Effect of layer geometry and stiffness on the fields of normal stresses in multilayer beams under asymmetric bending

A mathematical model is suggested for calculating the bending stiffness and fields of normal stresses (strength) at any point in the cross section of a multilayer beam. It is found that the structure of the scalar field of normal stresses allows one to solve some optimization problems with multivariant parameters. The method is illustrated with an example of two-layer beams. The results of an investigation into the strength and stiffness of two-layer beams, with a geometric and (or) stiffness asymmetry, in asymmetric bending are presented. The kinetics of bending stiffness and strength in relation to variations in the geometric parameters of cross sections and in the ratio of elastic moduli of layers is examined. It is established that the normal stresses in multilayer beams under asymmetric bending considerably depend on the location of the flexural center, neutral plane, and bending stiffnesses relative to the principal axes of cross sections of the beams.     

边界条件对梁类结构系统刚度的影响

车身骨架是由大量梁类结构单元构成的.除了这些梁的截面形状和尺寸外,边界条件也对系统刚度影响很大.讨论了各种边界条件和载荷模式下的梁系统的合成刚度.基于两端固支的均匀截面梁的弯曲和扭转刚度,研究了各联结刚度的大小对系统刚度的贡献,并绘制了相应的影响曲线.最后,通过上述解析公式和有限元法计算了某汽车仪表板横梁系统的实际弯曲和扭转刚度.文中获得的静态刚度公式对其它梁类结构也适用。     

Stress State of Multilayer Structural Elements Subjected to Static Loading and Optimization of Laminated Beams and Bars

The results of FEM investigation of the triaxial stress state in multilayer structural elements subjected to axial and bending loads are presented. The distribution regularities of the stiffness and stresses or strains depending on the geometric and mechanical characteristics of layers and their position in the cross section of beams and bars are examined. The optimization of these elements is carried out using the dependences of the Bareisis—Paulauskas method and the Optim-98 computer program created by the present authors. As the optimization criteria, the strength, stiffness, mass, and cost of the structural elements are considered.     

Resistance of three-layered structures to static and cyclic bending

Conclusions The above studies of two types of three-layer structural elements showed that the types have different resistances to static deformation in bending. Regardless of the materials, the use of structures which are symmetrical in regard to stiffness makes it possible to obtain a stiffness and strength for the structure which are 10–15% lower than the stiffness and strength of the external plates if the thickness of the latter does not account for more than 25% of the thickness of the structure. This finding, in turn, permits a substantial reduction in the weight of the structure by the use of a lower-density material for the internal layer. Resistance to static bending is determined mainly by the resistance of the structure to shear stresses. The mechanism of fatigue fracture differs appreciably from the fracture mechanism in static deformation. Regardless of the thickness of the structural elements, fatigue fracture for both types of structure occurs as a result of the acting normal compressive stresses. The endurance limit of the hybrid structure is determined by the fatigue resistance of the external layers, and its value is nearly equal to the resistance of the pure materials.Presented at the Sixth All-Union Conference on the Mechanics of Polymer and Composite Materials (Riga, November, 1986).Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 878–882, September–October, 1986.     

Formation of Saddle-Shaped Composite Sheets

Based on the deformation model of an unbalanced multilayer composite, changes in bending curvatures of sheet-type composites with nonsymmetric structure relative to the midplane of the sheet, depending on the moisture of layers, are predicted. The bending curvatures of saddle-shaped sheets of wood-based composites are calculated with regard to the physical and mechanical properties, geometrical dimensions, orientation, and distribution of layers. The analytical results are compared with the bending curvatures found experimentally for a four-layered unbalanced composite made of birch veneer. The applied calculation model enables us to determine the values of bending curvatures of saddle-shaped wood composite sheets, which can be used in elaborating the technological recommendations.     

Bending features of a sandwich panel with asymmetric structure under local loads

The bending characteristics of a composite panel with asymmetric layered structure under local surface loads are obtained. A refined version of the applied theory is developed using the analytical solution of the bending problem of a sandwich plate with arbitrary asymmetric structure under a point load. Local effects are investigated within the limits of a discrete model allowing for the specific character of elastic properties of a soft filler. The advantages of the solution are expressions of bending characteristics — layer curvatures, displacements, and stresses — in a closed form. It is shown that these characteristics can vary several times depending on the asymmetry parameters of the structure. Degeneration peculiarities of the solution, stemming from the slipping of layers or, otherwise, their rigid linking by the Kirchoff—Love hypothesis, as well as from account of the transverse shear and compression of the normal, are examined in line with the degeneration of geometric and physical parameters of the discrete model adopted. The results obtained are illustrated by curves and surfaces for the characteristics studied.Submitted for the 11th International Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000).Institute of Polymer Mechanics, Latvian University, Riga, LV-1006 Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 717–742, November–December, 1999.     

Buckling of laminated plates - A simple finite element based on higher-order theory

A four-noded rectangular element with seven degrees of freedom at each node is developed for buckling analysis of laminated plate structures having any number of layers with a constant thickness of individual layers. The displacement model is so chosen that it can explain adequately the parabolic distribution of transverse shear stresses and the non-linearity of in-plane displacements across the thickness. A geometrical stiffness matrix is developed using in-plane stresses. A wide range of plates from thick to thin are examined under uniaxial loading conditions. The results are compared with the existing analytical and numerical solutions. The present formulations confirm its applicability for buckling analysis of a wide range of plates.     

Finite element construction for simulating delamination of sandwich composite plates and masses

Displacements and transverse normal stresses in sandwich plates and masses have been approximated by the Ambartsumyan iterative approach to constructing mathematical models of the stress-strain state of sandwich structures. A linear distribution of the displacements in the sandwich structure is set up as the first step of the iterative process, while in the subsequent steps the displacement approximations with higher-order polynomials are obtained. The approximation of the compression stresses is based on Hooke's law using the expression of the tangential displacements in the second step and the normal displacements in the third step of the iterative process. Two shear functions are introduced. The finite element is rectangular and has four nodes. The number of degrees of freedom of finite elements is independent of the quantity of the layers that may be orthotropic. The finite element allows us to simulate delamination by a thin low-modulus interlayer. In doing so, the quantity of the layers increases, while the order of the resolving set of equations does not grow. A number of numerical experiments were carried out. It has been shown that the delamination can greatly increase the level of the stresses in the structure. This effect is especially significant for thin structures. The stresses are somewhat lower when taking into account the interlaminar friction.Submitted to the 10th International Conference on Mechanics of Composite Materials (Riga, April 20–23, 1998).Ukrainian Transport University, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 251–263, March–April, 1998.     

碳纤维/树脂基复合材料曲壁蜂窝夹芯结构的三点弯曲性能北大核心CSCD

为研究碳纤维复合材料(CFRP)曲壁蜂窝结构在三点弯曲载荷作用下的承载特性与失效模式,对不同芯层高度、面板厚度的结构进行了理论预报、数值模拟及试验.首先,根据夹芯结构的主要失效模式,提出了相应的理论预报公式,并绘制了失效机制图;其次,建立了CFRP曲壁蜂窝夹芯结构的有限元仿真模型,对其在三点弯曲载荷作用下的典型失效行为进行模拟;最后,通过模压成型工艺制备了不同尺寸的CFRP曲壁蜂窝夹芯结构,并将试验结果与理论、模拟结果进行比较.结果表明,蜂窝夹芯结构承载能力与芯层高度、面板厚度密切相关,结构芯层及面板刚度随其尺寸的减小而下降,导致结构失效模式由面芯脱黏失效变为面板压溃失效.     

Analysis of Compression Stress of a Filler and Bending of a Sandwich Panel under Multipoint Loading

The distribution of transverse stresses in the midlayer of a composite sandwich panel under multipoint loading is investigated. The stresses averaged across the thickness of a soft filler are estimated using a discrete model. Finite expressions for the compression of the filler along the length of the panel are derived by means of superposition of the local effects from the bending of face layers under an infinite system of transverse point forces constant across the panel thickness. The effects of compression and transverse extension of the filler, in the case of a high distribution frequency of these forces, i.e., when the distance between the forces is comparable to the panel thickness, are revealed. Compression of the panel by two systems of forces applied symmetrically or nonsymmetrically to the upper and lower faces is considered. The bending characteristics in the cases of loading with point forces and piecewise distributed loads are compared. The formulas obtained are used to determine the length of a small region on the panel surface for which the local effects from the distributed pressure and the point force are equivalent. The corresponding estimates are obtained in a closed form. The analysis, carried out with varied parameters of the structure, allows us to elucidate the peculiarities of the effect of discontinuous loads on the design characteristics in the local zones, using finite expressions derived by the operational method.     

TRAPEZOIDAL PLATE BENDING ELEMENT WITH DOUBLE SET PARAMETERS

Using double set parameter method, a 12-parameter trapezoidal plate bending element is presented. The first set of degrees of freedom, which make the element convergent, are the values at the four vertices and the middle points of the four sides together with the mean values of the outer normal derivatives along four sides. The second set of degree of freedom, which make the number of unknowns in the resulting discrete system small and computation convenient are values and the first derivatives at the four vertices of the element. The convergence of the element is proved.     

Thermoelastoplastic Bending of Complexly Reinforced Plates

A problem on the transverse-longitudinal bending of reinforced plates of variable thickness under a thermal-force loading is formulated. A qualitative analysis of the problem is carried out, and a way of its linearization is indicated. Calculations of isotropic and metal composite plates subjected to the transverse or transverse-longitudinal bending showed that their bearing capacity in the elastoplastic bending is a number of times (occasionally, by an order of magnitude) greater than in the elastic bending. The heating of the plates sharply decreases their resistance to the bending in the case of elasticity and affects it only slightly in the case of elastoplasticity. In the elastoplastic bending, the bearing capacity of the metal composite plate a number of times exceeds that of isotropic plates made of conventional structural metal alloys. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 715–742, November–December, 2005.     

Evolutionary structural optimization for problems with stiffness constraints

This paper presents a simple evolutionary procedure based on finite element analysis to minimize the weight of structures while satisfying stiffness requirements. At the end of each finite element analysis, a sensitivity number, indicating the change in the stiffness due to removal of each element, is calculated and elements which make the least change in the stiffness; of a structure are subsequently removed from the structure. The final design of a structure may have its weight significantly reduced while the displacements at prescribed locations are kept within the given limits. The proposed method is capable of performing simultaneous shape and topology optimization. A wide range of problems including those with multiple displacement constraints, multiple load cases and moving loads are considered. It is shown that existing solutions of structural optimization with stiffness constraints can easily be reproduced by this proposed simple method. In addition some original shape and layout optimization results are presented.     

Mixed Micromechanics and Continuum Damage Mechanics Approach to Transverse Cracking in [S, 90n]s Laminates

The stiffness reduction in [S, 90 _n ] _s laminates due to transverse cracking in 90-layers is analyzed using the synergistic continuum damage mechanics (SCDM) and a micromechanics approach. The material constants involved in the SCDM model are determined using the stiffness reduction data for a reference cross-ply laminate. The constraint efficiency factor, which depends on the stiffness and geometry of neighboring layers, is assumed to be proportional to the average crack opening displacement (COD). The COD as a function of the constraint effect of adjacent layers and crack spacing is described by a simple power law. The crack closure technique and Monte Carlo simulations are used to model the damage evolution: the 90-layer is divided into a large number of elements and the critical strain energy rate G _c having the Weibull distribution is randomly assigned to each element. The crack density data for a [0₂/90₄] _s cross-ply laminate are used to determine the Weibull parameters. The simulated crack density curves are combined with the CDM stiffness reduction predictions to obtain the stiffness versus strain. The methodology developed is successfully used to predict the stiffness reduction as a function of crack density in [±/90₄] _s laminates.     

Effect of building structure on the distribution of random bending strength of multilaminate reinforced plastics

A probabilistic structural model has been constructed for predicting the bending strength distribution in multilaminate reinforced plastics. The number and random strength/elastic properties of the layers or repeating structural elements are parameters of the structure. The random properties of the repeating structural elements are characterized by the scheme and geometry of the unidirectional layers. Two failure conditions have been analyzed: multistep failure caused by successive failure of separate layers and failure caused by the failure of the weakest component. The effect of the number of layers and the instability of the strength and elastic properties on the basic statistical properties of the bending strength was analyzed numerically for typical structures and for both failure conditions. The quantitative significance of the size effect determined by the thickness of the plastic was investigated. The main theoretical results were checked experimentally for unidirectional reinforced carbon plastic under uniaxial bending. The experimental distribution agrees well with the predicted distribution.Translated from Mekhanika Kompozitnykh Materialov, Vol. 29, No. 3, pp. 336–344, May–June, 1993.     

周期性正弦凸起凹凸板等效刚度的研究

针对构造正交各向异性周期性正弦凸起结构凹凸板的等效刚度问题，根据经典弹性薄板理论，基于对单胞结构力学特性分析和单胞结构在板宏观结构上周期性均匀化分布的特点，推导了正弦凸起凹凸板的等效刚度解析公式.以四边简支周期性正弦凸起结构凹凸板为例，将该文计算结果与有限元模拟结果进行对比，验证了该文等效刚度的合理性和精确性.最后，分析了正弦凸起凹凸板几何参数对等效刚度特性的影响，给出了结构几何参数与等效刚度之间的关系.结果表明:应用该文方法可以有效计算周期性正弦凸起凹凸板的等效刚度；由于凹凸板在构造上的几何结构变化，与基础平板相比其弯曲刚度和抗扭刚度都有明显的提升.该研究结果对凹凸板静力学和动力学的进一步研究以及实际工程应用具有指导意义.     

轻质热防护系统多层材料组合结构的热应力分析

提出了轻质热防护系统外面板使用多层结构的概念,设计了2种热防护材料组合构成的3种铺层方案．通过模拟飞行器再入大气层时受到的机械和热载荷条件,数值计算得到了层间剪切力、底部温度和y方向位移．计算结果发现,层间剪切力发生在边缘部位且呈反对称分布;选用高热导率和高热容材料能够减少材料内的温度梯度,进而有效地降低结构的热应力和热变形;在均匀温度场情况下,两种材料的热膨胀系数之差越小,则层间剪切力越小．该研究表明不同的材料组合和铺层次序的多层结构,可以满足不同设计要求,具有优化设计潜力．     

Finite elements based on consistently assumed stresses and displacements

Finite element stiffness matrices are derived using an extende Hellinger-Reissner principle in which internal displacements are added to serve as Lagrange multipliers to introducethe equilibrium constraint in each element. In a consistent formulation the assumed stresses are initially unconstrained and complete polynomials and the total displacements are also complete such that the corresponding strains are complete in the same order as the stressesSeveral examples indicate that resulting properties for elements constructed by this consistent formulation are ideal and are less sensitive to distortions of element geometries. The method has been used to find the optimal stress terms for plane elements, 3-D solids, axisymmetric solids, and plate bending elements.     

Research in the Limiting Efficiency of Plastic Deformation of Multilayer Tension (Compression) Bars

A method is proposed for calculating the limiting loads of multilayer bars in plastic deformation. The regularities of changes in the efficiency factor (EF) of plastic deformation in relation to the elastic moduli of the materials of which the bars are composed, the stress which causes the material plasticity, and variations in the cross-sectional areas of layers are examined. Mathematical expressions are derived which allow one to easily calculate the EF for arbitrary values of variable parameters and to find the limiting values of EF when one of the parameters is changed. The equivalent tension diagram for a multilayer bar is obtained in relation to the strength and stiffness of materials and their number and cross-sectional areas.     

Application of fuzzy sets to transient analysis of space structures

An application of fuzzy sets, in conjunction with finite elements, to the transient analysis of a precision-deployable space structure is presented. The structural members are modeled by using beam finite elements, and the structure's latch joint is modeled by using a spring–damper–Coulomb friction element. Two types of transient response simulations are performed: slow transient load–deflection response and transient impulse response. The first simulation is used to evaluate the stiffness and buckling loads at the structure's tip. The second simulation is used to evaluate the structure's natural frequencies, mode shapes and the precision of the final shape. For each simulation the possibility distributions of various response quantities are obtained. Fuzzy sets are used to represent three beam properties, namely: damping coefficient, bending stiffness, and axial stiffness; as well as two joint parameters: Coulomb friction force and damping coefficient. Fuzzy set techniques provide an insight into the range of possible responses associated with the combined selected variations in the system parameters.