The stiffness matrix of cracked finite elements: Introducing shortcomings in applying stiffness approach and proposing a solution

Several cracked finite elements have been introduced and implemented by researchers to analyze structures with cracks. In the formulation of these elements, the elastic strain energy caused by the crack is added to the elastic strain energy of the non-cracked element. Afterwards, the required relations are obtained by using Castigliano's second theorem (flexibility approach) or the first one (stiffness approach). In this paper, the two above-mentioned formulations of cracked elements are discussed. As a result, this study demonstrates shortcomings in applying stiffness approach. Therefore, this paper introduces a step by step method to improve the utilization of Castigliano's first theorem in producing the stiffness matrix of cracked elements. Finally, the topics discussed in this paper are numerically examined.