Creep deformation of beams under compression and bending stresses

Methods for calculating the creep strain of beams or plates in compression were considered by several authors (e.g., see [1–3]).The goal of calculations is to determine the function describing the deflection increase and the time in which the deflection attains the maximum admissible value.In this paper, we consider the creep strain with regard to the common action of compression and bending stresses.     

Fatigue failure of polyethylene methacrylate in adhesive assembly under unsymmetrical bending

In this paper, a special bending fatigue experiment was firstly performed to investigate the fatigue behavior of polyethylene methacrylate in adhesive assembly. Fatigue lifetime property (S–N curve) was obtained. Finite element calculations on the whole structures also gave the same results with the testing. Based on the experimental data and finite element analysis, a local stress law of predicting bending fatigue lifetime was put forward. The predication lifetime for the polyethylene methacrylate agreed well with the experimental results. Following the strain energy density (SED) criterion was applied to predict the crack initiation and growth path of the adhesive assembly. The predicted results were in good agreement with the optical microscopy (OM) failure image of the failure specimen. SEM image of fracture further showed that there were lots of parallel fatigue lines with perpendicularity to the direction of crack, and an obvious boundary from the crack propagation failure to final brittle fracture.     

混凝土三点弯曲梁软化特性的数值计算方法

本文推导了基于虚拟裂纹模型的混凝土梁荷载位移全曲线的计算公式,研究了不同网格剖分、不同的拉伸软化曲线模型和不同的预制裂纹长度等对荷载位移全曲线的影响。计算结果表明,其结果不受网格划分的影响,通过与实验结果对比,证明了其有效性。     

Measurement of stress distribution in sandwich beams under four-point bending

The bending-stress distributions through thickness in sandwich-composite beams are different from those obtained by conventional composite-beam theory because of the shear effect of the core, especially when the ratio of elastic moduli of face to core (k=E _f /E _c ) is large. Accordingly, the stress distributions in sandwich beams of composite materials with various combinations of face and core materials subjected to four-point bending are analyzed by introducing the multilayer-buitup theory. The bending stiffnesses of face and core, and the relative displacement between both faces are taken into consideration in the analysis. Photoelastic measurements were carried out on model specimens having four differentk-values and the applicable ranges of the two theories are discussed on the basis of the experimental results. It is shown that the experimental-stress distributions in sandwich-composite beams havingk-values larger than 120 can be well explained by the multilayer-builtup theory. The ratio of the coupling moment due to the axial forces in the two faces to the applied total moment, which denotes the sandwich structural efficiency, can be well estimated by the multilayer-builtup theory. The availability of this simple onedimensional theory should be useful in the structural design of sandwich beams with a small-core rigidity.     

Buckling life estimation of circular tubes under cyclic bending

In this paper, an experimental investigation of the degradation and buckling of circular tubes subjected to cyclic bending is discussed. The machinery specimens (with different diameter-to-thickness ratios but the same inside diameter) and method of testing (cyclic bending) in this study were the same as the ones used by Lee et al. (Lee, K.L., Pan, W.F., Kuo, J.N., 2001. The influence of the diameter-to-thickness ratio on the stability of circular tubes under cyclic bending. International Journal of Solids and Structures 38, 2401–2413.) for 316L stainless steel circular tubes. The experimental investigation was extended to different outside and inside diameters of the same circular tubes subjected to cyclic bending. Based on the experimental findings, the empirical formulation proposed by Lee et al. (2001) was modified so that it can now be used to simulate the relationship between the prescribed curvature and the number of cycles necessary to produce buckling. In addition, it was found that the experimental curve of the ovalization and the number of cycles necessary to produce buckling could be divided into three stages – an initial, secondary and tertiary stage. An empirical relationship, similar to the Bailey–Norton creep formulation, was proposed for simulating the aforementioned curve for the initial and secondary stages in this study. The derived empirical relationship was in good agreement with the experimental data.     

The unsymmetrical bending of cantilever rectangular plates

This paper discusses the problems of the unsymmetrical bending of cantilever rectangular plates under various loads by the energy method. We illustrate numerous calculating examples such as the plates which are subjected by the concentrated forces or concentrated couples unsymmetrically on free sides and corner points and by a uniformly or nonuniformly distributed loads unsymmetrically on free edges and so forth.     

Inelastic deformations of stainless steel leaf springs-experiment and nonlinear analysis

A stainless steel leaf spring is designed and constructed followed by its performance evaluation by experiment and non-linear analysis so that an insight into the optimum use of material can be made. Cantilever beams of uniform strength, popularly termed as leaf springs, undergo much larger deflections in comparison to a beam of constant cross-section; that needs inclusions of geometric non-linearity for rigorous analysis. This study deals with such a cantilever beam, but takes into account the material non-linearity as well. Experiments were conducted for such a cantilever beam, with highly non-linear stress-strain curves. In addition to the experiment, a computer code in ‘C’ has been developed using the Runge-Kutta technique for the purpose of simulation. Effective modulus-curvature relations are obtained from the non-linear stress-strain relations for different sections of the beam and used for the analysis. It is seen that non-linear stress-strain curve governs the bending of the beam. Importantly, non-linear analysis shows the stresses are not so high as predicted by the linear theory without end-shortening. Moreover, the tensile and compressive stresses are different in magnitude and both decrease along the span. Experimental load-deflection curves are found to be initially concave upward but, non-linear and convex upward at a high load. Comparison of the numerical results with the available experimental results from another research group and theory shows excellent agreement verifying the soundness of the entire numerical simulation scheme.     

Stress-concentration factors for semielliptical notches in beams under pure bending

Elliptical notches in rectangular beams under pure bending are examined photoelastically. Stress-concentration factors due to a single elliptical notch are obtained for wide ranges of 2a/h andd/h, where 2a, d, andh are the width of notch, depth of notch, and depth of beam, respectively. In particular, the geometries of the optimum elliptical notches producing the least stress concentrations are obtained. Almost the whole elliptical boundary of these notches are stressed to the same peak, which indicates that these notches will probably produce the least stress concentrations among all notches, elliptical or nonelliptical. The graphs herein will enable the designers to find the stress-concentration factors of elliptical notches and to pick out the geometry of the optimum notch which will give the least stress concentration for any given values of 2a/h andd/h. Stresses and the stress-concentration factors at the bottom of the beam opposite the notch are also obtained. These stresses, though smaller in magnitude, are of an opposite sign to the peak stress at the notch. For brittle materials, a smaller tensile stress may be more critical than a large compressive stress; therefore, these stress-concentration factors are also given.     

纵横载荷作用下功能梯度梁的弯曲和过屈曲

基于一阶非线性梁理论和物理中面概念,导出了纵横向载荷作用下功能梯度材料(FGM)梁非线性弯曲和过屈曲问题的控制方程,并获得了该问题的精确解;据此解研究了梯度材料性质、外载荷、横向剪切变形以及边界条件等因素对功能梯度材料梁非线性力学行为的影响,分析中假设功能梯度材料性质只沿梁厚度方向,并按成分含量的幂指数函数形式变化。结果表明:纵横载荷共同作用下,功能梯度梁的弯曲构形将有无限多个;随着梯度指数的增大,梁的变形减小,临界载荷升高;随着长高比的增大,横向剪切变形的影响减小。     

Large deformations of nonlinear viscoelastic and multi-responsive beams

This study presents analyses of deformations in nonlinear viscoelastic beams that experience large displacements and rotations due to mechanical, thermal, and electrical stimuli. The studied beams are relatively thin so that the effect of the transverse shear deformation is neglected, and the stretch along the transverse axis of the beams is also ignored. It is assumed that the plane that is perpendicular to the longitudinal axis of the undeformed beam remains plane during the deformations. The nonlinear kinematics of the finite strain beam theory presented by Reissner [27] is adopted, and a nonlinear viscoelastic constitutive relation based on a quasi-linear viscoelastic (QLV) model is considered for the beams. Deformation in beams due to mechanical, thermal, and electric field inputs are incorporated through the use of time integral functions, by separating the time-dependent function and nonlinear measures of field variables. The nonlinear measures are formulated by including higher order terms of the field variables, i.e. strain, temperature, and electric field. Responses of beams under mechanical, thermal, and electrical stimuli are illustrated and the effects of nonlinear constitutive relations on the overall deformations of the beams are highlighted.     

Lateral-torsional buckling of simply supported beams under uniform bending and axial tensile force

In this paper, the effect of a centrally applied external axial tensile load on the lateral-torsional buckling resistance of simply supported I-beams under uniform bending acting in the plane of maximum rigidity is studied. A linear and a nonlinear analysis are performed. Following the linear analysis, an expression for the critical moment of lateral-torsional buckling is presented in which the influence of the axial tensile force is included. There is an upper limit of this force over which the equilibrium in the deformed state is not possible. In the nonlinear analysis, the nature of the critical state is studied, considering the initial part of the post-buckling path. It is concluded that this critical state is associated with a stable symmetrical bifurcation point. Nevertheless, the post-buckling path is very shallow; therefore, the beam cannot exhibit practically post-buckling strength. The paper is supplemented by a representative example.     

Stress-concentration factors for multiple semielliptical notches in beams under pure bending

Uniformly spaced semielliptical notches in rectangular beams under pure bending are examined photoelastically. Minimum stress-concentration factors, produced by multiple elliptical notching of beams, are obtained for wide ranges of notch width, semiminor elliptical axis, notch depth, notch pitch and depth of beam. In particular, the geometries of the optimum elliptical notch producing the least stress concentration are obtained for a practical range of parameters. Stress-concentration factors for beams with multiple semi-elliptical notches are compared to those for beams with single semielliptical notches and to those for beams with semicircular notches. The maximum reduction of stress-concentration factor for beams with multiple semielliptical notches is to approximately 37 percent of the stress-concentration factor for beams with single semielliptical notches. Within the range of parameters investigated, the stress-concentration factor for beams with multiple semielliptical notches was 15 to 37 percent less than that for multiple semicircular notches.     

A Curvature-ovalization measurement apparatus for circular tubes under cyclic bending

In this paper, the authors report the design and construction of a new measurement apparatus, along with the result of its testing. This apparatus can be placed at the midspan of the circular tube specimen and is suitable for simultaneous experimental determinations of the tube curvature and ovalization of the tube cross section. For testing the newly designed apparatus, the tube specimen of AISI 304 stainless steel was bent cyclically. It has been shown that the actual curvature and ovalization of the circular tube subjected to bending can be directly measured by the apparatus.     

Inelastic responses of a two-bar system with temperature-dependent elastic modulus under cyclic thermomechanical loadings

This paper is concerned with the elastic plastic response of a two-bar system with temperature-dependent elastic coefficients under cyclic thermomechanical loadings. Such materials are characterized by lack of results concerning the asymptotic behaviors and conditions for shakedown occurrence. This study shows that the considered simple structure is sufficiently complex to experience different periodic long-term behaviors as in classical elastoplasticity. In order to understand how Melan–Koiter method works for such materials, the evolution of the structure’s response until the stabilization of the plastic strain (‘shakedown’) or the asymptotic dissipative behavior (‘alternating plasticity’ or ‘ratcheting’) is analytically addressed and the Bree diagram is then constructed. The main result of this work is that the residual stress and strain fields are time-dependent even when shakedown occurs. Besides, we proved that Halphen’s conjecture (Halphen, 2005) giving a sufficient condition for shakedown occurrence is not a necessary condition. Finally, numerical results performed by an incremental finite element procedure are presented.     

Energy-minimizing deformations of elastic sheets with bending stiffness

Necessary conditions for energy-minimizing deformations are derived for a theory of sheets in which the strain energy function depends on the second derivatives of the deformation as well as its first derivatives. All of these conditions are extensions of well-known necessary conditions in classical calculus of variations. The interpretation of some of these conditions as material stability conditions is explained.     

Generalized variational principles of symmetrical elasticity problem of large deformations

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Experimental study on crack curving propagation in bending beams under impulsive load

Dynamic fracture behaviour of crack curving in bent beams has been investigated. In order to understand the propagation mechanism of such cracks under impact, an experimental method is used that combines dynamic photoelasticity with dynamic caustics to study the interaction of the flexural waves and the crack. From the state change of the transient stresses in polymer specimen, the curving fracture in the impulsively loaded beams is analyzed. The dynamic responses of crack tips are evaluated by the stress intensity factors for the cracks running in varying curvature paths under bending stress wave. The project supported by the National Natural Science Foundation of China and the Scientific Commission of Yunnan Province of China     

Evaluation of cyclic plasticity models in ratcheting simulation of straight pipes under cyclic bending and steady internal pressure

This paper evaluates seven cyclic plasticity models for structural ratcheting response simulations. The models evaluated are bilinear (Prager), multilinear (Besseling), Chaboche, Ohno–Wang, Abdel Karim–Ohno, modified Chaboche (Bari and Hassan) and modified Ohno–Wang (Chen and Jiao). The first three models are already available in the ANSYS finite element package, whereas the last four were implemented into ANSYS for this study. Experimental responses of straight steel pipes under cyclic bending with symmetric end rotation history and steady internal pressure were recorded for the model evaluation study. It is demonstrated that when the model parameters are determined from the material response data, none of the models evaluated perform satisfactorily in simulating the straight pipe diameter change and circumferential strain ratcheting responses. A detailed parameter sensitivity study with the modified Chaboche model was conducted to identify the parameters that influence the ratcheting simulations and to determine the ranges of the parameter values over which a genetic algorithm can search for refinement of these values. The refined parameter values improved the simulations of straight pipe ratcheting responses, but the simulations still are not acceptable. Further, improvement in cyclic plasticity modeling and incorporation of structural features, like residual stresses and anisotropy of materials in the analysis will be essential for advancement of low-cycle fatigue response simulations of structures.