Structural problems

In a static analysis the lowest mode of the structure usually dominates the response. Knowing the frequency and, correspondingly, the period of the lowest mode, you can estimate the time required to obtain the proper static response. To illustrate the problem of determining the proper loading rate, consider the deformation of a side intrusion beam in a car door by a rigid cylinder, as shown in Figure 1. The actual test is quasi-static.

Figure 1. Rigid cylinder impacting beam.

The response of the beam varies greatly with the loading rate. At an extremely high impact velocity of 400 m/s, the deformation in the beam is highly localized, as shown in Figure 2. To obtain a better quasi-static solution, consider the lowest mode.

Figure 2. Impact velocity of 400 m/s.

The frequency of the lowest mode is approximately 250 Hz, which corresponds to a period of 4 milliseconds. The natural frequencies can be calculated easily using the eigenfrequency extraction procedure in Abaqus/Standard. To deform the beam by the desired 0.2 m in 4 milliseconds, the velocity of the cylinder is 50 m/s. While 50 m/s still seems like a high impact velocity, the inertial forces become secondary to the overall stiffness of the structure, and the deformed shape—shown in Figure 3—indicates a much better quasi-static response. While the overall structural response appears to be what we expect as a quasi-static solution, it is usually desirable to increase the loading time to 10 times the period of the lowest mode to be certain that the solution is truly quasi-static. To improve the results even further, the velocity of the rigid cylinder could be ramped up gradually—for example, using a smooth step amplitude curve—thereby easing the initial impact.

Figure 3. Impact velocity of 50 m/s.