### Reality is Nonlinear

Structural analysis is nonlinear in nature because the real-world is nonlinear. However, sometimes the structural engineers can do good approximations by approaching the problem with a linear analysis and obtaining good results. But also, it may yield to wrong results if significant nonlinear behaviors occur in the structure.

In the beginning, in the 1970s, most FEA software did not have nonlinear capabilities. This feature was implemented around 1977, when database technology was introduced. To obtain a solution, a nonlinear analysis requires an iterative and incremental process. However, the first implementations of nonlinear capabilities did not use automatic methods, and the user’s intervention was required at every iteration.

Nowadays, the very advanced capabilities of nonlinear solvers are based on algorithms that use automated iteration methods, with convergence criteria. To understand and use finite element nonlinear solutions, the FEA analyst must understand the deep interaction and mutual enrichment that exist between the physical aspects of a problem and its mathematical formulation.

### The Three Types of Nonlinearity

if a continuous body undergoes large deformations, the strain-displacement relations become nonlinear. Moreover, under large deformations, the stiffness of the system will change with deformation, making the problem nonlinear.**Geometric nonlinearity:**if a material does not follow Hooke’s law, nonlinear material models must be used.**Material nonlinearity:**the most frequent boundary nonlinearities are encountered in contact problems.**Boundary nonlinearity:**

### Nonlinear Geometric

**{P} = [K(u)]{u}**

### Nonlinear Material

**{P} = [K(u)]{u}**

### Boundary Nonlinearity

**{P(u)} = [K(u)]{u}**

### What is a Nonlinear System?

*In a linear analysis, it is assumed that the response of the structure (deformation, internal loads, or stresses, etc.) is linearly proportional to the applied loads. However, in real life, this response may not be linearly proportional to the applied load and then the structure must be analyzed using nonlinear assumptions. In linear static analysis, the stiffness [K] of the analyzed structure is assumed to be constant.*

**does not satisfy the principle of superposition.***using the deformed structure’s configuration, after each incremental load application.*

**the stiffness matrix is updated**### Characteristics of a Nonlinear System

**LOAD-DISPLACEMENT RELATION****STRESS-STRAIN RELATION****SCALABILITY****SUPERPOSITION****INITIAL STATE OF STRESS****LOAD HISTORY****REVERSIBILITY****SOLUTION SETTINGS**

The stiffness of the analyzed structure is not constant and varies with the applied loads. The displacements are large (translations and rotations) and are not related to the original stiffness of the structure.

Stresses and strains are not related to a linear function.

The results of a nonlinear analysis cannot be scaled.

The principle of superposition cannot be applied. If a load *P1* produces a displacement *d1* and a load *P2* a displacement *d2*, then the load *P1 + P2* will not cause a displacement *d1 + d2.*

The initial state of stress (residual stresses, temperature, pre-stressing) may be extremely important in the overall response.

The structure’s response is related to the load history: it is influenced by the loading sequence. When several subcases are applied in sequence in the structure, the end of a subcase is the initial condition for the next subcase.

The deformation of the structure is not fully reversible once the applied loads are removed.

The external loads are applied in small increments, and iterations are performed to ensure that equilibrium is satisfied at each load increment. Solution monitoring by the user is required to ensure convergence. The computing time is usually large. While linear problems always have a unique solution, a nonlinear problem might not. In fact, the iterative and incremental processes used to solve nonlinear problems may not converge and may even produce an incorrect solution at convergence.

### Do You Really Need to Conduct a Nonlinear Analysis?

A nonlinear analysis requires more resources in terms of disk space and computing time, so it is important to ensure that it is really necessary.

If you are planning to run a nonlinear analysis, you should answer a few simple questions to decide whether you really need to, or whether a linear analysis will suffice. If you answer "yes" to one of the following questions, you should go with a nonlinear analysis:

*Does the structure deform significantly?**Does the structure exhibits stress stiffening (tension-bending coupling in a membrane under pressure for example)?**Do the stresses exceed the proportional limit?**Does the stiffness of the structure change when a loading is applied?**Do you expect to capture contact conditions (engaged or disengaged) between some components of you model?*

### What Do you Need to Learn to Perform Nonlinear Analysis?

For the FEA learning process of nonlinear analysis, it is important to learn in detail the three types of nonlinearity: geometric, material and contact. It is strongly recommended to learn them separately.

It is also important to learn how the FE solvers compute nonlinear problems. The FE analyst must understand the various methods used by the solvers in order to select the proper method which better suit to her/his problem.

* "Chapter 21 – Nonlinear Static Analysis"* &

*from*

**"Chapter 23 – Nonlinear Buckling Analysis"***cover all the aspects of nonlinear analysis for solving solid mechanics and structural problems.*

**“Practical Finite Element Analysis for Mechanical Engineers – First Edition”**These chapters cover in detail the methods used to solve the three common nonlinearities mentioned above. The methods, guidelines and recommendations which permit to build reliable nonlinear analysis are presented and practical examples illustrate each type of nonlinearity.

See below, the detailed table of content for these two chapters.

### Table of Content of Chapters About Nonlinear Analysis

**Select a chapter below**

**Chapter 21 - NONLINEAR STATIC ANALYSIS**

**21.1 OVERVIEW**

**21.2 WHAT IS A NONLINEAR SYSTEM?**

**21.3 CHARACTERISTICS OF A NONLINEAR ANALYSIS**

21.3.1 LOAD-DISPLACEMENT RELATION

21.3.2 STRESS-STRAIN RELATION

21.3.3 SCALABILITY

21.3.4 SUPERPOSITION

21.3.5 INITIAL STATE OF STRESS

21.3.6 LOAD HISTORY

21.3.7 REVERSIBILITY

21.3.8 SOLUTION SETTINGS

**21.4 GEOMETRIC NONLINEARITY**

21.4.1 SOURCES OF GEOMETRICAL NONLINEARITY

21.4.2 HOW DOES NONLINEAR GEOMETRY WORK?

21.4.3 DO YOU REALLY NEED A NONLINEAR GEOMETRIC ANALYSIS?

21.4.4 THE FOLLOWER LOAD CONCEPT

21.4.5 SMALL OR LARGE STRAIN?

21.4.6 EXAMPLE OF GEOMETRIC NONLINEARITY

**21.5 MATERIAL NONLINEARITY**

21.5.1 YIELD CRITERIA

21.5.2 HARDENING RULES

21.5.3 MATERIAL MODELS

21.5.4 ENGINEERING STRESS-STRAIN OR TRUE STRESS-STRAIN?

21.5.5 HOW DOES NONLINEAR MATERIAL WORK?

21.5.6 DO YOU REALLY NEED A NONLINEAR MATERIAL ANALYSIS?

**21.6 BOUNDARY NONLINEARITY**

21.6.1 LOAD VARIATION

21.6.2 CONSTRAINT VARIATION

21.6.3 CONTACTS

**21.7 CHOOSING THE RIGHT ELEMENTS FOR A NONLINEAR ANALYSIS**

**21.8 HOW DO FEA SOFTWARE COMPUTE NONLINEAR PROBLEMS?**

21.8.1 CHARACTERIZATION AND FORMULATION OF A NONLINEAR PROBLEM

21.8.2 NEWTON-RAPHSON METHOD

21.8.3 MODIFIED NEWTON-RAPHSON METHOD

21.8.4 NEWTON-RAPHSON METHOD EXAMPLES

21.8.5 COMPUTATIONAL METHODS IN NONLINEAR ANALYSIS

21.8.6 EQUILIBRIUM PATH AND CRITICAL POINTS

21.8.7 ADAPTIVE SOLUTION STRATEGIES

21.8.8 STIFFNESS MATRIX UPDATE STRATEGIES

21.8.9 CHOOSING THE INCREMENTAL LOAD STEP

21.8.10 ARC-LENGTH METHODS

21.8.11 LINE SEARCH PROCEDURES

21.8.12 CONVERGENCE CRITERIA

21.8.13 HOW TO DEAL WITH CONVERGENCE ISSUES

21.8.14 SUMMARY OF ITERATIVE SOLUTION SCHEMES

21.8.15 HOW TO SELECT THE RIGHT ITERATIVE SOLUTION SCHEME

21.8.16 SUMMARY OF THE NONLINEAR SOLUTION STRATEGY

**21.9 GENERAL RECOMMENDATIONS FOR NONLINEAR ANALYSIS**

21.9.1 UNDERSTAND THE NONLINEAR FEATURES

21.9.2 UNDERSTAND YOUR PROBLEM AND STRUCTURAL BEHAVIOR

21.9.3 UNDERSTAND THE DIFFERENCE BETWEEN A LINEAR SUBCASE AND A NONLINEAR SUBCASE

21.9.4 SIMPLIFY YOUR MODEL

21.9.5 USE AN ADEQUATE MESH AND ELEMENT TYPES

21.9.6 APPLY LOADING GRADUALLY

21.9.7 READ THE OUTPUT

21.9.8 NUMBER OF INCREMENTS

21.9.9 CONVERGENCE PROBLEMS

21.9.10 KEEP AN EYE ON YOUR MATERIAL DEFINITION

**21.10 COMMON MISTAKES IN NONLINEAR ANALYSIS**

**21.11 EXAMPLES OF NONLINEAR STATIC ANALYSIS**

21.11.1 GEOMETRIC NONLINEARITY AND HISTORY PATH

21.11.2 CUMULATIVE EFFECT OF A NONLINEAR ANALYSIS

21.11.3 INFLUENCE OF THE INCREMENTAL LOAD STEP ON RESULTS

21.11.4 MATERIAL NONLINEARITY: ELASTOPLASTIC PLATE

21.11.5 HIGHLY NONLINEAR PROBLEM

**Chapter 23 - NONLINEAR BUCKLING ANALYSIS**

**23.1 OVERVIEW**

**23.2 WHY PERFORM A NONLINEAR BUCKLING ANALYSIS?**

**23.3 THE STABILITY PATH AND THE CONVERGED SOLUTION**

**23.4 NONLINEAR BUCKLING PROCEDURE**

**23.5 POST-BUCKLING**

**23.6 ESSENTIAL STEPS IN NONLINEAR BUCKLING ANALYSIS**

**23.7 EXAMPLES OF NONLINEAR BUCKLING ANALYSIS**

23.7.1 NONLINEAR BUCKLING OF A CURVED PANEL

23.7.2 SNAP-THROUGH: NEWTON-RAPHSON VS ARC-LENGTH

### ABOUT THE AUTHOR

**Dominique Madier**

He is the author of the book “Practical Finite Element Analysis for Mechanical Engineer”: 650+ pages about the best practical methods and guidelines for the development and validation of finite element models.