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Since the mid-1960s when the forms of curved shell finite elements were originated, including those pioneered by Professor Gallagher, the published literature on the subject has grown extensively. The first two present authors and Liaw presented a survey of such literature in 1990 in this journal. Professor Gallagher maintained an active interest in this subject during his entire academic career, publishing milestone research works and providing periodic reviews of the literature. In this paper, we endeavor to summarize the important literature on shell finite elements over the past 15 years. It is hoped that this will be a befitting tribute to the pioneering achievements and sustained legacy of our beloved Professor Gallagher in the area of shell finite elements. This survey includes: the degenerated shell approach; stress-resultant-based formulations and Cosserat surface approach; reduced integration with stabilization; incompatible modes approach; enhanced strain formulations; 3-D elasticity elements; drilling d.o.f. elements; co-rotational approach; and higher-order theories for composites. Copyright © 2000 John Wiley & Sons, Ltd.

An efficient method, using equivalent plate model, is developed for studying the static and vibration analyses of general built-up wing structures composed of skins, spars, and ribs. The model includes the transverse shear effects by treating the built-up wing as a plate following the Reissner-Mindlin theory, the so-called First-order Shear Deformation Theory (FSDT). The Ritz method is used with the Legendre polynomials being employed as the trial functions. This is in contrast to previous equivalent plate model methods which have used simple polynomials, known to be prone to numerical ill-conditioning, as the trial functions. The present developments are evaluated by comparing the results with those obtained using MSC/NASTRAN, for a set of examples. These examples are: (i) free-vibration analysis of a clamped trapezoidal plate with (a) uniform thickness, and (b) non-uniform thickness varying as an airfoil, (ii) free-vibration and static analyses (including skin stress distribution) of a general built-up wing, and (iii) free-vibration and static analyses of a swept-back box wing. The results obtained by the present equivalent plate model are in good agreement with those obtained by the finite element method.

To accurately model the structural response of e ber-reinforced polymer-matrix composites, it is necessary to account for viscoelastic behavior of the polymer matrix. Therefore, the geometrically linear dynamic analysis capability of a recently developed triangular e at shell element for elastic composite structures is extended here to include thermoviscoelasticity. The e at shell element is the combination of the discrete Kirchhoff theory plate bending element and a membrane element similar to the Allman triangle, but derived by transforming the linear strain triangle element. Linear viscoelastic composite materials are modeled, resulting in the relaxation moduli being expressed as Prony series. Hygrothermorheologically simple materials are considered for which a change in the hygrothermal environment results in a horizontal shifting of the relaxation moduli curves on a log timescale, in addition to the usual hygrothermal loads. The resulting hereditary integral terms are evaluated using a recursion relationship requiring only the previous two solutions. The Newmark method is used to incorporate the inertia terms. Numerical examples are presented to demonstrate the accuracy of the present formulation when compared with correspondence principle solutions and results available in the literature.

A procedure based on time finite element method to obtain the second-order sensitivity of the transient response with respect to various system parameters is developed and applied to single- and two-degree-of-freedom structural dynamic systems. The proposed method directly differentiates the algebraic equations, resulting from the time finite element formulation, with respect to system parameters. Computational algorithms are given, and detailed derivations are presented. The second-order sensitivity analysis has been used for approximation schemes for efficient solution of design optimization problems. Also, the results can be applied to gradient-based parameter identification schemes. The results are found to be in excellent agreement with those obtained using the finite difference method. The main advantages of the time finite element approach are the ease of use and the numerical efficiency with which the sensitivity of the transient response can be obtained.

Free vibration of generally laminated plates having arbitrarily located point supports is studied via the Rayleigh ‐Ritz method. Chebychev polynomials are chosen as the trial functions and essential boundary conditions along the edges are imposed by means of appropriate large artie cial springs. This allows for the edges of the plate to be free, simply supported, or clamped. An elastically restrained point is represented by the appropriate spring constant value and by allowing this spring constant to become large, a point support is represented. The e rst-order shear deformation theory is utilized in the development of the equations of motion, along with the linear stress /strain relationship. Verie cation of the method is carried out through comparison with published results for point-supported plates. After the method is verie ed, a brief study of generally laminated plates having point supports is carried out. The method developed is simple and efe cient, making it perfectly suited to applications that require multiple calculations, such as optimization, and can be applied to analyze joined-wings and truss-braced wings.

Control of thermal deformations of a thin hexagonal spherical mirror segment using discrete and distributed actuators is presented. To determine the effectiveness of the actuators in controlling the thermal deformations of the mirror segment, a comparative study is conducted using two different models of the mirror‐ actuator system: 1) the mirror mounted on kinematic supports and controlled by piezoelectric strips bonded to the rear surface of themirrorand 2 )themirrormounted on force actuators, which are used to supportthemirroraswell as to control the surface deformations of the mirror. The performance of evenly distributed strips and that of strips placed at near-optimal locations obtained using heuristic integer programming are also compared. Both the force actuators and the piezoelectric strips are found to be equally effective in controlling the surface deformations of the mirror. A major drawback of the force actuators is the increase in the overall weight of the system, which is undesirable for space applications. On the other hand, the piezoelectric strips are very lightweight, and hence a large number of such strips can be used to control the surface distortions of the mirror, without imposing a weight penalty. The piezoelectric strips appear to be promising candidates for static shape control of e exible structures in space. Nomenclature b = side of graphite/epoxy plate, m d33 = piezoelectric strain constant, m/V E = root mean square error of the mirror surface distortions f j = control inputs (force applied to the force actuators or the voltage applied to the piezoelectric strips ) h = thickness of graphite/epoxy plate, m m = number of nodes of the e nite element model n = number of force actuators or piezoelectric strips u = correction to the transverse displacement W = middee ection of graphite/epoxy plate, m a ij = ine uence coefe cients w = deformed shape (transverse displacement) of the mirror

With advanced subsonic transports and military aircraft operating in the transonic regime, it is becoming important to determine the effects of the coupling between aerodynamic loads and elastic forces. Because aeroelastic effects can signie cantly impact the design of these aircraft, there is a strong need in the aerospace industry to predict these interactions computationally. Such an analysis in the transonic regime requires high-e delity computational e uid dynamics (CFD) analysis tools, due to the nonlinear behavior of the aerodynamics, and high-e delity computational structural dynamics (CSD) analysis tools. Also, there is a need to be able to use a wide variety of CFD and CSD methods to predict aeroelastic effects. Because source codes are not always available, it is necessary to couple the CFD and CSD codes without alteration of the source codes. In this study, an aeroelastic coupling procedure is developed to determine the static aeroelastic response of aircraft wings using any CFD and CSD code with little code integration. The procedure is demonstrated on an F/A-18 stabilator using NASTD (an in-house McDonnell Douglas CFD code )and NASTRAN. In addition, the AeroelasticResearch Wing is used fordemonstration with ENSAERO (NASA Ames Research Center CFD code ) coupled with a e nite element wing-box code. The results obtained from the present study are compared with those available from an experimental study conducted at NASA Langley Research Center and a study conducted at NASA Ames Research Center using ENSAERO and modal superposition. The results compare well with experimental data.

This paper presents a simple and accurate global-local method to calculate the static response of simply-supported, composite plates with small cracks. The approach primarily involves two steps. In the first step a global approach, the Ritz method, is used to calculate the response of the structure. Displacement based Ritz functions for the plate without the crack are augmented with a perturbation function, which is accurate for isotropic plates only, to account for the crack. The Ritz solution does not accurately satisfy the natural boundary conditions at the crack boundary. Therefore, a second step, local in nature, is taken in which a small area in the vicinity of the crack and encompassing other points of singularities, is discretized using a fine finite element mesh. The displacement boundary conditions for the local region are obtained from the global Ritz analysis. The methodology is applied for the local model including the crack. The proposed approach resulted in accurate prediction of stresses, with 85% savings in computational time and 92% in data storage for the analysis of composite flat panels.

An updated Lagrangian formulation of a three-node flat triangular shell element is presented for geometrically nonlinear analysis of laminated plates and shells. The flat shell element is obtained by combining the discrete Kirchhoff theory plate bending element and a membrane element that is similar to the Allman element but a derivative of the linear strain triangular element. Results are presented for large-rotation static response analysis of a cantilever beam under end moment, cylindrical shell under pinching and stretching loads, a hemispherical shell under pinching and stretching loads, and a ring plate under a line load; for dynamic response analysis of a cylindrical panel; and for thermal postbuckling analysis of an imperfect square plate and a cylindrical panel. To estimate the accuracy of the present formulation in predicting the nonlinear response of large flexible structures, static analysis of an apex-loaded circular arch is performed. The arch is a building block of a large inflatable structure. The results are in good agreement with those available in the existing literature and those obtained using the commercial finite element software ABAQUS, demonstrating the accuracy of the present formulation. HE two most widely adopted approaches in the finite element analysis of shells are use of curved shell elements based on a suitable shell theory and approximation of the curved structure by an assemblage of flat shell elements in which the membranebending coupling is brought about as a result of material anisotropy and transformation of the element stiffness matrices computed in a local coordinate system to the global coordinate system prior to assembly. The curved shell elements can be computationally very expensive, especially in the case of nonlinear analysis, because of the complexity of the formulation and the need to compute the curvature information. Flat shell elements are more attractive because of their simplicity and the ease with which they can be built from alreadyexisting familiar membrane and plate bending elements. Though a large number of elements are required to accurately model curved structures, the analysis is computational ly less expensive because of extremely simple formulation. Updated Lagrangian formulations have been predominantly used in the flat shell formulations available in the existing literature. In an updated Lagrangian formulation, all of the variables are referred to a known configuration, the reference configuration, which is updated continuously during the deformation process. If the rigid-body modes are removed from the total or incremental displacements, the resulting deformational translations and rotations are very small and hence a linearized incremental formulation1 can be used. In a linearized incremental formulation, the stresses are computed using linear strain-displacement relations. The tangent stiffness matrix contains only the linear stiffness matrix and the initial stress matrix. The stiffness matrices resulting from the nonlinear terms in the strain-displacement relations are neglected, resulting in very economical analysis. If the rigid-body modes are not removed, all nonlinear terms in the strain-displacement relations will have to be considered for computing the stresses and the tangent stiffness matrix. A three-node flat triangular shell element was introduced by Argyris et al.2 for nonlinear elastic stability problems. This formula

Four types of orthogonal polynomials-Legendre, Chebyshev, Hermite, and integrated Legendre-are evaluated as basis functions in the time finite element method to solve initial value problems governed by second-order differential equations. Condition numbers of the augmented stiffness matrix for the selected problems are estimated by increasing the number of polynomial terms in the expansion. Results for the CPU time using an IBM 3090-300E/VF computer and the estimated condition numbers, using IMSL subroutine DLFCRG, for each of four basis functions are presented. The augmented stiffness matrix for the case of integrated Legendre polynomials is found to have the best condition number behavior, and that for the Hermite polynomial the worst. The integrated Legendre polynomials require the most CPU time, whereas the Chebyshev polynomials require the least.

The vibro-acoustic experiments are performed at Structural Acoustics Loads and Transmission (SALT) facility of Langley Research Center to measure the sound radiation from and transmission loss (TL) of two panels with straight and curvilinear stiffeners, respectively. The focus of the work addressed in this article is to validate the Vibro-acoustic analysis methods implemented in a multi-objective design optimization framework, EBF3PanelOpt. The stiffened test panels with six stiffeners have been designed using EBF3PanelOpt for turbulent boundary layer (TBL) excitation. To build the confidence in finite element models, free vibration tests of the manufactured stiffened panels are also performed. The excited experimental mode shape and eigen frequencies match quite well with the finite element analysis results for the free vibration test. A medium density fiberboard and aluminum frame is used to clamp the stiffened panels in the SALT transmission loss window. Due to strong coupling of the stiffened test panels with the frame, a finite element model of the stiffened panels with frame is developed. To capture the effect of the bolted joints in the test setup, bush elements with optimal values of rotational and translational stiffnesses are used. Measurements are also performed to estimate the damping in the stiffened panels with the rame. The numerical prediction of sound radiation from and TL of stiffened panels is in fair agreement with the measured values for both the panels.

Manufacturing defects, which occur often in cellular cores of sandwich structures, lead to buckled cells in the core and thus to degraded constitutive core properties. Defects in the core cells may occur due to unintentional tool impacts onto the panel during manufacturing, handling processes, or defects in the manufactured core itself. The degraded constitutive properties of the core lead to local stiffness degradations of the sandwich panel, which in turn may impact the capability of nearby inserts; inserts are often used for component mounting to the panel or as joints used for joining the sandwich panel with another panel. The aim of this paper is to: a) Understand the effect of core density on the critical buckling load/buckling capability of the unit cell for each of the three core shapes: hexagonal, square and triangular, b) Understand the effect of core height (core thickness within the sandwich panel) on the critical buckling load/buckling capability of the unit cell for each of the three core shapes, and c) Investigate the degradation in the constitutive properties of the cellular core due to permanent deformation resulting from transverse compression. Detailed finite element models for five cell sizes and twenty one core densities were created for each of the following cellular core shapes: hexagonal, square and triangular. The critical buckling load is calculated for all twenty one core densities and the five cell sizes under consideration for the hexagonal, square and triangular cellular cores. Additionally, the critical buckling load is calculated for four core heights from 0.25" to 1.00" core height with 0.25" increments to study the effect of the core height on the critical buckling load of the unit cell. The degradation of constitutive properties of the hexagonal, square and triangular cellular cores due to buckling under transverse compression loading for the 1/8′′ cell size and core density of 3.1 pcf is studied for the 0.5′′ core height cells. All cellular cores are assumed to be manufactured of corrugated Al-5056.

STIFFENERS attached to composite panels may significantly increase the overall buckling load of the resultant stiffened structure. First, buckling analysis of a composite panel with attached longitudinal stiffeners under compressive loads is performed using Ritz method with trigonometric functions. Results are then compared to those from ABAQUS FEA for different shell elements. The case of composite panel with one, two, and three stiffeners is investigated. The effect of the distance between the stiffeners on the buckling load is also studied. The variation of the buckling load and buckling modes with the stiffeners’ height is investigated. It is shown that there is an optimum value of stiffeners’ height beyond which the structural response of the stiffened panel is not improved and the buckling load does not increase. Furthermore, there exist different critical values of stiffener’s height at which the buckling mode of the structure changes. Next, buckling analysis of a composite panel with two straight stiffeners and a crack at the center is performed. Finally, buckling analysis of a composite panel with curvilinear stiffeners and a crack at the center is also conducted. ABAQUS is used for these two examples and results show that panels with a larger crack have a reduced buckling load. It is shown also that the buckling load decreases slightly when using higher order 2D shell FEM elements.

A global/local framework for multidisciplinary optimization of an aircraft wing structure has been developed. The concept of curvilinear stiffening members (spars, ribs and stiffeners) has been applied in the wing structural optimization. A global optimization framework EBF3WingOpt, which integrates the static aeroelastic, flutter and buckling analysis, has been implemented for exploiting the optimal design at the global level. The wing internal structure is optimized using curvilinear spars and ribs (SpaRibs). A two-step optimization approach, which consists of topology optimization with shape design variables and size optimization with thickness design variables, is implemented in EBF3WingOpt. A local panel optimization framework EBF3PanelOpt, which includes stress and buckling evaluation criteria, is performed to optimize the local panels for further structural weight saving. The local panel model is extracted from the global finite element model. The boundary conditions are defined on the edges of local panels using the displacement fields obtained from the global model analysis. The local panels are optimized to satisfy stress and buckling constraints. Stiffened panel with curvilinear stiffeners is implemented in EBF3PanelOpt to improve the buckling resistance of the local panels. The optimization of stiffened panels has been studied and integrated in the local optimization. The global-local optimization framework incorporates global optimization framework EBF3WingOpt and local optimization framework EBF3PanelOpt in an iterative manner. The global-local framework is developed using MATLAB and Python programming to integrate several commercial software: MSC.PATRAN for pre and post processing, MSC.NASTRAN for finite element analysis. The global-local optimization methodology is applied for NASA Common Research Model (CRM) wing. The results have shown the advantage of the multidisciplinary globallocal optimization framework in structural weight saving.

The study focuses on the development of a simple and accurate global/local method for calculating the static response of stepped, simply-supported, isotropic and composite plates with circular and elliptical cutouts. The approach primarily involves two steps. In the first step a global approach, the Ritz method, is used to calculate the response of the structure. Displacement based Ritz functions for the plate without the cutout are augmented with a perturbation function, which is accurate for uniform thickness plates only, to account for the cutout. The Ritz solution does not accurately satisfy the natural boundary conditions at the cut-out boundary, nor does it accurately model the discontinuities caused by abrupt thickness changes. Therefore, a second step, local in nature is taken in which a small area in the vicinity of the hole and encompassing other points of singularities is discretized using a fine finite element mesh. The displacement boundary conditions for the local region are obtained from the global Ritz analysis. The chosen perturbation function is reliable for circular cutout in uniform plates, therefore elliptical cutouts were suitably transformed to circular shapes using conformal mapping. The methodology is then applied to the analysis of composite plates, and its usefulness successfully proved in such cases. The proposed approach resulted in accurate prediction of stresses, with considerable savings in CPU time and data storage for composite flat panels.

Flutter analysis of a general wing is carried out using ® nite element structural modeling (MSC/NASTRAN), a subsonic kernel function unsteady aerodynamics and a V ± g type of solution (FAST). The shape of the wing cross section is generated by parameterization using the Joukowski transformation, so that shapes of varying thickness and camber can be obtained by varying the parameters. The CQUAD4 (quadrilateral) and CTRIA3 (triangular) membrane-bending shell elements, the CBAR beam element and the CSHEAR shear panel element from MSC/ NASTRAN have been used in this study. The validationof ® nite element modeling is done by performing dynamic analysis of an AGARD swept wing model. The sensitivity of utter speed of the wing to shape parameters, namely, aspect ratio, area, taper ratio, and sweep angle is obtained using a central difference scheme in conjunction with automatic differentiation (ADIFOR). The sensitivity of utter speed to modal parameters, namely, natural frequency, generalized mass, and generalized aerodynamic forces is computed using ADIFOR.

The time finite element method (TFM) is employed for parametric identification of nonlinear structural dynamic systems. An advantage of TFM is the ease with which one can calculate the sensitivity of the transient response with respect to various design parameters, a key requirement for gradient-based parameter identification schemes. The method is simple, because one obtains the sensitivities of the response to system parameters by differentiating the algebraic equations, not original differential equations. These sensitivities are used in the Levenberg-Marquardt iterative direct method to identify parameters for nonlinear single- and two-degree-of-freedom systems. The measured response was simulated by integrating the example nonlinear systems using the given values of the system parameters. The accuracy and the efficiency of the present method are compared to a previously available approach that employs a multistep method to integrate nonlinear differential equations. It is seen, for the same accuracy, that the present approach requires fewer data points.

The free vibration of curvilinearly stiffened doubly curved shallow shells is investigated by the Ritz method. Base on the first order shear deformation shell theory and Timoshenko’s 3-D curved beam theory, the strain and kinetic energies of the stiffened shells are introduced. Numerical results with different geometrical shells and boundary conditions, and different stiffener locations and curvatures are analyzed to verify the feasibility of the presented Ritz method for solving the problems. The results show good agreement with those using the FE method.Copyright © 2013 by ASME

Traditional flying qualities metrics use first-order descriptors of the ’classical’ aircraft modes such as the phugoid, short-period, Dutch roll, etc. These modes are often difficult to distinguish from one another in modern aircraft whose dynamics are usually coupled, leaving designers to develop lower order equivalent systems as approximations to the real aircraft’s behavior. Flying qualities metrics typically include limits on the modal parameters of the lower order equivalent system—constraints that are conducive to classical control techniques and pole placement strategies. On the other hand, modern optimal or robust control techniques directly address Multiple Input Multiple Output (MIMO) systems, with coupled dynamics, and they function well with higher-order measures of system behavior like state covariances or energy norms. This work presents a new method of approximating the traditional flying qualities requirements for piloted aircraft through a flight condition dependent recasting of these requirements as upper bounds on the state variances. A linear matrix inequality feasibility problem is developed to compute a control law which stabilizes the system for the given flight condition, while satisfying both the state variance and actuator constraints. The method is demonstrated through a horizontal tail sizing study of a transport aircraft, which shows that a much smaller tail than is conventional can satisfactorily fulfill the flying qualities constraints.

omparison among the equivalent-continuum (homogeneous) constitutive law for Hexagonal, square and triangular cores to be used in sandwich panels is presented. The cellular core homogeneous properties are calculated using detailed finite element models of representative unit cell for each core shape. A strain energy based homogenization scheme that was first presented by Hohe and is based on the concept that for every cellular material, there exists a homogeneous material that is of the same shape and behaves identically to the cellular structure when it is subject to the same loading and boundary conditions is also be used to calculate the constitutive law for the same core shapes and the same core densities. A detailed comparison between the homogenized constitutive law obtained using the detailed finite element models and that obtained using Hohe’s approach is presented herein. The strain energy based homogenization scheme was previously verified by Soliman et. al. [1] for a non-corrugated square shape core. In this paper, the hexagonal and square core shapes are made of corrugated Aluminum 5056 sheets. The variation of the mechanical behavior of the homogeneous structure equivalent to the cellular core with respect to the relative density and cell size of the cellular core is analyzed and equations for direct calculation of the constitutive behavior of the three types of cellular core is presented. Additionally, different finite element modeling techniques of the cellular cores within a 10”× 10” panel with simply supported boundary condition is also presented. The finite element modeling techniques presented are: a) detailed modeling of the cellular core within a sandwich panel, b) 2D modeling of the equivalent continuum within a sandwich panel using the MAT8 NASTRAN ® card to represent the core homogenized material , and c) 3D modeling of the equivalent continuum within a sandwich panel using the MAT9 NASTRAN ® card to represent the core homogenized material. The modeling techniques will be compared in terms of the displacement field of sandwich panels consisting of the different types of cellular core and subject to uniform pressure.

Motivated by a need for understanding the damage tolerance of unitized structures fabricated using the modern additive manufacturing process, fracture analysis of curvilinearly stiffened panels, unitized structures, with different crack lengths were performed by using a global-local finite element method under three different load cases: a) shear, b) normal, and c) combined load cases. It was observed that, in most of the cases, 85% data storage space and the same amount in CPU time requirement could be saved using the global-local finite element method compared to the standard global finite element analysis. It was also observed that the fracture mode in panels with different crack lengths was essentially Mode-I under the normal load case; Mode-II under the shear load case; and again Mode-I under the combined load case. Under the maximum combined loading condition, the largest effective stress intensity factor of the panel with a crack of recommended size was very smaller than the critical stress intensity factor. Therefore, considering the critical stress intensity factor, the stiffened panel was an optimum design satisfying damage tolerance constraints.

Fiber reinforced composite materials are increasingly being used in aircraft and automotive industries due to their outstanding mechanical properties of very high specific strength and stiffness, better corrosion and fatigue resistance and tailoring ability. During manufacturing and processing, these material structures are subjected to discontinuities in the form of holes, notches, and/or cracks. The presences of these discontinuities in the structures influence the stress distribution, and the stresses around a crack tip increase extraordinarily. In such a condition, it is necessary to consider fracture mechanics parameters known as critical stress intensity factor or toughness of the materials against crack growth under external loadings. To accurately predict the fracture response in terms of stress

Free vibration of plate structures of various material systems has been vastly studied. Transverse shear can play an important role even in thin laminated structures and therefore must be included in the analysis. A method applicable to the study of generally laminated, thick, skew, trapezoidal plates is developed as there appears to be limited information available in this area. Using Chebychev polynomials as displacement functions in the Rayleigh-Ritz method, the first-order shear deformation theory is utilized to account for the transverse shear effects. All appropriate inertia terms are retained. Boundary conditions are enforced by the appropriate use of distributed linear and rotational springs along the edges. Thus, the method is applicable to any supported quadrilateral plate but is applied to cantilever plates in this study. After the validity of the method is established through comparison to published results, an extensive study is made of thick, symmetrically laminated and unsymmetrically laminated, cantilever, skew, trapezoidal plates.

A piecewise, panel-based, pressure representation is incorporated to improve the fidelity of an existing wing aeroelastic model, including the sensitivity of the wing static aeroelastic response with respect to various shape parameters. The formulation is quite general and is easily adaptable to any aerodynamics and structural analysis codes desired. An interface code written earlier to transfer information between aerodynamic and structures modules using a global pressure representation has been modified to improve the representation of the calculated pressures using piecewise representations. These improvements are designed to overcome some drawbacks of the earlier representation as well as provide a more accurate representation of the aerodynamic loads to the structures module. A variety of refinements has been attempted to study their effectiveness. A program to combine the local sensitivities, such as the sensitivity of the stiffness matrix or the aerodynamic kernel matrix, into global sensitivity derivatives is developed. Results with excellent accuracy have been produced for large integrated quantities such as wingtip deflection and trim angle of attack. In all cases, accuracy is dependent on the size of the derivatives relative to the size of the quantity itself. The piecewise pressure approaches improved the fidelity of the model and the accuracy of the results at little or no additional computational expense.

Finite element static, free vibration and thermal analysis of thin laminated plates and shells using a three noded triangular flat shell element is presented. The flat shell element is a combination of the Discrete Kirchhoff Theory (DKT) plate bending element and a membrane element derived from the Linear Strain Triangular (LST) element with a total of 18 degrees of freedom (3 translations and 3 rotations per node). Explicit formulations are used for the membrane, bending and membrane-bending coupling stiffness matrices and the thermal load vector. Due to a strong analogy between the induced strain caused by the thermal field and the strain induced in a structure due to an electric field the present formulation is readily applicable for the analysis of structures excited by surface bonded or embedded piezoelectric actuators. The results are presented for (i) static analysis of (a) simply supported square plates under doubly sinusoidal load and uniformly distributed load (b) simply supported spherical shells under a uniformly distributed load, (ii) free vibration analysis of (a) square cantilever plates, (b) skew cantilever plates and (c) simply supported spherical shells; (iii) Thermal deformation analysis of (a) simply supported square plates, (b) simply supported-clamped square plate and (c) simply supported spherical shells. A numerical example is also presented demonstrating the application of the present formulation to analyse a symmetrically laminated graphite/epoxy laminate excited by a layer of piezoelectric polyvinylidene flouride (PVDF). The results presented are in good agreement with those available in the literature.

The flutter speeds for a wing are obtained using lifting surface unsteady aerodynamics using modules form a system of programs called the flutter analysis system (FAST). Flutter speeds are obtained using a V-g type of solution. The derivatives of the flutter speed with respect to shape parameters of the wing are calculated using ADIFOR

In a test-fixture that the authors were using, steel tabs adhesively bonded to an aluminum panel debonded before the design load on the real test panel was fully applied. Therefore, studying behavior of adhesive joints for joining dissimilar materials was deemed to be necessary. To determine the failure load responsible for debonding of adhesive joints of two dissimilar materials, stress distributions in adhesive joints as obtained by a nonlinear finite element model of the test-fixture were studied under a gradually increasing compression-shear load. It was observed that in-plane stresses were responsible for the debonding of the steel tabs. To achieve a better understanding of adhesive joints of dissimilar materials, finite element models of adhesive lap joints and Asymmetric Double Cantilever Beam (ADCB) were studied, under loadings similar to the loading faced by the test-fixture. The analysis was performed using ABAQUS, a commercially available software, and the cohesive zone modeling was used to study the debonding growth.Copyright © 2012 by ASME

A modular multidisciplinary analysis and optimization framework tool has been built with the goal of optimally designing a supersonic aircraft. This paper addresses the specic challenge of designing an ecient long range supersonic bomber aircraft. This framework includes all the disciplines normally required for supersonic aircraft design and it also includes disciplines specically required by an advanced military aircraft that is tailless and has embedded engines. Results from running the analysis framework for a B-58 supersonic bomber test case are presented as a validation of the methods employed.

This paper describes the development of an innovative hybrid optimization strategy that employs an evolutionary algorithm technique combined with a Proper Orthogonal Decomposition-based reduced order design-space scheme. The proper orthogonal decomposition method of extracting dominant modes from an ensemble of candidate congurations is used for the design space order reduction wherein the reduced number of dominant POD coecients act as the new shape design variables instead of large number of actual physical design variables. The snapshot of candidate population is updated iteratively using tness-driven evolutionary algorithm technique of health-based retention and modication. Established with the goal of developing a robust optimization framework, the proposed strategy capitalizes on the advantages of evolutionary algorithm as well as PODbased reduced order modeling, while overcoming the shortcomings inherent with these techniques. When linked with a streamlined multi-disciplinary optimization framework, this hybrid optimization strategy oers a computationally ecient methodology for problems having a high level of complexity with challenging design space that has a large number of physical design variables. The developed framework is demonstrated for its robustness on a non-conventional supersonic tailless air vehicle wing shape optimization problem.

The increased complexity of next generation aircraft and the resulting highly-coupled interactions between aircraft systems has popularized Multidisciplinary Design Optimization (MDO) as an aircraft design tool. However, MDO e orts routinely neglect aircraft stability and control analyses beyond simple static assessments. Not only does this limit the potential of the MDO to identify an optimal design, but it also fails to assess the MILSPEC or FAR ying qualities requirements the aircraft must achieve. In an e ort to bring ying qualities and dynamic performance into the MDO of a tailless E cient Supersonic Air Vehicle, an H1 control methodology combined with pole placement constraints is adopted to automate the feedback control system design and assess the aircraft’s performance. As a preliminary investigation, and to assist in software development, the longitudinal dynamics of a relaxed static stability transport aircraft are studied, and the path forward to implementation on a supersonic tailless aircraft is outlined.

This paper presents an extension of the Kulfan Class Shape Transformation (CST) method for the parameterization of aircraft component geometry. The VT-CST code represents a practical implementation of the CST method in an object oriented C++ framework for use within a Multidisciplinary Design Optimization of a tailless supersonic aircraft. Extensions to the method include the generation of a blended wing-fuselage, cowls and inlet ramps for embedded engines, automatic handling of centerline continuity, shape normalization and scaling, fuselage area-ruling capabilities, and control surfaces, among others. Examples demonstrating the utility of the method are given.

The exibility of current and next generation aircraft can lead to coupling between the ight dynamics and the aeroelastic modes, particularly the interaction between the short-period and the rst wing bending modes. Development of ight control laws without considering the structural dynamics can lead to suboptimal performance. The present work is focused on examining these interactions to facilitate the design of a subscale vehicle whose dynamics will su ciently capture the interactions present in the fullscale vehicle. A parametrization of the aircraft, and the resulting nondimensional groups, are discussed which would provide complete similitude according to Buckinghamtheory. However, this complete similitude is unfeasible, so partial similitude is investigated using two approaches. First, the classical approximations of the ight dynamic modes are used to reduce the order of the coupled model, and consequently the number of scaling parameters required to maintain the physics of the system. The second approach uses sensitivity of the response to errors in the aircraft's nondimensional parameters. Both methods give a consistent set of nondimensional parameters which do not have signi cant in uence on the aeroservoelastic interaction. These parameters do not need to be scaled, thus leading to a viable scaled model.

Various multi-disciplinary analysis optimization (MDAO) tools have been developed by the aeronautical industry to enhance the aircraft design process. These tools usually function as stand alone applications and encompass the analysis tools needed for evaluating the structural responses, the constraints, and the objective functions. Although, this type of framework can be greatly advantageous in improving both the overall structural performance and the e ciency of the design process, the current lack of versatility limits the use of current MDAO tools to speci c applications. A new MDAO philosophy has recently been introduced by NASA to overcome these limitations. This framework is based on the development of a \central executive" computational unit capable of communicating with external analysis tools and collect the information needed for the optimization design. The advantage of the NASA MDAO tool over the classic schemes is the possibility to add and interchange disciplines depending on the speci c requirements of the design, enhancing the versatility and enabling the use of multidelity analysis tools. The purpose of this research is to integrate three new components into the NASA MDAO central executive unit. The three components are brie y described as follow: a) The automatic re-meshing tool for the generation of the SpaRibs and the nite element model, b) The hybrid optimization scheme that integrates non-gradient based optimization methods as genetic algorithm (GA) and particle swarm optimization (PSO) with gradient based optimization methods for enhanced Research Engineer, Aerospace and Ocean Engineering, AIAA Student Member. yResearch Scientist, Aerospace and Ocean Engineering, AIAA Senior Member. zMitchell Professor, Aerospace and Ocean Engineering, AIAA Associate Fellow. xPresident, 9489 E. Ironwood Square Dr., Scottsdale, 85258, AZ, AIAA Senior Member. {Member of Technical Sta

High aspect-ratio aircraft require di erent design modeling from those with traditional moderate aspect ratios. High aspect-ratio aircraft are generally more exible structures than the traditional wing; therefore, they require modeling methods capable of handling a exible structure even at the preliminary design stage. An investigation of utter properties for a two-dimensional airfoil was conducted to approximate utter characteristics for thin, high aspect ratio wings in application to the Truss-Braced Wing project at Virginia Tech. Assumptions are made to maintain linearity of the model for the purposes of this study. A three degree-of-freedom model was developed from the standard two-degree-of-freedom model to investigate airfoil parameters that a ect utter in thin, high aspect ratio wings. The study shows that these model parameters will have a negative e ect on the utter velocity, frequency, and mode shapes under certain conditions. A simple beam model is developed to study a three dimensional wing system; this study focuses on a strut-braced wing con guration optimized for minimum take-o gross weight. Flutter suppression methods based on ballast mass placement are investigated for the three-dimension strut-braced wing con guration. Sensitivity studies for ballast mass and acceleration based active control methods are shown.

This paper describes the development of an innovative ’Proper Orthogonal Decompositionbased Reduced Order Design Scheme’ method to reduce the number of shape design variables needed for an e cient geometry optimization of air vehicles, while maintaining the generality of the design space. This POD-RODS method is based on the Proper Orthogonal Decomposition technique of extracting POD modes from a family of candidate con gurations and constructing a reduced order design space, in which the POD coe cients act as the new shape design variables. The candidate con gurations can be selected based on a designer’s past experience and entailing intuition. Established with the goal of developing a prototypical design process for unconventional air vehicle shape design, the POD-RODS method, when linked with a streamlined multi-disciplinary optimization framework, o ers a computationally e cient and cost e ective methodology for the conceptual design and con guration development of air platforms. The geometry optimization scheme developed herein is demonstrated for its robustness on a supersonic tailless air vehicle, which o ers a medium level of complexity and a much challenging design space to fully utilize the POD-RODS method.

A two part research study has been completed on the topic of compression after impact (CAI) of thin facesheet honeycomb core sandwich panels. The research has focused on both experiments and analysis in an effort to establish and validate a new understanding of the damage tolerance of these materials. Part one, the subject of the current paper, is focused on the experimental testing. Of interest are sandwich panels, with aerospace applications, which consist of very thin, woven S2-fiberglass (with MTM45-1 epoxy) facesheets adhered to a Nomex honeycomb core. Two sets of specimens, which were identical with the exception of the density of the honeycomb core, were tested. Static indentation and low velocity impact using a drop tower are used to study damage formation in these materials. A series of highly instrumented CAI tests was then completed. New techniques used to observe CAI response and failure include high speed video photography, as well as digital image correlation (DIC) for full-field deformation measurement. Two CAI failure modes, indentation propagation, and crack propagation, were observed. From the results, it can be concluded that the CAI failure mode of these panels depends solely on the honeycomb core density.

A two part research study has been completed on the topic of compression after impact (CAI) of thin facesheet honeycomb core sandwich panels. The research has focused on both experiments and analysis in an effort to establish and validate a new understanding of the damage tolerance of these materials. Part 2, the subject of the current paper, is focused on the analysis, which corresponds to the CAI testings described in Part 1. Of interest, are sandwich panels, with aerospace applications, which consist of very thin, woven S2-fiberglass (with MTM45-1 epoxy) facesheets adhered to a Nomex honeycomb core. Two sets of materials, which were identical with the exception of the density of the honeycomb core, were tested in Part 1. The results highlighted the need for analysis methods which taken into account multiple failure modes. A finite element model (FEM) is developed here, in Part 2. A commercial implementation of the Multicontinuum Failure Theory (MCT) for progressive failure analysis (PFA) in composite laminates, Helius:MCT, is included in this model. The inclusion of PFA in the present model provided a new, unique ability to account for multiple failure modes. In addition, significant impact damage detail is included in the model. A sensitivity study, used to assess the effect of each damage parameter on overall analysis results, is included in an appendix. Analysis results are compared to the experimental results for each of the 32 CAI sandwich panel specimens tested to failure. The failure of each specimen is predicted using the high-fidelity, physicsbased analysis model developed here, and the results highlight key improvements in the understanding of honeycomb core sandwich panel CAI failure. Finally, a parametric study highlights the strength benefits compared to mass penalty for various core densities.

Future aerospace vehicles like hybrid wing/body, truss-braced wing, and “double bubble” would have pressurized noncircular fuselage structures and complex wing geometry. Traditional aircraft designs have led to the confidence and experience of designing such structures using the knowledge base built over the years and the resulting rules of thumb. However, there is a lack of experience of load calculations and design of complex, multifunctional, aircraft structural concepts for future aerospace vehicles. Designing such structures will require a physics-based optimization framework. Therefore, a new optimization framework, EBF3PanelOpt, is being developed. Commercial software MD-PATRAN (geometry modeling and mesh generation) and MD-NASTRAN (finite-element analysis) are integrated in EBF3PanelOpt framework using the Python programming environment to design stiffened panels with curvilinear stiffeners. Currently, EBF3PanelOpt optimizes the stiffened panel with curvilinear blade stiffeners, where the loads ar...

We first review Puck and Schurmann’s progressive failure criteria for three-dimensional deformations of unidirectional fiber-reinforced composites, and then simplify these for plane stress deformations. These criteria have been implemented as an USDFLD subroutine in the commercial software, ABAQUS/Standard. Several example problems have been analyzed with the developed software, and for some problems computed results have been compared with those obtained by using the CLPT for stress analysis and Puck and Schurmann’s criteria for failure analysis. A close agreement between the two set of results verifies the implementation of the failure criteria in the software, ABAQUS. These examples illustrate that the ultimate failure load is significantly higher than that when the failure initiates first which is usually called the first ply failure load.

The stress-intensity factors of cracked box beams under torsion, geometrically linear and nonlinear twist buckling, and crack propagation under combined loading are discussed. This study is motivated by a previous work which derived a closed-form expression of Mode II stress-intensity factor for thin-walled beams with a centered longitudinal crack and subjected to torsion. Naturally, the way in which Mode I stress-intensity factors are influenced by torsion is of interest. The influence of parameters like crack length, crack angle, width and depth of beam, wall thickness, stiffener size on Mode I and Mode II stress intensity factors have been studied using the finite element method and represented the data in a surrogate model using two approaches: a) a Fourier-series based correction factor, and b) an artificial neural network. The torsional-buckling problem and crack propagation are also of interest. Specifically, the former is studied based on both geometrically linear and nonlinear theory for the critical loading and the load-twist angle relation; the crack propagation study focuses on both the limit loadings and the angle in which the initial crack growth occures. The results, which are based on finite element analysis of a box beam in the presence of a crack, indicate, that even for a tiny crack, the crack grows prior to buckling.

The vibrations and buckling of continuously supported beams resting on elastic foundation impact the design of aircraft structures, highway pavements, and railroad tracks. This article demonstrates the efficacy of Chebychev polynomials in the Rayleigh Ritz method when studying the torsional vibrations and buckling of beams on elastic foundation. Simple polynomials and orthogonal functions serve as displacement functions to illustrate the advantages of orthogonal functions. Fixed-fixed and fixed-simply supported represent the two treated sets of boundary conditions. The functions are chosen so that, most often, the kinematic boundary conditions are satisfied. In some cases, however, a penalty-type approach is used in which appropriate springs with large stiffness coefficients simulate the kinematic boundary conditions. The numerical results for natural frequencies and buckling loads are obtained and compared to the values of other researchers; good agreement is observed.