Vibratory Characteristics of Timoshenko Beams with Arbitrary Number of Cracks

An analytical approach is proposed for determining vibratory characteristics of cracked Timoshenko beams. Based on the rotational spring model for describing the local flexibility induced by a crack and the developed fundamental solutions, the frequency equation for a Timoshenko beam with any kind of two end supports and an arbitrary number of cracks can be established from a second-order determinant. The decrease in the determinant order can lead to significant savings in the computational time.     

Analytical Approach for Detection of Multiple Cracks in a Beam

An analytical approach for the detection of a beam with multiple cracks is presented in this article. The method is based on the bending vibration theory of Euler-Bernoulli beam and the cracks are treated as massless rotational springs, by which the cracked beam is separated into a number of segments of perfect beams. By using the nontrivial solution condition of the vibration mode of the beam elements, specially using the transfer matrix method for the multiple cracks detection, the crack identification equation of the cracked beam is obtained explicitly, which is a function of natural frequencies, the locations, and depths of the cracks. Since the natural frequencies of a cracked beam can be measured through many of the structural testing methods, then the relations of the locations and the depths of the cracks can be determined explicitly from the identification equation of a cracked beam which geometrical and physical parameters as well as the boundary conditions are given. The results of some examples are shown and the present method is validated with the existing and measured experimental data. The detection for other types of beams with different number of cracks and various boundary conditions can also be obtained by a similar procedure.     

Formulation of Cracked Beam Element for Structural Analysis

In this paper, a finite element for a cracked prismatic beam is developed. This element may be used in any matrix structural analysis. This paper details the derivation of the interpolation functions for a cracked Timoshenko beam finite element. These shape functions for rotational and translational displacements are also used to develop the consistent mass matrix for the cracked beam element. The crack effect on the stiffness matrix, as well as on the consistent mass matrix, is investigated and graphically represented.     

Buckling of Multiwalled Carbon Nanotubes Using Timoshenko Beam Theory

A Timoshenko beam model is presented in this paper for the buckling of axially loaded multiwalled carbon nanotubes surrounded by an elastic medium. Unlike the Euler beam model, the Timoshenko beam model allows for the effect of transverse shear deformation which becomes significant for carbon nanotubes with small length-to-diameter ratios. These stocky tubes are normally encountered in applications such as nanoprobes or nanotweezers. The proposed model treats each of the nested and concentric nanotubes as individual Timoshenko beams interacting with adjacent nanotubes in the presence of van der Waals forces. In particular, the buckling of double-walled carbon nanotubes modeled as a pair of double Timoshenko beams is studied closely and an explicit expression for the critical axial stress is derived. The study clearly demonstrates a significant reduction in the buckling loads of the tubes with small length-to-diameter ratios when shear deformation is taken into consideration.     

High-Accuracy Analysis of Beams of Bimodulus Materials

Recently, several authors have treated the problems of bending of beams of bimodulus materials. The present paper, applies Levinson beam theory, which includes shear deformation and warping of the cross section, to bending analysis of thick rectangular beams with bimodulus materials. Many numerical results are obtained by use of the transfer matrix approach and compared with the methods of Bernoulli-Euler beam theory, Timoshenko beam theory, and Levinson beam theory with various boundaries. Also, the neutral-surface location and displacements for beams of bimodulus materials are calculated.     

Analytical Solution for Fracture Analysis of CFRP Sheet–Strengthened Cracked Concrete Beams

Fiber-reinforced polymer (FRP) composite materials have been widely used in the field of retrofitting. Theoretical analysis of FRP plate- or sheet-strengthened cracked concrete beams is necessary for estimating service reliability of the structural members. In previous studies, the effect of a perfectly bonded FRP plate or sheet was equivalent to a cohesive force acting at the bottom of crack to delay the crack propagation in concrete and reduce the crack width. However, delamination between FRP and cracked beam is inevitable due to interfacial shear stress concentration at the bottom of crack. The intention of this paper is to present an analytical solution for fracture analysis of carbon FRP (CFRP) sheet–strengthened cracked concrete beams by considering both vertical crack propagation in concrete and interfacial debonding at CFRP-concrete interface. The interfacial debonding is modeled as the interfacial shear crack propagation in this paper. Four different stages are discussed after initial cracking state of the concrete. At the first stage, only fictitious crack propagation occurs in the concrete. At the second stage, macrocrack propagates in the concrete without interfacial debonding. At the third stage, both vertical macrocrack propagation in the concrete and horizontal shear crack propagation at the CFRP-concrete interface occur in the strengthened beam. The tensile stress in the CFRP sheet and interfacial shear stress along the span are formulated based on the deformation compatibility condition at the CFRP-concrete interface at this stage. Finally, macroshear crack propagates at the interface until the CFRP sheet is completely peeled out from the beam, and then the member is fractured. The applied load is determined as a function of the referred two crack lengths at different stages. At the beginning, the applied load increases to one peak value with the full propagation of fictitious crack at the first stage. At the third stage, the applied load is improved to another peak value due to the relatively high cohesive effect of the CFRP sheet. Then the two peak values are determined by the Lagrange multiplier method. The validity of the proposed analytical solution is verified with the experimental results and numerical simulations. It can be concluded that the proposed analytical solution can predict the load-bearing capacity of CFRP sheet-strengthened cracked concrete beams with reasonable accuracy.     

Planar Bending of Sandwich Beams with Transverse Loads off the Centroidal Axis

Conventional analysis methods for beams do not distinguish between transverse loads that are applied at the beam centroidal axis and those acting either above or below the centroidal axis. In contrast, this paper formulates a sandwich beam finite element solution which models the effect of load height relative to the centroidal axis. Towards this goal, the governing equilibrium equations and associated boundary conditions are derived based on a Timoshenko beam formulation for the core material. Special shape functions satisfying the homogeneous form of the equilibrium equations are derived and subsequently used to formulate exact stiffness matrices. By omitting the stiffness terms related to the faces, the formulation for a homogeneous Timoshenko beam can be recovered. Also, the Euler–Bernouilli counterpart of the formulation is recovered as a limiting case of the current Timoshenko beam formulation. Effects of load height relative to the centroid are observed to have similarities with those induced by axial forces in beam-columns. For a simply supported beam, downward acting loads located below the centroidal axis are found to induce a stiffening effect while those acting above the centroidal axis are found to induce a softening effect, resulting in higher transverse displacements.     

Asymptotic Derivation of Shear Beam Theory from Timoshenko Theory

A systematic reduction of Timoshenko beam theory to shear beam theory is presented and compared to a parallel reduction to Euler–Bernoulli theory. The agreement between Timoshenko and shear theories is seen to improve as the ratio of Young’s modulus to shear modulus increases, as the mode number increases, and as the beam becomes fatter, which are the opposite trends for agreement between Timoshenko and Euler–Bernoulli theories.     

Measurement of the Timoshenko Shear Stiffness. I: Effect of Warping

Fiber-reinforced polymer (FRP) composite beams are increasingly finding use in construction. Due to their lower stiffness relative to steel sections, the design of FRP structures is usually deflection controlled. Furthermore, shear deformation can be significant in FRP beams, thus, requiring the use of the Timoshenko beam theory to estimate deflections. However, the Timoshenko shear stiffness can be difficult to measure. Part of the measurement error has been attributed to shear warping effects. It has been hypothesized that warping restraints at loading points and supports increase the apparent shear stiffness to a degree that is significant at relatively short spans, e.g., L/h<10 to 15. In this study, the influence of warping on short to moderate length FRP beams under various types of loading and boundary conditions is considered using finite-element analysis. In particular, a commercially available thin-walled FRP beam was investigated. The results suggest that warping has a negligible effect for thin-walled beams at reasonable spans, i.e., L/h>5. On the contrary, the effective shear stiffness is found to decrease at shorter span lengths. This is the first of two papers in a series.     

Fundamental Frequency Testing of Reinforced Concrete Beams

Tests were conducted to measure the fundamental frequencies of reinforced concrete beams. Beams were tested prior to load application and after they had been loaded to various fractions of their ultimate moment capacity. Dynamic testing was performed in an unloaded state in both the direction of loading and in the direction perpendicular to loading. Resulting fundamental frequencies were used to determine the dynamic flexural stiffness (EdI) relative to the undamaged flexural stiffness. Results show that fundamental frequency tests can effectively measure decreases in dynamic flexural stiffness caused by flexural cracking. However, the effective moment of inertia in the relaxed state is not accurately predicted by American Concrete Institute recommendations for computing static beam deflections. Equations were developed to describe the effective flexural stiffness of unloaded, cracked beams. A relative dynamic flexural stiffness value of 70 provides a conservative prediction that a beam has failed by being loaded to its ultimate moment capacity.     

Closed-Form Solution for Reinforced Timoshenko Beam on Elastic Foundation

This paper suggests a method for obtaining closed-form solutions for a reinforced Timoshenko beam on an elastic foundation subjected to any pressure loading. A particular solution is obtained for uniform pressure loading at any location of the beam. This solution can be used to calculate settlement, rotation, tension, bending moment, and shear force of the beam. A parametric study is carried out to investigate the effects of geosynthetic shear stiffness and tension modulus and the location of the pressure loading. Results are presented and discussed.     

Continuous Dynamic Model for Tapered Beam‐Like Structures

Distributed parameter modeling may offer a viable alternative to the finite‐element approach for modeling large flexible space structures. In addition, the introduction of the transfer matrix method to the continuum modeling process provides a very useful tool to facilitate the distributed parameter model applied to more complex configurations. This paper proposes a piecewise continuous Timoshenko beam model, which was used for the dynamic analysis of a tapered beam‐like structure. Instead of the arbitrarily assumed shape functions used in a finite‐element analysis, the closed‐form solution of the Timoshenko beam equation was used. Application of the transfer matrix method was used to relate all elements as a whole. Using the corresponding boundary conditions and compatibility equations, a characteristic equation for the global tapered beam was produced and natural frequencies derived. The results from this analysis were compared to those obtained from a conventional finite‐element analysis. While comparable results are obtained, the piecewise continuous Timoshenko beam model significantly decreased the number of modal elements required.     

Concentrations of Pressure Between an Elastically Supported Beam and a Moving Timoshenko-Beam

The present paper is concerned with the motion of an elastically supported beam that carries an elastic beam moving at constant speed. This problem provides a limiting case to the assumptions usually considered in the study of trains moving on rail tracks. In the literature, the train is commonly treated as a moving line-load with space-wise constant intensity, or as a system of moving rigid bodies supported by single springs and dampers. In extension, we study an elastically supported infinite beam, which is mounted by an elastic beam moving at a constant speed. Both beams are considered to have distributed stiffness and mass. The moving beam represents the train, while the elastically supported infinite beam models the railway track. The two beams are connected by an interface modeled as an additional continuous elastic foundation. Here, we follow a strategy by Stephen P. Timoshenko, who showed that a beam on discrete elastic supports could be modeled as a beam on a continuous elastic Winkler (one-parameter) foundation without suffering a substantial loss in accuracy. The celebrated Timoshenko theory of shear deformable beams with rotatory inertia is used to formulate the equations of motion of the two beams under consideration. The resulting system of ordinary differential equations and boundary conditions is solved by means of the powerful methods of symbolic computation. We present a nondimensional study on the influence of the train stiffness and the interface stiffness upon the pressure distribution between train and railway track. Considerable pressure concentrations are found to take place at the ends of the moving train.     

Multicrack Detection on Semirigidly Connected Beams Utilizing Dynamic Data

The problem of crack detection has been studied by many researchers, and many methods of approaching the problem have been developed. To quantify the crack extent, most methods follow the model updating approach. This approach treats the crack location and extent as model parameters, which are then identified by minimizing the discrepancy between the modeled and the measured dynamic responses. Most methods following this approach focus on the detection of a single crack or multicracks in situations in which the number of cracks is known. The main objective of this paper is to address the crack detection problem in a general situation in which the number of cracks is not known in advance. The crack detection methodology proposed in this paper consists of two phases. In the first phase, different classes of models are employed to model the beam with different numbers of cracks, and the Bayesian model class selection method is then employed to identify the most plausible class of models based on the set of measured dynamic data in order to identify the number of cracks on the beam. In the second phase, the posterior (updated) probability density function of the crack locations and the corresponding extents is calculated using the Bayesian statistical framework. As a result, the uncertainties that may have been introduced by measurement noise and modeling error can be explicitly dealt with. The methodology proposed herein has been verified by and demonstrated through a comprehensive series of numerical case studies, in which noisy data were generated by a Bernoulli–Euler beam with semirigid connections. The results of these studies show that the proposed methodology can correctly identify the number of cracks even when the crack extent is small. The effects of measurement noise, modeling error, and the complexity of the class of identification model on the crack detection results have also been studied and are discussed in this paper.     

Structural Integrity Assessment of Welded Pressure Vessel Produced of HSLA Steel

Welded joint is a critical region of a welded structure.The operational safety of welded pressure equipment mostly depends on the behaviour of loaded weldments.Safety of welded structure is dependent on the properties of welded joint as whole and of its constituents (parent metal,heat affected zone and weld metal).In this paper the behaviour of welded joint cracked constituents is considered.Structural integrity assessment procedure is applied to welded pressure vessel produced of high-strength low-alloyed steel,operating at-40°C,comparing crack driving force and material crack resistance,using path-independent contour J-integral as fracture mechanics parameter.The comparison of crack driving force,expressed by J-integral and material resistant curve,J-R curve,provide the possibility to determine the extent of the stable crack as well as the critical crack size for its final fast propagation and also to assess the structural integrity of a cracked pressure vessel.     

Fictitious Crack Propagation in Fiber-Reinforced Concrete Beams

A nonlinear cracked hinge model is developed, aimed at the analysis of the bending fracture of fiber-reinforced concrete beams. The model is based on the fracture mechanics concepts of the fictitious crack model with a bilinear stress-crack opening relationship. Closed-form solutions are presented for the moment-rotation relationship of the hinge as a crack propagates, and the case of a nonzero normal force is covered. Special shapes of the stress-crack opening relationship are treated separately. These shapes are the so-called drop-linear and the drop-horizontal for which simplified expressions are obtained. The applicability of the hinge model is demonstrated through analysis of the bending fracture process in the case of a three-point bending beam and an infinitely long beam on a Winkler foundation, the latter analysis comprising the effect of a constant tensile normal force.     

Second-Order Stiffness Matrix and Loading Vector of a Beam-Column with Semirigid Connections on an Elastic Foundation

The second-order stiffness matrix and corresponding loading vector of a prismatic beam–column subjected to a constant axial load and supported on a uniformly distributed elastic foundation (Winkler type) along its span with its ends connected to elastic supports are derived in a classical manner. The stiffness coefficients are expressed in terms of the ballast coefficient of the elastic foundation, applied axial load, support conditions, bending, and shear deformations. These individual parameters may be dropped when the appropriate effect is not considered; therefore, the proposed model captures all the different models of beams and beam–columns including those based on the theories of Bernoulli–Euler, Timoshenko, Rayleigh, and bending and shear.The expressions developed for the load vector are also general for any type or combinations of transverse loads including concentrated and partially nonuniform distributed loads. In addition, the transfer equations necessary to determine the transverse deflections, rotations, shear, and bending moments along the member are also developed and presented.     

Comparative Modeling Study of Reinforced Beam on Elastic Foundation

In this paper, governing ordinary differential equations are derived for a reinforced Timoshenko beam on an elastic foundation. An analytical solution is obtained for a point load on an infinite Timoshenko beam on elastic foundation. Special attention is drawn to the location, tension, and shear stiffness of reinforcement and its influence on settlement/deflection of the beam and reinforcement tension force. A finite element (FE) model is established for the same infinite beam problem. Results from the TB model (Timoshenko beam on elastic foundation) are compared with results from the FE model and from the PB model (the Winkler model, based on the pure bending beam theory). It is found that results from the proposed TB model are, in general, in good agreement with results from the FE model as compared with results from the PB model. The TB model is better than the PB model in considering the shear deformation of the beam. This TB model is particularly useful in modeling a reinforced beam with or without considering the reinforcement shear stiffness. The TB model has practical applications in modeling geosynthetics∕fiber-glass reinforcement of foundation soils or pavement.     

考虑钢筋约束效应的开裂混凝土梁的自由振动

从柔度系数和裂纹应力强度因子的基本关系出发,基于Timoshenko梁理论,建立了考虑钢筋约束效应的开裂混凝土梁模型,得到了含裂纹简支梁的固有频率特征方程．通过数值分析,讨论了不同裂纹长度和深度以及钢筋约束效应对梁前三阶固有频率的影响．研究结果表明,开裂混凝土梁在钢筋的约束效应下,其固有频率大于不计及钢筋约束效应的情况,且裂纹深度和位置的变化会对结构的固有频率产生影响．     

Vibration Studies of Tsing Ma Suspension Bridge

A three-dimensional dynamic finite element model is established for the Tsing Ma long suspension Bridge in Hong Kong. The two bridge towers made up of reinforced concrete are modeled by three-dimensional Timoshenko beam elements with rigid arms at the connections between columns and beams. The cables and suspenders are modeled by cable elements accounting for geometric nonlinearity due to cable tension. The hybrid steel deck is represented by a single beam with equivalent cross-sectional properties determined by detailed finite element analyses of sectional models. The modal analysis is then performed to determine natural frequencies and mode shapes of lateral, vertical, torsional, longitudinal, and coupled vibrations of the bridge. The results show that the natural frequencies of the bridge are very closely spaced; the first 40 natural frequencies range from 0.068 to 0.616 Hz only. The computed normal modes indicate interactions between the main span and side span, and between the deck, cables, and towers. Significant coupling between torsional and lateral modes is also observed. The numerical results are in excellent agreement with the measured first 18 natural frequencies and mode shapes. The established dynamic model and computed dynamic characteristics can serve further studies on a long-term monitoring system and aerodynamic analysis of the bridge.