Prebuckling Stress Distribution Research Articles

Instability characteristics of variable stiffness laminated composite curved panels under non-uniform periodic excitation

The instability characteristics of variable stiffness laminated composite (VSLC) shell panels subjected to non-uniform in-plane periodic excitations are explored in the present article. The non-constant orientation of fiber angle in the VSLC panel induces non-uniform stress distributions within the shell panel under uniform as well as non-uniform in-plane loadings. Initially, the pre-buckling stress distributions within the shell panels are evaluated by minimizing the membrane strain energy. Subsequently, a displacement based Ritz method is employed to obtain the matrix representation of the governing equations for static and dynamic instability problems. Bolotin’s method is used to determine the dynamic instability regions. The numerical results depicted in the article show the influence of fiber angle orientations, non-uniform distribution of in-plane loadings and damping on the dynamic instability characteristics of VSLC shell panels.

Analytical approach to elastic stability problems of plates with different boundary conditions subjected to combined bending, shear and patch loading

The elastic stability of plates with simply supported edges (SSSS) and with two edges clamped and two simply supported (CSCS) under combination of bending, shear and patch load is investigated.General analytical approach for elastic buckling analysis of plates with different boundary conditions, subjected to any combination of in-plane loads, is presented. Exact solutions for pre-buckling stress distributions and double Fourier series for deflection functions within Ritz energy method guarantees accuracy of results and enabled very complex stability analysis.Combined in-plane loads action is very common during incremental bridge lunching and different stress conditions can occur in web girders. All analyses conducted so far regarding such problems are related to simply supported plate (SSSS). However, web girder is elastically restrained to the certain point depending on flanges rigidity and authors considered interesting to investigate behavior of plate with two opposite edges restrained and other two simply supported (CSCS) under same load conditions as simply supported plate.First part of the paper deals with patch load problem for plates with two different boundary conditions and evaluation of input data for bending-patch load (M-F), shear-patch load (V-F) and bending-shear-patch load (M-V-F) interaction curves and surfaces definition. Second part is dedicated to presentation, analysis and comparison of the interaction curves and surfaces for the cases of simply supported plate (SSSS) and plate with two clamped and two simply supported edges (CSCS), in order to obtain realistic bucking load ratios in accordance with boundary conditions.

Pre-buckling and Buckling Analysis of Variable-Stiffness, Curvilinearly Stiffened Panels

This work illustrates a fast numerical approach for analyzing the pre-buckling and buckling response of innovative panel configurations for aerospace structures. Specifically, the approach allows composite panels with variable-stiffness skins and stiffened by curvilinear stringers to be studied in a fast yet accurate manner. The method of Ritz is applied in conjunction with first-order theories for modeling the skin and the stiffeners, the former described referring to Mindlin plate theory, the latter to Timoshenko beam model. Due to the excellent convergence properties of the approach, pre-buckling stress distributions can be captured with reduced effort. Similarly, accurate buckling predictions can be obtained with relatively few degrees of freedom, much less with respect to typical models relying upon the use of finite elements. Results are presented for a number of test cases from the literature, illustrating the potential of the proposed approach as a mean for performing preliminary studies with reduced computational effort.

An analytical solution for buckling of plates with circular cutout subjected to non-uniform in-plane loading

This paper investigates the buckling of isotropic plates with circular cutout subjected to non-uniform in-plane loading. The buckling load is calculated in two steps. In the first step, the prebuckling stress distribution is computed. The existence of cutout causes the solution of stress distribution to be nontrivial. In this paper, a novel analytical method is presented for calculating the stress distribution. This method is based on expansion of the stress function in polar coordinates and using a boundary integral to satisfy the boundary conditions at plate edges. In the second step, the obtained stress distribution is used to calculate the buckling load from the Ritz method. In this method, The displacement field is defined according to the first order shear deformation theory and the characteristic beam functions are used for approximation. The effect of cutout size, plate’s aspect ratio, different uniaxial and biaxial loading profiles, i.e. constant, parabolic and cosine loading and different boundary conditions on the buckling load is studied.

Stability and failure analyses of delaminated composite plates subjected to localized heating

A layerwise B-spline finite element formulation is used to study the stability and first-ply failure of composite plates with embedded delaminations and subjected to localized thermal heating over the plate surface with uniform temperature rise through its thickness. The step functions are used to represent jump discontinuities in displacement fields at the zone of delamination between layers of composite plates. As the applied thermal loading is nonuniform, in the first step the prebuckling stress distribution within the plate is evaluated by solving the thermoelasticity problem. Subsequently, thermal buckling temperature is computed using the plate prebuckling stress distribution. Postbuckling response of the plate is obtained considering the von Kármán type geometric nonlinearity. In the delaminated region, virtual springs are added to prevent interpenetration of lower and upper sublaminates. The localized thermal load at which the first-ply failure of the lamina occurs has been detected by Tsai-Wu quadratic interaction criterion. The effects of the area of heating region, delamination size and its position in the thickness direction and plate boundary condition on the composite plate critical buckling temperature and on first-ply failure load are reported. The present model can capture global, local and combined global–local buckling mode shapes.

The role of n-3 polyunsaturated fatty acids in the formation of the psychophysiological status of ski racers

A layerwise B-spline finite element formulation is used to study the stability and first-ply failure of composite plates with embedded delaminations and subjected to localized thermal heating over the plate surface with uniform temperature rise through its thickness. The step functions are used to represent jump discontinuities in displacement fields at the zone of delamination between layers of composite plates. As the applied thermal loading is nonuniform, in the first step the prebuckling stress distribution within the plate is evaluated by solving the thermoelasticity problem. Subsequently, thermal buckling temperature is computed using the plate prebuckling stress distribution. Postbuckling response of the plate is obtained considering the von Kármán type geometric nonlinearity. In the delaminated region, virtual springs are added to prevent interpenetration of lower and upper sublaminates. The localized thermal load at which the first-ply failure of the lamina occurs has been detected by Tsai-Wu quadratic interaction criterion. The effects of the area of heating region, delamination size and its position in the thickness direction and plate boundary condition on the composite plate critical buckling temperature and on first-ply failure load are reported. The present model can capture global, local and combined global–local buckling mode shapes.

GBT buckling analysis of generally loaded thin-walled members with arbitrary flat-walled cross-sections

This paper deals with extending the domain of applicability of a recently developed Generalised Beam Theory (GBT) formulation intended to perform elastic linear buckling analyses of thin-walled members (i) exhibiting arbitrary flat-walled cross-sections (including those combining closed cells and open branches), and (ii) acted by general loadings. These loadings, which include transverse forces acting away from the member shear centre axis, are termed “general” in the sense that they may involve the presence of pre-buckling stress distributions associated with any possible combination of all the stress tensor membrane components (σxx, σss and τxs, for a plane stress state), including cell shear flows – therefore, all the relevant geometrically non-linear effects need to be taken into consideration. After briefly presenting the main concepts and procedures involved in the development and implementation of the above GBT formulation, this same formulation is employed to analyse the buckling behaviour of beams with different types of cross-section geometry (containing closed cells) and exhibiting different loading and support conditions. In particular, they consist of (i) a RHS cantilever acted by two tip point loads, (ii) a closed-flange I-section simply supported beam subjected to a uniformly distributed load and (iii) a two-cell RHS section cantilever acted by tip transverse forces and couples. In all cases, the loads are applied both along the shear centre axis and also along axes parallel to it and located at the beam top and bottom surfaces. The results presented and discussed, which consist of pre-buckling stress fields, buckling curves and buckling mode shapes, are obtained by means of the newly released code GBTul 2.0 and validated by means of the comparison with shell finite element values obtained with the code Ansys.

Semi-analytical approach for thermal buckling and postbuckling response of rectangular composite plates subjected to localized thermal heating

A semi-analytical solution of thermal buckling and postbuckling equilibrium paths of rectangular composite plates subjected to localized thermal heating over the plate area with uniform temperature rise through its thickness is reported here. The plate is either simply supported or clamped for out-of-plane displacements and immovable for in-plane displacements. The thermal buckling of the composite plate subjected to localized thermal loading is solved in two steps as the prebuckling stress distributions within the plate are not known a priori. In the first step the explicit analytical expressions for these in-plane prebuckling stress distributions within the composite plate are developed by solving the thermoelasticity problem using Airy’s stress function. Using the computed in-plane stress distributions within the plate, the plate nonlinear stability equations are formulated using a variational principle. First-order shear deformation theory incorporating von-Karman geometric nonlinearity is used to model the plate. The generalized differential quadrature method in conjunction with the modified Newton–Raphson method is used to solve the nonlinear governing partial differential equations for nonlinear thermal stability of composite plates. The influence of aspect ratio, area of heated region, and the boundary conditions on the critical buckling temperature and equilibrium paths are examined.

Non-linear vibration analysis of laminated composite circular cylindrical shells

The non-linear transverse dynamic response of laminated composite simply supported circular cylindrical shell subjected to periodic radial point loading and static axial partial loading is studied in this paper considering von Kármán type of non-linearity. The applied partial edge loading is represented by Fourier series and the exact prebuckling stress distribution within the cylindrical shell is computed by solving the in-plane elasticity problem. Donnell’s shell theory incorporating first order shear deformation, in-plane and rotary inertia is used to model the cylindrical shell. Galerkin’s method is used to reduce the governing partial differential equations to a set of non-linear ordinary differential equations. These equations are solved using Incremental Harmonic Balance (IHB) method to obtain frequency-amplitude responses for free and forced vibration. The numerical results illustrate the effects of number of layers, static partial preloading, and initial geometric imperfections on the non-linear forced vibration of laminated composite cylindrical shells.

Nonlinear Stability Characteristics of Composite Cylindrical Panel Subjected to Non-Uniform In-Plane Mechanical and Localized Thermal

The non-linear stability analysis of composite cylindrical panel subjected to concentrated in-plane mechanical and localized thermal loadings are reported here. The buckling of composite panel subjected to concentrated in-plane mechanical loading/ localized thermal loading is solved in two steps as the prebuckling stress distribution within the panel is not known a priori. In the first step, the semi-analytical expressions for the pre-buckling stresses within the composite cylindrical panel under in-plane mechanical/ thermal loadings are developed by solving in-plane elasticity/thermoelasticity problem. Subsequently, using these in-plane stresses within the cylindrical panel, the governing equations for nonlinear stability of layered composite panel are formulated using variational principle. The cylindrical panel is modeled based on Donnell’s shallow shell theory considering higher order shear deformation theory and incorporating von-Karman geometric nonlinearity. The Galerkin’s method is used to solve the non-linear governing partial differential equations. The influence of different types of mechanical and thermal loadings, initial geometric imperfections, and radius of curvature on the postbuckling equilibrium paths is investigated.

Effects of Partial Edge Loading and Fibre Configuration on Vibration and Buckling Characteristics of Stiffened Composite Plates

In this work, the influence of uniaxial and biaxial partial edge loads on buckling and vibration characteristics of stiffened laminated plates is examined by using finite element method. As the initial pre-buckling stress distributions within an element are highly non-uniform in nature for a given loading and edge conditions, the critical loads are evaluated by dynamic approach. Towards this, a nine-node heterosis plate element and a compatible three-node beam element are developed by employing the effect of shear deformation for both the plate and the stiffeners respectively. In the structural modeling, the plate and the stiffener elements are treated separately, and then the displacement compatibility is maintained between them by using a transformation matrix. Effect of different parameters such as loaded edge width, position of loads, boundary conditions, ply-orientations and stiffener factors are considered in this study. Buckling results show that the uniaxial loaded stiffened plate with around (+30º/-30º)2 layup can withstand higher load irrespective of boundary conditions and loading patterns, whereas the maximum load resisting layup for the bi-axially loaded stiffened plate is purely dependent on edge conditions and loading patterns.

Nonlinear stability and dynamics of composite skew plates under nonuniform loadings using differential quadrature method

The buckling, postbuckling and postbuckled vibration behaviour of composite skew plates subjected to nonuniform inplane loadings are presented here. The skew plate is modelled using first order shear deformation theory accounting for von-Kármán geometric nonlinearity and initial geometric imperfections. The different types of nonuniform loads that have been considered in this study are concentrated load, partial load and parabolic load. The explicit analytical expressions for prebuckling stress distributions within composite skew plate subjected to three different types of nonuniform inplane loadings are developed by solving plane elasticity problem using Airy's stress function approach. It is observed that the inplane normal stress distributions within the skew plate due to above nonuniform loadings do not become uniform even at mid-section. The generalized differential quadrature (GDQ) method is used to solve the nonlinear governing partial differential equations. It is observed that the postbuckled load carrying capacity of skew plate under concentrated loading is the lowest compared to other nonuniform and uniform loadings.

Buckling analysis of partially protected cold-formed steel channel-section columns at elevated temperatures

This paper presents a numerical investigation on the buckling behaviour of plasterboard protected CFS channel-section columns subjected to axial loading when exposed to fire on its one side. The work involves temperature-dependent pre-buckling stress analysis and buckling analysis. Two non-uniform temperature distributions with linear and nonlinear temperatures along the web and constant temperatures in flanges and lips have been included in this paper. Bernoulli beam theory has been used in the pre-buckling stress analysis with considering the effects of temperature on strain and mechanical properties. The buckling analysis is performed using combined finite strip analysis and classical Fourier series solutions, in which the mechanical properties are considered to be temperature dependent. The results show that the temperature distributions have a significant influence on both the pre-buckling stress distribution and slenderness of CFS column members. It is also found that the thermal bowing effect improves the buckling performance of intermediate CFS members, but deteriorates that of long CFS members. This paper gives a better understanding of the effect of non-uniform temperatures on the buckling behaviour of CFS columns, and further extends the potential application of the finite strip method to the buckling analysis of CFS members at elevated temperatures.

Lateral–torsional buckling of cold-formed channel sections subject to combined compression and bending

This paper presents an analytical study of the flexural buckling and lateral–torsional buckling of cold-formed steel channel section beams subject to combined compression and bending about their major and minor axes. For channel section beams a bending about the minor axis creates a non-symmetric pre-buckling stress distribution, which has a significant influence on the lateral–torsional buckling of the beams. This kind of feature has not been discussed in the existing literature. The focus of this present study is the interaction between the compression load and the bending moments about the major and minor axes. It has been found that for a section subject to combined compression and the major-axis bending the bending moment will decrease the critical compression load, although the critical value of the largest compressive stress in the section actually increases with the applied bending moment. However, for a section subject to combined compression and the minor-axis bending the effect of the bending moment on the critical compression load depends on the direction of bending applied. For bending that creates a compressive stress in the lips the bending moment will reduce the critical compression load. However, for bending that creates a compressive stress in the web the bending moment has almost no influence on the critical compression load.

Optimization of Variable-Stiffness Panels for Maximum Buckling Load Using Lamination Parameters

With the large-scale adoption of advanced fiber placement technology in industry, it has become possible to fully exploit the anisotropy of composite materials through the use of fiber steering. By steering the composite fibers in curvilinear paths, spatial variation of stiffness can be induced resulting in beneficial load and stiffness distribution patterns. One especially relevant area in which fiber steering has proved its effectiveness is in improving buckling loads of composite panels. Previous research used predefined forms of fiber angle variations and the coefficients of these analytic expressions were used as design variables. Alternatively, the local ply angles were used as design variables directly. In this paper, a framework is developed to treat the most general approach that considers the largest possible design space. The use of lamination parameters efficiently defines stiffness variation over a structural domain with the minimum number of variables. A conservative reciprocal approximation scheme is introduced. The inverse buckling factor is expanded linearly in terms of the in-plane stiffness and in terms of the inverse bending stiffness. The new approximation scheme is convex in lamination parameter space. Numerical results demonstrate improvements in excess of 100% in buckling loads of variable-stiffness panels compared to the optimum constant stiffness designs. Buckling load improvements are attributed primarily to in-plane load redistribution, which is confirmed both by the prebuckling stress distribution as well as by comparing the performance of designs optimized with variation of both in-plane and bending stiffness to those optimized with only bending stiffness variation. A tradeoff study between in-plane stiffness and buckling performance is also presented and shows the benefits of variable-stiffness design in enlarging the design possibilities of composite panels.

Thermo-elastic stability behavior of laminated cross-ply elliptical shells

In this work, thermo-elastic stability behavior of laminated cross-ply elliptical cylindrical shells subjected to uniform temperature rise is studied employing the finite element approach based on higher-order theory that accounts for the transverse shear and transverse normal deformations, and nonlinear in-plane displacement approximations through the thickness with slope discontinuity at the layer interfaces. The combined influence of higher-order shear deformation, shell geometry and non-circularity on the prebuckling thermal stress distribution and critical temperature parameter of laminated elliptical cylindrical shells is examined.

Strength of Stiffened Flanges in Horizontally Curved Box Girders

Longitudinal stiffeners are often attached to increase the buckling strength of thin-walled box girder flanges. The minimum required rigidity for longitudinal stiffeners for curved box girder flanges is given by the AASHTO Guide specifications for horizontally curved steel girder highway bridges. However, this requirement is simply adopted from the current AASHTO specifications for straight stiffened flanges. The validity of this requirement has been questioned in a series of recent studies. The effect of important design parameters on the minimum required stiffener rigidity is investigated numerically in this study by examining the prebuckling stress distribution and elastic and inelastic buckling stresses of horizontally curved stiffened flanges. In order to characterize and quantify the analytically collected data, a series of parametric studies were performed. A new equation for the minimum required rigidity for the longitudinal stiffeners is derived from regression analyses. Through the evaluation of a few selected case studies and a design example, the validity and reliability of the proposed new equation is demonstrated.

Lateral-Torsional Buckling of Post-Local Buckled Fibrous Composite Beams

Interactive buckling of fibrous composite I-section beams is discussed in this paper. As a result of a linear combination of local and lateral-torsional buckling modes, which may arise when the critical loads of these two types of buckling are nearly the same, a large degradation in the failure load capacity may occur. In this study, the flanges are considered as beam elements when the member bends laterally under the prebuckling stress distribution and as plate elements for normal bending and twisting; the web is analyzed as a plate element. Within the framework of the Rayleigh-Ritz method, a displacement function formulation is employed using compatibility conditions at the junction between flange and web for unifying the analytical treatment and reducing the problem size. The effect of the combined in-plane stresses, produced by longitudinally varying stresses due to the applied transverse loads, on the critical load also is discussed.

Buckling-load interaction in tailored composite plates

The concept of stiffness tailoring is investigated for improved buckling resistance of rectangular composite plates subject to general in-plane combined loading. The tailoring approach involves the piecewise-uniform redistribution of a portion of the given material with given orientations to create beneficial stiffness distributions across the planform of the plate. The resulting local nonuniformities in thickness and in membrane and bending stiffnesses combine to change the buckling response of the plate. Finite element models are used to calculate the prebuckling stress distribution and the buckling loads. The weight and average membrane shear stiffness are essentially unaffected by the tailoring. Optimum tailoring geometries are determined for a wide range of shear and transverse tension and compression loadings combined with longitudinal compression loading. Maximum improvements in the buckling loads of over 700% are shown possible with tailoring for some loading combinations compared with uniform plates. In general, the benefit of tailoring is greatest with low shear loads and high transverse tensile loads.

Buckling of polar and rectilinearly orthotropic annuli under uniform internal or external pressure loading

The problem of the out of plane buckling of polar orthotropic annuli under uniform internal or external pressure loading has been investigated with the assumption of axisymmetry by numerous researchers. In this paper, the buckling of polar and rectilinearly orthotropic annuli subjected to internal or external pressure loading is examined without restriction to axisymmetric buckling modes. The buckling analyses are performed in both the polar orthotropic and rectilinearly orthotropic cases by the Rayleigh-Ritz method. In the rectilinearly orthotropic cases, the pre-buckling solution is obtained via Galerkin's method as no known exact solution exists, while in the polar orthotropic case the pre-buckling stress distributions used are exact. It is found in both the polar orthotropic and the rectilinearly orthotropic annuli that restriction to axisymmetric buckling modes in the externally pressurized case does not produce the lowest buckling load over a significant range of annuli aspect rations. Example results for commercially used fiber/matrix systems are presented and discussed, and it is concluded that the unwarranted assumption of axisymmetry actually produces nonminimum buckling loads.