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2015 ANSYS, Inc. April 14, 2015 1
ANSYS Mechanical Introduction to Structural Nonlinearities
16.0 Release
Lecture 1: Overview
2015 ANSYS, Inc. April 14, 2015 2
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.
2015 ANSYS, Inc. April 14, 2015 3
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.
2015 ANSYS, Inc. April 14, 2015 4
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
2015 ANSYS, Inc. April 14, 2015 5
... 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.
2015 ANSYS, Inc. April 14, 2015 7
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).
2015 ANSYS, Inc. April 14, 2015 8
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)
2015 ANSYS, Inc. April 14, 2015 10
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
{
2015 ANSYS, Inc. April 14, 2015 11
... 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.
2015 ANSYS, Inc. April 14, 2015 12
... 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.
2015 ANSYS, Inc. April 14, 2015 13
... 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
2015 ANSYS, Inc. April 14, 2015 14
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!
2015 ANSYS, Inc. April 14, 2015 15
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.
2015 ANSYS, Inc. April 14, 2015 16
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.
2015 ANSYS, Inc. April 14, 2015 17
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.