Partial Edge Loading Research Articles

Buckling Behavior of Laminated Shell Panels Under Linearly Variable Edge Load

This paper reports a detailed numerical investigation on the buckling aspects of laminated composite shell panels of three forms (cylindrical, spherical and hyperbolic paraboloid) with five support conditions exposed to linearly varying in-plane edge load employing eight nodded isoparametric finite element formulation. The impacts of different parameters including ply orientation, load factor, shell forms, aspect ratio, modulus ratio, curvature ratio, width-to-thickness ratio and angle of lamination on the buckling load of shell panels are examined. It is found that the various parameters addressed in this study have a remarkable impact on the buckling phenomena of laminated composite shell panels. Further, a comparison is made showing the effects of five types of compressive edge loads like uniform, triangular, parabolic, partial edge load and point load on the buckling phenomena of laminated shell panels with respect to five support conditions.

Dynamic and buckling response modelling of functionally graded graphene oxide powder(GOP) reinforced cylindrical panels : Influence of GOP grading and non-uniformly, partially distributed edge loads.

Influence of non-uniform, partial edge loads and graded reinforcement of graphene oxide powder (GOP) on vibration and buckling behaviour of cylindrical panel is illustrated. A semi-analytical approach based on higher order shear deformation theory and Galerkin’s approach is used to obtain the results. Initially, the membrane stress resultants due to the applied edge loads are developed through Airy’s stress function expansion. Then the stress resultants are evaluated through the minimisation of strain energy. Followed by this, equations of motion are obtained based on Hamilton’s principle and the stress resultants. Influence of loading of GOP quantity, nature of grading of GOP, nature of non-uniform and partial edge loads on critical buckling coefficient and fundamental frequency and mode shapes are investigated. It is noted that the buckling and vibration characteristics are sensitive to the nature of GOP grading, GOP loading and nature of variation in edge loads. Furthermore, the fundamental buckling mode is not always typical (1, 1) mode and (2, 1) mode is observed as the buckling mode according to the variation in aspect ratio and nature of edge loads. It is found that near critical buckling load, the fundamental vibration mode changes to (2,1) from (1,1) for parabolic and partial edge loading cases for the panels with aspect ratio higher than 1.3.

Buckling of partially-compressed laminated composite plates

The web buckling is one failure mode of web crippling for thin-walled fiber-reinforced plastic (FRP) composite beams subjected to transverse concentrated or distributed compression load from both the top and bottom bearing plates, and the web between the top and bottom flanges can be modeled as a plate with the rotationally-restrained (RR) and partially-compressed loaded edges and the simply-supported (SS) unloaded edges. Explicit solution and universal analytical charts/formulas for buckling analysis of such RRSS composite plates partially compressed along the transverse direction are obtained. The unique Heaviside function and trigonometric series expansion are used to express the partial edge load with effective load length in the governing equilibrium equation. The explicit solution is established for the orthotropic plates subjected to concentrated point compression load in the middle; while the generic buckling analytical curves are developed for infinitely long laminated composite plates transversely compressed partially at the middle of edges as well as near the corners. The accuracy of the present method is validated with available simple analytical solution, numerical finite element method and experimental data. The effect of rotational restraint stiffness, plate aspect ratio, the location and length of partial edge compression load on buckling behavior of laminated composite plates is investigated in the parametric study. A practical guideline for laminated composite plates with the elastic rotational restraints along the loaded edges and partially compressed along the transverse direction is proposed, and it can be further applied in discrete plate analysis to predict the web buckling strength (one failure mode in web crippling) of FRP composite structural shapes under transverse loading at the locations of load bearing or supports.

Effects of different interlaminar hybridization and cutout sizes on the vibration and buckling characteristics of fiber metal composite laminates under partial edge loads

Fiber reinforced polymer composites are quickly gaining market share in structural applications, but further growth is limited by their lack of toughness. Fiber hybridization is a promising strategy to toughen composite materials. By combining two or more fiber types, these hybrid composites offer a better balance in mechanical properties than non-hybrid composites. This study presents the effect of different interlaminar hybridization and cutout sizes on the vibration and buckling characteristics of hybrid fiber metal laminates (HFMLs) subjected to partial edge loads by developing finite element formulation. Since the stress distribution within the plate element is highly non-uniform in nature, the dynamic approach has been used to solve the buckling problems wherein two sets of boundary conditions are used, one for pre-buckling stresses and other one for buckling load calculations. A 9-noded heterosis plate element has been used to discretize the plate by taking into account the effect of shear deformation and rotary inertia. The present study consists of aluminum metal face sheets bound with four layered symmetric hybrid cross-ply laminate schemes. In this scheme, six different hybrid combinations have been considered. The performance of each hybrid combination is investigated under various loading combinations. Further, the effect of different parameters such as boundary condition, cutout ratio, width and position of localized edge loads, plate aspect ratio, and hybrid configurations are included in this work. It is revealed from the study that the hybrid configurations under different cutouts significantly affect the vibration and buckling characteristics under different localized edge loads.

Buckling behavior of GLARE panels subjected to partial edge loads

The advancement in FRP laminates leads to the development of fibre metal laminates (FMLs) commonly found in aircraft components. The mechanical properties of FMLs are superior as compared to that of bare fiber laminates because of the infused strength combination of both fibers and metals. In this work, the Glass fiber reinforced aluminum/epoxy laminates (GLARE) are considered to investigate the effect of fibre-orientation on vibration and buckling characteristics under the action of partial edge loads by using finite element approach. A 9-noded heterosis plate element has been used to discretize the plate by incorporating the effect of shear deformation. For a given GLARE with ply-orientation and loading, the stress distribution within an element is highly non-uniform in nature. Therefore, dynamic approach has been used to calculate the buckling loads. Here, two boundary conditions are used, in which one for pre-buckling stress calculation and other one for buckling loads. The effect of various parameters such as ply-orientation in GLARE, thickness, boundary conditions, aspect ratios of plate and loading width and its position are considered to investigate the buckling characteristics in detail.

Buckling analysis of laminated composite plate due to localised in plane loading

Laminated composite plates are inevitable parts of structure due to its superior properties than that of conventional material. Many times, these structural elements are subjected to the partial loadings during various applications which arises buckling in the elements. This necessitates proper analysis of these types of structural member for better safety and stability. In the present paper buckling analysis of laminated composite stiffened plate under in-plane localized edge loading is presented. Buckling load for various cases of partial edge loading are investigated for different fiber orientations and loading extent. Finite element method-based MATLAB program is developed for the present analysis. Validation study is done, which shows well agreement with the results already published.

Non-linear vibration response of functionally graded circular cylindrical shells subjected to thermo-mechanical loading

The analysis of the non-linear vibration response is carried out for functionally graded (FG) circular cylindrical shells subjected to thermal environment along with mechanical in-plane non-uniformly distributed loading along the edges and harmonic radial force. The temperature dependent material properties of the simply supported shell are assumed to vary in the radial direction according to power-law distribution. Based on the first-order shear deformation theory and von-Kármán type geometric nonlinearity, the strain-displacement relationships are established for circular cylindrical shells. The coupled governing equations of motion for functionally graded cylindrical shells are then derived using Hamilton’s principle. Employing Galerkin’s method, the coupled partial differential equations of motions are reduced to a set of non-linear ordinary differential equations. In order to obtain the free and forced vibration response of the FG shell, the incremental harmonic balance method, in conjunction with the arc-length method, is used. The non-uniform in-plane loading is converted to Fourier series and the pre-buckling analysis is performed to determine the stress distribution within the shell. The non-linear frequency-amplitude response is studied to examine the effects of volume fractions of the constituents, static partial edge loadings, thermal loads, and radial periodic loadings.

Dynamic instability of damped composite plates with embedded delaminations

A layerwise finite element formulation based on B-spline function is used to study dynamic instability of damped laminated composite plates with embedded delaminations in this article. The out-of-plane variation of in-plane displacement components are interpolated by Linear Lagrange interpolation function and the in-plane variation of in-plane and out-of-plane displacement fields are interpolated by B-spline functions. Delamination between layers of composite plate is modeled by jump discontinuities in displacement fields using Heaviside functions. In the delaminated region, virtual springs are added to prevent interpenetration of lower and upper sublaminates. The local buckling of thinner sublaminate is observed when delamination is placed near the top/bottom interface. The dynamic instability regions are computed by Bolotin's method. The effects of delamination lengths and its position on the natural frequencies, mode shapes, buckling loads and dynamic instability regions of the composite plates with various boundary conditions are examined. Both uniform and non-uniform periodic edge loading in the form of concentrated and partial edge loading are considered. Furthermore, the effect of damping on the dynamic instability regions of composite plate is studied. From the linear and nonlinear responses in the stable and unstable regions, various characteristics such as beats, its dependence on forcing frequency and damping are observed.

Stability and vibration analysis of CNT-Reinforced functionally graded laminated composite cylindrical shell panels using semi-analytical approach

The present paper examined the buckling, postbuckling and vibration characteristics of pre-buckled and post-buckled laminated CNT reinforced composite (CNTRC) cylindrical shell panel made up of single walled carbon nanotubes (SWCNTs) and isotropic matrix. The effective material properties of CNTRC panel are computed using extended rule-of mixture (ROM) method. Higher order shear deformation theory (HSDT) with von Kármán type of nonlinearity is adopted to model the CNTRC cylindrical shell panel. Four different boundary conditions are considered. Besides uniform loading, different types of non-uniform in-plane load distribution such as triangular, trapezoidal, parabolic and partial edge loadings are considered. The internal stress distribution within the shell panel due to applied non-uniform loadings is evaluated by prebuckling analysis. Subsequently, via Hamilton's principle the governing partial differential equations of CNTRC laminated cylindrical shell panel are derived. Employing Galerkin's method and by neglecting the inertia terms the partial differential equations are reduced to a set of non-linear algebraic equation for the static problem. However, for dynamic problem the partial differential equations are converted to a set of ordinary differential equations. Beside parametric study the obtained numerical results from the present semi-analytical study illustrates the effects of CNT volume fraction, CNT dispersion profile, non-uniform load distribution and boundary conditions on the stability and vibration characteristics of CNTRC cylindrical panel.

Stochastic dynamic stability analysis of composite curved panels subjected to non-uniform partial edge loading

Abstract The stochastic dynamic stability analysis of laminated composite curved panels under non-uniform partial edge loading is studied using finite element analysis. The system input parameters are randomized to ascertain the stochastic first buckling load and zone of resonance. Considering the effects of transverse shear deformation and rotary inertia, first order shear deformation theory is used to model the composite doubly curved shells. The stochasticity is introduced in Love's and Donnell's theory considering dynamic and shear deformable theory according to the Sander's first approximation by tracers for doubly curved laminated shells. The moving least square method is employed as a surrogate of the actual finite element model to reduce the computational cost. The results are compared with those available in the literature. Statistical results are presented to show the effects of radius of curvatures, material properties, fibre parameters, and non-uniform load parameters on the stability boundaries.

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.

Buckling response of laminated composite stiffened plates subjected to partial in-plane edge loading

ABSTRACTThis article presents the buckling analysis of laminated composite stiffened plates subjected to partial in-plane edge loading. The finite element method is used to carry out the analysis. The eight-noded isoparametric degenerated shell element with C0 continuity and first-order shear deformation and a compatible three-noded curved beam element are used to model the plate skin and the stiffeners, respectively. The eigen value analysis is carried out to track the buckling load. The convergence study is performed for some specific problems and the results are compared with the available results in the literature. It is observed that the convergence of results is very fast for this finite element model. Effect of different parameters like orientation of fibers, number of layers, and loading types are considered in the present investigation. It is also observed that all these parameters have significant effect on the buckling response of the composite stiffened plate.

Postbuckling and postbuckled vibration analysis of sandwich plates under non-uniform mechanical edge loadings

Postbuckling and postbuckled vibration characteristics of sandwich plate subjected to non-uniform in-plane loadings is investigated in this article. The core compressibility effects are considered in the model by assuming fourth and fifth order expansions for the transverse and tangential displacement of the core. The exact stress distributions within the panel are determined by panel prebuckling analysis for the applied parabolic and partial edge loadings. Multi-term Galerkin's method is employed to reduce the governing partial differential equation to a set of non-linear ordinary differential equations in the case of dynamic analysis and to a set of non-linear algebraic equations in the case of static stability analysis. Newton-Raphson method in conjunction with Riks approach is used to trace the postbuckling path. Free vibration frequencies of the sandwich plate about a static equilibrium state are obtained by solving the associated linear eigenvalue problem. From numerical study it is observed that the postbuckling path becomes more softening type with increase of aspect ratio. Moreover, unloaded fixed edge of SCSC sandwich plate gives more rigidity than unloaded simply supported edge of CSCS. The prebuckling and postbuckling frequency increases with increase in edge restraint, and in postbuckling range, frequency increases with increase of in-plane load.

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.

Linear and nonlinear parametric instability behavior of cylindrical sandwich panels subjected to various mechanical edge loadings

ABSTRACTLinear and nonlinear dynamic instability behavior of cylindrical sandwich panels subjected to combined static and dynamic nonuniform in-plane loadings is studied in this article. The core compressibility effects are considered in the model by assuming fourth and fifth order expansions for the transverse and tangential displacement of the core. The exact stress distributions within the panel are determined by panel prebuckling analysis for the applied parabolic and partial edge loadings. Galerkin's method is used to reduce the governing partial differential equations of the shell panel into a set of nonlinear ordinary differential equations. Dropping the nonlinear term, dynamic instability regions are obtained by solving the Mathieu-type differential equation by the method of Fourier series. The characteristics feature of the stable and unstable regions are investigated by linear and nonlinear time history responses and phase plots of the shell panel in those regions using Newmark's time integration. Incremental harmonic balance (IHB) method is used to study the nonlinear frequency amplitude responses of the cylindrical sandwich panels.

Dynamic stability of simply supported composite cylindrical shells under partial axial loading

The parametric vibration of a simply supported composite circular cylindrical shell under periodic partial edge loadings is discussed in this article. Donnell׳s nonlinear shallow shell theory considering first order shear deformation theory is used to model the shell. The applied partial edge loading is represented in terms of a Fourier series and stress distributions within the cylindrical shell are determined by prebuckling analysis. The governing equations of the dynamic instability of shells are derived in terms of displacements (u−v−w) and rotations(φx,φθ). Employing the Galerkin and Bolotin methods the dynamic instability regions are computed. Using the expression for the stress function derived in this paper, the pre-buckling stresses in the cylindrical shell due to partial loading can be calculated explicitly. Numerical results are presented to show the influence of radius-to-thickness ratio, different partial edge loading distributions and shear deformation on the dynamic instability regions. The linear and nonlinear responses in the stable and unstable regions are presented to bring out the characteristic features of the dynamic instability regions, such as the existence of beats, its dependence on forcing frequency and effect of nonlinearity on the response. The effect of dynamic load amplitude on the nonlinear response is also studied. It is found that for higher values of dynamic loading, the shell exhibits chaotic behavior.

Elastic buckling behavior of rectangular plates with holes subjected to partial edge loading

Cut-outs or openings are inevitable part of steel structural systems since they are required as access ports for mechanical and electrical systems or to reduce the amount of material that is used. When such perforated plates experience compression loads, they may buckle or show instability due to axial compression. In this study, the buckling of perforated square and rectangular plates subjected to in-plane compressive edge loading is investigated using the finite element method. To analyze the behavior of the plates the following four edge loading cases were applied; concentrated edge loading, asymmetric partial loading at the opposite edges, partial loading at the center of the opposite edges and partial loading at the two ends of the opposite edges. The plate aspect ratio, the length and location of the edge loading and diameter of the circular hole are taken as the variables that have a buckling effect on the behavior of the plate. The results show that the square plates are highly sensitive to buckling when the loading at the center of the plates.

Static and dynamic instability analysis of composite cylindrical shell panels subjected to partial edge loading

The postbuckling and dynamic instability behavior of simply supported composite cylindrical shell panels subjected to dynamic partial edge loadings and transverse patch loadings is studied in this paper considering von Kármán type of non-linearity. The stress distribution within the panel due to the applied partial edge loadings is evaluated by panel׳s membrane analysis. Subsequently using these stress distribution and via Hamilton׳s variational principle, the equations governing the instability behavior of shell panel are derived. Neglecting inertia terms, governing equations for the postbuckling analysis of panel are obtained. Galerkin׳s method is used in the solution procedure. It is observed from the postbuckling analysis that the cylindrical shell panel subjected to partial edge compression behaves as an imperfect shell panel as the partial edge compression in the x-direction induces tensile stress in the y-direction which makes the shell panel to deflect out-of-plane. It is also observed that by suitably adjusting the lamina number and lamina layup, the snap through behavior of shells can be altogether avoided. Dynamic instability regions of simply supported composite shell panels are traced by the method suggested by Bolotin. The linear and non-linear dynamic responses of the shell in stable and unstable regions are studied. This brings out various features of the instability problem such as, existence of beats and its dependence on forcing frequency and initial conditions, and effect of non-linearity on the response. It is found that for certain value of dynamic partial edge loading, the panel exhibits chaotic behavior.

Nonlinear Stability Analysis of Laminated Composite Simply Supported Circular Cylindrical Shells Subjected to Partial Axial Loading

AbstractThe present investigation deals with the nonlinear stability behavior of cross-ply laminated composite circular cylindrical shells subjected to partial and complete edge loading along with uniform external pressure. The shell is modeled using Donnell’s shell theory including the first-order shear deformation theory (FSDT). The analysis uses the simply supported boundary condition (at x=0, L Nxx=N¯xx, w=Mxx=v=ϕθ=0). The equations governing the nonlinear stability behavior of cylindrical shells are derived in terms of displacements (u-v-w) and rotations (ϕx, ϕθ). The applied partial edge loading is expressed in terms of Fourier series, and stress distributions within the cylindrical shell are determined by prebuckling analysis. The study uses multiterm Galerkin’s method along with the Newton-Raphson method to solve the governing partial differential equations of the shell nonlinear stability. With the help of numerical investigations, the authors present the number of modes required for the postbuck...

Dynamical Analysis of Stiffened Plates under Patch Loading

The vibration characteristics of stiffened plates with cutouts subjected to in-plane partial edge loadings at one end at the plate boundaries are studied using the finite element method. Buckling loads and vibration frequencies are determined for different cutout ratios and extent of partial edge loading at one end. In the structural modelling, the plate and the stiffeners are treated as separate elements where the compatibility between these two types of elements is maintained. The main elegance of the formulation lies in the treatment of the stiffeners. The stiffeners can be placed anywhere within the plate element, and need not be placed on the nodal lines. The vibration characteristics are discussed and the results are compared with those available in the literature. Numerical results are presented for a range of cutout to plate size from 0 to 0.8.