Optimal Design of Thin-Walled Cylinders of Variable Wall-Thickness Subject to Flexural and Torsional Stiffness Constraints

ABSTRACT

This paper considers the problem of determining center-line shape and wall-thickness distribution of a thin-walled cylinder of given center-line length that uses the minimum possible amount of material to achieve prescribed minimum stiffnesses in torsion and bending in a given plane. Necessary optimality conditions are derived and the solution is found partly in closed form and partly by numerical methods. Optimal solutions are presented for various stiffness ratios and compared with other admissible designs.     

Flexural testing of an epoxy oversized strand model under traction

Experimental flexural tests are conducted on an epoxy oversized 1×6 strand; the objectives of these tests are to measure the bending stiffness and the associated strains. Three sets of boundary conditions are applied. The bending stiffness is deduced from the transverse midspan deflection and is found to be dependent on the axial force and the moment distribution. The deduced bending stiffness is compared to values computed from the theoretical bending stiffnesses.     

Optimum Design of Multistory Multispan Frames for Prescribed Elastic Compliance∗

This paper presents an analytical method for minimum cost design of regular rectangular building frames for constrained elastic compliance. The method, consisting of a semi-inverse method and a design region extension procedure, is illustrated by two classes of exact solutions. It is shown that the relative story displacements are almost uniformly distributed in an optimally designed frame with almost uniform story heights and that the solutions enable one to calculate all maximum member-end stresses from member-end curvatures. It is suggested that the proposed design formulas may be utilized for design problems, subject to relative story displacement and stress constraints. Since these solutions are shown to be dependent upon the nature of the prescribed minimum stiffnesses, a method of finding a practically reasonable set of minimum stiffnesses is also presented.     

面内变刚度薄板弯曲问题的挠度?弯矩耦合神经网络方法

发展了一种求解面内变刚度功能梯度薄板弯曲问题的神经网络方法. 面内变刚度薄板弯曲问题的偏微分控制方程为一复杂的4阶偏微分方程, 传统的基于强形式的神经网络解法在求解该偏微分方程时可能会遇到难以收敛、边界条件难以处理的情况. 本文基于Kirchhoff薄板弯曲理论, 提出了一种直角坐标系下任意面内变刚度薄板弯曲问题的神经网络解法. 神经网络模型包含挠度网络与弯矩网络, 分别用于预测薄板的挠度与弯矩, 从而将求解4阶偏微分方程转换为求解一系列二阶偏微分方程组, 通过对挠度、弯矩试函数的构造可使得神经网络计算结果严格满足边界条件. 在误差的反向传播中, 根据本文提出的误差函数公式计算训练误差并结合Adam优化算法更新模型的内部参数. 求解了不同边界条件、形状的面内变刚度薄板弯曲问题, 并将所得计算结果与理论解、有限元解进行对比. 研究表明, 本文模型对于求解面内变刚度薄板弯曲问题具备适应性, 虽然模型中的弯矩网络收敛较挠度网络要慢, 但本文方法在试函数的构造上更为简单、适应性更强.      

On the influence of interfacial properties to the bending rigidity of layered structures

Layered structures are ubiquitous, from one-atom thick layers in two-dimensional materials, to nanoscale lipid bi-layers, and to micro and millimeter thick layers in composites. The mechanical behavior of layered structures heavily depends on the interfacial properties and is of great interest in engineering practice. In this work, we give an analytical solution of the bending rigidity of bilayered structures as a function of the interfacial shear strength. Our results show that while the critical bending stiffness when the interface starts to slide plastically is proportional to the interfacial shear strength, there is a strong nonlinearity between the rigidity and the applied bending after interfacial plastic shearing. We further give semi-analytical solutions to the bending of bilayers when both interfacial shearing and pre-existing crack are present in the interface of rectangular and circular bilayers. The analytical solutions are validated by using finite element simulations. Our analysis suggests that interfacial shearing resistance, interfacial stiffness and preexisting cracks dramatically influence the bending rigidity of bilayers. The results can be utilized to understand the significant stiffness difference in typical biostructures and novel materials, and may also be used for non-destructive detection of interfacial crack in composites when stiffness can be probed through vibration techniques.     

General Anisotropy Identification of Paperboard with Virtual Fields Method

This work extends previous efforts in plate bending of Virtual Fields Method (VFM) parameter identification to include a general 2-D anisotropic material. Such an extension was needed for instances in which material principal directions are unknown or when specimen orientation is not aligned with material principal directions. A new fixture with a multi-axial force configuration is introduced to provide full-field strain data for identification of the six anisotropic stiffnesses. Two paper materials were tested and their Q _ij compared favorably with those determined by ultrasonic and tensile tests. Accuracy of VFM identification was also quantified by variance of stiffnesses. The load fixture and VFM provide an alternative stiffness identification tool for a wide variety of thin materials to more accurately determine Q ₁₂ and Q ₆₆.     

The Explicit Expression of Internal Forces in Prismatic Membered Structures*

ABSTRACT

Analytic expressions for member forces in linear elastic redundant trusses have recently been given by the author. It was shown that the internal forces in a truss are the ratios of two multilinear homogeneous polynomials in the longitudinal stiffnesses of the elements of the structure. The order of the polynomials is equal to the number of nodal degrees of freedom of the structure. The number of terms of each polynomial is equal to the number of statically determinate stable substructures that can be derived from the original structure. It was shown that coefficients of the polynomials can be computed through the equilibrium equations and by enforcing global compatibility of deformations. This paper generalizes these results to the case of linear elastic structures, composed of uniform prismatic elements that have extensional, flexural, and torsional stiffness. This is done by replacing each bi-modal bending element with a unimodal moment element and a unimodal shear element. This allows the representation of deformation of the elements by six uncoupled basic deformation patterns in the case of structures in space, thus paving the way for truss-type analysis for general structures that are composed of uniform prismatic elements. As a result, multilinear polynomials that appear in expressions for stress resultants are the products of axial, bending, and torsional stiffnesses of subsets of the original structure. The number of terms appearing in the polynomials renders the exact analytic expressions intractable for practical engineering structures. However, the construct of the analytic equations may constitute a basis for writing approximate expressions for member forces in frames, explicitly in terms of rigidities of components of the structure. This paper describes such an approximate expression, with a reduced number of terms in the polynomials. The theory is illustrated with numerical examples of analysis of planar structures     

Axially compressed buckling of pressured multiwall carbon nanotubes

This paper studies axially compressed buckling of an individual multiwall carbon nanotube subjected to an internal or external radial pressure. The emphasis is placed on new physical phenomena due to combined axial stress and radial pressure. According to the radius-to-thickness ratio, multiwall carbon nanotubes discussed here are classified into three types: thin, thick, and (almost) solid. The critical axial stress and the buckling mode are calculated for various radial pressures, with detailed comparison to the classic results of singlelayer elastic shells under combined loadings. It is shown that the buckling mode associated with the minimum axial stress is determined uniquely for multiwall carbon nanotubes under combined axial stress and radial pressure, while it is not unique under pure axial stress. In particular, a thin N-wall nanotube (defined by the radius-to-thickness ratio larger than 5) is shown to be approximately equivalent to a single layer elastic shell whose effective bending stiffness and thickness are N times the effective bending stiffness and thickness of singlewall carbon nanotubes. Based on this result, an approximate method is suggested to substitute a multiwall nanotube of many layers by a multilayer elastic shell of fewer layers with acceptable relative errors. Especially, the present results show that the predicted increase of the critical axial stress due to an internal radial pressure appears to be in qualitative agreement with some known results for filled singlewall carbon nanotubes obtained by molecular dynamics simulations.     

PROPERTY DEGRADATION OF ANISOTROPIC COMPOSITE LAMINATES WITH MATRIX CRACKING (I)-CONSTITUTIVE RELATION DEVELOPING FOR (θ_m/90_n)_s CRACKED LAMINATES BY STIFFNESS PARTITION

The study on property degradation of damaged composite laminates is extendedto anisotropic laminates with matrix cracking. In (I) of the paper, an idea of "stiffnesspatition" is proposed to deal with the puzzle that the in-plane normal response iscoupled with the shear response of the laminates. For (θ_m/90_n.), laminates containingtransversely cracked layers under general in-plane loading, the constitutive relationsare derived and the effective stiffnesses are expressed as the function of crack density.     

Limit models in the analysis of three different layered elastic strips

The mechanical behavior of three types of laminated strips is investigated. They are made of three layers filled with homogeneous, isotropic and elastic materials; the upper and lower layer are called adherents, the middle layer is called adhesive. The first model studies a strip consisting of three layers made of materials with similar stiffness; the second one concerns with a strip in which the adhesive is soft; in particular, we suppose that the elastic stiffness of the middle layer is two orders of magnitude smaller than that of the upper and lower layers; the third case is a strip in which the core is thinner and stiffer than the two adherents: the elastic modula of the adherents are one order of magnitude bigger that those of the adhesive. After identifying a parameter of smallness ε (which measures the thickness and the stiffness of each layer), the limit of the solution when ε tends to zero has been considered. Afterwards, it has been shown that each solution of the simplified models verifies the so-called limit problems, written using a “weak” and a “strong” formulation. The existence and uniqueness of the solutions of each limit problem have been established. The strong convergence of the exact solutions towards the solution of the limit problem of the first model has been established, too.     

反对称斜交叠层板双向弯曲问题的横向基理论解

研究反对称斜交叠层板的双向弯曲问题。根据横向基理论，在厚度方向，给出了三个方向位移沿厚度的分布规律；在面内方向，采用升阶谱位移函数；然后，应用最小势能原理导出叠层板的静力控制方程。通过对多种层数和铺层角的层板力学响应计算分析表明：在典型位置的应力分布，本文解与三维弹性解符合很好，并有较好的收敛性。     

Elastic waves guided by a material interface

The propagation of interfacial small-amplitude waves along a rectilinear thin film separating two pre-stressed, incompressible, elastic media is addressed. The film is modelled as a material surface possessing its own mass density and normal and flexural stiffnesses. It is shown that these features induce dispersion as the obtained secular equations are polynomials of the second degree in the wavenumber when bending stiffness is absent (membrane-like interface), and of the fourth degree otherwise (plate-like interface). In both case, beyond the modified Stoneley mode, a bending mode for the interface, an additional propagating wave can exist, with amplitude polarized along the interface (extensional mode). The associated bifurcation problem is analyzed with focus on the effects of compressive residual forces at the interface. The buckling strain of a compressed metal layer embedded in an elastomeric medium is computed also with an exact approach, to provide the range of validity of the proposed simplified model of material interface.     

The in-plane stiffnesses of a honeycomb core including the thickness effect

Summary The subject of the consideration is the contribution of a regular honeycomb core to the effective in-plane stiffnesses of a sandwich structure. Due to the coupling of the core displacements with those of the sandwich face sheets, the stiffness contribution of the core is not proportional to its total thickness, as could be expected. The corresponding thickness effect is investigated by means of an appropriate closed-form approach. In doing so, the total elastic core strain energy is calculated based on an adequately chosen displacement representation. Further on, the resultant effective stiffnesses are derived as a function of the total core thickness. A comparative computation of the effective stiffnesses by finite element analysis gives good agreement. Received 5 August 1997; accepted for publication 23 October 1997     

Optimal Elastic Design for Prescribed Maximum Deflection

ABSTRACT

For the problems of the optimal elastic design with prescribed maximum deflection, a variational formulation is proposed, with reference to the one-or two-dimensional bending structures.

Necessary optimum conditions are found, and the physical features of the optimal solutions are discussed for the “absolute” minimum cost problems, and, when dealing with beams, for the solutions with piece-wise constant design function.

Some examples are solved by using numerical methods that are directly derived from the variational formulation.     

The mechanics of pyramids

The elastostatic fields and free-vibration characteristics are studied for a class of continuous solids in the shape of homogeneous or layered pentahedral pyramids. The pyramids possess an arbitrary number of linear elastic layers containing dissimilar elastic constants. The layers are assumed to be perfectly bonded in the out-of-plane direction of the pyramid. The change in elastic stiffness across each interface requires a model that allows for a jump in displacement gradient at these location. A discrete-layer representation is used that combines the Ritz method with polynomial in-plane approximations with one-dimensional Lagrangian interpolation polynomials in the thickness direction. The free-vibration characteristics are examined for a variety of isotropic and anisotropic materials and representative bulk stiffness estimates are given for homogeneous pyramids under static deformation.     

零泊松比手风琴蜂窝等效模量

柔性蒙皮是变形机翼和风力机叶片的关键组成部分。一维变形的柔性蒙皮不仅要求其支撑结构具有良好的面内变形和面外承载能力,还需要具有零泊松比特性。手风琴蜂窝具有零泊松比特性,可用作一维变形柔性蒙皮支撑。为全面分析其面内外弹性变形特性,综合考虑结构的内力弯矩、轴力和剪力,通过卡氏第二定理对其x向等效弹性模量和x-y面内等效剪切模量进行了推导;利用最小余能原理和最小势能原理确定了x-z面内的等效剪切模量;此外还推导了其y和z向的等效弹性模量以及y-z面内的等效剪切模量;然后通过ANSYS有限元仿真对等效模量理论公式进行了验证;最后将本文理论模型与现有模型进行了比较。结果表明,理论公式和有限元仿真吻合较好,在结构设计时采用较大的斜梁高度系数h和斜梁间距系数g,较小的厚度系数t以及较大的竖直梁厚度系数η,有望获得具有较小面内刚度和较大面外刚度的手风琴蜂窝结构。该结果可用于手风琴蜂窝面内外等效模量的快速预测,为一维变形柔性蒙皮的结构设计提供相应的参考。此外,本文理论模型相比传统模型更为精确且具有更加广泛的应用范围。     

硬夹心矩形夹层板的整体稳定性分析

摘要：本文在Reissner型理论给出的位移模式基础上,修正其软夹心假设,考虑夹心层面内刚度,给出了硬夹心夹层板的几何方程、物理方程,建立了硬夹心夹层板结构在面内纵向载荷作用下的平衡微分方程,并对方程进行了简化,通过理论计算得到了四边简支条件下硬夹心矩形夹层板整体失稳临界载荷的解析解,并分别计算了夹心层材料的弹性模量 、厚度 、泊松比 对硬夹心夹层板临界载荷的影响,结果证明,对于硬夹心夹层结构,夹心层面内刚度对硬夹心夹层板整体失稳临界载荷的影响较大,考虑其面内刚度是必要的。     

层状地基任意形状刚性基础动力响应求解

提出了基于积分变换、对偶方程与精细积分算法求解多层地基任意形状刚性基础的动力刚度问题. 首先在频率波数域内圆柱坐标体系中利用圆形微元的对称与反对称特性建立多层地基中格林影响函数的波动方程,然后将应力和位移关系表示成对偶形式进行精细积分求解以提高计算精度和稳定性. 再将任意形状刚性基础与地基的交界面离散化为一系列圆形微元,利用格林影响函数建立其平动与转动动力刚度的矩阵方程. 该求解方法高效、准确并且计算稳定,适于任意复杂多层地基任意形状基础动力刚度的计算.      

高强钢波纹夹芯结构的力学性能研究

本文利用ABAQUS仿真软件分析了高强钢波纹夹芯结构在三点弯曲以及面内压缩时的力学响应。将计算结果与实验值和理论值进行了对比,验证了数值模拟的准确性。分析了芯板材料属性（高强钢DP900和普通低碳钢DC01）以及波纹芯板排列方向(横向、纵向)对夹芯结构力学性能的影响。结果表明DP900高强钢波纹芯板夹芯结构抗弯强度是DC01低碳钢夹芯结构的2.39倍,抗压强度是低碳钢夹芯结构的1.40倍,纵向波纹夹芯结构比横向波纹夹芯结构弯曲刚度高11.63倍。     

Derivation of equations for laminate plates and shells

Summary The basic principles of reducing the three-dimensional equations for laminate plates and shells to two-dimensional ones have been analyzed here. The equations for laminate plates and shells have been derived on the basis of the concept of a uniform stress-strain state of a thin-walled element, and without involvement of kinematic hypotheses. Two possible approaches have been considered, one without and one with the transverse shearing stress taken into account. In the first case the resulting equations are the classical equations based on the Kirchhoff-Love hypothesis., In the second case the resulting equations are more general than equations of the Timoshenko kind. Referred elasticity constants are defined which characterize the stiffnesses, longitudinal, flexural, and tangential of a stack of layers. It has been found that the correction introduced by taking into account the transverse shearing stressesis determined by the ratio of longitudinal stiffness to shear modulus as well as by the ratio of longitudinal to transverse stiffness.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 14, No. 10, pp. 3–21, October, 1978.