Finite element analysis of bimodulus composite stiffened thin shells of revolution

This paper is a sequel to the work published by the first and third authors[l] on stiffened laminated shells of revolution made of unimodular materials (materials having identical properties in tension and compression). A finite element analysis of laminated bimodulus composite thin shells of revolution, reinforced by laminated bimodulus composite stiffeners is reported herein. A 48 dot doubly curved quadrilateral laminated anisotropic shell of revolution finite element and it's two compatible 16 dof stiffener finite elements namely: (i) a laminated anisotropic parallel circle stiffener element (PCSE) and (ii) a laminated anisotropic meridional stiffener element (MSE) have been used iteratively.The constitutive relationship of each layer is assumed to depend on whether the fiberdirection strain is tensile or compressive. The true state of strain or stress is realized when the locations of the neutral surfaces in the shell and the stiffeners remain unaltered (to a specified accuracy) between two successive iterations. The solutions for static loading of a stiffened plate, a stiffened cylindrical shell. and a stiffened spherical shell, all made of bimodulus composite materials, have been presented.     

Laminated composite structures by continuum-based shell elements with transverse deformation

In this paper a first-order shear/fourth-order transverse deformation theory of laminated composite shells is presented. A nonlinear continuum-based (degenerated 3D) finite element model with a strain/stress enhancement technique is developed in such a way that the nonzero surface traction boundary conditions and the interlaminar shear stress continuity conditions are all satisfied identically. Analytical integration through the shell thickness is performed. The resultants of the stress integrations are expressed in terms of the laminate stacking sequence. Consequently, the shell laminate characteristics in the normal direction can be evaluated precisely and the computational cost of the overall analysis is reduced. The numerical results are compared with analytical solutions and other finite element solutions to demonstrate the effectiveness of the theory and the computational procedure developed herein.     

Analysis of laminated shells of revolution with laminated stiffeners using a doubly curved quadrilateral finite element

A finite element analysis of laminated shells of revolution reinforced with laminated stifieners is described here-in. A doubly curved quadrilateral laminated anisotropic shell of revolution finite element of 48 d.o.f. is used in conjunction with two stiffener elements of 16 d.o.f. namely: (i) A laminated anisotropic parallel circle stiffener element (PCSE); (ii) A laminated anisotropic meridional stiffener element (MSE).These stifiener elements are formulated under line member assumptions as degenerate cases of the quadrilateral shell element to achieve compatibility all along the shell-stifiener junction lines. The solutions to the problem of a stiffened cantilever cylindrical shell are used to check the correctness of the present program while it's capability is shown through the prediction of the behavior of an eccentrically stiffened laminated hyperboloidal shell.     

Exploiting symmetries in the modeling and analysis of tires

Three basic types of symmetry (and their combinations) exhibited by tire response are identified. A simple and efficient computational strategy is presented for reducing both the size of the model and the cost of the analysis of tires in the presence of symmetry-breaking conditions (e.g., unsymmetry of the tire material, geometry, and/or loading). The strategy is based on approximation of the unsymmetric response of the tire with a linear combination of symmetric and antisymmetric global approximation vectors (or modes).The three main elements of the computational strategy are as follows: (1) use of three-field mixed finite element models having independent shape functions for stress resultants, strain components, and generalized displacements, with the stress resultants and the strain components allowed to be discontinuous at interelement boundaries; (2) use of operator splitting (additive decomposition of some of the matrices and vectors in the finite element model) to delineate the symmetric and antisymmetric contributions to the response; and (3) successive use of the finite element method and the classic Rayleigh-Ritz technique to substantially reduce the number of degrees of freedom. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed with the Rayleigh-Ritz technique.The proposed computational strategy is applied to three quasi-symmetric problems of tires, namely, (1) linear analysis of anisotropic tires through the use of semianalytic finite elements, (2) nonlinear analysis of anisotropic tires through the use of two-dimensional shell finite elements, and (3) nonlinear analysis of orthotropic tires subjected to unsymmetric loading. In the first two applications, the anisotropy (nonorthotropy) of the tire is the source of the symmetry breaking; in the third application, the quasi-symmetry is due to the unsymmetry of the loading. The effectiveness of the proposed computational strategy is also demonstrated with numerical examples, and its potential for handling practical tire problems is outlined.     

Simple and effective elements based upon Timoshenko–Mindlin shell theory

The simple and effective mixed models are developed for the analysis of multilayered anisotropic Timoshenko–Mindlin-type shells. The effects of the transverse shear and transverse normal strains, and laminated anisotropic material response are included. The precise representation of rigid body motions in the displacement patterns of curved shell elements is considered. This consideration requires the development of the strain–displacement equations of the Timoshenko–Mindlin-type theory with regard to their consistency with the rigid body motions. The fundamental unknowns consist of six displacements and eleven strains of the face surfaces of the shell, and 11 stress resultants. The element characteristic arrays are obtained by using the Hu–Washizu mixed variational principle. Numerical results are presented to demonstrate the high accuracy and effectiveness of the developed mixed models and to compare their performance with other finite-element models reported in the literature.     

Sensitivity analysis for the dynamic response of thermoviscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric, fully-coupled, thermoviscoplastic response of shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of large-strain, transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be viscoplasticity with strain hardening and thermal hardening, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used for the shell equations with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. The energy-balance equation is solved using a Galerkin procedure, with the temperature as the fundamental unknown.Spatial discretization is performed in one dimension (meridional direction) for the momentum and constitutive equations of the shell, and in two dimensions (meridional and thickness directions) for the energy-balance equation. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) for the momentum equation; a predictor-corrector version of the trapezoidal rule is used for the energy-balance equation; and an explicit scheme consistent with the central difference method is used to integrate the constitutive equations. The sensitivity coefficients are evaluated by using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to the thickness, mass density, Young's modulus, two of the material parameters characterizing the viscoplastic response and the three parameters characterizing the thermal response. Time histories of the response and sensitivity coefficients are presented, along with spatial distributions of some of these quantities at selected times.     

Stress analysis of ribbed flanges

In this paper an attempt is made to study the role of stiffening ribs on the behaviour of flanges. Shell theory is resorted to in developing the ribbed shell equations. A shell element with stiffening ribs in the meridional direction is obtained by superposing the stress resultants of a grid shell with the stress resultants of a isotropic shell element. A case of unequal bolt loads is considered by expressing them as Fourier series in the circumferential direction. The effect of varying the rib parameters like rib height, number, etc. is studied. The finite-difference method of solution is adopted. The behaviour of a ribbed flange is found to be similar to that of a taper hub flange.     

Sensitivity analysis for the dynamic response of viscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric response of viscoplastic shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. Spatial discretization is performed using finite elements, with discontinuous stress resultants across element interfaces. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) with an implicit constitutive update. The sensitivity coefficients are evaluated using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading, and a circular plate subjected to impulsive loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to the thickness, mass density, Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients are presented, along with spatial distributions of these quantities at selected times.     

Nonlinear dynamic analysis of curved beams

A computational procedure is presented for predicting the dynamic response of curved beams with geometric nonlinearities. A mixed formulation is used with the fundamental unknowns consisting of stress resultants, generalized displacements and velocity components. The governing semidiscrete finite element equations consist of a mixed system of algebraic and differential equations. The temporal integration of the differential equations is performed by using an explicit half-station central difference method. A procedure is outlined for lumping both the flexibilities and masses of the mixed model, thereby uncoupling all the equations of the system. The advantages of the proposed computational procedure over explicit methods used with the displacement formulation are discussed. The effectiveness and versatility of the proposed approach are demonstrated by means of numerical examples.     

Thin shell and surface crack finite elements for simulation of combined failure modes

In this study we present a new approach to analyse cracked shell structures subjected to large geometric changes. It is based on a combination of a rectangular assumed natural deviatoric strain thin shell finite element and an improved linespring finite element. Plasticity is accounted for using stress resultants. A power law hardening model is used for shell and linespring material. A co-rotational formulation is employed to represent nonlinear geometry effects. With this, one can carry out nonlinear fracture mechanics assessments in structures that show instabilities due buckling (local/global), ovalisation and large rigid body motion. By numerical examples it is shown how geometric instabilities and fracture compete as governing failure mode.     

Analysis of laminated shells with laminated stiffeners using rectangular shell finite elements

A finite element analysis of laminated shells reinforced with laminated stiffeners is described in this paper. A rectangular laminated anisotropic shallow thin shell finite element of 48 d.o.f. is used in conjunction with a laminated anisotropic curved beam and shell stiffening finite element having 16 d.o.f. Compatibility between the shell and the stiffener is maintained all along their junction line. Some problems of symmetrically stiffened isotropic plates and shells have been solved to evaluate the performance of the present method. Behaviour of an eccentrically stiffened laminated cantilever cylindrical shell has been predicted to show the ability of the present program. General shells amenable to rectangular meshes can also be solved in a similar manner.     

Non-linear strain-displacement equations exactly representing large rigid-body motions. Part III. Analysis of TM shells with constraints

The precise representation of arbitrarily large rigid-body motions in the displacement patterns of curved Timoshenko-Mindlin-type (TM) shell elements has been considered in Part I of the present work. In Part II it has been developed an enhanced mixed finite element formulation that allows using load increments that are much larger than possible with existing geometrically exact displacement-based shell element formulations. In this paper the developed formulation is employed to solve frictionless contact problems for TM shells undergoing finite deformations and interacting with rigid bodies. The contact conditions are incorporated into the assumed stress-strain TM shell formulation by applying a perturbed Lagrangian procedure with the fundamental unknowns consisting of 6 displacements and 11 strains of the bottom and top surfaces of the shell, 11 conjugate stress resultants and the Lagrange multiplier, associated with a nodal contact force, through using the non-conventional technique. The efficiency and accuracy of the proposed finite element formulation are demonstrated by means of several numerical examples.     

Analysis of Nonlinear Multibody Systems with Elastic Couplings

This paper is concerned with the dynamic analysis of nonlinear multibody systems involving elastic members made of laminated, anisotropic composite materials. The analysis methodology can be viewed as a three-step procedure. First, the sectional properties of beams made of composite materials are determined based on an asymptotic procedure that involves a two-dimensional finite element analysis of the cross-section. Second, the dynamic response of nonlinear, flexible multibody systems is simulated within the framework of energy-preserving and energy-decaying time-integration schemes that provide unconditional stability for nonlinear systems. Finally, local three-dimensional stresses in the beams are recovered, based on the stress resultants predicted in the previous step. Numerical examples are presented and focus on the behavior of multibody systems involving members with elastic couplings.     

Extension of the Euler–Bernoulli model of piezoelectric laminates to include 3D effects via a mixed approach

In this paper a coupled Euler–Bernoulli model of laminated piezoelectric beams is proposed. It is characterized by accounting for the influence of 3D distribution of mechanical stresses and strains through corrected electromechanical constitutive equations. In particular, the hypothesis of vanishing transverse (width direction) normal stress typical of standard beam models is weakened by imposing vanishing stress resultants. This integral condition is enforced by adopting a mixed variational principle and Lagrange multiplier method. Explicit expressions for the beam constitutive coefficients are given and the sandwich and bimorph piezoelectric benders are studied in details. The model is assessed through comparisons with standard models and 3D finite element results, showing an important enhancement of standard beam theories.     

An error analysis approach for laminated composite plate finite element models

Errors in laminated composite plate finite element models occur at both the individual element level and at the discretization level. This paper shows that parasitic shear causes individual element errors and that its sources must be eliminated if numerically and physically correct results are to be provided by the finite element analysis. In addition, discretization errors occur when the behavior of the continuum is represented by a finite number of degrees of freedom. A procedure to estimate discretization errors in laminated composite plate finite element models and guide refinement, in order to achieve an acceptable level of accuracy, is developed. The error estimator built is based on the energy norm of the error in stress resultants.     

Nonlinear dynamic analysis of quasi-symmetric anisotropic structures

An efficient computational strategy is presented for the nonlinear dynamic analysis of quasi-symmetric anisotropic structures. A mixed formulation is used with the fundamental unknowns consisting of stress resultants, generalized displacements and velocity components. The governing semi-discrete finite element equations consist of a mixed system of algebraic and ordinary differential equations. The temporal integration of the differential equations is performed by using an explicit half-station central difference method. The three key elements of the strategy are:

• 1.(a) use of mixed finite element models with independent shape functions for the stress resultants, generalized displacements and velocity components and with the stress resultants allowed to be discontinuous at interelement boundaries
• 2.(b) operator splitting, or additive decomposition of the different arrays in the governing equations into the contributions to a symmetrized response plus correction terms
• 3.(c) application of a preconditioned conjugate gradient technique to generate the unsymmetric response of the structure, at each time step, as the sum of symmetric and antisymmetric modes, each obtained using approximately half the degrees of freedom of the original model. The preconditioning matrix is taken to be the matrix associated with the symmetrized response.The effectiveness of the proposed strategy is demonstrated by means of a numerical example and the potential of the proposed strategy for solving more complex nonlinear problems is discussed.

     

Mixed least-squares finite element model for the static analysis of laminated composite plates

This paper presents a mixed finite element model for the static analysis of laminated composite plates. The formulation is based on the least-squares variational principle, which is an alternative approach to the mixed weak form finite element models. The mixed least-squares finite element model considers the first-order shear deformation theory with generalized displacements and stress resultants as independent variables. Specifically, the mixed model is developed using equal-order C⁰ Lagrange interpolation functions of high p-levels along with full integration. This mixed least-squares-based discrete model yields a symmetric and positive-definite system of algebraic equations. The predictive capability of the proposed model is demonstrated by numerical examples of the static analysis of four laminated composite plates, with different boundary conditions and various side-to-thickness ratios. Particularly, the mixed least-squares model with high-order interpolation functions is shown to be insensitive to shear-locking.     

Thermal postbuckling of thin-walled composite stiffeners

A study is made of the thermal postbuckling response of composite stiffeners subjected to prescribed edge displacement and a temperature rise. The flanges and web of the stiffeners are modeled by using two-dimensional plate finite elements. A mixed formulation is used with the fundamental unknowns consisting of the generalized displacements and the stress resultants of the plate. A reduction method is used in conjunction with mixed finite element models for determining the postbuckling response of the stiffeners. Sensitivity derivatives are evaluated and used to study the effects of variations in the different lamination and material parameters of the stiffeners on their postbuckling response characteristics. Numerical studies are presented for anisotropic stiffeners with Zee and channel sections.     

Transverse shear stiffness of laminated anisotropic shells

The additional constitutive equations required by transverse shear deformation theory of anisotropic heterogeneous shells are derived without the usual assumption of thickness distribution for either transverse shear stresses or strains. The derivation is based on Taylor series expansions about a generic point for stress resultants and couples which identically satisfy plate equilibrium equations. These equations give the in-surface stress resultants and couples in terms of the transverse shear stress resultants at the point and arbitrary constants, which may be interpreted as redundant “forces”. Starting from these expressions, we derive statically correct expressions (in terms of the transverse shear stress resultants and redundants) of the following variables: (1) in-surface stresses, using the stretching-bending constitutive equations and the Kirchhoff distributions of in-surface strains, (2) transverse shear stresses, by integration in the normal direction of the three-dimensional equilibrium equations, and (3) the area density of transverse shear strain energy, by integration in the normal direction of the corresponding volumetric density. Finally, by applying Castigliano's theorem of least work, the shear strain energy is minimized with respect to the redundants, thereby leading to the desired constitutive equations. Corresponding transverse shear stiffnesses are presented for several laminated walls, and reasonable agreement is obtained between transverse shear deformation plate theory using these stiffnesses and exact three-dimensional elasticity solutions for the problem of cylindrical bending of a plate.