2. Cook Membrane Problem

The Cook membrane problem is a classical test case for linear static analysis named after the author, R. D. Cook[1] who first reported it. The structure consist of a trapezoidal surface in the x-y plane (see figure below and mdl input file for dimensions and material constants). The structure is clamped along the edges E4 and it is loaded by a edge traction load along the edges E2 with a total load of 1 in the y direction.

Cook membrane problem: EPATCH model. P3 is the sampling point for y displacements (left). Deformed shape with amplitude for finest mesh and boundary of underformed mesh (right).

Figure 4. Cook membrane problem: EPATCH model. P3 is the sampling point for y displacements (left). Deformed shape with amplitude for finest mesh and boundary of underformed mesh (right).


The test is run with Q4.S.MITC.E4, Q8.S.MITC, Q9.S.MITC, and T3.S.MITC shell elements, comparing the global y-displacement at point P3 to the reported solution of 23.91:

Cook membrane problem: Convergence behaviour.

Figure 5. Cook membrane problem: Convergence behaviour.




[1] Cook, R. D.; Improved two-dimensional finite element; ASCE J. Struct. Div., ST9, 1851-1863 (1974).