Analysis of plates stiffened by parallel beams

In this paper a general solution for the analysis of plates stiffened by parallel beams subjected to an arbitrary loading is presented. According to the proposed model, the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. The aforementioned integrated tractions result in the loading of the beams as well as the additional loading of the plate. Their distribution is established by applying continuity conditions in all directions at the interfaces. The analysis of both the plate and the beams is accomplished on their deformed shape taking into account second‐order effects. Six boundary value problems with respect to the plate transverse deflection, to the plate inplane displacement components, to the beam transverse deflections, to the beam axial deformation and to the beam non‐uniform angle of twist are formulated and solved using the analog equation method (AEM), a boundary element method (BEM)‐based method employing a boundary integral equation approach. The solution of the aforementioned plate and beam problems, which are non‐linearly coupled, is achieved using iterative numerical methods. The adopted model describes better the actual response of the plate beams system and permits the evaluation of the shear forces at the interfaces in both directions, the knowledge of which is very important in the design of prefabricated ribbed plates. The evaluated lateral deflections of the plate–beams system are found to exhibit considerable discrepancy from those of other models, which neglect inplane and axial forces and deformations. Copyright © 2006 John Wiley & Sons, Ltd.     

浮子卷套水箱设计及其内部运动与传热传质特性研究

许多热泵系统水箱中存在冷热水直接接触造成的不可逆热损失,为减小此项损失。设计制作了一种新型浮子卷套水箱,该水箱中的浮子卷套结构将冷热水分隔在两个腔中,避免了冷热水直接接触。实验测试了水箱内部浮子的运动特性、供热水时水温变化过程和水箱上下腔之间的传质特性;实验结果表明浮子卷套可在水箱中上下自由运动,浮子静止时漂移量小、运动时速度稳定;浮子卷套结构有效减小了水箱内部热损失;但目前设计的浮子卷套水箱上下腔之间仍存在少量传质渗漏,浮子卷套的材料和制作需进一步改进。     

Hydroelastic analysis of plates and some approximations

The paper concerns the modelling of very large pontoon-type floating structures by thin beams and plates of shallow draft, excited by regular waves. It is shown how the classical theory of hydroelasticity, involving the concepts of added mass and damping associated with the structural responses, may be reconciled with more recent formulations. In the latter, coupled equations for displacement and total hydrodynamic pressure are solved directly, without the breakdown into diffraction and radiation problems. A numerical model is adopted based on a Galerkin approach, and the nature of the various components of hydrodynamic loading on a shallow draft beam is investigated. The approach is then extended to the case of thin plate in waves, where the hydrodynamic effects are fully three dimensional.     

Nonlinear vibration of smart circular functionally graded plates coupled with piezoelectric layers

Nonlinear vibration analysis of thin circular pre-stressed functionally graded (FG) plate integrated with two uniformly distributed piezoelectric actuator layers with an initial nonlinear large deformation are presented in this paper. Nonlinear governing equations of motion are derived based on classical plate theory (CPT) with von-Karman type geometrical large nonlinear deformations. A nonlinear static problem is solved first to determine the initial stress state and pre-vibration deformations of the plate that is subjected to in-plane forces and applied actuator voltage. By adding an incremental dynamic state to the pre-vibration state, the differential equations that govern the nonlinear vibration behavior of pre-stressed piezoelectrically actuated circular FG plate are derived. An exact series expansion method is used to model the nonlinear electro-mechanical vibration behavior of the structure. Control of the FG plate’s nonlinear deflections and natural frequencies using high control voltages are studied and their nonlinear effects are evaluated. In a parametric study the emphasis is placed on investigating the effect of varying the applied actuator voltage as well as gradient index of FG plate on the dynamic characteristics of the structure.     

考虑芯层横向变形的粘弹性复合材料夹层板结构的声振特性分析

夹层板结构具有很高的比强度和比刚度。若芯层采用粘弹性阻尼材料，夹层板结构还具有良好的隔振和隔声特性，因此在工程结构中得到广泛应用。以往的夹层板理论大多忽略了芯层的横向正应变和横向正应力，在分析芯层较厚的夹层板或者夹层结构的高频振动问题时由于不能体现芯层的横向压缩变形，往往显得不够合理。针对这一不足，构造了一个复合材料夹层板单元：夹层板的上下面板采用基于一阶剪切变形理论的Mindlin假定以及层合板理论进行分析；采用文献[6，7]中提出的Timoshenko层合厚梁理论构造了单元每边的转角和剪应变场，消除了Mindlin板单元当板厚变小时的剪切锁死问题；假定芯层的位移沿厚度方向线性变化，并用上下面板的自由度表示，最终形成以上下面板自由度表示的系统总的运动方程。该单元不仅考虑了芯层的横向剪切变形，还考虑了芯层的横向压缩变形。数值计算结果表明：无论对于静力问题、动力问题还是声辐射等问题，考虑芯层的横向压缩变形是合理的，也是有必要的。     

Hydroelastic Behavior of a Floating Ring-Shaped Plate

The interaction between incident surface water waves and floating elastic plate is studied. This paper considers the diffraction of plane incident waves on a floating flexible ring-shaped plate and its response to the incident waves. An analytic and numerical study of the hydroelastic behavior of the plate is presented. An integro-differential equation is derived for the problem and an algorithm of its numerical solution is proposed. The representation of the solution as a series of Hankel functions is the key ingredient of the approach. The problem is first formulated. The main integro-differential equation is derived on the basis of the Laplace equation and thin-plate theory. The free-surface elevation, plate deflection and Green’s function are expressed in polar coordinates as superpositions of Hankel and Bessel functions, respectively. These expressions are used in a further analysis of the integro-differential equation. The problem is solved for two cases of water depth infinite and finite. For the coefficients in the case of infinite depth a set of algebraic equations is obtained, yielding an approximate solution. Then a solution is obtained for the general and most interesting case of finite water depth analogously in the seventh section. The exact solution might be approximated by taking into account a finite number of the roots of the plate dispersion relation. Also, the influence of the plate’s motion on wave propagation in the open water field and within the gap of the ring is studied. Numerical results are presented for illustrative purposes.     

C0 finite element discretization of Kirchhoff's equations of thin plate bending

An alternative formulation of Kirchhoff's equations is given which is amenable to a standard C⁰ finite element discretization. In this formulation, the potential energy of the plate is formulated entirely in terms of rotations, whereas the deflections are the outcome of a subsidiary problem. The nature of the resulting equations is such that C⁰ interpolation can be used on both rotations and deflections. In particular, general classes of triangular and quadrilateral isoparametric elements can be used in conjunction with the method. Unlike other finite element methods which are based on three-dimensional or Mindlin formulations, the present approach deals directly with Kirchhoff's equations of thin plate bending. Excellent accuracy is observed in standard numerical tests using both distorted and undistorted mesh patterns.     

Optimization of composite plates with piezoelectric stiffener-actuators under in-plane compressive loads

Stiffeners which are used to strengthen a plate can be constructed of piezoceramic materials and subsequently used as piezo actuators to improve the load carrying capacity of the plate. In the present study, a fibre composite plate with initial imperfections and under in-plane compressive loads is studied with a view towards minimizing its deflection using the piezo actuators and the fibre orientations. Piezoceramic stiffeners are bonded symmetrically on the top and bottom of the plate and deployed as actuators. Two cases of electric fields, namely, the in-phase and out-of-phase voltages are applied to the actuators. The presence of initial deflections leads to deformation under the in-plane compressive loads which should be less than the critical buckling load. Two cases of initial imperfections are considered, and the first one is the deterministic initial deflections which are known a priori and as such they are given as input parameters for the problem. In the second case the initial deflections are uncertain and they have to be obtained according to a given criterion. In the present study they are determined to produce the least favourable initial deflection (largest deflection) at a given point and the solution is obtained by convex modelling. The effect of the actuators, the ply angles and the voltage are studied and their effects on the transverse deflection are investigated. A performance index involving the L₂ norm of the deflections is minimized using the piezo effect and as well as the ply angles the optimal values of which are determined for various problem parameters.     

Springback of circular clamped armor steel plates subjected to spherical air-blast loading

The development of protective elements (plates) that are impulsively loaded requires comprehensive experiments in which the transient dynamic and residual deflections are measured. Whereas the residual plastic deflections are easily measured upon completion of the experiment, transient deflection measurements involve more elaborate setups. Relying on the residual deflection only can be misleading when one tries to assess the peak transient deflection which poses a real hazard to the protected subject. In this study, the peak transient and the residual deflections are compared for a clamped circular armor steel plate subjected to large close-range spherical air-blast loading. The difference between the maximal transient and the residual plastic deflection is addressed here as the elastic springback.     

Expansion formulae for wave structure interaction problems with applications in hydroelasticity

Alternate derivations of the expansion formulae for wave structure interaction problems are obtained in case of water of infinite depth and utilized to analyze the hydroelastic behavior of large floating structures. Considering the boundary value problem associated with Laplace equation having higher order boundary condition on the horizontal boundary and a Dirichlet type boundary condition on the vertical boundary in a quarter plane, Fourier sine transform is applied in the horizontal direction to convert the problem to a Sturm-Liouville type boundary value problem associated with non-homogeneous ordinary differential equation (ODE) in the transformed variable. Finally, inverting the transformed functions and applying the regularity criterion of the transformed function, the required expansion formula is derived. The expansion formula thus derived is extended to deal with similar boundary value problems having Neumann type boundary condition. The expansion formulae are applied to (i) analyze oblique scattering of flexural gravity waves by an articulated floating elastic plate and (ii) study the effect of compression on the oblique scattering of flexural gravity waves by a line discontinuity in a large floating ice sheet in water of infinite depth, which find applications in marine technology and arctic engineering, respectively. The present derivations of the expansion formulae are very simple and straightforward and can be easily used to study a large class of problems in the area of fluids and structures in mathematical physics and engineering.     

Thermomechanical bending analysis of functionally graded sandwich plates with both functionally graded face sheets and functionally graded cores

The bending response of functionally graded material (FGM) sandwich plates subjected to thermomechanical loads is investigated using a four-variable refined plate theory. A new type of FGM sandwich plate, namely, both FGM face sheets and an FGM hard core, is considered. Containing only four unknown functions, the governing equations are deduced based on the principle of virtual work and then these equations are solved via the Navier approach. Analytical solutions are obtained to predict the deflections and stresses of simply supported FGM sandwich plates. Benchmark comparisons of the solutions obtained for a degradation model (functionally graded face sheets and homogeneous cores) with ones computed by several other theories are conducted to verify the accuracy and efficiency of the present approach. The influences of volume fraction distribution, geometrical parameters, and thermal load on dimensionless deflections and normal and shear stresses of the FGM sandwich plates are studied.     

平面波激励矩形板辐射声计算中奇异积分处理方法研究

平面波垂直入射均匀薄板为一常见的透声问题;如果从振动声辐射角度来讲,这也是一个平面波激励板振动的声辐射问题,两种分析思路均已有经典方法可以采用,计算结果也理应一致。前者计算容易,但后者由于积分中含有奇点所以计算比较复杂。文章首先对有限薄板被激振动和声辐射的已有研究结果进行了回顾,重点处理了矩阵求解和奇点积分问题,给出了求解奇点积分的具体方法,对奇点的影响进行了分析,最后编写程序进行计算。计算结果表明,基于简正模态的计算方法在给定算例下的结果与理论分析结果接近。这表明基于振动模态分析的方法是可行的。此方法技术难度较大,但其可用于研究简支薄板的声场结构,以及曲面结构声辐射等复杂问题的求解。     

方板弹性突跳的数值模拟——第二部分:计算结果与实验结果的比较

本文的第一部分给出了分岔问题的计算方案和稳定性判据的数值方法,并对一个大型有限元程序进行了改进,本文的第二部分使用这一程序模拟方板在后屈曲过程中的弹性突跳(snap-through)现象。 对四边夹支(clamped support)边界条件下、受单向面内压力作用的方板在后屈曲过程中的力学行为进行了大范围数值追踪,确定了其中稳定的平衡状态,这些稳定的平衡状态可以模拟弹性突跳的整个过程,包括加载和卸载过程中的弹性突跳。可以解释卸载过程中第二屈曲形态跳回到第一屈曲形态后,横向挠度与加载过程中第一屈曲形态的横向挠度方向相反的现象[1]。计算结果和前人的实验结果[1]进行了比较。     

Regular wave impact onto an elastic plate

A computational analysis of an elastic plate dropped against regular long waves is presented. The problem is considered within the linear potential-flow theory. The liquid flow is two-dimensional and the plate is modelled by an Euler beam. The analysis is based on the normal-mode method with hydroelastic behavior of the plate being of main interest. Different impact conditions are considered to study the dependence of the total energy of the plate–liquid system on impact geometry and plate properties. The contributions of kinetic and potential energies to the total energy are analyzed. It is shown that the kinetic part of the system energy is small at the instant of time when bending stresses in the beam approach their maximum values. Estimations of both the total energy and the maximum of bending stresses are presented. Most of the calculations are performed for the conditions of experiments carried out in MARINTEK. A range of the problem parameters is also considered, to reveal peculiarities of the unsteady interaction between a falling elastic plate and surface waves.     

穿孔式双浮体装置水动力及波能转换特性研究

为进一步解决波能转换装置向深水环境推进过程中存在能量转换效率问题,在现有双浮体点吸式波能装置基础上考虑穿孔阻尼板,提出新的穿孔双浮体带支撑立柱的结构形式。基于线性微幅波假设,通过特征函数展开和边界匹配的势流半解析方法,并结合多自由度振动理论,探索穿孔阻尼板对获能系统水动力、运动响应及俘获宽度比的影响。计算结果表明,阻尼板开孔会降低浮子及阻尼板自身受到的波浪激励力,浮子、阻尼板以及它们之间的耦合辐射作用力会随着阻尼板开孔半径增大而减小;阻尼板的开孔半径增大能有助于浮子与阻尼板的相对响应振幅;阻尼板会使系统出现两个耦合共振频率,在较小共振频率处的最优俘获宽度比均随着阻尼板开孔半径的增加先增大后减小。研究结果可为深水波浪能利用的工程应用提供理论基础,为后续振荡浮子波浪能发电装置优化提供依据。     

Non‐linear finite element modal approach for the large amplitude free vibration of symmetric and unsymmetric composite plates

A theoretical analysis is presented for the large amplitude vibration of symmetric and unsymmetric composite plates using the non‐linear finite element modal reduction method. The problem is first reduced to a set of Duffing‐type modal equations using the finite element modal reduction method. The main advantage of the proposed approach is that no updating of the non‐linear stiffness matrices is needed. Without loss of generality, accurate frequency ratios for the fundamental mode and the higher modes of a composite plate at various values of maximum deflection are then determined by using the Runge–Kutta numerical integration scheme. The procedure for obtaining proper initial conditions for the periodic plate motions is very time consuming. Thus, an alternative scheme (the harmonic balance method) is adopted and assessed, as it was employed to formulate the large amplitude free vibration of beams in a previous study, and the results agreed well with the elliptic solution. The numerical results that are obtained with the harmonic balance method agree reasonably well with those obtained with the Runge–Kutta method. The contribution of each linear mode to the maximum deflection of a plate can also be obtained. The frequency ratios for isotropic and composite plates at various maximum deflections are presented, and convergence of frequencies with the number of finite elements, number of linear modes, and number of harmonic terms is also studied. Copyright © 2005 John Wiley & Sons, Ltd.     

高温下张弦梁结构力学特征解析解

在实腹钢梁或钢桁架下部张拉预应力高强钢索形成预应力张弦梁结构,然而钢索的预应力对温度变化敏感,因此火灾是危及张弦梁结构安全性的重要灾害,求解高温下结构的力学特征解析解是进一步研究其火灾安全性能的理论基础。考虑高温下钢材材性的衰减、非线性热应变以及几何形变的耦合,建立张弦梁结构高温下梁单元与索单元建立基本平衡微分方程,分别推导出在均布荷载作用下张弦梁结构在均匀温度场及非均匀温度场中,结构跨中挠度、预应力索水平张力和撑杆轴力的解析解;并且基于张弦梁结构力学特征解析解,对比分析了张弦梁结构热力耦合数值分析模型的可靠性,可为后继设计张弦梁受火试验方案及研究其受火破坏机理提供参考。     

Experimental and theoretical study on large ductile transverse deformations of rectangular plates subjected to shock load due to gas mixture detonation

The main purpose of this study is to show that metal plate forming by direct application of gas mixture detonation loads can be considered as an alternative high‐velocity forming method for structures instead of a conventional one. Therefore, in this investigation, a series of experimental tests have been conducted on aluminium alloy and mild steel plates with different thicknesses to examine large ductile transverse deformations of rectangular plates with clamped edge conditions subjected to gas mixture detonation loading. The main aim of the experimental section is to investigate the effects of predetonation pressures of acetylene (C₂H₂) and oxygen (O₂) gasses and different mixture ratios on the dynamic response of specimens. The permanent deflections of plates have widely varied from 21.66 up to 56.31 mm. In theoretical analysis, according to an upper bound solution and energy method, theoretical models have been presented by assuming a zero‐order Bessel function of the first kind in the x and y directions for a transverse displacement profile to predict permanent deflections. To account for material strain rate sensitivity, a Cowper–Symonds model has been used, whereas the material coefficients of this equation are constant values or functions of plate thickness. A comparison of the present models with Jones' theoretical model shows that a good agreement with experimental results can be obtained when constant values are used for material coefficients in the Cowper–Symonds equation.     

外包钢板-高延性混凝土组合连梁抗震性能试验研究

为改善小跨高比连梁的抗震性能,采用高延性混凝土（HDC）代替混凝土,设计了2个外包钢板-HDC组合连梁、1个外包钢板-混凝土组合连梁和1个内置钢板-HDC组合连梁试件。通过拟静力试验,研究其破坏形态、变形能力及耗能能力。试验结果表明:采用HDC代替混凝土可提高外包钢板组合连梁的变形能力和耐损伤能力;HDC与钢腹板的协同工作性能较好,有利于钢腹板抗剪作用的发挥;外包钢板-HDC组合连梁的耗能能力明显高于外包钢板-混凝土组合连梁和内置钢板组合连梁;外包钢板-HDC组合连梁的剪压比设计值为0.65~0.70,其剪压比明显高于内置钢板组合连梁。因此,采用外包钢板-HDC组合连梁,可提高小跨高比连梁的剪压比限值,解决连梁设计中剪压比超限的问题。     

应用移动质量法与分形维数实现梁损伤检测

以简支裂纹梁为例,利用移动质量法结合分形维数方法实现结构损伤检测。由于质量块在振动梁不同位置时对梁固有频率的影响不同,所以当质量块沿着梁长度方向移动时可以得到一组梁－质量块系统的固有频率曲线,通过分析这组固有频率的分形维数曲线确定梁的损伤位置及其损伤程度。随后讨论附加质量块大小和分形方法参数选择对损伤检测精确性的影响。数值计算结果表明,该方法能准确的识别裂纹位置和损伤程度。该方法的主要优点在于不需要结构的模态信息,并可以在随机激励下实现。