ABSTRACT
The global stiffness matrix (the assembled stiffness matrix K(g) of the two-element member) is the direct sum of the element stiffness matrices:
Generally, In this notation, the strain energy in the two elements can be written in the form
(10.21)
The total strain energy of the two elements is Finally, notice that K(g) is singular: the sum of the rows is the zero vector, as is the
sum of the columns. In this form, an attempt to solve the system will give rise to “rigidbody motion.” To illustrate this reasoning, suppose, for simplicity’s sake, that k(e)=k(e+1) in which case equilibrium requires that Pn=Pn+2. If computations were performed with perfect accuracy, the equation would pose no difficulty. However, in performing
computations, errors arise. For example, Pn is computed as and
Computationally, there is now an unbalanced force, εn+1−εn. In the absence of mass, this, in principle, implies infinite accelerations. In the finite-element method, the problem of rigid-body motion can be detected if the output exhibits large deformation.
