Section2 Module4 Static Analysis (1)

8/12/2019 Section2 Module4 Static Analysis (1)

http://slidepdf.com/reader/full/section2-module4-static-analysis-1 1/24

© 2011 AutodeskFreely licensed for use by educational institutions. Reuse and changes require a note indicating

that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunity

Education Community

Static Analysis:Static Analysis





Education Community

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.

Section II – Static Analysis

Module 4 – Static Analysis





Education Community

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.







Education Community

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







Education Community

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







Education Community

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







Education Community

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







Education Community

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







Education Community

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







Education Community

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







Education Community

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







Education Community

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







Education Community

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.







Education Community

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







Education Community

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.







Education Community

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.



Close up of

mesh without

pressure.





Education Community

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





© 2011 AutodeskFreely licensed for use by educational institutions. Reuse and changes require a note indicatingthat content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunityEducation Community

Run 1 – Analysis Log

Iteration Number

Convergence

parameter for

each iteration

This iteration

converged in 5

iterations

Each load increment required five iterations.







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.







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.







Run 2 – Analysis Log

Iteration Number

Convergenceparameter for

each iteration

Note that only six iterations were required

with one load increment.





© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicatingthat content has been modified from the original, and must attribute source content to Autodesk.


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.







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.







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.



Section2 Module4 Static Analysis (1)

Documents

Transcript of Section2 Module4 Static Analysis (1)