CE4011
FINITE ELEMENT METHODS
IN STRUCTURAL ENGINEERING
Munidasa P. Ranaweera
Sri Lanka Institute of Information Technology
Nonlinear Analysis
01 May 2017
NONLINEAR PROBLEMS
These are problems involving nonlinearities
due to nonlinear materials, large
displacements and changes in boundary
conditions, etc.
Principle of superposition cannot be used in
these cases.
These are normally solved using incremental
procedures
NONLINEARITIES
1.Material Nonlinearities
– Nonlinear elastic (rubber like material)
s
e
– Plastic (Mild steel, Aluminium)
s
e
– Creep( time dependent deformation under
constant stress) e
t
NONLINEARITIES
2. Geometric Nonlinearities
– Large displacements (cables, buckling,
etc.)
• Equilibrium equations must be written w.r.t.
deformed geometry
• Change in load direction (inflating membranes)
If material is elastic, can solve as optimization problems using PMPE
NONLINEARITIES
3. Boundary Nonlinearities
– Contact problems
• Closing gap
• Sliding contact (friction)
P
d
P
d
A NONLINEAR PROBLEM
A Steel Bar Bending Over a Cylinder
Has all three types of nonlinearities (material, geometric, boundary)
NONLINEAR PROBLEMS THAT CAN BE
HANDLED BY SAP2000
STATIC
Buckling
P-Delta Analysis
Tension or Compression only Materials
Staged Construction
Push Over Analysis
DYNAMIC
Non-linear Time History Analysis
NONLINEAR STATIC ANALYSIS WITH SAP2000
1. Buckling & P-Delta Analysis
This option accounts for the effect of axial load
upon the transverse bending behaviour of
elements.
The stiffness of the element is modified to take
the axial load into account.
D
H
P
Compression
reduces the
stiffness
Tension
increases the
stiffness
Refer to Demos:
https://www.youtube.com/watch?v=iKINDydUwSM
http://www.youtube.com/watch?v=YzWocAtK03I
Frame & Loading
Deformed Shapes
Support Reactions
Shear Force Diagrams
Bending Moment Diagrams
NONLINEAR STATIC ANALYSIS WITH SAP2000
2. Tension or Compression Only Material
(e.g Soil – compression only)
These can be modelled by using FRAME
elements where the compression/tension
limit is set to zero. These elements will be
activated in a nonlinear analysis only.
If a linear analysis is done with these elements
they behave as normal elements.
Eg: Beam on Elastic Foundation - Linear Analysis
Material capable of taking Tension and Compression
50 kN
80 mm
Beam (80x80 mm) – L = 2 m, E=200 GPa, v = 0.3
Elastic foundation
ko = 0.25 MPa/mm,
200 mm
Spring
MODEL
0.5k1
k1
k = 80 ko = 20 N/mm/mm
Spring stiffness k1 = 200x20 = 4000 N/mm
k1
0.5k1
+0.5 mm
-8 mm
Deflections
Reactions
BMD
Eg: Beam on Elastic Foundation – Nonlinear Analysis
Material capable of taking Compression Only
50 kN
80 mm
Beam (80x80 mm) – L = 2 m, E=200 GPa, v = 0.3
No Tension Material
ko = 0.25 MPa/mm,
k = 80 ko = 20 N/mm/mm.
For prop of height 100 mm , A = 2 mm2 , to give k = 20 N/mm/mm
Deflection
AFD
-25 mm
200 mm
beam
A
props
A
0.5A
225 mm
100 mm
No-tension
prop
MODEL
0.5A
NONLINEAR STATIC ANALYSIS WITH SAP2000
3. Staged Construction.
Staged construction analysis takes into
account the stages and the sequence of the
construction process.
This is especially useful in high-rise buildings,
where accumulation of displacements is
important.
When one adds a storey as a stage, SAP will
put the new nodes in the added stage, that
are not in the already constructed stages, in
the position of the unstressed initial model.
(For a demo refer to: http://www.youtube.com/watch?v=_Pkh624_jxc)
Two-Bay 3 Story Frame – Normal Analysis
Staged Construction
Staged Construction
Summary of Results
Quantity
Normal Analysis
Staged Construction
Maximum Deflection
0.200 in
0.379 in
Maximum Axial Force
2.70 kip
2.70 kip
Maximum Shear Force
0.667 kip
0.912 kip
Maximum Bending Moment
68.0 kip in
95.2 kip in
NONLINEAR STATIC ANALYSIS WITH SAP2000
4. Push Over Analysis
A static non-linear analysis under
permanent vertical loads and gradually
increasing lateral loads.
The analysis will indicate any
premature failure or weakness in the
structure.
(Refer paper by Habibullah A & Pyle S for a SAP2000 application)
NONLINEAR DYNAMIC ANALYSIS WITH SAP2000
Non-linear Time History Analysis
For structural systems which are primarily
linear elastic, but which have a limited
number of predefined nonlinear elements.