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Static Analysis:Static Analysis
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Objective
The objective of this module is to introduce the methods usedto 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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Governing Equations
The governing equations for a static finite element analysis canbe written as
.unbT
Ru K
The tangent stiffness matrix, , has three components
.321 K u K K K T
u
T K
321 and, K u K K
Where are the linear, displacement, andstress dependent contributions.
are the displacement increments.
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Governing Equations
is the unbalanced load array. It is the difference betweentwo arrays.
is an array of external forces acting on the nodes. Thisarray is obtained from the external virtual work term.
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 is zero.
int R F Rex t unb
unb R
ex t F
int R
unb
R
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Graphical Illustration
The solution of this equation can
be illustrated graphically for a
single degree-of-freedom system.
Point 1 lies on the solution pathand is in equilibrium.
Point 1 can be at any configuration
that is in equilibrium.
Point 2 is the desired solution
point and is also in equilibrium.
Point A is an estimate for point 2
based on the tangent stiffness and
displacement increment, u.
int R F u K ext T
ex t F
u
1u
2uu
Slope = KT
int R
DesiredSolution
Point
1
2 A
unb
R
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Graphical Illustration
The displacement increment, u,can be found by inverting the
tangent stiffness matrix
The total displacement for point
A is
If the solution path is linear,
points A and 2 will be coincident
and point 2 would be in
equilibrium.
ex t F
u
1u
2uu
Slope = KT
int R
Desired
SolutionPoint
1
2 A
.int
1 R F K u
ext T
Au
.1 uuu A
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Iterative Solution
In the case of a material orgeometric non-linearity, Point A
will only provide an
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 trueequilibrium configuration at Point
2.
ex t F
u
1u
2uu
Slope = KT
int R
Desired
SolutionPoint
1
2 A
Au
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Newton-Raphson Method
The derivation of the governing equation
was based on the Newton-Raphson method.
There are two fundamental iteration methods that can be usedwith 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.
unbT
Ru K
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Full Newton-Raphson Iteration
A full Newton-Raphson iterationuses a new tangent stiffness
matrix based on the latest
estimate of the stresses,
displacements, and material
properties along with an
updated internal restoring force.
A sequence of new estimates is
obtained until the error isdetermined to be acceptable.
ex t F
u
1u
2uu
Slope = KT
int R
1
2 A
Au
B
unb Rerror
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Modified Newton-Raphson Iteration
A modified Newton-Raphson
method uses a previously
factored tangent stiffness matrix
along with an updated internal
restoring force.
A sequence of new estimates isobtained 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.
ex t F
u
1u
2u
u
Slope = KT
int R
1
2 A
Au
unb Rerror
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Convergence
Both the full and modifiedNewton-Raphson iterations can
be applied repeatedly until
convergence is achieved.
The driver behind both methods
is the unbalanced load that is the
error between the desired
equilibrium point and the
current estimate.
Either the equilibrium error ordisplacement change can be
used to determine convergence.
For example, an error tolerancebased on the ratio of the most
recently computed displacement
increment to the sum of all
displacement increments for the
current load increment is
.RatioError
Reference
Current
uu
uu
T
T
ConvergedToleranceRatioError
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Simulation Iteration Controls
Newton-Raphson
Iterations
Modified Newton-
Raphson Iterations
Combination of full and
modified Newton-
Raphson iterations
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.
Control parameters
used with the
Combined Newton
Option
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Simulation Convergence Tolerance
User can select type of
convergence criteria to
use
Default displacement
convergence tolerance
Use default convergence
tolerance if checked
Simulation allows the user to change the type of convergence criterion used andthe associated convergence tolerance.
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Solution Methods
Both of the Newton-Raphsoniteration methods requires the
solution of the equation
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.
For example, the Choleskyfactorization method
decomposes the tangent
stiffness matrix into lower and
upper triangular matrices
The lower triangular matrix has
only non-zero elements on or
below the diagonal, while theupper triangular matrix only has
non-zero terms on or above the
diagonal.
.unbT
Ru K
.T T
L LU L K
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Iterative Methods
Iterative methods are based onan additive decomposition of the
stiffness matrix
The governing equation then
becomes
.U L K T
unb
RuU L
or
.1 iunbi uU Ru L
If an initial guess is made for thedisplacement 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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Example Problem
The iteration and convergence character of nonlinear solutions will bedemonstrated 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-sidenodes to improve the bending
response of the brick elements.
The elements are generated with a
1/16 inch absolute mesh size.
It is subjected to gravity and a 2 psipressure on its top surface.
The material is 6061-T6.
Section II – Static Analysis
Module 4 – Static Analysis
Page 16
Close up of
mesh without
pressure.
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Run 1 – Analysis Parameters
Load is applied in five increments
A maximum of 10
iterations per loadincrement will be
performed
A displacement-based
tolerance ratio of
0.0001 will indicate
that equilibrium has
been achieved
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Run 1 – Analysis Log
Iteration Number
Convergence
parameter for
each iteration
This iteration
converged in 5
iterations
Each load increment required five iterations.
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Run 1 - Results
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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Run 2 – Analysis Parameters
Load is applied in one increment.
A maximum of 10
iterations per loadincrement will be
performed.
A displacement-based
tolerance ratio of
0.0001 will indicate
that equilibrium has
been achieved.
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Run 2 – Analysis Log
Iteration Number
Convergenceparameter for
each iteration
Note that only six iterations were required
with one load increment.
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Run 2 - Results
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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Example Summary
Both of the runs presentedobtained similar answers for
different combinations of load
increments and iterations.
Both runs used a full Newton-
Raphson iteration.
A modified Newton-Raphson
iteration had trouble converging
for this problem.
Although not shown, a fullNewton-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 bestmethod for a particular problem.
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Module Summary
This module has provided anintroduction to the solution
methods used in static analysis.
Full and modified Newton-
Raphson equations are
presented and illustrated.
The driver behind static solution
methods is the unbalanced load
vector that approaches zero as
the solution approachesequilibrium.
The methods presented areapplicable to linear and non-
linear problems involving either
material or geometric non-
linearities.
The solution for a linear system
simply converges in one iteration
whereas the solution for a non-
linear system requires multiple
iterations.
Section II – Static Analysis
Module 4 – Static Analysis
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