A Bending-Gradient model for thick plates,Part II: Closed-form solutions for cylindrical bending of laminates

In the first part (Lebée and Sab, 2010a) of this two-part paper we have presented a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called Bending-Gradient plate theory is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case when the plate is homogeneous. Moreover, we demonstrated that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In this paper, the Bending-Gradient theory is applied to laminated plates and its predictions are compared to those of Reissner–Mindlin theory and to full 3D (Pagano, 1969) exact solutions. The main conclusion is that the Bending-Gradient gives good predictions of deflection, shear stress distributions and in-plane displacement distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

A Bending-Gradient model for thick plates. Part I: Theory

This is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In part two (Lebée and Sab, 2011), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner–Mindlin theory and to full 3D Pagano’s exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

A kinematic FE limit analysis model for thick English bond masonry walls

Two-wythes masonry walls arranged in English bond texture were often used in the past as bearing panels in seismic area. On the other hand, earthquake surveys have demonstrated that masonry strength under horizontal actions is usually insufficient, causing premature collapses of masonry buildings, often ascribed to out-of-plane actions. Furthermore, many codes of practice impose for new brickwork walls a minimal slenderness, which for instance is fixed by the Italian O.P.C.M. 3431 equal to 12 for artificial bricks and 10 for natural blocks masonry.For the above reasons, the analysis at failure of English bond brickwork walls under out-of-plane actions is a topic that deserves consideration, despite the fact that almost the totality of the studies of masonry at failure is devoted to running bond arrangements. Furthermore, it must be noted that an approach based on the analysis of running bond texture – in comparison with English bond pattern – is not suitable for the investigation of the behavior of bearing panels.In this framework, in the present paper, a Reissner–Mindlin kinematic limit analysis approach is presented for the derivation of the macroscopic failure surfaces of two-wythes masonry arranged in English bond texture. In particular, the behavior of a 3D system constituted by infinitely resistant bricks connected by joints reduced to interfaces with frictional behavior and limited tensile/compressive strength is identified with a 2D Reissner–Mindlin plate. In this way, assuming both an associated flow rule for the constituent materials and a finite subclass of possible deformation modes, an upper bound approximation of macroscopic English bond masonry failure surfaces is obtained as a function of macroscopic bending moments, torsion and shear forces.Several examples of technical relevance are treated both at a cell level and at a structural level, addressing the differences in terms of collapse loads and failure surfaces due to different textures and constituent laws for joints. Finally, two meaningful structural examples consisting of a panel in cylindrical flexion and a masonry slab constrained at three edges and out-of-plane loaded are discussed. A detailed comparison in terms of deformed shapes at collapse and failure loads between a 2D FE Reissner–Mindlin limit analysis approach and a full 3D heterogeneous FE model shows the reliability of the results obtained using the kinematic identification approach proposed.     

基体开裂的正交复合材料对称层合板的粘弹性力学行为

本文讨论了含有均匀基体裂纹的正交复合材料对称层合板的线性粘弹性力学行为.采用二维剪切滞后模型并对其层间剪应力在厚度方向进行线性假设分布,求得层合板的平均应力应变的线弹性解,利用等效约束模型和经典层合理论可得到层合板因为含有基体裂纹而所引起的刚度退化现象.在弹性-粘弹性对应原理的基础上对其层合板的线粘弹性进行了讨论和研究.结果表明:层合板的松弛模量和蠕变泊松比随着时间的增加而减少,到达稳态后其值基本上是恒值.并跟Zocher的解析解和有限元数值解作了比较,发现结果非常吻合.     

A Reissner–Mindlin limit analysis model for out-of-plane loaded running bond masonry walls

Earthquake surveys have demonstrated that the lack of out-of-plane strength is a primary cause of failure in many traditional forms of masonry. Moreover, bearing walls are relatively thick and, as a matter of fact, many codes of practice impose a minimal slenderness for them, as for instance the recent Italian O.P.C.M. 3431 [2005. Ulteriori modifiche ed integrazioni all’OPCM 3274/03 (in Italian) and O.P.C.M. 3274, 20/03/2003, Primi elementi in materia di criteri generali per la classificazione sismica del territorio nazionale e di normative tecniche per le costruzioni in zona sismica (in Italian)], in which the upper bound slenderness is fixed respectively equal to 12 for artificial bricks and 10 for natural blocks masonry. In this context, a formulation at failure for regular assemblages of bricks based both on homogenization and Reissner–Mindlin theory seems particularly attractive. In this paper a kinematic limit analysis approach under the hypotheses of the thick plate theory is developed for the derivation of the macroscopic failure surfaces of masonry out-of-plane loaded. The behavior of a 3D system of blocks connected by interfaces is identified with a 2D Reissner–Mindlin plate. Infinitely resistant blocks connected by interfaces (joints) with a Mohr–Coulomb failure criterion with tension cut-off and compressive cap are considered. Finally, an associated flow rule for joints is adopted. In this way, the macroscopic masonry failure surface is obtained as a function of the macroscopic bending moments, torsional moments and shear forces by means of a linear programming problem in which the internal power dissipated is minimized, once that a subclass of possible deformation modes is a priori chosen. Several examples of technical relevance are presented and comparisons with previously developed Kirchhoff–Love static [Milani, G., Lourenço, P.B., Tralli, A., 2006b. A homogenization approach for the limit analysis of out-of-plane loaded masonry walls. J. Struct. Eng. ASCE (in press)] and kinematic [Sab, K., 2003.Yield design of thin periodic plates by a homogenisation technique and an application to masonry walls. C.R. Mech. 331, 641–646] failure surfaces are provided. Finally, two meaningful structural examples are reported, the first concerning a masonry wall under cylindrical flexion, the second consisting of a rectangular plate with a central opening out-of-plane loaded. For both cases, the influence of the shear strength on the collapse load is estimated.     

A homogenized Reissner–Mindlin model for orthotropic periodic plates: Application to brickwork panels

This paper describes a new procedure for the homogenization of orthotropic 3D periodic plates. The theory of Caillerie [Caillerie, D., 1984. Thin elastic and periodic plates. Math. Method Appl. Sci., 6, 159–191.] – which leads to a homogeneous Love–Kirchhoff model – is extended in order to take into account the shear effects for thick plates. A homogenized Reissner–Mindlin plate model is proposed. Hence, the determination of the shear constants requires the resolution of an auxiliary 3D boundary value problem on the unit cell that generates the periodic plate. This homogenization procedure is then applied to periodic brickwork panels.A Love–Kirchhoff plate model for linear elastic periodic brickwork has been already proposed by Cecchi and Sab [Cecchi, A., Sab, K., 2002b. Out-of-plane model for heterogeneous periodic materials: the case of masonry. Eur. J. Mech. A-Solids 21, 249–268 ; Cecchi, A., Sab, K., 2006. Corrigendum to A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork [Int. J. Solids Struct., vol. 41/9–10, pp. 2259–2276], Int. J. Solids Struct., vol. 43/2, pp. 390–392.]. The identification of a Reissner–Mindlin homogenized plate model for infinitely rigid blocks connected by elastic interfaces (the mortar thin joints) has been also developed by the authors Cecchi and Sab [Cecchi A., Sab K., 2004. A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork. Int. J. Solids Struct. 41/9–10, 2259–2276.]. In that case, the identification between the 3D block discrete model and the 2D plate model is based on an identification at the order 1 in the rigid body displacement and at the order 0 in the rigid body rotation.In the present paper, the new identification procedure is implemented taking into account the shear effect when the blocks are deformable bodies. It is proved that the proposed procedure is consistent with the one already used by the authors for rigid blocks. Besides, an analytical approximation for the homogenized shear constants is derived. A finite elements model is then used to evaluate the exact shear homogenized constants and to compare them with the approximated one. Excellent agreement is found. Finally, a structural experimentation is carried out in the case of masonry panel under cylindrical bending conditions. Here, the full 3D finite elements heterogeneous model is compared to the corresponding 2D Reissner–Mindlin and Love–Kirchhoff plate models so as to study the discrepancy between these three models as a function of the length-to-thickness ratio (slenderness) of the panel. It is shown that the proposed Reissner–Mindlin model best fits with the finite elements model.     

Similarity solutions for power-law and exponentially stretching plate motion with cross flow

Laminar boundary layers generated by power-law plate stretching with cross flows are studied. Only the stretching solutions of Banks [10] are considered, those being bounded by exponentially stretched plates. In one case the cross flow is generated by a uniform transverse stream far above the stretching plate or a wall moving with uniform transverse velocity. Two other cases deal with cross flows generated by transverse shearing motions of the surface. Possible two parameter solutions appear, but here we present two one-parameter families of cross flow solutions generated by transverse plate shearing motion. Streamwise and transverse shear stresses and velocity profiles are displayed in graphical form.     

A model for the through-thickness elastic–plastic behaviour of paper

An elastic–plastic material model for the out-of-plane mechanical behaviour of paper is presented. This model enables simulation the elastic–plastic behaviour under high compressive loads in the through-thickness direction (ZD). Paper does not exhibit a sharp transition from elastic to elastic–plastic behaviour. This makes it advantageous to define critical stress states based on failure stresses rather than yield stresses. Moreover, the failure stress in out-of-plane shear is strongly affected by previous plastic through-thickness compression. To cover these two features, a model based on the idea of a bounding surface that grows in size with plastic compression is proposed. Here, both the bounding and the yield surfaces are suggested as parabolas in stress space. While the bounding surface is open for compressive loads, the yield surface is bordered by the maximum applied through-thickness compression.     

Q235钢板对半球形头弹抗侵彻特性

利用轻气炮进行了半球形头杆弹正撞击单层板和等厚接触式三层板的实验, 得到了这两种结构靶体的初始-剩余速度曲线以及弹道极限。采用ABAQUS/EXPLICIT数值模拟软件对杆弹撞击金属板的过程进行了数值模拟研究, 通过对比数值模拟和实验结果, 验证了数值模拟材料模型和参数的有效性。研究了靶体结构对抗侵彻特性的影响, 并分析了弹体对靶体的撞击过程。研究结果表明:多层板的弹道极限高于等厚单层板。单层板主要失效模式为剪切, 而多层板的主要失效模式为整体的蝶形变形和局部的盘式隆起。对于多层板, 靶板具体的失效模式与其在靶中位置相关。     

A computational homogenization approach for the yield design of periodic thin plates. Part I: Construction of the macroscopic strength criterion

The purpose of this paper is to propose numerical methods to determine the macroscopic bending strength criterion of periodically heterogeneous thin plates in the framework of yield design (or limit analysis) theory. The macroscopic strength criterion of the heterogeneous plate is obtained by solving an auxiliary yield design problem formulated on the unit cell, that is the elementary domain reproducing the plate strength properties by periodicity. In the present work, it is assumed that the plate thickness is small compared to the unit cell characteristic length, so that the unit cell can still be considered as a thin plate itself. Yield design static and kinematic approaches for solving the auxiliary problem are, therefore, formulated with a Love–Kirchhoff plate model. Finite elements consistent with this model are proposed to solve both approaches and it is shown that the corresponding optimization problems belong to the class of second-order cone programming (SOCP), for which very efficient solvers are available. Macroscopic strength criteria are computed for different type of heterogeneous plates (reinforced, perforated plates,…) by comparing the results of the static and the kinematic approaches. Information on the unit cell failure modes can also be obtained by representing the optimal failure mechanisms. In a companion paper, the so-obtained homogenized strength criteria will be used to compute ultimate loads of global plate structures.     

Theoretical and numerical investigation of HF elastic wave propagation in two-dimensional periodic beam lattices

The elastic wave propagation phenomena in two-dimensional periodic beam lattices are studied by using the Bloch wave transform. The numerical modeling is applied to the hexagonal and the rectangular beam lattices, in which, both the in-plane （with respect to the lattice plane） and out-of-plane waves are considered. The dispersion relations are obtained by calculating the Bloch eigenfrequencies and eigenmodes. The frequency bandgaps are observed and the influence of the elastic and geometric properties of the primitive cell on the bandgaps is studied. By analyzing the phase and the group velocities of the Bloch wave modes, the anisotropic behaviors and the dispersive characteristics of the hexagonal beam lattice with respect to the wave prop- agation are highlighted in high frequency domains. One im- portant result presented herein is the comparison between the first Bloch wave modes to the membrane and bend- ing/transverse shear wave modes of the classical equivalent homogenized orthotropic plate model of the hexagonal beam lattice. It is shown that, in low frequency ranges, the homog- enized plate model can correctly represent both the in-plane and out-of-plane dynamic behaviors of the beam lattice, its frequency validity domain can be precisely evaluated thanks to the Bloch modal analysis. As another important and original result, we have highlighted the existence of the retro- propagating Bloch wave modes with a negative group veloc- ity, and of the corresponding ＂retro-propagating＂ frequency bands.     

Effect of loading conditions on regimes of solid-phase transformations in a plate

A two-dimensional mathematical model of propagation of chemical conversion in a plate under mechanical loading is proposed. The problem is numerically studied for three variants of loads and displacements on the plate surfaces: clamped plate, uniaxial tension, and shear. As the fields of temperatures, concentrations, and stresses are coupled, the conversion modes under different loading conditions are found to be different. By means of a parametric study of the model, it is demonstrated that the value of internal stresses may be substantially higher than the external load.     

Vibration and buckling of laminated sandwich plates having interfacial imperfections

The vibration and buckling characteristics of sandwich plates having laminated stiff layers are studied for different degrees of imperfections at the layer interfaces using a refined plate theory. With this plate theory, the through thickness variation of transverse shear stresses is represented by piece-wise parabolic functions where the continuity of these stresses is satisfied at the layer interfaces by taking jumps in the transverse shear strains at the interfaces. The transverse shear stresses free condition at the plate top and bottom surfaces is also satisfied. The inter-laminar imperfections are represented by in-plane displacement jumps at the layer interfaces and characterized by a linear spring layer model. It is quite interesting to note that this plate model having all these refined features requires unknowns only at the reference plane. To have generality in the analysis, finite element technique is adopted and it is carried out with a new triangular element developed for this purpose, as any existing element cannot model this plate model. As there is no published result on imperfect sandwich plates, the problems of perfect sandwich plates and imperfect ordinary laminates are used for validation.     

Bending of orthotropic plates resting on Pasternak’s foundations using mixed shear deformation theory

The mixed first-order shear deformation plate theory(MFPT) is employed to study the bending response of simply-supported orthotropic plates.The present plate is subjected to a mechanical load and resting on Pasternak’s model or Winkler’s model of elastic foundation or without any elastic foundation.Several examples are presented to verify the accuracy of the present theory.Numerical results for deflection and stresses are presented.The proposed MFPT is shown simplely to implement and capable of giving satisfactory results for shear deformable plates under static loads and resting on two-parameter elastic foundation.The results presented here show that the characteristics of deflection and stresses are significantly influenced by the elastic foundation stiffness,plate aspect ratio and side-to-thickness ratio.     

Elastic analyses of heterogeneous hollow cylinders

Two different kinds of heterogeneous elastic hollow cylinders are studied in the present paper. One is a multi-layered cylinder with different values in different layers for both elastic modulus and Poisson’s ratio. Another is an elastic hollow cylinder with continuously graded material properties. By introducing two recursive algorithms, the extrusion stresses between two neighbor layers in the multi-layered cylinder submitted to uniform pressures on the inner and outer surfaces can be simply determined. Then the exact solutions of the multi-layered structure can be found based on Lamé’s solution. For the hollow cylinder with continuously graded properties, the displacement method is used. Both Whittaker equation and hyper-geometric equation are derived and successfully solved, and then the exact solutions are found. The results obtained in the present paper are compared with the numerical solutions and good agreements are found. At the end of the present paper, some inherent properties of these two different kinds of heterogeneous elastic hollow cylinders are presented and discussed. The results obtained in the present paper are useful in the design and analysis for composites reinforced by unidirectional fiber layers.     

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。     

Local elastodynamic stresses in the unit cell of a woven fabric composite

Summary A theoretical study of the local elastodynamic stresses of woven fabric composites under dynamic loadings is presented in this article. The analysis focuses on the unit cell of an orthogonal woven fabric composite, which is composed of two sets of mutually orthogonal yarns of either the same fiber (nonhybrid fabric) or different fibers (hybrid fabric) in a matrix material. Using the mosaic model for simplifying woven fabric composites and a shear lag approach to account for the inter-yarn deformation, a one-dimensional analysis has been developed to predict the local elastodynamic and elastostatic behavior. The initial and boundary value problems are formulated and then solved using Laplace transforms. Closed form solutions of the dynamic displacements and stresses in each yarn and the bond shearing stresses at the interfaces between adjacent yarns are obtained in the time domain for any type of in-plane impact loadings. When time tends to infinity, the dynamic solutions approach to their corresponding static solutions, which are also developed in this article. Solutions of certain special cases are identical to those reported in the literature. Lastly, the dynamic stresses and bond shearing stresses of plain weave composites subjected to step uniform impacts are presented and discussed as an example of the general analytical model. Received 3 May 1999; accepted for publication 22 September 1999     

Higher order zig-zag theory for fully coupled thermo-electric–mechanical smart composite plates

A higher order zig-zag plate theory is developed to refine the prediction of the mechanical, thermal, and electric behaviors fully coupled. Both in-plane displacement and temperature fields through the thickness are constructed by superimposing linear zig-zag field to the smooth globally cubic varying field. Smooth parabolic distribution through the thickness is assumed in the out-of-plane displacement field in order to consider transverse normal deformation. Linear zig-zag form is adopted in the electric potential. The layer-dependent degrees of freedom of displacement and temperature fields are expressed in terms of reference primary degrees of freedom by applying interface continuity conditions as well as bounding surface conditions of transverse shear stresses and transverse heat fluxes. Thus the proposed theory is not only accurate but also efficient. Through the numerical examples of coupled and uncoupled analysis, the accuracy and efficiency of the present theory are demonstrated. The present theory is suitable in the predictions of fully coupled behaviors of thick smart composite plate under mechanical, thermal, and electric loads combined.     

A multi-scale framework for layered composites with thermo-rheologically complex behaviors

A multi-scale modeling framework is proposed for generating the effective nonlinear thermo-viscoelastic responses of multi-layered and thick-section composites. The modeling framework is demonstrated for multi-layered composite systems consisting of two alternating fiber-reinforced polymeric layers: unidirectional fiber (roving) and continuous filament mat (CFM). The viscoelastic behavior of the polymeric matrix is considered as stress-dependent and thermo-rheologically complex. Two simplified 3D micromechanical models with periodic boundary conditions are formulated for the roving and CFM. In addition, a sublaminate model is developed for the 3D effective homogenized through-thickness response of the roving and CFM layers. This framework provides an effective time-dependent material response while simultaneously recognizing the corresponding deformation at the microconstituents under overall thermo-mechanical loadings at the macro-level. Stress correction algorithms are developed at each scale to enhance computational efficiency and accuracy. Short-term (30 min) creep tests on off-axis multi-layered specimens are also conducted under combined stresses and temperatures to calibrate in-situ fiber and matrix properties and verify the predictions of the multi-scale framework. A time-shifting method is applied to create long-term material behaviors from the available short-term creep data. Verification of the multi-scale material framework is also done for the overall long-term responses. Finally, long-term structural analyses under thermo-mechanical loadings are conducted by integrating the multi-scale material framework to finite element (FE) structural analyses.     

Linear coupled bending and extension of an unbalanced bonded repair

An unbalanced repair is a composite patch bonded to one side of a cracked structure for the purpose of preventing or reducing damage growth in the substrate. A single-sided repair offsets the load path within the structure, inducing out-of-plane bending. This bending increases the stress intensity in the underlying crack and causes adhesive peel stresses and bending of the repair which can, relative to a repair that is restrained against bending, lead to early failure. In this article the authors correct the analysis of Wang and Rose [Wang, C.H., Rose, L.R.F., 1997. On the design of bonded patches for one-sided repair. In: Proceedings of the 11th International Conference on Composite materials, Gold Coast, Australia, vol. 5, pp. 347–356] developed by using an energy analysis of a single-sided or unbalanced repair applied to a very long-crack, to comply with Maxwell’s reciprocal theorem and to account for transverse normal and shear stresses at the crack tip and the accompanying shear deflections. The authors then develop closed-form equations useful for bonded composite repair design and damage tolerance assessment of cracks of arbitrary length by developing a new method for interpolation between this long-crack limit and a short-crack limit based on the stress intensity and crack face displacements for an unreinforced crack. The interpolation method is then tested against an advanced line-spring model that has been created by using a 6th order generalized plane strain plate formulation in extension and a new 8th order formulation in bending, thus allowing for the inclusion of transverse shear and normal stresses. The closed-form equations are found to be accurate when compared to the line-spring model, and to provide reasonable results when compared to a three-dimensional finite element model of a bonded repair. Inaccuracies are shown to exist principally in the determination of the nominal stresses in the vicinity of the crack.