Reduced integration and the shear‐flexible beam element

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Abstract
A clearer insight into the ‘shear locking’ phenomenon, which appears in the development of C0 continuous element using shear‐flexible or penalty type formulations, is obtained by a careful study of the Timoshenko beam element. When a penalty type argument is used to degenerate thick elements to thin elements, the various approximations of the shear related energy terms act as different types of constraints and, depending on the formulation, two types of constraints which are classified as true or spurious may emerge. The spurious constraints, where they exist, are responsible for the ‘shear locking’ phenomenon, and its manifestation and elimination is demonstrated in a very simple example. The source of difficulty is shown to be the mathematical operations involved in the various shape function definitions and subsequent integration of functionals. It is seen that formulations that ensure only true constraints in the extreme penalty limit cases display far superior performance in the thick element situation as well, and thus guidelines for the development of efficient elements are drawn. A similar type of behaviour is observed in a shallow curved beam element and here ‘inplane locking’ can be eliminated by selective integration to obtain an improved curved beam element. However, ‘inplane locking’ does not cause a spurious constraint as the error quickly vanishes with the reduction of element size for a reasonable radius of curvature conforming with shallow shell theory.

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