Bending of a Viscoelastic Timoshenko Cracked Beam Based on Equivalent Viscoelastic Spring Models

Considering the transverse crack as a massless viscoelastic rotational spring, the equivalent stiffness of the viscoelastic cracked beam is derived by Laplace transform and the generalized Dirac delta function. Using the standard linear solid constitutive equation and the inverse Laplace transform, the analytical expressions of the deflection and rotation angle of the viscoelastic Timoshenko beam with an arbitrary number of open cracks are obtained in the time domain. By numerical examples, the bending results of the analytical expressions are verified with those of the FEM program. Additionally, the effects of the time, slenderness ratio, and crack depth on the bending deformations of the different cracked beam models are revealed.

Static repair of multiple cracked beam using piezoelectric patches

This paper addresses the problem of repairing multiple cracked beams subjected to static load using piezoelectric patches. First, the problem is formulated and solved analytically for the case of two cracks that results in ratio of restoring moments produced by employed piezoelectric patches. Since the ratio is dependent only on crack positions but not their depth, the result obtained for case of two cracks has been extended for the case of multiple cracks. This proposition is then validated by finite element simulation where repairing piezoelectric patches are replaced by mechanical moment load equivalent to the restoring bending moments produced by the piezoelectric patches. The excellent agreement between analytical solution and numerical simulation results in case of single and double cracks allows making a conclusion that a piezoelectric patch could productively repair a cracked beam by producing a restoring moment due to its piezoelectricity. Thus, the problem of repairing multiple cracked beam using piezoelectric patches is solved.

Modal analysis of cracked beam with a piezoelectric layer

Piezoelectric material was employed first as sensor/actuator for structural control and then it has got an effective use for structural health monitoring and repairing damaged structures. In this report, modal analysis of cracked beam with piezoelectric layer is carried out to investigate effect of crack and piezoelectric layer thickness on natural frequencies of the structure and output charge generated in the piezoelectric layer by vibration modes. Governing equations of the coupled structure are established using the double beam model and two-spring (translational and rotational) representation of crack and solved to obtain the modal parameters including the output charge associated with natural modes acknowledged as modal piezoelectric charge (MPC). Numerical examples have been examined for validation and illustration of the developed theory.

Nonlinear vibration analysis of a cantilever beam with a breathing crack and bilinear behavior

Nonlinear phenomena widely occur in practical engineering applications. A typical example in aerospace structures is the creation of a breathing crack that opens and closes under cyclic loads, which causes bilinear behavior in the structural response. Late detection of such cracks can lead to a catastrophic failure that results in extensive structural damage. Therefore, analyzing the behavior of the structure because of the presence of a breathing crack is very important and needs to be investigated in detail. In this article, the nonlinear response of a single-degree-of-freedom nonlinear cantilever beam with a transverse breathing crack and bilinear behavior was studied. To investigate the nonlinear behavior, bilinear functions of the beam stiffness and nonlinear geometric stiffness were converted to polynomial functions. The proposed model is validated by comparing the time history responses of the approximated polynomials with the bilinear model of the cracked beam. Moreover, by considering damping changes because of the presence of the breathing crack, the nonlinear behavior was investigated. The results indicated that the proposed method is sensitive to the presence of a breathing crack. Also, the nonlinearity increases with an increase in the crack depth and location ratios associated with the jump phenomenon in the vibration response of the cracked beam.

Evaluation and Repair of Cracks on Statically Loaded Beams Using Piezoelectric Actuation

In the present work, damage produced by a crack in a statically loaded beam is first evaluated. Subsequently, an attempt is made to repair the effect of the crack by attaching a piezoelectric patch to the beam as an actuator. Static analysis of PZT patched cracked beam along with rotational spring is performed using Ritz method. Subsequently, a finite element analysis is performed by using ABAQUS 6.12 to collate the analytical results. It is shown in the study that when PZT patch is subjected to external electric field, it yields a local reactive moment, which counters the crack effects. An equation is procured in order to compute the required actuation voltage for repairing of cracks. A parametric study is performed for various boundary conditions and loading patterns. It is distinctly noticed that the technique nullifies the discontinuity in slope curve which develops due to a crack.

Dynamic analysis of multiple cracked Timoshenko beam under moving load–analytical method

When cracks start to surface in the surrounding areas of the structure, they create a local softness zone and influences on the dynamic response of the structure. The beams are more susceptible to shear and flexural cracks because of being subjected to shear and bending stress. In this study, the dynamic response of the single-span and multi-span damped beam under moving load with multi-crack and elastic boundary condition is studied based on Timoshenko’s theory. The Green’s function method is used to calculate the dynamic response of the cracked beam. In addition, the Green’s function method provides a solution for the differential equations. Moreover, the effects of the crack on the essential characteristics of the multi-span beams, especially the natural frequencies, are investigated. In this study, crack by itself is modeled in different situations and its effect on the behavior of the beam is analyzed. Also, the elastically restrained beam is modeled and its effect on the behavior of the beam is assessed. Because of the fact that the Euler–Bernoulli theory is also used in most beams, in this study, the results of the numerical examples are compared with the Euler–Bernoulli theory. Several examples are analyzed for a better understanding of the Timoshenko cracked beam.