Performance of a fiber reinforced polymer web core skew bridge superstructure. Part I: field testing and finite element simulations

An analytical method is presented to investigate hydrothermal effects on locally buckling for an elliptical delamination near the surface of cylindrical laminated shells. The critical strain of non-linear buckling for a locally elliptic delamination of cylindrical laminated shells is obtained by considering transverse displacements of the elliptically sub-laminated shells. The stacking sequence of sub-laminated shells may be multilayer and asymmetric. The geometrical axis of sub-laminated shells may be arbitrary. The Young's modulus and the thermal and humidity expansion coefficients of the material are considered as functions of temperature change in base-laminated shells. The critical strains of locally buckling for cylindrical laminated shells subjected to hydrothermal effects are extracted for different stacking sequences, and different radii of base-laminated shells, by applying Rayleigh–Ritz method based on second variation of potential energy. From results carried out, it is found that the critical strain from the non-linear buckling for a locally elliptic delamination near the surface of a cylindrical laminated shell presents a lower value than that from linear considerations.     

Research on nonlinear torsional buckling and post-buckling of eccentrically stiffened functionally graded thin circular cylindrical shells

This paper is presented to solve the nonlinear buckling and post-buckling problem of functionally graded stiffened thin circular cylindrical shells only under torsion by the analytical approach. The shells are reinforced by rings and stringers attached to their inside and the material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. Theoretical formulations based on the smeared stiffeners technique and the classical shell theory with the geometrical nonlinearity in von Karman sense are derived. Approximate three-term solution of deflection is chosen more correctly and the explicit expression to finding critical load and post-buckling torsional load–deflection curves are given. The effects of various parameters and the effectiveness of stiffeners on the stability of shell are shown.     

受轴压黏弹性层合圆柱壳的极值点蠕变失稳

基于Donnell薄壳理论, 采用准弹性方法, 分析了含初始几何缺陷黏弹性层合圆柱壳极值点形式的延迟失稳特征。由轴向缩短量的突然增加定义失稳临界时间, 对玻璃纤维/环氧树脂层合圆柱壳进行了数值计算。结果表明: 圆柱壳存在临界时间趋于无穷的持久临界荷载; 表征延迟失稳程度的瞬时弹性临界荷载与持久临界荷载之差值随着初始几何缺陷的增加而减小; 边界条件、 铺设方式对延迟失稳的影响机制可通过对应弹性层合圆柱壳的缺陷敏感性分析来考察。      

Elliptically delaminated buckling near the surface of piezoelectric laminated shells under electric and thermal loads

This paper reports the result of an investigation into the local buckling for the delimitation of an elliptic shape near the surface of piezoelectric laminated shells, where local delaminated sub-shells may be monolayer and multiplayer, and the geometrical axes of elliptically delaminated sub-shells are inconsistent with the geometrical main axes of piezoelectric laminated base-shells. From example calculations, the effects of the geometrical and physical parameters, stacking models of piezoelectric laminated base-shells, elliptically laminated sub-shells, and coupled electric and thermal loads on the local buckling for elliptically delaminated sub-shells near the surface of piezoelectric laminated base-shells are described and discussed.     

On the linear stability of eccentrically stiffened functionally graded annular spherical shell on elastic foundations

The study deals with the formulation of governing equations of eccentrically stiffened functionally graded materials annular spherical shells resting on elastic foundations and based upon the classical shell theory and the smeared stiffeners technique taking into account geometrical nonlinearity in Von Karman-Donnell sense. The annular spherical shells are reinforced by eccentrically longitudinal and transversal stiffeners made of full metal or full ceramic depending on situation of stiffeners at metal-rich side or ceramic-rich side of the shell respectively. Approximate solutions are assumed to satisfy the simply supported boundary condition and Galerkin method is applied to obtain closed-form relations of bifurcation type of buckling loads. Numerical results are given to evaluate effects of inhomogeneous, dimensional parameters, outside stiffeners and elastic foundations to the buckling of structures.     

Vibration of composite-material cylindrical shells with ring and/or stringer stiffeners

Free vibration is analyzed for thin-walled circular cylindrical shells constructed of composite materials and provided with ring and/or stringer stiffeners. Numerical results are presented for graphite-epoxy shells: unstiffened and stiffened with rings only, stringers only, and both rings and stringers.     

Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment

The main aim of this paper is to investigate the nonlinear buckling and post-buckling of functionally graded stiffened thin circular cylindrical shells surrounded by elastic foundations in thermal environments and under torsional load by analytical approach. Shells are reinforced by closely spaced rings and stringers in which material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. The elastic medium is assumed as two-parameter elastic foundation model proposed by Pasternak. Based on the classical shell theory with von Karman geometrical nonlinearity and smeared stiffeners technique, the governing equations are derived. Using Galerkin method with three-term solution of deflection, the closed form to find critical torsional load and post-buckling load–deflection curves are obtained. The effects of temperature, stiffener, foundation, material and dimensional parameters are analyzed.     

反对称正交铺层剪切圆柱壳在外压下的屈曲和后屈曲

给出了反对称正交铺层剪切圆柱壳广义大挠度Donnell 型方程, 并运用位移型摄动技术构造出该圆柱壳在均匀外压作用下的后屈曲渐近级数解。考虑到边界效应对中短圆柱壳的影响及边值问题摄动解的一致性, 详细研究了该圆柱壳端部边界层方程和奇异摄动解, 以便与中部正则摄动解相匹配。文中同时给出一些典型例子并讨论了横向剪切变形、Batdo rf 数、弹性模量比和初始几何缺陷对圆柱壳屈曲与后屈曲性态的影响。比较显示, 横向剪切变形对圆柱壳的屈曲与后屈曲有重要影响。      

Buckling loads of CFRP composite cylinders under combined axial and torsion loading – experiments and computations

During the years 1994 through 1999, a European research project under the title “Design and Validation of Imperfection-Tolerant Laminated Shells” (DEVILS) was carried out. In this project, 11 European partners were involved. A goal of the project was an analytical and experimental study of the buckling behavior of thin-walled carbon fibre reinforced polymer (CFRP) laminated shells under combined axial and torsion loading. An additional aim was to compose a guideline for the dimensioning of such shells. This paper deals with the experimental and the analytical work conducted by DLR (Institute of Structural Mechanics, Braunschweig), ETH Zürich (former Institute of Lightweight Structures and Ropeways) and EMPA Dübendorf (Department of Polymers/Composites) in that project. The study was aimed at the determination of buckling loads of circular cylindrical shells of different laminate lay-ups. Nine shells were tested at DLR in Braunschweig for axial compression and at EMPA in Dübendorf under axial load and under combined axial compression and superimposed torsion. To determine the geometrical quality, the internal and the external surfaces of the specimens were mapped. ETH used photogrammetry and laser scanning prior to loading, while EMPA applied coordinate measurements for the unloaded shells and Moiré projection to monitor the lateral deflection of the cylindrical wall during loading and after buckling. At DLR, strain measurements were performed to assess regularity of the load distribution throughout the loading. The investigation showed that buckling loads of cylinders which are imperfection-sensitive under axial loading may not be so sensitive to combined loads. Furthermore, it was found that the stiffness eccentricity of the laminate played a significant role on the magnitude of the axial buckling load, while for combined loads this effect was somewhat reduced. This paper contains the results of those tests and also the comparison with results of analytical investigations and FE modeling; the obtained data can be used as benchmark reference.     

Dynamic buckling of a laminated composite stringer–stiffened cylindrical panel

The present study deals with the “dynamic buckling” of a laminated composite stringer–stiffened curved panel. The “dynamic buckling”, in the present study, is concerned with the unbounded lateral response of the panel, which is subjected to an axial impact load.In reinforced panels with widely spaced adequately stiff stringers, the structure may pass through two major states before its total collapse: buckling of the panel skin between stiffeners and buckling of the stiffeners themselves. This study focuses on the lowest buckling load of the stringer–stiffened panel, which is, buckling of the panel skin between stiffeners.The analysis of the laminated composite stringer–stiffened cylindrical panel was performed by using the commercial ANSYS finite element software. The model simulates the structure and its associated boundary conditions. The boundary conditions simulate the stringer–stiffened cylindrical panel as a part of a fuselage. The static buckling analysis was performed using the eigenvalue buckling approach to determine the static critical load. Modal analysis was used to calculate the first natural frequency and corresponding mode shape of the structure. Nonlinear transient dynamic analysis was used to determine the dynamic critical load. In the transient dynamic analysis the Newmark method with the Newton–Raphson scheme were used.In the present study, the equation of motion approach was applied. By this approach, the equations of motion were numerically solved for various load parameter values (loading amplitude and loading duration) to obtain the system response. Special attention was given to the neighborhood of loading durations corresponding to the period of the lowest bending frequency of the skin.For each load duration, the dynamic buckling load was calculated using a load versus lateral displacement curve generated by the ANSYS code.The results were plotted on a dynamic load amplification factor (DLF) graph. The DLF is defined, as the ratio of the dynamic buckling to the static buckling of the panel. For loading periods in the neighborhood of the lowest natural frequency of the panel, the DLF was less than unity. It means that, for those particular loading periods, the dynamic buckling load is lower than the static one.     

Bending,buckling and free vibration of non-homogeneous composite laminated cylindrical shells using a refined first-order theory

The bending, buckling and free vibration problems of non-homogeneous composite laminated cylindrical shells are considered. Hamilton–Reissner's mixed variational principle is used to deduce a consistent first-order theory of composite laminated cylindrical shells with non-homogeneous elastic properties. The governing equations with their required boundary conditions are derived without introducing any shear correction factors. Numerical results for the transverse deflections, stresses, natural frequencies and critical buckling loads are presented to show the advantages of this theory. The influences of the non-homogeneity and thickness ratio on the shell structural response are investigated. The study concludes that the inclusion of the non-homogeneity effect is required, even if it is weak, for predicting the actual structural response of the shells.     

Dynamic stability of stiffened laminated composite plates and shells subjected to in-plane pulsating forces

The dynamic stability of laminated composite stiffened or non-stiffened plates and shells due to periodic in-plane forces at boundaries is investigated in this paper. A three-dimensional (3-D) degenerated shell element and a 3-D degenerated curved beam element are used to model plates/shells and stiffeners, respectively. The characteristic equations to find the natural frequencies, buckling loads and their corresponding mode shapes are obtained from the finite element equation of motion. Then, the method of Hill's infinite determinants or the method of multiple scales is applied to analyse the dynamic instability regions. Numerical results are presented to demonstrate the effects of various parameters, such as skew angle, lamination scheme, stiffened scheme, in-plane force type and curvature of cylindrical shell, on the dynamic stability of stiffened and non-stiffened plates and shells subjected to in-plane pulsating forces at boundaries.     

具有初始几何缺陷加劲板的动态屈曲

针对工程中常用的加劲板,研究了动态屈曲的求解方法。将加劲板分为母板与加劲肋两个部分考虑,其中母板按经典薄板理论计算,加劲肋视为Euler梁。假定加劲板的位移,利用Hamilton原理结合系统能量和振型叠加法建立了加劲板的动态屈曲特征方程。最后,选择四边简支加劲板进行数值分析,分析中考虑初始几何缺陷的影响,并讨论了初始几何缺陷、加劲肋的数量及其刚度的变化对动态屈曲临界荷载的影响。结果表明：一阶模态的初始几何缺陷对加劲板的临界荷载影响很大,而增加加劲肋的数量及其刚度可以提高加劲板的抗动态屈曲能力。研究结果也为加劲板的结构设计方法提供一定的参考。     

Calculation of the collapse load of an axially compressed laminated composite stringer-stiffened curved panel–An engineering approach

The effective width method that is widely applied for the analysis of isotropic planar stringer-stiffened panels has been extended to laminated composite stringer-stiffened circular cylindrical panels. The approach was modified and adapted to handle curved composite structures.

Panels stiffened by blade type stiffeners, J-form stiffeners and T-form stiffeners were considered in the present study. Bending buckling of the stiffeners, their torsional buckling, combined bending and torsion buckling and local buckling of the stringers were accounted for in the investigation. Using the proposed extended effective width method, a MATLAB based software code TEW¹ was developed and implemented. To validate this code, predictions obtained by it were compared with experimental results and with finite element calculations. Good agreement between the present proposed method, experiments and finite element simulations was found, thus yielding an efficient, simple to apply and fast engineering code to be used in design and optimization stages.     

On the solution of eigenvalue problems of laminated non-homogeneous orthotropic circular shells with clamped edges subjected to hydrostatic pressure

This article presents a method to study the free vibration and stability of laminated homogeneous and non-homogeneous orthotropic cylindrical, truncated and complete conical shells of general staking with clamped edges under a hydrostatic pressure. Based on the Love first approximation theory, the basic relations, the modified Donnell-type stability and compatibility equations have been obtained for laminated orthotropic truncated conical shells, the material properties of which vary piecewise continuously in the thickness direction. To solve this problem an unknown parameter λ was included in the approximation functions. Applying Galerkin methods, the buckling pressures and fundamental natural frequencies of laminated homogeneous and non-homogeneous orthotropic conical shells are obtained. The parameter λ which is included in the obtained formulas is obtained from the minimum conditions of critical stresses and frequencies. The different generalized values are obtained for the parameter λ for buckling pressures and frequencies of cylindrical shells, truncated and complete conical shells. The appropriate formulas for single-layer and laminated cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Finally, the influences of the degree of non-homogeneity, the number and ordering of layers and the variations of conical shell characteristics on the critical hydrostatic pressure and natural frequencies are investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

复合材料层合扁锥壳的冲击屈曲

研究了计及横向剪切的对称铺设正交异性复合材料层合扁锥壳在三角脉冲载荷作用下的非线性动力屈曲问题；通过在复合材料层合扁锥壳非线性稳定性的基本方程中增加横向转动惯量项并引入三角脉冲冲击力，得到了层合扁锥壳的冲击控制方程，采用Galerkin方法得到以顶点挠度表达的冲击响应方程，并用Runge—Kutta方法进行数值求解，应用B—R准则确定冲击屈曲的临界荷载；讨论了壳体几何参数、物理参数和边界条件对复合材料层合扁锥壳冲击屈曲的影响。     

Buckling analysis of rectangular composite plates with rectangular cutout subjected to linearly varying in-plane loading using fem

A numerical study is carried out using finite element method, to examine the effects of square and rectangular cutout on the buckling behavior of a sixteen ply quasi-isotropic graphite/epoxy symmetrically laminated rectangular composite plate [0°/+45°/-45°/90°]_2s, subjected to various linearly varying in-plane compressive loads. Further, this paper addresses the effects of size of square/rectangular cutout, orientation of square/rectangular cutout, plate aspect ratio(a/b), plate length/thickness ratio(a/t), boundary conditions on the buckling bahaviour of symmetrically laminated rectangular composite plates subjected to various linearly varying in-plane compressive loading. It is observed that the various linearly varying in-plane loads and boundary conditions have a substantial influence on buckling strength of rectangular composite plate with square/rectangular cutout.     

Optimum buckling design of composite stiffened panels using ant colony algorithm

Optimal design of laminated composite stiffened panels of symmetric and balanced layup with different number of T-shape stiffeners is investigated and presented. The stiffened panels are simply supported and subjected to uniform biaxial compressive load. In the optimization for the maximum buckling load without weight penalty, the panel skin and the stiffened laminate stacking sequence, thickness and the height of the stiffeners are chosen as design variables. The optimization is carried out by applying an ant colony algorithm (ACA) with the ply contiguous constraint taken into account. The finite strip method is employed in the buckling analysis of the stiffened panels. The results shows that the buckling load increases dramatically with the number of stiffeners at first, and then has only a small improvement after the number of stiffeners reaches a certain value. An optimal layup of the skin and stiffener laminate has also been obtained by using the ACA. The methods presented in this paper should be applicable to the design of stiffened composite panels in similar loading conditions.     

环加肋圆柱壳屈曲分析的有限条法

导出了有限条法分析环加肋圆柱壳在静水压力作用下总体屈曲的计算格式，将环加肋圆柱壳作为一个构造上的正交各向异性壳处理，推导了考虑环向加肋影响后有限条元的正交各向异性弹性矩阵。对两端简支环加助圆柱壳总体屈曲临界载荷的计算表明，本文方法计算结果与解析解符合良好。有限条法是分析密加肋圆柱壳屈曲问题的有效数值方法。     

一类双壳结构振动特性的半解析法

以Hamilton正则方程的半解析法为基础,为一类双壳结构的振动特性提出了一种新的数学模型。基本步骤:(1)独立地建立内外壳和连接筋的线性方程组;(2)考虑到内外壳和连接筋的界面上的应力和位移的连续性,联立内外壳和连接筋的方程,从而得到全结构的方程组。主要优点是:采用了同一种Hamiltonian等参元离散壳和连接筋,结构的转动惯性、剪切变形等因素都得到了考虑,而且不限制壳的厚度和筋的高度;该方法象一般的有限元法一样适应复杂的边界条件和由多种材料构成的结构。本文的方法可推广用来研究加筋复合材料或加筋压电材料层合壳及相应的双壳结构的动力学问题。