Exact Solution of the Elastica with Transverse Shear Effects

A new derivation of Euler's Elastica with transverse shear effects included is presented. The elastic potential energy of bending and transverse shear is set up. The work of the axial compression force is determined. The equation of equilibrium is derived using the variation of the total potential. Using substitution of variables an exact solution is derived. The equation is transcendental and does not have a closed form solution. It is evaluated in a dimensionless form by using a numerical procedure. Finally, numerical examples of laminates made of composite material (fiber reinforced) and sandwich panels are provided.

Non-linear analysis of the thermo-electro-mechanical behaviour of shear deformable FGM plates with piezoelectric actuators

This paper investigates the non-linear bending behaviour of functionally graded plates that are bonded with piezoelectric actuator layers and subjected to transverse loads and a temperature gradient based on Reddy's higher-order shear deformation plate theory. The von Karman-type geometric non-linearity, piezoelectric and thermal effects are included in mathematical formulations. The temperature change is due to a steady-state heat conduction through the plate thickness. The material properties are assumed to be graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The plate is clamped at two opposite edges, while the remaining edges can be free, simply supported or clamped. Differential quadrature approximation in the X-axis is employed to convert the partial differential governing equations and the associated boundary conditions into a set of ordinary differential equations. By choosing the appropriate functions as the displacement and stress functions on each nodal line and then applying the Galerkin procedure, a system of non-linear algebraic equations is obtained, from which the non-linear bending response of the plate is determined through a Picard iteration scheme. Numerical results for zirconia/aluminium rectangular plates are given in dimensionless graphical form. The effects of the applied actuator voltage, the volume fraction exponent, the temperature gradient, as well as the characteristics of the boundary conditions are also studied in detail. Copyright (C) 2004 John Wiley Sons, Ltd.

Detailed Measurements during a Transient Front in a Small Subtropical Estuary

High-resolution measurements of velocity and physio-chemistry were conducted before, during and after the passage of a transient front in a small subtropical system about 2.1 km upstream of the river mouth. Detailed acoustic Doppler velocimetry measurements, conducted continuously at 25 Hz, showed the existence of transverse turbulent shear between 300 s prior to the front passage and 1300 s after. This was associated with an increased level of suspended sediment concentration fluctuations, some transverse shear next to the bed and some surface temperature anomaly.

Large amplitude vibration of thermo-electro-mechanically stressed FGM laminated plates

This paper presents a large amplitude vibration analysis of pre-stressed functionally graded material (FGM) laminated plates that are composed of a shear deformable functionally graded layer and two surface-mounted piezoelectric actuator layers. Nonlinear governing equations of motion are derived within the context of Reddy's higher-order shear deformation plate theory to account for transverse shear strain and rotary inertia. Due to the bending and stretching coupling effect, a nonlinear static problem is solved first to determine the initial stress state and pre-vibration deformations of the plate that is subjected to uniform temperature change, in-plane forces and applied actuator voltage. By adding an incremental dynamic state to the pre-vibration state, the differential equations that govern the nonlinear vibration behavior of pre-stressed FGM laminated plates are derived. A semi-analytical method that is based on one-dimensional differential quadrature and Galerkin technique is proposed to predict the large amplitude vibration behavior of the laminated rectangular plates with two opposite clamped edges. Linear vibration frequencies and nonlinear normalized frequencies are presented in both tabular and graphical forms, showing that the normalized frequency of the FGM laminated plate is very sensitive to vibration amplitude, out-of-plane boundary support, temperature change, in-plane compression and the side-to-thickness ratio. The CSCF and CFCF plates even change the inherent hard-spring characteristic to soft-spring behavior at large vibration amplitudes. (C) 2003 Elsevier B.V. All rights reserved.

Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading

In this paper, we examine the postbuckling behavior of functionally graded material FGM rectangular plates that are integrated with surface-bonded piezoelectric actuators and are subjected to the combined action of uniform temperature change, in-plane forces, and constant applied actuator voltage. A Galerkin-differential quadrature iteration algorithm is proposed for solution of the non-linear partial differential governing equations. To account for the transverse shear strains, the Reddy higher-order shear deformation plate theory is employed. The bifurcation-type thermo-mechanical buckling of fully clamped plates, and the postbuckling behavior of plates with more general boundary conditions subject to various thermo-electro-mechanical loads, are discussed in detail. Parametric studies are also undertaken, and show the effects of applied actuator voltage, in-plane forces, volume fraction exponents, temperature change, and the character of boundary conditions on the buckling and postbuckling characteristics of the plates. (C) 2003 Elsevier Science Ltd. All rights reserved.

Development of beam and plate finite elements based on the absolute nodal coordinate formulation

The focus of this dissertation is to develop finite elements based on the absolute nodal coordinate formulation. The absolute nodal coordinate formulation is a nonlinear finite element formulation, which is introduced for special requirements in the field of flexible multibody dynamics. In this formulation, a special definition for the rotation of elements is employed to ensure the formulation will not suffer from singularities due to large rotations. The absolute nodal coordinate formulation can be used for analyzing the dynamics of beam, plate and shell type structures. The improvements of the formulation are mainly concentrated towards the description of transverse shear deformation. Additionally, the formulation is verified by using conventional iso-parametric solid finite element and geometrically exact beam theory. Previous claims about especially high eigenfrequencies are studied by introducing beam elements based on the absolute nodal coordinate formulation in the framework of the large rotation vector approach. Additionally, the same high eigenfrequency problem is studied by using constraints for transverse deformation. It was determined that the improvements for shear deformation in the transverse direction lead to clear improvements in computational efficiency. This was especially true when comparative stress must be defined, for example when using elasto-plastic material. Furthermore, the developed plate element can be used to avoid certain numerical problems, such as shear and curvature lockings. In addition, it was shown that when compared to conventional solid elements, or elements based on nonlinear beam theory, elements based on the absolute nodal coordinate formulation do not lead to an especially stiff system for the equations of motion.

Development of finite elements for analysis of biomechanical structures using flexible multibody formulations

The absolute nodal coordinate formulation was originally developed for the analysis of structures undergoing large rotations and deformations. This dissertation proposes several enhancements to the absolute nodal coordinate formulation based finite beam and plate elements. The main scientific contribution of this thesis relies on the development of elements based on the absolute nodal coordinate formulation that do not suffer from commonly known numerical locking phenomena. These elements can be used in the future in a number of practical applications, for example, analysis of biomechanical soft tissues. This study presents several higher-order Euler–Bernoulli beam elements, a simple method to alleviate Poisson’s and transverse shear locking in gradient deficient plate elements, and a nearly locking free gradient deficient plate element. The absolute nodal coordinate formulation based gradient deficient plate elements developed in this dissertation describe most of the common numerical locking phenomena encountered in the formulation of a continuum mechanics based description of elastic energy. Thus, with these fairly straightforwardly formulated elements that are comprised only of the position and transverse direction gradient degrees of freedom, the pathologies and remedies for the numerical locking phenomena are presented in a clear and understandable manner. The analysis of the Euler–Bernoulli beam elements developed in this study show that the choice of higher gradient degrees of freedom as nodal degrees of freedom leads to a smoother strain field. This improves the rate of convergence.

Design of laminated piezocomposite shell transducers with arbitrary fiber orientation using topology optimization approach

Sensor and actuator based on laminated piezocomposite shells have shown increasing demand in the field of smart structures. The distribution of piezoelectric material within material layers affects the performance of these structures; therefore, its amount, shape, size, placement, and polarization should be simultaneously considered in an optimization problem. In addition, previous works suggest the concept of laminated piezocomposite structure that includes fiber-reinforced composite layer can increase the performance of these piezoelectric transducers; however, the design optimization of these devices has not been fully explored yet. Thus, this work aims the development of a methodology using topology optimization techniques for static design of laminated piezocomposite shell structures by considering the optimization of piezoelectric material and polarization distributions together with the optimization of the fiber angle of the composite orthotropic layers, which is free to assume different values along the same composite layer. The finite element model is based on the laminated piezoelectric shell theory, using the degenerate three-dimensional solid approach and first-order shell theory kinematics that accounts for the transverse shear deformation and rotary inertia effects. The topology optimization formulation is implemented by combining the piezoelectric material with penalization and polarization model and the discrete material optimization, where the design variables describe the amount of piezoelectric material and polarization sign at each finite element, with the fiber angles, respectively. Three different objective functions are formulated for the design of actuators, sensors, and energy harvesters. Results of laminated piezocomposite shell transducers are presented to illustrate the method. Copyright (C) 2012 John Wiley & Sons, Ltd.

A new insight in stress recovery tecniques and hessian reconstruction methods

Stress recovery techniques have been an active research topic in the last few years since, in 1987, Zienkiewicz and Zhu proposed a procedure called Superconvergent Patch Recovery (SPR). This procedure is a last-squares fit of stresses at super-convergent points over patches of elements and it leads to enhanced stress fields that can be used for evaluating finite element discretization errors. In subsequent years, numerous improved forms of this procedure have been proposed attempting to add equilibrium constraints to improve its performances. Later, another superconvergent technique, called Recovery by Equilibrium in Patches (REP), has been proposed. In this case the idea is to impose equilibrium in a weak form over patches and solve the resultant equations by a last-square scheme. In recent years another procedure, based on minimization of complementary energy, called Recovery by Compatibility in Patches (RCP) has been proposed in. This procedure, in many ways, can be seen as the dual form of REP as it substantially imposes compatibility in a weak form among a set of self-equilibrated stress fields. In this thesis a new insight in RCP is presented and the procedure is improved aiming at obtaining convergent second order derivatives of the stress resultants. In order to achieve this result, two different strategies and their combination have been tested. The first one is to consider larger patches in the spirit of what proposed in [4] and the second one is to perform a second recovery on the recovered stresses. Some numerical tests in plane stress conditions are presented, showing the effectiveness of these procedures. Afterwards, a new recovery technique called Last Square Displacements (LSD) is introduced. This new procedure is based on last square interpolation of nodal displacements resulting from the finite element solution. In fact, it has been observed that the major part of the error affecting stress resultants is introduced when shape functions are derived in order to obtain strains components from displacements. This procedure shows to be ultraconvergent and is extremely cost effective, as it needs in input only nodal displacements directly coming from finite element solution, avoiding any other post-processing in order to obtain stress resultants using the traditional method. Numerical tests in plane stress conditions are than presented showing that the procedure is ultraconvergent and leads to convergent first and second order derivatives of stress resultants. In the end, transverse stress profiles reconstruction using First-order Shear Deformation Theory for laminated plates and three dimensional equilibrium equations is presented. It can be seen that accuracy of this reconstruction depends on accuracy of first and second derivatives of stress resultants, which is not guaranteed by most of available low order plate finite elements. RCP and LSD procedures are than used to compute convergent first and second order derivatives of stress resultants ensuring convergence of reconstructed transverse shear and normal stress profiles respectively. Numerical tests are presented and discussed showing the effectiveness of both procedures.

Static analysis of functionally graded cylindrical and conical shells or panels using the generalized unconstrained third order theory coupled with the stress recovery

A 2D Unconstrained Third Order Shear Deformation Theory (UTSDT) is presented for the evaluation of tangential and normal stresses in moderately thick functionally graded conical and cylindrical shells subjected to mechanical loadings. Several types of graded materials are investigated. The functionally graded material consists of ceramic and metallic constituents. A four parameter power law function is used. The UTSDT allows the presence of a finite transverse shear stress at the top and bottom surfaces of the graded shell. In addition, the initial curvature effect included in the formulation leads to the generalization of the present theory (GUTSDT). The Generalized Differential Quadrature (GDQ) method is used to discretize the derivatives in the governing equations, the external boundary conditions and the compatibility conditions. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. In this way, the six components of the stress tensor at a point of the conical or cylindrical shell or panel can be given. The initial curvature effect and the role of the power law functions are shown for a wide range of functionally conical and cylindrical shells under various loading and boundary conditions. Finally, numerical examples of the available literature are worked out.

Influence of imperfect interfaces on bending and vibration of laminated composite shells

This paper is devoted to modeling elastic behavior of laminated composite shells, with special emphasis on incorporating interfacial imperfection. The conditions of imposing traction continuity and displacement jump across each interface are used to model imperfect interfaces. Vanishing transverse shear stresses on two free surfaces of a shell eliminate the need for shear correction factors. A linear theory underlying elastostatics and kinetics of laminated composite shells in a general configuration is presented from Hamilton's principle. In the special case of vanishing interfacial parameters, this theory reduces to the conventional third-order zigzag theory for perfectly bonded laminated shells. Numerical results for bending and vibration problems of laminated circular cylindrical panels are tabulated and plotted to indicate the influence of the interfacial imperfection. (C) 2000 Elsevier Science Ltd. All rights reserved.

Vibration of symmetrically laminated thick super elliptical plates

This paper reports a free vibration analysis of thick plates with rounded corners subject to a free, simply-supported or clamped boundary condition. The plate perimeter is defined by a super elliptic function with a power defining the shape ranging from an ellipse to a rectangle. To incorporate transverse shear deformation, the Reddy third-order plate theory is employed. The energy integrals incorporating shear deformation and rotary inertia are formulated and the p-Ritz procedures are used to derive the governing eigenvalue equation. Numerical examples for plates with different shapes and boundary conditions are solved and their frequency parameters, where possible, are compared with known results. Parametric studies are carried out to show the sensitivities of frequency parameters by varying the geometry, fibre stacking sequence, and boundary condition. (C) 1999 Academic Press.

Finite element analysis of the statics of and vibrations in axisymmetric shells

The finite element process is now used almost routinely as a tool of engineering analysis. From early days, a significant effort has been devoted to developing simple, cost effective elements which adequately fulfill accuracy requirements. In this thesis we describe the development and application of one of the simplest elements available for the statics and dynamics of axisymmetric shells . A semi analytic truncated cone stiffness element has been formulated and implemented in a computer code: it has two nodes with five degrees of freedom at each node, circumferential variations in displacement field are described in terms of trigonometric series, transverse shear is accommodated by means of a penalty function and rotary inertia is allowed for. The element has been tested in a variety of applications in the statics and dynamics of axisymmetric shells subjected to a variety of boundary conditions. Good results have been obtained for thin and thick shell cases .

A finite-strain solid–shell using local Löwdin frames and least-squares strains

A finite-strain solid–shell element is proposed. It is based on least-squares in-plane assumed strains, assumed natural transverse shear and normal strains. The singular value decomposition (SVD) is used to define local (integration-point) orthogonal frames-of-reference solely from the Jacobian matrix. The complete finite-strain formulation is derived and tested. Assumed strains obtained from least-squares fitting are an alternative to the enhanced-assumed-strain (EAS) formulations and, in contrast with these, the result is an element satisfying the Patch test. There are no additional degrees-of-freedom, as it is the case with the enhanced-assumed-strain case, even by means of static condensation. Least-squares fitting produces invariant finite strain elements which are shear-locking free and amenable to be incorporated in large-scale codes. With that goal, we use automatically generated code produced by AceGen and Mathematica. All benchmarks show excellent results, similar to the best available shell and hybrid solid elements with significantly lower computational cost.