Arc-Length & Path-Following Methods

Tracing unstable equilibrium paths through limit points and snap-back — load control, displacement control, arc-length (Riks) methods, solver behaviour and when each is appropriate.

Article 36Imperfections & Nonlinear Buckling9 min read

bucklingarc-lengthpath followingRikssnap-throughsnap-backload controldisplacement controlsolver

The problem of unstable equilibrium paths

A nonlinear buckling analysis traces the equilibrium path of the structure — the relationship between the load and the displacement. For a structure with an unstable post-buckling path (a thin shell, a shallow arch), the path goes through a limit point (the load reaches a maximum) and then continues into a region where the load decreases with increasing displacement. The path may also exhibit snap-back, where the displacement reverses (decreases) while the load also decreases. To trace this path, the solver must be able to follow the equilibrium through the limit point and the unstable region. Standard load-controlled or displacement-controlled solvers cannot do this — they fail at the limit point. The arc-length method was developed to solve this problem.

Load control

In load control, the applied load is the control variable and is increased monotonically by the solver. At each load level, the solver finds the equilibrium displacement. Load control works well for stable equilibrium paths (monotonically increasing load with displacement). At a limit point, the load cannot increase further — the structure cannot carry more load. The solver fails to converge at the next load increment because there is no equilibrium at that load. Load control gives the limit-point load (the last converged load) but cannot trace the post-limit path. Load control is adequate when only the collapse load is needed and the post-buckling path is not of interest.

Displacement control

In displacement control, a characteristic displacement (typically at the point of load application) is the control variable and is increased monotonically. The load is computed as the reaction. Displacement control can trace past a limit point where the load drops with increasing displacement (a snap-through), because the displacement continues to increase even as the load decreases. However, displacement control cannot trace past a snap-back point, where the displacement reverses — the solver is trying to increase the displacement but the equilibrium path requires the displacement to decrease. Displacement control is appropriate for structures with a monotonic displacement response (snap-through without snap-back), such as many shallow arches and frames.

Arc-length (Riks) method

The arc-length method, originally developed by Riks and later refined by Crisfield and others, uses the arc length along the equilibrium path as the control variable. The arc length is a combination of the load parameter and the displacement vector, so the solver can advance along the path regardless of whether the load or the displacement is increasing or decreasing. The arc-length constraint is an additional equation added to the equilibrium equations, making the system solvable at limit points and snap-back points. The arc-length method can trace the full equilibrium path, including unstable sections, and is the standard method for nonlinear buckling analysis of structures with unstable post-buckling paths.

Snap-through and snap-back

Two types of unstable behaviour are encountered on the equilibrium path. Snap-through (a limit point) is where the load reaches a maximum and then decreases with continued increasing displacement — the structure jumps dynamically from the pre-limit to the post-limit configuration if the load is held constant. Snap-back is where, after a limit point, the displacement reverses direction while the load continues to decrease — the structure would jump back to a lower displacement at a lower load. Snap-back is characteristic of very unstable structures (thin shells, certain frames) and is the harder case to trace. The arc-length method can handle both; load control handles neither; displacement control handles snap-through but not snap-back.

Solver behaviour and parameters

The arc-length solver requires additional control parameters compared to load or displacement control: the initial arc-length size, the maximum and minimum arc-length, the reference load (used to normalise the load parameter in the arc-length constraint), and the number of iterations per step. The initial arc-length is chosen to give a reasonable first increment; the solver then adjusts the arc-length automatically based on convergence (increasing it if convergence is fast, decreasing it if convergence is slow). Near a limit point, the arc-length is reduced to resolve the sharp turn in the path. If the solver fails near a limit point, reducing the minimum arc-length and the initial arc-length usually helps. The reference load should be the total buckling-critical load, so the load parameter is normalised to a value near 1 at the collapse.

When arc-length methods are useful

Arc-length methods are essential for tracing the equilibrium path of structures with unstable post-buckling behaviour — thin shells, shallow arches and caps, certain frames and trusses with snap-through or snap-back. They are needed when the engineer wants to understand the full post-buckling response, not just the collapse load: the post-buckling reserve (how much the load drops after the limit point), the secondary bifurcations (mode changes on the post-buckling path), and the energy absorption (the area under the load-displacement curve). For structures with a stable post-buckling path (flat plates), displacement control is often sufficient and simpler. For structures where only the collapse load is needed and the post-buckling path is not of interest, load control gives the collapse load (as the last converged load) but without the post-buckling detail.

Practical guidance

The choice of path-following method depends on the structure and the information needed. For a collapse-load-only assessment of a structure with a stable post-buckling path, load control is adequate. For a structure with snap-through (limit point, no snap-back), displacement control traces the post-buckling path. For a structure with snap-back or for a full post-buckling path trace, the arc-length method is required. In all cases, the solver parameters (increment size, convergence tolerance) must be tuned for the problem — default parameters may fail near limit points. A practical approach is to start with load control to find the approximate collapse load, then switch to arc-length to trace the post-buckling path with parameters tuned to the approximate collapse load.

For thin-shell buckling, always use an arc-length (Riks) method. Load control will fail at the limit point and give only the collapse load, not the post-buckling path. The post-buckling path shape — the steepness of the load drop, the presence of snap-back — is critical for understanding the failure mode and the energy absorption.

Branch selection and secondary bifurcations

An arc-length solver follows an equilibrium branch, but that is not a guarantee that it will follow the physically relevant branch after a bifurcation. Symmetry, the initial imperfection and numerical perturbations influence which path is selected. Structures with several close modes may undergo secondary bifurcations in which the deformation pattern changes after initial buckling. To investigate this, the analyst can seed alternative imperfection shapes, perturb the solution near the bifurcation or compare results from different path-following controls. A smooth numerical trace should therefore be accompanied by deformation-mode inspection throughout the solution. The objective is to understand the sequence of structural states, not merely to obtain a complete-looking load-displacement curve.

When a dynamic solution may be more representative

Snap-through and snap-back are inherently dynamic physical events if the structure is loaded through an unstable region without an ideal quasi-static control device. Arc-length analysis is useful for mapping the underlying static equilibrium path, including unstable states that may not be observable during a real event. If the engineering question concerns the transient jump, impact with stops or energy released during snap-through, an implicit or explicit dynamic analysis may be more appropriate. The loading rate should then be chosen to distinguish quasi-static structural behaviour from inertial effects. Static path following and dynamic simulation answer different questions and can be used together: one identifies the equilibrium landscape, while the other predicts how the structure moves across it.