3.6.3: Analyze Nonlinear WIP

Linear structural behaviour means that if one changes the external loads by a factor $f$ also the physical response quantities (displacements, cross section forces, stresses, …) change by that factor. This has the pleasant effect, that the impact of different loads can be superimposed. Thus it is not necessary to recalculate the model for each possible combination of external loads. For real structures the assumption of linear behaviour is an approximation – a good one in many cases. There are two mayor sources of non-linearity:

Physical non-linearity: Comes into play when materials leave the linear elastic range (e.g. concrete that cracks, steel that yields, …)
Geometric non-linearity: Takes effect when
- lateral displacements get so large, that their effect on the axial (in case of e.g. beams) or in-plane (think of shells) deformation can not be neglected any more,
- a nodal rotation $\alpha$ reaches such a value, that the difference between $\alpha$ and $\tan(\alpha)$ gains importance.

The “Analyze Nonlinear WIP”-component lets one deal with geometric non-linearity. It is work-in-progress. This means that especially for shells the algorithms may not converge within acceptable time for some structures. If however a result is returned, then it is sound.

With the “Analyze Nonlinear WIP”-component one can chose from three variants of iterative solution algorithms. Each of these has different benefits and liabilities which will be explained below. The algorithms are based on the assumption of small strains, but allow arbitrarily large displacements.

The target of all three algorithms is to find a displacement state, where the external loads and the internal forces are in equilibrium. Starting from a known initial displacement state, one has to guess how the structure deforms under the given loads. This guess leads to a second displacement state where the internal and external forces usually do not match. The remaining imbalance forms the basis of a next prediction regarding the change of displacements and so on. Equilibrium is reached when the residual-force or change of displacements falls below a given threshold. The three algorithms offered by the “Analyze Nonlinear WIP”-component differ in how they predict the displacement increments.

Dynamic Relaxation

Fig. 3.5.3.1 shows a cantilever beam with a bending moment load about the local y-axis at its tip. It consists of 20 beam elements. For calculating its response the “DynamicRelaxation”-option is used. This algorithm predicts the next move of a structure based on the direction of the residual forces acting on each node. It is a robust procedure which converges to equilibrium quite reliably but sometimes needs a large number of iterations to do so. This component offers the following input-plugs:

During a non-linear calculation lots of things can happen. In order to get an idea about why and where something went wrong, the DR variant of the “DynamicRelaxation”-option produces the following output:

Newton-Raphson Method

In practice, dynamic relaxation(DR) procedures are used for highly non-linear problems like numerical crash-tests of cars, bolts being shot into a wall, …. The reason is, that implementing non-linear effects as DR code is relatively easy. This ease of implementation comes at the cost of high computational effort: Many iterations are necessary to reach equilibrium with acceptable accuracy. The way out of this is to invest more effort in a better prediction of the displacement increments. In DR-methods the residual forces at the nodes form the basis of predicting the next position of a node. Methods like the Newton-Raphson- or Arc-Length-method use a stiffness matrix for producing displacement predictions. There the computational cost per iteration is higher, but the number of iterations can be made much smaller as compared to DR-methods. With a consistent stiffness matrix quadratic convergence can be achieved under optimal conditions. This means that for the iterative displacement- and force-errors the number of zeros after the decimal separator doubles in each iteration. For the “Analyze Nonlinear WIP”-component this is not yet the case and one reason for the “work in progress”-label. Details on the Newton-Raphson- or Arc-Length methods can be found in [6] on page 102 ff. and 214 ff..

Fig. 3.5.3.2 shows the same cantilever beam as before, this time analyzed with the “NewtonRaphson”-option. The Newton-Raphson variant of the “Analyze Nonlinear WIP”-component comes with nearly the same input- and output-plugs as the DR-version. The only difference is the missing “StepSizeFac”-input. Since Newton-Raphson procedures have the same limitation with respect to unstable structures as DR-methods, an interval halving strategy for closing in on limit-points is applied as before.

Arc-Length Method

For many structures reaching a first point of instability is not yet the end of the story. Especially thin plate and shell structures show large load bearing reserves when considering their post-buckling behavior. The Arc-Length-method can be used for these kinds of situations. Fig. 3.5.3.3 shows the calculation of a truss structure which snaps through from an unstable state to a stable post-buckling configuration.

The first two inputs of the “Arclength”-component have the same meaning as before. Here a description of how the rest of the input-plugs controls the solution process:

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