On the load distribution of thin-walled beams subjected to bending with respect to the cross-section distortion

The paper deals with the estimation of the load distribution under which distortion of the cross-sections of thin-walled beams subjected to bending cannot occur. It is assumed that beam walls are hinged along their longitudinal edges. Beams with closed and open rectangular cross-sections, with three or two cells, with two or one axis of symmetry are considered. It is shown that the problem can be treated by two equivalent beams, defined by the shear flow zero points of the beam with rigid cross-section. The beam load must be distributed in the plane of beam walls, proportionally to the cross-section moments of inertia of the equivalent beams. Some illustrative examples are given.     

Nonuniform elastic torsion and flexure of members with asymmetric cross-section

In a previous paper, the authors considered the linear response of a solid prismatic elastic member to a torque which varied along the length of the member. The cross-section considered was only restricted by the requirement of two axes of symmetry. That restriction is removed in the present work, but its removal implies that the response to torque will generally now involve bending, so the treatment has been extended to allow arbitrary loading in torsion/flexure.It transpired that a satisfactory analysis is incompatible with the concept of a “shear centre”, and the existence of a shear centre, as part of a rigorous theory or as an approximation, is clarified in the paper.     

Elastic lateral-torsional buckling of circular arches subjected to a central concentrated load

An arch under an in-plane central concentrated radial load is subjected to combined axial compressive and bending actions. When these combined axial compressive and bending actions reach a certain value, the arch may suddenly deflect laterally and twist out of its plane of loading and fail in a lateral-torsional buckling mode. This paper derives analytical solutions for the elastic lateral-torsional buckling load of pin-ended circular arches that are subjected to a central concentrated load, using the principle of stationary potential energy in conjunction with the Rayleigh-Ritz method. Analytical solutions of the buckling load for in-plane fixed and out-of-plane pin-ended arches and for the case of the load acting above or below the shear centre are also derived. The analytical solutions are compared with results of a commercial finite element package ANSYS and a finite element code developed by authors elsewhere for arches with different slendernesses, included angles, and cross-sections. The agreement between the analytical solutions and the finite element results is very good.     

Torsional bridge setup for the characterization of integrated circuits and microsensors under mechanical shear stress

We present a torsional bridge setup for the electro-mechanical characterization of devices integrated in the surface of silicon beams under mechanical in-plane shear stress. It is based on the application of a torsional moment to the longitudinal axis of the silicon beams, which results in a homogeneous in-plane shear stress in the beam surface. The safely applicable shear stresses span the range of ±50 MPa. Thanks to a specially designed clamping mechanism, the unintended normal stress typically stays below 2.5% of the applied shear stress. An analytical model is presented to compute the induced shear stress. Numerical computations verify the analytical results and show that the homogeneity of the shear stress is very high on the beam surface in the region of interest. Measurements with piezoresistive microsensors fabricated using a complementary metal-oxide-semiconductor process show an excellent agreement with both the computational results and comparative measurements performed on a four-point bending bridge. The electrical connection to the silicon beam is performed with standard bond wires. This ensures that minimal forces are applied to the beam by the electrical interconnection to the external instrumentation and that devices with arbitrary bond pad layout can be inserted into the setup.     

Thermoelastic lateral-torsional buckling of fixed slender beams under linear temperature gradient

The thermal expansions and rotations that result from a linear in-plane temperature gradient field are fully restrained at the ends of a fixed beam. These restrained expansions and rotations will produce internal bending and compressive actions in the beam, and these actions increase with an increase of the temperature differential and average temperature of the linear temperature gradient field. When these actions reach critical values, the fixed beam may bifurcate from its primary equilibrium state to a buckled equilibrium configuration. This paper presents a systematic treatment of classical buckling analysis for thermoelastic lateral-torsional buckling and for in-plane thermoelastic flexural buckling of a fixed beam of doubly symmetric open thin-walled cross-section that is subjected to a linear temperature gradient field over its cross-section. It is shown that the effective centroid and shear centre, rather than the geometric centroid and shear centre, should be used in formulating the thermoelastic prebuckling and buckling analysis and that the effects of temperature on the buckling resistance need to be considered. The thermoelastic lateral-torsional buckling of a fixed beam under a linear temperature gradient field is more complicated than its mechanical counterpart for uniform bending or for uniform compression, and iterative methods are needed to obtain accurate solutions.     

Extension of the Hill (1948) yield criterion to the case of prismatic monoclinic symmetry

The theory of the invariant representation for tensor functions in first illustrated by providing a general form of the Hill (1948) orthotropic yield criterion. It is then applied to derive a quadratic yield equation for the case of prismatic monoclinic symmetry, which is induced by simple shear deformation. This new criterion can in turn be approximated by an orthotropic one by choosing the ‘best’ symmetry axes. The above equations are then used to derive the angular dependences of the uniaxial yield stress and strain rate ratio in the plane of a prismatic monoclinic sheet. Finally, it is shown on an example that they are able to predict the axial stresses occurring during torsion testing with a fairly good accuracy.     

Nonlinear forced vibrations and static deformations of 3D beams with rectangular cross section: The influence of warping, shear deformation and longitudinal displacements

The geometrically nonlinear vibrations of beams with rectangular cross section, which may experience longitudinal, torsional and bending deformations, in any plane, are investigated by the p-version finite element method. Two main models are used: one of them is based on Bernoulli-Euler theory for beam bending and the other is based on Timoshenko theory; both models assume that the cross section rotates as a rigid body and is free to warp in longitudinal direction, as in Saint-Venants’s theory for torsion. The geometrical nonlinearity is taken into account by considering Green’s strain tensor and the importance of the longitudinal displacements of quadratic order, which are most often neglected in the strain-displacement relation, is here examined. Generalized Hooke’s law is used and the equation of motion is derived by the principle of virtual work. The importance of the warping function is analysed for different rectangular cross sections, and it is shown that its consideration can be fundamental to obtain correct results. Then, it is shown that the linearization of the trigonometric functions related with the twist angle, which is usually applied in the displacement field in models based on Saint-Venant’s hypothesis, should be done in the strain-displacement relations. Comparisons of the models for 3D beams based on Bernoulli-Euler’s and Timoshenko’s theories is presented. It is shown that if the rotations along the transverse axes are directly approximated by the respective derivatives of the transverse displacement functions, as is assumed in the model presented here that is based on Bernoulli-Euler’s theory, the additional shear stresses that appear when the bending and torsion motions are coupled, lead to wrong results. Finally, taking into account accuracy and simplicity, a model is chosen and employed to investigate the nonlinear forced vibrations of beams using direct integration of the equations of motion in the time domain. Examples of bending-torsion couplings due to the nonlinear terms are presented in dynamical problems.     

Elastoplastic response and recoil of lattice structures under hyperbolic hardening

Elastoplastic response and recoil analyses for hexagonal honeycomb lattice structures are presented when hardening is described by a hyperbolic law. By exploiting the translational symmetry of the problem, the analysis is reduced to that of a thin beam under combined bending and axial loading coupled with the kinematics of lattice deformation and its relationship with cell wall deformation. A closed-form solution for the load-curvature relationship of a beam with rectangular cross-section is obtained. A systematic study of beam response, as the stress-strain curve of the constituent material approaches an ideal elastic-perfectly plastic law, is presented. The analysis is then applied to an infinite honeycomb sheet under remote tensile load to obtain the apparent non-linear structural response. Apparent recoil of such a lattice material upon unloading is also calculated in closed form, when unloading is assumed to take place along a linear stress-strain curve. The analytical results are in excellent agreement with the numerical calculations.

     

Bending and torsion of anisotropic cantilevers

A mathematical formulation is developed for bending and torsion of an inhomogeneous cantilever constructed of anisotropic composite materials. It is shown that anisotropic beams do not have a constant rate of twist. Hence the shear centre requires redefinition, and design for specified rate of twist is possible. A numerical solution is obtained using the finite element method, and the lateral or pinching stresses are shown to be two order of magnitude smaller than the other stresses. Since most practical composite material beams are constructed from nearly symmetric laminates, this confirms that traditional orthotropic stressing yields quite reasonable results.     

Bending of long, rectangular-section bonded rubber blocks

Readily-calculable explicit closed-form representations are determined for the bending stiffness of, and stresses within, generally loaded bonded rubber blocks of long, rectangular cross-section. They satisfy exactly the stated governing equations and conditions based upon the classical theory of elasticity. From these, more detailed investigations are given for the especially interesting loading cases corresponding to simple bending, cantilever loading and apparent shear. It is observed that the deformed profiles of the side surfaces are not in general parabolic as has been usually assumed previously. In simple bending an improved approximate elementary expression is deduced for the couple needed to create a specified rotation of the loaded end of the block. Similarly, in apparent shear a more realistic simple estimate is derived from the exact expression for the ratio of the true to the apparent shear modulus.     

A simple model for stability of a long axial surface crack in a thin walled cylinder

Growth in the thickness direction of a long axial surface crack at the inner surface of a thin walled cylinder has been analyzed for loads generated by internal pressure and a thermal gradient through the wall thickness. Plane-strain deformations have been considered. It has been assumed that the cracked cross-section is fully plastic, but that the plastic zone width in the circumferential direction is very small. The cracked cross-section transmits a normal force and a bending moment, which have been considered as external forces on an equivalent cut ring element, to compute the deformation of the cracked cross-aection. An analytical expression has been derived for the crack-opening-displacement, as a function of the loads and the crack depth. Stable and unstable crack growth have been investigated on the basis of a critical crack-opening-displacement and a smoothly rising crack-opening R-curve. The condition for unstable crack growth depends primarily on the magnitude of the internal pressure. A thermal gradient by itself is less likely to cause unstable crack propagation.     

Analytical studies for buckling and vibration of weld-bonded beam shells of rectangular cross-section

An approach to analytical solution is presented for vibration and buckling of thin-walled tubular beam shells typical of automotive structures, which are fabricated by joining sheet metal stampings along the two longitudinal edges with periodic spot welds, adhesive bonding, or combination of spot welds and bonding, known as weld bonding. Solutions are obtained for such beam shells of rectangular cross-section with two opposite ends simply supported. The beam shell is modeled as an assembly of the constituent walls and Levy-type formulation is used to obtain a series solution for the transverse displacement of each of the walls. The challenge of expressing the discrete point support conditions at the spot welds by a continuous function is addressed using the flexibility function approach used in literature. The flexibility function, used earlier to represent the flexibility distribution along weld-bonded edges of rectangular plates with periodic spot welds, is used here. The characteristic equations are obtained by satisfying the displacement, slope, shear, and moment equilibrium at the mating edges of the walls including the two weld-bonded edges and the compatibility conditions at the spot-weld locations. This approach to analytical solution, described here for thin-walled beam shells of rectangular cross-section, can be suitably adopted for more general cross-sections and joints along non-symmetric edges. A parametric study is undertaken to show the effect of aspect ratio of the beam shell, adhesive joint parameters, and the number of spot welds on the elastic buckling loads and the natural frequencies. Such parametric studies can be of use to designers in arriving at an optimal joint configuration of weld-bonded beam shells from buckling and vibration considerations.     

旋转变截面扭梁的变形应力分析

本文研究了在剪切变形和转动的影响下,在轴向变形、弯曲变形和剪切变形的作用下的旋转变截面扭梁的单元刚度矩阵。假设扭转角、宽度和厚度沿着梁的长度方向都是线性变化的。给出了转速和轮毂半径对于梁的变形和应力的影响：转速增大时能增加轴线方向的变形和应力,但是却减小弯曲变形、剪切变形以及由它们产生的应力；而轮毂半径主要是影响梁的轴线方向的变形和应力,对弯曲变形和剪切变形的影响比较小。     

Stochastic bending–torsion coupled response of axially loaded slender composite-thin-walled beams with closed cross-sections

The stochastic bending–torsion coupled response of axially loaded slender composite beams with solid or thin-walled closed cross-sections are investigated by using normal mode method in conjunction with receptance method. The classical composite beam theory with shear deformation and rotary inertia ignored is employed and the effects of bending–torsion coupling and axial force are included in the present formulations. The theoretical expressions for the displacement response of axially loaded slender composite beams subjected to concentrated or distributed stochastic excitations with stationary and ergodic properties are derived. The proposed method is illustrated by its application to two particular examples to study the effects of bending–torsion coupling and axial force on the stochastic response of the composite beams.     

State space elastostatics of prismatic structures

This paper provides an exposition of the problem of a prismatic elastic rod or beam subject to static end loading only, using a state space formulation of the linear theory of elasticity. The approach, which employs the machinery of eigenanalysis, provides a logical and complete resolution of the transmission (Saint-Venant's) problem for arbitrary cross-section, subject to determination of the Saint-Venant torsion and flexure functions which are cross-section specific. For the decay problem (Saint-Venant's principle), the approach is applied to the plane stress elastic strip, but in the transverse rather than the axial direction, leading to the well-known Papkovitch–Fadle eigenequations, which determine the decay rates of self-equilibrated loading; however, extension to other cross-sections appears unlikely. It is shown that only a repeating zero eigenvalue can lead to a non-trivial Jordan block; thus degenerate decay modes cannot exist for a prismatic structure.     

An improved closed-form solution to interfacial stresses in plated beams using a two-stage approach

Shear stress and normal stress in the thickness direction at interfaces (referred to as interfacial shear and transverse normal stresses, respectively, hereafter) have played a significant role in understanding the premature debonding failure of beams strengthened by bonding steel/composite plates at their tension surfaces. Due to the occurrence of dissimilar materials and the abrupt change of cross-section, the stress distribution at plate ends becomes singular and is hence considerably complicated. Extensive experimental and analytical analyses have been undertaken to investigate this problem. Large discrepancies have been found from various studies, particularly from experimental results due to the well-acknowledged difficulty in measuring interfacial stresses. Numerical analyses, e.g. 2-D or 3-D finite-element analysis (FEA), may predict accurate results, but they demand laborious work on meshing and sensitivity analysis. Analytical solutions, in particular those in a closed form, are more desirable by engineering practitioners, as they can be readily incorporated into design equations. This paper reports an improved closed-form solution to interfacial stresses in plated beams using a two-stage approach. In this solution, beams and bonded plates can be further divided into a number of sub-layers to facilitate the inclusion of steel bars or multiple laminae. Thermal effects may also be considered by using equivalent mechanical loads, i.e. equivalent axial loads and end moments. Numerical examples are presented to show interfacial stresses in concrete or cast iron beams bonded with steel or FRP plates under mechanical and/or thermal loads. The effect of including steel reinforcements with various ratios in the RC beam on interfacial stresses is also investigated. Compared with previously published analytical results, this one improves the accuracy of predicting the transverse normal stresses in both adhesive-beam and plate-adhesive interfaces and the solution is in a closed form.     

Analytical Higher-Order Model for Flexible and Stretchable Sensors

The stretchable sensor wrapped around a foldable airfoil or embedded inside of it has great potential for use in the monitoring of the structural status of the foldable airfoil.The design methodology is important to the development of the stretchable sensor for status monitoring on the foldable airfoil.According to the requirement of mechanical flexibility of the sensor,the combined use of a layered flexible structural formation and a strain isolation layer is implemented.An analytical higher-order model is proposed to predict the stresses of the strain-isolation layer based on the shear-lag model for the safe design of the flexible and stretchable sensors.The normal stress and shear stress equations in the constructed structure of the sensors are obtained by the proposed model.The stress distribution in the structure is investigated when bending load is applied to the structures.The numerical results show that the proposed model can predict the variation of normal stress and shear stress along the thickness of the strain-isolation(polydimethylsiloxane)layer accurately.The results by the proposed model are in good agreement with the finite element method,in which the normal stress is variable while the shear stress is invariable along the thickness direction of strain-isolation layer.The high-order model is proposed to predict the stresses of the layered structure of the flexible and stretchable sensor for monitoring the status of the foldable airfoil.     

Computed determination of the coefficient of bending stresses on the teeth of gear wheels under skewness

An analytical method is given to determine the coefficient of bending stresses in the teeth of gear wheels under skewness caused by elastic deformations and inaccuracies of the manufacturing and mounting of the elements of the gearing. The problem of bending a beam with a finite length on an elastic base under the effect of an arbitrary load was solved. Experimental data of the attenuation of deflections along the length of a cantilever plate were shown to be in satisfactory agreement with the computed results for deflections of a beam with a finite length on an elastic base. Formulas to deter-mine the coefficient of the concentration of bending stresses in the teeth of gear wheels were obtained. The dependences obtained allow us to develop a distribution diagram for variations of the coefficient of the concentration of bending stresses along the length of the teeth under skewness.     

Deflections of Timoshenko beam with varying cross-section

Closed form solutions are presented for bending beams with linearly and (in the binomial form) parabolically varying depth and for bending beams with linearly varying width along the beam's length. The solutions are developed taking into account the shear deformation of the beam. The solutions are achieved, in an original way, by transforming the fourth-order differential equations with variable coefficients into fourth-order differential equations with constant coefficients. Though the solutions presented refer to three recurrent variations in the beam cross-section shape, the procedure outlined can be applied to beams with binomial variation (with any exponent) in the depth or width of the cross-section. Moreover, the solutions can be achieved for polynomial, exponential and sinusoidal load conditions. The solutions can be utilized to obtain the stiffness factors and the flexibility coefficients of beams in the analysis of frames. Closed form solutions for longitudinal displacements are also presented. The analytical solutions are applied to four recurrent beams commonly used in civil engineering practice and a comparison with a numerical procedure is made.     

A spatial closed-form nonlinear stiffness model for sheet flexures based on a mixed variational principle including third-order effects

The sheet flexure is commonly used to provide support stiffness in flexure mechanisms for precision applications. While the sheet flexure is often analyzed in a simplified form, e.g. by assuming planar deformation or linearized stiffness, the deformation in practice is spatial and sufficiently large that nonlinear effects due to the geometric stiffness are significant.This paper presents a compact analytical model for the nonlinear stiffness characteristics of spatially deforming sheet flexures under general 3-D load conditions at moderate deformations. This model provides closed-form expressions in a mixed stiffness and compliance matrix format that is tailored to flexure mechanism analysis. The effects of bending, shear, elongation, torsion and warping deformation are taken into account, so that the stiffness in all directions, including the in-plane lateral support direction, is modeled accurately. The model is verified numerically against beam and shell-based finite elements. The approach for deriving closed-form solutions in a nonlinear context is detailed in this paper. The Hellinger–Reissner variational principle with a specific physically motivated set of low-order interpolation functions is shown to be well-suited to the geometrically nonlinear analysis of flexures.An extension of the derivation approach to the nonlinear closed-form analysis of general flexure mechanisms consisting of multiple sheet flexures connected in parallel is presented. This is demonstrated with the case of a spatially deforming parallelogram flexure mechanism and a cross-hinge flexure mechanism.