Static Analysis:
Static Analysis
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Section II – Static Analysis
Objective
Module 4 – Static Analysis
Page 2
The objective of this module is to introduce the methods used
to solve static problems where inertia or time-dependent
material effects are not important.
The solution methods will build on material presented in Modules 1
through 3.
 The methods are based on the Newton-Raphson method and are
applicable to the solution of non-linear geometric or material
problems.
 The solution of problems governed by linear equations is treated as a
special case of the more general non-linear methods.

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Section II – Static Analysis
Governing Equations
Module 4 – Static Analysis
Page 3

The governing equations for a static finite element analysis can
be written as
KT u  Runb.

The tangent stiffness matrix, KT  , has three components
KT   K1   K 2 u   K3  .

Where K1 , K 2 u  and K3   are the linear, displacement, and
stress dependent contributions.
u
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are the displacement increments.
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Section II – Static Analysis
Governing Equations
Module 4 – Static Analysis
Page 4
Runb is the unbalanced load array. It is the difference between
two arrays.
Runb  Fext  Rint 
Fext  is an array of external forces acting on the nodes.
This
array is obtained from the external virtual work term.
Rint  is an array of node forces associated with the stresses
inside the body. This array is obtained from the internal
virtual work term.
At equilibrium the two arrays are equal and Runb  is zero.
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Section II – Static Analysis
Graphical Illustration
Module 4 – Static Analysis
Page 5
KT u  Fext  Rint 





Slope = KT
The solution of this equation can Fext
be illustrated graphically for a
single degree-of-freedom system.
Point 1 lies on the solution path
and is in equilibrium.
Point 1 can be at any configuration R 1
int
that is in equilibrium.
Point 2 is the desired solution
point and is also in equilibrium.
Point A is an estimate for point 2
u1 u
based on the tangent stiffness and
displacement increment, u.
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Desired
Solution
Point
A 2
Runb
u
u2
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Section II – Static Analysis
Graphical Illustration
Module 4 – Static Analysis
Page 6

The displacement increment, u, Slope = KT
can be found by inverting the
tangent stiffness matrix
Fext
A 2
u  KT  Fext  Rint .
1

The total displacement for point
A is
u A  u1  u.

Desired
Solution
Point
If the solution path is linear,
points A and 2 will be coincident
and point 2 would be in
equilibrium.
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Rint
1
u
u1 u u A u2
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Section II – Static Analysis
Iterative Solution
Module 4 – Static Analysis
Page 7


Slope = KT
In the case of a material or
geometric non-linearity, Point A
will only provide an
Fext
approximation to the equilibrium
configuration at Point 2.
A numerical method is necessary
that will take the information
available and obtain an improved
estimate that is closer to the true
equilibrium configuration at Point
2.
Rint
A 2
Desired
Solution
Point
1
u
u1 u u A u2
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Section II – Static Analysis
Newton-Raphson Method
Module 4 – Static Analysis
Page 8

The derivation of the governing equation
KT u  Runb

was based on the Newton-Raphson method.
There are two fundamental iteration methods that can be used
with this method:
First is a full Newton-Raphson iteration,
 Second is a modified Newton-Raphson iteration.



These two methods can be used individually or in combination.
Each iteration method can also be used in combination with a line
search algorithm based on the method of steepest descent used in
optimization theory.
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Section II – Static Analysis
Full Newton-Raphson Iteration
Module 4 – Static Analysis
Page 9

A full Newton-Raphson iteration
uses a new tangent stiffness
matrix based on the latest
estimate of the stresses,
displacements, and material
properties along with an
updated internal restoring force.
Slope = KT
A 2
Fext
error  Runb
Rint
B
1

A sequence of new estimates is
obtained until the error is
determined to be acceptable.
u1
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uu
uA 2
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Section II – Static Analysis
Modified Newton-Raphson Iteration
Module 4 – Static Analysis
Page 10



A modified Newton-Raphson
method uses a previously
factored tangent stiffness matrix
along with an updated internal
restoring force.
A sequence of new estimates is
obtained until the error is
determined to be acceptable.
This method uses reduced
computational effort associated
with forming and factoring the
tangent stiffness matrix, but
generally requires more
iterations.
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Slope = KT
A 2
Fext
error  Runb
Rint
1
u1 u u u2
A
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Section II – Static Analysis
Convergence
Module 4 – Static Analysis
Page 11



Both the full and modified
 For example, an error tolerance
Newton-Raphson iterations can
based on the ratio of the most
be applied repeatedly until
recently computed displacement
convergence is achieved.
increment to the sum of all
displacement increments for the
The driver behind both methods
is the unbalanced load that is the current load increment is
error between the desired
Current
T
u u
equilibrium point and the
Error Ratio 
.
Reference
T
current estimate.
 u  u
Either the equilibrium error or
displacement change can be
Error Ratio  Tolerance  Converged
used to determine convergence.
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Section II – Static Analysis
Simulation Iteration Controls
Module 4 – Static Analysis
Page 12
 Simulation enables the user to select the type of equilibrium iteration to be used in
an analysis. Simulation also provides a line search option for each type of iteration.
Combination of full and
modified NewtonRaphson iterations
Newton-Raphson
Iterations
Modified NewtonRaphson Iterations
Control parameters
used with the
Combined Newton
Option
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Section II – Static Analysis
Simulation Convergence Tolerance
Module 4 – Static Analysis
Page 13
 Simulation allows the user to change the type of convergence criterion used and
the associated convergence tolerance.
User can select type of
convergence criteria to
use
Use default convergence
tolerance if checked
Default displacement
convergence tolerance
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Section II – Static Analysis
Solution Methods
Module 4 – Static Analysis
Page 14

Both of the Newton-Raphson
iteration methods requires the
solution of the equation

KT u  Runb.


Matrix inversion of the tangent
stiffness matrix is not efficient
and finite element programs rely
on factorization methods or
iteration methods.
Factorization methods
decompose the matrix into
multiplicative components.
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For example, the Cholesky
factorization method
decomposes the tangent
stiffness matrix into lower and
upper triangular matrices
KT   LU   LL .
T

The lower triangular matrix has
only non-zero elements on or
below the diagonal, while the
upper triangular matrix only has
non-zero terms on or above the
diagonal.
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Section II – Static Analysis
Iterative Methods
Module 4 – Static Analysis
Page 15

Iterative methods are based on
an additive decomposition of the
stiffness matrix

KT   L U .

The governing equation then
becomes

L  U u  Runb
or
Lui1  Runb U ui .
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
If an initial guess is made for the
displacement increment on the
right hand side of the equation,
an improved estimate can be
found by solving the left hand
side.
The additive decomposition of
the tangent stiffness matrix
takes less time than the
multiplicative decomposition.
However, iterations are required
as a trade-off.
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Section II – Static Analysis
Example Problem
Module 4 – Static Analysis
Page 16

The iteration and convergence character of nonlinear solutions will be
demonstrated with a cantilevered beam subjected to gravity and a pressure
load. The pressure load will stay normal to the surface as it deforms.
The beam is 0.125 inch thick, 1
inch wide, and 12 inches long. It
uses brick elements with mid-side
nodes to improve the bending
response of the brick elements.
The elements are generated with a
1/16 inch absolute mesh size.
Close up of
mesh without
pressure.
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It is subjected to gravity and a 2 psi
pressure on its top surface.
The material is 6061-T6.
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Section II – Static Analysis
Run 1 – Analysis Parameters
Module 4 – Static Analysis
Page 17
Load is applied in five increments
A maximum of 10
iterations per load
increment will be
performed
A displacement-based
tolerance ratio of
0.0001 will indicate
that equilibrium has
been achieved
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Section II – Static Analysis
Run 1 – Analysis Log
Module 4 – Static Analysis
Page 18
Iteration Number
Convergence
parameter for
each iteration
This iteration
converged in 5
iterations
Each load increment required five iterations.
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Section II – Static Analysis
Run 1 - Results
Module 4 – Static Analysis
Page 19
Contour plot of Von Mises
stress superimposed on
deformed shape of the
structure.
The maximum stress is
58.2 ksi.
Note the neutral axis
running down the side of
the beam.
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Section II – Static Analysis
Run 2 – Analysis Parameters
Module 4 – Static Analysis
Page 20
Load is applied in one increment.
A maximum of 10
iterations per load
increment will be
performed.
A displacement-based
tolerance ratio of
0.0001 will indicate
that equilibrium has
been achieved.
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Section II – Static Analysis
Run 2 – Analysis Log
Module 4 – Static Analysis
Page 21
Iteration Number
Convergence
parameter for
each iteration
Note that only six iterations were required
with one load increment.
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Section II – Static Analysis
Run 2 - Results
Module 4 – Static Analysis
Page 22
Contour plot of Von Mises
stress superimposed on a
deformed shape of the
structure.
The maximum stress is
58.6 ksi which compares
well with 58.2 ksi obtained
from Run 1.
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Section II – Static Analysis
Example Summary
Module 4 – Static Analysis
Page 23



Both of the runs presented
obtained similar answers for
different combinations of load
increments and iterations.
Both runs used a full NewtonRaphson iteration.
A modified Newton-Raphson
iteration had trouble converging
for this problem.
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


Although not shown, a full
Newton-Raphson iteration with
Line Search required more
iterations than the standard full
Newton-Raphson iteration.
The type of iteration and its
performance depends on the
problem.
Experience and trial and error is
required to determine the best
method for a particular problem.
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Section II – Static Analysis
Module Summary
Module 4 – Static Analysis
Page 24



This module has provided an
introduction to the solution
methods used in static analysis.
Full and modified NewtonRaphson equations are
presented and illustrated.
The driver behind static solution
methods is the unbalanced load
vector that approaches zero as
the solution approaches
equilibrium.
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

The methods presented are
applicable to linear and nonlinear problems involving either
material or geometric nonlinearities.
The solution for a linear system
simply converges in one iteration
whereas the solution for a nonlinear system requires multiple
iterations.
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