Bending and buckling of inflatable beams: Some new theoretical results

The non-linear and linearized equations are derived for the in-plane stretching and bending of thin-walled cylindrical beams made of a membrane and inflated by an internal pressure. The Timoshenko beam model combined with the finite rotation kinematics enables one to correctly account for the shear effect and all the non-linear terms in the governing equations. The linearization is carried out around a pre-stressed reference configuration which has to be defined as opposed to the so-called natural state. Two examples are then investigated: the bending and the buckling of a cantilever beam. Their analytical solutions show that the inflation has the effect of increasing the material properties in the beam solution. This solution is compared with the three-dimensional finite element analysis, as well as the so-called wrinkling pressure for the bent beam and the crushing force for the buckled beam. New theoretical and numerical results on the buckling of inflatable beams are displayed.     

充气梁的有限元分析

采用包括Timoshenko动力学、有限转动和小应变的虚功原理定义充气梁弯曲和屈曲的离散非线性计算公式。随后推导出预应力参考构型的线性计算公式,从而得出新的充气梁有限元分析方法。刚度矩阵包括剪切系数和内压力。采用三节点梁单元建模分析悬臂梁的弯曲和屈曲、带箍环面的变形以及承受放射状压力的环面屈曲。采用这种梁元计算的结果与采用三维膜单元有限元计算结果很相近。结合充气梁的冲撞力或者蠕变压力概念,讨论了数值计算结果的有效性。     

Deflections of highly inflated fabric tubes

Inflatable beams made of modern textile materials with important mechanical characteristics can be inflated at high pressure. The aim of the paper is to present experimental, analytical and numerical results on the deflections of highly inflated fabric tubes submitted to bending loads. Experiments are displayed and we show that tube behaviour looks like that of inflatable panels (Thin-Walled Struct. 40 (2002) 523–536). Equilibrium equations are once again written in the deformed state to take into account the geometrical stiffness and the following forces. The influence of the shear stress cannot be neglected and Timoshenko’s beam theory is used. A new inflatable tube theory is established and simple analytical formulas are given for a cantilever-inflated tube. Comparisons between analytical and experimental results are shown. A new inflatable finite tube element is constructed by use of algebraic operations, because the compliance matrix of the cantilever beam is not symmetric. Comparisons between experimental, analytical and numerical results prove the accuracy of this beam theory and on this new finite element for solving problems on the deflections of highly inflated tubes.     

Inflation and bending of an orthotropic inflatable beam

An inflatable beam is an airtight structure made of a soft technical fabric and subjected to an internal pressure which gives it a final cylindrical shape, a pre-stress in the membrane and a bearing capacity. Against all appearances, it is not a standard beam and it requires a specific formulation in order to take account of the internal pressure which plays a key role in its mechanical response.This work deals with inflatable beams made of orthotropic materials. The first part of the paper is concerned with the inflation of the membrane tube, an important stage which is often neglected so far in the literature. As preliminaries of the bending problem studied in the next part of the paper, the constitutive law related to the inflated state of the tube – not the natural state – is investigated. It will be shown that the constitutive law related to the inflated pre-stressed state is not the same as the constitutive law related to the natural state. Expressions of the material coefficients involved in the former constitutive law will be established from the material coefficients defined on the natural reference configuration which are the only ones supposed to be known. The second part of the paper deals with the bending of the inflatable beam. The Timoshenko beam kinematics will be chosen because of the significant shear effect in the tube wall and the problem will be formulated in finite deformations in order to account for all the nonlinear effects, in particular the action due to the internal pressure which is a follower load. The nonlinear system of equations obtained will then be linearized around the pre-stressed configuration and will result in a more tractable linear system. The proposed formulation allows a comprehensive study of the influence of the internal pressure on the geometry and material properties of orthotropic inflatable beams. The analytical results will be compared with numerical results obtained from a nonlinear membrane finite element code.     

Coupled static and dynamic buckling of thin walled beam by spline finite element

For the coupled static and dynamic buckling of thin walled beam subjected to various forces, such as axial force, uniform bending moment, and bending moment due to concentrated and distributed lateral forces, the spline finite element method is employed to obtain the dynamic stiffness matrix. Second order effects of the axial force and moment are considered. A doubly symmetric cantilever beam with uniform cross-section is investigated. Extensive static and dynamic interaction diagrams are plotted. The effects of warping rigidity, torsional rigidity, axial tension and compression on moment buckling, moment on axial buckling compression, higher buckling modes are discussed in detail. The spline finite element method is proved to be very efficient for the present problem and many interaction diagrams can be plotted easily. Some new results are presented. The methodology is based on finite element formulation and therefore it can be easily extended to analyze structural frames.     

Coupled stability analysis of thin-walled composite beams with closed cross-section

For the coupled stability analysis of thin-walled composite beam with closed cross-section subjected to various forces such as eccentric constant axial force, end moments, and linearly varying axial force, the efficient numerical method to evaluate the element stiffness matrix is newly presented based on the homogeneous form of simultaneous ordinary differential equations. The general bifurcation type of buckling theory for thin-walled composite box beam is developed based on the energy functional corresponding to semitangential rotations and semitangential moments. The coupled stability equations including variable coefficients and the force–displacement relationships are derived from the energy principle and explicit expressions for displacement functions are presented based on power series expansions of displacement components. The element stiffness matrix is evaluated by applying member force–displacement relationships to these displacement functions. In addition, the finite element model based on the cubic Hermitian interpolation polynomial is presented. In order to verify the accuracy and validity of this study, numerical solutions are presented and compared with the finite element solutions using the Hermitian beam elements and the available results from other researchers. Particularly, the influence of the eccentricity and the force ratio of axial forces, the fiber orientation, and the boundary conditions on the buckling behavior of composite box beam are parametrically investigated. Also the emphasis is given in showing the phenomenon of buckling mode change.     

Analytical buckling analysis of an inflatable beam made of orthotropic technical textiles

An analytical approach was considered to study the buckling and the behavior of an inflatable orthotropic beam subjected to uniform compression loads under different boundary conditions. In order to assess the stability of inflatable structures, it is necessary to evaluate the critical load of the inflatable components in their pressurized configurations. First, a 3D inflatable orthotropic beam model based on the Timoshenko's kinematics was briefly introduced: the nonlinearities (finite rotation, follower forces) were included in this model. The nonlinear equilibrium equations were derived from the total Lagrangian form of the virtual work principle: the linearized equations were then obtained. By solving these linearized equations, an analytical expression of the critical buckling load was obtained. This critical buckling load was investigated through several load cases with several boundary conditions. The discrepancy due to the orthotropic character between the present model and the isotropic models found in the literature was evaluated, as well as the influence of the inflation pressure and the fabric mechanical properties on the value of critical load. The buckling mode shapes were also determined. To check the limit of validity of the results, the wrinkling load was also presented in every case.     

Free vibration analysis of inflatable beam made of orthotropic woven fabric

The free vibration of inflatable beams was studied using the dynamic stiffness method. A 3D Timoshenko beam with a homogeneous orthotropic woven fabric (OWF) was considered. Using the usual total Lagrangian form of the virtual work principle, the model took the geometric nonlinearities and the inflation pressure follower force effect into account. The nonlinear equilibrium equations were then linearized around the prestressed reference configuration. The exact dynamic stiffness matrix was developed by directly solving the governing differential equations of a 3D loaded inflatable beam in a free vibration. The effects of the inflation pressure, fabric mechanical properties and the boundary conditions on the natural frequencies and mode shapes of the inflatable beams were demonstrated. The proposed model was validated favorably through its comparison with a 3D thin shell finite element model and an isotropic fabric model found in the literature.     

Numerical investigation of inelastic buckling of steel–concrete composite beams prestressed with external tendons

The ultimate resistance of a continuous composite beam is governed by either distortional lateral buckling or local buckling, or an interactive mode of the two which is sharply different from the torsional buckling mode in a bare steel beam. A finite element model is developed and based on the proposed FE model, inelastic finite element analysis of composite beams in negative bending is investigated, considering the initial geometric imperfection and the residual stress patterns and the FE results are found agree well with the test results. Parametrical analysis is carried out on the prestressed composite beams with external tendons in negative bending. Factors that influence load carrying performance and buckling moment resistance of prestressed composite beams are analyzed, such as initial geometric imperfection, residual stress in steel beams, force ratio, which is defined as the extent of prestressing force and negative reinforcement in the beams, as well as the slenderness ratios of web, flange, and beams. By varying cross-section parameters, 25 groups of composite beams under negative uniform bending with initial geometric imperfection, residual stress as well as different force ratios, 200 beams in total are studied by means of the FE method. The computed buckling moment ratios are drawn against the modified slenderness proposed by the authors and compared with the Chinese Codified steel column design curve. It is demonstrated that the tentative design method based on the Chinese Codified design curve can be used in assessment of buckling strength of composite beams in a term of the modified slenderness defined.     

单梁托四墙的梁式转换的有限元分析

利用通用有限元程序ALGOR中的三维块体线弹性单元 ,对两条墙肢并排被支承在同一条柱上的梁式转换进行了实例分析。计算模式中考虑了竖向压力、剪力和弯矩三种外力的共同作用 ,并将ALGOR的应力结果积分为内力 (扭矩、轴力、剪力和弯矩 ) ,绘制成转换梁的内力图。通过对转换梁的内力进行分析 ,得出了这种转换梁的受力特点 ,包括梁上剪力墙对转换梁扭转的约束 ,转换梁的轴力对弯矩的影响等     

Elastic stability analysis of loosely fitted thin liners – A proposed simplified procedure and evaluation of existing solutions

The paper reports the results of an extensive numerical simulation to estimate the elastic buckling pressure and the corresponding thrust and bending moment induced in loosely fitted thin liners. The study is conducted numerically using a two-dimensional (2D) non-linear finite element model that accounts for the effects of large deformations on the stability of loosely fitted liners. The finite element results together with a non-linear multi-variant regression analysis are used to develop simplified non-dimensional formulae that provide the critical pressure, thrust and bending moments, at buckling, based on the liner’s geometry and material properties.In addition, several other analytical and numerical solutions for the same problem are revisited and critically reviewed. For comparison purposes, two of such methods are extended to allow for the calculation of the thrust and bending moment developed at the most heavily stressed point in the loosely fitted liners at the onset of buckling. In general, the comparison reveals the appropriateness of the proposed regression models in predicting the critical pressure and the associated thrust and bending moment induced in imperfect loosely fitted liners. Developed formulae provide designers with a simple and reliable means for ensuring stability and safety of such special type of structures.     

Elastic lateral-torsional buckling of tapered I-girder with corrugated webs

This paper deals with the elastic lateral-torsional buckling (LTB) strength of tapered I-girders with corrugated webs under two types of loading conditions: uniform moment and moment gradient with various end restraint conditions. A finite element (FE) program using beam elements is developed to study LTB behaviors. The results from this program are compared with those from the commercial software ABAQUS using shell elements. From the comparisons, it is found that the developed FE program’s results agree well with the results from ABAQUS. For design purpose, the closed-form equations for the critical buckling moment of the tapered I-girder with corrugated webs under uniform moment and moment gradient with four types of end restraint conditions: simply supported, warping fixed, lateral bending fixed, and completely fixed are proposed based on the results from the developed FE program. From the numerical investigations, the new design equations give reasonably accurate results. These equations increase efficiency in bridges and buildings design.     

Eigenvalue analysis of flexural-torsional buckling of angle-bar stiffened plates

Equilibrium method is employed to analyze interaction of flexural and torsional buckling of angle-bar stiffened plates. The cross sectional areas of the angle-bar stiffened plate for these two modes are different (hybrid beam). In flexural buckling mode, angle-bar and attached plate buckle together, however in torsional buckling mode only angle-bar would be buckled. Basic equations of equilibrium for flexural and torsional buckling modes of angle-bar stiffened plates are deduced based on hybrid beam concept and new strain distribution assumption for sideway bending of stiffeners. Elastic buckling stress of different angle-bar stiffened plates are calculated and compared with finite element method and those available in the literature. It is shown that present method has very good agreements with finite element method for isolated rigid angle-bar and isolated rigid angle-bar with rigid attached plate. For isolated rigid angle-bar with pin connected to rigid attached plates it has better agreement than previously proposed method.     

对预装体外钢筋的钢-混凝土组合梁弹塑性屈曲的数值分析

连续组合梁的极限承载力由侧向畸变屈曲、局部屈曲,或者由两者的相互作用决定,这一点与纯钢梁的扭转屈曲破坏模式完全不同。利用有限元模型对负弯矩区的组合梁进行了弹塑性有限元分析,其中考虑了初始几何缺陷和残余应力,最终发现有限元分析结果与试验结果吻合良好。另外,对负弯矩区的预装体外钢筋的组合梁进行了参数研究。分析了影响组合梁承载能力和屈曲弯矩抗力的因子,如初始几何缺陷、钢梁的残余应力、力比值、预应力范围、负力矩钢筋、板、翼缘和梁的宽厚比。利用有限元法对负弯矩作用下的25组共200根具有不同截面参数、初始几何缺陷、残余应力和不同力比值的组合梁进行了分析。将计算出的屈曲弯矩率与改进宽厚比的曲线与中国规范中钢柱设计曲线进行了对比。结果证实:对于改进了的宽厚比值,本文依据中国规范的设计曲线推导出来的设计方法可以用于对组合梁屈曲强度的评估。     

交叉加劲钢板深梁弹性屈曲分析

以交叉加劲钢板深梁为研究对象,利用有限元软件ANSYS分析其弹性屈曲性能,讨论了抗弯刚度比、跨高比、钢板深梁厚度对其弹性屈曲性能的影响;考虑钢板深梁在钢框架的弯剪受力特性,根据板的经典理论建立了交叉加劲钢板深梁屈曲荷载计算公式,提出了等效屈曲系数。结果表明:交叉加劲钢板深梁的临界屈曲荷载随抗弯刚度比增大而增大,但达到门槛刚度比后,增大幅度急剧减小,得到门槛刚度比约为10;临界屈曲荷载随跨高比和板厚的减小而减小,等效屈曲系数随板厚减小而增大;等效屈曲系数与跨高比关系曲线由二次抛物线形向波浪形渐变,交叉加劲钢板深梁受力特性由剪切主导向弯曲主导过渡。     

压弯钢构件在循环荷载作用下的非线性弯扭屈曲

本文根据非线性有限元的基本理论,采用基于修正的拉格朗日法描述的八节点超参数壳体单元,考虑几何非线性和材料非线性,结合混合强化本构关系,编制了非线性有限元计算程序。对压弯钢构件在循环荷载作用下的弹塑性弯扭屈曲进行计算机模拟,系统分析了H形截面压弯钢构件的滞回特性,以及翼缘宽厚比、腹板高厚比、构件轴压比和长细比对压弯钢构件弯扭屈曲的影响规律。通过对计算结果的比较,提出了强烈地震条件下,H形截面压弯钢构件在循环荷载作用下保持整体稳定,满足延性要求,具有较高屈曲后强度的长细比和板件宽厚比限值。     

Deflections of inflatable fabric panels at high pressure

Inflatable structures made of modern textile materials with important mechanical characteristics can be inflated at high pressure (up to a several hundreds of kPa). They can be used as strong building elements thanks to their mechanical strength. The aim of the paper is to present experimental and analytical studies on the behaviour of inflated fabric panels at high pressure and submitted to bending loads. It is shown that inflatable structures cannot be viewed as ordinary plates or beams, because their deformation pattern is quite different. Experiments show that their behaviour depends on the applied load, the inflation pressure, and the constitutive law of the fabrics. Equilibrium equations are written in the deformed state to take into account the influence of geometrical stiffness and the following forces. A Timoshenko’s beam theory must be used because sections of the panels do not satisfy the usual Bernoulli’s beam theory. A new inflatable beam theory is developed. Wrinkling loads are derived from equilibrium equations. Deflections satisfy the fact that the compliance of the inflatable panel is the sum of the beam compliance and of the yarn compliance. Comparisons between the results of our modelling and experimental results are shown and prove the accuracy of this theory on the mechanical strength of inflatable structures at high pressure.     

Experimental characterization and finite element analysis of inflated fabric beams

An airbeam is a high-strength fabric sleeve with a highly flexible internal bladder that can be used as a load-bearing beam or arch when inflated. Due to their fabric construction, airbeams are inherently thin-walled structures that are prone to local buckling. In this study, airbeams were tested in bending at different inflation pressures to quantify their load–deformation response and the effect of inflation pressure on response. Tension–torsion tests of the airbeam fabric were conducted to estimate the fabric shear modulus, and the bend test results were used in conjunction with Timoshenko beam theory to estimate the fabric elastic modulus. Three-dimensional membrane finite element (FE) models were then used to predict the beam load–deformation response given these moduli. The FE models successfully predicted localized fabric buckling and softening of load–deflection response. Comparison of FE model-predicted load–deflection response with beam theory shows that conventional beam theory is accurate prior to local buckling of the airbeam fabric. The FE model and test results indicate that the consideration of work done by pressure under deformation-induced volume changes may increase beam capacity beyond previously derived theoretical limiting values.     

Buckling formulation for shear deformable thin-walled members—II. Finite element formulation

The variational principle developed in a companion paper is adopted to formulate a new finite element for the buckling analysis of members of doubly symmetric cross-sections. A series of examples demonstrate the convergence characteristics of the element and its applicability to a wide variety of problems. The examples include column flexural buckling, lateral–torsional buckling of beams subjected to moment gradients, buckling of members under combinations of axial force and bending moment, and eccentrically supported structural members. In all cases, the validity of the solutions is assessed through comparisons to well-established closed-form and/or other established numerical solutions.     

屈曲约束钢板剪力墙受力性能分析及盖板简化计算方法

内嵌钢板与混凝土盖板之间相互作用机理较为复杂,难以通过简单方法确定屈曲约束钢板墙连接螺栓拉力与混凝土盖板弯矩.为此,采用有限元软件ABAQUS建立屈曲约束钢板剪力墙计算模型,并通过与模型试验对比验证其可靠性.通过非线性有限元数值模拟,分析钢板高厚比、盖板与钢板厚度比以及盖板与钢板间隙等参数对应力分布、屈曲变形和滞回性能...