Buckling of sandwich beams with compliant interfaces

Buckling of elastic sandwich beams is analyzed accounting for the compliance of the interfaces between the skin and core. A relation between tractions and displacement jumps across the interfaces characterizes the interfacial compliance. Timoshenko co-rotational beam elements are used to discretize each layer of the sandwich. The dependence of the bifurcation load on the stiffness of the core and on the interfacial compliance are illustrated by considering examples of a sandwich beam with two sets of boundary conditions. It is shown that the load at bifurcation buckling is sensitive to the compliance of the interfaces and that a sufficiently large interfacial compliance can significantly decrease the bifurcation load.     

夹层板系统碰撞性能数值仿真分析技术

为促进夹层板系统（Sandwich Plate System,SPS）在船舶耐撞结构设计中的应用,用Abaqus分析SPS在碰撞载荷下的数值模型化技术,包括夹芯层和面板的建模方式、连接形式和网格尺寸．根据该技术研究SPS在碰撞冲击载荷作用下的力学性能,如结构损伤变形、碰撞力和结构吸能等．结果表明,SPS建模采用壳一体混合模型（即上、下面板采用壳单元,夹芯层采用体单元）较合理;夹芯层与面板之间采用绑定连接较合理;SPS具有良好的耐撞性能．     

Natural frequencies of vibration of cantilever sandwich beams

Theoretical and experimental studies were made in obtaining the natural frequencies of cantilever sandwich beams subjected to only gravity forces. The method of minimizing the total energy of the system was used for determining the frequencies. A vibration system made by Unholtz-Dickie was utilized to set the beam in vibration. Resonance occurred when the frequency of the shaker coincided with the natural frequency of the beam. The resonance frequencies were measured by transducers mounted at various locations on the beam. A total of sixteen beams of various lengths, thickness and core density were tested.

It was found that the natural frequency of a cantilever sandwich beam depends largely upon the thickness, length, core density and stiffness of the beam. In addition, the natural frequency has a nonlinear variation with the mode and for any particular mode, the value of the frequency increases as the length of the beam decreases.

Design factors were developed based upon the ratios of the theoretical frequencies of homogeneous beams having the same thicknesses and stiffnesses of that of sandwich beams and of the frequencies experimentally determined for similar sandwich beams.     

Minimum material cost design of five-layer sandwich beams

Welded aluminium box beams have very low vibration damping capacity. Regarding damping it is better to use a three-layer beam constructed from two rectangular hollow sections and a rubber layer glued between them. These three-layer beams have good damping capacity, but the dynamic deflection is large due to shear deformation of the rubber layer. To decrease this deflection two fiber-reinforced plastic layers are used. The minimum material cost design is worked out for such five-layer sandwich beams using the Rosenbrock Hillclimb mathematical programming method. Constraints on stress and local buckling are considered. A comparison is made between the optimized versions of a simple welded aluminium beam, three-layer and five-layer beams. It is shown that the deflection of the five-layer beams is smaller than that of the three-layer ones, but the cost of the five-layer beams is greater.     

Optimal design of box-section sandwich beams subject to moving load

Minimum weight designs of sandwich beams subject to moving load are performed as a function of prescribed load index, where the stiffness and strength constraints are both considered. The dynamic properties of sandwich beams are detected and validated through finite element simulation. Five different topologies are optimized and compared. Numerical results show that the topology of a hollow tube with sandwich walls comprising a lattice tetrahedral truss core is the most weight efficient, whereas the topology of a hollow tube with monolithic walls and a foam-filled core is the heaviest. The failure mechanisms that govern the optimal designs are also evaluated. The optimal results exhibit great sensitivity to the load moving velocity when it approaches the critical velocity of the sandwich beams. The optimization for sandwich beams under three-point bending is implemented for comparison and special dynamic characteristics of sandwich beams subject to moving load are discovered.     

Stiffness matrix for sandwich beams with thick anisotropic laminated faces

A finite element capability is described for the analysis of sandwich beams with thick unbalanced laminated faces. Particular attention is focused on the effects of bending-membrane coupling in the faces. The stiffness matrix is developed using displacement functions generated from explicit solution of the governing differential equations.     

An accurate approximate formula for the natural frequencies of sandwich beams

The flexural beam approximation commonly used for evaluating the vibration frequencies of sandwich beams is shown to be of limited applicability. A formula proposed for evaluating the fundamental frequencies of cantilever sandwich beams is generalised for higher modes and other boundary conditions, and is shown to be in very good agreement with experimental and theoretical values.     

Application of folded plate analysis to bending,buckling and vibration of multilayer orthotropic sandwich beams and panels

A unified analysis method based on two-dimensional elasticity theory is outlined for evaluation of bending, buckling and vibration of multilayer orthotropic sandwich beams and panels. The effects of initial geometric imperfections are included. It is shown that beams or panels deforming under conditions of plain stress or plane strain may be treated as special instances of folded-plate structures using computer programs which are now widely available. Examples are given, including evaluation of stress contours in a sandwich panel under patch load and analysis of overall and local (face-wrinkling) buckling modes in sandwich panels with stiff and soft cores.     

A discrete layer beam finite element for the dynamic analysis of composite sandwich beams with integral damping layers

A discrete layer finite element is presented for the dynamic analysis of laminated beams. The element uses C⁰ continuous linear and quadratic polynominals to interpolate the in-plane and transverse displacement field, respectively, and is free from the effects of shear locking. Modal frequencies and damping are estimated using both the modal strain energy method and the complex modulus method. A forced response version of the model is also presented. The model predictions are compared with experimental data for composite sandwich beams with integral damping layers. Four damping configurations are considered, a constrained layer treatment, a segmented constrained layer treatment and two internal treatments.     

An optimal control approach to the design of vibrating elastic-viscoelastic sandwich beams

This work features the application of an optimal control algorithm to a new class of continuous one-dimensional structural design problems. A sandwich beam of rectangular cross-section is considered. It has a variable-thickness core and two equal variable-thickness cover layers and is subjected to harmonic forced vibrations. The objective is to distribute both the core and layer mass so as to minimize a measure of dynamic compliance for forced steady-state vibration and fixed material volumes. Either or both materials may be viscoelastic. Any constitutive relation may be used provided it is linear and time-invariant.The design problem is formulated as an optimal control problem. The resulting problem contains ten state variables, two control functions, four control parameters, and six terminal state constraints. Simple transformations are used to treat the minimum-gage constraints. A conjugate gradient/gradient projection optimal control algorithm is then used to obtain numerical solutions. Several optimal beam designs are presented and compared for a variety of problem parameter values.     

折叠式夹层结构压皱性能数值仿真

为研究网格尺寸、加载速率及仿真试件的单元数和单元长度等对夹层板压皱性能的影响,用Abaqus对U-I形折叠式夹层板的压皱性能进行数值仿真,提出精度高、效率高的数值仿真模型化技术,分析不同变形模式和夹层结构尺寸对压皱性能的影响.结果表明:变形模式是决定夹层板压皱性能的关键因素;不同的结构参数组合会形成不同的变形模式,进而影响结构的压皱载荷和吸能特性;存在较优的夹芯壁厚、高度、间距和夹角的尺寸组合使夹层结构的吸能效率较优.     

Numerical paraxial approximation for highly relativistic beams

This paper is concerned with the numerical simulation of highly relativistic beams. We developed a particle-in-cell code for highly relativistic beams, based on the paraxial Vlasov-Maxwell formulation of Laval et al. This formulation follows the beam in a speed-of-light frame. It is fourth order accurate in the small characteristic velocity of the beam. The formulation is simpler than standard particle-in-cell codes in the lab frame or in the beam frame and gives a fast and easy to implement algorithm. Numerical examples illustrate the accuracy of the method.     

旋转粘弹性夹层梁非线性自由振动特性研究

对旋转粘弹性夹层梁的非线性自由振动特性进行了分析.基于Kelvin-Voigt粘弹性本构关系和大挠度理论,建立了旋转粘弹性夹层梁的非线性自由振动方程,并使用Galerkin法将偏微分形式振动方程化为常微分振动方程.采用多重尺度法对非线性常微分振动方程进行求解,通过小参数同次幂系数相等获得微分方程组,并通过求解方程组及消除久期项来获得旋转粘弹性夹层梁非线性自由振动的一次近似解.用数值方法讨论了粘弹性夹层厚度、转速和轮毂半径对梁固有频率的影响.结果表明:固有频率随转速增大而增大,随夹层厚度增大而减小,随轮毂半径的增大而增大.     

Refined sandwich model for the vibration of beams with embedded shear piezoelectric actuators and sensors

This work extends a previously presented refined sandwich beam finite element (FE) model to vibration analysis, including dynamic piezoelectric actuation and sensing. The mechanical model is a refinement of the classical sandwich theory (CST), for which the core is modelled with a third-order shear deformation theory (TSDT). The FE model is developed considering, through the beam length, electrically: constant voltage for piezoelectric layers and quadratic third-order variable of the electric potential in the core, while mechanically: linear axial displacement, quadratic bending rotation of the core and cubic transverse displacement of the sandwich beam. Despite the refinement of mechanical and electric behaviours of the piezoelectric core, the model leads to the same number of degrees of freedom as the previous CST one due to a two-step static condensation of the internal dof (bending rotation and core electric potential third-order variable). The results obtained with the proposed FE model are compared to available numerical, analytical and experimental ones. Results confirm that the TSDT and the induced cubic electric potential yield an extra stiffness to the sandwich beam.     

Nonlinear dynamic analysis of curved beams

A computational procedure is presented for predicting the dynamic response of curved beams with geometric nonlinearities. A mixed formulation is used with the fundamental unknowns consisting of stress resultants, generalized displacements and velocity components. The governing semidiscrete finite element equations consist of a mixed system of algebraic and differential equations. The temporal integration of the differential equations is performed by using an explicit half-station central difference method. A procedure is outlined for lumping both the flexibilities and masses of the mixed model, thereby uncoupling all the equations of the system. The advantages of the proposed computational procedure over explicit methods used with the displacement formulation are discussed. The effectiveness and versatility of the proposed approach are demonstrated by means of numerical examples.     

变厚度智能硬夹心板振动分析

研究由形状记忆合金与普通钢材制成的硬夹心板的振动控制方法,求出硬夹心夹层板的平衡方程,并分析了变厚度智能夹心板的振动问题.算例表明该方法对于夹心板的振动能够有效地控制.     

Numerical modeling of softening hinges in thin Euler–Bernoulli beams

This paper presents new finite elements for thin Euler–Bernoulli beams that incorporate the softening hinges observed at failure. The proposed methods rely crucially on the identification of the classical notion of inelastic hinge with strong discontinuities of the generalized displacements describing the beam’s deformation. The development of a multi-scale framework that effectively incorporates the localized dissipative mechanisms associated with these discontinuous solutions into the large-scale problem of the beam, or general frame system, defines a crucial step undertaken here. This framework defines the setting of its numerical implementation by finite elements enhanced with the singular strains corresponding to the discontinuities. A general procedure is presented that leads, in particular, to finite elements free of stress-locking and that resolve exactly the kinematics of the hinge. Several numerical simulations are presented to illustrate the performance of the proposed methods.     

Stiffness analysis of locally-buckled thin-walled continuous beams

A direct iterative numerical method is presented for predicting the post-local-buckling response of thin-walled continuous structures. Nonlinearities due to local buckling and non-linear material properties are accounted for by the nonlinear moment-curvature relations of the section derived with the aid of effective width concept. Since the effective width of the compression element decreases as the stress borne by the element edge increases, the effective flexural rigidity of the cross-section varies along the member length depending upon the magnitude of the moment at the section. In the post-buckling range, the member is treated as a nonprismatic section. For continuous thin-walled structures, it is further complicated by the fact that the bending moment distribution throughout the structure and the member stiffnesses are interdependent. The proposed direct iterative solution scheme includes a stiffness matrix method of analysis in conjunction with a numerical integration procedure for evaluating the member stiffnesses. The method is employed to analyze continuous beams in the post-buckling range. Using the moment distribution of an elastic prismatic continuous beam based on the nonbuckling analysis as a first approximation, it has been found that the iterative solution scheme converges rapidly.An excellent agreement has been obtained between the results based on the method presented and from an earlier study for continuous beams. The stiffness formulation is direct and is well suited for the analysis of continuous thin-walled structures.     

Nonlinear finite element analysis of curved beams

Mixed curved-beam finite elements are developed for the geometrically nonlinear analysis of deep arches. The analytical formulation is based on a form of the nonlinear deep-arch theory with the effects of transverse shear deformation and bending-extensional coupling included. The fundamental unknowns consist of the six internal forces and generalized displacements of the arch. The generalized stiffness matrix is obtained by using a modified form of the Hellinger-Reissner mixed variational principle. Numerical studies are presented to demonstrate the high accuracy of the solutions obtained by the mixed models and to show that their performance is considerably less sensitive to variations in the arch geometry than that of the displacement models.