Computational analysis of multi-stepped beams and beams with linearly-varying heights implementing closed-form finite element formulation for multi-cracked beam elements

• Slender multi-stepped beams and beams with linearly-varying heights are modelled.
• The stiffness matrix of a beam having an arbitrary number of cracks is being derived.
• The closed analytical form is derived at without the implementation of shape functions.
• Apparent influences of all cracks on the stiffness matrix and the load vector.
• Analytical form of computational model’s exact transverse displacements’ functions.

The model where the cracks are represented by means of internal hinges endowed with rotational springs has been shown to enable simple and effective representation of transversely-cracked slender Euler–Bernoulli beams subjected to small deflections. It, namely, provides reliable results when compared to detailed 2D and 3D models even if the basic linear moment–rotation constitutive law is adopted.This paper extends the utilisation of this model as it presents the derivation of a closed-form stiffness matrix and a load vector for slender multi-stepped beams and beams with linearly-varying heights. The principle of virtual work allows for the simple inclusion of an arbitrary number of transverse cracks. The derived at matrix and vector define an ‘exact’ finite element for the utilised simplified computational model. The presented element can be implemented for analysing multi-cracked beams by using just one finite element per structural beam member. The presented expressions for a stepped-beam are not exclusively limited to this kind of height variation, as by proper discretisation an arbitrary variation of a cross-section’s height can be adequately modelled.The accurate displacement functions presented for both types of considered beams complete the derivations. All the presented expressions can be easily utilised for achieving computationally-efficient and truthful analyses.