Thick Composite Shells Research Articles

Dynamic analysis of cracked thick composite shells by the Boundary Element Method

This article presents a numerical formulation based on the Boundary Element Method for the transient dynamic analysis of cracked thick symmetrical composite shells. The integral formulation uses the static fundamental solutions for thick orthotropic symmetric plates and the anisotropic plain elasticity fundamental solution. Domain integrals associated to distributed loads, curvature and inertial terms are evaluated employing the Radial Integration Method. The crack was modeled using the sub-region method. The developed formulation is implemented computationally and validated through the analysis of several proposed examples. The obtained results demonstrate the validity and robustness of the developed formulation.

K-space analysis of complex large-scale meta-structures using the Inhomogeneous Wave Correlation method

Abstract An inverse wavenumber identification tool is used to characterize the vibration behavior of three structures and meta-structures with different complexity levels: a plane steel panel, a curved thick composite sandwich shell and a stiffened aluminum aircraft sidewall panel. Bare structures are first studied and then equipped with spatially distributed small-scale resonators, leading to meta-structures. For the two curved panels, tests are conducted under diffuse acoustic field and point mechanical excitations. For each studied case, the effect of the industrially-oriented small-scale resonators is highlighted using frequency and wavenumber analysis, showing general attenuation of the vibration level and even band gaps occurrence. The complex wavenumber identification allows also estimating the structural loss factor in the composite sandwich panel, while the multi-modal behavior is captured in the aluminum aircraft sidewall panel.

Free edge stress prediction in thick laminated cylindrical shell panel subjected to bending moment

• A Galerkin based Layer-wise formulation is presented for thick composite panel subjected to bending moment. • An analytical solution is presented for governing equations of the panel considering free edge conditions. • Free edge stresses in bending of thick panel with general layer stacking is studied for the first time. • The predictions satisfy the free edge and free surface conditions. • The effect of geometrical parameters and layer stacking are investigated. A thick composite cylindrical shell panel with general layer stacking is studied to investigate the free edge and 3D stresses in the panel which is subjected to pure bending moment. To this aim, a Galerkin based layerwise formulation is presented to discretize the governing equation of the panel to ordinary differential equations. Employing a reduced displacement field for the cylindrical panel, the governing equations for thick panel are developed in terms of displacements and a set of coupled ordinary differential equations is obtained. The governing equations are solved analytically for free edge boundary conditions and applied pure bending moment. The accuracy of numerical results is examined and the distribution of interlaminar and in-plane stresses is studied. The free edge stresses are studied and the effect of radius to thickness ratio, width to thickness ratio and layer stacking on the distribution of stresses is investigated. The focus of numerical results is on the prediction of boundary layer and free edge stress distribution.

Numerical Analysis on Process-Induced Residual Stress in Thick Semi-Cylindrical Composite Shell Using a State-Dependent Viscoelastic Model

A two-step state-dependent analysis model corresponding to the flow state before gelation and the viscoelastic state after gelation was proposed for the prediction of the residual stresses developed in polymer matrix composites (PMCs) during cure. The evolution of the viscoelasticity of PMCs during cure was analyzed based on the non-dimensional viscoelasticity parameters 1/Dem, and the gelation time was determined at the condition of Min(1/Dem) = 102. The proposed model was used to predict the process-induced residual stress and distortion in a thick semi-cylindrical composite shell. By comparing the predictions with those of the original cure-dependent viscoelastic model, the proposed model was verified and the computing time and memory storage were significantly reduced. The results show that during the cure of thick semi-cylindrical composite shell, the residual stresses caused by the cure shrinkage at high cure temperature are quickly relaxed, and most of the residual stress develops during cool-down. The spring-in deformation of the semi-cylindrical shell when release from the mould is mainly caused by the significant hoop stress gradient in the thickness direction developed during cure. Obvious warpage is not present in the composite semi-cylindrical shell due to the high bending stiffness.

Three-dimensional buckling and free vibration analyses of initially stressed functionally graded graphene reinforced composite cylindrical shell

The buckling and free vibration of initially stressed functionally graded cylindrical shell reinforced with non-uniformly distributed graphene platelets (GPLs) are investigated using the state-space formulation based on three-dimensional elasticity theory. The shell is under an axial initial stress and composed of multilayers with GPLs uniformly dispersed in each individual layer but its weight fraction changing layer-by-layer along the thickness direction. The modified Halpin-Tsai model and rule of mixtures are employed to evaluate the effective elastic properties of the GPL-reinforced shell. Analytical buckling and frequency solutions are obtained for simply supported shells. Numerical results are presented for functionally graded GPL-reinforced cylindrical shells with five GPL dispersion patterns (GPL-UD, GPL-V, GPL-A, GPL-X, and GPL-O). The effects of GPL weight fraction, dispersion pattern, geometry, and size as well as the influence of initial stress on the buckling and free vibration characteristics of the shell are discussed in detail. It is found that the addition of a small amount of GPLs significantly increases the critical buckling stress and natural frequencies. The GPL-X pattern outperforms other patterns for thin composite shells while the uniform pattern GPL-UD works better for thick composite shells.

Recent developments in investigation on buckling and post buckling responses of laminated composite shells

This paper presents the state of art in the field of composite shells with focus on stress analysis, buckling and post‐buckling responses under mechanical and thermal loading. Many researchers have investigated the buckling and post‐buckling responses of cylindrical, conical, spherical, thin and thick composite shells with single and doubly curved shells under mechanical and thermal loading. From the literature it can be inferred that hydrostatic pressure, axial compression and thermal environment influence the buckling behavior of laminated composite shell significantly. An effort is taken to review the research on buckling and post‐buckling responses of cylindrical and conical shaped shells under mechanical and thermal loading and their behavior during buckling and post‐buckling conditions. POLYM. COMPOS., 39:4231–4242, 2018. © 2017 Society of Plastics Engineers

On the Theory of Functionally Graded Moderately Thick Composite Shells

In this paper, exact solutions for double curved functionally graded shells subjected to static and dynamic distributed loads are presented for the first time. Shear deformable shell theory is used and the governing equations for the laminated and functionally graded shells with respect to the middle surface are presented in the Laplace transform domain. The exact solutions of the laminated and functionally graded shells should serve as benchmark solutions for numerical methods.

Numerical analysis of free vibration of cross-ply thick laminated composite cylindrical shells by continuous element method

This paper presents the vibration analysis of thick laminated composite cylindrical shells by a new approach using the Continuous Element Method (CEM). Based on the analytical solutions for the differential equations of thick composite cylindrical shell taking into account shear deflection effects, the dynamic transfer matrix is built from which natural frequencies are easily calculated. A computer program is developed for performing numerical calculations and results from specific cases are presented. Numerical results of this work are compared with published analytical and Finite Element Method (FEM) results. Through different examples, advantages of CEM are confirmed: reduced size of model, higher precision, reduced time of computation and larger range of studied frequencies.

Dynamic stiffness matrix of continuous element for vibration of thick cross-ply laminated composite cylindrical shells

This paper investigates the vibration analysis of thick laminated composite cylindrical shells by a continuous element constructed from the dynamic stiffness matrix. Based on the analytical solutions of the differential equations of thick composite cylindrical shell taking into account the shear deflection effects, the dynamic stiffness matrix has been built from which natural frequencies are easily calculated. A computer program is developed for performing the numerical calculations and results from specific case studies are presented. Results on natural frequencies and harmonic responses of this work are compared to published analytical and Finite Element Method (FEM) results. Numerical examples demonstrate that the proposed model can be used to solve efficiently the vibration analysis of thick cross-ply composite cylindrical shell with many advantages: size of model reduced, time of computing saved, higher precision, direct acquisition of harmonic response and suitable for the analysis of medium and high frequency range.

First-Ply Strength of Thick Circular Cylindrical GRP Composite Shell Subjected to Static External Pressure via Finite Element Simulation and Analytical Approaches

One-ply thick externally pressurized GFRP cylinders were analyzed using computational numerical simulation and analytical method. The main part of the work was to study the critical external pressure to avoid failure in the laminated shells by employing various phenomenological failure criteria. Comparison between ABAQUS FEM simulation results and analytical solutions using MATLAB programming codes proved high accuracy of the analytical approach. Various fiber angles stacking sequences were chosen for optimization analysis. The results did not show any increase in stability as the layer number increased. The results also showed high correlation with corresponding failure criteria. The analytical solutions gave higher accuracy as the number of plies and the total laminate thicknesses were reduced.

Free vibration of thick laminated cylindrical shells with different boundary conditions using general differential quadrature

Equations of motion with required boundary conditions for deep and thick cylindrical composite shells are shown using two First-order Shear Deformation Theories. The difference between them is the inclusion of the effects of curvature in the evaluation of stiffness parameters. Equations of motion with stress resultants lead to a system of fifteen first-order differential equations for dynamics of cylindrical shells. These equations are solved numerically using the General Differential Quadrature method for free vibration of isotropic, cross-ply, angle-ply and general lay-up cylindrical shells with six types of different boundary conditions using the aforementioned theories. The first five frequency parameters and mode shapes which are obtained from both theories are compared with the available results in the literature and those obtained using a three-dimensional finite element analysis to test the accuracy of the shell theories presented here.

Static analysis of thick laminated shells with different boundary conditions using GDQ

Equilibrium equations and the associated boundary conditions for doubly curved, relatively deep and thick composite shells are shown. Two First Order Shear Deformation theories (FSDTs) are used. The first one uses plate stiffness parameters for thick shells and the other includes the effect of curvature in the calculation of stiffness parameters. Equilibrium equations are put together with the equations of stress resultants to arrive at a system of seventeen first order differential equations. These equations are solved numerically with the aid of General Differential Quadrature (GDQ) method for isotropic, cross-ply, angle-ply and general lay-up cylindrical shells with six types of different boundary conditions using above mentioned theories. Results obtained using both theories are compared with the available results in literature and those obtained using a three-dimensional (3D) analysis to test the accuracy of the shell theories presented here.

Static and vibration analyses of thick deep laminated cylindrical shells using 3D and various shear deformation theories

Equations of motion with required boundary conditions for doubly curved deep thick composite shells are shown using two formulations. The first is based upon the formulation that was presented initially by Rath and Das [1] and followed by Reddy [2]. In this formulation, plate stiffness parameters are used for thick shells, which reduced the equations to hose applicable for shallow shells. The second formulation is based upon that of Qatu [3,4]. In this formulation, the stiffness parameters are calculated using exact integration (and/or terms truncated to a specific order) of stress resultant equations. In addition, Qatu considered the radius of twist in his formulation. In both formulations, first order polynomials for in-plane displacements in the z-direction are utilized allowing for the inclusion of shear deformation and rotary inertia effects (first order shear deformation theory or FSDT). Exact static and free vibration solutions for isotropic and symmetric and anti-symmetric cross-ply cylindrical shells for different length-to-thickness and length-to-radius ratios are obtained using the above theories. Results of both theories are compared with those obtained using a three-dimensional (3D) analysis to test the accuracy of the shell theories presented here.

Higher order zig-zag theory for smart composite shells under mechanical-thermo-electric loading

A higher order zig-zag shell theory based on general tensor formulation is developed to refine the predictions of the mechanical, thermal, and electric behaviors. All the complicated curvatures of surface including twisting curvatures can be described in a geometrically exact manner in the present shell theory because the present theory is based on the geometrically exact surface representation. The in-surface displacement fields are constructed by superimposing the linear zig-zag field to the smooth globally cubic varying field through the thickness. Smooth parabolic distribution through the thickness is assumed in the out-of-plane displacement in order to consider transverse normal deformation and stress. The layer-dependent degrees of freedom of displacement fields are expressed in terms of reference primary degrees of freedom by applying interface continuity conditions as well as bounding surface free conditions of transverse shear stresses. Thus the proposed theory has only seven primary displacement unknowns and they do not depend upon the number of layers. To assess the validity of present theory, the developed theory is evaluated under the thermal and electric load as well as under the mechanical load of composite cylindrical shells. Through the numerical examples, it is demonstrated that the proposed smart shell theory is efficient because it has the minimal degrees of freedom. The present theory is suitable in the predictions of deformation and stresses of thick smart composite shells under the mechanical, thermal, and electric loads combined.

Linear static and dynamic analysis of laminated composite plates and shells using a 4-node quasi-conforming shell element

Quasi-conforming formulation of 4-node stress resultant shell element is presented for the linear static and dynamic analysis of composite plates and shells. The element formulations use interrelated displacement-rotation interpolations making it applicable for moderately thick and thin composite shells. The formulation also includes drilling degrees of freedom, allowing the modeling of stiffened composite plates and shells. The transverse shear stiffness is defined by an equilibrium approach instead of using the shear correction factor. The stiffness matrices for the present elements are explicitly expressed and the stresses are taken accurately at the nodal points. A lot of numerical tests were carried out for the validation of present 4-node composite shell element and the results are in good agreement with the references.

Effects of Transverse Stitching and Hygrothermal Environment on Composite Wing T-Joints

The present paper deals with the effects of transverse stitching and hygrothermal environment at the web-skin interface of composite wing T-joints that are made of carbon fibre composite (CFC) materials. The T300 carbon fibres are considered for the transverse stitching. The statistical bundle strength based on the Weibull distribution function is used for the estimation of the joint strength. A thick composite shell finite element is specially developed taking into account the hygrothermal effects and is used for a detailed finite element analysis of composite T-joints. T-joints may be exposed to moisture and temperature during their service life. Under such circumstances moisture and temperature diffuse into T-joints. As a consequence, the stiffness and strength properties of the laminates are degraded. Moreover, since the laminas in the laminate cannot deform freely, residual stresses are introduced. All the normal and shear stresses are computed and the failure loads are also computed using various failure criteria to get better insight into the behaviour of laminated T-joints. Additional stresses due to hyghrothermal forces are also computed. The T-joint is supported along the length and a pull-out force is applied on the top of the web. It is found that the web-skin interface where load transfer is out of plane, is the critical zone for failure due to high transverse normal and shear stresses. Also, higher temperature and moisture levels introduce additional transverse normal and shear stresses that make the zone more critical. The present study also shows how the transverse stitching at the web-skin interface zone adds to the transverse strengthening of the interface laminates and helps the T-joints to sustain higher loads.

Finite Element Analysis of Composite Wing T-Joints

The present paper is concerned with numerical studies of composite T-joints which are made of carbon fibre composite (CFC) materials. CFC materials are used due to their high load carrying capacity and lightweight. The composite T-section is formed at the spar-lower skin joint section in aircraft wings. The behaviour of such joints is very complex. In this study, a brief account of the problem is first presented and areas of principal interest which most influence joint behaviour are identified. A thick composite shell finite element is specially developed and it is then used for a detailed finite element analysis of composite T-joints. Though several types of forces act on the spar, in the present case, a pull-out force is applied on the top of the web to determine strength and to study the behaviour of T-joints. Parametric variations account for two different stacking sequences and variations in filler area and radius of curvature at the web/skin interface. Accuracy of the present finite element analysis is established by comparing present results with the published ones. Numerical results obtained through parametric variations are presented and discussed to understand the joint behaviour.

Anisotropy Related “Spring-in” of Angled Composite Shells

During the manufacture of curved or angled shell composite structures, the enclosed angle of such structures tends to be reduced after cure. This phenomenon is referred to as “spring-in”. It is believed that such distortion is caused mainly by the significant difference between the in-plane coefficient of thermal expansion (CTE) and the through-thickness CTE. This might result in a larger out-of-plane contraction than the in-plane contraction during the time that a composite structure is cooling down from the curing temperature. In this paper, a 3-D Finite Element Analysis procedure was developed to predict “spring-in” resulting from anisotropy for both thin and thick angled composite shell structures. The results of the FE analysis were evaluated together with those from the analytical study and experimental investigation conducted by Jain at the Cooperative Research Centre for Advanced Composite Structures (CRC-ACS) in Australia. It was concluded, based on these results, that the FE model gives more accurate results than the analytical model, particularly for thicker composite shells. The corner radius effect, shell thickness effect, and lay-up effect on “spring-in”, together with the effect of tool/part interaction on the total distortion were also discussed. Comparison was made with the experimental results reported by Radford and Rennick.

Numerical Analysis of the Efficiency of Using Theories of Thin and Thick Composite Shells in Stress Concentration Problems

The feasibility of overcoming “cutoff” effects in four versions of the theory employing classical and mixed functionals is analyzed for sample problems involving the stress concentration near a hole in composite spherical shells with different thicknesses and transverse shear rigidities. Recommendations on the choice of theory are given depending on the shell parameters. A class of “not very thick” shells with a low shear rigidity is found for which all the versions of the theory yield an acceptable computational accuracy.

BOUNDARY-VALUE PROBLEMS OF THE THEORY OF THIN AND NONTHIN ORTHOTROPIC SHELLS WITH ACCOUNT OF NONLINEARLY ELASTIC PROPERTIES AND LOW SHEAR RIGIDITY OF COMPOSITE MATERIALS

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.