Problem. The centre deflection of a square plate is followed as it gets thinner. The load is scaled with the cube of the thickness, so the Kirchhoff deflection is the same at every thickness, and a computed deflection that stays at this value shows that the element does not lock.
Reference document. In the thin limit the reference is the value from Kirchhoff plate theory.
In the thin plate limit the centre deflection of a simply supported square plate takes this value.
Numerical model. The plate is modelled with eight by eight quadrilateral shell elements, with the vertical translation held at the edge nodes. The thickness ratio is swept in four steps from one tenth to one ten-thousandth.
Simply supported square plate, locking check at the thin limit
Published source. Kirchhoff plate theory, simply supported square plate
| Quantity | Condition | Reference | Computed | Deviation |
|---|---|---|---|---|
| Centre deflection | t/a = 0.001 · mesh = 8×8 | -0.112251 | -0.111703 | 0.489% |
| Centre deflection | t/a = 0.0001 · mesh = 8×8 | -0.112251 | -0.111701 | 0.490% |
The deflection on the vertical axis is normalised by the plate rigidity, so thin plate theory says it stays at 0.00406 whatever the thickness. In shear locking an element produces shear strain that does not exist as the plate gets thinner, behaves stiffer than it is, and the curve drops towards zero. The MITC4 curve stays flat at this value from t/a = 1/10 to 1/10,000 and rises only at the thickest end, by the shear deformation.