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Shell element, plate bending

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ExampleReferenceRecordsLargest deviation
Simply supported square plate, thin plate limitPublished source42.43%
Thick plate, shear deformationPublished source30.0591%
Clamped square plate, uniformly distributed loadPublished source11.20%

Problem. The centre deflection of a square plate under uniform pressure is compared with plate theory. A thin simply supported plate, a thick plate and a clamped plate are covered.

Reference document. The Navier series is used for the simply supported plate and Timoshenko & Woinowsky-Krieger, Theory of Plates and Shells, Table 35, for the clamped plate. For the thick plate the reference is the Mindlin-Kirchhoff relation.

The plate bending stiffness is proportional to the cube of the thickness.

D=E t312 (1−ν2)D = \dfrac{E\,t^3}{12\,(1-\nu^2)}

The centre deflection of a simply supported square plate is computed with sixty terms of the Navier series.

wc=16 q a4π6D∑m,n odd(−1)(m−1)/2 (−1)(n−1)/2m n (m2+n2)2w_{c} = \dfrac{16\,q\,a^4}{\pi^6 D}\sum_{m,n\ \mathrm{odd}}\dfrac{(-1)^{(m-1)/2}\,(-1)^{(n-1)/2}}{m\,n\,(m^2+n^2)^2}

The centre deflection of a clamped square plate is taken from Timoshenko & Woinowsky-Krieger Table 35.

wmax=0.00126 q a4Dw_{\mathrm{max}} = 0.00126\,\dfrac{q\,a^4}{D}

For a thick plate the ratio of the Mindlin solution to the Kirchhoff solution is given by this relation.

wMwK=1+3.02 (t/a)2(1−ν) κ\dfrac{w_M}{w_K} = 1 + 3.02\,\dfrac{(t/a)^2}{(1-\nu)\,\kappa}

Numerical model. The plate is modelled with quadrilateral shell elements (MITC4). The in-plane translations and the rotation about the node normal are held at every node. A simply supported edge holds the vertical translation. In the cases marked in the table the rotation about the axis normal to the edge is held as well. A clamped edge holds the vertical translation and both rotations.

Simply supported square plate, thin plate limit

Published source. Navier series with sixty terms, Kirchhoff plate theory

QuantityConditionReferenceComputedDeviation
Centre deflectionmesh = 4×4 · t/a = 0.001-0.112316-0.1095852.43%
Centre deflectionmesh = 8×8 · t/a = 0.001-0.112316-0.1117030.546%
Centre deflectionmesh = 16×16 · t/a = 0.001-0.112316-0.1121670.132%
Centre deflectionmesh = 32×32 · t/a = 0.001-0.112316-0.1122830.0296%

Thick plate, shear deformation

Published source. The Navier deflection times the Mindlin-Kirchhoff relation

wM=wK(1+3.02 (t/a)2(1−ν)κ)w_M = w_K\left(1 + 3.02\,\dfrac{(t/a)^2}{(1-\nu)\kappa}\right)
QuantityConditionReferenceComputedDeviation
Centre deflectionmesh = 24×24 · t/a = 0.01 · support = uz and edge-normal rotation held-0.112366-0.11230.0591%
Centre deflectionmesh = 24×24 · t/a = 0.025 · support = uz and edge-normal rotation held-0.00720856-0.007204330.0587%
Centre deflectionmesh = 24×24 · t/a = 0.05 · support = uz and edge-normal rotation held-0.000908708-0.000908190.0569%

Clamped square plate, uniformly distributed load

Parameters. mesh = 16×16 · t/a = 0.025

Published source. Timoshenko & Woinowsky-Krieger, Theory of Plates and Shells, Table 35

QuantityReferenceComputedDeviation
Centre deflection 0.00126 qa4/D0.00126\,qa^4/D-0.00222953-0.00225621.20%
The horizontal axis is the number of elements per edge and the vertical axis the relative deviation of the centre deflection from the Navier solution. Each time the mesh is doubled the deviation falls to about a quarter, so the points run parallel to the dashed reference slope. This behaviour is called second-order convergence.The horizontal axis is the number of elements per edge and the vertical axis the relative deviation of the centre deflection from the Navier solution. Each time the mesh is doubled the deviation falls to about a quarter, so the points run parallel to the dashed reference slope. This behaviour is called second-order convergence.510205e-40.0010.0020.0050.010.02Convergence to the Kirchhoff solution for a thin plateElements per edge|w(FEM)/w(Navier) − 1|MindFEM, MITC4Second-order convergence, reference slope

The horizontal axis is the number of elements per edge and the vertical axis the relative deviation of the centre deflection from the Navier solution. Each time the mesh is doubled the deviation falls to about a quarter, so the points run parallel to the dashed reference slope. This behaviour is called second-order convergence.

Kirchhoff theory neglects shear deformation, so as the plate gets thicker its actual deflection exceeds the Kirchhoff value. The red curve gives this increase from Mindlin theory, and the model holding uz and the rotation about the axis normal to the edge follows it. When only uz is held, extra deformation forms in a strip about one plate thickness wide along the edge and the deflection stays above the theory.Kirchhoff theory neglects shear deformation, so as the plate gets thicker its actual deflection exceeds the Kirchhoff value. The red curve gives this increase from Mindlin theory, and the model holding uz and the rotation about the axis normal to the edge follows it. When only uz is held, extra deformation forms in a strip about one plate thickness wide along the edge and the deflection stays above the theory.00.0250.050.07511.05Shear deformation in a thick plate, with the support model separated outt / aw / w(Kirchhoff)Mindlin shear correction, theoryMindFEM, uz and edge-normal rotation heldMindFEM, only uz held

Kirchhoff theory neglects shear deformation, so as the plate gets thicker its actual deflection exceeds the Kirchhoff value. The red curve gives this increase from Mindlin theory, and the model holding uz and the rotation about the axis normal to the edge follows it. When only uz is held, extra deformation forms in a strip about one plate thickness wide along the edge and the deflection stays above the theory.