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ANSYS Mechanical Introduction to Structural Nonlinearities

16.0 Release

Lecture 1: Overview

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Chapter Overview

In this chapter, an overview of the basics of nonlinear finite-

element analysis (FEA) is presented:

A. What is Nonlinear Behavior?

B. Types of Nonlinearities

C. Nonlinear solution using linear solvers

D. Nonlinear FEA issues

The purpose is to give you an understanding of the fundamental nature of nonlinear FEA.

The capabilities described in this section are generally applicable to Structural licenses and above.

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A. What is nonlinear behavior?

Recall, in the 1600s, Robert Hooke discovered a simple linear relationship between force (F) and displacement (u), known as Hookes Law:

F = Ku

The constant K represents structural stiffness.

A linear structure obeys this linear relationship.

A common example is a simple spring:

K

F

u K

F

u

Linear structures are well-suited to finite-element analysis, which is based on linear matrix algebra.

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Significant classes of structures do not have a linear relationship between force

and displacement.

Because a plot of F versus u for such structures is not a straight line, such

structures are said to be nonlinear.

The stiffness is no longer a constant, but varies as you progress through the load path

KT (tangent stiffness) represents the tangent to the force deflection curve at a particular point in the load path.

... What is nonlinear behavior?

F

u

KT

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... What is nonlinear behavior? A structure is nonlinear if the loading causes significant changes in stiffness.

Typical reasons for stiffness change are:

Strains beyond the elastic limit (plasticity)

Large deflections, such as a loaded fishing rod

Changing Status (Contact between two bodies, Element birth/death)

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B. Types of Nonlinearities

There are three main sources of nonlinearities:

Geometric nonlinearities: If a structure experiences large deformations, its

changing geometric configuration can

cause nonlinear behavior.

Material nonlinearities: A nonlinear stress-strain relationship, such as metal plasticity shown on

the right, is another source of nonlinearities.

Contact: A changing status nonlinearity, where an abrupt change in stiffness may occur when

bodies come into or out of contact with each

other.

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Types of Nonlinearities

Of course, all three types of nonlinearities can be encountered in combination.

Mechanical can readily handle combined nonlinear effects.

Rubber Boot Seal

An example of nonlinear geometry (large strain and large deformation), nonlinear material (rubber), and changing status nonlinearities (contact).

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B. Nonlinear solution using linear solvers

How does Mechanical solve for a changing stiffness?

In a nonlinear analysis, the response cannot be predicted directly with a set of linear equations.

However, a nonlinear structure can be analyzed using an iterative series of linear approximations, with corrections.

Mechanical uses an iterative process called the Newton-Raphson Method. Each iteration is known as an equilibrium iteration.

F

u Displacement

Load

1

2 3

4 A full Newton-Raphson iterative analysis for one increment of load. (Four iterations are shown.)

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Nonlinear solution using linear solvers

In the Newton-Raphson Method, the total load Fa is applied in iteration 1. The result is x1. From the displacements, the internal forces F1 can be calculated. If Fa F1, then the system is not in equilibrium. Hence, a new stiffness matrix (slope of dotted line) is calculated based on the current conditions. The difference of Fa - F1 is the out-of-balance or residual forces. The residual forces must be small enough for the solution to converge.

This process is repeated until Fa = Fi. In this example, after iteration 4, the system achieves equilibrium and the solution is said to be converged.

Fa

x

1

2

3 4

Newton-Raphson Method

F1

x1

The actual relationship between load and displacement (shown with a blue dotted line) is not known beforehand. Consequently, a series of linear approximations with corrections is performed. This is a simplified explanation of the Newton-Raphson method (shown as solid red lines)

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Nonlinear solution using linear solvers

The difference between external and internal loads, {Fa} - {Fnr}, is called the

residual. It is a measure of the force imbalance in the structure.

The goal is to iterate until the residual becomes acceptably small; that is, until the solution is converged.

When convergence is achieved, the solution is in equilibrium, within an acceptable tolerance.

{Fa} {Fnr}

Fa

Fnr

u

{

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... Nonlinear solution using linear solvers

F

u Displacement

Load

ustart

Diverging!

F

u Displacement

Load

ustart

Converged

Starting outside the

radius of convergence Starting inside the radius

of convergence

The Newton-Raphson method:

Is not guaranteed to converge in all cases!

Will converge only if the starting configuration is inside the radius of convergence.

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... Nonlinear solution using linear solvers

Two techniques can help you obtain a converged solution:

F

u ustart

Apply load incrementally to move

the target closer to the start

F1

F

u ustart

Use convergence-enhancement

tools to enlarge the radius of

convergence

Mechanical combines both strategies to obtain convergence.

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... Nonlinear solution using linear solvers As a general rule, sudden changes to any aspect of a system will cause

convergence difficulties.

With this in mind, it is useful to understand how loads are managed

Load steps differentiate changes in general loading.

In the Figure at the bottom right, Fa and Fb are loadsteps.

Substeps apply the loads in an incremental fashion

Because of the complex response, it

may be necessary to apply the load

incrementally. For example, Fa1 may be

near 50% of the Fa load. After the load

for Fa1 is converged, then the full Fa load

is applied. Fa has 2 substeps while Fb

has 3 substeps in this example

Equilibrium iterations are the corrective solutions to obtain a converged substep

In this example, the iterations between the dotted lines indicate equilibrium iterations.

Fa

xa

Fb

xb

Fa1

Fb2

Fb1

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C. Nonlinear FEA Issues Three main issues arise whenever you do a nonlinear finite element analysis:

Obtaining convergence Balancing expense versus accuracy Verification

It takes care and skill to juggle these three issues successfully!

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Nonlinear FEA Issues

Obtaining convergence

Usually your biggest challenge.

Solution must start within the radius of convergence.

The radius of convergence is unknown!

If solution converges, the start was within the radius.

If solution fails to converge, the start was outside the radius.

Trial-and-error is sometimes required.

Experience and training reduce your trial-and-error effort.

Difficult problems might require many load increments, and many iterations at

each load increment, to reach convergence.

When many iterations are required, the overall solution time increases.

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Nonlinear FEA Issues

Balancing expense versus accuracy

All FEA involves a trade-off between expense (elapsed time, disk and memory requirements) and accuracy.

More detail and a finer mesh generally lead to a more accurate solution, but require more time and system resources.

Nonlinear analyses add an extra factor, the number of load increments, which affects both accuracy and expense. More increments =improve the accuracy, with increase the expense.

Other nonlinear parameters, such as contact stiffness (discussed later), can also affect both accuracy and expense.

Use your own engineering judgment to determine how much accuracy you need, how much expense you can afford.

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Nonlinear FEA Issues Verification

In a nonlinear analysis, as in any finite-element analysis, you must verify your results.

Due to the increased complexity of nonlinear behavior, nonlinear results are generally more difficult to verify.

Sensitivity studies (increasing mesh density, decreasing load increment, varying other model parameters) become more expensive.

Stress

Mesh Density

Typical Sensitivity Study

Later chapters will provide modeling tips for different nonlinear situations.