An Investigation in Finite Element Theory of Shear Locked Elements

This report shows the theoretical background for some of the problems associated with shear loaded beams using the finite element method. The report develops the theory behind both shell and brick elements on a reduced order by using beam and membrane elements. The cause of the shear-locking phenomenon is presented, as are common methods of avoiding these effects. The two presented methods of defeating shear locking are adding a bubble mode to an element, and creating the stiffness matrix with reduced order Gaussian integration. The reasoning behind each solution to the locking problem is explained and demonstrated both mathematically and through solutions to an example problem. After the validity of the particular elements is verified, general guidelines for which element type should be used are given.