On the derivation of symbolic form of stiffness matrix and load vector of a beam with an arbitrary number of transverse cracks

This paper considers derivation of the stiffness matrix and the load vector due to a uniform transverse load for an already-known simplified computational model of a slender beam having an arbitrary number of transverse cracks.The principle of virtual work allows for the coefficients of the stiffness matrix and the load vector to be given in clear and closed analytical forms which enable faster and straightforward evaluation. However, since the derivation approach excludes information about the transverse displacement distributions between the nodes the alternatives for the determination of transverse displacements within the finite element are thus further discussed to complete the analysis of multi-cracked beams. Also these results are given in clear and closed analytical forms.The presented stiffness matrix is ideal for modeling any flexural cracks of beams and columns near supports and joints with other structural elements which is, for example, required in earthquake engineering, where the European earthquake engineering design code EC8 requires the cracks to be included in the analysis of concrete elements. Furthermore, as the newly-presented form of stiffness matrix makes the influence of the depths and locations of the cracks to the flexural bending deformation more recognizable that may also open new possibilities in the identification of cracks.

We model a slender beam, having an arbitrary number of transverse cracks. All developed expressions are derived at in a very plain and straightforward manner. The derivation approach excludes implementation of shape functions. The related stiffness matrix and load vector are given in closed analytical forms. Presented expressions enable simple computation of accurate transverse displacements.