具有夹层的正交异性复合材料圆锥壳体的非线性轴对称屈曲分析

研究均布载荷作用下边缘可移夹支具有正交异性复合材料表层和软夹心的夹层圆锥壳非线性轴对称屈曲问题.利用变分原理导出具有正交各向异性表层夹层圆锥壳在均匀外压作用下的非线性轴对称屈曲问题的基本方程,采用修正迭代法[8]求得了可移夹支边界条件下壳体临界屈曲载荷的解析表达式,对几种典型的纤维增强复合材料表层夹层圆锥壳给出了数值结果和图表,讨论了几何参数和物理参数对临界屈曲载荷的影响。     

普通卷边槽钢的弹性畸变屈曲荷载

畸变屈曲是控制冷弯型钢稳定设计的重要屈曲模式之一。该文以理论研究成果为基础,结合应用有限条屈曲分析程序,对冷弯卷边槽钢在偏心受压和纯弯曲下的弹性畸变屈曲荷载进行了深入细致的分析。首先基于对轴心受压和纯弯曲两种极端荷载情况下畸变屈曲荷载的参数分析,提出了屈曲临界半波长度和考虑翼缘自身剪切与畸变影响的修正参数的近似计算公式;接着对腹板提供给翼缘板的转动约束刚度进行了分析讨论,提出了精度较高的近似计算公式;利用以上结果,最后提出了冷弯卷边槽钢在偏心受压和纯弯曲下的弹性畸变屈曲荷载简化计算公式。简化公式具有足够的精度、便于手算、实用性强,可供工程设计人员和各国冷弯薄壁型钢设计规范修订相应内容时参考。     

Delamination buckling of FRP layer in laminated wood beams

An analytical solution for predicting delamination buckling and growth of a thin fiber reinforced-plastic (FRP) layer in laminated wood beams under bending is presented. Based on a strength-of-materials approach, displacement functions for a delaminated beam under four-point bending are derived. Using force and displacement compatibility conditions, an explicit form relating the applied transverse load with the delamination buckling load is established. An explicit form of the strain-energy release rate is presented to study the delamination growth in beams under bending. The analytical solution is evaluated using experimental data for glued-laminated timber (glulam) beams reinforced with a thin fiber-reinforced plastic composite on the compression face. The delamination growth in bending is shown to behave differently to that of the in-plane loading case.     

Optimum design of fiber-reinforced composite cylindrical skirts for solid rocket cases subjected to buckling and overstressing constraints

In order to increase the flight range of aerospace vehicles and the efficiency of solid rocket motors, designers attempt to reduce the weight of solid rocket motors. A skirt is a potential element for weight reduction in rocket motors as it leads to reduction of the total weight of solid rocket motor. Due to its significance for solid rocket motors, the objective of this paper is to investigate the optimal design of a fiber-reinforced composite cylindrical skirt subjected to a buckling strength constraint and an overstressing strength constraint under aerodynamic torque and axial thrust. The present optimal design problem involve in determining the best laminate configuration to minimize the weight of the cylindrical skirt. To find the optimal solution accurately and quickly, the hybrid genetic algorithm (HGA) is employed in this work. Buckling strength and overstressing strength of the fiber-reinforced composite cylindrical skirt are analyzed using classical laminate theory and elastic stability theory of thin shells. The Tsai-Wu failure criterion is employed to assess the first ply failure, and an overstressing load level factor is introduced to describe the failure strength. In addition, a buckling load factor is introduced to describe the buckling strength. Due to the critical issue of buckling strength, the effects of the design parameters on the buckling strength are investigated in this work. Finally, a practical design example of the proposed fiber-reinforced composite cylindrical skirt is investigated using the present analysis procedure. Results reveal that the fiber-reinforced composite cylindrical skirt laminated symmetrically with both cross-ply layers [0/90°] and angle-ply layers [+45/−45°] can sustain a great buckling load. Furthermore, the buckling strength of the skirt shell laminated with equal-hybrid between the angle-ply layers and the cross-ply layers is greater than that of the skirt shell laminated with over-weighted hybrid between the angle-ply layers and the cross-ply layers. Results provide a valuable reference for designers of aerospace vehicles.     

An analysis of compressive stability of elastic solids containing a crack parallel to the surface

An analysis is presented for compressive stability of elastic solids containing a crack parallel to the free surface based on the mathematical theory of elasticity. Basic buckling equations derived from the mathematical theory of elasticity are employed and are reduced to a system of homogeneous Cauchy-type singular integral equations by means of Fourier integral transform. The integral equations are solved numerically by utilizing Gauss-Chebyshev integral formulae. Numerical results for buckling loads are presented for various geometrical parameters and are compared with those obtained from classical theory of beam-plate stability based on the Kirchhoff assumption. The comparison of both results shows that the buckling loads obtained from the classical theory of beam-plate stability are much larger than those obtained from the mathematical theory of elasticity, referring to which the limitations of the classical theory applied to the present buckling problem are discussed. A simple but accurate approximate method for estimating the buckling load is developed by the use of the elastic support coefficient obtained from the present analysis and the Euler formula derived from the classical theory for the case of elastically supported ends. Finally, the numerical results of the buckling wave shapes and the Mode I and II stress factors, which cannot be obtained from the classical theory, are presented.     

斜卷边冷弯薄壁槽钢轴压柱弹性畸变屈曲计算

畸变屈曲是冷弯薄壁卷边槽钢的一种重要屈曲模式,弹性畸变屈曲临界应力是目前一些主要规范计算畸变屈曲的关键。该文首先推导出畸变屈曲模式下,构件纵向翘曲位移的显式计算公式,从而简化计算过程;然后基于经典的薄板理论,计算平面内位移,确定构件平面内的变形,在现有研究基础上得到斜卷边冷弯薄壁槽钢轴压柱弹性畸变屈曲计算公式;最后与有限条法、近似模型计算法进行对比,验证了该文计算方法的正确和有效性。该文中的弹性畸变屈曲计算公式简单,便于手算,可供工程设计人员参考。     

冷弯薄壁卷边槽钢梁的弹性畸变屈曲荷载简化计算公式

以既有理论研究成果为基础,结合应用广义梁理论,对冷弯薄壁卷边槽钢在绕截面强轴弯矩作用下的弹性畸变屈曲荷载进行了深入细致的分析。首先基于45个截面冷弯薄壁卷边槽钢梁畸变屈曲荷载的参数分析,提出了屈曲临界半波长度的近似计算公式;接着对腹板提供给翼缘板的转动约束刚度进行了分析,提出了精度较高的线性近似计算公式;依据畸变屈曲特征方程,提出了考虑翼缘自身剪切与畸变影响的修正参数的近似计算公式;利用以上结果,最后提出了冷弯薄壁卷边槽钢在绕强轴弯矩作用下的弹性畸变屈曲荷载简化计算公式。简化公式具有足够的精度,便于手算,实用性强,可供工程设计人员和各国冷弯薄壁型钢设计规范修订相应内容时参考采用。     

Postbuckling Analysis of Shear-Deformable Composite Laminated Plates on Two-Parameter Elastic Foundations

Postbuckling analysis is presented for a simply supported, shear-deformable, composite laminated plate subjected to uniaxial compression and resting on a two-parameter (Pasternak-type) elastic foundation. The initial geometric imperfection of the plate is taken into account. Two cases of in-plane boundary conditions are considered. The formulations are based on Reddy’s higher-order shear-deformation plate theory, including plate–foundation interaction. The analysis uses a deflection-type perturbation technique to determine buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performance of perfect and imperfect, antisymmetric angle-ply and symmetric cross-ply laminated plates resting on Pasternak-type elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The effects played by foundation stiffness, transverse shear deformation, the character of the in-plane boundary conditions, plate aspect ratio, total number of plies, fiber orientation, and initial geometric imperfections are studied.     

Exact elasticity solution for the vibration of functionally graded anisotropic cylindrical shells

An exact elasticity solution is presented for the free and forced vibration of functionally graded cylindrical shells. The functionally graded shells have simply supported edges and arbitrary material gradation in the radial direction. The three-dimensional linear elastodynamics equations, simplified to the case of generalized plane strain deformation in the axial direction, are solved using suitable displacement functions that identically satisfy the boundary conditions. The resulting system of coupled ordinary differential equations with variable coefficients are solved analytically using the power series method. The analytical solution is applicable to shallow as well as deep shells of arbitrary thickness. The formulation assumes that the shell is made of a cylindrically orthotropic material but it is equally applicable to the special case of isotropic materials. Results are presented for two-constituent isotropic and fiber-reinforced composite materials. The homogenized elastic stiffnesses of isotropic materials are estimated using the self-consistent scheme. In the case of fiber-reinforced materials, the effective properties are obtained using either the Mori–Tanaka or asymptotic expansion homogenization (AEH) methods. The fiber-reinforced composite material studied in the present work consists of silicon-carbide fibers embedded in titanium matrix with the fiber volume fraction and fiber orientation graded in the radial direction. The natural frequencies, mode shapes, displacements and stresses are presented for different material gradations and shell geometries.     

A theory for physically nonlinear elastic fiber-reinforced composites

A three-dimensional constitutive relationship for the fiber-reinforced composite with a nonlinearly elastic matrix material is derived. The composite is modeled by a medium consisting of thin nonlinear matrix layers alternating with effective linearly elastic fibrous layers. Shear wave propagations are investigated. The formation of shock wave, its stability and growth behaviors are discussed.     

An energy method for calculating the stiffness of aligned short-fiber composites

A procedure is presented for calculating the longitudinal elastic modulus of a composite possessing dilute concentrations of aligned cylindrical fibers. For a dilute concentration of fibers, the modulus can be written in terms of the perturbation that a single fiber induces in the displacement field of an otherwise homogeneous body subjected to remote uniaxial tension. A boundary value problem, referred to as the auxiliary problem, is formulated which yields the perturbed displacement field directly. The longitudinal modulus can then be expressed in terms of the potential energy of the auxiliary problem; thus, energy-based approximate methods (e.g. FEM) can be exploited to calculate the modulus accurately. With a small modification, this procedure is shown to be applicable to the case of fibers which are imperfectly bonded to the matrix. Moduli for a few representative cases are calculated using a finite element method. Finally, a new means of defining and computing the ineffective length of a fiber is introduced.     

Scattering of elastic waves by spherical inclusions with applications to low frequency wave propagation in composites

Scattering of plane elastic waves by a spherical inclusion is considered. A unified method of solution is presented which treats compressional and shear incidence on a similar basis. Explicit results are given for Rayleigh scattering. We apply the results of the single scattering problem to the propagation of low frequency waves in a composite containing a dilute concentration of spherical inclusions. Explicit formulae are given for the effective wave speeds and attenuations when the inclusions are voids. Both the compressional and shear wave speeds decrease initially as a function of frequency.     

冷弯薄壁卷边Z形钢梁的弹性畸变屈曲荷载

该文以既有理论研究成果为基础,结合应用广义梁理论,对两端简支的冷弯薄壁卷边Z形钢梁在绕截面强轴弯矩作用下的弹性畸变屈曲荷载进行了深入细致的分析。通过对45个截面的参数分析,将原先适用于冷弯薄壁卷边槽钢梁的参数计算公式推广应用于Z形钢梁,并根据Z形钢截面的几何特性对相应参数进行了重新推导和修改。利用以上结果,最后提出了冷弯薄壁卷边Z形钢梁在绕强轴弯矩作用下的弹性畸变屈曲荷载简化计算公式。简化公式具有足够的精度、便于手算、实用性强,可供工程设计人员和各国冷弯薄壁型钢设计规范修订相应内容时参考采用。     

纵向变厚度(LP)矩形钢板弹性屈曲系数的理论分析

纵向变厚度(LP)钢板因适应现代化建筑超高层及大跨度的发展使用需求,将广泛应用于工程结构中。因LP钢板几何形状的特殊性,其局部屈曲变形更容易发生在靠近薄端的位置,并且钢板局部稳定性能与相同受力状态下的等厚度钢板差异较大,需要进行深入研究。基于能量原理,采用Galerkin和Rayleigh-Ritz法(GRM),对单向均匀受压荷载作用下四种不同边界条件下的矩形LP钢板的弹性屈曲系数的计算公式进行了理论推导,并采用ANSYS有限元软件验证了公式的正确性。最后得到了不同厚度放大系数下,四种边界条件下的矩形钢板的弹性屈曲系数与钢板长宽比的关系曲线图,进一步为LP钢板的工程应用提供理论指导和设计依据。     

Elastic–plastic thermomechanical response of composite co-axial cylinders

A new method is developed in this paper to deal with the thermomechanical response of continuous fiber-reinforced composites. Treating the matrix as an elastic-perfectly plastic solid, the analytical formulae of the deformations and stresses of the matrix are obtained from the plasticity theory, axisymmetric equilibrium equation, and stress–strain and strain–displacement relations. The fiber is taken to be an anisotropic, elastic material, and the formulae calculating its deformations and stresses are also presented. The boundary conditions and the consistence of deformations and stresses between the fiber and matrix, and between elastic and plastic regions of the matrix are employed to determine the unknown constants in the analytical formulae. With the developed method, the thermomechanical stress distributions in composites reinforced with circumferentially orthotropic, radially orthotropic and transversely isotropic fibers are investigated, and how the elastic-perfectly plastic property and different materials of the matrix affect the thermomechanical response of the composites is discussed. For the thermomechanical loads and composite systems given in this paper, the elastic-perfectly plastic property of the matrix can reduce the compressive stresses in the fiber, and the tensile circumferential and axial stresses in the matrix. A strong matrix can raise the compressive stresses in the fiber, and the tensile circumferential and axial stresses in the matrix.     

Bimodal optimization of a compressed rotating rod

Summary. By using Pontryagins maximum principle we determine the shape of the lightest compressed rotating rod, stable against buckling. It is assumed that boundary conditions correspond to clamped ends. This boundary condition leads to a bimodal optimization problem. The necessary conditions of optimality, given by Eq. (17), are derived. In the special case when the rod is not rotating, this optimality conditions reduce to the previously obtained condition for the compressed rod only. It is shown that the cross-sectional area as a function of arc-length is determined from the solution of a nonlinear boundary value problem. A first integral for the system of equations is constructed. The optimal cross-sectional area and the post-buckling shape are determined by numerical integration. The stability of the bifurcation branches is investigated by use of the energy method.     

Elastic properties of an orthotropic binary fiber-reinforced composite with auxetic and conventional constituents

Closed-form expressions for the nine effective elastic constants of a binary fiber-reinforced composite with transversely isotropic constituents with positive (conventional) and negative (auxetic) Poisson’s ratio are considered. Such formulae were obtained by means of the asymptotic homogenization method and were verified numerically with an independent finite element model. The overall properties display explicit dependence on (i) the properties of the constituents, (ii) the volume fraction or radius of inclusion and (iii) the array periodicity. They are finally obtained by solving a normal infinite symmetric linear system of algebraic equations by truncation to a relatively small order term. This allows a fast solution and low computation cost. The overall orthotropy of the elastic properties is obtained by varying the distance between the fibers in two of the principal directions leading to different spacial aspect ratio for fiber distribution. In addition to this, an analytical relation between the effective properties based on the symmetry of the stiffness tensor is introduced. With the previous elements, we present reliable predictions for auxetic and conventional composites of this kind wherein a significant enhancement in Young’s modulus is found in a composite with an auxetic matrix reinforced by conventional fibres. Finally, we compute auxeticity windows (i.e., intervals of volume fraction where the composite is auxetic) when the fibres are auxetic. It is reported that spacial fiber aspect ratio plays a key role in the composite auxetic behavior.     

Stress-intensity factors of r-cracks in fiber-reinforced composites under thermal and mechanical loading

This paper is concerned with the problem of the calculation of stress-intensity factors at the tips of radial matrix cracks (r-cracks) in fiber-reinforced composites under thermal and/or transverse uniaxial or biaxial mechanical loading. The crack is either located in the immediate vicinity of a single fiber or it terminates at the interface between the fiber and the matrix. The problem is stated and solved numerically within the framework of linear elasticity using Erdogan's integral equation technique. It is shown that the solutions for purely thermal and purely mechanical loading can simply be superimposed in order to obtain the results of the combined loading case. Stress-intensity factors (SIFs) are calculated for various lengths and distances of the crack from the interface for each of these loading conditions. The behavior of the SIFs for cracks growing towards or away from the interface is examined. The role of the elastic mismatch between the fibers and the matrix is emphasized and studied extensively using the so-called Dundurs' parameters. It is shown that an r-crack, which is remotely located from the fiber, can either be stabilized or destabilized depending on both the elastic as well as the thermal mismatch of the fibrous composite. Furthermore, Dundurs' parameters are used to predict the exponent of the singularity of the crack tip elastic field and the behavior of the corresponding SIFs for cracks which terminate at the interface. An analytical solution for the SIFs is derived for all three loading conditions under the assumption that the elastic constants of the matrix and the fiber are equal. It is shown that the analytical solution is in good agreement with the corresponding numerical results. Moreover, another analytical solution from the literature [15], which is based upon Paris' equation for the calculation of stress-intensity factors, is compared with the numerical results and it is shown to be valid only for extremely short r-cracks touching the interface. The numerical results presented are valid for practical fiber composites with r-cracks close to or terminating at the interface provided the matrix material is brittle and the crack does not interact with other neighboring fibers. They may be applied to predict the transverse mechanical behavior of high strength fiber composites.     

Effective constitutive equations for fiber-reinforced viscoplastic composites exhibiting anisotropic hardening

Effective elastic-viscoplastic stress-strain relations are derived for fiber-reinforced composites whose constituents are elastic-viscoplastic materials displaying anisotropic hardening. The derivation is based on a recently developed high-order continuum theory with microstructure for the modeling of viscoplastic composites, and is generalized here to incorporate anisotropic hardening effects. A specific reduction of the theory gives the effective rate-dependent elastic-plastic behavior of the composite which exhibits plastic anisotropy. In the special case of perfectly elastic constituents, the approximate overall moduli of the fiber-reinforced composite are obtained. Rate-dependent average stress-strain curves are given for numerous modes of cyclic loading of the composite. The effective behavior of periodically bilaminated viscoplastic composites is determined as a special case.     

正弯矩区组合梁腹板局部稳定性研究

对正弯矩区组合梁腹板局部稳定性进行研究,首先要建立板件在拉、弯、剪共同作用下的相关屈曲方程。该文在前人研究的基础上,根据能量方法研究了四边简支板件在拉、弯、剪共同作用下的弹性相关屈曲方程,并利用通用有限元软件ANSYS建模进行计算,验证了理论推导结果,最终提出简化公式。随后在相关方程的基础上,推导出相应的弹性高厚比限值。并将该限值与未考虑弯拉作用的限值进行对比,结果有明显提高,多数情况下可放宽26%以上。该文成果为进一步研究正弯矩区组合梁腹板考虑残余应力和初始缺陷的弹塑性屈曲提供基 础。