Fea theory

Introduction to Finite Element Analysis
(FEA) or Finite Element Method (FEM)

The Finite Element Analysis (FEA) is a
numerical method for solving problems of
engineering and mathematical physics.
Useful for problems with complicated
geometries, loadings, and material properties
where analytical solutions can not be
obtained.
Finite Element Analysis (FEA) or Finite
Element Method (FEM)

The Purpose of FEA
Analytical Solution
• Stress analysis for trusses, beams, and other simple
structures are carried out based on dramatic simplification
and idealization:
– mass concentrated at the center of gravity
– beam simplified as a line segment (same cross-section)
• Design is based on the calculation results of the idealized
structure & a large safety factor (1.5-3) given by experience.
FEA
• Design geometry is a lot more complex; and the accuracy
requirement is a lot higher. We need
– To understand the physical behaviors of a complex
object (strength, heat transfer capability, fluid flow, etc.)
– To predict the performance and behavior of the design;
to calculate the safety margin; and to identify the
weakness of the design accurately; and
– To identify the optimal design with confidence

Brief History
Grew out of aerospace industry
Post-WW II jets, missiles, space flight
Need for light weight structures
Required accurate stress analysis
Paralleled growth of computers

Common FEA Applications
Mechanical/Aerospace/Civil/Automotive
Engineering
Structural/Stress Analysis
 Static/Dynamic
 Linear/Nonlinear
Fluid Flow
Heat Transfer
Electromagnetic Fields
Soil Mechanics
Acoustics
Biomechanics

Complex Object Simple Analysis
(Material discontinuity,
Complex and arbitrary geometry)
Discretization
Real
Word
Simplified
(Idealized)
Physical
Model
Mathematical
Model
Discretized
(mesh)
Model

Discretizations
Model body by dividing it into an
equivalent system of many smaller bodies
or units (finite elements) interconnected at
points common to two or more elements
(nodes or nodal points) and/or boundary
lines and/or surfaces.

Elements & Nodes - Nodal Quantity

Feature
Obtain a set of algebraic equations to
solve for unknown (first) nodal quantity
(displacement).
Secondary quantities (stresses and
strains) are expressed in terms of nodal
values of primary quantity

Displacement Stress
Nodes
Strain

Examples of FEA – 1D (beams)

Advantages
Irregular Boundaries
General Loads
Different Materials
Boundary Conditions
Variable Element Size
Easy Modification
Dynamics
Nonlinear Problems (Geometric or Material)
The following notes are a summary from “Fundamentals of Finite Element Analysis” by David V. Hutton

Principles of FEA
The finite element method (FEM), or finite element analysis
(FEA), is a computational technique used to obtain approximate
solutions of boundary value problems in engineering.
Boundary value problems are also called field problems. The field
is the domain of interest and most often represents a physical
structure.
The field variables are the dependent variables of interest governed
by the differential equation.
The boundary conditions are the specified values of the field
variables (or related variables such as derivatives) on the boundaries
of the field.

For simplicity, at this point, we assume a two-dimensional case with a
single field variable φ(x, y) to be determined at every point P(x, y) such
that a known governing equation (or equations) is satisfied exactly at every
such point.
-A finite element is not a differential element of size dx × dy.
- A node is a specific point in the finite element at which the value of the
field variable is to be explicitly calculated.

The values of the field variable computed at the nodes are used to
approximate the values at non-nodal points (that is, in the element interior)
by interpolation of the nodal values. For the three-node triangle example,
the field variable is described by the approximate relation
φ(x, y) = N1(x, y) φ1 + N2(x, y) φ2 + N3(x, y) φ3
where φ1, φ2, and φ3 are the values of the field variable at the nodes, and
N1, N2, and N3 are the interpolation functions, also known as shape
functions or blending functions.
In the finite element approach, the nodal values of the field variable are
treated as unknown constants that are to be determined. The interpolation
functions are most often polynomial forms of the independent variables,
derived to satisfy certain required conditions at the nodes.
The interpolation functions are predetermined, known functions of the
independent variables; and these functions describe the variation of the
field variable within the finite element.
Shape Functions

Again a two-dimensional case with a single field variable φ(x, y). The
triangular element described is said to have 3 degrees of freedom, as three
nodal values of the field variable are required to describe the field variable
everywhere in the element (scalar).
In general, the number of degrees of freedom associated with a finite
element is equal to the product of the number of nodes and the number of
values of the field variable (and possibly its derivatives) that must be
computed at each node.
φ(x, y) = N1(x, y) φ1 + N2(x, y) φ2 + N3(x, y) φ3
Degrees of Freedom

A GENERAL PROCEDURE FOR
FINITE ELEMENT ANALYSIS
• Preprocessing
– Define the geometric domain of the problem.
– Define the element type(s) to be used (Chapter 6).
– Define the material properties of the elements.
– Define the geometric properties of the elements (length, area, and the like).
– Define the element connectivities (mesh the model).
– Define the physical constraints (boundary conditions). Define the loadings.
• Solution
– computes the unknown values of the primary field variable(s)
– computed values are then used by back substitution to compute additional, derived variables, such as
reaction forces, element stresses, and heat flow.
• Postprocessing
– Postprocessor software contains sophisticated routines used for sorting, printing, and plotting
selected results from a finite element solution.

Stiffness Matrix
The primary characteristics of a finite element are embodied in the
element stiffness matrix. For a structural finite element, the
stiffness matrix contains the geometric and material behavior
information that indicates the resistance of the element to
deformation when subjected to loading. Such deformation may
include axial, bending, shear, and torsional effects. For finite
elements used in nonstructural analyses, such as fluid flow and heat
transfer, the term stiffness matrix is also used, since the matrix
represents the resistance of the element to change when subjected
to external influences.

LINEAR SPRING AS A FINITE ELEMENT
A linear elastic spring is a mechanical device capable of supporting axial
loading only, and the elongation or contraction of the spring is directly
proportional to the applied axial load. The constant of proportionality
between deformation and load is referred to as the spring constant, spring
rate, or spring stiffness k, and has units of force per unit length. As an
elastic spring supports axial loading only, we select an element coordinate
system (also known as a local coordinate system) as an x axis oriented
along the length of the spring, as shown.

Assuming that both the nodal displacements are zero when the spring is
undeformed, the net spring deformation is given by
δ= u2 − u1
and the resultant axial force in the spring is
f = kδ = k(u2 − u1)
For equilibrium,
f1 + f2 = 0 or f1 = − f2,
Then, in terms of the applied nodal forces as
f1 = −k(u2 − u1)
f2 = k(u2 − u1)
which can be expressed in matrix form as
or
where
is defined as the element stiffness matrix in the element coordinate system (or local
system), {u} is the column matrix (vector) of nodal displacements, and { f } is the
column matrix (vector) of element nodal forces.
Stiffness matrix for one spring element

The equation shows that the element stiffness matrix for the linear spring element
is a 2 × 2 matrix. This corresponds to the fact that the element exhibits two nodal
displacements (or degrees of freedom) and that the two displacements are not
independent (that is, the body is continuous and elastic).
Furthermore, the matrix is symmetric. This is a consequence of the symmetry of
the forces (equal and opposite to ensure equilibrium).
Also the matrix is singular and therefore not invertible. That is because the
problem as defined is incomplete and does not have a solution: boundary
conditions are required.
{F} = [K] {X}
with
known unknown

SYSTEM OF TWO SPRINGS
Free body diagrams:
These are internal forces
These are external forces

To begin assembling the equilibrium equations describing the behavior of the
system of two springs, the displacement compatibility conditions, which relate
element displacements to system displacements, are written as:
Writing the equations for each spring in matrix form:
And
therefore:
Superscript refers to element
Here, we use the notation f ( j )
i to represent the force exerted on element j at node i.

Expand each equation in matrix form:
Summing member by member:
Next, we refer to the free-body diagrams of each of the three nodes:

Final form:
Where the stiffness matrix:
Note that the system stiffness matrix is:
(1) symmetric, as is the case with all linear systems referred to orthogonal coordinate
systems;
(2) singular, since no constraints are applied to prevent rigid body motion of the
system;
(3) the system matrix is simply a superposition of the individual element stiffness
matrices with proper assignment of element nodal displacements and associated
stiffness coefficients to system nodal displacements.
(1)

(first nodal quantity)
(second nodal quantities)

Example with Boundary Conditions
Consider the two element system as described before where Node 1 is attached to a
fixed support, yielding the displacement constraint U1 = 0, k1= 50 lb/in, k2= 75 lb/in,
F2= F3= 75 lb for these conditions determine nodal displacements U2 and U3.
Substituting the specified values into (1) we have:
Due to boundary condition

Example with Boundary Conditions
Because of the constraint of zero displacement at node 1, nodal force F1 becomes an
unknown reaction force. Formally, the first algebraic equation represented in this
matrix equation becomes:
−50U2 = F1
and this is known as a constraint equation, as it represents the equilibrium condition
of a node at which the displacement is constrained. The second and third equations
become
which can be solved to obtain U2 = 3 in. and U3 = 4 in. Note that the matrix
equations governing the unknown displacements are obtained by simply striking out
the first row and column of the 3 × 3 matrix system, since the constrained
displacement is zero (homogeneous). If the displacement boundary condition is not
equal to zero (nonhomogeneous) then this is not possible and the matrices need to be
manipulated differently (partitioning).

Truss Element
The spring element is also often used to represent the elastic nature of supports for
more complicated systems. A more generally applicable, yet similar, element is an
elastic bar subjected to axial forces only. This element, which we simply call a bar or
truss element, is particularly useful in the analysis of both two- and three-
dimensional frame or truss structures. Formulation of the finite element
characteristics of an elastic bar element is based on the following assumptions:
1.The bar is geometrically straight.
2.The material obeys Hooke’s law.
3.Forces are applied only at the ends of the bar.
4.The bar supports axial loading only; bending, torsion, and shear are not
transmitted to the element via the nature of its connections to other elements.

Truss Element Stiffness Matrix
Let’s obtain an expression for the stiffness matrix K for the beam element. Recall
from elementary strength of materials that the deflection δ of an elastic bar of
length L and uniform cross-sectional area A when subjected to axial load P :
where E is the modulus of elasticity of the material. Then the equivalent spring
constant k:
Therefore the stiffness matrix for one element is:
And the equilibrium equation in matrix form:

Truss Element Blending Function
An elastic bar of length L to which is affixed a uniaxial coordinate system x with its
origin arbitrarily placed at the left end. This is the element coordinate system or
reference frame. Denoting axial displacement at any position along the length of the
bar as u(x), we define nodes 1 and 2 at each end as shown and introduce the nodal
displacements:
u1=u (x=0) and u2 = u(x = L).
Thus, we have the continuous field variable u(x), which is to be expressed
(approximately) in terms of two nodal variables u1 and u2. To accomplish this
discretization, we assume the existence of interpolation functions N1(x) and N2(x)
(also known as shape or blending functions) such that
u(x) = N1(x)u1 + N2(x)u2

To determine the interpolation functions, we require that the boundary values of u(x)
(the nodal displacements) be identically satisfied by the discretization such that:
u1=u (x=0) and u2 = u(x = L).
lead to the following boundary (nodal) conditions:
N1(0) = 1 N2(0) = 0
N1(L) = 0 N2(L) = 1
As we have two conditions that must be satisfied by each of two one-dimensional
functions, the simplest forms for the interpolation functions are polynomial forms:
N1(x) = a0 + a1x
N2(x) = b0 + b1x

Where the polynomial coefficients are to be determined via satisfaction of the
boundary (nodal) conditions. Application of conditions yields a0 = 1, b0 = 0 ,
therefore a1 = −(1/L) and b1 = x/L. Therefore, the interpolation functions are
N1(x) = 1 − x/L
N2(x) = x/L
Therefore the final expression of the blending function is:
u(x) = (1 − x/L)u1 + (x/L)u2
And in matrix form:
This is the displacement field in terms of nodal variables.
u2
u1
u1 contribution
u2 contribution
total interpolation

Truss Element Example
Figure depicts a tapered elastic bar subjected to an applied tensile load P at one end
and attached to a fixed support at the other end. The cross-sectional area varies
linearly from A0 at the fixed support at x = 0 to A0/2 at x = L. Calculate the
displacement of the end of the bar (a) by modeling the bar as a single element
having cross-sectional area equal to the area of the actual bar at its midpoint along
the length, (b) using two bar elements of equal length and similarly evaluating the
area at the midpoint of each, and compare to the exact solution.

Truss Element Example Solution a)
For a single element, the cross-sectional area is 3A0/4 and the element
“spring constant” and element equation are:
and
Applying the constraint condition U1 = 0, we find for U2 as the
displacement at x = L

Truss Element Example Solution b)
Two elements of equal length L/2 with associated nodal displacements. For element
1, A1 = 7A0/8 so
while for element 2, we have
Since no load is applied at the center of the bar, the equilibrium equations for the
system of two elements is:
Applying the constraint condition U1 = 0 results in

Truss Element Example Solution b)
Adding the two equations gives
and substituting this result into the first equation results in
Comparing the displacement for the three solution at x = L:
a)
b)
c) Exact solution

Truss Element Example Solution Comparison
Deflection
Node
u(x)
Shape function for interpolated values: u(x) = (1 − x/L)u1 + (x/L)u2

Truss Element Example Solution Comparison
Stress
Node
For stress results are much different, discontinuous for FEA and highly dependent on number of nodes

Beam Element
The usual assumptions of elementary beam theory are
applicable here:
1. The beam is loaded only in the y direction.
2. Deflections of the beam are small in comparison to the
characteristic dimensions of the beam.
3. The material of the beam is linearly elastic, isotropic, and
homogeneous. The beam is prismatic and the cross section
has an axis of symmetry in the plane of bending.

Beam Element
The equation shows that the element stiffness matrix for the beam element is a 4 ×
4 matrix. This corresponds to the fact that the element exhibits four degrees of
freedom and that the displacements are not independent (that is, the body is
continuous and elastic).
Furthermore, the matrix is symmetric. This is a consequence of the symmetry of the
forces and moments (equal and opposite to ensure equilibrium).
Also the matrix is singular and therefore not invertible. That is because the
problem as defined is incomplete and does not have a solution: boundary
conditions are required.

Beam Element Shape Function and
Stiffness Matrix
Shape function:
With
And the Stiffness Matrix:

Way of Stacking Blocks/Elements
• Compatibility requirement: ensures that the
“displacements” at the shared node of adjacent
elements are equal.
• Equilibrium requirement: ensures that elemental
forces and the external forces applied to the
system nodes are in equilibrium.
• Boundary conditions: ensures the system satisfy
the boundary constraints and so on.

Limitations
of Regular
FEA
Software
• Unable to handle
geometrically
nonlinear - large
deformation problems:
shells, rubber, etc.

Interpolation Functions for General
Element Formulation
In finite element analysis, solution accuracy is judged in terms of
convergence as the element “mesh” is refined.
There are two major methods of mesh refinement.
In the first, known as h-refinement, mesh refinement refers to the process
of increasing the number of elements used to model a given domain,
consequently, reducing individual element size.
In the second method, p-refinement, element size is unchanged but the
order of the polynomials used as interpolation functions is increased.
The objective of mesh refinement in either method is to obtain sequential
solutions that exhibit asymptotic convergence to values representing the
exact solution.

Fea theory

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