Introduction to Finite
Element Analysis for
Structure Design
Dr. A. Sherif El-Gizawy
Elasticity Principles
F : Applied Force & A : Area
l : Initial Length
l
A
Stress =  = F/ A
dl
dl= displacement (deformation)
F
Strain =  = dl/l
Stress-Strain Relation
(Hooke’s Law)
Elastic Deformation Zone
Modulus of Elasticity =
E = /
 = E * x


 in the x-direction

 = /E

Plastic Work (Deformation Energy)
Plastic Work/ Unit Volume = dW = F x dl/Volume
=F x dl/ (A x l) =  x 
dW =  x 
3D Stress-Strain Relationship
x = 1/E*((x- (y + z))
Where
x : normal strain along x direction
 : Poisson Ration
Shear Strain, xy = xy / G

: Shear Stress
G : Shear Modulus of Elasticity
xy
Effective Stress (Von-Mises)
1
 = ((x- y)2 + (y- z)2 + (z- x)2)1/2
2
when  reaches a certain value (yield stress),
the applied stress state will cause yielding
Effective Strain
 =(
2
(x2+
3
y2+ z2))1/2
FEM Solution for Structural Design
 = E X  (Hooke’s Law)
F = A X(E X ) = A X E (l /l)
F = (A X E/l) X l
This is an analogy to spring
Force with
A x E/l = Element Stiffness
= Keq
F = Keq
X
l
l
l
A
l
l
K
F
F
F = Keq
X
l
Fsp = K x l
 Element Stiffness = A x E/l = K
eq
 The Applied force F is given.
 Deformation (deflection or displacement)
l = F / Keq 
 = Strain = l /l (calculated) 
 = Stress = E x 
Introduction to the Finite Element Method
•Typically, for the structural stress analysis, it is required to
determine the stresses and deformation (strains) throughout
the structure which is in equilibrium and is subjected to
applied loads.
•The finite element method involves modeling of the
structure using small units (finite elements).
•A displacement function is associated with each finite
element. The followings are the steps used in finite element
method. This will be followed by illustration of the
application of these steps on springs and plane stress
cases.
Developing a Model for Finite Element Analysis
The problem to be solved is specified in a) the physical
domain and b) the discretized domain used by FEA
Line Element
Two-dimensional Elements
Three-dimensional Elements
Axisymmmetric Element
Step 1. Discretize and Select Element Types
Divide the structure into an equivalent system of finite
elements with associated nodes.
The simplest line elements, Fig.1.a has two nodes, one at each
end.
The basic two-dimensional elements, Fig. 1.b are loaded by
forces in their own plane (plane stress). They are triangular or
quadrilateral elements.
The common three dimensional elements, Fig.1.c, are
tetrahedral and hexahedral (brick) elements. They are used to
perform three dimensional stress analysis in 3-D solid bodies.
Developing a Model for Finite Element Analysis
Overall steps in FEA software
Step 2. Select a Displacement Function
•Choose a displacement function within each element
using the nodal values of the element. Linear,
quadratic, and polynomials are frequently used
functions.
Step 3. Define the Stress/Strain
Relationships
= dl/l
 = E
Step 4.
Derive the Element Stiffness Matrix and Equations
•The stiffness matrix and element equations relating nodal
forces and displacements are obtained using force equilibrium
conditions or the principle of minimum potential energy.
Step 5. Assemble the Element Equations to obtain the Global
Equations
Step 6. Solve for the Unknown Displacements
Step 7. Solve for the Element Strains and Stresses
Step 8.
Interpret the Results
•The final goal is to interpret and analyze the results for use in
the design process.
Von Mises stress in ¼
model of thin plate
under tension using 1st
order elements
A disaster waiting to
happen using first order
elements
Approximation of stress function in a model
A mesh of solid
tetrahedral (4 nodes) helements
A mesh of tetrahedral pelements produced by
MECHANICA
Two common convergence measures using p-elements
Finite Element Analysis
by Pro- Mechanica
Steps in FEA using Pro-Mechanica
 Step 1: Draw part in Pro-Engineer
 Step 2: Start Pro-Mechanica
 Step 3: Choose the Model Type
 Step 4: Apply the constraints
 Step 5: Apply the loads
 Step 6: Assign the material
 Step 7: Run the Analysis
 Step 8: View the results by post-processing
Step 1: Creation of the part
 Use Protrusion by Sweep to create this
part (bar.prt)
Step 2: Starting Pro-Mechanica
 In Pro-Engineer window, go to
Applications Mechanica to start ProMechanica.
 The part (bar.prt) will be loaded in ProMechanica with a new set of icons for
Structural, thermal Analysis
Step 3: Choosing the model type
 In Mechanica menu, select
 Structure  Model  Model Type
 Four different models can be created:




3D Model
Plane Stress
Plane Strain
2D Axisymmetric
 We will select 3D Model
Step 4: Applying the Constraints
 Create a new constraint by
 Model Constraints New Surface
 Give a name for the constraints
(fixed_face) and select the surface to be
constrained
 Specify the constraints (in our case will be
fixed for all degrees of freedom)
 Preview and press Ok
Step 5: Applying the loads
 Similar to Constraints, create a new load




by Model  Load New  Surface
Give a name for the applied load (endload)
Select the surface where the load will be
applied
Specify the loads (Fx:500, Fy:-250, Fz:0)
Preview and press Ok
Step 6: Assigning the material
 Model  Materials
 A window will pop up with the list of ProMechanica materials. Add the required
material and then assign the material to
the part.
 Click on Edit if any change in material
properties are to be made.
 Press Ok
Modeled part with constraints and
loads
Step 7: Running the Analysis
 In Mechanica menu, select Analysis
 Select File  New Static in “Analysis and
Design Studies” dialog box and give a name for
the analysis (bar).
 The constraints and loads are automatically
loaded.
 In Convergence tab, select Quick Check to
check for errors and then select Multi-pass
adaptive for the reliable and accurate results.
Change the order of the polynomial and
percentage of convergence as required.
 Finally, click on Run icon to start the analysis
(click on Display Study Status to view the current
status and completion of the analysis)
Step 8: Viewing the results
 For post-processing, select Results from ProMechanica window
 A new window will open, and click on “Insert a
New Definition” icon. In the dialog box, select
the folder where the analysis is saved.
 Select Fringe as Display type, Stress as
Quantity and von-mises as the stress
component to display
 Similarly, other quantities can be displayed in
one window.
Post-processing Results