A simply supported cross frame consists of eight beams that have a square section (0.125 x 0.125 meter).

Case Description

A simply supported cross frame consists of eight beams that have a square section (0.125 x 0.125 meter). The frame is 10 m x 10 m. Consider modes in the XY plane only (constrain the back face against translation in the Z direction). To simulate a pinned constraint at the end of a beam element, the solid model includes a cylindrical groove at the end of each spoke (at the centerline of the cross section). A radial pin constraint is applied to the face of each groove.

Calculate 20 modes. The benchmark comparison is performed on the first 16 modes only. Calculating 20 modes improves the accuracy of the highest few modes for comparison.

Study Type

• Modal Frequencies

Mesh Parameters

• Mesh Type = Solid, Tetrahedral
• Element Order = Parabolic
• Mesh Size = 31.25 mm, absolute (producing four elements through the thickness)
• Adaptive Mesh Refinement: None

Material Properties

• Modulus of elasticity = 2.0 X 105 MPa
• Density = 8000 kg/m3
• Poisson's ratio (v) = 0.3

Constraints

• Z-direction constrained at the back (-Z) face
• Pin constraint with only radial translation prevented. Apply to the half-cylinder groove at the center of each end of the square beam

Comparison of Results

The first 16 frequencies correspond to the benchmark as follows:

Natural frequencies	F1	F2	F3	F4	F5	F6	F7	F8
NAFEMS (Beam Elements)	11.336 Hz	17.709 Hz	17.709 Hz	17.709 Hz	17.709 Hz	17.709 Hz	17.709 Hz	17.709 Hz
Fusion (Solid Elements)	11.33 Hz	18.38 Hz	18.41Hz	18.41 Hz	18.48 Hz	18.48 Hz	18.51 Hz	18.51 Hz
% Difference	0.053 %	3.79 %	3.96 %	3.96 %	4.35 %	4.35 %	4.52 %	4.52 %
Natural frequencies	F9	F10	F11	F12	F13	F14	F15	F16
NAFEMS (Beam Elements)	45.345 Hz	57.390 Hz	57.390 Hz	57.390 Hz	57.390 Hz	57.390 Hz	57.390 Hz	57.390 Hz
Fusion (Solid Elements)	45.91 Hz	59.39 Hz	59.44 Hz	59.44 Hz	59.46Hz	59.46 Hz	59.66 Hz	59.66 Hz
% Difference	1.25 %	3.48 %	3.57 %	3.57 %	3.61 %	3.61 %	3.96 %	3.96 %

Reference

NAFEMS Publication Test No. FV2, The International Association for the Engineering Analysis Community, "NAFEMS Background To Benchmarks", G.A.O Davies, R.T. Fenner, R.W. Lewis, June, 1992.