Alexander Tessler

An improved plate theory of {1,2}-order for thick composite laminates

Abstract A new two-dimensional laminate plate theory is developed for the linear elastostatic analysis of thick composite plates. The theory employs equivalent single-layer assumptions for the displacements, transverse shear strains, and transverse normal stress. The inplane and transverse displacements are respectively linear and quadratic expansions through the laminate thickness, where the low-order expansion coefficients correspond to the variables of Reissners first-order shear-deformable theory. The transverse shear strains and transverse normal stress are assumed to be quadratic and cubic respectively through the thickness ; they are expressed in terms of the kinematic variables of the theory by means of a least-squares compatibility requirement for the transverse strains and explicit enforcement of exact traction boundary conditions on the top and bottom plate surfaces. Application of the virtual work principle results in the loth-order equations of equilibrium and associated Poisson boundary conditions. A major advantage of this theory over other higher-order theories lies in its perfect suitability for finite element approximation : simple C 0 continuous interpolations for the kinematic variables of the first-order theory (and, optionally, C −1 interpolations for the two higher-order displacements) are needed to formulate effective and robust two-dimensional plate elements capable of full three-dimensional ply-by-ply strain and stress recovery within the framework of general-purpose finite element codes. In assessing the predictive capability of the theory, analytic solutions for the problem of cylindrical bending are derived and compared with exact three-dimensional elasticity results and those of the earlier version of the {1,2}-order plate theory.

A {3,2}-Order Bending Theory for Laminated Composite and Sandwich Beams

A higher-order bending theory is derived for laminated composite and sandwich beams thus extending the recent {1,2}-order theory to include third-order axial effects without introducing additional kinematic variables. The present theory is of order {3,2} and includes both transverse shear and transverse normal deformations. A closed-form solution to the cylindrical bending problem is derived and compared with the corresponding exact elasticity solution. The numerical comparisons are focused on the most challenging material systems and beam aspect ratios which include moderate-to-thick unsymmetric composite and sandwich laminates. Advantages and limitations of the theory are discussed.

Analysis of thick sandwich construction by a {3,2}-order theory

Abstract This study is an extension of the {1,2}-order plate theory to a higher order {3,2} theory. Based on the equivalent single-layer assumptions, the in-plane and transverse displacement components are expressed as cubic and quadratic expansions through the thickness of the sandwich construction. Also, the transverse stress component is assumed to vary as a cubic function through the thickness. Utilizing Reissner’s definitions for kinematics of thick plates, the displacement components at any point on the plate are approximated in terms of weighted-average quantities (displacements and rotations) that are functions of the in-plane coordinates. The undetermined coefficients defining the in-plane and transverse displacement fields are then expressed in terms of the weighted-average displacements and rotations and their derivatives by directly employing Reissner’s definitions and enforcing the zero transverse-shear-stress conditions on the upper and lower surfaces of the sandwich panel. The coefficients defining the transverse stress component are obtained by requiring the transverse strain component, which is expressed in terms of the unknown coefficients of the transverse stress component from a mixed constitutive relation, to be the least-squares equivalent of the kinematic definition of the transverse strain component. The resulting expressions for the unknown coefficients of the transverse stress component are related to resultant strains and curvatures defined from kinematic relations. The equations of equilibrium and boundary conditions of the sandwich plate based on the {3,2}-higher-order theory are derived by employing the principles of virtual displacements. The robustness and accuracy of this {3,2}-order plate theory are established through comparisons with exact solutions available in the literature. The finite element implementation of the present {3,2}-order plate theory is also discussed.

A variational method for finite element stress recovery and error estimation

Abstract A variational method for obtaining smoothed stresses from a finite element derived non-smooth stress field is presented. The method is based on minimizing a functional involving discrete least-squares error plus a penalty constraint that ensures smoothness of the stress field. An equivalent accuracy criterion is developed for the smoothing analysis which results in a C1-continuous smoothed stress field possessing the same order of accuracy as that found at the superconvergent optimal stress points of the original finite element analysis. Application of the smoothing analysis to residual error estimation is also demonstrated.

An improved variational method for finite element stress recovery and a posteriori error estimation

Abstract A new variational formulation is presented which serves as a foundation for an improved finite element stress recovery and a posteriori error estimation. In the case of stress predictions, interelement discontinuous stress fields from finite element solutions are transformed into a C 1 -continuous stress field with C 0 -continuous stress gradients. These enhanced results are ideally suited for error estimation since the stress gradients can be used to assess equilibrium satisfaction. The approach is employed as a post-processing step in finite element analysis. The variational statement used herein combines discrete-least squares, penalty-constraint, and curvature-control functionals, thus enabling automated recovery of smooth stresses and stress gradients. The paper describes the mathematical foundation of the method and presents numerical examples including stress recovery in two-dimensional structures and built-up aircraft components, and error estimation for adaptive mesh refinement procedures.

Nonlinear analysis of laminates through a mindlin-type shear deformable shallow shell element

Abstract This study presents a nonlinear analysis with application to a doubly curved shallow shell element free of ‘locking’. The ‘locking’ phenomenon is eliminated by explicitly determining the shear and membrane correction factors. The element formulation utilizes the Reissner-Mindlin and Marguerre theories. The analysis of thin and moderately thick composite shells undergoing large displacements and rotations is achieved by using the corotational form of an updated Lagrangian formulation. The validity of the analysis is established by correlating present results with various benchmark cases that involve large displacements and rotations, as well as elastic stability.

An inverse finite element method for beam shape sensing: theoretical framework and experimental validation

Shape sensing, i.e., reconstruction of the displacement field of a structure from surface-measured strains, has relevant implications for the monitoring, control and actuation of smart structures. The inverse finite element method (iFEM) is a shape-sensing methodology shown to be fast, accurate and robust. This paper aims to demonstrate that the recently presented iFEM for beam and frame structures is reliable when experimentally measured strains are used as input data.The theoretical framework of the methodology is first reviewed. Timoshenko beam theory is adopted, including stretching, bending, transverse shear and torsion deformation modes. The variational statement and its discretization with C0-continuous inverse elements are briefly recalled. The three-dimensional displacement field of the beam structure is reconstructed under the condition that least-squares compatibility is guaranteed between the measured strains and those interpolated within the inverse elements.The experimental setup is then described. A thin-walled cantilevered beam is subjected to different static and dynamic loads. Measured surface strains are used as input data for shape sensing at first with a single inverse element. For the same test cases, convergence is also investigated using an increasing number of inverse elements. The iFEM-recovered deflections and twist rotations are then compared with those measured experimentally. The accuracy, convergence and robustness of the iFEM with respect to unavoidable measurement errors, due to strain sensor locations, measurement systems and geometry imperfections, are demonstrated for both static and dynamic loadings.

Structural Analysis Methods for Structural Health Management of Future Aerospace Vehicles

Two finite-element-based, full-field computational methods and algorithms for use in Structural Health Management (SHM) systems are reviewed. Their versatility, robustness, and computational efficiency make them well suited for real-time, large-scale space vehicle, structures, and habitat applications. The methods may be effectively employed to enable real-time processing of sensing information, specifically for identifying three-dimensional deformed structural shapes as well as the internal loads. In addition, they may be used in conjunction with evolutionary algorithms to design optimally distributed sensors. These computational tools have demonstrated substantial promise for utilization in future SHM systems.

A Robust and Consistent First-Order Zigzag Theory for Multilayered Beams

In this paper a recently developed refinedfirst-order zigzag theory for multilayered beams is reviewed from a fresh theoretical perspective. The theory includes the kinematics of Timoshenko beam theory as its baseline. The use of a novel piecewise-linear zigzag function provides a more realistic representation of the deformation states of transverse-shear-flexible multilayered beams. Though the formulation does not enforce full continuity of the transverse-shear stresses across the beam’s depth, yet it is robust in the sense that transverse-shear correction factors are not required to yield accurate results. The new theory is variationally consistent, requires only C 0-continuity for kinematic approximations, and is thus perfectly suited for developing computationally efficient finite elements.

Equivalent single-layer theory for a complete stress field in sandwich panels under arbitrarily distributed loading

Abstract In single-layer theory, the displacement components represent the weighted-average through the thickness of the sandwich panel. Although discrete-layer theories are more representative of sandwich construction than the single-layer theories, they suffer from an excessive number of field variables in proportion to the number of layers. In this study, utilizing Reissner’s definitions for kinematics of thick plates, the displacement components at any point on the plate are approximated in terms of weighted-average quantities (displacements and rotations) that are functions of the in-plane coordinates. The equations of equilibrium and boundary conditions of the sandwich panel are derived by employing the principle of virtual displacements. The solution for an arbitrarily distributed load is obtained by employing Fourier series representations of the unknown field variables. This single-layer theory is validated against an analytical solution for simply supported square sandwich panels under pressure over a small region on the face sheet and is also compared with previous single-layer plate theories.