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.     

Transverse-stiffener requirements for the post-buckling behaviour of a plate in shear

The optimum stiffener rigidity in a transversely-stiffened web under pure shear is sought in the post-critical range. An energy-based large-deflection scheme is employed to study the post-critical strength of the system up to the initiation of material yielding for the usual range of plate geometries and varying stiffener rigidities. Several criteria for “optimum” rib stiffness are assessed including stiffener deflection, the effective halving of the plate length and, finally, the strength of the plate itself. Tentative conclusions are reached on the basis of comparing optimal rib rigidities for the two instances of elastic and post-buckling regimes.     

Flexural–torsional post-buckling analysis of thin-walled elements with open sections

The post-buckling analysis of thin-walled elements under compression is investigated. A nonlinear model is developed by using nonlinear relationships between curvatures and bending moments. Warping and shortening effects are considered in the torsion equilibrium equation. Based on Galerkin's method, a nonlinear algebraic system is obtained for simply supported boundary conditions. The three resulting equations in bending and torsion are highly coupled and the Newton–Raphson algorithm with displacement control is adopted for the solution. The post-buckling equilibrium curves are obtained for various sections shapes, such as bisymmetric and monosymmetric sections. The importance of the shortening effect is outlined.     

Finite element based coupled mode initial post-buckling analysis of a composite cylindrical shell

This paper presents a finite element formulation of Koiter's initial post-buckling theory using a multi-mode approach. Initial post-buckling theory provides direct information about the imperfection sensitivity of a structure under compressive loading, and is also the basis of a nonlinear reduced order model. The objective of the present work is to illustrate the capability of the implementation for buckling analysis of shell structures including modal interaction. A coupled mode initial post-buckling analysis for a composite cylindrical shell under axial compression, including the effect of a nonlinear pre-buckling state, has been carried out using a small number of representative modes. For small imperfection amplitudes the limit-point buckling loads obtained with the reduced order model compare reasonably well with full model nonlinear analysis, illustrating that a fast prediction of the coupled mode response of imperfect shells is possible using the approach proposed.     

An analytical study on post-buckling behaviour of imperfect sandwich panels subjected to uniform thermal stresses

The governing equations for determining thermal buckling of imperfect sandwich plates are developed using the large deflection theory and considering first shear deformation principles. The equations are then solved via an analytical method comprising infinite summation of trigonometric series and defining in-plane and out-of-plane displacement. After verification of the proposed method and convergence studies, the effects of parameters like plate aspect ratio, material properties and layer setting angles on buckling stress and the post-buckling path are studied. Numerical results show that the number of trigonometric terms does not affect the buckling thermal stress or the post-buckling path of sandwich plates having face layer setting angles less than 15°. Results also show that the buckling stress and post-buckling path for complementary layer setting angles are similar and that the rate of variation of bifurcation points for different face layer setting angles of sandwich panels strongly depend on material properties, especially on the face and core stiffness. Furthermore, for aspect ratios greater than five, buckling thermal stress remains about constant.     

Semi-analytical models for the post-buckling analysis and ultimate strength prediction of isotropic and orthotropic plates under uniaxial compression with the unloaded edges free from stresses

This paper introduces two semi-analytical models developed for the nonlinear analysis of stability of isotropic and orthotropic plates under uniaxial compression. The possibility of considering fully free in-plane displacements at longitudinal edges (or unloaded edges) is the innovation of these models over existing models, where these displacements are always assumed constrained to remain straight. Contributions for the large deflection theory of plates related to the derivation of analytical solutions for the Airy stress function which satisfy Marguerre׳s equations for isotropic and orthotropic plates are presented. Namely, the extension of the Coan and Urbana solution for isotropic plates in order to consider all the terms of the unknown amplitudes of the out-of-plane displacements and the derivation of a solution for orthotropic plates. Comparisons between the semi-analytical model and nonlinear finite element model results are presented in order to discuss the effect of in-plane displacement boundary conditions on behaviour and strength of plates similar to bottom flanges used in steel box girder bridges. This study shows that the semi-analytical models have a clear potential to provide accurate solutions, requiring only a short computer time. It is also shown that the in-plane displacement boundary conditions for the unloaded edges significantly influence the behaviour and strength of plates and this problem cannot be neglected in the definition of the design rules.     

Simplified amplification factors representing material and geometric inelasticity in frame instability

An approach for assessing the inelastic, in-plane, collapse load of framed structures using amplification factors is extended to a broader range of material inelasticity. This manual method is based on the amplification factor approach for elastic, rectangular frames in the Canadian steel design code, but allows for material and geometric nonlinearity of both rectangular and non-rectangular frames on a consistent basis. Improved accuracy is achieved with a linear variation of the incremental amplification factor between a lower limit at the onset of inelastic behaviour and an upper limit at which plastic collapse occurs. A method is demonstrated for notionally separating the effects of material and geometric nonlinearity in experimental measurements or advanced finite element analyses. This is used for validating the proposed inelastic models for a steel frame and a reinforced concrete frame.     

Elastic post-buckling analysis of compressed plates using a two-strip model

This paper shows that a simple two-strip model can be used to describe the post-buckling behaviour of simply supported rectangular plates subjected to uni- and biaxial compression, provided the ratio of the post-buckling to pre-buckling stiffness of the perfectly flat plate is known. The formulas for this two-strip model are derived in two ways: by using a force method, and by using an energy method. The two-strip model is compared to other available solutions in literature and it is shown that the model can be regarded as a modification of the lower bound plate analysis method.     

Considerations on the numerical analysis of initial post-buckling behaviour in plates and beams

In this work, an exceedingly simple examination of the von Kármán theory for the buckling analysis of thin plates is carried out. It is shown that the von Kármán assumption, based on the interaction between axial and flexural deformation, is intrinsically less “robust” in capturing the target phenomenology than the usual treatment of Euler inextensible strut. Finally, the same observation is carried out in the case of a hinged cantilever, which constitutes the simplest structural example of a similar kind.     

A numerical and experimental investigation on composite stiffened panels into post-buckling

The structural behaviour of composite stiffened flat panels under axial compression is here investigated up to collapse. The panel configuration is designed to buckle once the limit load is reached and to work in post-buckling until the ultimate load. The design phase is based on the use of four different kinds of finite element analyses: eigenvalue, non-linear static with modified Riks’ method and both implicit and explicit dynamic analyses. Once the final configuration is identified, two specimens are manufactured. The initial geometrical imperfections are measured and analyzed, then axial compression tests are performed until collapse. As foreseen by the numerical analyses, experimental results prove the ability of the panels designed to work in the post-buckling field until collapse which takes place due to the failure of the stiffener blades. Finally, the measured initial imperfections are included in the model significantly increasing the numerical–experimental correlation.     

Evaluation of seismic behaviour of steel special moment frame buildings with vertical irregularities

This paper concentrates on investigating the seismic behaviours of vertically irregular steel special moment frame (SMF) buildings by comparison with the regular counterpart. All buildings in this study were assumed to be located in Los Angeles, and subjected to 20 earthquake ground motions with a seismic hazard level of 2% probability of exceedance in 50 years. The beam–column connections of the buildings were modelled to consider the panel zone deformation. In addition, a ductile connection model accompanied by strength degradation was incorporated to the analysis programme in an effort to obtain more accurate response results. Three types of irregularities (mass, stiffness and strength irregularity) specified as vertical irregularities in the IBC 2000 provisions were imposed to the original building. Nonlinear static and dynamic analyses were performed, and the confidence levels of which the performance objective will be satisfied were calculated as well. The effects of different irregularity types and levels on the seismic behaviours of the buildings were investigated and discussed in terms of the height‐wise distribution of storey drifts, maximum storey drift demands, global collapse storey drift capacities and confidence levels. Copyright © 2010 John Wiley & Sons, Ltd.     

Lateral post-buckling analysis of thin-walled open section beams

Thin-walled beams with open sections are studied using a nonlinear model. This model is developed in the context of large displacements and small deformations, by accounting for bending-bending and bending-torsion couplings. The warping and shortening effects are considered in the torsion equilibrium equation. The governing coupled equilibrium equations obtained from Galerkin’s method are solved by a Newton–Raphson iterative process. It is established that the buckling loads are highly dependent on the pre-buckling deformations of the beam. The bifurcated branches are unstable and strongly influenced by shortening effects. Some comparisons are presented with the solutions commonly used in linear stability, like in the standard European steel code (Eurocode 3). The regular solutions appear to be very conservative, especially for I sections with large flanges.     

The computational post-buckling analysis of fuselage stiffened panels loaded in shear

Fuselage panels are commonly fabricated as skin-stringer constructions, which are permitted to locally buckle under normal flight loads. The current analysis methodologies used to determine the post-buckling response behaviour of stiffened panels relies on applying simplifying assumptions with semi-empirical/empirical data. Using the Finite Element method and employing non-linear material and geometric analysis procedures it is possible to model the post-buckling behaviour of stiffened panels without having to place the same emphases on simplifying assumptions or empirical data. Previous work has demonstrated that using a commercial implicit code, the Finite Element method can be used successfully to model the post-buckling behaviour of flat riveted panels subjected to uniform axial compression. This paper expands the compression modelling procedures to flat riveted panels subjected to uniform shear loading, investigating element, mesh, idealisation and material modelling selection, with results validated against mechanical tests. The work has generated a series of guidelines for the non-linear computational analysis of flat riveted panels subjected to uniform shear loading, highlighting subtle but important differences between shear and compression modelling requirements.     

Analytical calculation of local buckling and post-buckling behavior of isotropic and orthotropic stiffened panels

A methodology for the analytical assessment of local buckling and post-buckling behavior of isotropic and orthotropic stiffened plates is presented. The approach considers the stiffened panel segment located between two stiffeners, while the remaining panel is replaced by equivalent transverse and rotational springs of varying stiffness, which act as elastic edge supports. A two-dimensional Ritz displacement function (pb-2 Ritz) is utilized in the solution of the local buckling problem of isotropic and laminated symmetric composite panels with arbitrary edge boundary conditions. The buckling analysis of the segment provides an accurate and conservative prediction of the panel local buckling behavior. Consequently, the developed methodology is extended in the prediction of the post-buckling response of stiffened panels of which the skin has undergone local buckling. Of high importance for the calculation of the post-buckling behavior is the selection of appropriate boundary conditions for the structural members analyzed. A comparison of the present methodology results to respective finite element (FE) results has shown a satisfactory agreement.     

A semi-analytical model for local post-buckling analysis of stringer- and frame-stiffened cylindrical panels

A fast semi-analytical model for the post-buckling analysis of stiffened cylindrical panels is presented. The panel is comprised of a skin (shell) and stiffeners in both longitudinal (stringers) and circumferential direction (frames). Local buckling modes are considered where the skin may buckle within a bay and may induce rotation of the stiffeners. Stringers and frames are considered as structural elements and are thus not ‘smeared’ onto the skin. Large out-of-plane deflections and thus non-linear strain–displacement relations of skin and stiffeners are taken into account. The displacements of skin and stiffeners are approximated by trigonometric functions (Fourier series). First, a linear buckling eigenvalue analysis is carried out and some combination of buckling eigenmodes is chosen as imperfection. Then the load history is started and the Fourier coefficients are determined by minimizing the stiffened panel's energy at each load level. A curve-tracing algorithm, the Riks method, is used to solve the equations. The present model can be used to assess the post-buckling behavior of stiffened panels, for example, aircraft fuselage sections.     

Mechanical and thermal post-buckling analysis of FGM rectangular plates with various supported boundaries resting on nonlinear elastic foundations

Mechanical and thermal post-buckling analysis is presented for FGM rectangular plates resting on nonlinear elastic foundations using the concept of physical neutral surface and high-order shear deformation theory, and investigations on post-buckling behavior of FGM rectangular plates with two opposite simply supported edges and other two opposite clamped edges are also new. Approximate solutions of FGM rectangular plates are given out using multi-term Ritz method, and influences played by different supported boundaries, foundation stiffnesses, thermal environmental conditions and volume fraction index are discussed in detail. It is worth noting that the effect of nonlinear elastic foundation is small at the pre-buckling and initial post-buckling state and is significant with increasing deflection at the deep post-buckling state. Especially, comparisons of post-buckling for FGM rectangular plates resting on nonlinear elastic foundations with movable simply supported edge subjected to compression acting on the geometric middle surface and the physical neutral surface are innovative, and may be helpful to clarify typical mistakes in literature.     

Post-buckling analysis of corrugated panels in the presence of multiple interacting modes

In this paper, the effect of the interaction between one overall buckling mode and many local modes on the post-buckling behaviour of uniformly compressed corrugated sheets is analysed by means of the general theory of elastic stability. The analysis is restricted to third order terms of the energy expansion and, therefore, it can be fruitfully applied to the investigation of structures with asymmetric post-buckling behaviour only. Initial imperfection effect is taken into account. By using the finite strip method, an extensive parametric analysis is performed.     

带L形横撑的大跨CFST桁式拱桥稳定性分析

为研究新型L形横撑在大跨度钢管混凝土(CFST)桁式拱桥稳定中的有效性,以合江三桥为工程背景,采用数值计算和理论分析相结合的方式对其受力性能和稳定性进行比较分析,并利用正交试验和方差分析方法对L形横撑在大跨度CFST桁式拱桥稳定中的显著性进行了检验。结果表明：①对于L形拓扑横撑类型,当K形斜撑设置在主拱肋上弦且靠近拱脚一侧时,主拱的稳定性最好;②与常用的K形、X形和米字形横撑相比,L形横撑对主拱最大轴力、最大面内弯矩、最大面外弯矩、最大应力和竖向变形的影响均较小,仅对主拱水平变形有一定影响;③当各参数在合理的取值范围内时,使用L形横撑代替常用的K形、X形或米字形横撑,不会显著降低大跨度CFST桁式拱桥的分支点和极值点稳定性,工程师在该类拱桥的稳定设计中可以放心选用。     

The control of the post-buckling response in thin composite plates using smart technology

Restoration forces, associated with embedded activated pre-strained shape memory alloy wires, have successfully been employed to enhance the post-buckling behaviour of various laminated plate structures. An extensive experimental and numerical programme has been conducted, the results of which will be presented. The manufacturing methodology of the hybrid SMA/carbon/epoxy plates is outlined. Such specimens feature 0.4-mm diameter shape memory alloy wires located within tubing at desired locations. Numerical thermal analysis has been employed to predict the non-uniform temperature profile, attributed to shape memory alloy activation through resistive heating, within the laminates. Structural finite element analysis has been employed to determine the hybrid plates' adaptive response while under the influence of a uniaxial compressive load in excess of its critical buckling value. It is shown that, utilising the considerable control authority generated, even for a small actuator volume fraction, the out-of-plane displacement of the post-buckled laminates can be significantly reduced. Such displacement alleviation allows for load redistribution away from the specimens' unloaded edges. With the increase in use of composite materials within aerospace platforms, it is envisaged that the hybrid adaptive SMA/laminate configuration will extend the operational performance over conventional materials and structures, particularly when the structure is exposed to an elevated temperature.