Four-Beam Model for Vibration Analysis of a Cantilever Beam with an Embedded Horizontal Crack

As one of the main failure modes, embedded cracks occur in beam structures due to periodic loads. Hence it is useful to investigate the dynamic characteristics of a beam structure with an embedded crack for early crack detection and diagnosis. A new four-beam model with local flexibilities at crack tips is developed to investigate the transverse vibration of a cantilever beam with an embedded horizontal crack; two separate beam segments are used to model the crack region to allow opening of crack surfaces. Each beam segment is considered as an Euler-Bernoulli beam. The governing equations and the matching and boundary conditions of the four-beam model are derived using Hamilton＇s principle. The natural frequencies and mode shapes of the four-beam model are calculated using the transfer matrix method. The effects of the crack length, depth, and location on the first three natural frequencies and mode shapes of the cracked cantilever beam are investigated. A continuous wavelet transform method is used to analyze the mode shapes of the cracked cantilever beam. It is shown that sudden changes in spatial variations of the wavelet coefficients of the mode shapes can be used to identify the length and location of an embedded horizontal crack. The first three natural frequencies and mode shapes of a cantilever beam with an embedded crack from the finite element method and an experimental investigation are used to validate the proposed model. Local deformations in the vicinity of the crack tips can be described by the proposed four-beam model, which cannot be captured by previous methods.

Fracture characteristics and ductility of cracked concrete beam post-strengthened with CFRP Sheet

The present paper concerns the fracture characteristics and ductility of cracked concrete beam externally bonded with carbon fiber-reinforced polymer (CFRP) sheet as well as the integration behaviors between CFRP/concrete interfacial debonding and concrete cracking.Three-point bending tests were carried out on the CFRP-strengthened cracked concrete beams with varying specimen depth and initial crack length.A straingauge method was developed to monitor the crack initiation and propagation in concrete,and the CFRP/concrete interfacial bonding behaviors,respectively.Clip gauges were used to measure crack mouth opening displacement (CMOD) and the deflection at midspan.Experimental results revealed that CFRP-strengthened specimen shows a higher load capacity under the same deformation level and a better inelastic deformation capacity compared with the unstrengthened one.For there are two manifest peak values in the obtained load versus displacement curve,the ductility of CFRP-strengthened concrete beams were investigated using index expressed as area ratio on the load versus displacement curve.The calculated results indicated that the contribution from CFRP sheet to the ductility improvement of specimen is notable when the deflection at midspan exceeded 10.5 times the first-crack deflection.

AN APPROXIMATE ANALYSIS OF FREQUENCY RESPONSE OF A CRACKED HINGEDHINGED BEAM

A new method based on a modified line-spring model is developed for evaluating the natural frequencies of vibration of a cracked beam.This model in conjunction with the Euler-Bernoulli beam theory,modal analysis and linear elastic fracture mechanics is applied to obtain an approximate characteristic equation of a cracked hinged-hinged beam.By solving this equation the natural frequencies are determined for different crack lengths in different positions.The results show good agreement with the solutions through finite element analysis.The present method may be extended to analyze other cracked complicated structures with various boundary conditions.

AN APPROXIMATE METHOD OF RESPONSE ANALYSIS OFVIBRATIONS FOR CRACKED BEAMS

In this paper a method based on a line-spring model was proposed to analyze approximately vibration responses of cracked beams. The method is conjunction with the Euler-Bernoulli beam theory, modal analysis and fracture mechanics principle was applied to derive a characteristic equation for the cracked beam vibration. As application examples, natural frequency responses for a cracked hinged-hinged beam and a cracked cantilever beam were examined. It was shown that the present solutions obtained are quite in agreement with the solutions or experimental results in available references.

High resolution ultrastructure imaging of fractures in human dental tissues

Human dental hard tissues are dentine, cementum, and enamel. These are hydrated mineralised composite tissues with a hierarchical structure and versatile thermo-mechanical properties. The hierarchical structure of dentine and enamel was imaged by transmission electron microscopy （TEM） of samples prepared by focused ion beam （FIB） milling. High resolution TEM was carried out in the vicinity of a crack tip in dentine. An intricate ＂random weave＂ pattern of hydroxyapatile crystallites was observed and this provided a possible explanation for toughening of the mineralized dentine tissue at the nano-scale. The results reported here provide the basis for improved understanding of the rela- tionship between the multi-scale nature and the mechanical properties of hierarchically structured biomaterials, and will also be useful for the development of better prosthetic and dental restorative materials.

FREE VIBRATION ANALYSIS OF SIMPLY SUPPORTED BEAM WITH BREATHING CRACK USING PERTURBATION METHOD

In this paper, a new analytical method for vibration analysis of a cracked simply supported beam is investigated. By considering a nonlinear model for the fatigue crack, the governing equation of motion of the cracked beam is solved using perturbation method. The solution of the governing equation reveals the superhaxmonics of the fundamental frequency due to the nonlinear effects in the dynamic response of the cracked beam. Furthermore, considering such a solution, an explicit expression is also derived for the system damping changes due to the changes in the crack parameters, geometric dimensions and mechanical properties of the cracked beam. The results show that an increase in the crack severity and approaching the crack location to the middle of the beam increase the system damping. In order to validate the results, changes in the fundamental frequency ratios against the fatigue crack severities are compared with those of experimental results available in the literature. Also, a comparison is made between the free response of the cracked beam with a given crack depth and location obtained by the proposed analytical solution and that of the numerical method. The results of the proposed method agree with the experimental and numerical results.

A modeling method for vibration analysis of cracked beam with arbitrary boundary condition

This paper establishes a cracked Timoshenko beams model to investigate the vibration behavior based on the ultraspherical polynomials.Timoshenko beam theory is applied to model the free vibration analysis of the cracked beam and the numerical results are obtained by using ultraspherical orthogonal polynomials.The boundary conditions of both ends of the cracked beam are modeled as the elastic spring and the beam is divided into two parts by the crack section,and continuous conditions at the connecting face are modeled by the inverse of the flexibility coefficients of fracture mechanics theory.Ignoring the influence of boundary conditions,displacements admissible functions of cracked Timoshenko beam can be set up as ultraspherical orthogonal polynomials.The accuracy and robustness of the present method are evidenced through comparison with previous literature and the results achieved by the finite element method(FEM).In addition,the effects of flexibility coefficient on the natural frequencies are also investigated by using the proposed method.Numerical examples are given for free vibration analysis of cracked beams with various boundary conditions,which may be provided as reference data for future study.

A semi-analytical method for forced vibration analysis of cracked laminated composite beam with general boundary condition

In this paper,a semi-analytical method for the forced vibration analysis of cracked laminated composite beam(CLCB)is investigated.One computational model is formulated by Timoshenko beam theory and its dynamic solution is solved using the Jacobi-Ritz method.The boundary conditions(BCs)at both ends of the CLCB are generalized by the application of artificial elastic springs,the CLCB is separated into two elements along the crack,the flexibility coefficient of fracture theory is used to model the essential continuous condition of the connective interface.All the allowable displacement functions used to analyze dynamic characteristics of CLCB are expressed by classical Jacobi orthogonal polynomials in a more general form.The accuracy of the proposed method is verified through the compare with results of the finite element method(software ABAQUS is used in this paper).On this basis,the parametric study for dynamic analysis characteristics of CLCB is performed to provide reference datum for engineers.

Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bemoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method （DTM） has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.