1.1.6 Snap-through of a shallow, cylindrical roof under a point load

Product: ABAQUS/Standard  

This example illustrates the use of the modified Riks method to obtain the unstable static equilibrium response of an elastic shell structure that exhibits snap-through behavior. The shell in this case is a shallow, cylindrical roof, pinned along its straight edges and loaded by a point load at its midpoint. Since the example has been studied by several authors, comparison with those published results provides verification of this type of analysis. An illustration of the volume proportional damping stabilization capability is also shown as an alternative to the Riks method.

Problem description

Modeling and solution control

The roof is assumed to deform in a symmetric manner, so one quadrant is discretized, as shown in Figure 1.1.6–1. We use two regular 6 × 6 meshes of shell elements, one of type S4R5 (4-node elements with one integration point) and one of type S4R (finite membrane strain shell element), and an 8 × 8 mesh of triangular shell elements of type S3R. In addition, two regular 6 × 6 meshes of continuum shells are provided, one of type SC6R (finite membrane strains, in-plane continuum shell wedge) and one of type SC8R (finite membrane strains, hexahedron continuum shell). No mesh convergence studies have been performed, but the comparison of the results given by these meshes with published numerical solutions suggests that, at least with respect to load-deflection behavior, these meshes give reasonably accurate results.

When using the modified Riks method, the load magnitude and suggested initial increment size should provide a reasonable estimate for the sense and magnitude of the first increment in load. It is known that the critical load for this case will not exceed 750 N. With an initial time step of .025 for a time period of 1.0, we give a load of 3000 N. This implies an initial load increment of about 75 N on the entire roof. Furthermore, we are not interested in post-snap behavior much beyond the magnitude of the critical load, so we terminate the analysis when a load proportionality factor of 0.06 has been reached. This corresponds to a total load on the entire roof of 720 N. In this problem the static equilibrium load actually reverses direction as the roof goes through an unstable snap. The modified Riks algorithm is able to track such load reversals. Gauss integration is used for the shell cross-section.

When using the volume proportional damping capability, a total load of 1332 N is applied, which is roughly equivalent to the load at which the Riks method analysis stops. The initial load increment is 10 percent of the total load. This algorithm does not capture load reversals; when such reversals would occur, the structure accelerates and the increased velocity produces enough viscous forces to balance the externally applied load. As a result, the external load stays almost constant during the unstable part of the deformation.

Results and discussion

Input files

References

Figures

Figure 1.1.6–1 Shallow cylindrical roof under point load.

Figure 1.1.6–2 Load-displacement response for shallow cylindrical shell.

Figure 1.1.6–3 Comparison of solutions for shallow cylindrical roof.

Figure 1.1.6–4 Comparison of Riks and stabilized solutions for shallow cylindrical roof.