Overview

Test Case

This test case models a simply-supported beam with both supports subjected to vertical motion. The motion is defined in terms of a seismic displacement response-spectrum, with frequencies varying from 0.1 to 10.0 Hz and a displacement amplitude of 0.44 in. The beam has a length of 240 in and cross-section properties defined below. The objective is to validate the natural frequency, the maximum deflection, and the maximum bending stress of the beam. Figure 196 illustrates the domain dimensions and boundary conditions.

This problem is also presented in test case VM70 in the Mechanical APDL Verification Manual.

Figure 196: Schematic of the test case, including domain geometry, main dimensions, and boundary conditions

Analysis Assumptions and Modeling Notes

The natural frequency of a simply supported beam can be calculated as:

(48)

where

is the beam length

is the Young's modulus

is the moment of inertia

and is the mass per unit length

The mass per unit length is calculated as the product of the material density ρ and the cross-sectional area A. The natural frequency of the current system is 6.09793 Hz. For a simply-supported beam subjected to support motion, the maximum deflection can be calculated as:

(49)

where is the maximum displacement of the beam support. For the current test case, the maximum deflection is 0.56023 in. The maximum bending stress can be calculated as:

(50)

where is the maximum moment of the beam and is the maximum cross-section dimension measured from the neutral axis (h/2 in the current problem). Therefore, the maximum bending stress is 2.01586 ⋅ 10⁴ psi.

Figure 197: Model setup in LS-DYNA application of the seismic analysis of a simply supported beam

One part is defined to represent the beam, being meshed with 1D beam elements. These elements use Belytschko-Schwer resultant beam formulation (ELFORM=2) with cross-sectional properties defined from the table above. The beam elements use a linear elastic material card (*MAT_ELASTIC) with material properties defined from the table above. The keyword *BOUNDARY_SPC_NODE is used to define the pinned-roller conditions of both end nodes, while *BOUNDARY_SPC_NODE_SET is used to constrain the motion of all nodes in XZ plane. The keywords *CONTROL_IMPLICIT_GENERAL (IMFLAG=1) and *CONTROL_IMPLICIT_EIGENVALUE (NEIG=1) are used to activate the implicit eigenvalue analysis with one eigenvalue to be extracted. The keywords *FREQUENCY_DOMAIN_RESPONSE_SPECTRUM and *DATABASE_FREQUENCY_BINARY_D3SPCM are used to activate and create the binary output file for the response spectrum analysis generated due to the excitation of the base.

Results Comparison

The visualization of the fundamental mode can be performed by reading the d3eigv file, generated for the modal analysis. The relative displacement of the nodes can be observed in the d3spcm file, generated for the response spectrum analysis. To quantify the error between the theoretical and LS-DYNA results, the natural frequency, maximum deflection, and maximum bending stress of the simply supported beam are calculated with their relative errors and shown in the following table. Since beam stresses cannot be directly obtained with the current element formulation, the maximum bending moment was obtained and used to calculate the maximum bending stress. This comparison verifies the agreement between the natural frequencies, maximum deflections, and maximum bending stresses.