Analysis of a few sample problems helps illustrate the basics of the nonlinear approach.

Associate Professor Department of Civil and Environmental Engineering University of Rhode Island Kingston, R.I. Edited by Leslie Gordon • Simplified load-deflection cases can help explain nonlinear (structural) FEA. |

Which designers need nonlinear FEA and why? Actually, projects ranging from designing a paper clip to a high-rise can benefit from the technique. A linear FEA study somewhat simulates reality only when a number of restricting conditions are met. These include that all deflections are infinitesimally small and that stresses and strains induced by loads remain small enough to let their linear relationship (Hooke’s law) hold.

The truth is that most, if not all, problems are nonlinear from the get-go. Or they become nonlinear at higher load levels. For example, problems characterized by large deflections (compared to the problem’s dimensions) necessarily involve geometric nonlinearity. And when the relationship between stress and strain becomes nonlinear (e.g., plasticity), then the problem involves material nonlinearity.

To become familiar with some of the issues encountered with a nonlinear (structural) FEA, consider a few simplified load-deflection cases.

In *Progressive load-deflection curve*, the stiffness of the system rises with the load. Hence, it could be argued that a linear analysis (dashed line) is conservative for it errs on the safe side. So the advantage of a nonlinear analysis in a case like this lies in the ability to more realistically account for the system’s strength at higher load levels and, in so doing, build lighter structures.

Not so with regard to *Load-deflection curve with snap-through and Load-deflection curve with snap-back*, where a linear analysis clearly misses the target because it fails to detect the system’s unstable regions (dashed red lines). In these regions, even a small change in load magnitude could lead to significantly different deformation patterns.

For a basic rundown on the math the so-called and computational issues associated with solving a nonlinear problem, consider a simple coil spring. Its load-deflection curve resembles *Progressive load-deflection curve*, and is described by an equation such as:

(*k*_{0} + *k*_{1}• *x* + *k*_{2} • *x*^{2} +...)• *x* = *f*

where the *k*’s = stiffness parameters, *x* = (unknown) deflection, and *f* = (known) force in the spring.

Because the equation cannot be solved directly, an iterative scheme must be devised to obtain an approximate solution. This means the load is applied in increments. In each step, a new solution is generated using the scheme, which is described by these two equations:

*k*(*x _{old}*) • Δ

*x*= Δ

*f*+ r ;

*x*=

_{new}*x*+ Δ

_{old}*x*

where Δ*f* = force increment, Δ*x* = resulting deflection change, *k* = nonlinear stiffness (typically referred to as tangent stiffness), and *r* = unbalanced force resulting from treating the nonlinear problem as a sequence of linear problems.

In each step, iterations are performed until r is made sufficiently small (compared to a certain predefined tolerance). Generalizing the equations for the case of a system with many DOFs is straightforward and leads to:

*K*(*x _{old}*) • Δ

*x*= Δ

*f*+

*r*;

*x*=

_{new}*x*+ Δ

_{old}*x*

where the scalar variables have been replaced by vectors and matrices.

For the load-incrementation procedure to work, all the equations must be solvable for the entire load-deflection path. The closer calculations get to the system’s unstable regions (if such regions exist), the more difficult it becomes to obtain a convergent solution. (In mathematical jargon, this is referred to as ill-conditioning of algebraic equations.) Hence, in situations like those depicted in *Load-deflection curve with snapthrough and Load-deflection curve with snap-back*, a load-incrementation scheme would not be suitable for tracing the entire diagram.

With respect to a snap-through problem, a commonly used remedy is to resort to a displacement- incrementation scheme in which a suitably chosen system deflection becomes the step-wise imposed independent variable and the load is treated as a dependent variable. But this approach does not always work. Among the various alternative techniques, the so-called arc-length incrementation (Riks algorithm) is perhaps the most popular. (For details, consult with the theory manual of an advanced nonlinear FEA code such as Abaqus or Ansys.)

**Solving nonlinear problems **Consider a few nonlinear examples and the steps to complete the analysis. Examples represent the situations shown in

*Progressive load-deflection curve, Load-deflection curve with snap-through, and Load-deflection curve with snap-back*. Ansys was the software of choice, but users of other nonlinear codes such as Abaqus would benefit as well.

**Problem without unstable regions **A cantilever beam undergoing large deflections (rotations) under the action of an end moment as shown in

*Deformed shapes of beam under end moment*is perhaps the simplest geometrically nonlinear problem.

The moment is applied in five increments until it reaches its maximum value, *M*_{max}. For the initially straight beam to deform into a full circle, the maximum moment must be chosen according to the formula:

where *EI* = the beam’s flexural rigidity and *L* = the beam’s length.

An inspection of the three load-deflection curves shown in *Load-deflection curves for beam under end moment* reveals that the end rotation, θ, is a linear function of the end moment, *M*. The other deflections, *u* and *v*, depend nonlinearly on *M*. Results were obtained using the Ansys two-noded STIFF3 beam element. The default settings were used for the convergence tolerances.

The situation changes when the analysis is carried out using four-noded shell elements instead of beams. Specifically, when the mesh shown in *Mesh consisting of fournoded shell elements* is used along with the default convergence tolerances, the solution does not converge.

To remedy the situation, users can either change the convergence tolerances or force the structure into cylindrical bending by way of coupled DOF, as shown in *Using coupled DOF*. This approach causes the solution to rapidly converge.

The large deflection behavior of a cantilever beam subject to an end force or a uniform pressure is more complicated than the previous example because the load deflections are nonlinear for all three end-point deflections, as shown in *Deflection behavior of a cantilever beam*. STIFF3 beam elements were again used to discretize the structure, and the load was applied in five increments.

**Problems with snap-through instability **As shown in

*von Mises truss*, a two-member structure is undergoing large deflections under axial loading only.

*Williams toggle*shows a situation in which axial loading and bending are present. This difference notwithstanding, both structures display similar nonlinear behaviors as illustrated by their load deflection diagrams, as shown in

*Similar nonlinear behaviors*. Despite its simplicity, the von Mises truss illustrates the importance of properly choosing the solution controls in a nonlinear analysis.

When a standard load-incrementation scheme is employed, the obtained load-deflection curve is incomplete because it does not include the unstable (“snap-through”) portion of the diagram. The “line search” option must be turned on to force the software to continue the analysis beyond the first instability point.

**Problems with snap-back instability **The simple frame (bending stiffness

*EI*= 1.44e6) as shown in

*Lee’s frame*—

*geometry and loading*exhibits a highly nonlinear behavior at large load levels. To obtain a converged solution for the entire load-deflection diagram, it is necessary to invoke the “arclength” algorithm.

A similar but considerably more complicated example that also exhibits snap-back instability is the shallow, asymmetrically supported circular arch (R = 100 in., *EI* = 1e6) depicted in *Hingedfixed circular arch — geometry and loading*.

**What’s the upshot? ** Although in all the cases, it was possible to trace the entire load-deflection diagram in a single run through proper choice of the solution controls, the approach does not always work. Sometimes it is only possible to get the complete solution through multiple runs in which the results from a previous analysis are used to initiate a restart-based analysis.