夹层板弯曲问题的Hoff理论的解析解

本文根据Ｈｏｆ提出的夹层板理论和胡海昌、柳春图对该理论的简化理论，提出了矩形夹层板弯曲问题的解析解的一般格式，它可以用于求解各种边界条件下矩形夹层板弯曲问题的Ｈｏｆ理论的解析解。从而，使得考虑表层抗弯刚度的矩形夹层板弯曲问题求解格式化。     

Non-homogeneous response of cross-ply laminated elastic plates using a higher-order theory

A consistent formulation for the bending of cross-ply laminated composite plates that possess non-homogeneous elastic properties is presented. Based on a third-order shear deformation plate theory, the governing equations are obtained using the principle of virtual work. With the help of the small parameter method, a wide variety of results are presented for the symmetric and antisymmetric analysis of non-homogeneous rectangular laminated plates. The influence of non-homogeneity, lamination schemes, aspect ratio and material anisotropy on the deflections and stresses is investigated. The new results for non-homogeneous response of composite plates should serve as bench marks for future comparisons.     

Edge effects of uniformly loaded cross-ply composite laminates

Interlaminar stresses resulting from bending of rectangular cross-ply composite laminates are determined using a layer wise laminate theory. Two types of laminates are considered. First a fully simply supported laminate subjected to bi-directional bending is analyzed. The results obtained from this theory are compared with those of the published three-dimensional elasticity solutions to verify the validity and accuracy of the present theory. Then laminates with two edges simply supported and the other two edges free are examined. The results indicate the presence of significant interlaminar stresses near the free edges.     

Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress

A microstructure-dependent nonlinear theory for axisymmetric bending of circular plates, which accounts for through-thickness power-law variation of a two-constituent material, is developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, temperature-dependent properties, and the von Kármán geometric nonlinearity. Classical and first-order shear deformation theories are considered in the study. The modified couple stress theory contains a material length scale parameter that can capture the size effect in a functionally graded material plate. The theories presented herein can be used to develop analytical solutions of bending, buckling, and free vibration for the linear case and finite-element models for the nonlinear case to determine the effect of the geometric nonlinearity, power-law index, and microstructure-dependent constitutive relations on linear and nonlinear response of axisymmetric analysis of circular plates.     

Nonlinear bending of thick beams laminated from bimodular composite materials

A higher-order shear deformation beam theory is presented to analyze geometrically nonliner bending of thick, rectangular beams made from bimodular materials. This is performed by using a mixed finite element model. Results are reported for maximum deflections of simply-supported and clamped-clamped beams under a uniformly distributed load. The effect of material bimodularity on the nonlinear transverse deflection is investigated.     

Plate spectral elements based upon Reissner–Mindlin theory

Plate bending spectral elements based upon the Reissner–Mindlin theory is developed for a rectangular plate. Numerical comparison studies for static and eigenvalue problems, demonstrate the excellent performance and robustness of the present method. The improvement in accuracy and computational efficiency in comparison with low order plate bending elements is remarkable and suggests that the proposed approach is an alternative to existing schemes.     

The resonance frequency of laminated piezoelectric rectangular plates

Based on the theory of elasticity and piezoelectricity, a dynamic model of laminated elastic-piezoelectric rectangular plates is considered. The bending equations are established and solved, taking the nonlinear behavior of the piezoelectric material into account. The resonance frequency of a laminated piezoelectric rectangular plate with four kinds of different boundary conditions is then investigated. The present results agree very well with the experimental findings and can be extended to practical applications, such as considering the effect of an epoxy package on the resonance of piezoceramic plate.     

A single variable shear deformable nonlocal theory for transversely loaded micro- and nano-scale rectangular beams

In this paper, a simple single variable shear deformable nonlocal theory for bending of micro- and nano-scale rectangular beams is presented. To incorporate small size effects, the theory uses Eringen’s nonlocal differential constitutive relations. The theory has only one fourth-order governing differential equation involving a single unknown variable. The governing equation and the expressions for the bending moment and shear force of the present theory are strikingly similar to those of nonlocal Euler-Bernoulli Beam Theory (EBT) formulated based on Eringen’s nonlocal elasticity theory. The theory assumes that the axial and lateral displacements have bending and shear components such that the bending components do not contribute towards shear force, and the shear components do not contribute towards bending moment. Also, the chosen displacement functions of the theory give rise to a realistic parabolic transverse shear stress distribution across the beam cross-section. Efficacy of the proposed theory is demonstrated through bending of simply supported, cantilever and clamped-clamped micro- and nano-scale beams of rectangular cross-section. The numerical results obtained by using the present theory are compared with those predicted by other nonlocal first-order and higher-order shear deformation beam theories. The results obtained are quite accurate.     

A new shear-flexible FRP-reinforced concrete slab element

In this paper, a simple shear-flexible rectangular layered FRP-reinforced concrete slab element is developed based on Mindlin–Reissner plate theory and Timoshenko’s composite beam functions for nonlinear finite element analysis of FRP-reinforced concrete slabs. The Timoshenko’s composite beam functions are developed for FRP-reinforced concrete slabs based on those for composite laminates. The plane displacement interpolation functions of a quadrilateral isoparametric element with drilling degrees of freedom are employed to describe the membrane effects and the bending effects of the slab element are represented by the rotation functions of the slab element derived from Timoshenko’s composite beam functions. Both geometric nonlinearity and material nonlinearity of the materials are included in the new element. The efficiency of the element for nonlinear finite element analysis of FRP-reinforced concrete slabs is validated by comparing the computed results of two numerical examples with those obtained from lab tests.     

Elasto‐plastic analysis of thin‐walled tube under cyclic bending

Abstract

The elasto‐plastic endochronic theory is extended to describe the material behavior of tubes under cyclic bending. Experimental data of cyclic bending of 6061‐T6 aluminum and 1018 steel was used for comparison. It is shown that by using the complete differential equations of the theory and by assuming a linear relation between the strain and the curvature, the theory is capable of predicting the moment‐curvature response.     

Natural frequencies of cracked functionally graded material plates by the extended finite element method

In this paper, the linear free flexural vibration of cracked functionally graded material plates is studied using the extended finite element method. A 4-noded quadrilateral plate bending element based on field and edge consistency requirement with 20 degrees of freedom per element is used for this study. The natural frequencies and mode shapes of simply supported and clamped square and rectangular plates are computed as a function of gradient index, crack length, crack orientation and crack location. The effect of thickness and influence of multiple cracks is also studied.     

Load‐bearing capacity of slender unreinforced masonry compression members under biaxial bending

Dedicated to Prof. Dr.‐Ing. Carl‐Alexander Graubner on the occasion of his 60th birthday Masonry members have to resist vertical loads and bending moments about the weak axis due to rotation of adjacent slabs. If the compression member is part of the bracing system, there are also bending moments about the strong axis. This paper deals with the load‐bearing capacity of biaxially eccentrically compressed unreinforced compression members with rectangular cross‐sections. For linear‐elastic material, the principles of an analytical model is presented, which considers geometrical and physical (cracking) non‐linearity. The deflections of the wall can be determined by using moment‐curvature relations, making possible the analytical analysis of compression members considering the effects of 2nd order theory. For a non‐linear stress‐strain relation, the calculation of the load carrying capacity of rectangular compression members under biaxial bending is complex and has to be determined numerically. The good accordance of the results of the analytical model with the numeric calculations is also shown for various eccentricities. In addition, a simplified proposal for the calculation of the load‐bearing capacity of biaxially eccentrically compressed unreinforced compression members is shown. The proposal is based on the load‐bearing capacity of uniaxially eccentrically compressed unreinforced compression members. Therefore it is possible to use the proposal considering existing models, for example according to Eurocode 2 or 6.     

Elastic-Plastic Analysis Under Proportional and Non-Proportional Loading Paths Using Deformation Plasticity Theory

Abstract

An estimation of elastic-plastic stresses and strains is presented for mechanical components, using pseudo elastic analysis based on the deformation theory of plasticity. Analysis of two applications, one under proportional loading and other under non-proportional loading paths using pseudo elastic finite element method, is presented. A rectangular plate with a hole under tension loading and a rectangular plate fixed at one end under bending-tension non-proportional loading are considered for analysis. Pseudo Elastic finite element analysis for proportional loading uses elastic solutions and varies material properties for elements in plastic zone to estimate elastic-plastic solution. A finite element code is developed based on pseudo elastic analysis method. An attempt is made to extend pseudo elastic analysis to analyze bending-tension non-proportional loading problem. Both applications in consideration are assumed to be of Von Mises material and follow isotropic hardening rule with elastic-linear hardening material model. Non-linear analysis of the plate with a hole under proportional tensile loading and that of rectangular plate under bending-tension non-proportional loading are performed in ANSYS and results are compared for validation and are observed to be in good agreement with present analysis.     

A nonlinear modified couple stress-based third-order theory of functionally graded plates

In this paper a general nonlinear third-order plate theory that accounts for (a) geometric nonlinearity, (b) microstructure-dependent size effects, and (c) two-constituent material variation through the plate thickness (i.e., functionally graded material plates) is presented using the principle of virtual displacements. A detailed derivation of the equations of motion, using Hamilton’s principle, is presented, and it is based on a modified couple stress theory, power-law variation of the material through the thickness, and the von Kármán nonlinear strains. The modified couple stress theory includes a material length scale parameter that can capture the size effect in a functionally graded material. The governing equations of motion derived herein for a general third-order theory with geometric nonlinearity, microstructure dependent size effect, and material gradation through the thickness are specialized to classical and shear deformation plate theories available in the literature. The theory presented herein also can be used to develop finite element models and determine the effect of the geometric nonlinearity, microstructure-dependent size effects, and material grading through the thickness on bending and postbuckling response of elastic plates.     

基于三维梁理论的复合材料层合管等效抗弯刚度

针对任意铺层形式和任意壁厚复合材料圆形层合管件,提出一种等效抗弯刚度的计算方法。此方法采用符合复合材料圆形管件梁真实变形的变形理论,考虑横向剪切变形,非均匀扭转效应,主、次挠曲效应和层合材料的三维弹性效应,按照壳壁中实际应力状态,建立了复合材料圆形管件等效抗弯刚度的计算模型。通过与4种铺层管件的三点弯曲实验结果以及经典层合板理论计算的等效抗弯刚度进行对比,验证了计算模型的正确性。通过退化与各向同性材料抗弯刚度的计算方法进行对比,分析了计算模型的适用性。      

基于哈密顿解法的厚板边界效应典型算例分析

在平板弯曲问题中存在边界效应现象。这个现象用Reissner厚板理论可以论证,而经典薄板理论则无法解释。边界效应现象在板的自由边界附近表现最为典型。该文首先选取均载作用下的三边简支,一边自由的矩形厚板作为典型问题,采用哈密顿解法求出解析解。然后结合这个典型问题,对边界效应现象进行分析,对其影响因素和影响范围进行讨论。     

On approximate solutions for the deformation of plane anisotropic beams

Plane deformation of anisotropic beams with narrow rectangular cross sections exhibits coupling of stretching, bending and transverse shearing. For anisotropic cantilever beams with a stiff end-cap under end forces and an end couple, assessments were made for approximate solutions by comparing these with numerically exact finite element (FE) solutions. Specific attention is given to point-wise or approximate satisfaction of the end-fixity conditions. As approximate methodologies, (i) the elementary polynomial form of Airy's stress function for the plane stress problem in a rectangular region, (ii) a Timoshenko-type beam theory, and (iii) the Bernoulli-Euler beam theory were selected. Among these, only the polynomial form of Airy's stress function violates the point-wise end-fixity conditions. Both the polynomial Airy stress function and the Timoshenko-type beam theory successfully model the effects of transverse shear deformation and the coupling of stretching and transverse deflection. Analytical solutions demonstrate that the normal shear coupling effect increases linearly with the thickness-to-span ratios in axial normal stress and axial displacement, while the coupling manifests quadratically in transverse displacement. The comparison of end displacements with the numerically exact FE solutions indicates that the polynomial form of Airy's stress function is no better than the Timoshenko-type beam theory. Similar conclusions were reached for the problem of uniformly loaded cantilever beams. It has been found that the accurate prediction of the deformation of thick anisotropic beams with significant normal-shear coupling requires the use of higher order theories.     

Material ratchetting of beam under steady axial load and cyclic bending

Abstract

The purpose of this paper is to extend the applicability of the complete differential elasto‐plastic constitutive equations of the endochronic theory to describe the material ratchetting of a beam under a steady axial load and cyclic bending. Based on the assumption of the linear distribution of the through‐thickness strain of a beam under cyclic bending, the stress‐strain behavior of each through‐thickness point can be found theoretically. Experimental data of a lead alloy under a steady axial load and cyclic bending from Hyde et al. are used for comparison with theoretical predictions. It is shown that the theoretical results are in agreement with the experimental data.     

三维四向编织复合材料的几何建模及刚度预报

提出了四步法三维四向矩形编织复合材料的单胞几何模型, 该模型考虑了空间纤维束的相互扭结和挤压而造成的纤维束弯曲和截面形状变化。基于刚度体积平均思想, 采用微元段直纤维的单向层板刚度分析方法建立了相应的刚度预报模型, 得到了材料的工程弹性常数, 数值结果表明了该模型的有效性;分析了工艺参数和纤维束横截面形状对弹性常数的影响规律, 得到了一些有益结论。     

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.