Non-linear buckling and postbuckling behavior of thin-walled beams considering shear deformation

The static stability of thin-walled composite beams, considering shear deformation and geometrical non-linear coupling, subjected to transverse external force has been investigated in this paper. The theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field (accounting for bending and warping shear) considering moderate bending rotations and large twist. This non-linear formulation is used for analyzing the prebuckling and postbuckling behavior of simply supported, cantilever and fixed-end beams subjected to different load condition. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results show that the beam loses its stability through a stable symmetric bifurcation point and the postbuckling strength is in relation with the buckling load value. Classical predictions of lateral buckling are conservative when the prebuckling displacements are not negligible and the non-linear buckling analysis is required for reliable solutions. The analysis is supplemented by investigating the effects of the variation of load height parameter. In addition, the critical load values and postbuckling response obtained with the present beam model are compared with the results obtained with a shell finite element model (Abaqus).     

Time-varying nonlinear dynamics of a deploying piezoelectric laminated composite plate under aerodynamic force

Using Reddy’s high-order shear theory for laminated plates and Hamilton’s principle, a nonlinear partial differential equation for the dynamics of a deploying cantilevered piezoelectric laminated composite plate, under the combined action of aerodynamic load and piezoelectric excitation, is introduced. Two-degree of freedom (DOF) nonlinear dynamic models for the time-varying coefficients describing the transverse vibration of the deploying laminate under the combined actions of a first-order aerodynamic force and piezoelectric excitation were obtained by selecting a suitable time-dependent modal function satisfying the displacement boundary conditions and applying second-order discretization using the Galerkin method. Using a numerical method, the time history curves of the deploying laminate were obtained, and its nonlinear dynamic characteristics, including extension speed and different piezoelectric excitations, were studied. The results suggest that the piezoelectric excitation has a clear effect on the change of the nonlinear dynamic characteristics of such piezoelectric laminated composite plates. The nonlinear vibration of the deploying cantilevered laminate can be effectively suppressed by choosing a suitable voltage and polarity.     

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.     

Free-edge stresses in antisymmetric angle-ply laminates in extension and torsion

The first-order shear deformation theory and the layerwise theory of laminated plates are employed to analyze the edge-effect problem of an antisymmetric angle-ply laminate subjected to arbitrary combinations of extensional and torsional loads. The first-order theory is used for predicting the unknown constant parameters appearing in the reduced displacement field of elasticity which, on the other hand, signify the global behavior of the laminate. A layerwise theory is then utilized to determine the local interlaminar stresses within the boundary-layer regions of laminates. In order to closely examine the behavioral characteristics of interlaminar stresses, various numerical examples are presented for different antisymmetric angle-ply laminates under an axial force and a torque.     

Control of mechanical deformation of a laminate by piezoelectric actuator taking into account the transverse shear

Summary The control of dynamic deformation of a laminate plate is conducted by applying electrical load to a piezoelectric actuator integrated into the laminate. The dynamic behavior of the laminate is analyzed in the paper taking into account the effect of transverse shear. The analytical model of the laminate is composed of fiber-reinforced laminae and piezoelectric layers constituting a symmetric cross-ply laminate rectangular plate with simply-supported egdes. It is subjected to mechanical and electrical loads acting on the piezoelectric actuator in order to compensate the effect of the mechanical loads. The behavior of the laminate is analyzed based on the first-order shear deformation theory. Closed-form solutions are obtained for the following quantities: (1) natural frequencies of the laminate plate, (2) weight functions for the deflections and rotations and (3) transient deflections due to loads varying arbitrarily with time. Illustrative examples are shown for the control of deflections caused by the mechanical loads by means of electrical voltage applied to the piezoelectric actuators.     

SMA纤维混杂层合梁的材料阻尼

研究一类由形状记忆合金（SMA）和普通纤维混杂而成的层合梁的阻尼特性,基于最大应变能理论提出SAM混杂层合梁的等效材料阻尼预测的数学模型,其中,单层材料的弹性性能和阻尼性能分别采用多胞模型及其阻尼细观力学分析模型确定,利用正交各向异性层合梁的铁木辛柯理论分析梁的变形,通过数值算例分析了SMA含量,纤维铺设角对梁的等效阻尼比的影响。     

Effect of damping on dynamic behavior of a piezothermoelastic laminate considering the effects of transverse shear

In this paper, we analyze dynamic behavior of a piezothermoelastic laminate considering the effect of damping due to interlaminar shear and the effect of transverse shear. The analytical model is a rectangular laminate composed of fiber-reinforced laminae and piezoelectric layers. The model is assumed to be a symmetric cross-ply laminate with all egdes simply supported and to be subjected to mechanical, thermal and electrical loads varying arbitrarily with time. Behavior of the laminate is analyzed based on the first-order shear deformation theory. The effect of damping due to interlaminar shear is incorporated into our analysis by introducing the interlaminar shear stresses which satisfy the Newton’s law of viscosity. Solutions of the following quantities are obtained: (1) natural frequencies of the laminate, (2) weight functions for the deflection and rotations and (3) unsteady deflection due to loads varying arbitrarily with time. Moreover, numerical examples of the solutions are shown to examine the effects of damping and transverse shear on dynamic behavior of the laminate and how the voltage applied to the laminate decreases the deflection due to mechanical or thermal loads.     

脱层梁屈曲的高阶剪切理论

脱层的存在将会大大降低层合结构的屈曲载荷。该文将含任意位置脱层的两端固支梁分成多段子层,用厚度的三次多项式模拟脱层梁屈曲时子层的轴向位移,利用变分原理和欧拉方程导出了脱层梁的屈曲方程和定解条件,并用状态空间方法进行求解。通过与一阶剪切理论和经典理论的比较,指出了它们各自的适用范围;考虑了脱层梁三种不同的屈曲模态。分析了脱层长度、深度、位置和材料的铺层方向对脱层梁屈曲载荷的影响;最后给出了多处简单脱层的屈曲分析。     

BUCKING AND FREE VIBRATION OF PIEZOELECTRIC/PIEZOMAGNETIC LAMINATED NANO-BEAM 1)

针对压电/压磁层合纳米梁屈曲、自由振动问题，基于非局部理论与正弦剪切型变形梁理论，建立了力学模型；利用哈密顿原理推导出层合梁运动方程与边界条件；通过数值解法求得层合梁临界屈曲载荷与自由振动频率。对数值结果分析可知：磁电弹夹层对压电/压磁层合纳米梁屈曲和自由振动的影响不能忽略；磁电弹夹层中压电或压磁材料的体积分数和夹层厚度为主要影响因素；分析得到的影响规律可为此类材料在工程中的应用提供理论参考。     

A nonlinear composite beam theory

Presented here is a general theory for the three-dimensional nonlinear dynamics of elastic anisotropic initially straight beams undergoing moderate displacements and rotations. The theory fully accounts for geometric nonlinearities (large rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvature and strain-displacement expressions that contain the von Karman strains as a special case. Extensionality is included in the formulation, and transverse shear deformations are accounted for by using a third-order theory. Six third-order nonlinear partial-differential equations are derived for describing one extension, two bending, one torsion, and two shearing vibrations of composite beams. They show that laminated beams display linear elastic and nonlinear geometric couplings among all motions. The theory contains, as special cases, the Euler-Bernoulli theory, Timoshenko's beam theory, the third-order shear theory, and the von Karman type nonlinear theory.     

Non-linear free periodic vibrations of variable stiffness composite laminated plates

Steady-state free vibrations, with large amplitude displacements, of variable stiffness composite laminated plates (VSCL) are analysed. The intentions of this research are: (1)?to find out how the natural frequencies and (mode) shapes evolve with the displacement amplitude in this new type of laminated composite material; (2)?to describe modal interactions in VSCL due to energy interchanges under the coupling induced by non-linearity; (3)?to compare the VSCL with traditional, constant stiffness, laminated plates. The VSCL of interest here have curvilinear fibres and the numerical analysis carried out is based on a recently developed p-version finite element with hierarchic basis functions. The element follows first-order shear deformation theory and considers Von Kármán??s non-linear terms. The time domain equations of motion are first reduced using the linear modes of vibration and then transformed to the frequency domain via the harmonic balance method. These frequency domain equations are solved by an arc-length continuation method.     

板结构计算模型的新发展

现代复合材料层合板具有高强和轻型的突出优点，从而在军工和民用等诸多领域发挥着重要作用。这种板结构的特点是随着纤维走向的不同，层间材料的物理-力学特性发生剧烈变化。沿板厚方向变形的梯度比较陡峭，并在层间结合面处发生强不连续，呈现zig-zag （锯齿状）现象。这导致横向剪应变在板的静态和动态响应中发生重要作用，不计横向变形的经典组合板计算模型CLPT难以适应现代多层板计算分析的需要。考虑横向剪切变形影响的板的计算模型得到重视和发展。需要指出，现有各种考虑剪切变形影响的计算模型虽然有了很大的发展，但在全面和准确性上仍然存在一定的不足，难以适应现代多层组合板横向力和物理性能多变的情况。模型预测的沿板厚方向位移和应力的变化规律难以通过严格的检验。本文提出的以比例边界有限元为基础的正交各向异性板的数值计算模型，同时可适用于各种薄板与厚板的分析，对现代复合材料层合板的分析具有特殊的优越性。所得到的板的位移、正应力和剪应力沿板厚方向的变化，与三维弹性理论的标准解高度吻合。数值算例进一步表明，随着层间纤维走向的变化，板内位移场和应力场沿板厚方向剧烈变化所呈现的锯齿现象均可以精准地进行模拟。据此，本文建议方法对现代板分析的广泛适应性和高度准确性得到了充分论证。     

Generalized layerwise mechanics for the static and modal response of delaminated composite beams with active piezoelectric sensors

A coupled linear layerwise laminate theory and a beam FE are formulated for analyzing delaminated composite beams with piezoactuators and sensors. The model assumes zig-zag fields for the axial displacements and the electric potential and it treats the discontinuities in the displacement fields due to the delaminations as additional degrees of freedom. The formulation naturally includes the excitation of piezoelectric actuators, their interactions with the composite laminate, and the effect of delamination on the predicted sensory voltage. The quasistatic and modal response of laminated composite Gr/Epoxy beams with active or sensory layers having various delamination sizes is predicted. The numerical results illustrate the strong effect of delamination on the sensor voltage, on through the thickness displacement and on the stress fields. Finally, the effect of delamination on modal frequencies and shapes are predicted and compared with previously obtained experimental results.     

Creep buckling and post-buckling analysis of the laminated piezoelectric viscoelastic functionally graded plates

The creep buckling and post-buckling of the laminated piezoelectric viscoelastic functionally graded material (FGM) plates are studied in this research. Considering the transverse shear deformation and geometric nonlinearity, the Von Karman geometric relation of the laminated piezoelectric viscoelastic FGM plates with initial deflection is established. And then nonlinear creep governing equations of the laminated piezoelectric viscoelastic FGM plates subjected to an in-plane compressive load are derived on the basis of the elastic piezoelectric theory and Boltzmann superposition principle. Applying the finite difference method and the Newmark scheme, the whole problem is solved by the iterative method. In numerical examples, the effects of geometric nonlinearity, transverse shear deformation, the applied electric load, the volume fraction and the geometric parameters on the creep buckling and post-buckling of laminated piezoelectric viscoelastic FGM plates with initial deflection are investigated.     

Dynamics of the geometrically non-linear analysis of anisotropic laminated cylindrical shells

A general approach, based on shearable shell theory, to predict the influence of geometric non-linearities on the natural frequencies of an elastic anisotropic laminated cylindrical shell incorporating large displacements and rotations is presented in this paper. The effects of shear deformations and rotary inertia are taken into account in the equations of motion. The hybrid finite element approach and shearable shell theory are used to determine the shape function matrix. The analytical solution is divided into two parts. In part one, the displacement functions are obtained by the exact solution of the equilibrium equations of a cylindrical shell based on shearable shell theory instead of the usually used and more arbitrary interpolating polynomials. The mass and linear stiffness matrices are derived by exact analytical integration. In part two, the modal coefficients are obtained, using Green's exact strain-displacement relations, for these displacement functions. The second- and third-order non-linear stiffness matrices are then calculated by precise analytical integration and superimposed on the linear part of equations to establish the non-linear modal equations. Comparison with available results is satisfactorily good.     

Vibration analysis of a stepped laminated composite Timoshenko beam

The goal of this study is to investigate the vibration characteristics of a stepped laminated composite Timoshenko beam. Based on the first order shear deformation theory, flexural rigidity and transverse shearing rigidity of a laminated beam are determined. In order to account for the effect of shear deformation and rotary inertia of the stepped beam, Timoshenko beam theory is then used to deduce the frequency function. Graphs of the natural frequencies and mode shapes of a T300/970 laminated stepped beam are given, in order to illustrate the influence of step location parameter exerts on the dynamic behavior of the beam.     

压电智能复合材料层合梁的力电耦合模型及层间应力分析

由于非凡的物理性能,石墨烯纳米片(GPL)被认为是最有吸引力的复合材料增强材料之一.GPL增强材料可以明显提高聚偏氟乙烯(PVDF)压电性能和力学性能.在力电载荷作用下,对含均匀石墨烯薄片增强(GSR)智能压电复合材料层合梁层间应力预测至关重要.若对受到力电耦合作用且层与层之间材料性能突变的压电层合梁层间剪切变形预测有误,则其层间应力过大可能导致层间失效.因此,论文提出一种适于分析此类问题且满足层与层之间相容性条件的有效力电耦合模型,用于含GSR致动器的复合材料层合梁层间应力分析.应用Reissner混合变分原理(RMVT),可以提高考虑力电耦合效应的横向剪应力预测精度.三维(3D)弹性理论和所选模型计算结果将用于评估所提梁模型性能.此外,还从力电载荷、压电层厚度、石墨烯体积分数和长厚比等方面对含GSR致动器复合材料层合梁力学响应特性进行了系统的研究.     

Modified shooting approach to the non-linear periodic forced response of isotropic/composite curved beams

Large amplitude periodic forced vibration of curved beams under periodic excitation is investigated using a three-noded beam element. The element is based on the higher-order shear deformation theory satisfying interlayer continuity of displacements and transverse shear stress, and top-bottom conditions on the latter. The periodic responses are obtained using shooting technique coupled with Newmark time marching and arc length continuation algorithm developed. The second order governing differential equations of motion are solved without transforming to the first order differential equations thereby resulting in a computationally more efficient algorithm. The effects of excitation amplitude, support conditions and beam curvature on the frequency versus response amplitude relation are highlighted. The typical frequency response curves for isotropic and cross-ply laminated curved beams are presented. Phenomenon of strong modal interactions is observed.     

A unified Chebyshev–Ritz formulation for vibration analysis of composite laminated deep open shells with arbitrary boundary conditions

In this paper, a unified Chebyshev–Ritz formulation is presented to investigate the vibrations of composite laminated deep open shells with various shell curvatures and arbitrary restraints, including cylindrical, conical and spherical ones. The general first-order shear deformation shell theory is employed to include the effects of rotary inertias and shear deformation. Under the current framework, regardless of boundary conditions, each of displacements and rotations of the open shells is invariantly expressed as Chebyshev orthogonal polynomials of first kind in both directions. Then, the accurate solutions are obtained by using the Rayleigh–Ritz procedure based on the energy functional of the open shells. The convergence and accuracy of the present formulation are verified by a considerable number of convergence tests and comparisons. A variety of numerical examples are presented for the vibrations of the composite laminated deep shells with various geometric dimensions and lamination schemes. Different sets of classical constraints, elastic supports as well as their combinations are considered. These results may serve as reference data for future researches. Parametric studies are also undertaken, giving insight into the effects of elastic restraint parameters, fiber orientation, layer number, subtended angle as well as conical angle on the vibration frequencies of the composite open shells.