Basic FEA Concepts
FEA Project Outline
START
Finite Element Analysis
1
2
3
Consider the physics of
the situation. Devise a
mathematical model.
Obtain approximate
results for subsequent
comparison with FEA.
Plan a finite element
discretization of the
mathematical model.
H
Physics
FEA
What is at fault,
Inadequate physical basis
for the model or a poor
finite element model?
4a
D2
D1
Are error
estimates
small? Does
mesh revision
do little to
alter the FEA
results?
No
Yes
4b
Revise the
finite
element
discretizatio
n
Solution:
Generate and
solve equations
of the finite
element model.
STOP
No
5
Preprocess:
Build the finite
element model in
the computer.
Yes
Are the FEA results free of obvious errors, such as disagreement with the intended
boundary conditions? Are FEA results physically reasonable? Do FEA results agree
reasonably well with predictions and approximations obtained by other means?
Adapted from RD Cook et al., Concepts and Applications of Finite Element Analysis, 4th ed., John Wiley & Sons, 2002
4c
Postprocess:
Output/display
computed results
for examination.
Mechanics Analysis
• Geometry
• Boundary Conditions
• Constitutive Relationships
• These basic elements the same whether the analysis is
“engineering”, theory of elasticity, continuum mechanics, or
finite element analysis
• The approachable problems are very different, with FEA being
the most general (but not necessarily the best)
Geometry
• Defines the spatial region over which the field
quantities are defined
• Defines the boundaries where surface tractions
(constraints and loads) are specified
• All problems are 3D, but certain problems lend
themselves to 2D or even 1D approaches
• Important FEA points:
– An “element” is much more than a geometric entity
– More geometric detail is not necessarily better
Boundary Conditions
• Points or surfaces where constraints are specified or
loads are applied
– Constraints are often displacement = known value points
– Loads are generally point forces or pressures
– Boundary conditions are also used to enforce symmetry in
reduced geometry problems (e.g. half, quarter models)
• Important FEA points:
– Rigid body motions must be prevented by the constraints
– It is very easy to over/under constrain a model
– Pressures end up being point (node) loads, check carefully
Constitutive Relationships
• The sophisticated term for “material properties”
• This is generally the most difficult and troublesome
part of advanced simulations
• Think more broadly than isotropic linear elasticity
– Anisotropy is common (e.g. fiber composite materials)
– Plasticity is common (what happens after yield?)
– Damage and Fracture are common
• Important FEA points:
– Many constitutive models are available
– Just because it runs, doesn’t mean it’s right
Our Introductory Problem
• Cross-sectional area:
– A = 400 mm2
• Modulus of Elasticity:
– E = 200 GPa
• Poisson’s Ratio:
– n = 0.32
• We are interested in:
– Stresses in the truss
members
– Deflections at the load
points
Getting Started
1
2
3
Consider the physics of
the situation. Devise a
mathematical model.
Obtain approximate
results for subsequent
comparison with FEA.
Plan a finite element
discretization of the
mathematical model.
1. Just mechanics going on here. Linear elasticity for now. No
friction. Small deflections.
2. You have some work to do here:
a)
b)
c)
Statics analysis to determine reaction forces and internal member forces
Simple F/A calculation for the dominant stress component
Castigliano’s approach to determine the deflections (this takes some work)
3. We need to consult our Element Library for this step
a)
b)
c)
Is this a 3D, 2D, or 1D problem?
Are there any sub-components, and are they 3D, 2D, or 1D?
What is the fundamental behavior of the sub-components?
Approximate Results
FABY
FABX
q
FAG
RAy
F1
F2
REy
RAy,REy  f F1,F2 

RAy
F   f q,F1,F2 
Stay organized, use symmetry …

Castigliano’s Theorem
For a truss, Castigliano’s Theorem becomes:
1 n N i 
P 
N i  Li

AE i1 P 
P  Displacement (in the direction of load) of point where load P acts
n = The number of members in the truss
N i  The force in the ith member of the truss
 N
P
i
 Partial derivative of ith member force with respect to the load P
L i  length of the ith member

To determine the partial derivatives, you must express the
member forces as functions of the truss loads …
Element Libraries
• Commercial Finite Element Codes have extensive element
collections (marc_2010_doc_volume_b.pdf)
• Marc, as of today, has 227 elements in the library
• Structural elements have 1D, 2D, and 3D geometry
• Geometry is not the only important consideration!
Truss Element
• Review Element 9 in the library
• This is the truss element introduced in ME 420/520
• It is 1D in geometry, thus we will have an element
property required (cross-sectional area)
• It allows no variation of element or material
properties along the element length
• This element does not have any bending stiffness
• It can be mixed together with other elements of
different materials or properties in the same truss