Multi-cracked Beams Research Articles

Time-Varying Reliability Analysis of Multi-Cracked Beams Considering Maintenance Dependence

Time-varying reliability models of multi-cracked beam structures are established in this paper, which provide a theoretical method for the safety evaluation of multi-cracked beam structures. The reliability models proposed in this paper consider the interaction between the complex statistical correlations between system parameters during system operation and maintenance correlations, which is a difficult problem in the time-varying reliability modeling that takes into account work mechanisms and maintenance behavior. In the proposed models, multiple cracked elements are regarded as a dependent series system. The stresses, crack extensions, and multiple failure modes between each element constitute the complex failure dependence of the system. The time-varying reliability models of a multi-cracked beam structure are established via the neural network method and failure dependence analysis. Moreover, the failure dependence coefficient is proposed to quantify the time-varying failure dependence. Based on the working principle of the beam structures and the maintenance mechanism for the cracked state of the beams, a time-varying system reliability mode considering the maintenance dependence is proposed. Furthermore, the maintenance dependence coefficient index is proposed to quantitatively measure the interaction between the maintenance dependence and the failure dependence. Finally, the validity of the model is verified through the Monte Carlo simulation method. In the numerical examples, the relationship between maintenance dependence and failure dependence is illustrated and the influences of the statistical characteristics of the maintenance characteristic parameters on the maintainability and failure dependence are analyzed.

An exact closed-form explicit solution of free transverse vibration for non-uniform multi-cracked beam

Non-uniformity and damage are the two primary subjects in studying the vibrations of the beam-type elements. An exact closed-form explicit solution for the transverse displacement of a non-uniform multi-cracked beam with any type of boundary conditions is introduced. The generalized functions and the distributional derivative concepts are adopted. Four fundamental functions are introduced. These functions make the boundary conditions' process and compute the frequency equation more convenient. By introducing the non-dimensional parameters, the non-dimensional motion equation of the damaged beam with an arbitrary count of cracks is derived, and its exact closed-form explicit solution is obtained based on the four introduced fundamental functions. The standard method of computing these functions is presented, and the closed-form of these functions is determined for eight cases like uniform and conical beams. The closed-form of the frequency equation and mode shapes of the non-uniform multi-cracked beam are derived for several boundary conditions. The influence of the count of cracks, their location and intensity, and the boundary conditions on the natural frequency and mode shape are assessed by running a numerical study. The first and second frequencies of a conical beam are computed to verify the obtained results by applying this newly presented closed-form solution and the Differential Quadrature Element Method. A good agreement is evident when the obtained results are compared.

An extensive comparative analysis on multi-cracked beams with uncertain damage

The present paper is devoted to the structural response analysis of multi-damaged beams with open cracks characterized by uncertain parameters. The generic crack is modeled as a linearly-elastic rotational spring whose stiffness (in turn related to the crack depth) and position are treated as uncertain variables. Following both probabilistic and non-probabilistic approaches, the uncertain parameters are described by random variables with assigned probability density functions (PDFs) as well as by uncertain-but-bounded variables defined in terms of lower and upper bounds. Taking full advantage of the closed analytical forms for all the beam response functions with an explicit dependence on the uncertain parameters the Probability Transformation Method (PTM) applied in the stochastic setting returns the PDFs of the response functions; within a non-probabilistic framework, a sensitivity-based (SB) method or a global optimization (GO) technique allow to evaluate their bounds. Through a broad numerical investigation, the response intervals are directly compared with the response PDFs, overcoming the common comparison in terms of the so-called confidence intervals related to a second-order stochastic analysis and providing a new perspective on the interpretation of the results.

Nonlinear vibration analysis of a cantilever beam with multiple breathing edge cracks

The nonlinear vibration analysis of cracked structures could be a suitable indicator for diagnosing defects. When breathing cracks appear on the beam, its behavior can be modeled as a bilinear oscillator. In this paper, the nonlinear vibration behavior of a cantilever beam with multiple breathing cracks is investigated. For this purpose, the multi-cracked beam’s stiffness is calculated by introducing the effective length for the local stiffness reduction in the vicinity of cracks. Furthermore, owing to the importance of the damping variation in a cracked beam, its exact value is evaluated through theoretical expressions. A continuous polynomial function with the Weierstrass approximation is exploited to simulate the bilinear behavior of breathing cracks. By performing nonlinear analysis using the perturbation technique, the cracked beam’s dynamic behavior is studied. Moreover, the sensitivity of the response to the crack parameters in the primary resonance of the structure is analyzed. Finally, the sensitivity of the beam’s nonlinear response to the different number of breathing cracks and various crack depths is investigated. It is shown that the beam with a higher number of cracks or deeper ones has obvious softening behavior, and hence a more significant jumping can appear in the response.

Response probability density function for multi-cracked beams with uncertain amplitude and position of cracks

This paper addresses the determination of the response of beams under static loads, in presence of multiple cracks. The crack is modeled as a rotational internal spring. Both the amplitude and position of the spring are taken as uncertain variables. In particular, its random position characteristic has been modeled as Poisson law distribution along the axial coordinate. By using the theory of generalized functions, the probability transformation method (PTM) has been easily applied to the beam equilibrium equations. The PTM, employing the space transformation laws of random vectors, allows determining the probability density function (PDF) of the system response directly and easily. The results of some numerical applications have confirmed the goodness of the proposed stochastic procedure.

Analysis of the associated stress distributions to the nonlinear forced vibrations of functionally graded multi-cracked beams

1. Sridhar R, Chakraborty A, Gopalakrishnan S. Wave propagation analysis in anisotropic and inhomogeneous uncracked and cracked structures using pseudospectral finite element method. International Journal of Solids and Structures 2006; 43(16):4997–5031. https://doi.org/10.1016/j.ijso.... CrossRef Google Scholar

An interval framework for uncertain frequency response of multi-cracked beams with application to vibration reduction via tuned mass dampers

The paper addresses the frequency response of beams in presence of open cracks with interval parameters. On adopting the standard Euler–Bernoulli beam theory, every crack is modelled as a linearly-elastic rotational spring whose stiffness and position are treated as uncertain-but-bounded parameters. A two-step method is proposed to calculate the bounds of all response variables. First, the sensitivity functions of the response are calculated as every uncertain parameter varies within the respective interval. Next, the bounds of the response are computed by either a sensitivity-based method or a global optimization technique, the former if the response is monotonic with respect to all uncertain parameters and the latter if the response is non-monotonic with respect to even one parameter only. The method relies on analytical forms for all response variables and the associated sensitivity functions. The applications focus on the frequency response of multi-cracked beams equipped with tuned mass dampers, showing potential and accuracy of the method.

Free vibration of axially loaded multi-cracked Timoshenko beams

Free vibration of axially loaded multi-cracked Timoshenko beams

Free vibration of multi-cracked beams

A new method is presented to study the free vibrations of multi-cracked beams with arbitrary boundary conditions. In the literature, the method based on changes in modal strain energy was used to perform the dynamic analysis of beams with just one crack. In this paper, the changes in modal strain energy are used iteratively to study the dynamic behavior of beams with multiple cracks. The iterative method consists in finding the dynamic frequencies in steps by considering the effects of cracks one by one. First, the beam is assumed intact, for it a single crack is taken into account via the method based on changes in modal strain energy. Then, this procedure is repeated iteratively by taking at steps (i+1), the frequencies obtained at step (i). The end is detected when the total number of cracks is reached. This developed iterative approach is used to analyze the effect of multiple open cracks on the modal parameters of a cantilever beam subjected to free vibration. The results are in good agreement with finite element and experimental methods. This developed method may also be used to generate training data for pattern recognition approaches to health monitoring.

Analytical solution of deflection of multi-cracked beams on elastic foundations under arbitrary boundary conditions using a diffused stiffness reduction crack model

Crack can significantly affect the performance of structures and is one of the crucial indicators of damage in structural health monitoring. In this paper, the deflection behaviors of Euler–Bernoulli beams with arbitrary open edge cracks under arbitrary elastic boundary conditions are investigated. A continuous diffused stiffness reduction crack model is implemented to simulate the cracks in beams, which can incorporate multiple cracks and consider the stiffness reduction effect in the vicinity of a crack. With the proposed diffused stiffness reduction model, the fourth-order differential equation governing the deflection behavior of the multi-cracked Euler–Bernoulli beam is constructed. The powerful variational iteration method is applied to obtain the analytical solution of the multi-cracked beams on elastic foundations. Five shape functions are introduced, based on which the deflection of the multi-cracked beam is proposed. Both the solutions corresponding to the general elastic boundary conditions and the conventional boundary conditions are presented explicitly. The deflection solution is benchmarked and verified against the literature, and encouraging agreements are obtained. Parametric studies are carried out to investigate the influences of crack position, crack ratio, stiffness of the elastic foundation, and boundary conditions on the deflection of the cracked beams. The proposed crack model and the deflection solution overcome some of the limitations in the literature.

Interval static analysis of multi-cracked beams with uncertain size and position of cracks

• Interval static analysis on cracked beams with uncertain size/position of cracks. • Uncertainty in size/position of cracks is handled by proper generalized functions. • Monotonicity of response is tested with respect to all uncertain parameters. • Bounds of monotonic response are obtained by sensitivity-based method. • Bounds of non-monotonic response are computed by global optimization technique. This paper deals with beams under static loads, in presence of multiple cracks with uncertain parameters. The crack is modelled as a linearly-elastic rotational spring and, following a non-probabilistic approach, both stiffness and position of the spring are taken as uncertain-but-bounded parameters. A novel approach is proposed to compute the bounds of the response. The key idea is a preliminary monotonicity test, which evaluates sensitivity functions of the beam response with respect to the separate variation of every uncertain parameter within the pertinent interval. Next, two alternative procedures calculate lower and upper bounds of the response. If the response is monotonic with respect to all the uncertain parameters, the bounds are calculated by a straightforward sensitivity-based method making use of the sensitivity functions built in the monotonicity test. In contrast, if the response is not monotonic with respect to even one parameter only, the bounds are evaluated via a global optimization technique. The presented approach applies for every response function and the implementation takes advantage of closed analytical forms for all response variables and related sensitivity functions. Numerical results prove efficiency and robustness of the approach, which provides very accurate bounds even for large uncertainties, avoiding the computational effort required by the vertex method and Monte Carlo simulation.

Analytical approach to coupled bending-torsional vibrations of cracked Timoshenko beam

The explicit expressions of steady-state responses of a coupled bending-torsional Timoshenko beam with cracks and damping effect subjected to external harmonic loadings are presented in this paper. The Green's functions method is employed to obtain the analytical solutions. The beam is split to several segments due to the existence of cracks. General Green's functions of each segment with unknown boundary constants are given by using Laplace transform method. A mixture line-spring is introduced to obtain the compatibility conditions of the cracked cross-section between the segments. Based on the transfer matrix method, those constants in the Green's functions for each segment are determined with those matching conditions, as well as boundary relationships and end boundary conditions. Then, the solutions of the whole cracked beam with one or multiple cracks can be expressed in terms of the piecewise functions. The solutions of the multi-cracked beam cover those of the beam with only one crack and the uncracked beam. The theory developed can be used for the beam with classical boundary conditions, but for illustration, results for a cantilever cracked beam with T shape cross-section are illustrated, emphasizing the effect of cracks on the solutions. Comparisons between results in the published literature and those obtained from the developed method show a good agreement, which confirms the validity and accuracy of the proposed approach. The influences of crack depth, location and number on the natural frequencies are discussed. The changes of the steady-state responses of beam are investigated due to the existence of crack. Moreover, the symmetric property of the Green's functions and damping effect on the amplitude of steady-state responses of the cracked beam are studied particularly.

Geometrically non-linear forced vibrations of fully clamped functionally graded beams with multi-cracksresting on intermediate simple supports

The objective of this work is to investigate the non-linear forced dynamic response at large vibration amplitudes of fully clamped functionally graded beams containing a multiple open edge cracks and resting on intermediate simple supports. The theoretical model is based on the Euler-Bernoulli beam theory and the crack rotational spring model. The functionally graded beam properties are supposed to vary continuously through the beam thickness. A homogenization procedure based on the neutral surface approach taking into account the presence of the crack is developed to reduce the problem examined to that of an equivalent isotropic homogeneous multi-cracked beam. In the non-linear analysis, harmonic motion is assumed and the discretized expressions for the beam total strain and kinetic energies are derived by applying Hamilton’s Principle, so as to reduce the problem to a non-linear algebraic system solved using an approximate explicit method previously applied to various non-linear structural vibration problems. Considering the forced vibration case, the non-linear frequency response functions are obtained numerically in the neighbourhood of the predominant non-linear mode shape using a single mode approach. The effects of crack, the beam material gradient and the applied harmonic force are presented and discussed.

On the response bounds of damaged Euler–Bernoulli beams with switching cracks under moving masses

The response of damaged Euler–Bernoulli beams with any number of unilateral cracks and subjected to one or more moving masses which remain always in contact with the beam are investigated in this paper. A flexibility switching crack model is employed here to represent a crack as either open or closed at a particular instant depending on the sign of the axial strain at the crack centre and taking into account the location of the crack either at the top or bottom of the beam. This model leads to closed-form solutions of the damaged beam mode shapes as a function of four integration constants, which can be computed by enforcing the boundary conditions, and a Boolean switching crack array which identifies the open cracks. Based on this switching crack model, an efficient computational procedure is proposed to tackle the problems of multiple masses moving on a multi-cracked beam with generic boundary conditions. The proposed procedure is applied to three beams with switching cracks under moving masses, and the results obtained are compared to the widely adopted always open cracks model, showing significant differences. In particular, it is shown that: (i) Flexible boundary conditions, multiple switching cracks and multiple moving masses can be easily accounted for with the proposed approach; (ii) the two limiting conditions of always open cracks and always closed cracks (undamaged) models do not always provide bounds on the dynamic response of a beam with switching cracks subject to moving masses; (iii) the side where each crack is located can largely affect the response.

Free Vibration of Axially Loaded Multi-Cracked Beams Using the Transfer Matrix Method

In this paper, an analytical method is developed to study the free vibration of multi-cracked, axially loaded beams with differing boundary conditions, namely, hinged-hinged, clamped-clamped, clamped-hinged, and clamped-free. The cracked beam system is modelled as a number of beam segments connected by massless rotational springs with sectional flexibility. Each segment is assumed to obey the Euler-Bernoulli beam theory. The characteristic equation of the cracked beam with differing boundary conditions, which is a function of the natural frequency, sizes and location of the cracks, and the physical parameters of the beam, as well as the corresponding mode shapes, is derived using a simple transfer matrix method. In this paper, a detailed parametric study is conducted to show the effects of cracks and axial load on vibrational properties of the cracked beam. The results obtained in this study agree well with analytical results available in the literature.

Linear and geometrically nonlinear free and forced vibrations of fully clamped multi-cracked beams

1. Ne D. Mechanical Behaviour of Materials Engineering Methods for Deformation Fracture and Fatigue. Prence-Hall Int; 1993. Google Scholar

Dynamic Response of Multi-cracked Beams Resting on Elastic Foundation

Cracks cause to change dynamic response of beams and make discontinuity in slope of the deflection of the beams. The dynamic analysis of the Euler-Bernoulli cracked beam on the elastic foundation subjected to the concentrated load is presented in this paper. The stiffness of the elastic foundation and elastic supports influence on vibrational characteristics of the cracked beam. The Dynamic Green Function is applied to solve the governing equation. Thus, the dynamic response of the cracked beam is determined by Laplace Transform method. The effects of depth and location of the crack on natural frequency and deflection of the cracked beam on an elastic foundation are evaluated. In order to demonstrate the capability of the present approach, several numerical examples are worked out and the results are discussed.

On the Bending Analysis of Multi-Cracked Slender Beams with Continuous Height Variations

We studied the implementation of a multi-stepped multi-cracked beam finite element for analysing beams of various heights’ variations along the length. The genuine continuous variation of height was modelled by an adequate series of steps and for the cracks, the model of internal hinges endowed with rotational springs was applied. As the increased number of steps simultaneously increases the possibility that the crack location and step location may coincide, the continuous conditions at the steps were rewritten to a more general form in order to cover this combination. The newly presented form is also capable of taking a transverse concentrated force into account. Numerical evaluations were performed for several structures under various loads and the transverse displacements, nodal reactions, and inner forces were studied. The results obtained are positioned side by side with the results from more detailed 2D finite element meshes. The comparisons show excellent matching of the beam’s response providing the mesh of steps was dense enough. The examples demonstrate the potential of the present approach proving that the model is suitable for achieving computationally-efficient and truthful analyses.

Geometrically nonlinear free and forced vibrations analysis of clamped-clamped functionally graded beams with multicracks

Geometrically nonlinear free and forced vibrations of clampedclamped Functionally Graded beams with multi-cracks, located at different positions, based on the equivalent rotational spring model of crack and the transfer matrix method for beams is investigated. The FG beam properties are supposed to vary continuously through the thickness direction. The theoretical model is based on the Euler-Bernoulli beam theory and the Von Karman geometrical nonlinearity assumptions. A homogenization procedure, taking into account the presence of the crack, is developed to reduce the problem examined to that of an equivalent isotropic homogeneous multi-cracked beam. Upon assuming harmonic motion, the discretized expressions for the total strain and kinetic energies of the beam are derived, and through application of Hamilton’s principle and spectral analysis, the problem is reduced to a nonlinear algebraic system solved using an approximate explicit method developed previously (second formulation) to obtain numerically the FG multi-cracked beam nonlinear fundamental mode and the corresponding backbone curves for a wide range of vibration amplitudes. The numerical results presented show the effect of the number of cracks, the crack depths and locations, and the volume fraction on the beam nonlinear dynamic response.

Closed-form solution based genetic algorithm software: Application to multiple cracks detection on beam structures by static tests

In this paper a procedure for the static identification and reconstruction of concentrated damage distribution in beam-like structures, implemented in a dedicated software, is presented. The proposed damage identification strategy relies on the solution of an optimisation problem, by means of a genetic algorithm, which exploits the closed form solution based on the distribution theory of multi-cracked beams subjected to static loads. Precisely, the adoption of the closed-form solution allows a straightforward evolution of an initial random population of chromosomes, representing different damage distributions along the beam axis, towards the fittest and selected as the sought solution. This method allows the identification of the position and intensity of an arbitrary number of cracks and is limited only by the amount of data experimentally measured. The proposed procedure, which has the great advantage of being robust and very fast, has been implemented in the powerful agent based software environment NetLogo, and is here presented and validated with reference to several benchmark cases of single and multi-cracked beams considering different load scenarios and boundary conditions. Sensitivity analyses to assess the influence of instrumental errors are also included in the study.