Buckling and ultimate strength criteria of stiffened shells under combined loading for reliability analysis

The work presented in this paper forms part of a broader task in establishing a guide to serve as technical documentation for buckling and ultimate strength assessment of various types of marine structural components using the best state-of-the-art knowledge for extreme environmental loading. This paper concentrates on buckling and ultimate strength assessment of ring stiffened shells and ring and stringer stiffened shells involving various modes of buckling and under various loading like axial compression, radial pressure and combined loading. Comparisons are made with screened test data, which have realistic imperfections and various radius to thickness ratio values in the range generally used in offshore structures. The statistical data of model uncertainty factors in terms of bias and coefficient of variation (COV) are calculated and may be used in a further reliability study. Comparisons are also made with the codified rules, API BUL 2U and DNV buckling strength of shells.     

Non-probabilistic reliability-based design optimization of stiffened shells under buckling constraint

Stiffened shells are affected by numerous uncertainty factors, such as the variations of manufacturing tolerance, material properties and environment aspects, etc. Due to the expensive experimental cost of stiffened shell, only a limited quantity of statistics about its uncertainty factors are available. In this case, an unjustified assumption of probabilistic model may result in misleading outcomes of reliability-based design optimization (RBDO), and the non-probabilistic convex method is a promising alternative. In this study, a hybrid non-probabilistic convex method based on single-ellipsoid convex model is proposed to minimize the weight of stiffened shells with uncertain-but-bounded variations, where the adaptive chaos control (ACC) method is applied to ensure the robustness of search process of single-ellipsoid convex model, and the particle swarm optimization (PSO) algorithm together with smeared stiffener model are utilized to guarantee the global optimum design. A 3 m-diameter benchmark example illustrates the advantage of the proposed method over RBDO and deterministic optimum methods for stiffened shell with uncertain-but-bounded variations.     

The buckling characteristics of some integrally formed bead stiffened composite panels

Composite panel stability can easily be improved by using vertical male beads. In this paper, new methods of stabilizing techniques used for the panels, webs and ribs of composite structures are studied. A parametric study is performed to assess the effects of important design considerations such as, bead length, number of beads, bead radius, bead depth and bead spacing on the initial buckling load of the panels. The results show that, there is an optimum bead spacing for each panel containing more than one bead which can be estimated using a simple equation. Integration of vertical beads with a length of less than 0.5 times the panel's length has no significant effect on the buckling load. There are no significant changes on the buckling loads of the beaded panels with bead depths greater than 0.6 times the bead radius. In this investigation, the instability of the nose and main ribs of a light airplane wing structure made of woven E-glass material and stiffened by P.V.C foam core and vertical male beads are also studied using experimental methods. The experimental results show that we can easily improve the buckling capability of the panels and webs by using vertical male beads instead of sandwiched construction. It is estimated that this would cause a weight reduction of about 50% and a manufacturing time reduction of about 50%.     

Compression tests of longitudinally stiffened plates undergoing distortional buckling

This paper describes a series of compression tests performed on longitudinally stiffened plates fabricated from a mild steel plate of thickness of 4.0 mm with nominal yield stress of 235.0 MPa. The stiffened plates with longitudinal stiffeners of a range of rigidities were tested to failure. The ultimate strengths and performances of the longitudinally stiffened plates in compression undergoing distortional buckling or interaction between local and distortional buckling were investigated experimentally and theoretically. The compression tests indicated that the critical buckling mode was dependent mainly on the rigidity of the longitudinal stiffeners and the width-to-thickness ratio of the sub-panels. A noticeable interaction between local and distortional buckling was also observed for some stiffened plates. A significant post-buckling strength reserve was shown for those sections with distortional buckling and for those sections showing interaction between local and distortional buckling. A limiting strength curve for distortional buckling of longitudinally stiffened plates was studied. Simple design strength formulas in the direct strength method are proposed to account for the distortional buckling and the interaction between local and distortional buckling of longitudinally stiffened plates. The strength curves were compared with the test and FE results conducted. The adequacy of the strength curve was confirmed. A set of conclusions on the buckling behavior of longitudinally stiffened plates was drawn from the experimental studies.     

Experimental studying of buckling of stringer cylindrical shells under axial compression

Results of tests on axial compression of small-sized quality steel cylinder shells strengthened by 24 and 36 longitudinal thin-walled stiffeners are presented. The shell length was varied. Shells both with inside and outside stiffening were tested at simply supported and clamped edges. The shell carrying capacity that was governed in the tests by overall buckling in the elastic range was compared with the estimated critical loads based on structural-orthotropic theory. The satisfactory quantitative correlation has been received only for the long simply supported shells with 36 inner stiffeners, which demonstrated insignificant effect of local undulation that preceded overall deflections. The experimental and the theoretical results differed significantly (twice as much) when the actual mechanism of lateral deflection caused by the intensive local undulation differed from the adopted model.     

A semi-analytical model for global buckling and postbuckling analysis of stiffened panels

A computational model for global buckling and postbuckling analysis of stiffened panels is derived. The loads considered are biaxial in-plane compression or tension, shear, and lateral pressure. Deflections are assumed in the form of trigonometric function series, and the principle of stationary potential energy is used for deriving the equilibrium equations. Lateral pressure is accounted for by taking the deflection as a combination of a clamped and a simply supported deflection mode. The global buckling model is based on Marguerre’s nonlinear plate theory, by deriving a set of anisotropic stiffness coefficients to account for the plate stiffening. Local buckling is treated in a separate local model developed previously. The anisotropic stiffness coefficients used in the global model are derived from the local analysis. Together, the two models provide a tool for buckling assessment of stiffened panels. Implemented in the computer code PULS, developed at Det Norske Veritas, local and global stresses are combined in an incremental procedure. Ultimate limit state estimates for design are obtained by calculating the stresses at certain critical points, and using the onset of yielding due to membrane stress as the limiting criterion.     

Buckling of simply supported rectangular plates under combined bending and compression using eigensensitivity analysis

In this paper, an approximate quadratic closed-form expression is presented for the critical buckling analysis of a plate subjected to combined bending and compression. The formula is developed by expanding the eigenvalue, the critical buckling load, for a plate under combined bending and compression in a Mauclaurin's series about a plate subjected only to compression. The general expression can be used for all combinations of simply supported and clamped rectangular plates boundary conditions. An explicit formula in terms of the plate aspect ratio R and plate load parameter α is evaluated for simply supported plates. Compared with the Rayleigh–Ritz method, this approximate expression provides an excellent comparison when the load parameter α1.52 for plate aspect ratio between 0.2R2.8.     

Elastoplastic buckling of circular annular plates under uniform in-plane loading

This paper deals with the elastoplastic buckling of a circular annular plate, with various axially symmetric boundary conditions and uniform axially symmetric in-plane radial loads on the inner and outer edge. The analysis is based on the standard linear buckling equations and the material behaviour is modelled by the small strain J₂ flow and deformation theories of plasticity where an elastic linear hardening rheological model of the material is considered. The solutions are obtained using the equilibrium approach where the governing differential equation is solved by the finite difference method which leads to the determination of eigenvalues of a homogeneous system of linear equations. Elastoplastic buckling loads for axially symmetric and asymmetric buckling shape modes with m waves in the circumferential direction are calculated and compared for both theories of plasticity. For one case, an experiment was performed and the results were compared with theoretical predictions.     

Buckling design of stiffened sheet-based inner core based on bifurcation analysis

In the structural design of the sandwich plate, the inner core plays a key role to have its maximum performance. A shaped pyramidal truss core is proposed in order to increase the strength and productivity of the sandwich core. In this paper, the design guidelines of the shaped pyramidal truss core, which is enhanced by forming a cross-section of an arc shape at the strut of the inner core, is described. The inner core is composed of a stiffened section and a transient section with a varying cross-section. The critical load for bifurcation in compressive instability is calculated using an analytical and FEM simulation. The analytical equation for the critical load of the shaped column is derived using the energy method. The various buckling modes (global, distortional, local) occur due to these effects. Therefore, complications induced by such effects must be taken into account in the design. Parametric studies for the stiffened core are conducted. The effect of geometric parameters is investigated for optimal design of the inner core and their influence have been discussed.     

The computational post buckling analysis of fuselage stiffened panels loaded in compression

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. Investigation of element, mesh, idealisation, imperfection and solution procedure selection has been undertaken, with results validated against mechanical tests. The research undertaken 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. The work has generated a series of guidelines for the non-linear computational analysis of flat riveted panels subjected to uniform axial compression.     

A simplified method for elastic large deflection analysis of plates and stiffened panels due to local buckling

A computational model for analysis of local buckling and postbuckling of stiffened panels is derived. The model provides a tool that is more accurate than existing design codes, and more efficient than nonlinear finite element methods. Any combination of biaxial in-plane compression or tension, shear, and lateral pressure may be analysed. Deflections are assumed in the form of trigonometric function series. The deformations are coupled such that continuity of rotation between the plate and the stiffener web is ensured, as well as longitudinal continuity of displacement. The response history is traced using energy principles and perturbation theory. The procedure is semi-analytical in the sense that all energy formulations are derived analytically, while a numerical method is used for solving the resulting set of equations, and for incrementation of the solution. The stress in certain critical points are checked using the von Mises yield criterion, and the onset of yielding is taken as an estimate of ultimate strength for design purposes.     

Compressive tests on stiffened panels of intermediate slenderness

Test results are presented of eight stiffened panels subjected to axial compression until collapse and beyond. The specimens are three-bay stiffened panels with associated plate made of very high tensile steel S690. The use of this very high strength steel led to the unconventional solution of using U stiffeners and this paper aims at understanding the difference of performance of this stiffener type as compared with the conventional ones. Four different configurations are considered for the stiffeners, which are made of mild or high tensile steel for bar stiffeners and mild steel for ‘L’ and ‘U’ stiffeners. The influence of the stiffener's geometry on the ultimate strength of the stiffened panels under compression is analyzed.     

Buckling of laminated composite stiffened panels subjected to in-plane shear: A parametric study

The presence of in-plane loading may cause buckling of stiffened panels. An accurate knowledge of critical buckling load and mode shapes are essential for reliable and lightweight structural design. This paper presents some parametric studies on simply supported laminated composite blade-stiffened panels subjected to in-plane shear loading. A total of 450 models were analyzed using ANSYS 7.1 and a database is prepared for different plate and stiffener combinations. Studies are carried out by changing the panel orthotropy ratio, stiffener depth, pitch length (number of stiffeners), smeared extensional stiffness ratio of stiffener to that of the plate and extensional stiffness to shear stiffness ratio of the plate. Based on the studies, few important parameters influencing the buckling behaviour are identified and guidelines for better stiffener proportioning are developed, which will be helpful for the designer.     

Lower bound to plastic load of cylinders with opening under combined loading

This paper proposes an analytical solution to obtain the lower bound limit load of cylindrical shells with circular opening under the action of combined axial force and bending moment at ends. The yield criterion proposed by Ivanov for plastic analysis of shells employing Von Mises criteria is used for general cylindrical shells. The method of solution is based on formulating the lower bound to limit load as a nonlinear mathematical programming problem. In this analysis Feasible Sequential Quadratic Programming (FSQP) is employed as an optimization technique to obtain the lower bound limit load. The lower bound results are presented in dimensionless form for a wide range of loading cases and geometric parameters of practical interest and compared with finite element elastic–plastic results which are found to be in good agreement.     

Numerical and experimental investigations on the compression behaviour of stiffened plates

The main objective of this paper is to present the results of the finite element method for non-linear analysis of stiffened plates subjected to axial compression load considering post-buckling behaviour up to collapse. For this purpose two series of well executed experimental data on longitudinally stiffened steel plates with and without transversal stiffeners subjected to uniform axial in-plane load carried out to study the buckling and post-buckling up to final failure have been chosen. The first series are those of Ghavami where the influences of stiffener cross-section of the type rectangular (R), L and T, their spacing and the presence of rigid transversal stiffeners have been studied. The second series of Tanaka & Endo, where the behaviour of stiffened plates having three and two flat bars for longitudinal and transversal stiffeners respectively were analysed. For the purpose a well-established commercially available Finite Element program ANSYS has been chosen. The selected element was SHELL43, which can trace the full-range, elastic-plastic behaviour of the stiffened plates. It is seen that the simulated results of FEM are in good consistency with the test results.     

Semi-analytical buckling strength analysis of plates with arbitrary stiffener arrangements

Buckling of plates with arbitrarily oriented, sniped stiffeners is studied. The main objective is to present and validate an approximate, semi-analytical computational model for such plates subjected to in-plane loading. Estimation of the buckling strength is made using the von Mises’ yield criterion for the membrane stress as the strength limit. The formulations derived are implemented in a Fortran computer code, and numerical results are obtained for a variety of plate and stiffener geometries. The model may handle complex plate geometries, by using inclined stiffeners to enclose irregular plate shapes. The method allows for a very efficient analysis. Relatively high numerical accuracy is achieved with low computational efforts. The results are, in most cases, found to be conservative compared to fully nonlinear finite element analysis results.     

Dynamic buckling of thin thermoviscoplastic cylindrical shell under radial impulsive loading

The dynamic plastic buckling of a homogeneous and isotropic thin thermoviscoplastic cylindrical shell loaded radially is studied analytically by analyzing the stability of its stressed/deformed configuration under superimposed infinitesimal perturbations. The wave number of the perturbation that maximizes its initial growth rate is assumed to determine the buckling mode. Cubic algebraic equations are obtained for both the maximum initial growth rate of perturbation and the corresponding wave number. The buckled shape of a cylindrical shell is found to match well with that observed experimentally. The sensitivity of the buckled shape to the impact velocity, the hardening modulus, and the material viscosity has been delineated. For axially restrained shells, it is found that for materials exhibiting strain rate hardening only the maximum initial growth rate of the perturbation and the corresponding wave number vary as and , respectively. For axially unrestrained cylindrical shells made of strain hardening only materials, the maximum initial growth rate of a perturbation and the corresponding wave number vary as and , respectively. Here is the mean value of the generalized stress, ρ the mass density, β the material viscosity, h the shell thickness, and R the mean radius of the shell.     

Numerical study on the permissible gap of intermittent fillet welds of longitudinally stiffened plates under in plane axial compression

Intermittent fillet welding of stiffeners to plates and its influence on the collapse behaviour of stiffened plates is investigated applying the finite element method. Special attention is paid to the modelling of the fillet welds at the plate-to-stiffener junction. Some available experimental results are simulated in order to obtain reliable numerical results. A series of numerical analyses of stiffened steel plates subjected to an in plane axial compressive load has been performed. Stiffened plates are selected from the deck structure of real sea-going ships and inland waterway vessels. Complete equilibrium paths are traced up to collapse for the non-linear elastoplastic response of stiffened plates. Finally a proposal is presented for the permissible gap of welds in intermittent fillet welding of stiffened plates.     

The characterisation of friction stir welding process effects on stiffened panel buckling performance

The introduction of advanced welding methods as an alternative joining process to riveting in the manufacture of primary aircraft structure has the potential to realise reductions in both manufacturing costs and structural weight. However, welding processes can introduce undesirable residual stresses and distortions in the final fabricated components, as well as localised loss of mechanical properties at the weld joints. The aim of this research is to determine and characterise the key process effects of friction stir welding assembly methods on stiffened panel static strength performance. Utilising experimentally validated finite element modelling methods, it is demonstrated that welding induced residual stresses have a significant influence, and that collapse behaviour is less sensitive to advanced process effects and process effect magnitudes than panel initial buckling behaviour.     

Large deflection orthotropic plate approach to develop ultimate strength formulations for stiffened panels under combined biaxial compression/tension and lateral pressure

This paper uses the large deflection orthotropic plate approach to develop the ultimate strength formulations for steel stiffened panels under combined biaxial compression/tension and lateral pressure loads, considering the overall (grillage) buckling collapse mode. The object panel has a number of one-sided small stiffeners in either one or both orthogonal directions. The stiffened panel is then modeled as an equivalent orthotropic plate, for which the various elastic constants characterizing structural orthotropy are determined in a consistent systematic manner using classical theory of elasticity. The panel edges are considered to be simply supported. The influence of initial deflections is taken into account. The membrane stress distribution inside the panel under combined uniaxial loading (in either longitudinal or transverse direction) and lateral pressure is analyzed by solving the nonlinear governing differential equations of large deflection orthotropic plate theory. It is presumed that the panel collapses when the most highly stressed boundary location yields, resulting in closed-form expressions for the ultimate strength of the stiffened panel. Based on the insights previously developed through numerical studies, the panel ultimate strength interaction formulation between biaxial loads, with lateral pressure regarded as a secondary load component is then proposed as a relevant combination of the two sets of panel ultimate strength formulations, i.e. one for combined longitudinal axial load and lateral pressure and the other for combined transverse axial load and lateral pressure. The validity of the proposed ultimate strength formulations is verified by a comparison with nonlinear finite element and other numerical solutions.