GBT formulation to analyse the behaviour of thin-walled members with variable cross-section

The generalised beam theory (GBT) provides a general solution for the linear/non-linear analysis of prismatic thin-walled structures, using bar elements capable of describing the cross-section rigid-body motions and distortions. Nowadays GBT is fully developed for thin-walled members having a large variety of constant cross-sections. This paper provides the extension of GBT for the special case of thin-walled members with variable open cross-section and the limits of its applicability.     

GBT formulation to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections

This paper presents the derivation, validates and illustrates the application of a Generalised Beam Theory (GBT) formulation developed to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections. Following a brief overview of the conventional GBT, one addresses in great detail the modifications that must be incorporated into its cross-section analysis procedure, in order to be able to handle the ‘branching’ points — they concern mostly issues related to (i) the choice of the appropriate ‘elementary warping functions’ and (ii) the determination of the ‘initial flexural shape functions’. The derived formulation is then employed to investigate the local-plate, distortional and global buckling behaviour of (i) simply supported and fixed asymmetric E-section columns and (ii) simply supported I-section beams with unequal stiffened flanges. For validation purposes, several GBT-based results are compared with ‘exact’ values, obtained by means of finite strip or shell finite element analyses.     

Generalised beam theory to analyse the buckling behaviour of circular cylindrical shells and tubes

A formulation of generalised beam theory (GBT) developed to analyse the elastic buckling behaviour of circular hollow section (CHS) members (cylinders and tubes) is presented in this paper. The main concepts involved in the available GBT are adapted to account for the specific aspects related to cross-section geometry. Taking into consideration the kinematic relations used in the theory of thin shells, the variation of the strain energy is evaluated and the terms are physically interpreted, i.e., they are associated with the geometric properties of the CHS. Besides the set of shell-type deformation modes, the formulation also includes axisymmetric and torsion deformation modes. In order to illustrate the application and capabilities of the formulated GBT, the local and global buckling behaviour of CHS members subjected to (i) compression (columns), (ii) bending (beams), (iii) compression and bending (beam-columns) and (iv) torsion (shafts), is analysed. Moreover, the GBT results are compared with estimates obtained by means of shell finite element analyses and are thoroughly discussed.     

Re-interpreting simultaneous buckling modes of axially compressed isotropic conical shells

Elastic stability of shell structures under certain loading conditions is characterized by a dramatically unstable postbuckling behavior. The presence of simultaneous ‘competing’ buckling modes (corresponding to the same critical buckling load) is understood to be largely responsible for such behavior. In this paper, within the framework of linear bifurcation eigenvalue analysis and Donnell shallow shell theory, the presence of simultaneous buckling modes in axially compressed isotropic cones is determined using the semi-analytical method of Galerkin. The results are presented in the plane of the dimensionless reciprocal meridional and circumferential buckling half wavelengths, and are compared with the locus of simultaneous buckling modes of axially compressed cylinders, described by the so-called Koiter circle. By using an optimizing procedure, it is shown that the cluster of simultaneous buckling modes in cones is well described by the Koiter circle of an equivalent cylinder with appropriate length and radius. Such optimizing values of length and radius allow us to gain some insight into the simplifying treatment of the buckling of cones through the concept of equivalent cylinder.     

Effects of openings of the buckling of cylindrical shells subjected to axial compression

Experimental and numerical methods are used to study the stability problem of cylindrical shells with cut-outs. The paper presents parametric research of the shape (square, rectangular, circular), the dimensions (axial and circumferential sizes, diameter) of the hole. The effect of the location and the number of the holes are also studied. The analysis indicates that the critical load is sensitive to the opening angle or circumferential size of the hole. The function (critical load-opening angle) is linear for large openings and independent of the geometrical imperfections of the shell. However for small openings, it is necessary to take into account the coupling between the initial geometrical imperfections and the openings. The linear approach does not fit because of the importance of the evolution of the displacements near the openings. These results will be used for the development of European rules.     

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.     

A study of elastic-plastic buckling of cylindrical shells under torsion

The elastic-plastic buckling of cylindrical shells under torsion is analysed with a deep thick-shell model under various boundary conditions. The word ‘deep’ means that in the general equations of equilibrium the three non-linear terms that involve the torsional force are all retained for the buckling analysis. In the Donnell-type shallow-shell theory, however, only one of such terms is retained. The word ‘thick’ means that in calculating strains and stress resultants the factor (1+z/R) is retained. This factor results from the trapezoid-like shape of the cross-section and is usually neglected in the thin-shell theory. For boundary conditions, not only the conventional geometrical boundary conditions, which are in terms of displacements and rotations, but also the mechanical boundary conditions, which are in terms of forces and moments, are considered. The numerical results of examples assess the effect of the additional non-linear terms, the effect of the factor (1+z/R), and the effect of the mechanical boundary conditions.     

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.     

Elastic-plastic collapse of non-uniform cylindrical shells subjected to uniform external pressure

The objective of this paper is to derive analytical solutions for the elastic buckling and plastic collapse pressures of a cylindrical shell with reduced thickness over part of its circumference. The section of reduced thickness is used to represent a corroded region in a pipe. The proposed solutions are extensions of Timoshenko's solutions for the elastic-plastic collapse of a linear elastic, perfectly plastic cylindrical shell subjected to uniform external pressure. A modified interaction formula for the fully plastic membrane forces and bending moments in the non-uniform cylinder has been proposed for plastic collapse. A parametric study shows that the elastic buckling pressure decreases smoothly with corrosion angle when the corrosion depth is less than 0.5t. When the corrosion depth is greater than 0.5t, the elastic buckling strength first decreases very rapidly with corrosion angle. Furthermore, the elastic buckling pressure decreases uniformly with corrosion depth when the corrosion angles are greater than 30°, while the elastic buckling strength decreases more rapidly at higher corrosion depths when corrosion angles are less than 30°. Another parametric study on a steel pipe shows that the initial and fully plastic yield pressures both decrease monotonically with corrosion depth for a given corrosion angle and imperfection.     

Global buckling analysis of plane and space thin-walled frames in the context of GBT

This paper reports research work concerning the use of Generalised Beam Theory (GBT) to analyse the global buckling behaviour of plane and space thin-walled frames. Following a brief overview of the main concepts and procedures involved in the performance of a GBT buckling analysis, one presents in detail the formulation and numerical implementation of a GBT-based beam finite element that includes only the first four (rigid-body) deformation modes — namely, one describes (i) the kinematical models developed to simulate the warping transmission at frame joints connecting two or more non-aligned U- and I-section members, (ii) the procedures adopted to handle the effects stemming from the non-coincidence of the member centroidal and shear centre axes (cross-sections without double symmetry), and (iii) the definition of joint elements, which involves providing a relation between the connected member GBT degrees of freedom and the joint generalised displacements. Finally, one presents and discusses numerical results that make it possible to illustrate the application and show the capabilities of the above GBT-based finite-element formulation and implementation. For validation purposes, the GBT-based results (critical buckling loads and mode shapes) are also compared with values yielded by shell (mostly) and beam finite element analyses carried out in the code ANSYS.     

Buckling of thin-walled conical shells under uniform external pressure

Shells are for the most part the deep-seated structures in manufacturing submarines, missiles, tanks and their roofs, and fluid reservoirs; therefore it is a matter of concern to bring about some basic regulations associated with the existing codes. Above all, truncated conical shells (frusta) and shallow conical caps (SCC) subjected to external uniform pressure when discharging liquids or wind loads are discussed closely in this paper concerning and thrashing out their empirical nonlinear responses along with envisaging numerical methods in contrast. The buckling aptitude of shells is contingent upon two leading geometric ratios of “slant-length to radius” (L/R) and “radius to thickness” (R/t). In this paper, developing six frusta and four shallow cap specimens and their relevant FE models, use is made of laboratory modus operandi to enumerate buckling elastic and plastic responses and asymmetric imperfection sensitivity, whose adequacy has been reckoned through comparisons with arithmetical and numerical data correspondingly. These obtained upshots were aimed at validating and generalizing the data for unstiffened truncated cones and SCC in full scale.     

Thermal buckling of FGM shells resting on a two-parameter elastic foundation

This paper focuses on the thermal buckling analysis of FGM shells resting on the two-parameter elastic foundation. Material properties of the constituents are graded in the thickness direction according to the power-law distribution. The surrounding elastic medium is modeled as an elastic foundation of the Pasternak-type. After giving the fundamental relations, the stability and compatibility equations of an FGM truncated conical shell subjected to thermal load and resting on a two-parameter elastic foundation have been derived. Critical temperature differences of FGM truncated conical shells with or without elastic foundations subjected to non-linearly distributed temperature across the thickness of the shells are obtained by solving eigenvalue problems. The appropriate formulas for FGM cylindrical shells with or without elastic foundations are found as a special case. In order to assure the accuracy of the present study, convergence properties of the critical temperature are examined in detail.     

Dynamic buckling of fiber composite shells under impulsive axial compression

The paper deals with dynamic buckling due to impulsive loading of thin-walled carbon fiber reinforced plastics (CFRP) shell structures under axial compression. The approach adopted is based on the equations of motion, which are numerically solved using a finite element code (ABAQUS/Explicit) and using numerical models validated by experimental static buckling tests. To study the influence of the load duration, the time history of impulsive loading is varied and the corresponding dynamic buckling loads are related to the quasi-static buckling loads. To analyse the sensitivity to geometric imperfections, the initial geometric imperfections, measured experimentally on the internal surface of real shells, are introduced in the numerical models. It is shown numerically that the initial geometric imperfections as well as the duration of the loading period have a great influence on the dynamic buckling of the shells. For short time duration, the dynamic buckling loads are larger than the static ones. By increasing the load duration, the dynamic buckling loads decrease quickly and get significantly smaller than the static loads. Since the common practice is to assume that dynamic bucking loads are higher than the static ones, which means that static design is safe, careful design is recommended. Indeed, taking the static buckling load as the design point for dynamic problems might be misleading.     

约束屈曲支撑在单层网壳减震控制中的应用研究

采用有限元分析程序ANSYS,分别对带有约束屈曲支撑(Bckling Restrained Braces,简称BRBs)的单层柱面网壳、单层球面网壳进行动力时程响应分析,分别考虑了水平及竖向地震作用.比较各种情况下网壳节点位移和杆件内力变化情况,综合研究了约束屈层支撑在四边支承单层柱面网壳和单层球面网壳的适用性,探讨了如何布置约束屈层支撑,分析了约束屈层支撑减震性能.     

Elastic/plastic buckling of a composite flat plate subjected to uniform edge compression

A theoretical approach, based on the plastic theory, has been developed for studying the elastic/plastic buckling behavior of a simply supported rectangular anisotropic plate subjected to edge compression. The said approach was employed to determine the critical buckling stresses of three different types of fiber reinforced composite plates, namely, carbon epoxy, glass epoxy and boron aluminum plates. Since the proposed approach is for a preliminary prediction of critical buckling loads of composite plates, the results obtained are deemed acceptable compared with those obtained by other authors experimentally.     

Plastic buckling of dented steel circular tubes under axial compression: An experimental study

This paper examines the effect of large local imperfections, known as dents, on the plastic buckling capacity of short steel tubes under axial compression. A total of 11 tests on such short columns were carried out. The specimens were indented through a separate process and the ultimate axial capacity was subsequently obtained through compression tests. Dent imperfections with various depths were introduced to different locations on the body of the specimens. Plastic buckling modes as well as the ultimate capacity of the specimens were thoroughly investigated. The adverse effect of such a local damage on the load carrying capacity was quantified for different values and types of imperfections.     

Local buckling of steel plates in rectangular CFT columns with binding bars

This paper presents a theoretical study on the elastic local buckling of steel plates in rectangular concrete-filled steel tubular (CFT) columns with binding bars under axial compression. It is assumed that the unloaded edges of the steel plate are elastically restrained against rotation, whereas the loaded edges are clamped. Based on the energy method, the formulas for elastic local buckling strength of the steel plate in rectangular CFT columns under axial compression are derived, which are calibrated with the experimental results. Then the formulas are employed to study the elastic local buckling of steel plates in rectangular CFT columns with binding bars under axial compression. It is shown that the binding bars can increase the local buckling coefficient, which results in improvement of the capacity of the steel plate to resist local buckling. Finally, appropriate spacing of binding bars, appropriate limitation for aspect ratio and corresponding appropriate limitation for width-thickness ratio are suggested for rectangular CFT columns with binding bars under axial compression.     

Multiobjective optimization of orthogonally stiffened cylindrical shells for minimum weight and maximum axial buckling load

In the present research, the weight and axial buckling optimization of orthogonally stiffened cylindrical shells is carried out by the Genetic Algorithm. Constraints include two nondimensional functions of weight and buckling load in such a way that the stiffened shell has no increase in the weight and no decrease in the buckling load with respect to the initial unstiffened shell. In analytical solution, the Rayleigh–Ritz energy procedure is applied and the stiffeners are treated as discrete members. The optimization is implemented for shells with simply supported end conditions stiffened by four shapes of stiffeners including rectangular-, cee-, I-, and hat-shaped ones. The results show that the I-section and rectangular-section stiffeners are, respectively, the most and the least efficient in designing stiffened cylindrical shells for minimum weight and maximum critical axial buckling load.     

New method of inelastic buckling analysis for steel frames

In general, the concept of bifurcation stability cannot be used to evaluate the critical load of typical steel frames that have geometric imperfections and primary bending moment due to transverse loads. These cases require a plastic zone or plastic hinge analysis based on the concept of limit-load stability instead. However, such analyses require large computation times and complicated theories that are unsuitable for practical designs. The present paper proposes a new method of inelastic buckling analysis in order to determine the critical load of steel frames. This inelastic analysis is based on the concept of modified bifurcation stability using a tangent modulus approach and the column strength curve. Criteria for an iterative eigenvalue analysis are proposed in order to consider the primary bending moment as well as the axial force by using the interaction equation for beam-column members. The validity and applicability of the proposed inelastic buckling analysis were evaluated alongside elastic buckling analysis and refined plastic hinge analysis. Simple columns with geometric imperfections and a four-story plane frame were analyzed as benchmark problems. The results show that the proposed inelastic buckling analysis suitably evaluates the critical load and failure modes of steel frames, and can be a good alternative for the evaluation of critical load in the design of steel frames.     

Combined axial and pressure buckling of end supported shells of revolution

A reduced stiffness theoretical analysis of the imperfection sensitive elastic buckling for end supported shells of revolution is extended to the case of arbitrary combinations of axial and radial pressure loading. Depending upon the shell and loading parameters, the potential reductions in load capacity due to imperfections are shown to involve two distinct forms of post-buckling loss of stiffness. Lower bounds in each of these regimes are provided by appropriate reduced stiffness models, and shown by comparisons with available test data to be reliable even for relatively perfect test models. By attributing reductions in load carrying capacity to weakened end support conditions, it is suggested that past interpretations of these tests may have underestimated the deleterious effects of initial imperfections.