•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.
Summary
In this paper, bending problem of a transversely cracked beam resting on elastic foundations is addressed by means of the finite element method. The paper covers the derivation of a new ...finite element, where the soil is modelled by classical Winkler soil model, and the cracked beam is represented by the simplified computational model, as already widely used for various analyses of such transversely cracked slender beams. The derivation of transverse displacements' interpolation functions for the transversely cracked slender beam and the stiffness matrix, as well as the corresponding load vector for the linearly distributed continuous transverse load are presented. All expressions are derived at in closed symbolic forms, and they allow easy implementation in the existing finite element software. The presented approach is ideal for the effortless modelling of cracked beams in conditions where neither information about the crack's growth nor the stresses at the crack's tip is required. Numerical examples covering several load situations are presented to support the discussed approach. The results obtained are further compared with the values from corresponding coupled differential equations as well as from a large 2D plane finite elements model, where a detailed description of the crack was accomplished. It is evident that the drastic difference in the computational effort is not reflected in any significant difference in the results between the models.
A new finite element with an additional degree of freedom at the crack location is derived for static bending analysis of a transversely cracked uniform slender beam. In the simplified computational ...model, which is based on Euler-Bernoulli's theory of small displacements, the crack is represented by a linear rotational spring connecting two elastic parts. The derivation of the transverse displacements, the coefficients of the stiffness matrix as well as the load vector for uniformly distributed load along the whole beam element was based on the utilization of polynomial interpolation functions of the fourth degree and all derived expressions were obtained in the closed form. The novelty of the new model, by comparison to the previously presented simplified finite element models, is that the transverse displacements functions obtained by utilization of the newly presented interpolation functions for the case of uniform continuous transverse load along whole beam element, as well as the functions of the bending moments and transverse forces, are accurate. The values obtained by the simplified model also exhibited good agreement in additional comparison with the results from more demanding and more detailed 2D models.
► We model a slender transversely-cracked beam resting on Winkler’s foundation. ► The derived finite element has an additional internal node at the crack’s location. ► The crack is modelled as a ...rotational spring connecting two non-cracked parts. ► The related stiffness matrix and load vector are derived at in closed symbolic forms. ► The presented expressions enable the computation of transverse displacements and inner forces.
This paper discusses transversely-cracked slender beams on elastic foundations. The bending problem of a cracked beam resting on Winkler’s soil is addressed by means of the finite element method. This paper thus covers the derivation of a new finite element, where the soil is modelled by classical Winkler’s soil model, and the cracked beam is represented by the simplified computational model, as widely-used for various analyses of transversely-cracked slender beams. The newly presented finite element has three nodes – one at each end of the finite element, and an additional internal node at the location of the crack. The stiffness matrix for the transversely-cracked slender beam, the corresponding load-vector for the linear continuous transverse load, as well as the derivation of transverse displacements’ interpolation functions are presented, and all expressions are derived at in closed symbolic forms.
The presented approach is ideal for the effortless modelling of cracked beams in conditions where neither information about the crack’s growth nor the stresses at the crack’s tip is required. A numerical example covering several load situations is briefly presented to support the discussed approach. The results obtained with the presented approach are further compared with the values from a large 2D finite elements model, where a detailed description of the crack was accomplished. It is evident that the drastic difference in the computational effort is not reflected in any significant difference in the results between the models.
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.
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.
This paper formulates the finite element of a beam with an arbitrary number of transverse cracks. The derivations are based on a simplified computational model, where each crack is replaced by a ...corresponding linear rotational spring, connecting two adjacent elastic parts. The stiffness and geometrical stiffness matrices thus take into account the effect of flexural bending deformation caused by the presence of the cracks.
The expressions for calculating the coefficients of stiffness and geometrical stiffness matrices, as well as the load vector of the element, are presented in closed forms.
Since the corresponding interpolation functions were implemented in the derivations, transverse displacements within the finite element can also be obtained.
Due to the fact that the number of parameters describing the cracked beam's structure is thus reduced to its minimum, it can be expected that this element could be efficiently implemented, not only in static and stability analysis, but also in inverse identification of cracks in beam-like structures.
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