On the integral equation method for buckling loads

An initial value method for the integral equation of the column is presented for determining the buckling load of columns. The differential equation of the column is reduced to a Fredholm integral equation. An initial value problem is derived for this integral equation, which is reduced to a set of ordinary differential equations with prescribed initial conditions in order to find the Fredholm resolvent. The singularities of the resolvent occur at the eigenvalues. Integration of the equations proceeds until the integrals become excessively large, indicating that a critical load has been reached. To check this method, numerical results are given for two examples, for which the critical load is well known. One is the Euler load of a simply supported beam, and the other case is the buckling load of a cantilever beam under its own weight. The advantage of this initial value method is that it can be applied easily to solve other nonlinear problems for which the critical loads are unknown. This approach will be illustrated in future papers.     

Hamilton力学下框筒结构剪滞翘曲位移模式研究北大核心CSCD

以等效连续化方法为基础,在Hamilton力学体系下进行框筒结构剪滞翘曲位移函数精度研究.选用不同类型的函数描述翼缘板的剪滞翘曲位移,考虑等效板的剪切变形以及纵向翘曲,得到不同位移函数下结构的总势能及对应的Lagrange函数.区别于传统变分法,该文在Hamilton力学体系下进行问题研究,导出框筒结构弯曲问题的Hamilton正则方程并利用精细积分法求解,进而计算出柱轴力并进行精度分析.算例验证结果表明:使用该方法分析框筒结构的剪力滞后效应是简单可行的;不同翘曲位移函数的选择对侧移计算结果影响不大,对轴力求解结果影响较大,二次抛物线最能反映等效翼缘板的实际翘曲位移;对比不同形式荷载作用下等效翼缘板中应力分布可知,随着外荷载合力作用点位置的升高,结构顶部负剪力滞后效应逐渐减弱至消失.     

阶梯柱屈曲的改进Fourier级数分析

该文对阶梯柱的弹性屈曲问题进行了研究.首先基于改进Fourier级数法采用局部坐标逐段建立阶梯柱的位移函数表达式,然后由带约束的势能变分原理得到含屈曲荷载的线性方程组,利用线性方程组有非零解的条件把问题转化为矩阵特征值问题得到临界载荷,最后讨论方法中的参数取值,并把结果与已有文献和有限元的结果比较,从而验证方法的精度.所提模型在阶梯柱的两端和变截面处引入横向弹簧和旋转弹簧,通过改变弹簧的刚度值模拟不同的边界.所提方法在工程设计中能比较精确地确定各种弹性边界条件下阶梯柱的临界载荷.     

An elastoplastic energy model for predicting the deformation behaviors of various structural components

Elastoplastic load-displacement relation is widely concerned in material testing. For isotropic-homogeneous and power-law hardening ductile materials, an elastoplastic energy model (EEM) correlating energy, load, displacement and uniaxial constitutive parameters is derived based on equivalent energy principle. An elastoplastic factor λ for engineering superposition of elastic displacement and plastic displacement is introduced and discussed, which makes the model applicable with more structural components (SCs). The model is verified with thirteen SCs used in materials testing and the results show a good agreement between model predictions and calculations from finite element analysis (FEA) under linear elastic, fully plastic and elastoplastic conditions. Some experimental results for ring-compression, spherical indentation and funnel tension are conducted to verify the model and a valid accordance is presented. Additionally, an explicit J-integral – load relation for classic cracked SCs is also derived based on the EEM and a good coincidence is observed during a comparison with results directly from FEA.     

Selection of optimum design for conical segmented aluminum tubes as energy absorbers: Application of MULTIMOORA method

The optimum design for energy absorbers requires considering different criteria. Therefore, the design process can benefit from Multiple-Attribute Decision-Making (MADM) approach. Thin-walled structures can be particularly employed as energy absorbers in automobiles to dissipate energy and protect passengers from the peak crush load caused by severe collisions. The present study investigates a thin-walled conical segmented aluminum tube which consists of several circular sections with different thicknesses. To achieve the optimum design, a number of conical segmented tubes were modeled by finite element method. Energy absorption, initial peak load, crush force efficiency (the ratio of mean load to maximum load), mass of the structure and deformation type as the conflicting objective functions were defined as attributes/criteria. The numeric logic (NL) technique was applied to determine the weight of various criteria. Furthermore, the MULTIMOORA method, as a well-accepted MADM model, was utilized to obtain the optimum design for a conical segmented tube. The design variables included wall thickness, lengths of sections, and taper angle. Drawing a comparison between the top rank conical segmented tube and the top rank conical simple tube revealed that the conical segmented tube could increase the efficiency of the crush load up to 23.8% and reduce the initial peak load and mass of the tube up to 54% and 17.54%, respectively.     

Crack opening displacement for two unequal straight cracks with coalesced plastic zones – A modified Dugdale model

The problem investigated is of an infinite plate weakened by two collinear unequal hairline straight quasi-static cracks. Uniform constant tension is applied at infinity in a direction perpendicular to the rims of the cracks. Consequently the rims of the cracks open in Mode I type deformation. The tension at infinity is increased to the limit such that the plastic zones developed at the two adjacent interior tips of cracks get coalesced. To arrest the crack from further opening normal cohesive variable stress distribution is applied on the rims of the plastic zones. Closed form analytic expressions are obtained for load bearing capacity and crack opening displacement (COD). An illustrative case is discussed to study the behavior of load bearing capacity and crack opening displacement with respect to affecting parameters viz. crack length, plastic zone length and inter crack distance between the two cracks. Results obtained are reported graphically and analyzed.     

初封载荷下封隔器胶筒表面自由变形特性分析

研究了封隔器胶筒在自由变形阶段受初封载荷作用下内、外表面发生位移变形的特性.依据连续介质力学理论，建立自由变形阶段的有限变形数学模型，给出了胶筒在初封轴向载荷下内、外表面径向变形的过程，得到胶筒非线性变形解析解.通过数值计算，在求解出胶筒外表面自由变形解析式的基础上，进一步分析了容易被忽略的胶筒内表面非线性变形规律和相关参数变化对其密封性能的影响.该变形特性分析可适用于不同型号的封隔器胶筒，为胶筒的密封和可靠性设计提供重要理论依据.     

轴向冲击下理想塑性直杆动态屈曲的过应力简化分析模型*

本文在理想塑性直杆的动态屈曲分析中引入应变率效应,得到相应的动力学微分方程,求出了屈曲半波长,临界载荷和屈曲时间的表达式.讨论了应变率效应对杆的塑性动态屈曲的影响.并与文[4]的理论和试验结果作了比较.     

Deformative properties of flexible plastic foams in compression

A calculation of the compression diagrams of flexible plastic foams during their long-time testing under a load is given on the basis of the proposed cell model, on the assumption that permanent set is related to buckling of the strands. The character of change of the nominal compressive strength and corresponding deformation as a function of the magnitude of permanent set is established. A marked change of the second section of the compression diagram (plateau), even to its degeneration, in the presence of considerable permanent set was found. A comparison of the experimental and calculated data showed their satisfactory correspondence.Vladimir Research Institute of Synthetic Resins. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 443–449, May–June, 1973.     

Novel closed – form solutions in buckling of inhomogeneous columns under distributed variable loading

In this study we consider buckling of columns with variable stiffness, under axially distributed loading varying polynomially. The objective is to obtain closed – form solutions for the buckling load. The problem is posed in inverse setting: determine the column’s stiffness, so that it has the given, polynomial, buckling mode. Four sets of boundary conditions are investigated. Some perplexing results are obtained, namely, that irrespective of boundary conditions, the critical load of the column is the same; this occurs in conjunction with the fact that the obtained distribution for stiffness is different for each set of boundary conditions.     

A higher-order gradient theory for modeling of the vibration behavior of single-wall carbon nanocones

A higher-order gradient theory is used to investigate the free vibration characteristics of single-wall carbon nanocones (SWCNCs). This atomistic–continuum model simulates SWCNCs at the atomistic level and links the deformation of the crystal lattice structure to that of the continuum field. The dependence of vibration frequencies of SWCNCs on apex angles, heights and top radii, as well as constraints, is studied under a developed mesh-free computational framework based on moving Kriging interpolation. It is found that the proposed model gives a good prediction of the MD simulation and Timoshenko beam model. Several kinds of SWCNCs were investigated and the results reveal that the apex angle markedly affects the vibration frequency. It is observed that the fundamental frequency increases as the top radius increases, until it reaches a critical value. The critical top radii are largely dependent on the constraints at the ends of the SWCNCs. It is also observed that for SWCNCs with different apex angles, the same fundamental frequency is obtained by an appropriate combination of height and top radius. As the top radius continues to increase, the change of fundamental frequency becomes smaller and smaller.     

On column generation approaches for approximate solutions of quadratic programs in intensity-modulated radiation therapy

This paper deals with numerical behavior of a recently presented column generation approach for optimization of so called step-and-shoot radiotherapy treatment plans. The approach and variants of it have been reported to be efficient in practice, finding near-optimal solutions by generating only a low number of columns. The impact of different restrictions on the columns in a column generation method is studied, and numerical results are given for quadratic programs corresponding to three patient cases. In particular, it is noted that with a bound on the two-norm of the columns, the method is equivalent to the conjugate-gradient method. Further, the above-mentioned column generation approach for radiotherapy is obtained by employing a restriction based on the infinity-norm and non-negativity. The column generation method has weak convergence properties if restricted to generating feasible step-and-shoot plans, with a “tailing-off” effect for the objective values. However, the numerical results demonstrate that, like the conjugate-gradient method, a rapid decrease of the objective value is obtained in the first few iterations. For the three patient cases, the restriction on the columns to generate feasible step-and-shoot plans has small effect on the numerical efficiency.     

An extended sequential limit analysis based on moving coordinates for pressurized spherical cap membranes with large shape change

Sequential limit analysis (SLA) is an effective and normally used method to calculate the plastic limit load of structures with large deformation. However, attribute to the assumption of neglecting the changing behavior of shape and plane stress direction, the conventional SLA method would be inaccurate in the plastic response prediction for the large-shape-change structures, especially for the pressurized spherical cap. This research develops a novel analytical model for the pressurized spherical cap based on the advanced SLA method, which features introducing the moving coordinate system and considering of the changing behavior of shape and plane stress direction into the conventional method. With the proposed method, the effects of geometry and material parameters on the plastic limit load are analyzed. Compared with the validating FE simulation results of ABAQUS software, the newly extended SLA method performances a more precise prediction of the load deflection response and plastic limit load than the normal one. Due to the limit yield degree and bending moment, the accuracy of the new model will increase with the increase of yield strength and radius. The larger the initial deflection of the pressurized spherical cap is, the smaller the relative error between the analysis results of advanced SLA and FE method is. Moreover, this newly proposed SLA remains effective and accurate within a wide range of the initial thickness-curvature radius ratio, especially for low elasticity modulus materials.     

Optimization of cyclindrical shells of fiber-reinforced composite materials

Problems of optimizing nonelastic circular shells are considered. The material of the shells is assumed to be a fiber-reinforced composite with fibers unidirectionally embedded in a relatively less stiff but ductile metallic matrix so that the material has the yield surface suggested by Lance and Robinson. The shell is subjected to an impulsive loading of short-time periods generating initial kinetic energy. During plastic deformation of the shell the initial kinetic energy is transformed into the plastic strain energy. The shell thickness is assumed to be piecewise constant. Various thicknesses and coordinates of the rings, where the thickness has jumps, are preliminarily unspecified. We look for a shell design for which the maximum residual deflection has a minimum value for the total weight given. The alternative problem of minimizing the shell weight for the maximum deflection given is also studied.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, Octobe, 1995.Tartu University, Estonia. Published in Mekhanika Kompozitnykh Materialov, No. 1, pp. 65–71, January–February, 1996.     

Geometrical super-elements for elasto-plastic shells with large deformation

This paper presents the application of the so-called Geometrical Elements Method (Lukasiewicz and Szyszkowski, 1974; Pogorelov, 1967) to the solution of elasto-plastic problems of shells. The approach is based on the observation that, during large deformations, the shell structure deforms in a nearly isometrical manner. Therefore, its deformed shape can be determined and analysed making use of the Gauss theorem according to which the Gaussian curvature of the isometrically deformed surface remains unchanged. The shell structure is subdivided into elements of two kinds: purely-isometrically deformed elements and quasi-isometrically deformed elements. The equilibrium of the whole structure is defined by the stationary value of the Hamiltonian function which requires the calculation of the strain energy in the elements. This can easily be obtained if we recognize that the isometrically deformed elements contain only bending energy. Using the method described, we are able to significantly the number of unknown values defining the shape of the deformed structure. The problem is reduced to the numerical evaluation of the minimum of a function of many variables. The elasto-plastic state of stress of the plastic material in the structure canbe determined by using the deformation theory of plasticity or the theory of plastic flow. Also, the strains and stresses in the plastic regions are the only functions of the assumed displacements field. The corresponding energy of the plastic deformation can easily be evaluated and added to the minimized functionals. For example, the elasto-plastic behaviour of a spherical shell under a concentrated load is studied. The solution obtained defines the large deformation behaviour and the motion of the plastic zones on the surface of the shell.     

Stress softening, strain localization and permanent set in the circumferential shear of an incompressible elastomeric cylinder

Corresponding author, email huntley{at}umich.edu A constitutive theory for elastomeric materials has recentlybeen developed according to which stress is generated by differentmicromechanisms at different levels of deformation. When thedeformation is small, the stress is given by the usual theoryof rubber elasticity. As the deformation increases, some junctionsof the macromolecular microstructure rupture. Junctions thenre-form to generate a new microstructure. The constitutive equationallows for continuous scission of the original junctions andformation of new ones as deformation increases. The macromolecularscission causes stress reduction. The formation of new microstructuresresults in permanent set on release of external load. The present work considers a hollow circular cylinder composedof such a material, also assumed to be incompressible and isotropic.The cylinder is fixed rigidly at its inner surface and undergoesaxisymmetric deformation due to a uniform axial moment appliedat the outer surface. There develops an outer zone of materialwith the original microstructure and an inner zone of materialhaving undergone macromolecular scission, separated by a cylindricalinterface, the radius of which increases with the rotation ofthe outer surface. The shear deformation distribution, moment-rotationresponse and permanent set on release of moment are determined.It is found that microstructural scission can lead to higherlevels of shear deformation near the inner surface of the cylinderthan in the case of purely elastic response. It is also seenthat a residual state of high shear deformation can arise ina thin layer of material at the inner boundary of the cylinder.     

Stability of Pflüger's column with imperfections

The stability of a column loaded with continuously distributed tangential load is analyzed. It is assumed that imperfections in the shape (initial deformation) and loading (a small concentrated force) are present. It is shown that the influence of the imperfections is modelled by a Whithney cusp.     

薄扁球壳受冲击载荷作用时动力屈曲的数值计算

本文在轴对称变形的假设前题下,通过将扩展了的Marguerre’s方程化为差分方程,对其上作用荷载面积不同时的固支弹性薄扁球壳的屈曲问题作了一些探讨.由此发现,固支薄扁球壳在外界冲击荷载作用下,在λ(=2[3(1-v~2)]~(1/4)(H/h)~(1/2))的某一范围内发生了跳跃屈曲,并得到轴对称跳跃屈曲荷载随壳体上荷载作用面积的增大而提高的结论.     

Exploration of the encased nanocomposites functionally graded porous arches: Nonlinear analysis and stability behavior

This paper explores the nonlinear stability mechanism of the functionally graded porous (FGP) arch reinforced by graphene nanocomposites. Both the pores and the nanocomposites are distributed symmetrically to the mid-surface of the arch but not uniformly in the cross-section so that the bending stiffness can be best improved. The arch is confined in an elastic medium with a radially-pointed concentrated load at the crown position. The confinement of the medium results in a symmetrical deformed shape of the arch, which can be described by an admissible displacement function. Associated with the thin-walled arch theory and the principle of minimum potential energy, analytical predictions are obtained to express the critical buckling load, as well as the hoop force and bending moment. Subsequently, a numerical model is developed to simulate the medium and the arch in ABAQUS software. By introducing the modified arc-length method, the equilibrium paths of the encased arch are traced. After comparison in terms of the critical buckling load and the equilibrium paths, it is found the numerical results are in good accordance with the analytical solutions. Finally, particular attention is paid to the parameters that may impact the buckling load, such as the porosity coefficient, the weight fraction, the central angle, the geometry of the Graphene platelets (GPLs) et al.     

非均匀法向荷载下半空间的二阶弹性效应问题

本文提供各向同性弹性半空间，在非均匀分布法向荷载下，二阶弹性效应的一个封闭形式解，运用积分变换方法，讨论了按Ｈｅｒｔｚ规律分布的荷载情形；导出了不可压缩各向同性弹性材料的极限解；算出了上述二阶弹性材料问题在ｚ方向的位移和法向应力数值。我们发现，与线弹性情形相比较，在二阶弹性材料中相应位移增大而法向应力减小。