Stress intensity factors for cracked T-sections and dynamic behaviour of T-beams

This paper presents an extension of a simple and convenient method proposed by Kienzler and Herrmann [An elementary theory of defective beams. Acta Mech 1986;62:37-46] to estimate the stress intensity factors of cracked beams and bars. This method is based on an elementary beam theory estimation of the strain energy release as the crack is widened into a fracture band. As an extension, the power of the simple beam theory analysis is demonstrated by application to cracked T-beams subjected to a bending moment, shear forces and a torsion. Moreover, the present work addresses the coupled bending-torsional vibration of cracked T-beams within the context of the dynamic stiffness matrix method of analysing structures.     

Strength of sandwich beams with mid-plane debondings in the core

Sandwich beams containing cracks in the mid-plane of the core are investigated. The cracked part is subjected to a constant remote shear stress field. Beams with different cross-section geometries and materials were analyzed by the finite element method (FEM) in order to compute the stress intensity factors at the crack tips. An analytical approach for estimation of the energy release rate, based on a potential energy calculation, is presented that agrees well with results from the FE analyses. Results from four-point bending tests with cracked beams show that the fracture load can be accurately predicted. The simplicity of the analytical model makes it possible to compute critical crack lengths and safety factors for various types of sandwich beams.     

Combined effect of shear and large deformation on stress intensity factors

An attempt has been made to study the influence of large deformation on the stress intensity factor in a cracked plate subjected to bending including shear deformation. It is assumed that singular terms for stress resultants and strains in the case of large deformation have the same angular distribution and order of singularity as in the case of a linear problem. With this in view the small deformation singular element has been used at the crack tip region surrounded by large deformation plate bending elements. The finite element analysis, based on total Lagrangian formulation combined with the modified Newton–Raphson technique, has been used to get numerical results. Several examples connected with large deformation of cracked plates subjected to bending are studied. Using the above technique stress intensity factors for linear and non-linear cases have been compared.     

Effect of defective holes on crack using the boundary element method

Boundary value problem of a plate with crack and defect such as the circular and/or elliptical holes is a multi-connected domain problem, this kind of problem is suitable for solving by boundary element method with its higher precision. The sub-region method is used in the paper, a center cracked plate subjected to remote tensile and shear loading is studied numerically. The effect of the circular hole on Mode-I and Mode-II stress intensity factors is studied assuming the plane strain condition, the results are more precise than that of using FEM. The effect of the elliptical hole on Mode-I stress intensity factors is studied as well and some significant results are obtained.     

Free vibration analysis of uniform and stepped cracked beams with circular cross sections

This paper presents a novel numerical technique applicable to analyse the free vibration analysis of uniform and stepped cracked beams with circular cross section. In this approach in which the finite element and component mode synthesis methods are used together, the beam is detached into parts from the crack section. These substructures are joined by using the flexibility matrices taking into account the interaction forces derived by virtue of fracture mechanics theory as the inverse of the compliance matrix found with the appropriate stress intensity factors and strain energy release rate expressions. To reveal the accuracy and effectiveness of the offered method, a number of numerical examples are given for free vibration analysis of beams with transverse non-propagating open cracks. Numerical results showing good agreement with the results of other available studies, address the effects of the location and depth of the cracks on the natural frequencies and mode shapes of the cracked beams. Modal characteristics of a cracked beam can be employed in the crack recognition process. The outcomes of the study verified that presented method is appropriate for the vibration analysis of uniform and stepped cracked beams with circular cross section.     

Stress intensity factors for curvilinear cracks loaded by bending and torsional moments

The geometry of a thin flat plate containing a curvilinear crack of finite size is investigated. The plate is loaded by uniform bending and torsional moments at infinite distance from the crack contour, while distributed normal bending moments and torques exist at the upper and lower crack surfaces. The bending stress intensity factors for the curvilinear crack are calculated on the basis of classical plate theory. The displacements and internal moments are represented by two complex analytic functions. Extra conditions are imposed to ensure the univalence of the displacements, which is not evident because the plate comprises a multiply connected domain due to the presence of the crack. A linearization with respect to the crack-curvature function has been performed and the bending stress intensity factors are calculated as the first-order solutions for slightly curved cracks. The results are illustrated with a few examples, such as uniform loading configurations and the geometry of a crack along a circular arc. The loading of thin flat plates by a combination of tensile forces and bending moments is also investigated. In analogy with the variation of the stress components over the cross section of the plate, two combined stress intensity factors are introduced having the same dependence on the perpendicular coordinate and being related to the symmetric and anti-symmetric stress intensity factors of the separate plane-stress and bending problems. The resulting energy release rate is shown to be in full agreement with known results in the literature.     

Boundary element analysis of curved cracked panels with adhesively bonded patches

A new boundary element formulation for analysis of curved cracked panels with adhesively bonded patches is presented in this paper. The effect of the adhesive layer is modelled by distributed body forces (i.e. two in‐plane forces, two moments and one out‐of‐plane force). A coupled boundary integral formulation of a shear deformable plate and two‐dimensional plane stress elasticity is used to determine bending and membrane forces along the adhesive layer taking into consideration the compatibility conditions in the patch area. Two numerical examples are presented to demonstrate the efficiency of the proposed method. It is shown that the out‐of‐plane bending behaviour and panel curvature have significant influence on the magnitude of the stress intensity factors. Copyright © 2003 John Wiley & Sons, Ltd.     

Analysis of cracked anisotropic plates using the hybrid finite element method

This paper presents a convenient and efficient method to obtain accurate stress intensity factors for cracked anisotropic plates. In this method, a complex variable formulation in conjunction with a hybrid displacement finite element scheme is used to carry out the stiffness and stress calculations of finite cracked plates subjected to general boundary and loading conditions. Unlike other numerical methods used for local analysis such as the boundary element method, the present method results in a symmetric stiffness matrix, which can be directly incorporated into the stiffness matrix representing other structural parts modeled by conventional finite elements. Therefore, the present method is ideally suited for modeling cracked plates in a large complex structure.     

New unloading compliance correlation for throughwall circumferentially cracked pipe to measure crack growth during fracture tests

Crack growth is generally measured during fracture experiment of specimen or component. The unloading compliance technique is commonly used for this purpose because of its simplicity. It infers the crack length from unloading compliance of cracked component. The pre‐requisite of this technique is the availability of an equation that correlates crack length with unloading compliance. While such correlations are available for compact tension and three‐point bend specimens, it is not available for big components such as pipe or pipe bend. Development of such a correlation for throughwall circumferentially cracked (TCC) straight pipe under bending, therefore, forms the objective of the present study. However, the challenge to develop such correlation for TCC pipe is that the equation should contain not only crack length as a function but also the current deformation or load level as a parameter. This is attributed to the fact that the circular cross section of the pipe ovalizes during deformation leading to change of bending stiffness of the cracked body. New compliance correlations have been proposed for TCC pipe under bending load considering these complexities. Elastic‐perfectly plastic material behaviour has been assumed to characterize the material stress–strain response. However, it has been shown that error due to this approximation with respect to the actual stress–strain behaviour is negligible if one chooses flow stress equal to average of yield and ultimate strength. The proposed correlations are expressed in terms of normalized parameters to make them independent of specific values of geometric dimensions such as radius, thickness and span length of four‐point bending loading system. Effectiveness of this normalization has also been verified by carrying out a sensitivity study.     

Comparison between solid and hollow reinforced concrete beams

Comparison between test results of seven hollow and seven solid reinforced concrete beams is presented. All of the fourteen beams were designed as hollow sections to resist combined load of bending, torsion and shear. Every pair (one hollow and one solid) was designed for the same load combinations and received similar reinforcement. The beams were 300 × 300 mm cross-section and 3,800 mm length. The internal hollow core for the hollow beams was 200 × 200 mm creating a peripheral wall thickness of 50 mm. The main variables studied were the ratio of bending to torsion which was varied between 0.19 and 2.62 and the ratio in the web of shear stress due to torsion to shear stress due to shear force which was varied between 0.59 and 6.84. It was found that the concrete core participates in the beams’ behaviour and strength and cannot be ignored when combined load of bending, shear and torsion are present. Its participation depends partly on the ratio of the torsion to bending moment and the ratio of shear stress due to torsion to the shear stress due to shear force. All solid beams cracked and failed at higher loads than their counterpart hollow beams. The smaller the ratio of torsion to bending the larger the differences in failure loads between the hollow and solid beams. The longitudinal steel yielded while the transverse steel experienced lower strain values.     

Fracture mechanics analysis of geometrically nonlinear shear deformable plates

This paper presents boundary integral equations for fracture mechanics analysis of geometrically nonlinear shear deformable plates. A radial basis function and dual reciprocity method are utilized to evaluate the derivative terms and the domain integrals that appear in the formulations, respectively. Numerical examples of the clamped and simply supported plates containing a center crack subjected to uniform transversal loadings are presented. Displacement extrapolation technique is used to compute the stress intensity factors (SIFs). Stress intensity factors of mode I for plate bending and membrane problems are presented. The normalized stress intensity factors in membrane significantly increase after few increments of the load while the normalized stress intensity factors in bending decrease. Less displacement and rotational constraints in cracked plates under uniform transversal loadings will raise the stress intensity factors. The bending stress intensity factors of a central crack in clamped square plate were found to be the highest values compared to those for clamped non-square plates.     

Computation of membrane and bending stress intensity factors for thin,cracked plates

The crack tip stress fields for plate bending and membrane loading problems are reviewed and the four stress intensity factors that determine these fields are defined. These four stress intensity factors arise from use of Kirchhoff plate theory to account for the bending loads and two dimensional plane stress elasticity to account for the membrane loads. The energy release rate is related to the stress intensity factors and to the stress resultants of plate theory. Virtual crack extension, nodal release and modified crack closure integral methods are discussed for computing components of the energy release rate from finite element analyses of cracked plates. Sample computations of stress intensity factors for single and mixed mode cases are presented for a crack in an infinite plate. Sample computations of stress intensity factors for a double edge notched tension-torsion test specimen are given as well.School of Civil and Environmental Engineering, Cornell University     

Boundary element analysis of flat cracked panels with adhesively bonded patches

Adhesively bonded patch repairs for cracked finite sheets are analysed by the boundary element method. The interaction between the plate and the patch on a repaired sheet is modelled as a distribution of forces which include in-plane, out-of-plane and two moment body forces. The coupled boundary integral formulations of shear deformable plate (Mindlin theory) and two-dimensional plane stress elasticity are presented. Stress intensity factors, three for the bending problem and two for the membrane problem, are evaluated from crack opening displacements. Several examples are presented to demonstrate the accuracy and efficiency of the proposed method. Comparison with two-dimensional solutions demonstrate the significance of the bending loads on the stress intensity factors.     

Stress intensity factors for cracked I-beams

We present simple, closed-form expressions for stress intensity factors for cracked I-beams subjected to a bending moment. The estimates are based on the elementary strength theory for cracked beams forwarded by Herrmann and co-workers, coupled with dimensional considerations and a finite element calibration. The expressions given here are valid for the case when the crack has propagated through the flange and into the web of the beam. The simple expressions are accurate to within 5% of detailed finite element calculations for the range of practical applicability. To further demonstrate the validity of the stress intensity factor expression, we measured fracture loads for cracked polymethyl methacrylate (PMMA) I-beams in four-point flexure. Using the failure loads and our expression for the stress intensity factor, we deduce the fracture toughness. The fracture toughness so-obtained results in excellent fracture correlation for the cracked I-beams.     

Stress intensity factors for cracked triangular cross‐section thin‐walled tubes

For one kind of finite‐boundary crack problems, the cracked equilateral triangular cross‐section tube, an analytical and very simple method to determine the stress intensity factors has been proposed based on a new concept of crack surface widening energy release rate and the principle of virtual work. Different from the classical crack extension energy release rate, the crack surface widening energy release rate can be defined by the G*‐integral theory and expressed by stress intensity factors. This energy release rate can also be defined easily by the elementary strength theory for slender structures and expressed by axial strains and loads. These two forms of crack surface widening energy release rate constitute the basis of a new analysis method for cracked tubes. From present discussions, a series of stress intensity factors are derived for cracked equilateral triangular cross‐section tubes. Actually, the present method can also be applied to cracked polygonal tubes.     

An accurate solution for the responses of circular curved beams subjected to a moving load

In this paper, an accurate and effective solution for a circular curved beam subjected to a moving load is proposed, which incorporates the dynamic stiffness matrix into the Laplace transform technique. In the Laplace domain, the dynamic stiffness matrix and equivalent nodal force vector for a moving load are explicitly formulated based on the general closed‐form solution of the differential equations for a circular curved beam subjected to a moving load. A comparison with the modal superposition solution for the case of a simply supported curved beam confirms the high accuracy and applicability of the proposed solution. The internal reactions at any desired location can easily be obtained with high accuracy using the proposed solution, while a large number of elements are usually required for using the finite element method. Furthermore, the jump behaviour of the shear force due to passage of the load is clearly described by the present solution without the Gibb's phenomenon, which cannot be achieved by the modal superposition solution. Finally, the present solution is employed to study the dynamic behaviour of circular curved beams subjected to a moving load considering the effects of the loading characteristics, including the moving speed and excitation frequency, and the effects of the characteristics of curved beams such as the radius of curvature, number of spans, opening angles and damping. The impact factors for displacement and internal reactions are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Influence of loading rate on shear fracture toughness for failure mode transition

When subjected to shear loading of sufficiently high rate, many materials do not fail by cracks, propagating at an angle of 70° with respect to the ligament, but by adiabatic shear bands, which extend nearly straight in the direction of the ligament. Work is reported on investigations for determining the dependence of the impact shear fracture toughness as a function of loading rate, in particular in the regime of failure mode transition from cracks to adiabatic shear bands. For achieving high rate shear conditions of loading, edge cracked specimens are asymmetrically impacted at the cracked edge by a projectile accelerated by an air gun. The resulting mode-II stress intensity factors and the times of onset of failure are determined by a specially developed strain gauge measuring technique. Results on shear fracture toughnesses with increasing loading rate are reported for two structural materials, a 1% chromium steel and a high strength aluminum alloy. Whereas decreasing fracture toughnesses are observed with increasing loading rate when failure occurs by tensile cracks, the fracture toughness increases with loading rate when failure occurs by adiabatic shear bands.     

层间滑移对摩擦型叠梁弯曲特性的影响

对叠梁滑移问题的计算,通常假定粘结层的滑移应变与剪应力成线性关系;但对于库仑摩擦型叠梁来说,只有当层间剪力超过层间最大抵抗摩擦力后,才发生层间滑移。为了更精确研究摩擦型叠梁的力学特性,该文首先推导了考虑层间摩擦力影响的截面剪应力计算公式,进而确定了摩擦型叠梁层间滑移展开的力学条件和路径;最后借助传递矩阵法求解了叠梁滑移展开的长度以及对叠梁构件力学特性的影响;与ANSYS的计算结果对比表明该文提出的算法准确高效;滑移效应会破坏截面的整体性,增大局部弯曲应力,结构计算时应根据实际滑移情况考虑滑移效应对结构受力的影响。     

SHEAR FATIGUE CRACK GROWTH: A LITERATURE SURVEY

A fatigue crack is often initiated by a localized cyclic plastic deformation in a crystal where the active slip plane coincides with the plane of maximum shear stress. Once a crack is initiated, the crack will propagate on the maximum shear plane for a while and, in the majority of the cases, will eventually change to the plane of the applied tensile stress. The “shear” and “tensile” modes of fatigue crack propagation are termed stage I and stage II fatigue crack growth. They are also known as mode II and mode I fatigue crack growth. However, the mechanism of the tensile mode fatigue crack propagation is shear in nature. Considerable progress has been made recently in the understanding of mode II fatigue crack growth. This paper reviews the various test methods and related data analyses. The combined mode I and mode II elastic crack tip stress field is reviewed. The development and the design of the compact shear specimen are described and the results of fatigue crack growth tests using the compact shear specimens are reviewed. The fatigue crack growth tests and the results of inclined cracks in tensile panels, center cracks in plates under biaxial loading, cracked beam specimens with combined bending and shear loading, center cracked panels and the double edge cracked plates under cyclic shear loading are reviewed and analyzed in detail.     

Crack-tip fields in anisotropic shells

Asymptotic crack-tip fields including the effect of transverse shear deformation in an anisotropic shell are presented. The material anisotropy is defined here as a monoclinic material with a plane symmetry at x ₃=0. In general, the shell geometry near the local crack tip region can be considered as a shallow shell. Based on Reissner shallow shell theory, an asymptotic analysis is conducted in this local area. It can be verified that, up to the second order of the crack tip fields in anisotropic shells, the governing equations for bending, transverse shear and membrane deformation are mutually uncoupled. The forms of the solution for the first two terms are identical to those given by respectively the plane stress deformation and the antiplane deformation of anisotropic elasticity. Thus Stroh formalism can be used to characterize the crack tip fields in shells up to the second term and the energy release rate can be expressed in a very compact form in terms of stress intensity factors and Barnett–Lothe tensor L. The first two order terms of the crack-tip stress and displacement fields are derived. Several methods are proposed to determine the stress intensity factors and `T-stresses'. Three numerical examples of two circular cylindrical panels and a circular cylinder under symmetrical loading have demonstrated the validity of the approach.