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.     

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.     

Impact response of rectangular and square stiffened plates supported on two opposite edges

Experimental drop weight impact tests have been performed to examine the dynamic response of small-scale stiffened plates struck laterally by a mass with a spherical indenter. The laboratory results are compared with numerical simulations. The plates stiffened with a flat bar or L profile are supported at two opposite edges and impacted at different velocities and locations along the span. The impact scenarios could represent incidents in marine structures, such as load actions due to dropped objects on decks. The experiments are conducted using a fully instrumented impact testing machine. The obtained force–displacement responses show a good agreement with the simulations performed by the LS-DYNA finite element solver. The finite element model includes defining the experimental boundary conditions so as to simulate small axial displacements of the specimen at the supports. This representation can be used to analyze the structural crashworthiness of similar marine structures under collision scenarios. The strain hardening of the material is defined using experimental data of quasi-static tension tests and the strain rate sensitivity is evaluated using standard coefficients of the Cowper–Symonds constitutive model. The results show that the plastic response of the specimens is highly sensitive to the amount of restraint provided at the supports. Furthermore, it is found that in most of the specimens the contribution of the stiffeners to the impact response is insignificant, since the ends of the stiffener are free at the unsupported edges and the specimens experience small axial displacements at the supports.     

The in-plane shear failure of transversely stiffened thin plates

This paper examines the response of stiffened plates with plain flat outstands when subjected to in-plane shear loading in the form of applied in-plane shear displacement. The buckling and post-buckling failure capabilities of thin plates subjected to in-plane shear, can, of course, be improved through the introduction of stiffening elements whose flexural and torsional rigidities can contribute significantly towards a more stabilised structural system. This paper details appropriate suitable finite element modelling strategies and procedures to enable the determination of the post-buckled failure response of the stiffened shear panels and to highlight the significant influence of the stiffeners. The modelling procedures are able to describe the complete loading history of the stiffened panel structures from the onset of initial buckling through the elastic post-buckling phase of behaviour involving the considerable interaction between plate and stiffener and then through initial material yielding and yield propagation to ultimate conditions and elasto-plastic unloading.     

Influence of corrosion on the ultimate compressive strength of steel plates and stiffened panels

This study concentrates on a comparison between steel plate and stiffened panels subject to localised corrosion. A finite element analysis is used to investigate the effect of random corrosion on the compressive strength capacity of marine structural units. Variables include the extent of corrosion; slenderness ratio and aspect ratio. A corrosion prediction model is incorporated to determine the thickness reduction with time. Corrosion-induced volume loss results in a greater reduction of ultimate strength for slender plates compared to stiffened panels, up to 45%, showing the structural element selection can strongly influence the accuracy of the estimated corrosion damage effect.     

Residual ultimate strength of cracked steel unstiffened and stiffened plates under longitudinal compression

This paper numerically deals with the influence of cracks (in terms of length and location) on the ultimate compressive strength characteristics of unstiffened and stiffened plate elements used in thin-walled structures. The cracks were presumed to be through-thickness, having no contact between their faces and no propagation was allowed. A series of nonlinear finite element analyses was conducted using ANSYS commercial finite element code in which the Newton–Raphson method has been employed to solve the nonlinear governing equations.This study indicates that the length of cracks and especially its location can significantly affect the ultimate strength characteristics of unstiffened and stiffened plate elements subjected to axial compressive action.     

Impact characteristics of stiffened plates penetrated by sub-ordnance velocity projectiles

A series of experiments are carried out to explore the impact characteristics of stiffened plates struck by the sub-ordnance velocity projectiles. Considering the relative position of impact point to the nearest stiffeners on target, six kinds of representative target plates with different stiffened styles are specially designed. Altogether sixteen target plates, respectively without, single- and cross-stiffened are tested and compared. Two kinds of projectiles, one with spherical nose and the other with truncated oval nose, are fired using a 25 mm refitted cannon. The initial velocities of projectiles range from 244 to 430 m/s. Experimental results show that the penetration process is greatly dependent on the nose shapes of projectiles. The tumbling of truncated oval-nosed projectiles is very prominent, which results in the great uncertainty of deformation and failure mode of target plates. While the spherical-nosed projectiles keep much more stable, plugging and ductile hole enlargement are the most distinct failure modes. In the end, some empirical formulas aimed at blunt-nosed projectiles are proposed, which could be helpful for the design of the protection structures.     

Optimal stiffener design of moderately thick plates under uniaxial and biaxial compression

In this paper, the optimal stiffener design of moderately thick plates under uniaxial and biaxial compression is investigated on the premise that the plate thickness and the required ultimate strength are given. As the theoretical basis of stiffener design, the ultimate strength formulations of weak stiffened thick panels under in-plane biaxial compression are first developed on the basis of large deflection orthotropic plate theory, in which the post-weld initial deflection is taken into account. The von Mises yield criterion is employed to determine the limit state of the panel, and the Nelder-Mead simplex algorithm is used to obtain the efficient solution of nonlinear differential equations. The optimization method presented is based on the stiffener design principles of the overall instability stress and of the working stress. In the optimization formulation, the numbers and geometric sizes of the stiffeners are defined as design variables; the weight ratio of stiffeners to plate is taken as a single objective function; requirements against overall buckling of the panel, local buckling of the plates between the stiffeners and local buckling of the stiffeners themselves are set as constraint functions. Results of both design examples and parameter studies show that, for moderately thick plates, the stiffener weight given by the proposed optimization method is much lower than the weight determined by the current stiffener design method on the premise of the same requirement of structural safety. Using the present optimization method to obtain the lightest and the most effective stiffener layout for moderately thick plates is proposed.     

The energy flow analysis in stiffened plates of marine structures

The transmission of vibrational energy flow in stiffened plates is analyzed by the structural intensity method. Three typical cases for ship and marine structures are selected as examples: a plate with one longitudinal stiffener under lateral area pressure excitation, a longitudinally stiffened plate under point force excitation, and a cross-stiffened plate under in-plane pressure excitation. The relationships between structural intensity and structural mode shapes and the effects of stiffeners on the changes of energy flow in plate are discussed. Finally, the potential application of structural intensity technique towards the design for stiffened plates of marine structures is explored and presented.     

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 behaviour of pin-ended flange elements in compression

Stowell's solution [1] for the buckling behaviour of flange elements in compression was premised on the assumption that the element was fixed against flexural rotations at the ends, a condition representing relatively thick elements for which the thickness dimension is adequate to prevent rotations. This paper presents a solution similar to Stowell's which is applicable to pin-ended flange elements. Aspects not considered in Stowell's work, such as the use of elliptic functions to describe the gradual change of mode shape from sinusoidal to essentially linear, and the gradual and asymptotic changes in axial rigidity in the post-buckling range are described in the paper. The paper also presents comparisons between the behaviour of pin-ended and fixed-ended flange elements. Finally, simple strength equations for flange elements in uniform compression based on the first yield criterion are derived.     

A new stiffened plate element for the analysis of arbitrary plates

In spite of the large number of finite elements developed so far, most of these lack in generality, and are found to be inadequate and inefficient in some way or other, when it comes to analyzing plates of arbitrary geometrical configurations. So far the isoparametric element has been the most successful among available elements because of its ability to model a curved boundary successfully. However, the shear-locking problem inherent in the isoparametric element makes it unsuitable for analyzing thin plates of arbitrary shapes. Though research has been conducted using reduced integration and stabilization to overcome the problem, the formulations either do not converge to the correct solution in the thin-plate limit or they make the stiffness matrix a singular one. In this paper, a four-noded stiffened plate element is developed. This has the advantages and elegance of an isoparametric element in modelling arbitrary shaped plates, but without the disadvantage of shear-locking phenomena. Though this element is a high-order element, only the usual degrees of freedom have been considered, and performance is superior to that of the low-order ones. The stiffened plate element has the feature of accommodating the arbitrary shape of the plate geometry, and the stiffener modelling has been done in a general manner, with the stiffener lying anywhere with arbitrary orientation, and not necessarily following the nodal lines. The new element has been successfully used for the static, free vibration and stability analyses of arbitrary bare and stiffened plates. The results are found to agree quite satisfactorily with those of previous investigators.     

Torsional buckling behaviour of stiffened cylinders under combined loading

In this study cylindrical boundary conditions for finite element analysis are formulated that allow torsional displacement and buckling of a sector of a cylinder of half axial height, and of a circumferential arc angle that will divide into 360°. Finite element tests are carried out on un-stiffened elastic cylinders to verify the method of analysis against classical elastic torsional buckling theory.Elastic–plastic limit point finite element tests are carried out on ring and stringer stiffened and stringer stiffened cylinders to investigate the effects of stiffeners on post-buckling behaviour in torsion.A stringer stiffened cylinder is subjected to many combinations of axial force and surface pressure in the elastic range of response and then tested to failure in torsion to investigate the effects of axial and surface pressure loads on the resistance to plastic collapse in torsion.     

Empirical formulations for estimation of ultimate strength of continuous stiffened aluminium plates under combined in-plane compression and lateral pressure

The present research was undertaken based on the results obtained by the same authors in a sensitivity study on the buckling and ultimate strength of continuous stiffened aluminium plates. Empirical expressions are developed for predicting ultimate compressive strength of welded stiffened aluminium plates used in marine applications under combined in-plane axial compression and different levels of lateral pressure. Existing data of the ultimate compressive strength for stiffened aluminium plates numerically obtained by the authors through the previously performed sensitivity analysis are used for deriving formulations that are expressed as functions of two parameters, namely the plate slenderness ratio and the column (stiffener) slenderness ratio. Regression analysis is used in order to derive the empirical formulations. The formulae implicitly include effects of the weld on initial imperfections, and the heat-affected zone.     

Accurate discrete modelling of stiffened isotropic and orthotropic rectangular plates

An analytical approach is presented here for simply supported blade-stiffened rectangular plates wherein important non-classical effects such as transverse shear deformation and rotary inertia are carefully accounted for. The analysis differs from full three-dimensional modelling of both the plate and the stiffener in that a plane stress idealization is used to model the kinematics of transverse bending of the stiffener while simple one-dimensional classical models are employed for lateral bending and torsion. Parametric studies are used to highlight the importance of non-classical effects in plate and stiffener kinematics and to finally lead to certain recommendations for accurate modelling of stiffened plates.     

Analysis and optimization of externally stiffened crush tubes

Nonlinear finite element analysis is used to investigate the quasi-static axial collapse response of cylindrical tubes which are externally stiffened by multiple identical rings. The rings divide the long tube into a series of short thin-walled tubes. It is assumed that the size and shape of integral stiffeners are controlled through a machining process. The effects of various geometric parameters such as wall thickness, ring spacing, ring thickness and width on the collapse response, crush force and energy absorption of monolithic, integrally stiffened steel tubes are studied and used as a general framework for a design optimization study. Through design and analysis of computer experiments, global metamodels are developed for the mean crush force and energy absorption, using the radial basis function approximation technique. Using both single- and multi-objective design optimization formulations, optimum designs for different response characteristics are found. The crush mode in the form of progressive collapse or buckling is found to heavily depend on the ratio of stiffener spacing to stiffener height as well as the ratio of wall thickness to stiffener thickness. The optimization results show the viability of externally stiffened tubes as efficient energy absorbers.     

Tailoring the elastic postbuckling response of thin-walled cylindrical composite shells under axial compression

The buckling of cylindrical shells has long been regarded as an undesirable phenomenon, but increasing interests on the development of active and controllable structures open new opportunities to utilize such unstable behavior. In this paper, approaches for modifying and controlling the elastic response of axially compressed laminated composite cylindrical shells in the far postbuckling regime are presented and evaluated. Three methods are explored (1) varying ply orientation and laminate stacking sequence; (2) introducing patterned material stiffness distributions; and (3) providing internal lateral constraints. Experimental data and numerical results show that the static and kinematic response of unstable mode branch switching during postbuckling response can be modified and potentially tailored.     

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.     

Tripping of thin-walled stiffeners in the axially compressed stiffened panel with lateral pressure

Tripping of stiffeners in stiffened panels under combined loads of axial force and lateral pressure is studied. Firstly, on the basis of the Vlasov's differential equation for torsional buckling of thin-walled bars, a generalized eigenvalue problem for tripping of stiffeners is derived by using the Galerkin's Method. Then the effect of the lateral pressure (dead load) to the critical axial stress (live load) upon tripping is investigated by solving the eigenvalue problem. The rotational restraint provided by the plate is taken into account. The effects of the compressive stress in the plate and the plate buckling mode are also discussed. Finally, an approximate equation to estimate the critical tripping stress with the effect of the lateral pressure is proposed. After some modifications, it can be applied in design rules for the purpose of checking the tripping strength of the stiffeners.     

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.