Overview

Test Case

A long beam with a rectangular cross-section is fixed at one end. A linearly elastic to perfectly plastic isotropic material model was used for the beam, with a Young's modulus (E) of 200 GPa, Poisson's ratio of 0.3, and Yield Stress (σ₀) of 1,300 MPa. A pure moment is applied to the free end such that the moment axis is parallel to the beam width. Determine the beam projective length and beam deflection at the loaded end.

Figure 235: Problem Sketch

Figure 236: Ansys Workbench Model

Analysis Assumptions and Modeling Notes

The closed form solutions were found using the method described by Tayyar (2016). The key equations are listed below

(38)

(39)

Results Comparison

Note that the two components in the last equation (i and j) refer to the y- and x-deflections, respectively, relative to Figure 235: Problem Sketch, of the free end of the beam (α(L)). The term 2σ₀0 I/h represents the moment at which plastic deformation begins and M₀ is the maximum moment capacity. For this configuration, the applied moment was selected to be 22,000 Nmm, which is between 2σ₀ I/h = 16,300 Nmm and M0 = 24,400 Nmm. Therefore, the beam projective length l = 49.52 mm and the beam deflection δ = 5.982 mm.

The LS-DYNA Explicit settings were selected to minimize solution running time but ensure that the output deflection reached a steady state free from vibrations.

The load application was defined over 0.3 seconds (moment was ramped from 0 to 22,000 Nmm over 0.2 seconds then constant for 0.1 seconds).

The Global Damping was defined to be 100. The element used was LS-DYNA Explicit Hex 1 with an element size of 0.5 mm.

Results Comparison values are reported for the Windows 11 platform.