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NAFEMS Test P18.FV73

Benchmark Summary

Revision History

• August 3, 2026 – Initial version

Objective

This test assesses shell element performance in finding the natural frequencies of a Cantilevered Thin Square Plate.

Geometry and Mesh

The original FV73 Test computes frequencies obtained for different mass DOF reductions via dynamic condensation. OpenSees does not perform mass reduction, so only Test 1 from FV73 is relevant for this test.

Boundary Conditions

• u_X=u_Y=u_Z= θ_Y= 0 along X=0

Material Properties

• Elastic modulus, E = 200e9 Pa
• Poisson ratio, ν = 0.3
• Density, ρ = 8000 kg/m³

OpenSees Elements Tested

The original FV73 Test uses an eight-noded semi-Loof thin shell element. This element formulation is not available in OpenSees, so the test compares the following shell element formulations to the reference solution.

• ShellMITC4
• ShellDKGQ
• ASDShellQ4

Note that each of these elements is a four-noded element.

Frequencies

Frequencies (Hz) along with percent difference from the reference solution.

Among the elements considered, ShellDKGQ agrees most closely with the NAFEMS reference solution. This element uses a thin plate formulation with a consistent mass matrix. Although ShellMITC4 uses a consistent mass matrix, the element formulation is thick plate. While ASDShellQ4 uses a thin plate formulation, its mass matrix is lumped. This benchmark alone does not prove one element formulation is better than the others.

Mode Shapes

The mode shapes returned by all three element formulations are shown below. The mode shapes are consistent between the formulations, taking into account differences in algebraic sign for some eigenvectors, and provide an additional qualitative verification beyond the natural frequencies.

OpenSees Script

import openseespy.opensees as ops

from numpy import pi,sqrt

# Units = N, m, sec

L = 10
t = 0.05
E = 200e9
v = 0.3
rho = 8000

Nmodes = 6

ops.wipe()
ops.model('basic','-ndm',3,'-ndf',6)

ops.node(1,0,0,0)
ops.node(2,L,0,0)
ops.node(3,L,L,0)
ops.node(4,0,L,0)

c = L/4 # Mesh size

ops.mesh('line',1,2,*[1,2],0,6,c)
ops.mesh('line',2,2,*[2,3],0,6,c)
ops.mesh('line',3,2,*[3,4],0,6,c)
ops.mesh('line',4,2,*[4,1],0,6,c)

ops.section('ElasticMembranePlateSection',1,E,v,t,rho)
# For `ASDShellQ4`, see note below
ops.mesh('quad',5,4,*[1,2,3,4],0,6,c,'ShellMITC4',1) # Or 'ShellDKGQ'

ops.fixX(0,1,1,1,0,1,0)

ops.analysis('Transient','-noWarnings')
w2 = ops.eigen(Nmodes)
f = sqrt(w2)/(2*pi)

OpenSees Changes

This test requires the following changes to OpenSees:

• Modify the mesh command to use ASDShellQ4.
  In openseespy <= 3.8.0.0, export the nodes and elements from the ShellMITC4 model then manually define ASDShellQ4 elements.

Verification Summary

Among the shell elements considered, ShellDKGQ is in closest agreement with the published reference frequencies. All three formulations give consistent mode shapes.

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