On shear lag effects in orthotropic composite beams

Based upon the linearised theories of the bending and stretching of thin plates, an analysis is presented for the interaction between non-prismatic beams and an orthotropic concrete plate. It is shown that an exponential representation for the steel beam profiles provides a suitable basis for studying interaction in continuous non-prismatic beams, and for deducing suitable effective widths of slab for design purposes. The influence of “elastic” shear connection modulus is studied, as well as the effect of the varying flexural rigidity of the steel beams. The dependence of interaction on shear connection modulus in continuous beams is demonstrated through deflexion and slip characteristics, and so also is the dependence of interaction on the severity of the variation of flexural rigidity. The solution can be specialised to the limiting case of prismatic steel beams and a concrete slab and also to the solutions of rectangular plates with certain edge conditions.     

Analytical solutions for the time-dependent behaviour of composite beams with partial interaction

This paper presents a generic modelling for the time-dependent analysis of composite steel–concrete beams with partial shear interaction that occurs due to the deformation of the shear connection. The time effects considered in this modelling are those that arise from shrinkage and creep deformations of the concrete slab, and these effects are modelled using algebraic representations such as those of the age-adjusted effective modulus method (AEMM) and the mean stress method (MS), which are viscoelastic models for concrete deformation that can be stated algebraically. The generic model lends itself to closed form solutions for the analysis of composite beams subjected to a generic applied loading under a variety of end conditions. In this paper, the generic model is applied for the time-dependent analysis of composite beams that are simply supported and encastré, and to a propped cantilever, that are subjected to uniformly distributed loading and shrinkage deformations. Various representations of the structural behaviour of these beams are given in closed form which can also be used to benchmark available modelling techniques, i.e. finite element and finite difference formulations, which require a spatial discretisation to be specified as well as the time discretisation to perform a time analysis.     

Generic non-linear modelling of a bi-material composite beam with partial shear interaction

Composite members composed of two materials joined by shear connection find widespread use in engineering infrastructure, in both traditional practice and innovative applications. Studies in the literature dating back nearly 60 years have elucidated the mechanics of the behaviour of these composite structural members in which the solution for the slip at the interface between the materials was determined by solving a linear differential equation. However, these solutions are based on a linear formulation of the strain-displacement relationship, and in some applications this relationship must be represented in non-linear form, so that the second order effects in the member can be quantified correctly. This paper presents such a study for a composite member with two materials, being typical of a steel-concrete composite beam in structural engineering. It quantifies the restraint of the member ends by longitudinal and rotational elastic springs, so that the axial tension developed is a function of the transverse loading, material properties, cross-sectional properties and the restraint stiffness. The problem is treated using minimisation of the total potential stored in the two members, the elastic shear connection at their interface, the restraints at the ends and the work done by the transverse forces, for which the differential equations for the deformations can be determined from routine variational calculus. The non-linear equation of equilibrium relating the external loading to the internal actions is stated in closed form by invoking the static and kinematic boundary conditions for the member. The solution is compared with closed form treatments derived elsewhere, and a representative member is analysed so that the influences of the non-linearity, end restraint stiffness and degree of partial shear interaction on its behaviour can be examined.     

A BEM solution to transverse shear loading of composite beams

In this paper a boundary element method is developed for the solution of the general transverse shear loading problem of composite beams of arbitrary constant cross-section. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson’s ratio and are firmly bonded together. The analysis of the beam is accomplished with respect to a coordinate system that has its origin at the centroid of the cross-section, while its axes are not necessarily the principal ones. The transverse shear loading is applied at the shear centre of the cross-section, avoiding in this way the induction of a twisting moment. Two boundary value problems that take into account the effect of Poisson’s ratio are formulated with respect to stress functions and solved employing a pure BEM approach, that is only boundary discretization is used. The evaluation of the transverse shear stresses is accomplished by direct differentiation of these stress functions, while both the coordinates of the shear center and the shear deformation coefficients are obtained from these functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of the proposed shear deformation coefficients compared with those obtained from a 3-D FEM solution of the ‘exact’ elastic beam theory is remarkable.     

Large displacement analysis of shear deformable composite beams with interlayer slips

This paper presents a novel geometric non-linear finite element formulation for the analysis of shear deformable two-layer beams with interlayer slips. We adopt the co-rotational approach where the motion of the element is decomposed into two parts: a rigid body motion which defines a local coordinate system and a small deformational motion of the element relative to this local coordinate system. The main advantage of this approach is that the transformation matrices relating local and global quantities are independent to the choice of the geometrical linear local element. The effect of transverse shear deformation of the layers is taken into account by assuming that each layer behaves as a Timoshenko beam element. The layers are assumed to be continuously connected and partial interaction is considered by considering a continuous relationship between the interface shear flow and the corresponding slip. In order to avoid curvature and shear locking phenomena, the local linear element is formulated using “exact” displacement shape functions derived from the closed-form solution of the governing equations of a two-layer beam element. Finally, three numerical applications are presented in order to assess the performance of the proposed formulation.     

梯形箱梁剪力滞后效应的精细化分析

引入剪滞翘曲应力自平衡条件的影响,考虑了剪切变形和剪滞效应等因素,设置了三个不同的剪滞纵向位移差函数以准确反映梯形箱梁不同宽度翼板的剪滞变化幅度,提出了一种能对工程中常用箱梁静力学特性分析的精确解法。本文以能量变分原理为基础建立了薄壁箱梁的弹性控制微分方程和自然边界条件,获得了相应广义位移的闭合解。算例中,分析了不同荷载形式、跨宽比和悬臂板长度等因素对箱梁静力学特性的影响,结果显示出引入剪滞翘曲应力自平衡条件的必要性。     

Free vibration analyses of axially loaded laminated composite beams based on higher-order shear deformation theory

The dynamic stiffness matrix method is introduced to solve exactly the free vibration and buckling problems of axially loaded laminated composite beams with arbitrary lay-ups. The Poisson effect, axial force, extensional deformation, shear deformation and rotary inertia are included in the mathematical formulation. The exact dynamic stiffness matrix is derived from the analytical solutions of the governing differential equations of the composite beams based on third-order shear deformation beam theory. The application of the present method is illustrated by two numerical examples, in which the effects of axial force and boundary condition on the natural frequencies, mode shapes and buckling loads are examined. Comparison of the current results to the existing solutions in the literature demonstrates the accuracy and effectiveness of the present method.     

Axial-flexural coupled vibration and buckling of composite beams using sinusoidal shear deformation theory

A finite element model based on sinusoidal shear deformation theory is developed to study vibration and buckling analysis of composite beams with arbitrary lay-ups. This theory satisfies the zero traction boundary conditions on the top and bottom surfaces of beam without using shear correction factors. Besides, it has strong similarity with Euler–Bernoulli beam theory in some aspects such as governing equations, boundary conditions, and stress resultant expressions. By using Hamilton’s principle, governing equations of motion are derived. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results for cross-ply and angle-ply composite beams are obtained as special cases and are compared with other solutions available in the literature. A variety of parametric studies are conducted to demonstrate the effect of fiber orientation and modulus ratio on the natural frequencies, critical buckling loads, and load-frequency curves as well as corresponding mode shapes of composite beams.     

Free vibration analysis of third-order shear deformable composite beams using dynamic stiffness method

The dynamic stiffness method is introduced to investigate the free vibration of laminated composite beams based on a third-order shear deformation theory which accounts for parabolic distribution of the transverse shear strain through the thickness of the beam. The exact dynamic stiffness matrix is found directly from the analytical solutions of the basic governing differential equations of motion. The Poisson effect, shear deformation, rotary inertia, in-plane deformation are considered in the analysis. Application of the derived dynamic stiffness matrix to several particular laminated beams is discussed. The influences of Poisson effect, material anisotropy, slenderness and end condition on the natural frequencies of the beams are investigated. The numerical results are compared with the existing solutions in literature whenever possible to demonstrate and validate the present method.     

基于剪切变形的矩形梁剪力滞求解方法

Timoshenko梁通过假设截面的剪切刚度和附加平均剪切转角变形的方式来近似修正初等梁中未考虑剪切变形能的问题,这与梁剪应力沿梁高变化的实际不符。本文基于材料力学剪应力计算式和相应的剪切变形理论,从剪切变形与梁的位移关系入手,导出矩形梁考虑剪切变形时的纵向位移沿梁高方向的函数关系式,证明该位移可分解为纯弯曲引起的位移和剪力引起的剪力滞翘曲位移之和。应用剪力滞广义坐标与广义力的概念,基于能量变分原理得到等截面梁剪力滞控制微分方程组及其通解形式。对均布荷载作用下矩形简支梁的算例分析表明,本文算法与弹性力学精确解对比,两者的应力和挠度剪力滞系数求解结果非常接近,本文算法有足够的精度,且比弹性力学简单。     

剪力滞效应对箱形梁挠度影响的研究

从剪力滞翘曲应力的轴向平衡条件出发,选取双室箱梁的合理翘曲位移函数,引入相应于剪力滞翘曲变形的惯性矩和惯性积等几何特性,用能量变分法建立薄壁箱梁剪力滞效应分析的控制微分方程。通过求解控制微分方程,导出集中荷载和均布荷载作用下简支箱梁和悬臂箱梁的挠度公式及有限梁段单元刚度矩阵,模型试验和ANSYS壳单元计算结果证实了其正确性。结合简支、悬臂和连续箱梁数值算例,具体分析剪力滞效应对箱梁挠度的提高程度。结果表明,无论在集中荷载还是均布荷载作用下,剪力滞效应对简支箱梁的挠度均有显著提高。在集中荷载作用下,剪力滞效应对连续箱梁挠度的提高可达14%;对于跨宽比约为4.0～6.0的简支箱梁,可将按初等梁计算的跨中挠度乘以提高系数1.05～1.11;计算悬臂箱梁的挠度时,一般可以忽略剪力滞效应的影响。     

箱形梁剪力滞和剪切效应引起的附加挠度分析

将箱形梁腹板剪切变形纳入初等梁挠曲变形,在全截面上引入剪力滞翘曲修正系数,重新定义了剪力滞翘曲位移模式。选取剪力滞效应引起的附加挠度为广义位移,计算外力势能时考虑剪力滞广义位移的影响,应用能量变分法建立了反映剪力滞和剪切效应的控制微分方程,并导出了均布荷载作用下简支箱梁和两跨连续箱梁剪力滞和剪切效应附加挠度的解析解。数值算例表明,本文方法计算的总挠度值与有限元数值解吻合良好,从而验证了本文方法的合理性。算例箱梁剪切附加挠度明显大于剪力滞附加挠度;简支箱梁跨中截面的剪切和剪力滞附加挠度分别占初等梁挠度的2.50%和1.97%,两跨连续箱梁距中支点9l/16截面分别占27.45%和16.87%。     

The growth of shear bands in composite microstructures

Shear band formation in materials with inhomogeneous and composite microstructures is influenced by factors that usually do not come into play in monolithic materials. Experiments and calculations have shown that inhomogeneities in material properties enhance the localization of deformation. This investigation concerns the propagation of shear bands in a two-phase tungsten composite under the conditions of nominally pure shear deformation. Finite element calculations are carried out to delineate the effects of different grain–matrix morphologies. In the numerical models, the initiation of shear bands is triggered by a notch, simulating the effect of defects such as microcracks and microvoids in materials. Calculations demonstrate that phase morphology, particle size and the relative location of initiation site have significant influences on the development of localized deformation. The work and energy evolutions are tracked for each constituent phase in the microstructures. In addition, the exchange of thermal energy through heat flow between the phases is analyzed. The results show that a strong correlation exists between the course of shear band propagation and the thermomechanical coupling between microscopic phases.     

框筒结构考虑负剪力滞效应的简化分析

在引入D值法分析框筒结构在水平荷载作用下的整体剪切变形和内力的基础上,提出了框筒结构各层同时存在正剪力滞和负剪力滞的简化假定,并假设了框筒各柱的轴向应变分布模式,对负剪力滞的影响因素进行了分析,得出了一些重要的结论。本文方法比等效平面框架法和等效连续体法更为简单、实用,可直接进行手算。通过与空间框架分析程序的比较,精度可供初步设计使用。     

Geometrical nonlinear analysis of thin-walled composite beams using finite element method based on first order shear deformation theory

Based on a seven-degree-of-freedom shear deformable beam model, a geometrical nonlinear analysis of thin-walled composite beams with arbitrary lay-ups under various types of loads is presented. This model accounts for all the structural coupling coming from both material anisotropy and geometric nonlinearity. The general nonlinear governing equations are derived and solved by means of an incremental Newton–Raphson method. A displacement-based one-dimensional finite element model that accounts for the geometric nonlinearity in the von Kármán sense is developed to solve the problem. Numerical results are obtained for thin-walled composite beam under vertical load to investigate the effects of fiber orientation, geometric nonlinearity, and shear deformation on the axial–flexural–torsional response.     

The composite damping capacity of sandwich cantilever beams

The governing equation of the flexural forced vibration of a cantilever sandwich beam excited by a sinusoidal displacement at the clamped end is developed by utilizing the conventional Hamilton's Principle. The effect of damping of the composite beam is incorporated into the elastic equation of motion by utilizing the Correspondence Principle of the linear viscoelastic theory. Several plots for different values of the composite damping factor are presented. For comparison, an experimental setup was utilized to test different composite beams. The variation of the experimental results with those derived theoretically seem to be in agreement within the frequency range of the first few modes.     

考虑剪力滞和剪切变形的薄壁箱梁自振特性分析

以能量变分原理为基础,综合考虑箱形梁满足应力自平衡的剪力滞、剪切变形和转动惯量等多重因素的影响,推导出箱梁的自由振动方程及自然边界条件。通过算例将本文解析计算结果与ANSYS有限元计算结果进行了比较。结果表明,两者计算结果吻合良好,论证了本文计算方法的正确。所得公式比以往箱形箱梁自振特性计算理论有一定发展,并得出了一些对工程设计有意义的结论;在剪力滞效应的作用下,箱形梁的固有频率减小幅度较大,不能忽略;剪力滞效应随频率阶次的升高而变大,随着跨宽比的减小而增大。     

The shear viscosity dependence on concentration,molecular weight,and shear rate of polystyrene solutions

The solution viscosity of narrow molecular weight distribution polystyrene samples dissolved in toluene and trans-decalin was investigated. The effect of polymer concentration, molecular weight and shear rate on viscosity was determined. The molecular weights lay between 5 10⁴ and 24 10⁶ and the concentrations covered a range of values below and above the critical valuec ^*, at which the macromolecular coils begin to overlap. Flow curves were generated for the solutions studied by plotting log versus log . Different molecular weights were found to have the same viscosity in the non-Newtonian region of the flow curves and follow a straight line with a slope of – 0.83. A plot of log ₀ versus logM _w for 3 wt-% polystyrene in toluene showed a slope of approximately 3.4 in the high molecular weight regime. Increasing the shear rate resulted in a viscosity that was independent of molecular weight. The sloped (log)/d (logM _w ) was found to be zero for molecular weights at which the corresponding viscosities lay on the straight line in the power-law region.On the basis of a relation between _sp and the dimensionless productc · [], simple three-term equations were developed for polystyrene in toluene andt-decalin to correlate the zero-shear viscosity with the concentration and molecular weight. These are valid over a wide concentration range, but they are restricted to molar masses greater than approximately 20000. In the limit of high molecular weights the exponent ofM _w in the dominant term in the equations for both solvents is close to the value 3.4. That is, the correlation between _sp andc · [] results in a sloped(log _sp)/d(logc · []) of approximately 3.4/a at high values ofc · [] wherea is the Mark-Houwink constant. This slope of 3.4/a is also the power ofc in the plot of ₀ versusc at high concentrations. a Mark-Houwink constant - B ₁,B ₂,B _n constants - c concentration (g · cm^–3) - c ^* critical concentration (g · cm^–3) - K, K constants - K _H Huggins constant - M molecular weight - M _c critical molecular weight - M _n number-average molecular weight - M _w weight-average molecular weight - n sloped(log _sp)/d (logc · []) at highc · [] - PS polystyrene - T temperature (K) - shear rate (s^–1) - critical shear rate (s^–1) - viscosity (Pa · s) - ₀ zero-shear viscosity (Pa · s) - _s solvent viscosity (Pa · s) - _sp specific viscosity - [] intrinsic viscosity (cm³ · g^–1) - dynamic viscosity (Pa · s) - | ^*| complex dynamic viscosity (Pa · s) - angular frequency (rad/s) - density of polymer solution (g · cm^–3) - ₁₂ shear stress (Pa) Dedicated to Prof. Dr. J. Schurz on the occasion of his 60^th birthday.Excerpt from the dissertation of Reinhard Kniewske: Bedeutung der molekularen Parameter von Polymeren auf die viskoelastischen Eigenschaften in wäßrigen und nichtwäßrigen Medien, Technische Universität Braunschweig 1983.     

The shear gage: For reliable shear modulus measurements of composite materials

A special strain gage called the shear gage was developed for composite materials testing with notched shear specimens. The shear-gage records the average shear strain across the entire test section between the notches of the losipescu and compact shear specimens rather than just sampling the shear strain over a small region in the center of the test section. Hence, the shear stress/strain response is obtained by dividing the average shear stress (load divided by the cross-sectional area between the notches) by the average shear strain. By placing gages on both faces of the specimen, accurate and repeatable shear-modulus measurements can be made without prior knowledge of the shear strain or stress distributions. This scheme essentially integrates the shear strain through the entire test section. Knowledge of other material properties is not required to accurately determine shear modulus values. The shear gage was tested on a variety of composite and isotropic materials resulting in more reliable shear modulus determination and less scatter than previously possible.