates’ relative
displacement.
Case (a)
Case (b)
Fric. Opt. = 1
36.067
36.067
Fric. Opt. = 1
46.124
Load Factor (per radian) LF • D/8
14.90 lb/rad
14.90 lb/rad
19.056 lb/rad
Analysis Time (seconds)
592 sec
671 sec
452 sec
In
de
x
Fric. Opt. = 2
Load Factor (per radian)
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
Figure NS80-2
In
de
x
Figure NS80-3
8-170
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Part 1 NSTAR / Nonlinear Analysis
NS81: Extension of a Blatz-Ko Hyperelastic Bar
TYPE:
Nonlinear static analysis, displacement control algorithm, Blatz-Ko hyperelastic
material model
NS81A)
PLANE2D plane stress elements
NS81B)
SOLID elements
REFERENCE:
J. K. Knowles and Eli Sternberg, “On the Ellipticity of the Equations of Nonlinear
Elastostatics for a Special Material,” Journal of Elasticity, Vol. 5, Nos. 3-4, 1975, pp.
341-361.
PROBLEM:
A Blatz-Ko hyperelastic bar is subjected to an axial load (see Figure NS81-1).
Determine the variations of nominal stress (force per unstressed area) versus
principal stretch.
MODELING HINTS:
A displacement control algorithm is used along the axial direction where the
maximum displacement equals the initial length (L).
GIVEN:
L
= 0.1 m
B
= 0.01 m
H
= 0.005 m
E
= 2.0732 Mpa
RESULTS:
The analytical solution for 2nd P.K. stress in the principal direction has the following
form:
In
de
x
S1 = µ λ1-2 (−λ1-2 + J)
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
where
J
= λ1 λ2 λ3
λi
= Stretch ratio in the i-th direction
λ1 = L/L0 (final length/initial length)
since
S2 = S3 = 0 → λ2 = λ3 = λ1-1/4
thus
S1 = µ λ1-2 (-λ1-2 + λ1-1/2)
The nominal stress is:
F/A = S1 λ1
where F and A represent the applied force and the initial area respectively. Figure
NS81-3 shows the nominal stress versus principal- stretch curve.
Figure NS81-1
E-E
L
P
In
de
x
PROBLEM SKETCH
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E-E
Part 1 NSTAR / Nonlinear Analysis
Figure NS81-2
D
I
S
P
L
A
C
E
M
E
N
T
/
L
o
DISPLACEMENT - TIME CURVE
Figure NS81-3
N
O
M
I
N
A
L
S
T
R
E
S
S
ANALYTICAL
COSMOS/M (PLAIN STRESS)
In
de
x
PRINCIPAL STRETCH
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Chapter 8 Verification Problems
NS82: Extension of a Blatz-Ko Hyperelastic Sheet
TYPE:
Nonlinear static analysis, displacement control algorithm, Blatz-Ko hyperelastic
material model, PLANE2D plane strain elements.
REFERENCE:
J. K. Knowles and Eli Sternberg, “On the Ellipticity of the Equations of Nonlinear
Elastostatics for a Special Material,” Journal of Elasticity, Vol. 5, Nos. 3-4, 1975, pp.
341-361.
PROBLEM:
An infinitely long sheet of Blatz-Ko hyperelastic material is subjected to a uniform
extension load (see Figure NS82-1). Determine the variation of nominal stress
versus principal stretch.
GIVEN:
L
= 10 cm
B
= 0.5 cm
E
= 2.0732 Mpa
MODELING HINTS:
The sheet was modeled with 10*1 PLANE2D-plane strain elements. Displacement
is controlled in the direction of the applied force with a maximum displacement
equal to the initial length (L).
RESULTS:
The analytical solution for 2nd P.K. stress in the principal direction has the following
form:
S1 = µ λ1-2 (-λ1-2 + J)
where
J
= λ1 λ2 λ3
λi
= Stretch ratio in the i-th direction
In
de
x
λ1 = L/L0 (final length/initial length)
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since
S2 = 0,
λ3 = 1 → λ2 = λ1-1/3
thus
S1 = µ λ1-2 (-λ1-2 + λ1-2/3)
The nominal stress:
F/A = S1 λ1
where F and A represent the applied force and the initial area respectively. Figure
NS81-2 shows the nominal stress versus principal- stretch curve.
Figure NS82-1
B
P
L
PROBMEM STRETCH
Figure NS82-2
N
O
M
I
N
A
L
S
T
R
E
S
S
ANALYTICAL
COSMOS/M (PLAIN STRAIN)
In
de
x
PRINCIPAL SKETCH
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
NS83: Biaxial Tension of a Blatz-Ko
Hyperelastic Sheet
TYPE:
Nonlinear static analysis, displacement control algorithm, Blatz-Ko hyperelastic
material model, PLANE2D-plane stress elements.
REFERENCE:
J. K. Knowles and Eli Sternberg, “On the Ellipticity of the Equations of Nonlinear
Elastostatics for a Special Material,” Journal of Elasticity, Vol. 5, Nos. 3-4, 1975, pp.
341-361.
PROBLEM:
A square sheet of Blatz-Ko hyperelastic material is subjected to a uniform extension
load in both X- and Y- directions (see Figure NS83-1). Determine the variation of
nominal stress versus principal stretch.
GIVEN:
L
= 10 cm
B
= 0.1 cm
E
= 2.0732 Mpa
MODELING HINTS:
The sheet was modeled with one PLANE2D plane stress element. Displacement is
controlled such that the maximum displacement equals the initial length (L).
RESULTS:
The analytical solution for 2nd P.K. stress in the principal direction has the following
form:
S1 = µ λ1-2 (-λ1-2 + J)
where
J
= λ1 λ2 λ3
λi
= Stretch ratio in the i-th direction
In
de
x
λ1 = L/L0 (final length/initial length)
8-176
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Part 1 NSTAR / Nonlinear Analysis
since
S2 = 0,
Figure NS83-1
λ1 = λ2 → λ3 = λ1-2/3
P
thus
S1 = µ λ1-2 (-λ1-2 + λ1-4/3)
The nominal stress:
F/A = S1 λ1
where F and A represent the applied
force and the initial area. Figure
NS83-2 shows the nominal stress
versus principal- stretch curve.
L
P
L
PROBMEM STRETCH
Figure NS83-2
N
O
M
I
N
A
L
S
T
R
E
S
S
ANALYTICAL
COSMOS/M (PLAIN STRESS)
In
de
x
PRINCIPAL STRETCH
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Chapter 8 Verification Problems
NS84: Pure Bending of a Stretched
Rectangular Membrane
TYPE:
Nonlinear static analysis, wrinkling membrane material model, small deflection,
coupling degree of freedom, PLANE2D plane stress and BEAM2D elements.
REFERENCE:
Richard K. Miller and coworkers, “Finite Element Analysis of Partly Wrinkled
Membranes,” Computers and Structures, Vol. 20, No. 1-3, pp 631-639, 1985.
PROBLEM:
Consider a rectangular membrane which is uniformly pretensioned with normal
stress σo in the y-direction and with axial load P = σo th in the x-direction, as shown
in Figure NS84-1. After pretensioning, an in-plane bending moment M is applied
along the edges as shown. As M is increased, a band of vertical wrinkles of width b
forms along the lower edge of the membrane as the normal strain εxx in this region
becomes compressive. Find the nonlinear moment-curvature curve.
GIVEN:
Figure NS84-1
L/2 = 10 in
h
σo
= 5 in
t
= 0.1 in
(thickness)
σo = 1 psi
P
= σ0 ht = 5 lb
E
= 3.E7 psi
ν
= 0.3
h
P
TAUT REGION
Y
M
P
WRINKLED REGION
b
M
X
MODELLING:
σo
In
de
x
A finite element
model of this problem (only half of the membrane) was created by using PLANE2D
plane stress elements as shown in Figure NS84-2. The symmetric boundary
conditions are applied to the left edge of the model. The right edge is attached to a
column of very stiff BEAM2D elements on which the external loads P and M are
applied. In order to satisfy the displacement compatibility, some degrees of freedom
are coupled along the interface (CPDOF).
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The time-curves of the pretensioning force P and σo and the bending moment M are
shown in Figure NS84-3.
COMPARISON OF RESULTS:
If K denotes the overall curvature of the membrane acting as a beam, then the
analytical solution of the moment-curvature relation is given by:
where
The relation shows that for excessively large loads with
> 1, the entire surface is
wrinkled and instability results. the membrane then collapses.
The numerical solution from COSMOSM is shown in Figure NS84-4 for
comparison. Note that the curvature K =θ/L where θ is the rotation of the stiff beam.
Figure NS84-2
PRETENSION
STIFF BEAM
P
M
In
de
x
PRETENSION
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
Figure NS84-3
PRETENSION
V
A
L
U
E
BENDING MOMENT
TIME
Figure NS84-4
ANALYTICAL
COSMOS/M
_
K
In
de
x
_
M
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NS85: Shape-finding and Loading Analysis
of a Suspended Membrane
TYPE:
Nonlinear static analysis, wrinkling membrane material model, large deflection,
prescribed non-zero displacement, SHELL4T membrane elements, auto-stepping,
geometry updating.
REFERENCE:
Hideki Magara and Kiyoshi Okamura, “A study on Modeling and Structural
Behavior of Membrane Structures,” Shells, Membranes and Space Frames,
proceedings IASS Symposium, Osaka, 1986, Vol. 2, pp. 161-168.
PROBLEM:
Modeling and mechanical characteristics of a suspended wrinkling membrane in the
reference configuration and under the external loads are studied. The surface
geometry of the membrane is in equilibrium under the specified boundary condition
prestresses. After the equilibrium is reached, the external snow loads are uniformly
distributed on the surface. Compare the numerical results to the experimental results.
GIVEN:
Surface geometry:
z = x2/400 + y2/400
= 160 in
thickness
= 0.1 in
E
= 2.47E3 psi
ν
= 0.39
prestress
= 2 psi
snow loads
= 520 lb
In
de
x
span
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
MODELLING:
A finite element model of this problem was created by using SHELL4T membrane
elements as shown in Figure NS85-1.
1.
Shape-finding analysis:
The analysis starts with a flat surface geometry. Prescribed displacements are
applied to the nodes along boundary-1 to -4 to satisfy the equation shown above.
An auto-stepping algorithm is used with a tiny initial time increment to control
the ill-conditioned stiffness in the beginning of the analysis. Several runs are
performed until the stress distribution is almost uniform, i.e., the discrepancy is
within one percent tolerance of 2.0 psi. It is noted that each run starts with a
surface geometry obtained from the previous run (A_NONLINEAR) without the
use of the RESTART option.
2.
Loading analysis:
Adopting the surface geometry from the shape-finding analysis, a pressure
loading is then applied to it as specified by the associated time curve. The surface
geometry is no longer updated at the end of analysis.
COMPARISON OF RESULTS:
1.
Shape-finding analysis:
As illustrated in Figure NS85-2, the curves representing the bracing of the model
show good agreement between analytical, experimental, and numerical results.
Figure NS85-3 shows the three dimensional view of the final shape of the
membrane.
2.
Loading analysis:
In
de
x
The load-displacement curve in the saddle point and the load-tension curves in
the bracing and hanging are shown in Figures NS85-4 and NS85-5 respectively.
It is noted that in the load-tension curves, an increase in loading results in an
increase in tension in hanging but a greater decrease in the bracing direction due
to a partially wrinkled region along the bracing. This phenomena is found in both
experimental and numerical results.
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Figure NS85-1
BOUNDARY - 4
HANGING
MEMBRANE
Y
Z
BOUNDARY - 1
X
BOUNDARY - 2
BRACING
BOUNDARY - 3
Figure NS85-2
Z
C
O
O
R
D
I
N
A
T
E
ANALYTICAL
EXPERIMENT
COSMOS/M
(inch)
ALONG Y = 0
In
de
x
X COORDINATE (inch)
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
Figure NS85-3
Main DEF Step:13 = 1
DEFORMED SHAPE WITH PRESTRESS = 2.0 PSI
Figure NS85-4
D
I
S
P
L
A
C
E
M
E
N
T
EXPERIMENT
COSMOS/M
Y
X
IN SADDLE
POINT
(inch)
In
de
x
LOAD ( PSI )
8-184
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Figure NS85-5
EXPERIMENT
COSMOS/M
HANGING
T
E
N
S
I
O
N
HANGING
Y
( PSI )
X BRACING
BRACING
In
de
x
LOAD ( PSI )
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
NS86: Nonlinear Elasticity of a Bar
TYPE:
Nonlinear static analysis, nonlinear elastic material model (unsymmetric behavior in
tension and compression), small deflection, displacement control algorithm.
NS86A)
TRUSS3D
NS86B)
BEAM3D
NS86C)
SPRING (axial)
NS86D)
PLANE2D (plane stress)
NS86E)
SOLID
NS86F)
SHELL4T
PROBLEM:
A nonlinear elastic bar is subjected to a uniform pressure loading in the x-direction
as shown in Figure NS86-1. The stress-strain curve is given in Figure NS86-2. Note
that the material shows stronger resistance in compression than in tension. the
displacement in the x-direction is controlled such that the path of loading follows the
curve as shown in Figure NS86-3. Verify the corresponding load factor for each
element type listed above.
GIVEN:
L
= 10 in
B
= 1 in
t
= 1 in (thickness)
ν
= 0.3 (required for 2- and 3-dimensional elements)
COMPARISON OF RESULTS:
In
de
x
Figure NS86-4 shows the load-displacement curve for PLANE2D plane stress
elements. Other types of elements show similar results. Remember that in the 2- and
3-dimensional elements, a ratio R is used to calculate the elastic modulus by
interpolation. R is defined as:
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Part 1 NSTAR / Nonlinear Analysis
In the current case, R = 1 for the time in the range of (0,2) and R = -1 for the time in
the range of (2,4).
Figure NS86-1
PRESSURE
B
L
PROBMEM SKETCH AND FINITE ELEMENT MODEL (PLANE2D)
Figure NS86-2
TENSION
S
T
R
E
S
S
COMPRESSION
(psi)
In
de
x
STRAIN
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
Figure NS86-3
D
I
S
P
L
A
C
E
M
E
N
T
TENSION
COMPRESSIO
N
(IN)
TIME
Figure NS86-4
L
F
A
C
T
In
de
x
UX (INCH)
8-188
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NS87: Rubber Cylinder Pressed
Between Two Plates
TYPE:
Nonlinear static analysis, Mooney-Rivlin hyperelastic material model, large strain
and large deflection, prescribed displacements, coupling degrees of freedom,
reaction force calculation, auto-stepping, frictionless contact, PLANE2D plane
strain, displacement-pressure (u/p) elements.
PROBLEM:
A plane strain rubber cylinder is pressed between two frictionless plates, see Figure
87-1. Determine the force-deflection curve for the cylinder and the distribution of
the von Mises stresses when the applied displacement equals one-half of the initial
diameter of the cylinder.
REFERENCE:
T. Sussman and K. J. Bathe, “A Finite Element Formulation for Nonlinear
Incompressible Elastic and Inelastic Analysis,” Computers & Structures, Vol. 26,
No. 1/2, pp. 357-409, 1987.
GIVEN:
Diameter of cylinder D
= 0.4 m
Mooney-Rivlin constants A = 0.293 MPa
Mooney-Rivlin constants B = 0.177 Mpa
Poisson’s ratio ν
= 0.4999 (K/G = 5000)
Prescribed displacement ∆
= 0.2 m
MODELLING:
In
de
x
A finite element mesh of 8-noded PLANE2D displacement-pressure elements are
used with 64 elements (Figure 87-2). Due to symmetry, only one quarter of the
cylinder is meshed. Along the outer surface of the cylinder, the gap elements are
generated in conjunction with a rigid frictionless target surface at the bottom. The
load is applied by prescribing the displacements at the top of the mesh. A
displacement of ∆ = 0.2 m is applied.
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
COMPARISON OF RESULTS:
The resulting force-deflection curve and von Mises stress band plot are shown in
Figure 87-3 and 87-4 respectively. The solutions by ADINA are enclosed for
comparison. They are in close agreement.
Figure NS87-1
∆
D = 0.4M
A = 0.293 MPA
B = 0.177 MPA
FRICTIONLESS
CONTACT
Pla in S tra in Rubbe r C ylinde r
Figure NS87-2
PRESCRIBED
DISPLACEMENT ∆ / 2
COMPRESSIO
N
In
de
x
TARGET SURFACE
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Figure NS87-3
F
O
R
C
E
ADINA
COSMOS/M
Y
(IN)
∆ (M)
In
de
x
Figure NS87-4
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
NS88: Torsion of a Rubber Cylinder
TYPE:
Nonlinear static analysis, Mooney-Rivlin hyperelastic material model, large strain
and large deflection, prescribed rotation, coupling degrees of freedom in local
coordinate system, reaction force calculation, auto-stepping, SOLID, displacementpressure (u/p) elements.
PROBLEM:
A solid rubber cylinder shown in Figure 88-1 is constrained in the axial direction and
is twisted by the applied moment. Determine the moment-rotation curve and the
axial force-rotation curve for the cylinder.
REFERENCE:
T. Sussman and K. J. Bathe, “A Finite Element Formulation for Nonlinear
Incompressible Elastic and Inelastic Analysis,” Computers & Structures, Vol. 26,
No. 1/2, pp. 357-409, 1987.
GIVEN:
Radius of cylinder R
= 0.05 m
Length of cylinder L
= 0.1 m
Mooney-Rivlin constant A
= 3.E5 Pa
Mooney-Rivlin constant B
= 1.5E5 Pa
Poisson’s ratio ν
= 0.4999 (K/G = 5000)
Prescribed rotation θ
= 0.1 rad
MODELING:
In
de
x
The mesh layout of 64 collapsed 8-node SOLID elements are shown in Figure 88-2.
The displacement-pressure formulation is used. On the far end of the cylinder, all of
the nodes are fixed. On the near end, the angular displacement is prescribed at the
center node of a beam. Rigid links are then placed between the beam and the nodes
on the near end of the cylinder to cause the entire cross-section to rotate by this
amount.
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COMPARISON OF RESULTS:
The moment-rotation curve and the axial force-rotation curve are shown in Figure
88-3 and Figure 88-4. Analytical solutions are enclosed for comparison.
Figure NS88-1
L
R
N
M
M
N
Figure NS88-2
Y
FIXED
BOUNDARY
Z
X
RIGID LINK TO
BEAM ELEMENTS
PRESCRIBED ROTATION
ATCENTER MODE
In
de
x
8 -N ode S olid Ele me nts
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
Figure NS88-3
ANALYTICAL SOLUTION
COSMOS/M
M
0
M
E
N
T
N-M
ROTATION (RAD)
ROTATION (RAD)
Figure NS88-4
A
X
I
A
L
ANALYTICAL SOLUTION
COSMOS/M
F
O
R
C
E
(N)
In
de
x
ROTATION (RAD)
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Part 1 NSTAR / Nonlinear Analysis
NS89: Thick-Walled Cylinder Subjected to a Constant Radial Displacement Rate at its Inner Wall
TYPE:
Nonlinear static analysis, elastoplastic material model, large strain, large deflection,
displacement-dependent-pressure loading, displacement control algorithm,
PLANE2D axisymmetric, displacement-pressure (u/p) elements.
PROBLEM:
A long, thick-walled cylinder, as shown in Figure 89-1, is subjected to a constant
radial displacement rate at the inner wall of the cylinder. The cylinder is made of an
elastoplastic, nearly incompressible material. Compare the numerical solutions
against the analytical solutions for the reaction pressure and the normal stress along
the radial direction at the point initially half-way through the cylinder wall versus
the radius of the inner wall.
REFERENCE:
W. Prager and P. G. Hodge, “Theory of Perfectly Plastic Solids,” John Willy and
Sons, New York, 1951.
GIVEN:
Initial inner radius A0= 10 mmInitial yield stress σyo= 50 MPa
Initial outer radius B0= 20 mmTangent modulus ET= 0 (perfect-plasticity)
Elastic modulus E
= 25000 MPsInner radius ratioA/A0= 3
Poisson’s ratio ν
= 0.499
In
de
x
MODELLING:
The finite element model is built using five 8-noded PLANE2D axisymmetric
elements. Plane strain boundary conditions in the cylinder axial direction are
applied. The mesh is also shown in Figure 89-1. The displacement-pressure (u/p)
formulation is employed because of the nearly incompressible condition for the
material. The cylinder is expanded by applying the internal pressure. The cylinder
reaches a limit state within the very small strain, after which the pressure decreases
rapidly as the cylinder expands. In order to handle such kind of instability, a
displacement control algorithm is used. The cylinder is finally expanded to three
times its initial radius (inner). Under such large deformation, the displacementdependent-pressure loading is employed and the finite strain plasticity theory is
applied to the element formulation.
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
COMPARISON OF RESULTS:
The analytical solution for the normal stress along the radial direction is:
where
As R0 = A0,
σxx = -P
where P is the reaction pressure.
Figure 89-2 shows the normalized reaction pressure versus the normalized inner
radius. The reaction pressure is the load factor computed from the displacement
control algorithm. Figure 89-3 shows the normalized normal stress in the radial
direction at the point initially half-way through the cylinder wall (R0/A0 = 1.5)
versus the normalized inner radius. Note that the large displacement increment in
Figure 89-2 and Figure 89-3. Nevertheless, the numerical solutions are in excellent
agreement with the analytical solutions.
Figure NS89-1
Y
Z
X
X
PRESSURE
Y
P
A
2A
In
de
x
2B
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Part 1 NSTAR / Nonlinear Analysis
Figure NS89-2
ANALYTICAL SOLUTION
COSMOS/M
−σ xx /K
A / AO
Figure NS89-3
ANALYTICAL SOLUTION
COSMOS/M
−σxx
/K
In
de
x
A / AO
COSMOSM Advanced Modules
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Chapter 8 Verification Problems
NS90: Pressurization of a Sphere
with Elastoplastic Material
TYPE:
Nonlinear static analysis, elastoplastic material model, large strain, large deflection,
displacement-dependent-pressure loading, arc-length method, local boundary
conditions.
NS90A)
SHELL4T elements
NS90B)
SOLID, U/P elements
NS90C)
TETRA4, U/P elements
NS90D)
TETRA10, U/P elements
PROBLEM:
The problem discussed in this example involves the pressurization of a sphere with
elastoplastic material behavior. The strains are very large so that for the elastoplastic
case, rigid plasticity analysis provides an accurate comparative result. Compare the
numerical solutions of SHELL4T, SOLID, TETRA4, and TETRA10 elements
against the analytical solutions for the reaction pressure versus the mean radius.
GIVEN:
Initial mean radius R0 = 0.1 m
Thickness t
= 0.001 m
Young’s modulus E
= 2.E11 N/m2
Poisson’s ratio ν
= 0.3
Initial yield stress σyo
= 2.5E8 N/m2
Tangent modulus ET
= 0 (perfect-plasticity)
MODELING:
In
de
x
The problem geometry is shown in Figure NS90-1. A sector of 3.6 x 1.0° (for
SHELL4T) and 1.0 x 1.0 degree (for SOLID, TETRA4, and TETRA10 elements)
are used for modeling. The meshes consist of 2 x 2 elements (for SHELL4T) and 2
x 2 x 4 (through the thickness) elements (for SOLID, TETRA4, and TETRA10) as
shown in Figure NS90-2A and Figure NS90-2B respectively. Axisymmetric
boundary conditions are applied. The sphere is expanded by applying the internal
pressure. The sphere reaches a limit state within the very small strain, after which
the pressure decreases rapidly as the sphere expands. In order to handle such kind of
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instability, an arc-length method is used. The sphere is finally expanded as twice of
its initial mean radius. Under such large deformation, the displacement-dependentpressure loading is employed and the finite strain plasticity theory is applied to the
element formulation.
COMPARISON OF RESULTS:
Figure NS90-3 shows the normalized reaction pressure versus the normalized radial
displacement. The reaction pressure is the load factor computed from the arc-length
method.
Figure NS90-1
3.6 DEGREE
THICKNESS
R
Figure NS90-2A
1.0 DEGREE
3.6 DEGREE
1.0
DEGREE
R
In
de
x
S H ELL4 T Ele me nts (2 x 2 )
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Figure NS90-2B
(b) SOLID Elements (2 x 2 x 4)
In
de
x
(c) TETRA4 Elements (2 X 2 X 4)
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(d) TETRA10 Elements (2 x 2 x 4)
Part 1 NSTAR / Nonlinear Analysis
Figure NS90-3
Exact Rigid-Plasticity
In
de
x
COSMOSM Results
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Chapter 8 Verification Problems
NS91: Upset Forging of a Cylindrical Billet
TYPE:
Nonlinear static analysis, elastoplastic material model, large strain and large
deflection, prescribed displacements, coupling degrees of freedom, reaction force
calculation, auto-stepping, sticking friction contact, PLANE2D and TRIANG
axisymmetric, displacement-pressure (u/p) elements.
REFERENCE:
G. G. Weber, A. M. Lush, A. Zavaliangos, and L. Anand, “An Objective TimeIntegration Procedure for Isotropic Rate-independent and Rate-dependent Elasticplastic Constitutive Equations,” International Journal of Plasticity, Vol. 6, pp. 701744, 1990.
PROBLEM:
As a simple metal-forming example, the prototypical problem of isothermal upset
forging of a cylindrical billet was solved as shown in Figure 91-1. The dies were
modeled as being rigid, with sticking friction acting to prevent sliding between the
billet and the die faces when they are in contact. This friction causes the billet to
barrel, with the material near the corners folding over to come in contact with the
dies. Consequently, this example problem exhibits the realistic features of
inhomogeneous deformation, with variable rates of straining at material points and
time varying die contact geometry.
GIVEN:
Diameter D
= 2 mm
Height H
= 3 mm
Young’s modulus E
= 25000 Mpa
= 0.3
= 50 MPa
Tangent modulus ET
= 0 (perfect-plasticity)
In
de
x
Poisson’s ratio ν
Initial yield stress σyo
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MODELING:
Figure 91-2 shows the finite element mesh containing 107 4-noded PLANE2D and
119 6-noded TRIANG axisymmetric, displacement-pressure elements. Near the
corner where the roll-over is expected to occur, the elements are triangular in shape
to accommodate this deformation mode. Symmetry in the problem allowed only a
quarter of the billet to be modeled. The die face was modeled as a rigid surface and
the external surface of the model was covered with gap elements to model the
contact conditions.
COMPARISON OF RESULTS:
Figure 91-2 also shows the deformed finite element mesh superposed on the
undeformed mesh after a height reduction of 60%. The billet is seen to have
expanded radially by a considerable amount. Five elements have folded over and
come in contact with the die. Figure 91-3 shows the history of total die force versus
die displacement for the node located at the center of the global coordinate system
(node 1 for PLANE2D and node 23 for TRIANG). Note that jumps in die force occur
in the calculated result whenever new nodes came in contact with the die.
Figure NS91-1
DIE FACE
AXIS
OUTER
SURFACE
In
de
x
MIDDLE
LINE
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Chapter 8 Verification Problems
Figure NS91-2
Original and deformed mesh using PLANE2D
axisymmetric elements
In
de
x
Original and deformed mesh using TRIANG
axisymmetric elements
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Figure NS91-3
REFERENCE
In
de
x
COSMOSM Results
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Chapter 8 Verification Problems
NS92: Comparison of Model Prediction with
Cyclic Uniaxial Compression Test Data
TYPE:
Non-linear static analysis using TRUSS3D element with concrete material model.
PROBLEM:
To compare the model results in a cyclic uniaxial compression test with some
experimental data.
REFERENCES:
Moussa, R. A., and Buyukozturk, O., “A Bounding Surface Model for Concrete,”
Nuclear Engineering and Design 121, pp. 113-125, 1990.
Soon, K. A., “Behavior of Pressure Confined Concrete in Monotonic and Cyclic
Loadings,” Thesis Submitted in Partial Fulfillment for the Degree of Doctor of
Philosophy, Department of Civil Eng. M.I.T., Cambridge, MA, June 1987.
MODELING HINTS:
The uniaxial test is modeled using a series of truss elements and applying
compression force at one end while constraining the other end in X-direction.
GIVEN:
Ultimate Compression Strength f'c
= 1000 N/cm2
Ultimate Strain εu
= 0.002
Area of Concrete
= 1 cm2
RESULTS:
In
de
x
Figure 92-1 shows the model results (solid line), the results from Reference 1
(circles), and the experimental results (solid squares) from Reference 2.
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NORMALIZED DISPLACMENT
Figure NS92-1
In
de
x
NORMALIZED FORCE
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Chapter 8 Verification Problems
NS93: Reinforced Concrete Truss Analysis
TYPE:
Non-linear static analysis using TRUSS2D element with concrete material model.
PROBLEM:
A two dimensional reinforced concrete truss. The truss is simply supported and
vertical concentrated forces are applied at the top cord. The load factor is required
such that the maximum vertical displacement is 3 cm.
MODELING HINTS:
The truss members have a typical R.C. cross-section of 0.3 x 0.3 m2 with 4 rebars
2.5 cm diameter each. The truss members are modeled using two TRUSS2D
elements superimposed on each other; the first is to simulate the concrete section and
the second to simulate the steel rebars. The steel and the concrete are assumed to be
fully bonded.
GIVEN:
Ultimate Compression Strength fc'
= 2000 T/m2
Ultimate Strain εu
= 0.002
E (Steel)
= 2.0E7 T/m2
Area of Concrete
= 0.09 m2
Area of Steel
= 4 x 4.9 E-4 m2 = 19.6 E-4 m2
RESULTS:
In
de
x
Figure 93-1 shows the linear and the nonlinear solution of the problem. Figure 93-2
shows the damage factor for each member.
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Figure NS93-1
F
O
R
C
E
DISPLACEMENT
In
de
x
Figure NS93-2
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Chapter 8 Verification Problems
NS94: Splitting Tension of Concrete Cylinder
TYPE:
Nonlinear static analysis using concrete material model.
PROBLEM:
Simulating the concrete split test on a cylinder.
REFERENCES:
Han, D. J. and Chen, W. F., “Constitutive Modeling in Analysis of Concrete
Structures,” Journal of Engineering Mechanics, Vol. 113, No. 4 April 1987.
MODELING HINTS:
Three different models are used in this problem. The three models are generated
using different types of elements (plane strain 4-node PLANE2D, 6-node TRIANG,
and SOLID element) as shown in Figure 94-1. In all cases the concrete material
model is chosen.
In order to simulate the concrete split test, the load is applied as a pressure on a width
0.5 inch. Only quarter of the problem is modeled due to the double symmetry of the
problem.
The results are compared with the finite element analysis using a five parameter
model developed by Han and Chen [Reference 1] as shown in Figure 94-2.
GIVEN:
Ultimate Compression Strength fc'
= 4.54 ksi
Ultimate Strain εu
= 0.0029
Cylinder Diameter
= 6 in
In
de
x
Load is applied on a 0.5 inch width strip
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Figure NS94-1
Figure NS94-2
1
2
PLANE 2D & SOLID
TRIANG
HAND & CHEN
In
de
x
F
O
R
C
E
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Chapter 8 Verification Problems
NS95: Stress Intensity Factor of Three
Point Bend Specimen
TYPE:
J-integral evaluation for a mode I crack, 8-node PLANE2D plane-stress elements,
linear elastic material, small deflection.
REFERENCE:
Gross and Strawley, 1972, obtained a solution in the form:
PROBLEM:
A simply supported beam with a through-thickness edge crack at the center is
subjected to a point load as shown in the Figure NS95-1. Determine the mode-I
stress intensity factor.
MODELING HINTS:
Due to symmetry, one half of the model is used for analysis. Symmetric boundary
conditions are enforced along the vertical axis of symmetry. The Finite element
mesh is illustrated in Figure NS95-2. Notice that in this case J paths are symmetric
(extends from 0 to π) and the crack axis makes a 90° angle with the global x axis.
COMPARISON OF RESULTS:
KI Solution
In
de
x
Reference
STAR Crack Element (800 elements)
NSTAR J-Integral (200 elements)
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Error
3.3541
—
3.05867
8.81%
3.236
3.5%
Part 1 NSTAR / Nonlinear Analysis
Figure NS95-1. Geometry and Properties
P/2 = 0.5
P
B
W = S/2
40
B = 0.5
E = 30E6
v = 0.28
a
S
80
S
P/2
P/2
Model for Analysis
Three Point Bend Test for Mode-1
Fracture Toughness
In
de
x
Figure NS95-2. Finite Element Mesh and a J-Path
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Chapter 8 Verification Problems
NS96: Slant-Edge-Cracked Plate, Evaluation of
Stress Intensity Factors by Using the J-Integral
TYPE:
J-integral evaluation for a combined mode crack, 8-node PLANE2D plane-stain
elements, linear elastic material, small deflection.
NS96A)
Calculate the total J-integral parameter
NS96B)
Calculate the individual J-integral parameters for modes I and II
REFERENCE:
Bowie, O. L., “Solutions of Plane Crack Problems by Mapping Techniques,” in
Mechanics of Fracture I, Methods of Analysis and Solutions of Crack Problems (Ed
G. C. Sih), pp. 1-55, Noordhoff, Leyden, Netherlands, 1973.
PROBLEM:
A rectangular plate with an inclined edge crack is subjected to uniform uniaxial
tensile pressure at the ends. The crack starts from the middle of one side and inclines
at an angle towards the opposite side. Evaluate the crack stress intensity factors.
MODELING HINTS:
While for evaluation of the total J-integral parameter, any reasonable mesh is
acceptable, to evaluate J values at modes I and II, a symmetric mesh with respect to
the crack axis is required. Figures NS96-2A and NS96-2B illustrate the mesh for
parts (A) and (B). Special attention was given to avoid merging of nodes along the
crack free surfaces, and to pick the proper node defining the start and end of a J path,
accordingly.
COMPARISON OF RESULTS:
Error
Reference
J Path Part (a)
J Path 1 Part (b)
J Path 2 Part (b)
Error
1.85
—
0.880
—
2.05
—
—
—
—
—
1.991
3%
1.79
3%
0.873
1%
1.99
3%
1.79
3%
0.883
0.3%
1.99
3%
In
de
x
E' = E/(1 - ν2)
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GIVEN:
σ
= 1 psi
H
= 2.5 inch
w
= 2.5 inch
a
= 1 inch
E
= 30 x 106 psi
ν
= 0.3
φ
= 45°
thickness=1 inch
Figure NS96-1. Geometry and
Properties for SlantEdge-Cracked Plate
Figure NS96-2. Finite Element Mesh and J-Paths
σ
h
φ
a
h
w
(A) Regular Mesh
In
de
x
σ
(B) Symmetric Mesh
about Crack Axis
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Chapter 8 Verification Problems
NS97: Single-Edge-Cracked Plate Subjected
to Remote Uniform Tension and a
Thermal Gradient Across the Plate Width
TYPE:
Nonlinear static analysis using 8-node PLANE2D plane stress elements, von Mises
elastoplastic model, small deflection, J-integral evaluation.
REFERENCE:
V. Kumar, B. I. Schumacher, M. D. German, “Development of a Procedure for
Incorporating Secondary Stresses in the Engineering Approach,” in Advances in
Elastic-Plastic Fracture Analysis, EPRI NP-3607, Project 1237-1, Section 7, August
1984.
PROBLEM:
A single-edge cracked plate (SECP) in plane stress (Figure NS97-1) is subjected to
a thermal gradient across its width given by:
T (x,t) = T (t) [125 + 400x – 100x2]
The plate is then subjected to a uniformly increasing pressure in the longitudinal
direction. Evaluate the J-integral parameter for different stages of solution as the
pressure is risen from zero to 15 ksi.
MODELING HINTS:
The uniaxial elastoplastic stress-strain curve (Figure NS97-2), beyond the yield
point, is defined by the Ramberg-Osgood stress-strain relationship:
ε/ε0 = σ/σ0 + α (σ/σ0)n
In
de
x
with α = 0.5, and n = 5 (ε0, σ0 are the yield stress and strain). The finite element
mesh, and the loading time histories are given in Figure NS97-3. Also, assuming that
the thermal gradients, in this case, do not have a considerable effect on the loading
proportionality, the von Mises yield criteria (based on the flow theory of plasticity)
is employed for the modeling of plasticity.
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COMPARISON OF RESULTS:
Figure NS97-1.Single-Edge Cracked
Plate (SECP) Under
Remote Uniform
Tension
∆/2
σ
T
GIVEN:
b
= 2 in
a/b
= 0.25
L/b
=4
E
= 30 x 103 ksi
Nu
= 0.3
Yield Stress
= 60 ksi
Tref
= 70°F
Thermal
Coefficient
of Expansion
= 7.3 x 10-6 in/in°F
T
8
Figures NS97-4A and NS97-4B
demonstrate the variations of
J-integral parameter with respect
to the applied pressure. The
J-integral value at zero pressure
is due to the effects of temperature
alone. The obtained graphs are
in good agreement with those given
in the reference.
2
L
1
y
X
a
c
L
σ
8
b
In
de
x
∆/2
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Chapter 8 Verification Problems
Figure NS97-1.NS97-2.Uniaxial Elastoplastic Stress-Strain Curve
(Ramberg-Osgood Model)
S
T
R
E
S
S
(PSI)
STRAIN
Figure NS97-3. Finite Element Mesh and a Few J-Paths
1
T0
t
1.E-5
60
σ ∞ (ksi)
0 1.E-5
60
t
In
de
x
Time historie s of σ ∞
a nd T 0 for pla ns stre ss S EC P
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Finite Ele me nt Me sh
a nd 4 J Pa ths
Part 1 NSTAR / Nonlinear Analysis
Figure NS97-4. J-integral Versus Applied Pressure (plotted on small scale)
J
I
N
T
E
G
R
A
L
in-Kip/in 2
PRESSURE (KSI)
Figure NS97-5.J-Integral Versus Applied Pressure (plotted on large scale)
(Plane Stress SECP, Mechanical and Thermal Loading)
J
I
N
T
E
G
R
A
L
in-Kip/in 2
In
de
x
PRESSURE (KSI)
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Chapter 8 Verification Problems
NS98: Circumferentially Cracked Cylinder
Subjected to a Uniform Axial Tension and
a Thermal Gradient in the Radial Direction
TYPE:
Nonlinear static analysis using 8-node PLANE2D Axisymmetric elements, von
Mises elastoplastic model, small deflection, J-integral evaluation.
REFERENCE:
V. Kumar, B. I. Schumacher, M. D. German, “Development of a Procedure for
Incorporating Secondary Stresses in the Engineering Approach,” in Advances in
Elastic-Plastic Fracture Analysis, EPRI NP-3607, Project 1237-1, Section 7, August
1984.
PROBLEM:
A cylinder containing an axisymmetric crack (Figure NS98-1) is first subjected to a
thermal gradient in the radial direction:
T (x,t) = T (t) [125 + 100x – 6.25x2]
Where x is the distance from the inner surface. The cylinder is then loaded by a
uniformly applied tensile pressure at its ends. Evaluate the J-integral parameter for
different stages of solution as the pressure is risen from zero to 27 ksi.
MODELING HINTS:
The elastic-plastic behavior is defined by the Ramberg-Osgood stress-strain law,
given in problem NS97. The finite element mesh, and the loading time histories are
shown in Figures NS98-2 and NS98-3. Also, assuming that the thermal gradients, in
this case, do not have a considerable effect on the loading proportionality, the von
Mises yield criteria (based on the flow theory of plasticty) is employed for modeling
of plasticity.
COMPARISON OF RESULTS:
In
de
x
Figure NS98-4 demonstrates the variation of J-integral parameter with respect to the
applied pressure. The J value at zero pressure is due to the effects of temperature
alone. The obtained graph is in good agreement with that given in the reference.
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GIVEN:
Figure NS98-1.Circumferentially
r
Cracked Cylinder
in Tension
Ri/b
= 10
a/b
= 0.25
Ri
= 80 inch
L
= 120 inch
E
3
= 30 x 10 ksi
Nu
= 0.3
Yield Stress
= 60 ksi
Tref
= 70° F
Thermal
Coefficient
of Expansion
= 7.3 x 10-6 in/in °F
ET
= 4.25 x 103 ksi
CL
T2
σ∞
Ri
T1
R0
L
b
a
c
L
σ∞
Figure NS98-2. Time Histories of σÏ and To
1
90
σ∞ (ksi)
T0
t
0
0
1.E-5
90
t
In
de
x
1.E-5
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Chapter 8 Verification Problems
Figure NS98-3
A
A
A
Finite Element Model
(not to scale)
2
J-INTEGRAL (IN-KIP/IN )
Figure NS98-4. J-Integral Versus Applied Pressure
In
de
x
PRESSURE (KSI
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Part 1 NSTAR / Nonlinear Analysis
NS99: Initial Interference Between Two Thick
Hollow Cylinders with Elastic-plastic Behavior
TYPE:
Plasticity, large displacement analysis using PLANE2D axisymmetric and contact
(node to line gap) elements, Thermal loading.
PROBLEM:
Similar to problem NS36, except that considerable plastic strains are developed
during the process of fitting. To fit one cylinder inside the other, the outer cylinder
is first heated 100 °F and then cooled 100 °F.
MODELING HINTS:
Here, the analysis is performed in two steps. First the outer cylinder is heated while
the gap element group is excluded from analysis. Next, the gap group option is
changed to bring the gaps back into consideration, temperatures are gradually
reduced back to normal, and the analysis is continued using the restart option.
In
de
x
Figure NS99-1
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Chapter 8 Verification Problems
GIVEN:
Modulus of Elasticity
= 30.E6 psi
Poisson's Ratio
= 0.3
Yield Stress
= 3.E5 psi
Tangent Modulus
= 1.E6 psi
Thermal Coefficient
= 0.001 1/°F
Figure NS99-2
Figure NS99-3
COMPARISON OF RESULTS:
Linear Material Small
Displacement
Elastoplastic Material Large
Displacement
Contact Location(radius)
21.5709 in.
21.4133 in.
Pressure at the Interface
57.231 ksi
29.486 ksi
In
de
x
* Notice the great reduction in interface pressure due to elastoplastic behavior.
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NS100: Detection of Buckling Load Based on
State of Deformation for a Cylindrical Shell
TYPE:
Large displacement analysis, using SHELL4 elements, Automatic Stepping, and
buckling analysis based on the deformed geometry.
PROBLEM:
A shallow cylindrical shell is subjected to pressure loading along the two flat edges.
Study the accuracy of the predicted buckling pressure by performing buckling
analyses at different levels of deformation.
MODELING HINTS:
Due to symmetry, a 4x4 mesh is used to model a quarter of the shell. Starting from
time zero, first a nonlinear analysis is performed to solve for a pressure loading of
0.1 N/mm2, followed by a buckling analysis. Next, the nonlinear analysis is restarted
to solve for pressure = 0.2 N/mm2, followed by a second buckling analysis. This
procedure is repeated several times, raising the pressure a magnitude 0.1 N/mm2
each time.
GIVEN:
= 2540. mm
Width
= 254. mm
Theta
= 0.1 rad
Shell Thickness
= 6.35 mm
Modulus of Elasticity
= 3102.75 N/mm2
Poisson's ratio
= 0.3
In
de
x
Radius
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Figure NS100-1
Figure NS100-2
CONCLUSIONS:
In
de
x
Figure NS100-3 shows the response at the center of the shell with respect to time (or
pressure). It is evident from this curve that there is no perfect buckling behavior for
this problem. While the pressure never drops, the graph demonstrates two states of
rising and falling of displacement-rates. The predicted buckling pressure at low
displacements is close to the point in between these two states (where the
displacement path changes). As deformations become larger, the evaluated buckling
pressure also becomes larger (A negative buckling pressure, in this case, indicates
that the buckling solution is invalid). Considering the fact that the buckling load
factor (eigenvalue) at the time of buckling must equal one, we conclude that no
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buckling has occurred during this analysis, which is also supported by the results of
the nonlinear analysis.
Figure NS100-3
Buckling
Parameter
(Eigenvalue)
Buckling Pressure
= p x Eigenvalue (N/mm2)
0.1
4.389
0.4389
0.2
2.213
0.4426
0.3
1.517
0.4551
0.4
1.281
0.5124
0.5
1.806
0.903
0.6
3.741
2.2446
0.7
-10.06
----
In
de
x
Pressure P
(N/mm)
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Chapter 8 Verification Problems
NS101: Plasticity in the State of
Pure Shear, Comparison of Tresca
and von Mises Yield Criterion
TYPE:
Plasticity, Tresca yield criterion, cyclic loading conditions, kinematic hardening.
NS101A)
PLANE2D Plane Stress Elements
NS101B)
Solid 8-node Elements
NS101C)
Combined Kinematic & Isotropic Hardening (RK=0.5)
PROBLEM
A square plate is subjected to in-plane pressure along two normal edges, while it is
fixed along the other two edges. The loading is such that a state of pure shear is
created throughout the plate (compression is applied on one edge while tension is
applied on the other one).
Obtain the response as the
maximum shearing stress in
the plate is raised to equal the
tensile yield stress, reversed to
reach the same magnitude in
the opposite direction, and
reversed back to the original
magnitude.
Figure TL10-1
Compare the solutions based
on Tresca and von Mises
criteria.
Compare also with the case of
combined kinematic and
isotropic hardening using
Tresca yield criterion.
MODELING HINTS:
In
de
x
Here, the principal directions
and the global directions
coincide. To study the
principal shearing stresses, a local Cartesian coordinate is defined by a 45° rotation
of the global coordinates; the stresses are requested to be output in this local system.
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Figure NS101-2. Pressure Time History
Given:
Modulus of Elasticity
=1.E6 psi
Poisson’s Ratio
= 0.3
Yield Stress
= 1.E3 psi
Tangent Modulus
= 1.E4 psi
COMPARISON OF RESULTS:
Yield Criteria
von Mises
Pressure at Start of Yielding
0.5 ksi
0.577 ksi
Maximum Displacement in
X- or Y-direction
0.7525 in
0.641 in
Maximum Shearing Strain
0.1496 in/in
0.1281 in/in
In
de
x
Tresca
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Chapter 8 Verification Problems
Figure
NS101-3. Response Based on Tresca Yield Criteri0n
RESPONSE BASED ON TRESCA YIELD CRITERION
Figure NS101-4. Maximum Shearing Stress in Plate
In
de
x
MAXIMUM SHEARING STRESS IN PLATE
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Figure NS101-5. Response Based on von Mises Yield Criterion
In
de
x
Figure NS101-6. Response Based on Tresca Yield Criterion ( Combined Kinematic &
Isotropic Hardeniing RK=0.5)
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NS102: THERMAL BUCKLING OF A SIMPLY SUPPORTED PLATE
TYPE:
Nonlinear static analysis, uniform thermal loading, initial imperfection (small
transverse point force).
NS102A)
Using Displacement Control
NS102B)
Using Arc-Length Control
REFERENCE:
Timoshenko and Woinosky-Krieger, “Theory of Plates and Shells,” McGraw-Hill
Book Co., 2nd Ed., pp.389.
PROBLEM:
Find the point of buckling and the post buckling behavior of a simply supported plate
due to uniform rise of temperature. The plate is restricted from in-plane motion
along two parallel supports (Figure NS102).
Figure NS102 Finite Element Model
rollers
uniform
temperature
rise
In
de
x
simply
supported
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MODELING HINTS:
A quarter of the plate is modeled using shell elements. A uniform thermal loading of
1.0 oF is applied as the (unit) temperature pattern.
A small transverse point force is used as an imperfection to enable a post buckling
solution. This force is not associated with time-curve 1, so that it can be given a fixed
value, independent of the load (temperature) factor.
First, a one-step force control solution is performed to obtain displacements under
the transverse force. During this phase, time-curve 1 is kept at zero value, and timecurve 2 (prescribing the transverse force) is raised to 0.01 (and kept constant
thereafter).
Next, the solution is restarted with the control changed to the Displacement/Arclength control. In the case of Displacement Control, time-curve 1 is used to define
displacement of the controlled degree of freedom (central node, transverse
direction). In the case of the Arc-length Control, time curves are not used.
GIVEN:
a
= 20. in
h
= 1. in
E
= 3.E4 psi
Nu
= 0.3
Alp
= 1.E-4 1/ oF
COMPARISON OF RESULTS:
The behavior of plate under temperature loading is similar to the behavior of plate
under in-plane uniform pressure along two parallel supports (without restraining the
in-plane motion). The in-plane normal stress at the buckling temperature is equal to
the stress under in-plane buckling pressure:
Pb
T b, rel = -----------------E × Alp
Where Tb,rel is the temperature relative to the temperature at buckling, and Pb is the
buckling in-plane pressure.
In
de
x
Figures NS102A-1 and NS102B-1 show the response at the center of plate versus
the applied temperature, using Displacement and Arc-length controls, respectively.
Good agreement with the buckling temperature, based on the reference, is observed
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Figure NS102A-1. Central Deflection Vs. Temperature Using Displacement Control
In
de
x
Figure NS102B-1. Central Deflection Vs. Temperature Using Arc-length Control
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NS103: FREE ROTATION OF A TRIANGLAR SECTION ABOUT One Tip
TYPE:
Nonlinear static analysis, prescribed displacement in cylindrical coordinate system
(Uy=R * Theta), Plane2d Plane-stress elements.
Figure NS103-1. Finite Element Model
PROBLEM:
A triangular section is fixed at one tip, and is rotated 360 degrees about that tip. Find
response by prescribing the angle of rotation for one of the free tips. Verify that the
structure remains stress-free throughout analysis.
MODELING HINTS:
As the structure undergoes large rotations (regardless of the fact that there are no
strains), a large displacement analysis is a must.
In
de
x
Here the rotation is defined by prescribing Uy (=R * Theta, Theta in Radians) for a
node, in a local cylindrical system that is centered at the fixed tip.
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RESULTS:
Figure NS103-2 shows displacements at different solution steps. Listing or plotting
of stresses at any step shows that stresses are negligible (too close to zero).
Figure NS103-2 Displaced Positions Due to Prescribed Rotations
In
de
x
.
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NS104: Buckling & Post Buckling of a Simply Supported Orthotropic Plate
TYPE:
Nonlinear static analysis, Orthotropic SHELL4 elements, large displacement.
Figure NS104-1. Finite Element Model
REFERENCE:
Timoshenko and Woinosky-Krieger, “Theory of Plates and Shells,” McGraw Hill
Book Company, 2nd Ed.
PROBLEM:
In
de
x
A simply supported plate is subjected to in-plane uniform pressure applied in the Xdirection. Investigate the drop in magnitude of the buckling pressure when the
material strength is lowered in the y-direction while it is kept constant in the Xdirection (Ey < Ex). Compare the results with the isotropic case (Ex = Ey).
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Figure NS104-2-A In-Plane Pressure
In-Plane Pressure (lb/in)
100
(0,0)
Time
1.0
Time
1.0
Figure NS104-2-B Central Force
Central Force
1.0
0.1
(0,0)
MODELING HINTS:
Due to symmetry, a quarter of the plate is modeled using SHELL4 elements. In order
to obtain the post buckling behavior, a small transverse force is applied at the center
of the plate (initial imperfection).
In
de
x
Given:
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a
= 20 in
h
= 1 in
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Nu
= 0.3
Ex
= 3.0E+004 psi
10 < Ey < 3.0E+004
COMPARISON OF RESULTS:
A number of cases are run for different values of Ey. The results are shown in Fig.
NS104-3.
Figure NS104-3 Variation of Buckling Pressure with Respect to Orthotropic Intensity
In
de
x
Also for the case when Ey / Ex = 0.5, a graph of the central displacement versus the
applied pressure is given by Fig. NS104-4
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In
de
x
Figure NS104-4 Buckling of a S.S.Orthotropic Plate Under In-Plane Pressure(Ey/
Ex=0.5)
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NS105: Piercing of a Thick-Walled Cylinder with
Residual Stresses from Cyclic Internal Pressure
TYPE:
Nonlinear static analysis, Death of elements, Plasticity, Kinematic Hardening,
PLANE2D axisymmetric elements.
PROBLEM:
A thick-walled cylinder is first subjected to cyclic internal pressure. At the end of
the loading cycle, due to plastic straining of the material, considerable residual
stresses are present in the cylinder (same as problem NS33).
Next, the cylinder is pierced into two cylinders circumferentially at a radius of 1.35”,
Investigate the stress redistribution and drop in the residual stresses after piercing in
both cyliders.
Figure NS105-1
MODELING HINTS:
8-noded PLANE2D axisymmetric elements are used to model the cylinder. To
model the piercing, one element (element no. 8, r = 1.33” to 1.38”) is removed from
the analysis (killed using the EKILL command).
SOLUTION PROCEDURES:
The solution is performed in 2 stages:
In
de
x
• During the first phase of the solution, all elements are considered to be alive.
Using the von Mises yield criteria with kinematic hardening, the cylinder is
subjected to a complete cycle of loading and unloading.
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• The solution is restarted with no external forces; the variation of results is
due only to the killing of element no. 8 prior to the restart of solution.
NOTE:
In order to plot stresses correctly for a solution step in which some elements were
killed, you need to create and activate an element selection set that excludes the
killed elements before activating a stress plot.
COMPARISON OF THE RESULTS
A comparison of the stresses in the cylinder segments, before and after piercing,
shows considerable variation in the magnitude and distribution of the residual
stresses.
In
de
x
Figure NS105-2 Residual Stresses in the Cylinder After One Cycle of Loading
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In
de
x
Figure NS105-3 Residual Stresses After the piercing of the Cylinder (Element no. 8
is dead)
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NS106: Assembly of Two Cantilever Beams, Attached by a) Welding b) Cables
TYPE:
Nonlinear static analysis, Birth of Elements, Large Deformation, Contact,
PLANE2D elements. The attachment is modeled by:
NS106A- PLANE2D elements (for welding)
NS106B- Nonlinear Spring elements (for cables)
PROBLEM:
Two cantilever beams stand 0.5 inches apart. First, the beams are forced closer near
the free ends, such that welding [cables for part (b)] can be applied to attach the
beams in that area. Next, the assembly is subjected to a moment at the attached joint.
Determine the response of the assembly and maximum stresses in the beams and the
weld [or cables]. Contact between the two beams need also be considered in the
analysis.
MODELING HINTS:
8-noded PLANE2D elements are used to model the beams. And:
NS106A- Two 8-node PLANE2D elements are used to model the welding.
NS106B- Five Nonlinear (tension-only) spring elements are used to model
the cables.
Contact is modeled by a node-to-line Gap group. Rigid beam elements are used on
the free sections to allow for direct application of moments.
NOTES:
In
de
x
The elements that are used to model the welding (or cables) are treated as nonexistent (killed) at the start of the solution.
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Figure NS106A-1 Finite Element Model of Two Cantilever Beams Attached by Welding
Cantilevers
0.5”
Figure NS106B-1 Finite Element Model of Two Cantilever Beams Attached by Cables
SOLUTION PROCEDURES:
In
de
x
The solution is performed in 3 stages:
1.
At the start of the solution, the attachment elements are not considered in the
analysis (EKILL command). vertical displacements are applied to the nodes to
bring the free ends together.
2.
The imposed displacements are released, while the “killed” elements are
brought to life (ELIVE command). A restart allows the assembly to regain
equilibrium under no external forces.
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3.
The solution continued while the end moments are gradually applied: from zero
to the desired magnitude.
COMPARISON OF RESULTS:
A comparison of the two cases shows that while the stresses maybe high in the
attachment material, under the same loading conditions, the welded assembly
deforms less than the tied-by-cable assembly. The reason is obvious, however, as the
latter is not capable of passing any shear stresses.
In
de
x
Figure NS106A-2 Deformation Plots for the Welded Assembly
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In
de
x
Figure NS106B-2 Deformation Plots for the Cable Tied Assembly
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NS107: Uniaxial Tests on a Nitinol Cube Specimen
TYPE:
Nonlinear static analysis, Nitinol superelastic material model, Large strain, Large
deflection, Force control method with cyclic prescribed displacements, Tetra4
elements.
NS107A) Using a linear flow rule
NS107B) Using an Exponential flow rule
NS107C) Constant Stress Flow
REFERENCE:
Auricchio, F., “A Robust Integration-Algorithm for a Finite-Strain Shape-MemoryAlloy Superelastic Model,” International Journal of Plasticity, vol. 17, pp. 971-990,
2001.
Figure NS107-1
PROBLEM:
Nitinol cube specimens (1x1x1 mm3), with different material properties, are
analyzed under uniaxial conditions.
A cyclic displacement is prescribed on (and normal to) one the faces, while the
boundary conditions are such that a uniaxial state of stress is maintained.
MODELING HINTS:
Fig. NS107-1
Due to symmetry, only a quarter of the cube (1.x.5x.5) is modeled. The finite
element mesh is illustrated in Figure NS107-1 and the displacement time history is
given in Figure NS107-2.
In
de
x
Also, in part (c), in order to obtain unloading at zero stress, the yield stress for
unloading is approximated to 1. MPa, since a zero yield stress is unacceptable.
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Fig. NS107-2
Figure
Given:
Ex = 50,000.
MPa
Nuxy = 0.3
Part (a) :
SIGT_S1= 520 MPa
SIGT_F1=600 MPa
SIGT_S2=300 MPa
SIGT_F2=200 MPa
SIGC_S1= 700 MPa
SIGC_F1=800 MPa
SIGC_S2=400 MPa
SIGC_F2=250 MPa
Part (b): Same as Part (a) with the addition of:
In
de
x
BETAT_1= 250 MPa
BETAT_2=20 MPa
BETAC_1=250 MPa
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BETAC_2=20 MPa
Part (c):
SIGT_S1= 500 MPa
SIGT_F1=500 MPa
SIGT_S2=1. MPa
SIGT_F2=1. MPa
SIGC_S1= 700 MPa
SIGC_F1=700 MPa
SIGC_S2=1. MPa
SIGC_F2=1. MPa
RESULTS:
In
de
x
The Stress-displacement graphs are constructed to show that they are in good
agreement with the response curves given in reference. Graphs of stress versus time
are also presented in Figures NS106-8.
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de
x
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In
de
x
(B)
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de
x
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de
x
Chapter 8 Verification Problems
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de
x
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In
de
x
Chapter 8 Verification Problems
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NS108: Three-Point Bending Test
TYPE:
Nonlinear static analysis, Nitinol superelastic material model, Large strain, Large
deflection, Displacement Control method, Tetra10 elements.
NS108A) Ultimate plastic strain =0.092
NS108B) Ultimate plastic Strain= 0.15
NS108C) Ultimate plastic strain =0.092 (Exponential Flow rule)
REFERENCE:
Auricchio, F., Taylor, R.L., and Lubliner, J., “Shape-Memory-Alloys:
Macromodeling and Numerical Simulations of the Superelastic Behavior,”
Computer Methods in Applied Mechanics and Engineering, vol. 146, pp. 281-312,
1997.
PROBLEM:
A three-point bending test is performed on a Nitinol wire of circular cross section
with diameter d=1.49 mm. The wire is 20 mm long and it is simply supported at both
ends.
• Obtain the graph of the applied force versus deflection for the mid-span
section of the beam.
• Verify that by increasing the ultimate plastic strain for the material a closer
match to the experimental results can be obtained.
MODELING HINTS:
• Due to symmetry, only half of the beam with half of the cross section is
modeled. The finite element mesh is illustrated in Figure NS108-1.
• The node for displacement control is selected to be same as the node where
In
de
x
the force is applied. This node is displaced, in the direction of the force, a
maximum of 5.2 mm, and then is brought back to zero.
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a)
Figure Figure NS108-1
– Finite Element Mesh
Given:
Ex = 60,000. MPa
Nuxy = 0.3
Part (a):
SIGT_S1=SIGC_S1= 637 MPa
SIGT_F1=SIGC_F1= 735 MPa
SIGT_S2=SIGC_S2= 367 MPa
SIGT_F2=SIGC_F2= 245 MPa
Eul = 0.092 mm/mm
Part (b): Same as Part (a), except:
Eul = 0.15 mm/mm
Part (c): SIGT_S1=SIGC_S1= 637 MPa
SIGT_F1=SIGC_F1= 918.5 MPa
In
de
x
SIGT_S2=SIGC_S2= 674 MPa
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SIGT_F1=SIGC_F1= 245 MPa,
BETAT_1=BETA_C1=204 MPa
BETAT_2=BETAC_2= 16.3 MPa
Eul = 0.092 mm/mm
RESULTS:
Figures NS108-2, NS108-3, and NS108-4 demonstrate the load-displacement
graphs for parts: (a), (b), and (c). Ns108-2, and NS108-4 show close agreement with
results given in reference. Figure NS108-3 shows better agreement with the
experimental data.
a)
In
de
x
Figure Figure NS108-2: Part (a)
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a)
In
de
x
Figure Figure NS108-3: Part (b)
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a)
In
de
x
Figure Figure NS108-4: Part (c)
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In
de
x
Nonlinear Dynamic Analysis
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ND1: Time History of a Cantilever
Beam with Tip Mass
TYPE:
Linear Dynamic Analysis, Elastic Material, Beam elements.
REFERENCE:
Biggs, J., “Introduction to Structural Dynamics,” McGraw-Hill, New York, 1964,
pp. 43-49.
PROBLEM:
Solve for the displacement time history of the free end of the cantilever shown in the
figure below. The driving force is a triangular load pulse, and the displacement is
required at time 0.085 sec.
Figure ND1-1
F
m
h
b
L
Cross Section
Problem Sketch
F
F1
1
2
1
t
Finite Element Model
t
d
Time Curve
In
de
x
GIVEN:
L
= 13 in
b
= 1.30 in
h
= 0.125 in
E
= 29 x 106 psi
m
= 0.001 lb sec2/in
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F1 = 10 lb
td
= 0.066 sec
ANALYTICAL SOLUTION:
Calculated data for theoretical values:
Iz
= bh3/12 = 0.000211588 in4
K
= 3EI/(L3) = 8.37878 lb/in
ω = (K/m)1/2 = -91.5357 rad/sec
The dynamic load factor (DLF) for t ≥ td is given by:
DLF
= 2/ω td [2 sin ω (t – td /2) – sin ωt – sin ω (t – td)] = -1.3178853
∆ stat
= F1/K = 1.1934905 in
∆ dynamic = ∆ sat x DLF = 1.57288418 in
COMPARISON OF RESULTS:
Displacement at the tip:
(Time step No. 170)
δ (in)
Theory
1.57288
COSMOSM
1.57290
In
de
x
Figure ND1-2
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Figure ND1-3
In
de
x
Figure ND1-4
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ND2: Transient Response of a Dropped
Container Using Gap elements
TYPE:
Nonlinear dynamic analysis, Truss Element, Concentrated Masses, Gap element.
REFERENCE:
Thomson, W. T., “Vibration Theory and Applications,” Prentice Hall, Inc.,
Englewood Cliffs, N.J., 1965. pp. 110-112.
PROBLEM:
A mass m is packaged in a box (M), and dropped through a height h. Obtain the
maximum amplitude attained by the mass m and the time at which it occurs.
ASSUMPTIONS:
To analyze the collision, a tensile gap element is used (it can only resist tension). A
stiff truss connects this gap to the ground. This truss is used to account for the
elasticity of the ground. A soft truss element is also used to avoid rigid body motion
of the masses. The mass of the box is large compared to m. The stiffness of gap (kg)
is large compared to k.
Figure ND2-1
M>>m
Y
1
Kgap>>k
h
k
d
Kgap, A
Gap Element
static
m
2
2
1
h
M
k
Mass = M
3
3
m
X
In
de
x
Problem Sketch
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GIVEN:
k
= 1973.92 lb/in
g
= 386 in/sec2
m
= 0.5 lb sec2/in
gap distance
= h = 1 in
ASSUME:
kg
= 1.5E7 lb/in
M
= 50 lb sec2/in
ksoft
= 1 lb/in
COMPARISON OF RESULTS:
Max. Displacement
at Node 3
Time of Occurrence
1.5506 in
0.1 sec
1.5487 in
0.095 sec
In
de
x
Theory
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ND3: Time History Analysis of a SDOF System
TYPE:
Truss Element, Concentrated Mass, Gap-friction.
REFERENCE:
“Elements of Vibration Analysis,” L. Meirovitch, McGraw-Hill, p. 21-23.
PROBLEM:
A mass spring system is placed on a surface with friction and is subjected to a step
loading. Determine the amplitude decay due to friction and the time at which the
system rests.
ASSUMPTIONS:
A compressive gap element with a gap distance of zero and a friction coefficient of
0.01 is used to model the surface. A soft truss element is used along with the gap
element to avoid singularity of the structure stiffness.
Figure ND3-1
Force
k, L
Fo
M
F(t)
µ
time
Load Time Curve
Problem Sketch
y
F
fric
M
F(t)
1
1
2
2
Gap Friction
Element
W
Free Body Diagram
µ , Kgap
3
In
de
x
Finite Element Model
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GIVEN:
F0
=lN
k
= l N/m
M
= l kg
W
= Mg = 10 N
Coefficient of friction
= µ = 0.01
THEORETICAL SOLUTION:
Based on the theory of vibration, an SDOF system subjected to a constant force
oscillates with a constant amplitude around the static equilibrium position. The
frequency of oscillations is equal to the natural frequency of the system. The effect
of friction is to reduce the amplitude of motion by:
Amplitude decay in one cycle = 4 f = 4 Ff /k = 4 x 0.1/1 = 0.4 m
where
Ff = µ W = 0.1 N
W = Mg = 10 N (weight)
While the response frequency remains the same.
COMPARISON OF RESULTS:
The magnitude of friction force at any time remains less than or equal to Ff = 0.1 N,
and it is opposite in direction to the velocity of mass M. The frequency of response
agrees with theory.
Amplitude Decay
Theory
COSMOSM
0.4 m
0.40111 m
0.4 m
0.39895 m
0.2 m
0.2004 m
15.70796 sec
15.7 sec
In
de
x
First Cycle
Second Cycle
Last Half Cycle
Time to Rest
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Figure ND3-2
In
de
x
Figure ND3-1
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ND4: Time History Analysis of a SDOF
System with COULOMB Damping
TYPE:
Truss Element, Concentrated Masses, Gap-Friction.
REFERENCE:
“Elements of Vibration Analysis,” L. Meirovitch, McGraw-Hill, p. 21-23.
PROBLEM:
A mass spring system is placed on a frictionless surface, and is subjected to a step
loading. Show that if a second mass is added on top of the first one, with a high
coefficient of friction between surfaces of the two masses, the two masses will
oscillate together. Also, investigate the amplitude and frequency of response of
masses and the magnitude of the friction force.
ASSUMPTIONS:
A compressive gap element with a gap distance of zero and a friction coefficient of
0.01 is used to model the surface between the two masses. A soft. truss element is
used along with the gap element to avoid singularity of the structure stiffness.
3
Figure ND4-1
m
Gap-Friction
Element
F(t)
k, L
µ
m
M
Fo
y
F(t)
Time
1
Problem Sketch
x
M
Load_Time Curve
1
2
In
de
x
Finite Element Model
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GIVEN:
k
= l. N/m
m
= 0.5 kg
M
= l. 5 kg
w
= mg = 5 N
F
=lN
Coefficient of friction= µ = 0.1
THEORETICAL SOLUTION:
The effect of friction in this problem is to increase the total mass of the system
which, in turn, increases the natural period. No loss of energy or change in amplitude
of oscillations occurs.
ω1 = [k / (M + m)]1/2 = 1 / (2)1/2 = 0.7071 rad/sec
In order for the two masses to move together, if the acceleration of either mass is
shown by a (t), then a friction force:
Fs(t) = m.a(t) = 0.5x a(t) < µ.w = 0.5 N
is required to produce this acceleration for mass m.
COMPARISON OF RESULTS:
The results obtained yield the following conclusions:
1.
The displacement, velocity and acceleration of the two masses are identical at
any time.
2.
The masses respond harmonically reaching a maximum displacement of two
times the static deflection (F/k=1).
3.
The response period, which is equivalent to the natural period of the system for
this case, is increased by a factor of
α = [(M + m) / M]1/2 = (2/15)1/2 = 1.547
The friction force applied to node 3 (mass m) is equal to 1/2 the acceleration of
the system at any time. The friction force applied to node 2 (mass M) has the
same magnitude as the one at node 3 in the opposite direction.
In
de
x
4.
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ND5: Time History Analysis of a SDOF System
TYPE:
Truss Element, Concentrated Masses, Gap-Friction.
ND5)
Newmark Method
ND5A
Finite Difference Method
REFERENCE:
“Elements of Vibration Analysis,” L. Meirovitch, McGraw-Hill, p. 21-23.
PROBLEM:
Consider problem ND4, this time assuming that mass M is placed on a surface with
friction. Show that if a second mass is added on top of the first one, with a high
coefficient of friction between surfaces of the two masses, the two masses will
oscillate together. Also, investigate the amplitude and frequency of response of
masses and the magnitude of the friction force.
ASSUMPTIONS:
Two gap elements are used to model the interface of the two masses and the lower
mass and the ground. Soft truss elements are used along with the gaps to avoid
singularity of the structure stiffness.
Figure ND5-1
GIVEN:
In
de
x
F
=lN
k = l N/m
M = l. 5 kg
W = Mg = 15 N
m
w = mg = 5 N
= 0.5 kg
µ1 = 0.1
µ2 = 0.005
COSMOSM Advanced Modules
8-273
Chapter 8 Verification Problems
THEORETICAL SOLUTION:
The effect of friction between the two masses is to increase the total mass of the
system which, in turn, increases the natural period. Some energy loss is caused by
the Friction between mass M and the ground, which causes and amplitude decay in
the harmonic response of the system.
ω1 = [k / (M + m)]1/2 = 1/(2)1/2 = 0.7071 rad/sec
In order for the two masses to move together, if the acceleration of either mass is
shown by a (t), then a friction force:
Fs(t) = m.a(t) = 0.5x a(t) < µ w = 0.5 N
is required in gap No.2 to
produce this acceleration.
Figure ND5-2
The amplitude decay in one
cycle is:
4 f = 4 Ff/k = 4 x 0.1/1.0 =
0.4 m
where:
Ff = µ2 = (W + w) = 0.1 N
COMPARISON OF
RESULTS:
The results obtained yield the following conclusions:
1.
The displacement, velocity and acceleration of the two masses are identical at
any time.
2.
The masses respond harmonically with an amplitude decay of 0.4 meters at each
cycle.
3.
The response period, which is equivalent to the natural period of the system for
this case, is increased by a factor of
In
de
x
α = [(M + m) / M)1/2 = (2/15)1/2 = 1.547
8-274
4.
The friction force applied to mass M from ground, always opposes the velocity
at this point.
5.
The friction force applied to mass M from mass m is opposite in direction and
equal in magnitude to 1/2 the acceleration of the system at any time.
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
ND6: Two DOF with Friction Dynamic Analysis
TYPE:
Truss Elements, Concentrated masses, Gap-Friction.
PROBLEM:
Investigate the Response of the symmetric system shown below when one of the
masses is excited by a step loading.
Figure ND6-1
F(t) = F o
k
k
k
M
M
1
2
Problem Sketch
y
Fs, F's = Friction Forces
F(t) = Fo
1
2
3
M
M
1
4
x
2
W
Fs
3
W
F's
5
6
Finite Element Model
In
de
x
GIVEN:
k
= l N/m
Fo
=lN
M
= 1 kg
W
= 10 N
Coefficient of friction
= µ = 0.005
ω1
= (k / M)1/2 = 1.0 rad/sec
ω2
= (3k / M)1/2 = 1.732 rad/sec
COSMOSM Advanced Modules
8-275
Chapter 8 Verification Problems
Figure ND6-2
In
de
x
Figure ND6-3
8-276
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
In
de
x
Figure ND6-4
COSMOSM Advanced Modules
8-277
Chapter 8 Verification Problems
ND7: Elastic-Plastic Small
Displacement Dynamic Analysis
TYPE:
Nonlinear Dynamic Analysis, Plasticity, 2D Isoparametric (8-node) Plane Stress
Elements.
ND7)
Newmark Method
ND7A)
Finite Difference Method
REFERENCE:
Nagarajan, S., Popov, E. P., “Elastic-Plastic Dynamic Analysis of Axisymmetric
Solids,” Computers & Structures, Vol. 4, pp. 1117-1134.
PROBLEM:
Evaluate the dynamic response of the simply supported beam shown subject to a
uniformly distributed step pressure as depicted in the figure. Use Newmark method
to carry out the time integration.
MODELING HINTS:
Only a quarter of the beam is modeled because of the symmetry. Due to the presence
of plasticity and heavy loads full integration is used to get reliable results.
SOLUTION PARAMETERS FOR ND7A:
For this case a large time step increment with 500 sub-steps (within each step) is
selected.
In
de
x
Convergence tolerance is set to 0.05 and automatic stepping is used.
8-278
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure ND7-1
0.75 Po psi
h
L
b
Problem Sketch
p
3
5
33
h/2
2 1
2
3
4
5
1
6
32
.75 Po
31
4
L/2
t
Finite Element Model
Time Curve
GIVEN:
E
= 3 x 107 psi
ν
= 0.3
L
= 30 in
h
= 2 in
b
= 1 in
ET = 0
σy = 5 x 104 psi
ρ
= 0.733 x 10-3 lb sec2/in4
In
de
x
po = Static Collapse Load = 444.44 lb/in
COSMOSM Advanced Modules
8-279
Chapter 8 Verification Problems
COMPARISON OF RESULTS:
Displacement at the middle of the span (in).
Figure ND7-2
In
de
x
Figure ND7-3
8-280
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
ND8: Large Displacement Dynamic
Response of a Cantilever Beam
TYPE:
Nonlinear Dynamic Analysis, Large Displacements.
ND8A)
2D Isoparametric (8 node) Plane Stress Elements
ND8B)
BEAM Elements
REFERENCE:
Bathe, K. J., Ozdemir, H., Wilson, E. L., “Static and Dynamic Geometric and
Material Nonlinear Analysis,” Report No. UC SESM 74-4, Structural Engineering
Laboratory, University of California Berkeley, California, February 1974.
PROBLEM:
Determine the dynamic response of the cantilever beam shown subject to the
uniformly distributed step pressure given in the figure. Use Newmark method to
carry out the time integration.
Figure ND8-1
p/2
h
p/2
L
b
Problem Sketch
5
3
28
2
2
1
3
p
27
5
4
1
26
4
ND8A • PLANE2 D Model
1
2
3
4
5
6
t
1
2
3
4
5
Time Curve
In
de
x
ND8B • Beam Model
Finite Element Models
COSMOSM Advanced Modules
8-281
Chapter 8 Verification Problems
GIVEN:
L
= 10 in
h
= 1 in
b
= 1 in
E
= 12,000 psi
ν
= 0.2
p
= 2.85 lb/in
ρ
= 1 x 10-6 lb sec2/in4
COMPARISON OF RESULTS:
In
de
x
Figure ND8-2
8-282
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
ND9: Large Displacement Dynamic
Analysis of a Spherical Shell
TYPE:
Nonlinear Dynamic Analysis, Large Displacements, 2D Isoparametric (8 node)
Axisymmetric Elements.
REFERENCE:
Bathe, K. J., Ozdemir, H., Wilson, E. L., “Static and Dynamic Geometric and
Material Nonlinear Analysis,” Report No. UC SESM 74-4, Structural Engineering
Laboratory, University of California Berkeley, California, February 1974.
PROBLEM:
Investigate the dynamic response of the spherical cap shown subject to a step load
applied at the apex. Use Newmark method to carry out the time integration.
Figure ND9-1
P
h
H
w
o
R
θ
p
Problem Sketch
P/2π
t
Time Curve
3
2
1
2
1
3
4
5
6
7
8
9
10
In
de
x
Finite Element Model
COSMOSM Advanced Modules
53
52
51
8-283
Chapter 8 Verification Problems
GIVEN:
R
= 4.76 in
H
= 0.0859 in
h
= 0.01576 in
θ
= 10.9°
E
= 10 x 106 psi
P
= 100 lb
ν
= 0.3
ρ
= 0.245 x 10-3 lb sec2/in4
COMPARISON OF RESULTS:
In
de
x
Figure ND9-2
8-284
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
ND11: Small Displacement Dynamic
Analysis of a Simply Supported Plate
TYPE:
Linear Dynamic Analysis, 3D Isoparametric (20 node) Elements.
REFERENCE:
Bathe, K. J., Ozdemir, H., Wilson, E. L., “Static and Dynamic Geometric and
Material Non-linear Analysis,” Report No. UC SESM 74-4, Structural Engineering
Laboratory, University of California Berkeley, California, February 1974.
PROBLEM:
Determine the dynamic response of a simply supported plate subject to a step load
at the center. Use Newmark method to carry out the time integration.
Figure ND11-1
P
10
P
0
0.006
t
51
Time Curve
2b
t
2a
Problem Sketch and Finite Element Model
MODELING HINT:
In
de
x
Due to symmetry only one quarter of the plate is modeled.
COSMOSM Advanced Modules
8-285
Chapter 8 Verification Problems
GIVEN:
a
= 20 in
b
= 30 in
t
= 1 in
E
= 3 x 104 psi
ν
= 0.25
ρ
= 3 x 10-4 lb sec2/in4
P
= 1 lb
COMPARISON OF RESULTS:
At the center of plate (node 51):
In
de
x
Figure ND11-2
8-286
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
ND12: SDOF System with Nonlinear Damping
Subjected to a Step Loading
TYPE:
Nonlinear dynamic analysis using Spring-damper Elements (gaps).
REFERENCE:
Ray W. Clough, Joseph Penzien, “Dynamic of Structures,” McGraw-Hill, New York
(1975).
PROBLEM:
Determine the response for:
ND12A)
Coulomb damping (p = 1)
ND12B)
Damping is proportional to a power of velocity less than 1 (p = 0.5)
ND12C)
Damping is proportional to a power of velocity greater than 1(p=1.5)
MODELING HINT:
In
de
x
The spring-damper is modeled using 2 gaps (one tensile and 1 compressive.) A soft
truss element is used along with the gap element to avoid singularity of the structure
stiffness.
COSMOSM Advanced Modules
8-287
Chapter 8 Verification Problems
Figure ND12-1
F
(lbs)
K
M
c, p
1.
00
time
Problem Sketch
Load - Time Curve
Y
4
1
F
2
1
2
X
3
Finite Element Model
GIVEN:
m
= 1 lbs sec/in2
k
= 1 lbs/in
c
= 0.1 lbs sec/in
COMPARISON OF RESULTS:
The exact theoretical results are evaluated for case a. This case can also be solved
using a modal time-history analysis with a modal damping of 0.05. In all cases, the
response frequency remains the same, while some differences are observed in the
maximum response values.
Theory)
Case a
Case a
Case b
Case c
First Maximum
First Minimum
Second
Maximum
Second Minimum
1.8547
1.8534
1.8317
1.8692
0.2699
0.2711
0.3205
0.2372
1.6239
1.6234
1.5435
1.6759
0.4669
0.4674
0.5779
0.3977
In
de
x
Displacement
Peaks
8-288
COSMOSM Advanced Modules
COSMOSM
Part 1 NSTAR / Nonlinear Analysis
ND13: Two-Degree of Freedom System
with Spring Dampers
TYPE:
Dynamic direct time integration analysis using spring-damper elements (gaps).
REFERENCE:
COSMOSM Advanced Dynamic Module (using Concentrated Dampers).
PROBLEM:
Determine and compare the response peaks for each mass
MODELING HINTS:
The spring-dampers are modeled using 6 gaps (3 tensile and 3 compressive). Soft
truss elements are used along with gap elements to avoid singularity of the structure
stiffness.
Figure ND13-1
F(t)
K
K
K
M
M
c
00
c
00
c
Problem Sketch
f(t) = F
1
2
3
4
5
2
3
X
1
6
7
4
8
In
de
x
Finite Element Model
COSMOSM Advanced Modules
8-289
Chapter 8 Verification Problems
GIVEN:
M = 1 lbs sec/in2
K
= 1 lbs/in
c
= 0.1 lbs sec/in
F
= 1 lbs
COMPARISON OF RESULTS:
Same problem is solved using a modal time-history analysis using truss and
concentrated damper elements. A comparison of the peaks of response for each mass
is given in the following table:
Node 2
Modal Time History
Time
Direct Integration
Time
Displ.
First Maximum
2.7
1.067
2.7
1.067
First Minimum
6.6
0.3014
6.6
0.3012
Second Maximum
9.4
1.017
9.4
1.017
Second Minimum
12.6
0.4233
12.6
0.423
Third Maximum
15.9
0.9005
15.9
0.9007
Third Minimum
18.9
0.4663
18.9
0.4659
Node 3
Modal Time History
Direct Integration
Time
Displ.
Time
Displ.
First Maximum
3.4
0.8349
3.4
0.8348
First Minimum
6.1
-0.06234
6.1
-0.06278
Second Maximum
9.7
0.6115
9.7
0.6115
Second Minimum
12.7
0.04310
12.7
0.04268
Third Maximum
15.7
0.5578
15.7
0.5580
Third Minimum
19.0
0.1434
19.0
0.1432
In
de
x
8-290
Displ.
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
ND14: SDOF System with Rayleigh Damping
Subjected to Base Excitation
TYPE:
Nonlinear dynamic analysis, TRUSS2D and Mass elements, Elastoplastic Material.
REFERENCE:
Mario, P., “Structural Dynamics,” Third Edition, Van Nostrand, 1991.
PROBLEM:
Determine the response of the mass.
Figure ND14-1
R
..
ug
15 K
u
K
- 1.215
2
1
Disp (in)
1.215
M
- 15 K
X
00
c
Problem Sketch
A (t)
20
0.45
1.2
1.4
2.0
t (sec)
1.1
In
de
x
- 10
COSMOSM Advanced Modules
8-291
Chapter 8 Verification Problems
GIVEN:
E (truss)
= 12.35 kips/in2
A (truss) = 1 in2
L (truss)
= 1 in
M
= 0.2 kips sec2/in
α
= Damping coefficient associated with stiffness matrix = 0.010
β
= Damping coefficient associated with mass matrix = 0.755
∆t
= Time increment = 0.10 sec
A(t)
= Time curve of the base excitation acceleration
Base acceleration multiplier in X-direction = – 5 (in/sec2)
In
de
x
Figure ND14-2
8-292
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
ND15: Wave Propagation in a Bar
TYPE:
Linear dynamic analysis, Truss Elements, Short Duration rectangular pulse, Finite
Difference technique.
REFERENCE:
Ray W. Clough, Joseph Penzien, "Dynamics of Structures," McGraw-Hill, p.364367.
PROBLEM:
A bar is subjected to a rectangular pulse with a duration of 0.001 seconds. Determine
the response of the bar and the maximum base reaction force. Compare results using
finite difference and Newmark.
Figure ND15-1
F
Finite Element Mesh
In
de
x
Figure ND15-2
COSMOSM Advanced Modules
8-293
Chapter 8 Verification Problems
GIVEN:
L
= 100 in
Ex
= 12000 psi
Density
= 0.001 lb sec/in/in3
CONCLUSIONS:
C = wave velocity = (Ex/Dens)1/2 = 3464.1 in/sec
T = travel time per unit distance = 1./C = 0.000288675 sec
Thus the time for the wave to return to it's original location:
T' = 2 * L * T
However, at this time, the wave hits in the opposite direction. It will take another T'
for the wave to reverse itself to it's original direction. Thus, the first period of motion
in time is two times T' (which also equals the first natural period of the bar):
COMPARISON OF RESULTS:
Similar results are obtained for response from finite difference and Newmark.
However, more steps where required to obtain an accurate base reaction force using
the Newmark technique. The following table shows a comparison of the maximum
reaction force from several runs.
Number of
Solution Steps
Maximum
Reaction Force
Run-Time
Finite DIfference
(ISUB=10)
1293
932.7 lb
170 sec
Newmark
1293
859.0 lb
180 sec
Newmark
2290
911.4 lb
300 sec
Newmark
5260
928.7 lb
520 sec
In
de
x
Dynamic
Integration
Technique
8-294
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Figure ND15-3
Figure ND15-4
In
de
x
Figure ND15-5
COSMOSM Advanced Modules
8-295
Chapter 8 Verification Problems
ND16: Dynamic Response of a Cantilever Beam
to Release of a Prescribed Tip Rotation
TYPE:
Nonlinear Dynamic Analysis, release of a prescribed rotation, Plasticity, Large
Displacement, BEAM2D Elements, beam-section definition.
ND16A)
Linear Elastic Analysis
ND16B)
Large Displacement, Elasto-Plastic Analysis
ND16C)
The tip rotation is released but the tip moment is kept active
PROBLEM:
A cantilever beam is first subjected to a prescribed tip rotation. The tip is then
suddenly released. Investigate the response of the beam due to the initial tip rotation.
MODELING HINTS:
This problem is solved in two steps. First the tip rotation is prescribed by performing
a static analysis (In the nonlinear case, the beam undergoes considerable plastic
deformation during this phase). Next the rotation is released and analysis is restarted
to perform a linear/nonlinear dynamic analysis.
In
de
x
Figure ND16-1
8-296
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
GIVEN:
Figure ND16-2
E
= 30E6 psi
n
=0
sy
= 5,000 psi
L
= 90 in
H
= 3 in
B
= 1 in
ET
= 3E6 psi
Density
= 0.001 lb
sec/in/in3
CONCLUSIONS:
In case A, the beam vibrates linearly about its undeformed position. In case B, the
beam undergoes considerable plastic deformation during the first (static) solution
phase. As a result, much of the dynamic response is damped out and the beam
oscillates about a deformed position. In case C, since the applied moment is kept
acting at the tip, no release takes place and no dynamic response is observed.
Figure ND16-4
In
de
x
Figure ND16-3
COSMOSM Advanced Modules
8-297
Chapter 8 Verification Problems
Figure ND16-5
Figure ND16-6
In
de
x
Figure ND16-7
8-298
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
ND17: Long Thick-Walled Cylinder Subjected to a
Rectangular Pressure Pulse
TYPE:
Nonlinear Dynamic analysis, Artificial Bulk Viscosity, Nonlinear Elasticity, Finite
Difference Technique, PLANE2D axisymmetric elements.
PROBLEM:
A long thick-walled cylinder is subjected to an internal pressure pulse. The material
of the cylinder is nonlinear elastic (Fig.ND17-2). Study the effects of artificial bulk
viscosity on the solution. Compare with cases of Rayleigh damping and no damping.
Figure ND17-1 The Finite Element Model of the Problem
MODELING HINTS:
4-noded PLANE2D axisymmetric elements are used to model the cylinder. Since the
cylinder is considered to be long, all displacements in the y-direction are fixed.
GIVEN:
Ri = 5 in
RO = 200 in
In
de
x
n = 0.35
r = 0.0001 lb.Sec2/in4
COSMOSM Advanced Modules
8-299
Chapter 8 Verification Problems
Artificial Bulk Viscosity Constants:
Co = 1.5
C1 = 0.06
C2 = 8012.3 in/Sec
Rayleigh Damping (used for comparison)
a = 0.
β = 1000. 1/Sec
Figure ND17-2 Stress-Strain Property Curve
COMPARISON OF RESULTS:
Figures ND17-3a/b/c show graphs of radial stresses at two different locations:
1. Radius =Ri,
2. Radius = (Ri +RO)/2.
Using
a) Artificial bulk viscosity
b) Rayleigh damping
c) No damping
In
de
x
Considering the use of Rayleigh damping, although it effectively damps the
oscillations following the peak, it also heavily damps the magnitude of the shock
front as the it moves outward in the radial direction. Applying Artificial Bulk
Viscosity to the analysis, the response following the peak is effectively damped,
while the intensity of the shock front is mostly preserved.
8-300
COSMOSM Advanced Modules
Part 1 NSTAR / Nonlinear Analysis
Thus by comparing the three cases, it is easy to conclude that the use of artificial bulk
viscosity helps to reduce the stress variations at the shock location, damps the
oscillations following the shock, while it also preserves the shock intensity (to a
sufficient degree) as the shock travels along the thickness of cylinder.
Figure ND17-3 (a)
In
de
x
Figure ND17-3 (b)
COSMOSM Advanced Modules
8-301
Chapter 8 Verification Problems
In
de
x
Figure ND17-3 (c)
8-302
COSMOSM Advanced Modules
Index
A
adaptive step adjustment 5-15
arc-length 1-5, 2-3, 5-3
arc-length control 3-12, 3-40, 5-2,
5-12, 7-3, 7-15, 8-160, 8-163, 8-198
automatic stepping 1-6, 5-15, 7-4,
8-61, 8-153, 8-160, 8-163, 8-165, 8181, 8-189, 8-192, 8-202, 8-225
automatic stepping algorithm 1-6,
5-15
automatic-adaptive stepping 3-12
automatic-stepping 8-121, 8-143,
8-148
auto-stepping 8-115
axisymmetric 3-2, 3-40, 5-17, 5-20,
8-74
B
bandwidth 4-1
base acceleration 8-292
base excitation acceleration 8-292
base motion 5-13, 7-8
beam-section-definition 8-43, 8138
In
de
x
BFGS 5-8, 8-7
Blatz-Ko hyperelastic 8-171, 8174, 8-176
Blatz-Ko strain energy density
function 3-15
bounding surface 3-37, 3-38
bulk relaxation 3-68
control technique 5-1, 7-3
controlled degree of freedom 7-14
convergence 5-5, 5-15
coupling 8-178, 8-189, 8-192, 8-202
C
crack tip 5-18, 5-20
cable-like structures 1-2
creep 1-6, 8-54, 8-55, 8-57
cable-type behavior 3-6, 3-7
creep analysis 1-2, 5-16, 7-13
Cauchy 3-25
creep constant 3-31, 7-13
Cauchy-Green deformation
creep curve 3-30
tensor 3-9
creep laws 3-30, 3-60
Cauchy-Green strain tensors 3-66 creep model 3-30, 3-32
classical power law for creep 3-30 creep strain 3-64
coefficient of friction 8-168
cyclic loading 8-228
compressible polyurethane foam
D
type rubbers 3-15
damage coefficient 3-37, 3-38
compression strength 3-37
compressive gaps 4-7
damping coefficient 8-292
concrete 8-206, 8-208, 8-210
damping matrix 5-13, 5-14
concrete model 3-37
deformation tensor 5-17
concrete ultimate strength 3-39
deformation-controlled loading 15
contact 1-4, 4-1, 4-3, 4-5, 4-6, 4-9,
deformation-dependent
4-12, 4-13, 8-168
pressure 8-115
contact (node to line gap) 8-80, 884, 8-90, 8-223
deviatoric strain 3-28
contact (node to surface gap) 8displacement control 3-40, 5-2, 582, 8-86
contact surface 4-11, 4-17
contactor 4-5, 4-8, 4-9, 4-11, 4-13
COSMOSM Advanced Modules
3, 7-14, 8-61, 8-72, 8-153, 8-156, 8163, 8-168, 8-171, 8-174, 8-176, 8186, 8-195
I-1
Index
displacement increment 5-15
displacement vector 4-4
displacement-pressure (u/p) 8-189,
8-192, 8-195, 8-202
displacement-pressure
formulation 3-19
divergence 5-10, 5-15
Drucker-Prager 8-110, 8-113
Drucker-Prager model 3-20, 3-21
dynamic analysis 5-14, 5-26, 7-8
E
effective strain 3-6
elastic creep analysis 3-30
elastoplastic 7-2, 7-16, 8-195, 8-198,
8-202, 8-291
elastoplastic model 5-16, 7-2
element group 7-17, 7-25
energy tolerance 5-10
equibiaxial test 3-67
equilibrium equations 5-4
equilibrium iterations 8-61, 8-66
experimental data 3-69, 3-70
F
fabric tension structures 3-35
failure criterion 3-3
failure index 3-3, 3-4
fitting problems 1-4
flexibility matrix 4-3
Flow Rule 3-28
flow theory of plasticity 5-21
force control 5-3, 5-12, 5-15, 8-7, 861, 8-165
In
de
x
force-controlled loading 1-5
frequencies 5-22
friction 4-2
friction force 4-7
8-35, 8-37, 8-268, 8-271, 8-273, 8275
gear-tooth contacts 1-4
Generalized Maxwell model 3-33
geometric nonlinear analysis 7-5,
7-7, 7-11
geometry updating 8-181
H
Huber-von Mises model 3-17, 3-18
hybrid method 4-3
hybrid technique 4-2
hydrodynamic 8-121, 8-126
hydrostatic 8-121, 8-126, 8-129
hydrostatic pressure 3-37
hyperelastic material 3-8, 8-74
I
incremental procedure 5-15
isotropic hardening rule 7-18
isotropic material model 3-1
iterative method 5-1, 5-4, 5-5, 5-8,
7-4
J
Jacobian 5-5
J-integral 3-32, 5-17, 5-18, 5-20, 521, 8-212, 8-214, 8-216, 8-220
J-segments 5-21
K
kinematic hardening 3-19, 8-140, 8228
kinematic hardening rule 8-65
Kirchhoff 3-25
L
Lagrange multiplier method 4-1
Lagrangian 3-25
Lagrangian strain tensor 3-65
G
gap 4-3, 4-5, 4-6, 4-12, 4-16, 8-59, 8- laminated composite material 3-3
large deflection 2-3, 8-72, 8-74, 8121, 8-266
115, 8-121, 8-126, 8-129, 8-133, 8gap displacements 4-4
135, 8-138, 8-168, 8-181, 8-189, 8gap force 4-4
192, 8-195, 8-198, 8-202
gap iterations 4-17, 5-15
large displacement 1-2, 2-2, 7-24,
gap-friction 8-27, 8-28, 8-29, 8-32,
I-2
COSMOSM Advanced Modules
8-15, 8-17, 8-19, 8-21, 8-25, 8-39, 841, 8-46, 8-48, 8-61, 8-165, 8-223, 8225, 8-281, 8-283, 8-296
large strain plasticity 3-66, 8-61
line search 5-1, 5-9, 7-8, 8-7
load curves 7-5
load multiplier 5-2, 7-15
local boundary conditions 8-115
locking 3-10
logarithmic strain 3-28
logarithmic strains 3-25
M
material model utility 3-43
material models 6-1
material nonlinearities 1-2
mixed finite element
formulation 3-8
MNR 5-7, 5-12, 7-8, 7-14, 7-15
Modified Newton-Raphson
(MNR) 3-40
Mooney-Rivlin 3-11, 3-14, 3-67, 369, 3-70, 8-168
Mooney-Rivlin hyperelastic 8-72,
8-90, 8-115, 8-189, 8-192
Mooney-Rivlin strain energy 3-9
N
Newmark-Beta 5-11, 7-8
Newton's iterative method 5-11
Newton-Raphson 3-11, 3-18, 3-21,
3-40, 5-5, 8-61, 8-72, 8-74, 8-90, 8110, 8-113, 8-117, 8-119
Nitinol 3-25
nodal displacements 4-2, 4-3
nodal forces 4-3
node-to-line contact 4-13
node-to-surface contact 4-10
nonlinear dynamic analysis 5-11
nonlinear elastic 8-92, 8-186
nonlinear SPRING 3-7
NR 5-7, 5-12, 7-8, 7-14, 7-15
numerical procedures 5-1
O
Ogden 3-67, 3-69, 8-74
Part 1 NSTAR / Advanced Dynamics Analysis
Ogden hyperelastic material 8-156
Ogden model 3-12, 3-14, 3-74
one-node gap 4-8, 4-12
orthotropic material model 3-2
out-of-balance load 5-5, 5-7, 5-10
P
penalty approach 3-8
penalty finite element
formulation 3-8
penalty method 4-1
penalty values 4-1
plane strain 3-2, 3-67
plane stress 3-2, 3-40
plastic analysis 8-3, 8-5, 8-7
plastic material models 3-65
plastic strain 3-64, 5-16
plasticity 1-2, 3-18, 3-23, 3-66, 8-11,
8-13, 8-61, 8-65, 8-69, 8-140, 8-223,
8-228, 8-278, 8-296
Poisson's ratio 3-5
postbuckling 2-3
preconditioning 5-1
prescribed displacement 8-165
prescribed non-zero
displacement 8-181
principal strain 3-64, 3-66
principal stretch ratio 3-9, 3-66
Q
Quasi-Newton (QN) 5-7, 5-8
In
de
x
R
Rayleigh damping 5-13, 7-8
residual load vector 5-8
restart flag 5-14
Riks method 8-163
rubber-like material 3-8, 3-12
rubber-like materials 3-12
S
secant material matrix 3-5
secant modulus 3-5
shape-finding analysis 3-35
shear relaxation 3-68, 3-78
shrink fit 4-8
slack 5-17
small deflection 8-110, 8-113, 8117, 8-119, 8-178, 8-186
snap-back 5-3, 8-160, 8-163
snap-through 5-3, 8-160, 8-163
snap-through buckling 1-2, 1-5
softening 1-2
spring-damper 8-287, 8-289
stiffening 1-2
stiffening behavior 1-2
stiffness matrix 4-1, 4-2, 5-5, 5-7
strain energy 3-8
strain hardening 3-38, 3-40
strain softening 3-37, 3-40
stress intensity 5-17
stress-strain curve 3-6, 3-18, 7-2, 718
stretch ratio 3-65
subroutine UMODEL 3-44
T
tangent modulus 7-18
tangential stiffness matrix 5-5, 5-6
target 4-5, 4-8, 4-9, 4-10, 4-11, 4-12,
4-13, 4-17
taut 5-17
temperature gradients 5-20
temperature-dependent material
properties 7-11, 8-107
temperature-dependent yield
stress 8-103
temperature-time shift 8-148
tension cracks 3-37
termination schemes 5-1, 5-9
thermal analysis 8-9, 8-50, 8-53, 8119, 8-148
thermal gradients 5-17
thermal loading 8-148, 8-223
thermal strain 3-64
thermoplastic analysis 3-18
thermo-plasticity 1-2, 8-103, 8-107
threaded connections 1-4
time curve 1-6, 5-14, 7-3, 7-20, 7-27
time-dependent material
properties 1-6
total Lagrangian formulation 2-2,
COSMOSM Advanced Modules
3-11, 3-14, 8-72, 8-74, 8-90
total strain 3-6, 3-64
Tresca yield criterion 8-228
Tresca-Saint Venant yield
criteria 3-21
Tsai-Wu failure 3-3, 3-4, 8-117, 8119
two-node gap 4-6, 4-17
types of strain output 3-64
U
U/P formulation 3-19
UMODEL 3-43, 3-44
uniaxial compression test 3-38, 339, 3-40
uniaxial creep law 3-31
uniaxial tension test 3-38
uniaxial test 3-67
unsymmetric behavior 8-186
updated Lagrangian
formulation 2-2
V
viscoelastic material model 5-16
viscoelastic model 3-30, 3-32, 3-77
viscoelasticity 3-67, 8-143, 8-148,
8-153
volumetric strain 3-6, 3-28
von Mises 7-22, 8-216, 8-220
von Mises Yield Criterion 8-228
von Mises yield criterion 3-19, 322, 7-16, 7-18
W
Wilson-Theta 5-11, 7-8
wrinkling membrane 3-35, 3-36, 517, 8-178, 8-181
X
X-Y plot 7-4
X-Y-plot 7-9, 7-15
Y
yield criterion 3-27
yield Stress 7-18
I-3