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Review of the Finite Element Method
▪ Discretization: create a finite element mesh
ME274 Introduction
▪ Apply boundary conditions
▪ Element level calculations: 𝒌𝒆 𝒅𝒆 = 𝒇𝒆
(variety of element types: springs, trusses, beams, 2d and 3d solids)
▪ Assembly and global solution: 𝑲 𝒅 = 𝒇
▪ Postprocessing, e.g. stress recovery
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Administrative Information
ME 160 or ME 273 Limitations
▪ Dr. Eduardo Chan, Eduardo.Chan@sjsu.edu
▪ Linearity
▪ Office Hours: Tuesday/Thursday 8:45-9:15pm
▪ Isotropic Elasticity
▪ Lectures on Tuesdays & Labs on Thursdays
▪ Static analyses
▪ Grading:
▪ Single parts (mainly)
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Homework - 25%: homeworks + lab assignments
Tests - 25%
Project – 20%
Final exam – 30%
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Today’s Agenda
What You Should Know…
▪ Review Basic FEA knowledge: discretization, assembly, solution, accuracy
▪ Good working knowledge of Ansys
▪ Limitations of basic FEA: linear, elasticity, isotropic, static, single part
▪ 2D & 3D modeling
– 2D: plane stress, plane strain and axisymmetry
– 3D: solids, shells, mixed modeling, and beams
▪ Advanced FEA topics:
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shell modeling and composites
assemblies
heat transfer
transient problems: structural dynamics, transient thermal
nonlinear problems, introduction to types of nonlinearities
▪ mesh controls
– different mesh controls to refine meshes (max element size, edge curvature ratio etc)
– isolation for exclusion to handle singularities
▪ Convergence of results
– convergence using different measures (not using auto convergence)
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1/27/2022
FEA Software Capabilities
Other physics: Heat Transfer
▪ Some of the Ansys analysis capabilities
▪ Example: Steady-State Heat Transfer
𝛁 ∙ −𝜿𝛁𝑻 = 𝑸
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linear statics and thermal analyses
shells and composites
assembly modeling
structural dynamics via modal analysis
transient thermal analysis
nonlinear static: contact and large displacement (proportional load)
material nonlinearity - hyperelasticity & plasticity
– convective boundary conditions = Newton's Law of cooling: q = h(T - T)
– nonlinear: h = h(T), radiation (not covered in this course)
𝑞=𝐾
𝜕𝑇
𝜕𝑛
T
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Introduction to Advanced Topics
Transient Analysis
▪ Shells “2½ D” elements:
▪ Structural Dynamics
𝑲 𝒅(𝒕) + 𝑪
w, z
Local
Normal
To surface
𝝏 𝒅(𝒕)
𝝏𝟐 𝒅(𝒕)
+ 𝑴
= 𝒇(𝒕)
𝝏𝒕
𝝏𝒕𝟐
– modal superposition - time analysis, frequency analysis, random analysis
– based on modal analyses
v, y
u, x
▪ Transient thermal (direct time solution)
▪ Composite and “laminate shell theory”
𝝏
𝝆𝒄𝑻 + 𝛁 ∙ −𝜿𝛁𝑻 = 𝑸
𝝏𝒕
Layer 1: orientation +450
– direct time solution
Layer 2: orientation -450
Layer 1: orientation +450
Layer 2: orientation -450
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Assembly Analysis
Prestressed Analyses
▪ Multiple parts with interfaces
▪ Prestressed Analysis:
– effect of superimposing a second loading condition on a body that’s already under stress
– “stress stiffness”
– cheap alternative to nonlinear analysis
▪ Types of Interfaces:
– “welded" or "glued", compatible mesh (default)
– free
– contact (nonlinear)
▪ Buckling
𝝀 𝑲𝝈 + 𝑲
𝒅 =𝟎
▪ Prestressed Static
𝑲𝝈 + 𝑲
𝒅 = 𝒇
▪ Prestressed Modal
−𝝎𝟐 𝑴 + 𝑲𝝈 + 𝑲
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𝒅 =𝟎
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Nonlinear Analysis
Nonlinear Analysis
▪ Assumptions of linearity:
– small displacement - negligible difference between initial and deformed positions
– small strain
– linear stress-strain relation
N
f
▪ Consequences of linear assumptions:
sliding contact
bearing contact
– solution proportional to load
– superposition, e.g. load set summing, modal analysis/superposition
•f <  N
•f >= N
Pressure
p1
p2
p1
p2
time
(non-proportional loading)
time
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Nonlinear Analysis
Numerical Consequences
▪ Governing D.E.:
▪ Governing equation:
𝒇𝒊𝒏𝒕(𝒅) = 𝒇𝒆𝒙𝒕(𝒅)
𝒇𝒊𝒏𝒕(𝒅) = 𝒇𝒆𝒙𝒕(𝒅)
conservative
(e.g. nonlinear elasticity)
▪ Requires nonlinear solution, e.g. Newton-Raphson method.
▪ Types of nonlinearities:
▪ In addition to accuracy concerns of linear solution (e.g. discretization error), there are
concerns of convergence of the nonlinear solution.
▪ Material:
– 𝝈 ≠ 𝑬𝜺
– hyperelasticity (nonlinear elasticity)
– plasticity
nonconservative (hysteritic)
(e.g. plasticity )
▪ Re-meshing may be required as mesh becomes distorted during solution.
▪ Geometric:
– large displacements
– large strains: 𝜺 ≠
𝝏𝒖
𝝏𝒙
– engineering vs true strains
– 𝝈=
𝑷
𝑨
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conservative loads or follower loads
initial or final A ?
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Nonlinear Analysis
Numerical Consequences (cont)
▪ Nonlinear Stress:
▪ Mesh that is accurate for linear solution may need to be refined for nonlinear solution
(plasticity, contact regions, boundary layers in fluid dynamics etc.)
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Nonlinear Example - Contact
Nonlinear Analysis
▪ Stress comparison: linear vs nonlinear
linear
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Nonlinear Analysis
▪ Example – Thin part with large rotation
▪ Thin part with large rotation:
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shape of deformed model not captured by linear analysis
linear analysis cannot predict self-contact clearance
linear stresses are off by about 11%
need to use nonlinear or large deformation analysis !!!
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Nonlinear Analysis
Nonlinear Analysis
▪ Displacement comparison: linear vs nonlinear
▪ Example - Thin plate with “geometric stiffness”
linear
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Nonlinear Analysis
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nonlinear
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Nonlinear Analysis
▪ Results comparison: linear vs nonlinear
linear
nonlinear
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Nonlinear Analysis
▪ Displacement vs Load Graph
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Nonlinear Analysis
▪ Thin plate with “geometric stiffness”
– linear analysis predicts maximum displacement 6 times larger than nonlinear results
– linear analysis predicts stresses more than twice as high as nonlinear results
– need to use nonlinear / large deformation analysis!!!
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