Warping Restraint Effects Research Articles

GRILLAGE ANALYSIS OF CELLULAR PLATE STRUCTURESOF VARYING DEPTH WITH INCL

A static analysis of cellular plate structures of linearly varying depth is presented by using thesimplified grillage method. The cellular plate structure in this study consists of top and bottomflange plates (non-parallel) separated by longitudinal webs having varying depth. A variationalformulation of thin-walled beam element with linearly varying depth has been carried out for thederivation of the stiffness matrix of the grillage members for various warping restraint effects. Theeffect of warping restraint in stiffness is included by two methods, firstly; the implicit method(through the torsional constant) and secondly; by the explicit method (as independent warpingdegree of freedom). Two differer,t approaches are used to derive the torsional stiffness matrix,namely:by the assumed displacement field and by the differential equation of equilibrium. Theeffect of secondary shear stresses from the warping restraint is also considercd.The varying cross section properties of the eguivalent grillage members are derived from theproperties of the actual cellular plate structure.Several numerical examples and applications are presented . The results are compared with thoseobtained from the flat plate / shell elements of MSCA.{ASTRAN program. Acceptable agreement isobtained.

Stall nonlinear aeroelastic effects and active control of thin-walled composite blade with piezoelectric patches embedded

Stall nonlinear flutter behavior and active control of anisotropic composite blade have been investigated based on piezoelectric actuation. The blade body comprises an elliptical composite host structure and a lightweight honeycomb skin structure. The composite blade with elliptical section is modeled as a thin-walled beam structure with circumferential asymmetric stiffness (CAS) configuration, exhibiting transverse shear deformation, warping restraint effect, and vertical bending, with structural tailoring implemented. The stall-induced flutter suppression and nonlinear aeroelastic effects of linearized dynamic response characteristics of blade incorporating stall nonlinear aerodynamic model are investigated. The nonlinear aeroelastic coupled equations are reduced to ordinary equations by the Galerkin method, with the aerodynamic force decomposed by strip theory. Active feedback control based on boundary moment control strategy is developed to enhance the vibrational behavior and dynamic response to aerodynamic excitation and stabilize structures that might be damaged in the absence of control. Various active feedback control schemes tuned by optimal proportional-derivative (PD) controller are implemented. The excellent control effect is not only confirmed by the time response with its vibration amplitude greatly suppressed but also confirmed by the frequency change and comparison. The research provides a way for rare study of amplitude suppression of stall nonlinear aeroelastic effect of thin-walled composite blade with CAS configuration and piezoelectric patches embedded.

Capacity and design of cold-formed steel warping-restrained beam-column elements

In current design standards, cold-formed steel (CFS) beam-column elements are generally designed by considering fully warping free behaviour in their supports, which means the benefit of warping-restrained boundary conditions is neglected. In addition, while a non-linear relationship governs the interaction of axial compression and bending, simplified linear expressions are prescribed in design standards, which may lead to unreliable designs. This paper aims to investigate the efficiency of the well-known Direct Strength Method (DSM) as well as the methods proposed by previous researchers for CFS warping-restrained beam-column members. The results of experimentally validated warping-restraint Finite Element (FE) models, considering material nonlinearity and geometric imperfections, are used as a benchmark. A total of 270 CFS elements with various lengths, thicknesses and cross-sectional dimensions are considered under ten different load eccentricity levels. The results are then employed to investigate the effects of warping-restrained boundary conditions as well as code recommended interaction curves on the efficiency of the existing methods to estimate the strength of CFS beam-columns with warping restraint. The results indicate that the estimated capacity of CFS beam-columns is significantly affected (up to 55%) by the warping restraint effects and the errors associated with using the simple linear interaction curve, depending on the element length and thickness. While the influence of warping restraint is generally less than 6%, it is demonstrated that, on average, all existing design methods underestimate the capacity of CFS beam-columns by at least 20%, which highlights the need to develop more accurate design methods for practical design purposes.

Distortion of steel tub-girder with top flange bracing considering warping restraint effect

During the construction stage of the steel-concrete composite box girder, the steel tub-girder has to support its weight, the self-weight of the wet concrete, and other construction loads. To improve the torsional stiffness of the steel tub-girder, the top flange bracing is often used to connect the top flanges to make “semi-closed” cross sections. The steel tub-girder with top flange bracing can form a closed shear flow under torsion, and thereby causing the distortion effect. However, the distortional behavior of this type of structure remains unclear. In this paper, the equivalent thickness of the fictitious top flange is first derived based on the restrained torsion theory. Subsequently, the errors of distortion loads given by the conventional load decomposition method which ignores the effect of warping restraint are derived, and are further verified by finite element analysis. It is demonstrated that the effect of warping restraint on the distortion is significant in the continuous tub-girder bridges with the top flange bracing. Therefore, the corresponding correction coefficients for the distortion loads caused by eccentric vertical loads and the girder axis curvature are proposed, and the simplified formulas for estimating these correction coefficients are established based on parametric analyses. Finally, the calculation method for the internal force of cross frames is modified and the accuracy of the proposed method is validated with results from finite element analysis.

Nonlinear Modal Interactions of Swept Composite Aircraft Wings Carrying External Stores

Modal interactions of swept aircraft wings carrying external stores in the presence of an internal resonance are investigated. The wing structure is modeled as a geometrically nonlinear composite thin-walled beam incorporating transverse shear strain and warping restraint effects. The anisotropy of the constituent material–induced bending-twist elastic coupling and the heavy external store–caused inertial couplings are accounted as well. The governing equations and the associated boundary conditions are derived via Hamiltons principle. The extended Galerkins method is adopted to semidiscretize the wing-store system, and then the nonlinear analysis is implemented based on the method of multiple scales. Based on the derived amplitude modulation equations under the condition of simultaneous internal and external resonances, steady-state solutions and their stability are critically discussed. Energy flow directions during modal interactions are investigated. In addition, conditions for the occurrence of saturation and jump phenomena are discussed.

Classical Flutter and Active Control of Wind Turbine Blade Based on Piezoelectric Actuation

The aim of this article is to analyze classical flutter and active control of single-cell thin-walled composite wind turbine blade beam based on piezoelectric actuation. Effects of piezoelectric actuation for classical flutter suppression on wind turbine blade beam subjected to combined transverse shear deformation, warping restraint effect, and secondary warping are investigated. The extended Hamilton’s principle is used to set up the equations of motion, and the Galerkin method is applied to reduce the aeroelastic coupled equations into a state-space form. Active control is developed to enhance the vibrational behavior and dynamic response to classical aerodynamic excitation and stabilize structures that might be damaged in the absence of control. Active optimal control scheme based on linear quadratic Gaussian (LQG) controller is implemented. The research provides a way for rare study of classical flutter suppression and active control of wind turbine blade based on piezoelectric actuation.

Geometrically nonlinear analysis of thin-walled open-section composite beams

A geometrically nonlinear model for general thin-walled open-section composite beams with arbitrary lay-ups under various types of loadings based on the classical lamination theory is presented. It accounts for all structural coupling coming from the material anisotropy and geometric nonlinearity. Nonlinear governing equations are derived and solved by means of an incremental Newton-Raphson method. The finite element model that accounts for the geometric nonlinearity in the von Karman sense is developed to solve the problem. Numerical results are obtained for thin-walled composite Z-beam and I-beam to investigate effects of geometric nonlinearity, fiber orientation and warping restraint on the flexural-torsional response.

Dynamics of Composite Aircraft Wings Carrying External Stores

This paper focuses on the free vibration and dynamic response to external time-dependent loads of aircraft wings modeled as thin-walled anisotropic composite beams carrying eccentrically located heavy stores. In this context, bending-twisting coupling induced by both the eccentric heavy stores arbitrarily located along the wing span and chord and by the anisotropy of the constituent material that is essential when dealing with aircraft wing problems was included in the approach of the problem. In addition to the anisotropy of constituent materials, transverse shear and warping restraint effects were also incorporated. The governing equations of the wing-store system and the related boundary conditions are derived via Hamilton's principle. To solve the eigenvalue/boundary problems, the extended Galerkin method is applied. Numerical simulations highlighting the implications of external stores coupled with the implementation of the structural tailoring on eigenfrequency and dynamic response to external time-dependent loads are supplied and pertinent conclusions are outlined.

Implications of warping restraint on statics and dynamics of elastically tailored thin-walled composite beams

Static response and free vibration of elastically tailored thin-walled beams accounting for the warping restraint effect are investigated via an exact solution methodology within the context of a refined beam model. Analytical results obtained from the restrained warping model are compared with those based on its Saint–Venant model counterpart. It is revealed that the beam slenderness and thickness ratio, as well as the elastic anisotropy, considered in conjunction with the warping restraint have profound effects on the static and dynamic response characteristics. It is also shown that even for anisotropic composite thin-walled beams with high slenderness ratios, warping restraint can still be significant, implying the inadequacy of merely considering the geometric aspects in the modeling of anisotropic composite thin-walled beams.

Aeroelastic Tailoring of Composite Wings Exhibiting Nonclassical Effects and Carrying External Stores

Structural and aeroelastic tailoring applied to advanced straight and swept aircraft wings carrying external stores is addressed. The wing structure is modeled as a laminated composite plate exhibiting flexibility in transverse shear and warping restraint effects. The equations of motion and boundary conditions are obtained via Hamilton's variational principle and application of generalized function theory. For a comprehensive representation of the stores, their static weights and inertia terms are considered. Three-dimensional modified strip theory aerodynamics is employed, and the obtained eigenvalue/boundary value problems are solved using the extended Galerkin method. The model is used to investigate the implications of external stores and ply angle orientation on divergence, free vibration, and flutter. Within the context of aeroelastic tailoring, the influence of external stores attached to the wing structure has to be considered during the preliminary aircraft design phases.

Variable twist torsion tests on carbon fibre composite beams

A detailed experimental programme of work is presented which examines the constrained torsional response of carbon fibre composite beams for the cantilevered configuration with torque applied at the free end. The behaviour of open-section beams and that of single-cell closed-section box beams is reported and discussed and the experimental findings are shown to corroborate simple engineering theoretical approaches. Tests have been carried out on zed and angle-section beams and thus the effects of primary and secondary warping restraint respectively on the torsional response of open-section beams are examined. The behaviour of zed-section beams is governed, in the main, by primary effects and that of angle-section beams is associated solely with secondary effects. The stress distributions along the beams and around the cross-sections are determined from the measured strains during test obtained from surface bonded strain gauges and comparison of these with theory is shown to be in good agreement. The essential differences in response to constrained torsion between open and closed-section beams are detailed in the paper and the importance of experimental tests in the analysis of composite structures is highlighted.

Static and Dynamic Aeroelasticity of Advanced Aircraft Wings Carrying External Stores

A study of the static and dynamic aeroelasticity of aircraft wings carrying external stores is presented. The wing structure is modeled as a laminated composite plate exhibiting flexibility in transverse shear and including warping restraint effects. The relevant equations of motion and boundary conditions are obtained via Hamilton's variational principle and application of generalized function theory. To achieve a realistic representation of the store influence on static and dynamic behavior of the system, static weights, as well as dynamic inertias of the stores, are considered. For the problem at hand, three-dimensional strip theory aerodynamics is employed. The obtained eigenvalue/boundary value problems are being solved by application of the extended Galerkin method. The solution procedure is used to investigate the implications of external stores on static aeroelastic response, divergence, free vibration, and flutter. Comparisons with the very few results highlighting the effects of underwing and tip stores on flutter instability are carried out, and excellent agreements with the present predictions are reported.

Finite Element Modeling of Open-Section Composite Beams with Warping Restraint Effects

A new finite element model is developed to capture spanwise torsion-related warping restraint effects in open-section composite beams. The new model features both twist and twist rate degrees of freedom, allowing for direct implementation of the kinematic boundary conditions associated with torsion-related warping restraints. A Vlasov cross-sectional model is used to determine the beam stiffness coefficients. The new model is validated against closed-form solutions for uniform composite I beams and detailed shell element solutions for more general spanwise geometries and loading conditions. A previous method of modeling spanwise torsion-related warping effects is reviewed and compared to the current model. Beams with spanwise taper, elastic couplings, and arbitrary warping boundary conditions are analyzed. Bending loads and distributed torsional loads are also investigated. Excellent correlation is observed between the current model, dosed-form solutions, and finite element solutions. For beams with uniform cross-sectional and warping restraints at both ends, the global increase in torsional stiffness due to torsion-related warping effects is independent of distributed loadings. The stiffness increase was found to depend on the cross-sectional geometry in the vicinity of the restraint. For extremely slender beams, the new torsion model yields the classic St. Venant torsion solution.

The behaviour of open and closed section carbon fibre composite beams subjected to constrained torsion

A simple engineering theoretical approach is presented in this paper which is able to predict the initial constrained torsional response of a specific class of thin-walled open-section and single-cell closed-section carbon fibre composite beams. The flat walls of the composite beams are symmetrically laminated about their own mid-planes and possess membrane orthotropy. The laminated flats are assembled in such a way that the stiffness distribution round the section is of a symmetrically disposed nature and thus the flanges of a composite box-section, for example, can have a different lay-up configuration to that of the section webs. Beams of this type are essentially uncoupled in their overall stiffnesses and thus it is possible to apply axial load or bending to the sections without inducing torsional behaviour. The analysis procedures for such beams will, of course, be considerably less complex in nature than those associated with beams of a more general lay-up configuration. Indeed, the analysis approach adopted in this paper simply makes use of the existing theories of torsion appropriate to isotropic construction and these are then suitably modified to account for the non-isotropic nature of the composite material. The torsional and warping rigidities for use in the analysis of the composite beams are thus duly determined through the use of the appropriate equivalent engineering elastic constants of the individual thin composite walls and the concept of effective thickness is employed to account for the different stiffnesses in the walls. In the paper some detailed attention is paid to the effects of primary and secondary warping restraint on the torsional response of open section beams and the distinct differences between sections whose behaviour is governed predominantly by primary effects and those whose response is associated solely with secondary effects are discussed. The stress systems set up in open-section and single-cell closed-section carbon fibre composite beams when subjected to torsion with variable twist are examined in the paper and in particular it is shown that although the shear flow due to primary warping restraint in open-section beams serves in part to equilibrate the applied torque, that in closed box section beams is completely self equilibrating. Comparisons are given in the paper between theory and experiment and between theory and finite element solutions and these are shown to give good agreement for the Z, angle and box section beams considered.

Finite element modeling of open-section composite beams with warping restraint effects

Finite element modeling of open-section composite beams with warping restraint effects

Nonclassical effects on divergence and flutter of anisotropic swept aircraft wings

A refined structural model of swept aircraft wings is presented and used to study the associated flutter and divergence instability in an incompressible gas flow. The structural model incorporates nonclassical effects such as anisotropy, heterogeneity, transverse shear, and warping inhibition. An exact solution methodology appropriate for parametric studies is used to specifically determine the effects of transverse shear and warping restraint on aeroelastic instabilities. The results reveal strong quantitative and qualitative implications for instability predictions. The solution methodology yields results that are in excellent agreement with an exact approach devised for a special case and provides a high-efficiency mathematical tool for dealing with flutter and divergence instability problems.

Comprehensive model of anisotropic composite aircraft wings suitable for aeroelastic analyses

A comprehensive plate-beam structural model suitable for aeroelastic analyses of aircraft wings made of anisotropic composite materials is developed. The equations governing the static and dynamic aeroelastic equilibrium of cantilevered swept-wing structures and the associated boundary conditions are derived by means of the Hamilton variational principle. These equations incorporate a number of effects: 1) anisotropy of the materials of constituent layers, 2) warping inhibition, 3) transverse shear flexibility, and 4) rotatory inertias. A uniform swept-wing model composed of a transversely isotropic material is considered to illustrate the coupled and separate effects of transverse shear deformation and warping restraint upon its divergence and static aeroelastic load distribution. An exact method based upon the Laplace integral transform technique is used to solve the above mentioned problems. The results displayed in this article reveal the importance of transverse shear and warping restraint effects in predicting more accurately the static aeroelastic response of swept-forward wings. However, for swept-back wings, these effects represent higher-order corrections to the classical theory.

Aeroelastic divergence of swept-forward composite wings including warping restraint effect

A powerful analytical approach to the divergence instability of laminated composite swept-forward wings is developed in the paper. The approach, based on the state space concept (used in conjunction with Jordan canonical form), enables one to solve exactly the equations governing the aeroelastic divergence of swept-forward composite wings, the warping restraint effect being incorporated into the analysis. The results obtained here emphasize the complex role played by the warping restraint effect in the divergence instability of swept-forward composite wings.

General formulation for the aeroelastic divergence of composite swept-forward wing structures

This paper formulates a simple algorithm that allows for the determination, in a closed form, of the divergence instability of advanced composite swept (back and forward) wing structures. The warping restraint effect is incorporated into the analysis and its influence on the associated static instability condition is put into evidence. In this sense, it is shown that, in contrast to the case of conventional metallic wings where the warping restraint effect has a stabilizing influence only (which is strongly manifested in the case of small-aspect-ratio wings and declines for moderate AR wings), its influence becomes more complex in the case of anisotropic composite wing structures. The principal goal of this study is the divergence behavior of swept composite-type wings to the warping restraint effect being included in the analysis. The numerical examples illustrate the complex role played by the warping restraint effect on the divergence instability of composite wings.

Sectorial properties of straight thin-walled beams

The sectorial method of analysis for thin-walled beams subjected to torsional loading offers many advantages. In particular it enables warping restraint effects due to non-uniform torsion to be incorporated into a general beam theory covering all solid, thick-walled and thin-walled beams of non-deformable cross-section. The distribution of warping restraint stresses around the section is defined in a similar way as for bending by a system of sectorial co-ordinates and several additional geometrical terms. A wider understanding and acceptance of this useful method of analysis is only hindered by difficulties in calculating the various sectorial functions. Solutions are readily available for open sections and regular single cell closed sections. For more complex sections, particularly asymmetric or multicellular sections or those with tapering walls, pierced walls or bracing, these calculations are often tedious. A general computer program is described which analyses any cross-section with open or closed parts. Examples are given of its application to the analysis of several straight thin-walled beams.