Analytical solutions for coupled tension-bending of nanobeam-columns considering nonlocal size effects

Considering nanoscale size effects and based on the nonlocal elastic stress theory, this paper presents an analytical analysis with closed-form solutions for coupled tension and bending of nanobeam-columns subject to transverse loads and an axial force. Unlike previous studies where many nanomechanical models do not yield analytical solutions and hence the size effects are studied by molecular dynamics simulation or other numerical means, the nonlocal nanoscale effects at molecular level unavailable in classical mechanics are investigated analytically here and first-known closed-form analytical solutions are presented. New higher-order differential governing equations in both transverse and axial directions and the corresponding higher-order nonlocal boundary conditions for bending and tension of nanobeam-columns are derived based on the variational principle approach. Closed-form analytical solutions for deformation and tension are presented and their physical significance examined. Examples conclude that the nonlocal stress tends to significantly increase the stiffness of a nanobeam-column, which are contradictory to the current belief using the nonlocal analysis that structural stiffness should be reduced but consistent with many non-nonlocal analyses including the strain gradient and couple stress theories that stiffness should be enhanced. The relationship between bending deflection, axial tension and nanoscale is presented, and also its significance on stiffness enhancement with respect to the design of nanoelectromechanical systems.     

Mixed finite element method for axisymmetric shells

A mixed variational formulation is used as basis for developing a mixed finite element method for axisymmetric shell. The independent unknowns of the method are the axial and radial displacement components, the rotation of the normal to the middle surface and the meridional bending stress couple. The basic element is a frustrum of curved meridian. General advantages of the mixed method are presented, one of which is the possibility of using piece-wise linear functions of the meridional arclength to represent the basic unknowns. Test results are presented for plate bending, transverse shear deformation, membrane behaviour, edge-zone bending, bending near the junction of two shells, convergence of the method and accuracy of middle surface curvature interpolation. Shell geometries considered are cylindrical, conical, spherical and ellipsoidal. Good results are obtained which should increase interest in the relatively less known and less tested mixed method as compared to the stiffness method.     

Local postbuckling of hat-stringer-stiffened composite laminated plates under transverse compression

In this paper, a closed-form method for the analysis of the local postbuckling behaviour of aircraft panels that are braced by hat-stringers is presented. The stiffened panels are loaded by transverse compression which is a load case that has been treated only scarcely in the open literature, and the corresponding buckling and postbuckling behaviour that eventually leads to failure of the panel is quite different to what is observed when a panel under longitudinal compression is considered. This contribution clarifies that the ultimate load bearing capacity of a stiffened panel with closed-profile stringers under transverse compression is governed by several consecutive stability cases. Firstly a closed-form approximate analysis method for the linear buckling analysis of the skin between two stringers taking the torsional stiffness of the hat-stringers into account is derived (stability case 1). Secondly, a simple Marguerre-type postbuckling analysis method is presented that accounts for the geometrically nonlinear behaviour of the panel skin after buckling (stability case 2) and enables a closed-form analysis of its effective width. Thirdly, the linear buckling analysis is adapted to the analysis of the panel skin under a stringer (stability case 3). It will be shown that stability cases 2 and 3 cannot be treated independently, but have to be considered interactively which necessitates an iterative procedure. The accuracy of the proposed analysis method, relying on the simplifying assumption of a perfectly flat plate rather than considering a cylindrically curved panel is established by comparison with results of accompanying geometrically nonlinear finite element calculations.     

Bending of simply-supported,antisymmetrically laminated rectangular plate under transverse loading

The deformation of antisymmetric angle-ply laminated plate under transverse loading is analysed using the bending theory of laminated plate presented by Ren.^1,2 By expanding the load in a double Fourier series, a closed-form solution is obtained for special types of simply-supported boundary condition. The results are compared with similar results from the classical laminated plate theory and Mindlin theory. This shows that the theory presented by Ren is suitable not only for cylindrical bending but also for general bending problems.     

Finite-element modelling and buckling analysis of anisogrid composite lattice cylindrical shells

The paper investigates the buckling behaviour of anisogrid composite lattice cylindrical shells under axial compression, transverse bending, pure bending, and torsion. The lattice shells are modelled as three-dimensional frame structures composed of curvilinear ribs subjected to the tension/compression, bending in two planes and torsion. The specialised finite-element model generation procedure (model generator/design modeller) is developed to control the orientation of the beam elements allowing the original twisted geometry of the curvilinear ribs to be closely approximated. The effects of varying the length of the shells, the number of helical ribs and the angles of their orientation on the buckling behaviour of lattice structures are examined using parametric analyses. Buckling of the lattice shells with cutouts is also analysed. The results of these studies indicate that the modelling approach presented in this work can be successfully applied to the solution of design problems.     

Postbuckling and free vibrations of composite beams

An exact solution for the postbuckling configurations of composite beams is presented. The equations governing the axial and transverse deformations of a composite laminated beam accounting for the midplane stretching are derived. The inplane inertia and damping are neglected, and hence the two equations are reduced to a single nonlinear fourth-order partial–integral–differential equation governing the transverse deformations. We find out that the governing equation for the postbuckling of symmetric or asymmetric composite beams has the same form as that of beams made of an isotropic material. Composite beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are considered. A closed-form solution for the postbuckling deformation is obtained as a function of the applied axial load, which is beyond the critical buckling load. To study the vibrations that take place in the vicinity of a buckled equilibrium position, we exactly solved the linear vibration problem around the first buckled configuration. Solving the resulting eigen-value problem results in the natural frequencies and their associated mode shapes. Both the static response represented by the postbuckling analysis and the dynamic response represented by the free vibration analysis in the postbuckling domain strongly depend on the lay-up of the laminate. Variations of the beam’s midspan rise and the fundamental natural frequency of the postbuckling domain vibrations with the applied axial load are presented for a variety of lay-up laminates. The ratio of the axial stiffness to the bending stiffness was found to be a crucial parameter in the analysis. This control parameter, through the selection of the appropriate lay-up, can be manipulated to help design and optimize the static and dynamic behavior of composite beams.     

Thermal effects on the bending response of fiber-reinforced viscoelastic composite plates using a sinusoidal shear deformation theory

Summary. The static thermo-viscoelastic responses of fiber-reinforced composite plates are investigated by the use of a refined shear deformation theory. In this theory, trigonometric terms are used for the displacements in addition to the initial terms of a power series through the thickness. The form of the assumed displacements of this theory is simplified by enforcing traction-free boundary conditions at the plate faces. No transverse shear correction factors are needed because a correct representation of the transverse shear strain is given. Using the method of effective moduli solves the equations governing the bending of simply supported fiber-reinforced viscoelastic composite plates. An exact closed-form solution is presented for plates subjected to nonuniform distributions of temperature. The validity of the present theory is demonstrated by comparison with solutions available in the literature. A wide variety of results are presented for the bending response of viscoelastic rectangular plates under thermal loads. The influences of plate aspect ratio, side-to-thickness ratio, thermal expansion coefficients ratio and constitutive and volume fraction parameters on the thermally induced response are studied.     

Bending analysis of antisymmetric angle-ply laminated plates including transverse shear effects

In this paper an approximate analysis is presented for the bending of rectangular, simply supported, antisymmetric angle-ply laminated plates subjected to sinusoidal transverse loads. A published refined shear deformation theory is used to obtain a closed-form solution. Numerical results are compared with those given by three-dimensional elasticity solutions, first-order shear deformation theory solutions and classical lamination theory solutions. The present theory gives a better estimate of the stresses and deflections.     

A bond-based peridynamic model considering effects of particle rotation and shear influence coefficient

The classical bond-based peridynamic (BPD) model is limited with a fixed Poisson's ratio. To overcome this limitation, an improved BPD model is proposed to analyze the deformation and crack propagation of microelastic brittle materials with emphasis on varying Poisson's ratios. In the proposed model, the bond is subjected to axial and transverse pairwise forces, and the particle rotation angle is added to eliminate the additional bending moment caused by transverse forces, which is a key factor to satisfy the balance of angular momentum exactly. As a result, the bond not only has axial displacement but also has transverse displacement and particle rotation. This extension in the displacement mode overcomes the limitation of the fixed Poisson's ratio in the classical BPD model. The simulation results demonstrate the precision of the improved BPD model and prove its ability to predict deformations and crack propagations.     

A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates

A higher-order theory which satisfies zero transverse shear stress conditions on the bounding planes of a generally laminated fibre-reinforced composite plate subjected to transverse loads is developed. The displacement model accounts for non-linear distribution of inplane displacement components through the plate thickness and the theory requires no shear correction coefficients. A C∘ continuous displacement finite element formulation is presented and the coupled membrane-flexure behaviour of laminated plates is investigated. The nodal unknowns are the three displacements, two rotations and two higher-order functions as the generalized degrees of freedom. The simple isoparametric formulation developed here is capable of evaluating transverse shears and transverse normal stress accurately by using the equilibrium equations. The accuracy of the nine-noded Lagrangian quadrilateral element is then established by comparing the present results with the closed-form, three-dimensional elasticity and other finite element available solutions.     

Combined effects of relative density and material distribution on the mechanical properties of metallic honeycombs

This paper presents the combined effects of relative density and material distribution on the elastic constants and the yield strengths of metallic honeycombs. Periodic regular hexagonal cell is employed as the structural model. Cell wall bending, transverse shear and axial stretching/compression are taken as the deformation mechanisms in the analysis. Closed-form solutions for the yield strengths and all the five independent elastic constants are obtained for honeycombs with cell walls of uniform thickness. For honeycombs with cell walls of non-uniform thickness, the closed-form solutions would be too lengthy to use in practical applications. We instead provide numerical results to show the combined effects of relative density and material distribution on the initial and full yield strengths and all the five independent elastic constants of metallic honeycombs. The results can serve as a guide for the optimal design of metallic honeycombs.     

轴向往复运动抽油杆柱在弯曲井眼内横向振动的仿真模型

抽油杆柱横向振动的研究是分析油井杆管偏磨现象的基础。该文将定向井抽油杆柱在弯曲井眼内的横向振动问题简化为具有初弯曲的纵横弯曲梁在油管约束下的横向振动问题。除考虑交变轴向载荷对抽油杆柱横向振动的激励外,首次提出弯曲井眼也是轴向往复运动抽油杆柱横向振动的主要激励。综合考虑弯曲井眼对轴向往复运动抽油杆柱横向振动的激励,以及交变轴向载荷对抽油杆柱横向振动的激励,应用弹性碰撞理论描述油管对抽油杆柱横向振动的约束,基于弹性体振动理论建立了具有初弯曲的抽油杆柱在弯曲井眼内的横向振动仿真模型。采用有限差分法与Newmark法实现了对定向井抽油杆柱横向振动的仿真计算。仿真结果表明:仿真结果与现场实际偏磨情况相符,说明在分析抽油杆柱横向振动时,轴向运动导致的弯曲井眼激励是一项不可忽略的因素;算例油井中,杆管偏磨的危险点主要出现在油井的造斜段与抽油杆柱的受压段。该文所建立定向井抽油杆柱横向振动仿真模型可用于指导抽油杆扶正器的优化配置。     

Limit loads and approximate J estimates for axial through-wall cracked pipe bends

This paper presents plastic limit loads and approximate J estimates for axial through-wall cracked pipe bends under internal pressure and in-plane bending. These loads and estimates are based on small strain finite element limit analyses using elastic-perfectly plastic materials. Geometric variables associated with the crack and pipe bend are systematically varied, and three possible crack locations (intrados, crown and extrados) are considered. Effects of the bend and crack geometries on plastic limit loads are quantified, and closed-form limit load solutions are given. Based on the proposed limit load solutions, a reference stress based the J estimation scheme for axial through-wall cracked pipe bends under internal pressure and in-plane bending is proposed.     

变截面压电层合梁自由振动分析

考虑压电材料的质量效应和刚度效应,将表面粘贴或埋入式压电悬臂梁看作变截面梁,研究压电材料对智能结构固有特性的影响。基于一阶剪切变形理论导出压电层合梁的抗弯刚度和横向剪切刚度,计及梁的剪切变形和转动惯量,采用Timoshenko理论推导变截面压电层合梁的频率方程。给出了T300/970压电层合梁和硬铝压电层合梁的前3阶固有频率,并和有限元结果、等截面梁的计算结果进行比较。计算表明,压电材料对压电结构固有频率和固有振型的影响显著,在以振动控制为目标的压电结构动力学建模过程中,有必要考虑压电材料的质量和刚度。     

Low-frequency eddy-current loss estimation in long conductors byusing the moment of inertia of cross sections

A simple method is presented to estimate low-frequency eddy-current losses in long nonmagnetic conductors of uniform cross section when subjected to axial or transverse magnetic field. The method involves calculation of the moment of inertia of the conductor cross section. In the case of transverse fields, the relationship obtained agrees with analytical results for cross sections which have a symmetry axis parallel to the incident field. For axial fields the proposed relationship agrees with the analytical expressions for circular and elliptical cross sections and gives results with reasonable accuracy for rectangular cross sections     

Load-carrying capacity of GFRP bars under combined axial force–transverse force loading

In several applications, glass fiber reinforced plastic (GFRP) bars are loaded with a transverse force while they simultaneously carry an axial force. Short-term tests of combined axial and transverse loading on assemblies consisting of one GFRP bar and two steel sleeves are described in this paper. The results show that the axial force capacity of the GFRP bar is affected only if the transverse force exceeds a certain threshold. A qualitative fracture criterion for combined axial force–transverse force loading is suggested. Further combined axial force–transverse force loading tests have been conducted on a specimen with a severe, forced 4° bending misalignment. These misaligned GFRP bars sustained more than 80% of the tensile resistance of the corresponding straight bars (without misalignment).     

A simplified approach for incorporating thickness stress in the analysis of sheet metal forming using shell elements

A simplified scheme for considering the thickness stress of shell elements induced by contact is presented which improves the accuracy of sheet metal forming analysis. The yield function formulated on the basis of plane stress conditions is modified to incorporate the effect of transverse normal stress induced by contact forces acting on shell elements and return mapping routine is used to update in‐plane stresses at each time step. The transverse normal stress distributions in the thickness direction are determined using the analytic solution of the cylindrical tube under the internal pressure. As numerical examples, uni‐axial compression, bi‐axial tension and bending tests are treated. The problem of cylindrical cup drawing is also calculated. Each result is compared with the results obtained by the analysis using ABAQUS continuum elements. Copyright © 2002 John Wiley & Sons, Ltd.     

Nonlinear analysis of laminated composite plates and shells including the effects of shear and normal deformation

A nonlinear higher-order theory for laminated composite plates and shells with an arbitrary number and sequence of layers is presented. The theory takes into account both transverse shear and normal deformation and considers the elasto-plastic behaviour of the composite materials. The results presented illustrate first the importance of modelling the nonlinear behaviour of the material especially at high levels of loading, and secondly the importance of modelling both transverse shear and normal compression.     

Dynamic instability of spinning launch vehicles modeled as thin-walled composite beams

This paper presents the results of the dynamic stability analysis of a flexible spinning launch vehicle subjected to thrust modeled as a thin-walled composite beam with circular cross section. Due to the presence of gyroscopic forces, we mainly aimed to find divergence and/or flutter instabilities and establish the stability boundaries of the spinning beam. For this purpose, we implemented a circumferentially uniform stiffness (CUS) layup configuration to exhibit the coupled motion of bending–bending–shear. The solution of the eigenvalue problem is handled by the extended Galerkin method (EGM), and we computed the results addressing the effects of various parameters such as spin speed, axial load, ply angle, aspect ratio and transverse shear on the dynamic stability of the beam. Insights revealed by this study contribute to the design of advanced aerospace structures modeled as thin-walled composite beams reflecting the potential influence of transverse shear and aspect ratio on dynamic stability characteristics. A notable contribution is that we show that divergence/flutter instabilities can be delayed or even avoided using the directionality property of composite materials.     

A non-equilibrium approach to processing Hopkinson Bar bending test data: Application to quasi-brittle materials

The aim of this paper was to describe a method of analysing the test data recorded during a Hopkinson Bar bending test. This three-point dynamic bending test was designed for testing the strength of materials under dynamic loads. It is carried out on a specimen consisting of a beam placed on two supports, which is subjected to an impact. The use of Hopkinson Bar as supports makes it possible to determine the forces and displacements at these points. An analytical solution for the transient response of a long beam subjected to a transverse impact was used to determine the impact force and the displacement. This procedure applies for the first few instants when the motions generated by the impact have not yet reached the supports, and the mechanical state of the specimen is identical to that of an unsupported beam. It is suitable for use with quasi-brittle materials for which failure occurs at the very beginning of the test. The material strength is determined at the time of failure, which is characterised by a sudden decrease in the bending moment. The results of a test in which a quasi-brittle material was loaded up to failure are presented and analysed as outlined above. The results obtained confirm the relevance of the present method.