1. 1-1 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Workbench - Mechanical Structural Nonlinearities Chapter 1 Overview

2. Workbench Mechanical – Structural Nonlinearities Overview 1-2 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual In this chapter, an overview of the basics of nonlinear finite-element analysis (FEA) is presented: A.What is “Nonlinear” Behavior?What is “Nonlinear” Behavior? B.Types of Nonlinearities C.Nonlinear solution using linear solversNonlinear solution using linear solvers D.Nonlinear FEA issuesNonlinear 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. – Exceptions will be noted accordingly Chapter Overview

3. Workbench Mechanical – Structural Nonlinearities Overview 1-3 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual A. What is nonlinear behavior? Recall, in the 1600s, Robert Hooke discovered a simple linear relationship between force (F) and displacement (u), known as Hooke’s 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.

4. Workbench Mechanical – Structural Nonlinearities Overview 1-4 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual... What is nonlinear behavior? 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, K; it becomes a function of applied load, K T (the tangent stiffness). F u KTKT

5. Workbench Mechanical – Structural Nonlinearities Overview 1-5 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual 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)... What is nonlinear behavior?

6. Workbench Mechanical – Structural Nonlinearities Overview 1-6 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual 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: Include effects of contact is a type of “changing status” nonlinearity, where an abrupt change in stiffness may occur when bodies come into or out of contact with each other. B. Types of Nonlinearities

7. Workbench Mechanical – Structural Nonlinearities Overview 1-7 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual Of course, all three types of nonlinearities are commonly encountered in combination. – Workbench 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). … Types of Nonlinearities

8. Workbench Mechanical – Structural Nonlinearities Overview 1-8 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual B. Nonlinear solution using linear solvers How does Workbench 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. – ANSYS 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.)

9. Workbench Mechanical – Structural Nonlinearities Overview 1-9 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual … Nonlinear solution using linear solvers – In the Newton-Raphson Method, the total load F a is applied in iteration 1. The result is x 1. From the displacements, the internal forces F 1 can be calculated. If F a  F 1, then the system is not in equilibrium. Hence, a new stiffness matrix (slope of red line) is calculated based on the current conditions. The difference of F a - F 1 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 F a = F i. In this example, after iteration 4, the system achieves equilibrium and the solution is said to be converged. FaFa x 1 2 3 4 Newton-Raphson Method F1F1 x1x1 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)

10. Workbench Mechanical – Structural Nonlinearities Overview 1-10 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual … Nonlinear solution using linear solvers The difference between external and internal loads, {F a } - {F nr }, 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. {F a }{F nr } FaFa F nr u {

11. Workbench Mechanical – Structural Nonlinearities Overview 1-11 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual... Nonlinear solution using linear solvers F u Displacement Load u start Diverging! F u Displacement Load u start 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.

12. Workbench Mechanical – Structural Nonlinearities Overview 1-12 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual... Nonlinear solution using linear solvers Two techniques can help you obtain a converged solution: F u u start Apply load incrementally to move the target closer to the start F1F1 F uu start Use convergence-enhancement tools to enlarge the radius of convergence Workbench Mechanical combines both strategies to obtain convergence.

13. Workbench Mechanical – Structural Nonlinearities Overview 1-13 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual... 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, F a and F b 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, F a1 may be near 50% of the F a load. After the load for F a1 is converged, then the full F a load is applied. F a has 2 substeps while F b has 3 substeps in this example – Equilibrium iterations are the corrective solutions to obtain a converged substep In the example on right, the iterations between the dotted lines indicate equilibrium iterations. FaFa xaxa FbFb xbxb F a1 F b2 F b1

14. Workbench Mechanical – Structural Nonlinearities Overview 1-14 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual 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!

15. Workbench Mechanical – Structural Nonlinearities Overview 1-15 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual … 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.

16. Workbench Mechanical – Structural Nonlinearities Overview 1-16 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual … 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 load increments will often improve the accuracy, but will also generally 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.

17. Workbench Mechanical – Structural Nonlinearities Overview 1-17 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved. April 30, 2009 Inventory #002659 Training Manual … 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 case-by-case details of modeling tips for different nonlinear situations.