简支及悬臂箱梁在角隅轴向荷载作用下的翼板纵向应力

基于能量变分原理,拟定轴向荷载作用下箱梁的纵向位移函数,得到关于翼板剪切变形引起的位移差函数的基本微分方程,继而推导出箱梁翼板纵向应力表达式,并首次得出角隅轴向荷载作用下翼板出现应力不均匀分布的荷载及边界条件。通过对一模型箱梁进行计算,并与通用有限元软件ANSYS壳单元计算结果进行比较,验证了该方法和所推导公式的正确性。研究结果表明,当作用于简支箱梁截面角隅处的轴向荷载（合力无偏心）为集中或分布荷载时,翼板不产生纵向应力不均匀现象;当作用于悬臂箱梁截面角隅处的轴向荷载（合力无偏心）为集中荷载时,翼板不产生纵向应力不均匀现象,而当荷载轴向分布时,翼板将产生纵向应力不均匀现象。实际工程中,横力弯曲使悬臂箱梁产生剪力滞效应,这种效应会与轴向分布荷载产生的效应叠加,设计时对此应予以充分考虑。     

THE SHEAR LAG EFFECT OF BOX BEAMS WITH VARYING LATERAL LOADING LOCATIONS 1)

对箱梁各翼板(顶板、悬臂板、底板)分设不同剪力滞广义纵向位移,其横向分布均取二次抛物线形式,并引入载荷横向位置参数η,以分析载荷横向变位对剪力滞效应的影响.运用能量变分原理,建立剪力滞控制微分方程,求解了简支梁和悬臂梁在均布载荷作用下的控制微分方程的解.算例分析表明:载荷横向变位改变直接承受载荷的翼板的正负剪力滞特性,对非直接承载翼板只改变其应力幅度;箱梁横向框架效应对直接承载翼板纵向应力的贡献远远大于剪切变形.与块体有限元分析结果较吻合,表明该算法能较准确分析载荷横向变位作用下箱梁剪力滞的变化规律.     

A higher-order theory for static and dynamic analyses of functionally graded beams

The higher-order theory is extended to functionally graded beams (FGBs) with continuously varying material properties. For FGBs with shear deformation taken into account, a single governing equation for an auxiliary function F is derived from the basic equations of elasticity. It can be used to deal with forced and free vibrations as well as static behaviors of FGBs. A general solution is constructed, and all physical quantities including transverse deflection, longitudinal warping, bending moment, shear force, and internal stresses can be represented in terms of the derivatives of F. The static solution can be determined for different end conditions. Explicit expressions for cantilever, simply supported, and clamped-clamped FGBs for typical loading cases are given. A comparison of the present static solution with existing elasticity solutions indicates that the method is simple and efficient. Moreover, the gradient variation of Young’s modulus and Poisson’s ratio may be arbitrary functions of the thickness direction. Functionally graded Rayleigh and Euler–Bernoulli beams are two special cases when the shear modulus is sufficiently high. Moreover, the classical Levinson beam theory is recovered from the present theory when the material constants are unchanged. Numerical computations are performed for a functionally graded cantilever beam with a gradient index obeying power law and the results are displayed graphically to show the effects of the gradient index on the deflection and stress distribution, indicating that both stresses and deflection are sensitive to the gradient variation of material properties.     

On the uniqueness of large deflections of a uniform cantilever beam under a tip-concentrated rotational load

The problem of a uniform cantilever beam under a tip-concentrated load, which rotates in relation with the tip-rotation of the beam, is studied in this paper. The formulation of the problem results in non-linear ordinary differential equations amenable to numerical integration. A relation is obtained for the applied tip-concentrated load in terms of the tip-angle of the beam. When the tip-concentrated load acts always normal to the undeformed axis of the beam (the rotation parameter, β=0) there is a possibility of obtaining non-unique solution for the applied load. This phenomenon is also observed for other rotation parameters less than unity. When the tip-concentrated load is acting normal to the deformed axis of the beam (β=1), many load parameters are obtained for a tip-angle with different deformed configurations of the beam. However, each load parameter corresponds to a tip-angle, which confirms the uniqueness on the solution of non-linear differential equations.     

Homogenization estimates for fiber-reinforced elastomers with periodic microstructures

This work presents a homogenization-based constitutive model for the mechanical behavior of elastomers reinforced with aligned cylindrical fibers subjected to finite deformations. The proposed model is derived by making use of the second-order homogenization method [Lopez-Pamies, O., Ponte Castañeda, P., 2006a. On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—theory. J. Mech. Phys. Solids 54, 807–830], which is based on suitably designed variational principles utilizing the idea of a “linear comparison composite.” Specific results are generated for the case when the matrix and fiber materials are characterized by generalized Neo-Hookean solids, and the distribution of fibers is periodic. In particular, model predictions are provided and analyzed for fiber-reinforced elastomers with Gent phases and square and hexagonal fiber distributions, subjected to a wide variety of three-dimensional loading conditions. It is found that for compressive loadings in the fiber direction, the derived constitutive model may lose strong ellipticity, indicating the possible development of macroscopic instabilities that may lead to kink band formation. The onset of shear band-type instabilities is also detected for certain in-plane modes of deformation. Furthermore, the subtle influence of the distribution, volume fraction, and stiffness of the fibers on the effective behavior and onset of macroscopic instabilities in these materials is investigated thoroughly.     

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.     

Large deflection of cantilever beams with geometric non-linearity: Analytical and numerical approaches

Non-linear shooting and Adomian decomposition methods have been proposed to determine the large deflection of a cantilever beam under arbitrary loading conditions. Results obtained only due to end loading are validated using elliptic integral solutions. The non-linear shooting method gives accurate numerical results while the Adomian decomposition method yields polynomial expressions for the beam configuration. With high load parameters, occurrence of multiple solutions is discussed with reference to possible buckling of the beam-column. An example of concentrated intermediate loading (cantilever beam subjected to two concentrated self-balanced moments), for which no closed form solution can be obtained, is solved using these two methods. Some of the limitations and recipes to obviate these are included. The methods will be useful toward the design of compliant mechanisms driven by smart actuators.     

Investigation of the Effect of Axial Loads on the Transverse Vibrations of a Vertical Cantilever with Variable Parameters

The method of influence function is applied to the solution of the boundary-value problem on the free transverse vibrations of a vertical cantilever and a bar subjected to axial loads. To demonstrate the capabilities of the method, a cantilever with the free end under two types of loading — point forces (conservative and follower) and a load distributed along the length (dead load) — is analyzed. A characteristic equation in the general form, which does not depend on the cantilever shape and on the type of axial load, is given. The Cauchy influence function depends on the cantilever shape and the type of axial load. As an example, a tapered cantilever subjected to conservative and follower forces and an elastically supported bar under the dead load are considered in detail. The characteristic equation derived allows one to evaluate the natural frequencies and the Euler critical loads. It is shown that the calculated natural frequencies and critical forces are in a good agreement with the exact values when several terms are retained in the characteristic series. The high accuracy of the method is also confirmed     

Compressive failure of fiber composites under multi-axial loading

This paper examines the compressive strength of a fiber reinforced lamina under multi-axial stress states. An equilibrium analysis is carried out in which a kinked band of rotated fibers, described by two angles, is sandwiched between two regions in which the fibers are nominally straight. Proportional multi-axial stress states are examined. The analysis includes the possibility of bifurcation from the current equilibrium state. The compressive strength of the lamina is contingent upon either attaining a load maximum in the equilibrium response or satisfaction of a bifurcation condition, whichever occurs first. The results show that for uniaxial loading a non-zero kink band angle β produces the minimum limit load. For multi-axial loading, different proportional loading paths show regimes of bifurcation dominated and limit load dominated behavior. The present results are able to capture the beneficial effect of transverse compression in raising the composite compressive strength as observed in experiments.     

Large deflections of non-prismatic nonlinearly elastic cantilever beams subjected to non-uniform continuous load and a concentrated load at the free end

This work studies large deflections of slender,non-prismatic cantilever beams subjected to a combined loading which consists of a non-uniformly distributed continuous load and a concentrated load at the free end of the beam.The material of the cantilever is assumed to be nonlinearly elastic.Different nonlinear relations between stress and strain in tensile and compressive domain are considered.The accuracy of numerical solutions is evaluated by comparing them with results from previous studies and with a laboratory experiment.     

Strength calculations and optimal weight design of multilayer shell-shaped composite products under a set of loads

The stress-strain state of axisymmetric multilayer shells is analyzed using kinematic and static hypotheses that allow for the transverse shear stresses satisfying the necessary equations of state, continuity conditions at the boundaries between the layers and given boundary conditions. A numerical solution of the problem of the stress-strain state for a multilayer bar is compared with the Lekhnitskii solution (for a cantilever beam loaded by a concentrated force and moment) to asses the applicability of the employed bending equations of multilayer shells. It is shown that these solutions are in good agreement. The problem of the initial fracture of the shells considered is formulated using phenomenological strength criteria for each layer. A coordinate-wise descent method in the unit interval is proposed to solve weight optimization problems for multilayer shells of composite materials under combined loading. Regions of safe operating loads and the optimal weight distribution of layer thicknesses are determined for a multilayer bar acted upon by a uniformly distributed load and concentrated force.     

Effects of reduced inflation pressure and vehicle loading on off-road traction and soil stress and deformation state

Field experiments on off-road vehicle traction and wheel–soil interactions were carried out on sandy and loess soil surfaces. A 14 T, 6 × 6 military truck was used as a test vehicle, equipped with 14.00-20 10 PR tyres, nominally inflated to 390 kPa. Tests were performed at nominal and reduced (down to 200 kPa) inflation pressures and at three vehicle loading levels: empty weight, loaded with 3.6 and 6.0 T mass (8000, 11,600 and 14,000 kg, respectively). Traction was measured with a load cell, attached to the rear of the test vehicle as well as to another, braking vehicle. Soil stress state was determined with the use of an SST (stress state transducer), which consists of six pressure sensors. Soil surface deformation was measured in vertical and horizontal directions, with a videogrammetric system. Effects of reduced inflation pressure as well as wheel loading on traction and wheel–soil interactions were analyzed. It was noticed that reduced inflation pressure had positive effects on traction and increased stress under wheels. Increasing wheel load resulted in increasing drawbar pull. These effects and trends are different for the two soil surfaces investigated. The soil surface deformed in two directions: vertical and longitudinal. Vertical deformations were affected by loading, while longitudinal were affected by inflation pressure.     

Spherically isotropic,elastic spheres subject to diametral point load strength test

This paper presents an analytic solution for the stress concentrations within a spherically isotropic, elastic sphere of radius R subject to diametral point load strength test. The method of solution uses the displacement potential approach together with the Fourier–Legendre expansion for the boundary loads. For the case of isotropic sphere, our solution reduces to the solution by Hiramatsu and Oka, 1966 and agrees well with the published experimental observations by Frocht and Guernsey (1953) . A zone of higher tensile stress concentration is developed near the point loads, and the difference between this maximum tensile stress and the uniform tensile stress in the central part of the sphere increases with E/E′ (where E and E′ are the Youngs moduli governing axial deformations along directions parallel and normal to the planes of isotropy, respectively) , G′/G (where G and G′ are the moduli governing shear deformations in the planes of isotropy and the planes parallel to the radial direction) , and ν̄/ν′ (where ν̄ and ν′ are the Poissons ratios characterizing transverse reduction in the planes of isotropy under tension in the same plane and under radial tension, respectively) . This stress difference, in general, decreases with the size of loading area and the Poissons ratio.     

Nonsymmetric buckle patterns in progressive plastic buckling

This paper presents the results of a series of experiments on the progressive plastic buckling of cylindrical shells under axial compressive load. It shows that, for shell bodies with anR/t less than 100, the normal axisymmetric ring buckling will develop into nonsymmetric patterns. We demonstrate that there exists also a class of shells within this thickness-radius range for which nonsymmetric plastic buckling always occurs without the prior formation of a ring. It appears from the limited number of tests made that, for a particularR, R/t, material and rate of loading, there is a critical value ofL, above which there is a high probability of the buckle pattern developing in a nonsymmetric fashion. It seems probable, too, that there are bands ofR/t for a particularL/R, R, material and rate of loading for which the buckle number will be constant. The experiments tend to indicate that the postbuckling efficiency of the shell decreases with increasing buckle number. The nonsymmetric patterns demonstrated appear to be inextensional deformations. They are very similar in character to the Yoshimura pattern which occurs as the limiting case for thin shells in axial compression and, under impact loading. Load-displacement histories are presented for some of the various modes of failure demonstrated.     

Solution of bending of cantilever rectangular plates under uniform surface-load by the method of two-direction trigonometric series

The bending of a cantilever rectangular plate is a very complicated problem in thetheory of plates.For a long time,there have been only approximate solutions for thisproblem by energy methods and numerical methods.since 1979,Prof.F.V.Chang of Tsing Hua University obtained,by the method ofsuperposition,a series of analytic solutions for cantilever rectangular plates under uniformload and concentrated load.In this paper,the two-direction trigonometric series is used to obtain the solution forthe bending of cantilever rectangular plates under uniform load.The obtained results arecompared with the results by the method of superposition.The comparison shows that theresults of these two methods are in good agreement,hence they are mutually confirmed to becorrect.     

Deformations of an elastic,internally constrained material. Part 1: Homogeneous deformations

The nonlinear elastic response of a class of materials for which the deformation is subject to an internal material constraint described in experiments by James F. Bell on the finite deformation of a variety of metals is investigated. The purely kinematical consequences of the Bell constraint are discussed, and restrictions on the full range of compatible deformations are presented in geometrical terms. Then various forms of the constitutive equation relating the stress and stretch tensors for an isotropic elastic Bell material are presented. Inequalities on the mechanical response functions are introduced. The importance of these in applications is demonstrated in several examples throughout the paper.This paper focuses on homogeneous deformations. In a simple illustration of the theory, a generalized form of Bell's empirical rule for uniaxial loading is derived, and some peculiarities in the response under all-around compressive loading are discussed. General formulae for universal relations possible in an isotropic elastic, Bell constrained material are presented. A simple method for the determination of the left stretch tensor for essentially plane problems is illustrated in the solution of the problem of pure shear of a materially uniform rectangular block. A general formula which includes the empirical rule found in pure shear experiments by Bell is derived as a special case. The whole apparatus is then applied in the solution of the general problem of a homogeneous simple shear superimposed on a uniform triaxial stretch; and the great variety of results possible in an isotropic, elastic Bell material is illustrated. The problem of the finite torsion and extension of a thin-walled cylindrical tube is investigated. The results are shown to be consistent with Bell's data for which the rigid body rotation is found to be quite small compared with the gross deformation of the tube. Several universal formulas relating various kinds of stress components to the deformation independently of the material response functions are derived, including a universal rule relating the axial force to the torque.Constitutive equations for hyperelastic Bell materials are derived. The empirical work function studied by Bell is introduced; and a new constitutive equation is derived, which we name Bell's law. On the basis of this law, we then derive exactly Bell's parabolic laws for uniaxial loading and for pure shear. Also, form Bell's law, a simple constitutive equation relating Bell's deviatoric stress tensor to his finite deviatoric strain tensor is obtained. We thereby derive Bell's invariant parabolic law relating the deviatoric stress intensity to the corresponding strain intensity; and, finally, Bell's fundamental law for the work function expressed in these terms is recovered. This rule is the foundation for all of Bell's own theoretical study of the isotropic materials cataloged in his finite strain experiments on metals, all consistent with the internal material constraint studied here.     

Effect of rotary inertia and shear on vibration and buckling of a double beam system under compressive axial loading

Free transverse vibration and buckling of a double-beam continuously joined by a Winkler elastic layer under compressive axial loading with the influence of rotary inertia and shear are considered in this paper. The motion of the system is described by a homogeneous set of two partial differential equations, which is solved by using the classical Bernoulli?CFourier method. The boundary value and initial value problems are solved. The natural frequencies and associated amplitude ratios of an elastically connected double-beam complex system and the analytical solution of the critical buckling load are determined. The presented theoretical analysis is illustrated by a numerical example, in which the effect of physical parameters characterizing the vibrating system on the natural frequency, the associated amplitude ratios and the critical buckling load are discussed.     

Dynamic analysis of 3-D beam elements including warping and shear deformation effects

In this paper the dynamic analysis of 3-D beam elements restrained at their edges by the most general linear torsional, transverse or longitudinal boundary conditions and subjected in arbitrarily distributed dynamic twisting, bending, transverse or longitudinal loading is presented. For the solution of the problem at hand, a boundary element method is developed for the construction of the 14 × 14 stiffness matrix and the corresponding nodal load vector of a member of an arbitrarily shaped simply or multiply connected cross section, taking into account both warping and shear deformation effects, which together with the respective mass and damping matrices lead to the formulation of the equation of motion. To account for shear deformations, the concept of shear deformation coefficients is used, defining these factors using a strain energy approach. Eight boundary value problems with respect to the variable along the bar angle of twist, to the primary warping function, to a fictitious function, to the beam transverse and longitudinal displacements and to two stress functions are formulated and solved employing a pure BEM approach that is only boundary discretization is used. Both free and forced transverse, longitudinal or torsional vibrations are considered, taking also into account effects of transverse, longitudinal, rotatory, torsional and warping inertia and damping resistance. Numerical examples are presented to illustrate the method and demonstrate its efficiency and accuracy. The influence of the warping effect especially in members of open form cross section is analyzed through examples demonstrating the importance of the inclusion of the warping degrees of freedom in the dynamic analysis of a space frame. Moreover, the discrepancy in the dynamic analysis of a member of a spatial structure arising from the ignorance of the shear deformation effect necessitates the inclusion of this additional effect, especially in thick walled cross section members.     

Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi elliptical functions

The paper provides an exact analytical solution for the equilibrium configurations of a cantilever rod subject to inclined force and tip moment acting on its free end. The solution is given in terms of Jacobi’s elliptical functions and illustrated by several numerical examples and several graphical presentations of shapes of deformed cantilevers. Possible forms of the underlying elastica of a cantilever are discussed in detail, and various simple formulas are given for calculating the characteristic dimensions of the elastica. For the case when a cantilever is subject only to applied force, three load conditions are discussed: the follower load problem, the load determination problem, and the conservative load problem. For all cases, either a formula or an effective procedure for determining the solution is provided. In particular, a new efficient procedure is given to determine all possible equilibrium shapes in the case of the conservative load problem.     

Numerical analysis of the compressive and shear failure behavior of rock containing multi-intermittent joints

The failure behavior of intermittent jointed rocks is dependent on joint configurations. Joint inclination angle and continuity factor determined the joint arrangement in a rectangular numerical sample that was established by using the particle flow code approach. To identify the differences in the failure processes of identical intermittent jointed samples, uniaxial compressive and shear loads were applied on each sample. The crack growth path presented the four typical crack coalescence patterns identified via compressive and shear numerical tests. The crack coalescence pattern was associated with joint slant angle and continuity factor. The observed crack coalescence patterns of every sample with the same inclination angle and continuity factor were partially identical under compressive and shear loading. The differences in the crack patterns of the compressive and shear failure processes were described and compared. Typical compressive and shear failure processes were illustrated. Four compressive and three shear failure modes were identified. The cracking location and number of cracks in each failure mode were different. Additionally, the contact force evolution among particles during shear and compressive loading was different and likely accounted for the differences in cracking patterns. Under compressive or shear loading, the contact force concentration in each sample underwent the following stages: uniform distribution before loading, concentrated distribution, and scattered distribution after failure.