Unstable Collapse and Postbuckling Analysis

Geometrically nonlinear static problems sometimes involve buckling or collapse behavior, where the load-displacement response shows a negative stiffness and the structure must release strain energy to remain in equilibrium. Several approaches are possible for modeling such behavior. One is to treat the buckling response dynamically, thus actually modeling the response with inertia effects included as the structure snaps. This approach is easily accomplished by restarting the terminated static procedure (Restarting an Analysis) and switching to a dynamic procedure (Implicit Dynamic Analysis Using Direct Integration) when the static solution becomes unstable. In some simple cases displacement control can provide a solution, even when the conjugate load (the reaction force) is decreasing as the displacement increases. Another approach would be to use dashpots to stabilize the structure during a static analysis. Abaqus/Standard offers an automated version of this stabilization approach for the static analysis procedures (see Static Stress Analysis, Quasi-Static Analysis, Fully Coupled Thermal-Stress Analysis, or Coupled Pore Fluid Diffusion and Stress Analysis).

Alternatively, static equilibrium states during the unstable phase of the response can be found by using the “modified Riks method.” This method is used for cases where the loading is proportional; that is, where the load magnitudes are governed by a single scalar parameter. The method can provide solutions even in cases of complex, unstable response such as that shown in Figure 1.

Figure 1. Proportional loading with unstable response.

The Riks method is also useful for solving ill-conditioned problems such as limit load problems or almost unstable problems that exhibit softening.

In simple cases linear eigenvalue analysis (Eigenvalue Buckling Prediction) may be sufficient for design evaluation; but if there is concern about material nonlinearity, geometric nonlinearity prior to buckling, or unstable postbuckling response, a load-deflection (Riks) analysis must be performed to investigate the problem further.

The Riks method uses the load magnitude as an additional unknown; it solves simultaneously for loads and displacements. Therefore, another quantity must be used to measure the progress of the solution; Abaqus/Standard uses the “arc length,” l, along the static equilibrium path in load-displacement space. This approach provides solutions regardless of whether the response is stable or unstable. See the Modified Riks algorithm for a detailed description of the method.

Since the loading magnitude is part of the solution, you need a method to specify when the step is completed. You can specify a maximum value of the load proportionality factor, ${\lambda }_{end}$, or a maximum displacement value at a specified degree of freedom. The step will terminate when either value is crossed. If neither of these finishing conditions is specified, the analysis will continue for the number of increments specified in the step definition (see Defining an Analysis).

The Riks method works well in snap-through problems—those in which the equilibrium path in load-displacement space is smooth and does not branch. Generally you do not need take any special precautions in problems that do not exhibit branching (bifurcation). Snap-through buckling analysis of circular arches is an example of a smooth snap-through problem.

The Riks method can also be used to solve postbuckling problems, both with stable and unstable postbuckling behavior. However, the exact postbuckling problem cannot be analyzed directly due to the discontinuous response at the point of buckling. To analyze a postbuckling problem, it must be turned into a problem with continuous response instead of bifurcation. This effect can be accomplished by introducing an initial imperfection into a “perfect” geometry so that there is some response in the buckling mode before the critical load is reached.

The Riks algorithm cannot obtain a solution at a given load or displacement value since these are treated as unknowns—termination occurs at the first solution that satisfies the step termination criterion. To obtain solutions at exact values of load or displacement, the solution must be restarted at the desired point in the step (Restarting an Analysis) and a new, non-Riks step must be defined. Since the subsequent step is a continuation of the Riks analysis, the load magnitude in that step must be given appropriately so that the step begins with the loading continuing to increase or decrease according to its behavior at the point of restart. For example, if the load was increasing at the restart point and was positive, a larger load magnitude than the current magnitude should be given in the restart step to continue this behavior. If the load was decreasing but positive, a smaller magnitude than the current magnitude should be specified.

A Riks analysis is subject to the following restrictions:

• A Riks step cannot be followed by another step in the same analysis. Subsequent steps must be analyzed by using the restart capability.

• If a Riks analysis includes irreversible deformation such as plasticity and a restart using another Riks step is attempted while the magnitude of the load on the structure is decreasing, Abaqus/Standard will find the elastic unloading solution. Therefore, restart should occur at a point in the analysis where the load magnitude is increasing if plasticity is present.

• For postbuckling problems involving loss of contact, the Riks method will usually not work; inertia or viscous damping forces (such as those provided by dashpots) must be introduced in a dynamic or static analysis to stabilize the solution.