Variational Asymptotic Method Research Articles

Theory of initially twisted, composite, thin-walled beams

An asymptotically correct theory for initially twisted, thin-walled, composite beams has been constructed by the variational asymptotic method. The strain energy of the original, three-dimensional structure is first rigorously reduced to be a two-dimensional energy expressed in terms of shell strains. Then the two-dimensional strain energy is further reduced to be expressed in terms of the classical beam strain measures. The resulting theory is a classical beam model approximating the three-dimensional energy through the first-order of the initial twist. Consistent use of small parameters that are intrinsic to the problem allows a natural derivation for all thin-walled beams within a common framework, regardless of whether the section is open, closed, or strip-like. Several examples are studied using the present theory and the results are compared with a general cross-sectional analysis, VABS, and other published results.

Mathematical construction of a Reissner–Mindlin plate theory for composite laminates

A Reissner–Mindlin theory for composite laminates without invoking ad hoc kinematic assumptions is constructed using the variational-asymptotic method. Instead of assuming a priori the distribution of three-dimensional displacements in terms of two-dimensional plate displacements as what is usually done in typical plate theories, an exact intrinsic formulation has been achieved by introducing unknown three-dimensional warping functions. Then the variational-asymptotic method is applied to systematically decouple the original three-dimensional problem into a one-dimensional through-the-thickness analysis and a two-dimensional plate analysis. The resulting theory is an equivalent single-layer Reissner–Mindlin theory with an excellent accuracy comparable to that of higher-order, layer-wise theories. The present work is extended from the previous theory developed by the writer and his co-workers with several sizable contributions: (a) six more constants (33 in total) are introduced to allow maximum freedom to transform the asymptotically correct energy into a Reissner–Mindlin model; (b) the semi-definite programming technique is used to seek the optimum Reissner–Mindlin model. Furthermore, it is proved the first time that the recovered three-dimensional quantities exactly satisfy the continuity conditions on the interface between different layers and traction boundary conditions on the bottom and top surfaces. It is also shown that two of the equilibrium equations of three-dimensional elasticity can be satisfied asymptotically, and the third one can be satisfied approximately so that the difference between the Reissner–Mindlin model and the second-order asymptotical model can be minimized. Numerical examples are presented to compare with the exact solution as well as the classical lamination theory and the first-order shear-deformation theory, demonstrating that the present theory has an excellent agreement with the exact solution.

Mathematical construction of an engineering thermopiezoelastic model for smart compositeshells**Part of this paper was presented at the 45th Structures, Structural Dynamics andMaterials Conference, Palm Springs, California, April 19–22, 2004.

An engineering model for composite piezoelectric shells under mechanical, thermal, andelectrical loads has been constructed mathematically using the variational-asymptoticmethod. This work presents a unique formulation of the nonlinear, three-dimensional,one-way coupled, thermopiezoelasticity problem having the combined merits of bothmathematical rigor and engineering simplicity. The variational-asymptotic method is usedto rigorously split the three-dimensional problem into two problems: a nonlinear,two-dimensional, shell analysis over the reference surface to obtain the global response,and a linear analysis through the thickness to provide both the generalized shellconstitutive model and recovery relations to approximate the original three-dimensionalfields. The asymptotically correct electric enthalpy obtained herein is cast into theReissner–Mindlin form to account for transverse shear deformation including thegeometrical refinement due to initial curvatures. Recovery relations have been provided toobtain accurate stress distribution through the thickness. The present model isimplemented into the computer program VAPAS. Results for several cases obtainedfrom VAPAS are compared with exact thermopiezoelasticity solutions, classicallamination theory, and first-order shear-deformation theory. An excellent compromisebetween efficiency and accuracy for analyzing piezoelectric composite shells has beenachieved.

A simple thermopiezoelastic model for smart composite plates with accurate stress recovery**Presented at the 44th Structures, Structural Dynamics and Materials Conference, Norfolk,VA, 7–10 April 2003.

A Reissner–Mindlin model for analyzing laminated composite plates including piezoelectriclayers under mechanical, thermal and electric loads has been constructed using thevariational-asymptotic method. The present work formulates the original nonlinear,three-dimensional, one-way coupled, thermopiezoelasticity problem allowing for arbitrarydeformation of the normal line and using a set of intrinsic variables defined on the referenceplane. The variational-asymptotic method is used to rigorously split the three-dimensionalproblem into two problems: a nonlinear, two-dimensional, plate analysis over the referenceplane to obtain the global deformation, and a linear analysis through the thickness toprovide both the two-dimensional generalized constitutive model and recovery relations toapproximate the original three-dimensional results. The obtained asymptoticallycorrect second-order electric enthalpy is cast into the form of the commonly usedReissner–Mindlin model to account for transverse shear deformation. The present theory isimplemented into the computer program VAPAS (variational-asymptotic plate and shellanalysis). Results for several cases obtained from VAPAS are compared with theexact thermopiezoelasticity solutions, classical lamination theory and first-ordershear–deformation theory for the purpose of demonstrating the advantages of the presenttheory and the use of VAPAS. The proposed theory can achieve an accuracy comparableto higher-order layerwise theories at the cost of a first-order shear–deformationtheory.

Asymptotic Approach for Thermoelastic Analysis of Laminated Composite Plates

A thermoelastic model for analyzing laminated composite plates under both mechanical and thermal loadings is constructed by the variational asymptotic method. The original three-dimensional nonlinear thermoelasticity problem is formulated based on a set of intrinsic variables defined on the reference plane and for arbitrary deformation of the normal line. Then the variational asymptotic method is used to rigorously split the three-dimensional problem into two problems: A nonlinear, two-dimensional, plate analysis over the reference plane to obtain the global deformation and a linear analysis through the thickness to provide the two-dimensional generalized constitutive law and the recovering relations to approximate the original three-dimensional results. The nonuniqueness of asymptotic theory correct up to a certain order is used to cast the obtained asymptotically correct second-order free energy into a Reissner–Mindlin type model to account for transverse shear deformation. The present theory is impleme...

Elasticity Solutions Versus Asymptotic Sectional Analysis of Homogeneous, Isotropic, Prismatic Beams

The original three-dimensional elasticity problem of isotropic prismatic beams has been solved analytically by the variational asymptotic method (VAM). The resulting classical model (Euler-Bernoulli-like) is the same as the superposition of elasticity solutions of extension, Saint-Venant torsion, and pure bending in two orthogonal directions. The resulting refined model (Timoshenko-like) is the same as the superposition of elasticity solutions of extension, Saint-Venant torsion, and both bending and transverse shear in two orthogonal directions. The fact that the VAM can reproduce results from the theory of elasticity proves that two-dimensional finite-element-based cross-sectional analyses using the VAM, such as the variational asymptotic beam sectional analysis (VABS), have a solid mathematical foundation. One is thus able to reproduce numerically with VABS the same results for this problem as one obtains from three-dimensional elasticity, but with orders of magnitude less computational cost relative to three-dimensional finite elements.

Asymptotically accurate 3-D recovery from Reissner-like composite plate finite elements

An accurate stress/strain recovery procedure for laminated, composite plates that can be implemented in standard finite element programs is developed. The formulation is based on an asymptotic analysis and starts from a three-dimensional, anisotropic elasticity problem that takes all possible deformation into account. After a change of variable, which introduces intrinsic two-dimensional description for the deformation of the reference plane, the variational asymptotic method is then used to rigorously split this three-dimensional problem into two reduced-dimensional problems: a nonlinear, two-dimensional analysis of the reference surface of the deformed plate (an equivalent single-layer plate model), and a linear, one-dimensional analysis of the normal-line element through the thickness. The latter is solved by a one-dimensional finite element method and provides a constitutive law between the generalized, two-dimensional strains and stress resultants for the plate analysis, and a set of recovering relations to approximately express the three-dimensional displacement, strain and stress fields in terms of two-dimensional variables determined from solving the equations of the plate analysis. The strain energy functional that is asymptotically correct through the second-order in the small parameters is then cast into the form of Reissner’s theory. Although it is not in general possible to construct an asymptotically correct Reissner-like composite plate theory, an optimization procedure is used to drive the present theory as close as possible to being asymptotically correct, while maintaining the simplicity and beauty of the Reissner-like formulation. A computer program based on the present procedure, called variational asymptotic plate and shell analysis, has been developed. Its utility is demonstrated by inserting the recovery procedure into the plate element of a general-purpose finite element code. Numerical results obtained for a variety of laminated, composite plates show that three-dimensional field variables recovered from the present theory agree very well with those from exact solutions.

Validation of the Variational Asymptotic Beam Sectional Analysis

The computer program VABS (Variational Asymptotic Beam Section Analysis) uses the variational asymptotic method to split a three-dimensional nonlinear elasticity problem into a twodimensional linear cross-sectional analysis and a one-dimensional, nonlinear beam problem. This is accomplished by taking advantage of certain small parameters inherent to beam-like structures. VABS is able to calculate the one-dimensional cross-sectional stiffness constants, with transverse shear and Vlasov refinements, for initially twisted and curved beams with arbitrary geometry and material properties. Several validation cases are presented. First, an elliptic bar is modeled with transverse shear refinement using the variational asymptotic method, and the solution is shown to be identical to that obtained from the theory of elasticity. The shear center locations calculated by VABS for various cross sections agree well with those obtained from common engineering analyses. Comparisons with other composite beam theories prove that it is unnecessary to introduce ad hoc kinematic assumptions to build an accurate beam model. For numerical validation, values of the one-dimensional variables are extracted from an ABAQUS model and compared with results from a one-dimensional beam analysis using cross-sectional constants from VABS. Furthermore, point-wise three-dimensional stress and strain fields are recovered using VABS, and the correlation with the three-dimensional results from ABAQUS is excellent. Finally, classical theory is shown to be insufficient for general-purpose beam modeling. Appropriate refined theories are recommended for some classes of problems.

Asymptotic generalization of Reissner–Mindlin theory: accurate three-dimensional recovery for composite shells

A rigorous and systematic dimensional reduction of a shell-like structure is undertaken. It starts with geometrically nonlinear, three-dimensional (3-D), anisotropic elasticity theory and takes advantage of small parameters associated with the geometry. This reduction is carried out using the variational asymptotic method and splits the 3-D problem into a linear, one-dimensional (1-D), through-the-thickness analysis and a nonlinear, two-dimensional (2-D), shell analysis. The 2-D equations are put into the form of a nonlinear Reissner–Mindlin shell theory, details of which are dealt with in a separate paper. The focus of this paper is on the through-the-thickness analysis, which is solved by a 1-D finite element method and which provides two useful pieces of information: a generalized 2-D constitutive law for the shell equations, and a set of recovery relations that can be used to express the 3-D field variables through the thickness in terms of 2-D shell variables calculated in the shell analysis. The resulting analysis can be incorporated into standard Reissner–Mindlin shell finite element codes. Numerical results are compared with the exact solution, and the excellent agreement validates the fidelity of this modeling approach.

Asymptotic construction of Reissner-like composite plate theory with accurate strain recovery

The focus of this paper is to develop an asymptotically correct theory for composite laminated plates when each lamina exhibits monoclinic material symmetry. The development starts with formulation of the three-dimensional (3-D), anisotropic elasticity problem in which the deformation of the reference surface is expressed in terms of intrinsic two-dimensional (2-D) variables. The variational asymptotic method is then used to rigorously split this 3-D problem into a linear one-dimensional normal-line analysis and a nonlinear 2-D plate analysis accounting for classical as well as transverse shear deformation. The normal-line analysis provides a constitutive law between the generalized, 2-D strains and stress resultants as well as recovering relations to approximately but accurately express the 3-D displacement, strain and stress fields in terms of plate variables calculated in the plate analysis. It is known that more than one theory may exist that is asymptotically correct to a given order. This nonuniqueness is used to cast a strain energy functional that is asymptotically correct through the second order into a simple “Reissner-like” plate theory. Although it is not possible in general to construct an asymptotically correct Reissner-like composite plate theory, an optimization procedure is used to drive the present theory as close to being asymptotically correct as possible while maintaining the beauty of the Reissner-like formulation. Numerical results are presented to compare with the exact solution as well as a previous similar yet very different theory. The present theory has excellent agreement with the previous theory and exact results.

On Timoshenko-like modeling of initially curved and twisted composite beams

A generalized, finite-element-based, cross-sectional analysis for nonhomogenous, initially curved and twisted, anistropic beams is formulated from geometrically nonlinear, three-dimensional (3-D) elasticity. The 3-D strain field is formulated based on the concept of decomposition of the rotation tensor and is given in terms of one-dimensional (1-D) generalized strains and a 3-D warping displacement that is obtained from the formulation, not assumed. The warping is found in terms of the 1-D strains via the variational asymptotic method (VAM). In this paper a Timoshenko-like model is presupposed for a beam with cross-sectional characteristic length h, wavelength of deformation given by l, and the magnitude of the radius of initial curvature and/or twist is taken to be of the order R. First, a solution for the asymptotically correct refinement of classical anisotropic beam theory for initially curved and twisted beams through O(h2/R2) is obtained. Next, the O(h2/l2) correction is computed. It is known that Timoshenko-like theory is not capable of capturing all the O(h2/l2) corrections for generally anisotropic beams. However, if all the O(h2/l2) terms are known, then the corresponding Timoshenko-like theory is uniquely defined. Numerical results are presented to illustrate the trends of the various classical (extension-twist, bending-twist, and extension-bending) and nonclassical couplings (extension-shear, bending-shear, and shear-torsion) as the initial twist and curvatures are varied.

Active Beam Cross-Sectional Modeling

A finite-element based analysis formodeling initially twisted and curved active composite beams with embedded anisotropic actuation is presented. It is derived from three-dimensional electroelasticity, where the original problem is reduced via the variational asymptotic method. The resulting cross-sectional analysis takes into consideration passive and active anisotropic and nonhomogeneous materials, and can model general (thin-walled, thick-walled, solid, build-up structure) cross-sectional geometries. The formulation requires neither the costly use of three-dimensional finite element discretization nor the loss of accuracy inherent to any simplified representation of the cross section. The developed formulation is numerically implemented in the Active Variational-Asymptotical Beam Section Analysis code (VABS-A), and several numerical and experimental tests cases are used to support validation of the proposed theory. Also, the effect of the presence of a foam core in originally hollow configurations is presented and counter-intuitive conclusions are discussed. The generality of the method and accuracy of the results increase confidence at the design stage that the active beam structure will perform as expected and, consequently, should lower costs from experimental tests and further adjustments.

On the modeling of integrally actuated helicopter blades

This paper presents an asymptotical formulation for preliminary design of multi-cell composite helicopter rotor blades with integral anisotropic active plies. It represents the first attempt in the literature to asymptotically analyze such active structure. The analysis is broken down in two parts: a linear two-dimensional analysis over the cross-section, and a geometrically non-linear (beam) analysis along the blade span. The cross-sectional analysis revises and extends a closed form solution for thin-walled, multi-cell beams based on the variational-asymptotical method, accounting for the presence of active fiber composites distributed along the cross-section of the blade. The formulation provides expressions for the asymptotically correct cross-sectional stiffness constants in closed form, facilitating design-trend studies. These stiffness constants are then used in a beam finite element discretization of the blade reference line. This is an extension of the exact intrinsic equations for the one-dimensional analysis of rotating beams considering small strains and finite rotations, and now taking account of the presence of distributed actuators. Subject to external loads, active ply induced strains, and specific boundary conditions, the one-dimensional (beam) problem can be solved for displacements, rotations, and strains of the reference line. Analytical and numerical studies are presented to compare the proposed theory against the previously established analytical models. Discrepancies are found for general blade cross-section and discussed herein in details, especially for the piezoelectric actuation components. Direct results of the present formulation are also compared with experimental data.

Obliqueness effects in asymptotic cross-sectional analysis of composite beams

Cross-sectional stiffness properties for general composite beams, along with the interior solution for warping, are derived in such a way as to provide the extra freedom to choose a cross-sectional plane that is not perpendicular to the local beam reference line. The present development treats prismatic as well as initially twisted and curved beams. The main purpose of allowing such a choice of coordinate systems is for the convenience of the analyst; one, of course, should not expect the final 3D results to change. The 3D strain field is derived for the corresponding nonorthogonal curvilinear set of coordinates which describe the undeformed and the deformed geometry of the beam. The analysis is carried out based on the variational-asymptotic method which is used to determine the warping field. For the development of the solution, it is shown that there is a fundamental difference between the solutions for a prismatic beam and a beam with initial twist and curvature, which in turn produces a limitation of the angle of obliqueness: the effect of obliqueness is treated exactly for a prismatic beam, while for the initially twisted and curved beam the obliqueness is regarded as a small parameter. The ultimate goal of the analysis is the determination of the cross-sectional stiffness matrix, which can then be used as input for the 1D problem. Then, using the recovering relations, one can recover the strain field over the entire cross section, once the previous problem has been solved. The validity of the method is demonstrated by recovery of accurate cross-sectional stiffnesses associated with a normal cross-sectional plane from those associated with an oblique cross-sectional plane, using a simple rotation-of-axes transformation.

On asymptotically correct Timoshenko-like anisotropic beam theory

This paper presents a finite element cross-sectional beam analysis capable of capturing transverse shear effects. The approach uses the variational-asymptotic method and can handle beams of general cross-sectional shape and arbitrary anisotropic material. The work builds on previous works which deal with development of the classical beam theory, which includes only extension, torsion, and bending. A Timoshenko-like formulation is sought to achieve a refined theory with simple boundary conditions. Apart from some simple special cases, it is shown that this problem is overdetermined. However, it is possible to obtain `the best possible' solution using the least-squares minimization technique. The geometrical meaning of the shear variable for this formulation is also derived. Results are found to be in good agreement with published results for the shear stiffness coefficients for both isotropic and anisotropic beams.

Asymptotic treatment of the trapeze effect in finite element cross-sectional analysis of composite beams

Early analyses for numerical cross-sectional analysis of anisotropic beams were based on linear elasticity theory. For more general treatment of non-linear phenomena, asymptotic formulations can serve as the basis for the numerical method, say a finite element method. When based on geometrically non-linear elasticity theory, an intermediate result of such analyses is frequently the splitting of the non-linear 3-D problem into a linear 2-D analysis of the cross section and a non-linear 1-D analysis along the beam. Thus, the published work to date cannot treat the so-called “trapeze effect”, because it stems from non-linearity of the cross-sectional analysis. Herein, a non-linear numerical cross-sectional analysis is presented, based on the variational-asymptotic method and capable of treating cross sections of arbitrary geometry and generally anisotropic material. This type of analysis is particularly important in rotating structures, such as helicopter rotor blades. Results from this model are compared with those available in the literature, both theoretical and experimental.

Non-classical effects in non-linear analysis of pretwisted anisotropic strips

The literature on classical analysis of anisotropic beams assumes that all 1D “moment strain” measures (i.e. twist and bending curvatures) are of the same order of magnitude, resulting in a linear cross-sectional analysis. The present paper treats the situation in which one or more of the 1D moment strain measures may be larger than the other(s), resulting in a non-linear cross-sectional analysis. This type of non-classical analysis is needed, for example, in problems where the trapeze effect is important, such as in rotor blades. As a precursor to complicated non-linear sectional analysis of arbitrary cross sections, a non-linear sectional analysis is presented for an anisotropic strip with small pretwist, based on the dimensional reduction of laminated shell theory to a non-linear one-dimensional theory using the variationalasymptotic method. Results obtained from this strip-beam analysis are compared with available theoretical and experimental results for a problem in which the trapeze effect is important. In order to demonstrate the usage of the results in the analysis of structures made of an arbitrary geometrical combination of pretwisted generally anisotropic strips, a closed-form expression is derived for the torsional buckling of a column with a cruciform cross section.

Asymptotic theory for static behavior of elastic anisotropic I-beams

End effects for prismatic anisotropic beams with thin-walled, open cross-sections are analyzed by the variational-asymptotic method. The decay rates for disturbances at the ends of prismatic beams are evaluated, and the most influential end disturbances are incorporated into a refined beam theory. Thus, the foundations of Vlasovs theory, as well as restrictions on its applicability, are obtained from the variational-asymptotic point of view. Vlasovs theory is proved to be asymptotically correct for isotropic I-beams. The asymptotically correct generalization of Vlasovs theory for static behavior of anisotropic beams is presented. In light of this development, various published generalizations of Vlasovs theory for thin-walled anisotropic beams are discussed. Comparisons with a numerical 3-D analysis are provided, showing that the present approach gives the closest agreement of all published theories. The procedure can be applied to any thin-walled beam with open cross-sections.

Derivation of Plate Theory Accounting Asymptotically Correct Shear Deformation

The focus of this paper is the development of linear, asymptotically correct theories for inhomogeneous orthotropic plates, for example, laminated plates with orthotropic laminae. It is noted that the method used can be easily extended to develop nonlinear theories for plates with generally anisotropic inhomogeneity. The development, based on variational-asymptotic method, begins with three-dimensional elasticity and mathematically splits the analysis into two separate problems: a one-dimensional through-the-thickness analysis and a two-dimensional “plate” analysis. The through-the-thickness analysis provides elastic constants for use in the plate theory and approximate closed-form recovering relations for all truly three-dimensional displacements, stresses, and strains expressed in terms of plate variables. In general, the specific type of plate theory that results from variational-asymptotic method is determined by the method itself. However, the procedure does not determine the plate theory uniquely, and one may use the freedom appeared to simplify the plate theory as much as possible. The simplest and the most suitable for engineering purposes plate theory would be a “Reissner-like” plate theory, also called first-order shear deformation theory. However, it is shown that construction of an asymptotically correct Reissner-like theory for laminated plates is not possible in general. A new point of view on the variational-asymptotic method is presented, leading to an optimization procedure that permits a derived theory to be as close to asymptotical correctness as possible while it is a Reissner-like. This uniquely determines the plate theory. Numerical results from such an optimum Reissner-like theory are presented. These results include comparisons of plate displacement as well as of three-dimensional field variables and are the best of all extant Reissner-like theories. Indeed, they even surpass results from theories that carry many more generalized displacement variables. Although the derivation presented herein is inspired by, and completely equivalent to, the well-known variational-asymptotic method, the new procedure looks different. In fact, one does not have to be familiar with the variational-asymptotic method in order to follow the present derivation.

On asymptotically correct linear laminated plate theory

The focus of this paper is the development of asymptotically correct theories for laminated plates, the material properties of which vary through the thickness and for which each lamina is orthotropic. This work is based on the variational-asymptotical method, a mathematical technique by which the three-dimensional analysis of plate deformation can be split into two separate analyses: a one-dimensional through-the-thickness analysis and a two-dimensional “plate” analysis. The through-the-thickness analysis includes elastic constants for use in the plate theory and approximate closed-form recovering relations for all three-dimensional field variables expressed in terms of plate variables. In general, the specific type of plate theory that results from this procedure is determined by the procedure itself. However, in this paper only “Reissner-like” plate theories are considered, often called first-order shear deformation theories. This paper makes three main contributions: first it is shown that construction of an asymptotically correct Reissner-like theory for laminated plates of the type considered is not possible in general. Second, a new point of view on the variational-asymptotical method is presented, leading to an optimization procedure that permits a derived theory to be as close to asymptotical correctness as possible. Third, numerical results from such an optimum Reissner-like theory are presented. These results include comparisons of plate displacement as well as of three-dimensional field variables and are the best of all extant Reissner-like theories. Indeed, they even surpass results from theories that carry many more generalized displacement variables.