Critical load of a bilayered orthotropic elastic–plastic conical shell with the change of the shell basic surface location

The paper presents the derivation of stability equations and the comparison of critical loads for an orthotropic elastic–plastic conical shell with proper location of the shell basic surface. Prandtl–Reuss plastic flow theory is accepted in the analysis. To derive the stability equations the variational method was accepted and Ritz method was used to solve the equations. Numerical results were obtained by the use of a special iterative algorithm of the elastic–plastic analysis implemented on the PC.     

An equivalent plasticity theory for jointed rock masses

A non-representative volume element (NRVE) approach to equivalent rock mass properties shows that the form of the elastic–plastic constitutive equations is the same for homogeneous material elements and multiple-material elements, subsequently homogenized. Thus the average stress and strain increments in an arbitrary jointed rock mass volume are related by {dσ}=([C*_ep]){dε} where σ is effective stress. The equivalent elastic-plastic properties matrix [C*_ep] is the sum of an equivalent elastic moduli matrix [C^*] and a plastic ‘correction’ matrix [C*_p, as usual. However, there are no equivalent plastic potentials Y^* or yield functions, failure criteria F^* or strengths. The equivalent elastic-plastic properties are constructed from the elastic moduli and strengths of the rock mass joints, the intact rock between and strain influence functions that relate local to overall average strains. Numerical examples that simulate laboratory-like tests on jointed rock cubes illustrate the approach.     

Large displacement stability analysis of elastic–plastic unsymmetrical sandwich cylindrical shells

The paper presents stability analysis of elastic–plastic, free supported sandwich cylindrical shell with unsymmetrical faces, loaded by longitudinal forces, transversal pressure, and shear. The J₂ incremental Prandtl–Reuss plastic flow theory constitutive relations were used in the analysis and it was assumed that geometrical strain-displacement relations are nonlinear ones. It was also assumed that the shell faces have different thicknesses and these are made of different isotropic compressible materials with linear stress-hardening. Active loading processes are accepted in the analysis, the stability equations are derived using strain energy formulation. Ritz method is used to solve the equations and an iterative computational algorithm was elaborated to get numerical results.     

An elastic–plastic analysis of large deflection of thin shell structure using a delta-sequence function

The analysis of thin shell under large deflection has complex problem associated with the geometrical and material nonlinearities, in which the solutions for stress and deformation are desired to obtain the same level accuracy. One of ordinary and powerful method for this subject is a finite element method that has generally inconvenience to be necessarily extensive calculation due to large number of freedom.This paper is concerned with the elastic–plastic analysis of thin shell structures by the hybrid method using a functional for the principle of modified complementary energy. For elastic–plastic materials, the numerical calculation could be well executed by the way that the stress distribution across the panel thickness is expressed as continuous function using the delta-sequence function. This approach introduces a considerable way for the reduction of computing volume. The proposed method was applied to discuss the resisting mechanism of thin shell structure under large deflection.     

Stability of orthotropic elastic–plastic open conical shells

The paper presents a derivation of the stability equation and the solution method of the problem for an orthotropic elastic–plastic open conical shell. The use is made of the constitutive relations of incremental plastic flow theory, elastic compressibility of the material, and Shanley concept of increasing load are taken into account in the consideration. A variation, strain energy method is used to derive the stability equation for bilayered open conical shell with nonuniform pre-critical stress distribution. The shell is free supported at the edges and the load acting the shell, in the form of longitudinal force and lateral pressure, is active one, i.e. unloading is not considered.     

A semi-analytical method for the elastic-plastic large deflection analysis of welded steel or aluminum plating under combined in-plane and lateral pressure loads

The aim of the present paper is to develop a semi-analytical method which can quickly and accurately compute the elastic–plastic large deflection response of welded steel or aluminum plating under a combination of biaxial compression/tension, biaxial in-plane bending, edge shear and lateral pressure loads, until the ultimate limit state is reached. The post-weld initial imperfections (i.e. initial deflection and residual stresses) are included in the method as parameters of influence. It is assumed that the plating is simply supported at all (four) edges which are kept straight. A unique feature of the developed method is that geometric nonlinearity associated with large deflection response of plating under combined loads is treated by analytically solving the nonlinear governing differential equations of the elastic large deflection plate theory, while material nonlinearity due to plasticity is dealt with implicitly by a numerical procedure. This approach reduces the magnitude of numerical computations, resulting in a saving of modeling effort and computing time. As another contribution, this paper investigates and discusses the ultimate strength characteristics of plating, by varying the plate properties and load combinations, based on elastic–plastic large deflection analysis using the developed method.     

A study of the influence of diameter and wall thickness of cylindrical tubes on their axial collapse

Axial compression experiments on aluminium cylindrical shells of diameter to thickness ratios (D/t) between 11.5 and 31.49 were conducted on a gravity drop hammer set up and Zwick machine. Typical histories of their deformation, variation of shell thickness along the fold length, inner and outer radii, folding parameter and size of fold, load–compression curves, energy absorbing capacity, initial peak load, and mean collapse loads obtained from the experiments are presented. Influence of the D/t values of the shell on their modes of collapse and energy absorption capacities are discussed. The shells are numerically simulated and analysed in detail by using the finite element code FORGE2. The material was modelled as rigid-viscoplastic. The experimental and computed results are compared. Typical contours of equivalent strain, equivalent strain rate, different stress components and velocity distribution are presented. The impact response of the shells is compared with their static response.     

Design by analysis of ductile failure and buckling in torispherical pressure vessel heads

Thin shell torispherical pressure vessel heads are known to exhibit complex elastic–plastic deformation and buckling behaviour under static pressure. In pressure vessel Design by Analysis, the designer is required to assess both of these behaviour modes when specifying the allowable static load. The EN and ASME boiler and pressure vessel codes permit the use of inelastic analysis in design by analysis, known as the direct route in the EN Code. In this paper, plastic collapse or gross plastic deformation loads are evaluated for two sample torispherical heads by 2D and 3D FEA based on an elastic-perfectly plastic material model. Small and large deformation effects are considered in the 2D analyses and the effect of geometry and load perturbation are considered in the 3D analysis. The plastic load is determined by applying the ASME twice elastic slope criterion of plastic collapse and an alternative plastic criterion, the Plastic Work Curvature criterion. The formation of the gross plastic deformation mechanism in the models is considered in relation to the elastic–plastic buckling response of the vessels. It is concluded that in both cases, design is limited by formation of an axisymmetric gross plastic deformation in the knuckle of the vessels prior to formation of non-axisymmetric buckling modes.     

Plastic buckling of complete toroidal shells of elliptical cross-section subjected to internal pressure

The plastic bifurcation buckling pressures of 60 internally-pressurised, perfect, complete toroidal shells of elliptical cross-section are given in the present paper, assuming elastic, perfectly plastic, material behaviour. The shell buckling programs employed in the computations were BOSOR 5 and INCA. Denoting the major-to-minor axis ratio by k, the numerical results show that the plastic buckling pressures are considerably lower than their elastic counterparts in the range 1.25≤k≤1.5 and are approximately equal to them for k=2.5. A limited study of the effects of non-axisymmetric initial geometric imperfections on the buckling pressures of the shells was also carried out using the INCA code. For the four cases studied the post-buckling behaviour was stable. This means that designers can use the buckling pressures given herein for perfect shells as a basis for their initial designs.     

Parametric stress distribution in shell-of-revolution sludge digesters of parabolic ogival form

Egg-shaped sludge digesters have become popular in relatively recent times owing to their superior functional performance and lower maintenance costs in comparison with conventional cylindrical digesters. These innovative structures are usually constructed as thin shells of revolution in concrete, designed to withstand principally the hydrostatic pressure loading from the contained liquid. As regards the precise shape of the egg shell, a number of mathematical shell surfaces may be envisaged, and the stress distribution will very much depend on the chosen form. In this paper, it is desired to explore the possible adoption of the parabolic ogival shell as a sludge digester. The stress distribution in such a shell is expressed in terms of a single governing parameter ξ, greatly facilitating the investigation. For various values of ξ covering the most practical range for egg-shaped digester shells, recommendations are made regarding the positioning of supports. Taking into account maximisation of tank capacity, minimisation of peak stress resultants in the shell, and ease of prestressing, the best range of ξ for parabolic ogival digester shells is identified. The overall conclusion is that from a structural and functional point of view, the parabolic ogival profile is suitable for adoption in the design of egg-shaped concrete sludge-digester shells.     

Recent research advances on ECBL approach.: Part I: Plastic–elastic interactive buckling of cold-formed steel sections

The objective of this two parts paper is to present some recent developments and applications of erosion of critical bifurcation load (ECBL) approach for the interactive buckling. Two different types of problems are analysed: (1) plastic–elastic interactive buckling which implements into the Ayrton–Perry interaction formula the plastic strength of the stub columns evaluated by means of local plastic mechanism analysis, and (2) elastic–elastic interactive buckling for members with perforations.The first part of the paper analyses the occurrence of local plastic mechanisms in cold-formed steel sections in compression, and how they can be implemented in the ultimate limit state analysis of the members. Actually, the failure of thin-walled cold-formed members in compression always occurs with a local plastic mechanism. Starting from this observation, the authors suggest to use in the interactive local-overall buckling analysis the sectional plastic mechanism strength instead of traditional ‘effective section’. The ECBL approach is used to implement the proposed interactive buckling model. Results are compared with those of other two recent methods, namely the direct strength method and plastic effective width approach. Relevant tests are used to evaluate the three methods. Comparisons with European and American design codes are also presented in the paper.     

Elastic buckling of barrelled shell under external pressure

The aim of the paper is to present a procedure for design of a family of shells of revolution of constant mass and, as a next step, of constant volume. As a reference a cylindrical shell is taken into consideration. By decreasing the value of the meridional radius of curvature R₁, which for cylindrical shell equals infinity, barrelled shells are created up to the spherical shell for which both meridional and circumferential radii of curvature are equal (R₁=R₂). A numerical example of using the presented procedure is considered. Then for the family of shells of revolution of constant mass a buckling analysis using FEM method is carried out. Results of the analysis show the relationship between the radius of curvature of the shell R₁ and the critical load p_cr in the case of uniform external pressure.     

Numerical and experimental investigations of the response of aluminum cylinders with a cutout subject to axial compression

Understanding how a cutout influences the load bearing capacity and buckling behavior of a cylindrical shell is critical in the design of structural components used in automobiles, aircrafts, and marine applications. Numerical simulation and analysis of moderately thick and thin unstiffened aluminum cylindrical shells (D/t=45, 450 and L/D=2, 5, 10), having a square cutout, subjected to axial compression were systematically carried out in this paper. The investigation examined the influence of the cutout size, cutout location, and the shell aspect ratio (L/D) on the prebuckling, buckling, and postbuckling responses of the cylindrical shells.An experimental investigation on the moderately thick-walled shells was also carried out. A good correlation was observed between the results obtained from the finite element simulation and the experiments. Furthermore, empirical equations, in the form of a ‘buckling load reduction factor’ were developed using the least square regression method. These simple equations could be used to predict the buckling capacities of several specific types of cylindrical shells with a cutout.     

Unified solution of limit loads of thick wall cylinder subject to external pressure considering strain softening

Based on the unified strength theory [1], a unified strength criterion for strain softening materials, such as concrete or rock, was derived, and the elastic and plastic limit loads of a thick-walled cylinder made of these materials subject to external pressure were also given. In addition, the influence of some factors on the limit loads of such cylinders as the ratio of the external radius to internal radius, r _b/r _a, the coefficient b, which reflects the effect of medium principal stress and the normal stress of the relevant surface on the material destroy degree, the ratio of tensile strength to compressed strength of the material, α, and the damage variable β were discussed in detail. Some examples were given and some meaningful results were obtained. __________ Translated from Journal of Hunan University (Natural Science), 2006, 33(2): 1–5 [译自: 湖南大学学报 (自然科学版)]     

The effect of elastic foundations on the nonlinear buckling behavior of axially compressed heterogeneous orthotropic truncated conical shells

The buckling problem of a heterogeneous orthotropic truncated conical shell subjected to an axial load and surrounded by elastic media is analyzed based on the finite deformation theory. Using von-Karman nonlinearity, the governing equations of elastic buckling of heterogeneous orthotropic truncated conical shells surrounded by elastic media are derived. The governing equations are solved using superposition and Galerkin methods and obtained expressions for upper and lower critical axial loads. The influences of elastic foundations, heterogeneity, orthotropy and geometric characteristics on the upper and lower critical loads of conical shells with and without elastic foundations are studied in detail.     

Buckling and postbuckling behaviors of imperfect cylindrical shells subjected to torsion

Buckling and postbuckling behaviors of imperfect cylindrical shell subjected to torsion are investigated. The governing equations are based on the Karman–Donnell-type nonlinear differential equations. A boundary layer theory of shell buckling is applied to obtain the analytic solutions that meet the boundary conditions strictly. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. Numerical results reveal that the current theory gives quite good estimates of the postbuckling paths of cylindrical shells. The effects of the geometric parameters on the buckling and postbuckling behaviors of the cylindrical shells are analyzed. It is confirmed that the postbuckling equilibrium paths of cylindrical shells subjected to torsion are unstable and the relatively shorter shells have higher postbuckling equilibrium paths. Finally, the effects of the initial imperfections on the buckling and postbuckling behaviors of the cylindrical shells are clarified. The illustrated results of the imperfect shells with different initial transverse deflections show that extremely small imperfections do indeed reduce the buckling loads and make the postbuckling equilibrium paths be lower. The buckling and postbuckling of cylindrical shells under torsion exhibit obvious imperfect sensitivity. Furthermore, the effects become greater following with the larger imperfections.     

Stress–strain–electrical resistance effects and associated state equations for uniaxial rock compression

This paper reports stress–strain–electric resistance experiments for diabase, limestone and marble containing NaCl solution during the whole process of uniaxial compression. We obtained the complete testing data for the stress–strain curve and the associated electrical resistance–strain curve. The change caused by internal cracking of the rock causes the corresponding variation of rock electrical resistance. There is a minimum value for all the electric resistance–strain curves, corresponding to the cracking stress σ_c or the initial cohesion c_i. Based on the experimental results and stochastic property analyses of the rock fracture variation, we put forward a group of state equations for rock in sections to express the characteristics of rock during the whole process of uniaxial compression. The three variables, stress, strain and electrical resistance, together with data-fitted parameters, α₁,α₂ and β, are contained in the equations. The equations are used to express the inelastic response which intensifies with the propagation of cracking.     

Transformed initial dent as a trigger of the post-buckling process

A procedure for determination of critical loads of imperfect shells is proposed with allowance for the technique of stability and load carrying capacity of incomplete shells in an inhomogeneous stress–strain state developed earlier in the monographs [Gavrylenko GD. Stability of ribbed cylindrical shells in inhomogeneous stress–strain state. Kiev: Nauk. Dumka; 1989. 176pp [in Russian]; Gavrylenko GD. Stability of ribbed shells of incomplete form. Ukraine: Institute of Mathematics of NAS; 1999. 190pp [in Russian]].New method of numerical estimation of carrying capacity of a shell is suggested and realized. Results from a joint theoretical and experimental investigation of the buckling of cylindrical shells containing localized dent damage are presented.     

Optimum design of laminated composite plates to maximize buckling load using MFD method

This paper presents optimal design of simply supported laminated composite plates subject to given in-plane static loads for which the critical failure mode is buckling. The objective function is to maximize the buckling load capacity of laminated plates and the fiber orientation is considered as design variable. The first-order shear deformation theory is used for the finite element analysis. In this paper, the effects of bending–twisting coupling are also included for the buckling optimization. The modified feasible direction method is used as an optimization method. Also, computer programs are coded in MATLAB and Golden Section method is adapted in this program for the optimal design of laminated plates for maximum buckling load. The effect of width-to-thickness ratio, aspect ratio, number of layers, material anisotropy, load ratios (N_y/N_x), uncertainties in material properties and functionally graded materials on the results is investigated and compared.