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A numerical analysis method for wind-induced response of structures is presented which is based on the pseudo-excitation method to significantly reduce the computational complexity while preserving accuracy. Original pseudo-excitation method was developed suitable for adoption by combining an effective computational fluid dynamic method which can be used to replace wind tunnel tests when finding important aerodynamic parameters. Two problems investigated are gust responses of a composite wing and buffeting vibration responses of the Tsing Ma Bridge. Atmospheric turbulence effects are modeled by either k–ω shear stress transport or detached eddy simulation. The power spectral responses and variances of the wing are computed by employing the Dryden atmospheric turbulence spectrum and the computed values of the local stress standard deviation of the Tsing Ma Bridge are compared with experimental values. The simulation results demonstrate that the proposed method can provide highly efficient numerical analysis of two kinds of wind-induced responses of structures and hence has significant benefits for wind-induced vibration engineering.

Hypersonic aircraft always face bad flight environment related to the coupling effects of high temperature and random vibration. Numerical analysis of thermoacoustic characteristics of components inside such an aircraft is presented in this paper. An actual aircraft cabin composite structure including a heat resistant layer, an adiabatic layer and metal frameworks is modeled with the finite element mesh generation method. Transient heat conduction of the hypersonic flight aircraft is analyzed based on a block-iterative coupling method which can consider the atmosphere-aircraft interaction. Then the Von Karman turbulence spectrum is employed to define the random vibration environment of the present aircraft. The thermodynamic response of the system is solved by the pseudo-excitation method which can improve the computation efficiency greatly. Finally, thermoacoustic characteristic of the cavity inside the aircraft cabin is analyzed when the transient heat conduction and random vibration due to atmosphere turbulence are both included. A numerical method is proposed based on a structure-acoustic coupling method which can use acoustic equations to simulate the propagation of the acoustic wave in the flow. It can be found from the computed results that the change of the temperature influences both the thermodynamic characteristic of the aircraft cabin and thermoacoustic characteristic of the component inside the present aircraft significantly. So the coupling effects of high temperature and random vibration on thermoacoustic characteristic of a hypersonic flight aircraft cannot be neglected. The proposed numerical analysis method in this paper can be widely applied to numerical investigations of thermoacoustic systems inside hypersonic aircraft.

This paper focuses on numerical simulations of bluff body aerodynamics with three-dimensional CFD (computational fluid dynamics) modeling, where a computational scheme for fluid-structure interactions is implemented. The choice of an appropriate turbulence model for the computational modeling of bluff body aerodynamics using both two-dimensional and three-dimensional CFD numerical simulations is also considered. An efficient mesh control method which employs the mesh deformation technique is proposed to achieve better simulation results. Several long-span deck sections are chosen as examples which were stationary and pitching at a high Reynolds number. With the proposed CFD method and turbulence models, the force coefficients and flutter derivatives thus obtained are compared with the experimental measurement results and computed values completely from commercial software. Finally, a discussion on the effects of oscillation amplitude on the flutter instability of a bluff body is carried out with extended numerical simulations. These numerical analysis results demonstrate that the proposed three-dimensional CFD method, with proper turbulence modeling, has good accuracy and significant benefits for aerodynamic analysis and computational FSI studies of bluff bodies.

Purpose The purpose of this paper is to present a new random vibration-based assessment method for coupled vehicle-track systems with uncertain parameters when subjected to random track irregularity. Design/methodology/approach The uncertain parameters of vehicle are described as bounded random variables. The track is regarded as an infinite periodic structure, and the dynamic equations of the coupled vehicle-track system, under mixed physical coordinates and symplectic dual coordinates, are established through wheel-rail coupling relationships. The random track irregularities at the wheel-rail contact points are converted to a series of deterministic harmonic excitations with phase lag by using the pseudo excitation method. Based on the polynomial chaos expansion of the pseudo response, a chaos expanded pseudo equation is derived, leading to the combined hybrid pseudo excitation method - polynomial chaos expansion method Findings The impact of uncertainty propagation on the random vibration analysis is assessed efficiently. According to GB5599-85, the reliability analysis for the stability index is implemented, which can grade the comfort level by the probability. Comparing to the deterministic analysis, it turns out that neglect of the parameter uncertainty will lead to potentially risky analysis results. Originality/value The proposed method is compared with Monte Carlo simulations, achieving good agreement. It is an effective means for random vibration analysis of uncertain coupled vehicle-track systems and has good engineering practicality

The time-delay discrete-time equations are derived directly from the differential vibration equations for structures in state space, with discrete parameters obtained via the precise integration method. Then a state vector expansion method is used to transform these equations into ones which do not contain time-delay terms explicitly. Next, the order of this system is substantially reduced by using the balanced reduction method to form a dominant subsystem which is based on the eigenmodes of the state subspace with highest controllability. The controllers are then designed by the discrete time-delay optimal control theory, which contains the control terms not only of the current state but also of a few previous states, in order to reflect the effect of time delays. By using the proposed method, the order of the controller is very substantially reduced without causing any essential difference in the control effect. A numerical example for a shear-type building is presented and it shows the effectiveness of the proposed method even when time delays are quite large.

A powerful new method for dealing with stationary random seismic response is described in detail. It is an extension of the early-stage method. The emphasis is focused on multi-support excitation problems. The main features of this method are: 1 The computational efficiency is much higher than for the conventional method, e.g. approximately 10,100, or even more times faster. 2 All cross-correlation items, whether between the support excitations or between the participant modes, are involved naturally. Practically, it is impossible for this method to neglect these items 3 The principles and algorithm are simple and easy for engineering application. Complex 3D FEM models with dozens of ground excitations can be easily analysed on microcomputers as small as IBM-pc/ AT. 4 The non-proportional damping matrix can be dealt with easily and accurately and only real eigenmodes are used for degree reduction.

This paper presents a robust H∞ output feedback control approach for structural systems with uncertainties in model parameters by using available acceleration measurements and proposes conditions for the existence of such a robust output feedback controller. The uncertainties of structural stiffness, damping and mass parameters are assumed to be norm-bounded. The proposed control approach is formulated within the framework of linear matrix inequalities, for which existing convex optimization techniques, such as the LMI toolbox in MATLAB, can be used effectively and conveniently. To illustrate the effectiveness of the proposed robust H∞ strategy, a six-story building was subjected both to the 1940 El Centro earthquake record and to a suddenly applied Kanai-Tajimi filtered white noise random excitation. The results show that the proposed robust H∞ controller provides satisfactory results with or without variation of the structural stiffness, damping and mass parameters.

Since 1970, the Wittrick–Williams algorithm has been applied with increasing sophistication in structural mechanics to guarantee that eigenvalues cannot be missed and are calculated accurately. The underlying theorem enables its application to any discipline requiring eigenvalues of self-adjoint systems of differential equations. Its value in mathematics was recently illustrated by studying Sturm–Liouville equations on large homogeneous trees with Dirichlet boundary conditions and n (≤43) levels. Recursive subsysteming was applied n −1 times to assemble the tree progressively from sub-trees. Hence, numerical results confirmed the recent theoretical bounds of Sobolev & Solomyak for n →∞. In addition, a structural mechanics analogy yielded a proof that many eigenvalues had high multiplicities determined by n and the branching number b . Inspired by the structural mechanics analogy, we now prove that all eigenvalues of the tree are obtainable from n substitute chains r (=1, 2, …, n ) which involve only r consecutively linked differential equations and which have only singlefold eigenvalues. Equations are also derived for the multiplicities these eigenvalues have for the tree. Hence, double precision calculations on a PC readily gave eigenvalues for n =10 6 and b =10, i.e. ≃10 999 999 linked Sturm–Liouville equations. Moreover, a simple equation is derived which gives all the eigenvalues of uniform trees with Dirichlet conditions at both ends, and band-gap spectra are numerically demonstrated and theoretically justified for trees with the Dirichlet conditions at either end replaced by Neumann ones. Additionally, even if each multiple eigenvalue would be counted as if it were singlefold, the proportion of eigenvalues that are multiple is proved to approach unity as n →∞.

There is a need for fire modeling tools capable of rapid simulation of fire growth and smoke spread in multiple compartments with complex ventilation. Currently available tools are not capable of simulation of complex ventilation arrangements in a timely manner. To address this problem, a new fire model called Fire and Smoke SIMulator (FSSIM) has been developed. FSSIM is a network model whose core thermal hydraulic routines are based on MELCOR. FSSIM capabilities include remote ignition, multilayer heat conduction, radiation streaming, arbitrarily complex HVAC systems, detection, suppression, oxygen-limited combustion, and simple control systems.

There is a need for fire modeling tools capable of rapid simulation of fire growth and smoke spread with complex ventilation in multiple compartments. Currently available tools are not capable of either the speed, the simulation of complex ventilation arrangements, and/or the ability to participate as a federate in a simulation environment. To address this, a new fire model was developed called FSSIM (Fire and Smoke SIMulator) [1]. FSSIM is a network model whose core thermal hydraulic routines are based on MELCOR [2]. FSSIM capabilities include remote ignition, multi-layer heat conduction, radiation streaming, HVAC systems, detection, suppression, oxygen and fuel limited combustion, and simple control systems. FSSIM was used to simulate four tests from the ex-Shadwell Submarine Ventilation Doctrine Tests. Excellent results are obtained in predicting the time-dependent temperature, visibility, and velocities.

When exact dynamic stiffness matrices are used to compute natural frequencies and vibration modes for skeletal and certain other structures, a challenging transcendental eigenvalue problem results. The present paper presents a newly developed, mathematically elegant and computationally efficient method for accurate and reliable computation of both natural frequencies and vibration modes. The method can also be applied to buckling problems. The transcendental eigenvalue problem is first reduced to a generalized linear eigenvalue problem by using Newton's method in the vicinity of an exact natural frequency identified by the Wittrick–Williams algorithm. Then the generalized linear eigenvalue problem is effectively solved by using a standard inverse iteration or subspace iteration method. The recursive use of the Newton method employing the Wittrick–Williams algorithm to guide and guard each Newton correction gives secure second order convergence on both natural frequencies and mode vectors. The second order mode accuracy is a major advantage over earlier transcendental eigenvalue solution methods, which typically give modes of much lower accuracy than that of the natural frequencies. The excellent performance of the method is demonstrated by numerical examples, including some demanding problems, e.g. with coincident natural frequencies, with rigid body motions and large-scale structures. Copyright © 2003 John Wiley & Sons, Ltd.

This paper presents a post-buckling analysis for prismatic plate assemblies made of isotropic materials. The structures are assumed to consist of a series of long flat strips rigidly connected together at their edges, subjected to longitudinal in-plane compressive load. The buckling load and corresponding buckling mode of the structure are first obtained as the results of transcendental eigenvalue problems, which arise when exact solutions to the member differential equations are used to form the stiffness matrix of the plate assemblies. The other post-buckling field functions are also obtained analytically as exact solutions to the member differential equations. Results for the load end-shortening and load–deflection relationships for long prismatic plate assembly examples are obtained and compared with results obtained by other authors.

The purpose of this paper is to derive and apply the relationships of the dynamic substructure synthesis method, based on the subregion variational principle of elastodynamics in discrete form. Two theorems are derived for discretized systems and are used to demonstrate the interface compatibility condition of dynamic substructure synthesis. This enables the dynamic substructure synthesis method to be derived from the subregion variational principle. A complete modal expansion is used to derive the residual modes for both the free and fixed interface methods. The alternative formulations of the hybrid, free interface, and fixed interface methods, as well as Guyan-Irons reduction, are shown by coordinate transformations to be degenerate cases. These cases are all reasonable representations of the Rayleigh-Ritz method, and if they use constraint modes and/or attachment modes, good convergence can be assured because the static modes are included in the substructure reduction. The relationships between these ...

Convergence to local optima rather than the global optimum is a common shortcoming of directed seareh (gradient-based) optimization techniques. Successive convergence on better local optima is an efficient way of finding the global optimum and is achieved by the proposed hybrid global optimization strategy, which combines ideas from the improving hit-and-run and the tunneling global optimization methods. The searching technique of improving hit-and-run is adapted to find suitable starting points to converge on other (better) local optima. The suitability of candidate starting points is confirmed by the concept of the tunneling method. The strategy presented has unique features due to the underlying structural theory being exact, in the sense of differential equations being solved for the member (or element) stiffnesses so that they are transcendental functions of the load factor. Hence the buckling eigenproblem for the entire structure is a transcendental one, instead of the linear one associated with the more usual finite element method formulations. Constrained minimum-mass aerospace panel stability design problems are used to demonstrate the efficiency and robustness of the hybrid global optimization strategy.

In this paper, the extended Kantorovich method originally proposed by Kerr is further extended to the bending problem of rectangular Mindlin plates. A substantial extension made in this study, in addition to the employment of the Mindlin theory, is the use of multiterm trial functions. This eventually requires an iterative procedure at each iteration step of which a set of simultaneous ordinary differential equations (ODEs) must be solved. Despite the use of a state-of-the-art general-purpose ODE solver, which is supposed to be able to solve the derived system of ODEs to a user specified set of error tolerances, numerical stability proved to be a prohibitive obstacle if left untreated, especially when more terms are used and for thin plates. To ensure numerical stability and to overcome the shear locking difficulty, an orthogonalization technique is proposed that maintains good independence of each term of the trial functions. This enables virtually any number of trial function terms to be accommodated with excellent numerical accuracy and stability. Several typical numerical examples given in the paper consistently show that the proposed method performs excellently even in extreme cases such as extremely thin plates and the steep variation of internal forces near the free edges.

A novel and versatile optimum-cost design approach that embraces the ideology of concurrent engineering is described. The approach is incorporated into optimum-design software to permit the determination of the least-cost design for prismatic assemblies of laminated composite plates, which occur in advanced aerospace construction. Comparisons between cost and mass optimum designs of different panel topologies for two proposed manufacturing processes are presented. The different panel topologies were obtained by varying the number of stiffeners and by using either blade or T stiffeners. In total, 40 different panels were optimized separately for manufacturing cost, mass, and total cost, i.e., including running costs. The running costs were included by associating a penalty function with the mass of the panels. The results show that the approach is extremely versatile and enables different topologies and manufacturing processes to be compared in detail.

Based on multi-scattering speckle theory, the speckle fields generated by plant specimens irradiated by laser light have been studied using a pointwise method. In addition, a whole-field method has been developed with which entire botanical specimens may be studied. Results are reported from measurements made on tomato and apple fruits, orange peel, leaves of tobacco seedlings, leaves of shihu seedlings (a Chinese medicinal herb), soy-bean sprouts, and leaves from an unidentified trailing houseplant. Although differences where observed in the temporal fluctuations of speckles that could be ascribed to differences in age and vitality, the growing tip of the bean sprout and the shihu seedling both generated virtually stationary speckles such as were observed from boiled orange peel and from localised heat-damaged regions on apple fruit. Our results suggest that both the identity of the botanical specimen and the site at which measurements are taken are likely to critically affect the observation or otherwise of temporal fluctuations of laser speckles.

VICONOPT is an optimum design program for prismatic plate assemblies based on exact methods of eigenvalue analysis. It has proved to be reliable and efficient in the optimum design of stiffened aerospace panels. Parametric design studies have been made for panels with longitudinal voids, skewed panels and variants of benchmark aircraft wing panels, including estimation of the effects of allowance for transverse shear deformation. Current developments include efficient methods for the design of multi-cell wing boxes, the use of random design moves to escape from a local optimum, a generic minimum cost design procedure and an exploration of parallel methods for optimum design.

Point connections representing spot welds, bolts, rivets, screws, pins, etc. have been included in the powerful panel analysis and design computer program VICONOPT. The exact stiffness matrices of component structures are coupled by using Lagrangian multipliers to represent these point connections between the structures. The new approach gives a very quick and accurate analysis when longitudinal lines of point connections are used in panels which are loaded only in compression. In this case only one point connection needs to be modelled and only certain response wavelengths contributing to the buckling mode are used. Accurate solutions are also obtained for shear loaded panels but at a much greater computational cost. Illustrative examples include comparisons between results for line and point connections, indicating to designers how many point connections are required to produce an effective line connection. These comparisons cover both pure axial compression and pure shear loading cases for both isotropic and anisotropic panels. They suggest that for many panels, manufacturing costs could be reduced by using fewer point connections, with minimal reduction in the stability of the panel. The work presented here is of particular importance to aircraft panels where point connections are extensively used to connect component structures together.

The approximate representation of an exact dynamic stiffness matrix K(rho) by a quadratic matrix formulation, A-rho B-rho(2)C, is studied theoretically in this paper. The matrix formulation is formed by expressing the elements of K(rho) as parabolic functions based on choosing three fixed values of the eigenparameter rho. The general bounding properties of the approximate eigenvalues provided by the quadratic matrix formulation are shown to exist, provided that the three fixed values are below the lowest pole of the nonlinear stiffness matrix and that the three coefficient matrices, A, B and C, are positive definite. It is shown theoretically in this paper that the approximate eigenvalues are either upper or lower bounds of the corresponding exact ones of the exact dynamic stiffness matrix.

This paper develops the dynamic orthogonal decoupling method in order to decouple damped gyroscopic structural dynamic systems. The Arnoldi method is extended to solve the generalised eigenvalue problem via a real Schur decomposition. Very interesting results are obtained, which show that this method cannot only obtain the eigenvalues of the system, but can also solve dynamic response of damped gyroscopic dynamic systems in a decoupled way. Compared to the conventional complex mode decoupling method, this method virtually uses only real number computation, remains valid when the corresponding eigenvalue problem is defective and does not require solution of the left eigenvector problem. The structure of the decoupled dynamic equation is very satisfying, because it enhances understanding of the combination of damping with gyroscopic effects. For undamped gyroscopic systems, the decoupling method proposed degenerates to that of Meirovitch.

This paper presents a parallel structural optimization method and its implementation on a distributed memory, multiple-instruction, multiple-data parallel computer. The proposed strategy does not require the formation or solution of global system equations. Instead, each processor of the parallel computer stores only the data relevant to one structural element or substructure and generates its local stiffness matrix. A parallel conjugate gradient procedure using the optimality criterion approach is employed to compute the displacements and modified design for each element (or substructure). An implementation of the parallel structural optimization method on a Transtech Paramid parallel computer is discussed, the speedup and efficiency of this method being demonstrated by numerical examples.

A new form of curve is presented that can be used to find the buckling loads and/or natural frequencies of axially loaded prismatic structures which consist of a series of thin, flat, rectangular plates that are rigidly connected together along their longitudinal edges and are simply supported at their ends. Such curves are given for four realistic panels. Illustrative calculations show how the curves can be used to find either the critical buckling stress, unloaded natural frequencies or natural frequencies in the presence of any specified axial load. The results are exact for structures composed of equally stressed isotropic or orthotropic plates and are shown to also be good approximations for the great majority of cases for which the component plates are unequally stressed and/or anisotropic. The results have been chosen to give designers insight, the means to spot trends, and guidance on how to choose and interpret computer runns for panels which are not similar enough to those in the paper for the curves presented to give sufficient accuracy.

This paper studies the numerical accuracy of the recently presented constrained substructuring method for solving transient problems in structural dynamics, including the solution of its system of combined algebraic and differential equations. To evaluate the method's computational errors, the variations of the period elongation and the amplitude decay with time step size and with the values of the prediction parameter γ of the method are investigated. This evaluation confirms that the method is often sufficiently accurate. Techniques for improving the accuracy of the solution scheme are presented and evaluated, leading to the conclusion that one of them is well worth using if additional accuracy is required.

A parallel algorithm based on a recent high precision direct (HPD) integration method is presented and evaluated in this paper. A mixed fine grain and coarse grain strategy for parallel computing has been developed. Examples show that this method has good accuracy and that high speedup and efficiency are obtained. A hybrid form involving two types of coarse parallelisation has also been shown to have advantages when a Fourier series form of the HPD method is used and the number of processors is not a factor of the number of terms in the series.

A method is presented for computing propagation constants of longitudinal free wave propagation in a longitudinally prismatic plate assembly which is supported or transversely stiffened at longitudinally periodic intervals. The method combines a special complex Fourier series approach with the Wittrick-Williams algorithm, taking into account the supporting structures or transverse stiffeners by using Lagrangian multipliers. It enables Fourier series based methods and computer programs for natural frequency analysis to be readily used instead to investigate free wave propagation problems. It can be extended to handle other types of periodic structures. Three numerical examples of plate assemblies of different cross-sections are given and lead to some useful conclusions.

A procedure is presented for the buckling analysis of prismatic skew plate assemblies subject to invariant in-plane stresses. Based on the exact solution of the plate differential equations, the method of Lagrangian multipliers is used to enforce the transverse skew boundaries by a sufficient number of point constraints. Analysis assumes that the plate is infinitely long and that supports repeat at bay length intervals, typifying the continuity found in aircraft wing construction. Following a brief derivation of the formulation adopted, results are presented and comparisons are made with other analyses for an unstiffened isotropic skew plate, subject to pure compression loading with both simply supported and clamped boundary conditions. Results for four benchmark stiffened panels, i.e. plate assemblies, incorporating composite material and combined loading are also given for a range of skew angles.

An approximate representation of a transcendental dynamic stiffness matrix K(rho) by a simple quadratic matrix pencil A-rho B-rho(2)C is studied in this paper. The matrix pencil is formed by expressing the elements of K as parabolic functions based on choosing three fixed values of the eigenparameter rho. General bounds on the exact eigenvalues of the transcendental eigenvalue problem provided by the quadratic matrix pencil are shown to exist, provided that the three fixed values are below the lowest pole of the transcendental stiffness matrix considered and that the three coefficient matrices are positive definite. Numerical examples illustrate and confirm these bounding properties. Furthermore, the bounding properties are extended to constrained dynamic stiffness matrices. e.g., matrices formed by using Lagrangian multipliers to couple individual stiffness matrices of several different responses when a single response does not satisfy the desired boundary conditions.

An alternative algorithm for finding exact natural frequencies or buckling loads from the transcendental, e.g. dynamic, stiffness matrix method is presented in this paper and evaluated by using the plate assembly testbed program VICONOPT. The method is based on the bounding properties of the eigenvalues provided by either linear or quadratic matrix pencils on the exact solutions of the transcendental eigenvalue problem. The procedure presented has five stages, including two accuracy checking stages which prevent unnecessary calculations. Numerical tests on buckling of general anisotropic plate assemblies show that significant time savings can be achieved compared with an earlier multiple determinant parabolic interpolation method.

An elegant and accurate method for determining the number of clamped-clamped natural frequencies of a bending-torsion coupled beam which lie below a specified frequency is presented. The method facilitates an easy and straightforward application of the Wittrick-Williams algorithm when finding natural frequencies of bending-torsion coupled beams, or structures which contain them. Numerical results are given for an illustrative example.

A parametric study is presented of the undamped free vibration of a one layer truss type circular dome, with terrestrial proportions. The dome is composed of 36 identical portions, but is supported at only 4, 6, 9, 12, 18 or 36 uniformly spaced points. Only one of the 36 portions is needed in the analysis, using the large space frame computer program BUNVIS-RG, which can analyse linearly repetitive or rotationally periodic structures ‘exactly’, e.g. without the usual finite element discretisation approximations. The study includes changing: the height to diameter ratio; the number of supports; the type of supports; and the type of joint from rigid to pinned connections. In addition, modal densities are found for a near optimum dome. The method used gives valuable insights into the exact mode shapes by showing that they are always composed entirely of a (usually short) finite series of circumferential sine waves.

Starting from equations analogous to those that are widely used for vibration of axially loaded Timoshenko beams, insights are obtained into the buckling behaviour of columns which deform in shear. These insights help in explaining and recognizing those combinations of end conditions for which a simple formula, originally derived by Engesser for clamped-free columns, can be used to predict exactly the effect of shear deformation on the lowest critical buckling loads of columns. The formula is P c = P 0 /(1 + P 0 / kAG ), where P c is the required buckling load, P 0 is the buckling load if shear deformation is ignored and kAG is the shear rigidity. Illustrative results are given for metal columns, but these are intended primarily as an introduction to the principal result. This consists of a single curve which can be applied to metal or composite columns. It represents Engesser's equation, and is expressed in terms of the effective length ratio of the column obtained by ignoring the shear deformations. The combinations of end conditions for which this curve is valid are explored and it is shown to be valid for many standard cases, including hinged-hinged, clamped-free and clamped-clamped columns. It is also valid if the hinged-hinged column has rotational springs of equal stiffness added at its ends. However, the curve is not valid for the common clamped-hinged case (or if the hinged-hinged column has rotational springs of unequal stiffnesses at its ends) for which it can give 20% or greater discrepancy from the correct curve. Such large differences occur in ranges which are of relevance for many composite columns, which is the context in which the results presented are expected to be most useful.

Analytical expressions for the coupled bending-torsional dynamic stiffness matrix terms of an axially loaded uniform Timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element. The symbolic computing package REDUCE has been used to generate an analytical expression for each of the dynamic stiffness terms in a concise form. For check purposes, numerical values of the dynamic stiffness matrix terms were obtained using the derived explicit expressions as well as by an alternative nonanalytical method based on matrix inversions and matrix multiplications. Stiffnesses obtained from both methods agreed with each other to machine accuracy. Application of the developed theory is discussed with particular reference to an established algorithm. The influence of axial force, shear deformation and rotatory inertia on the natural frequencies of a bending-torsion coupled beam with cantilever end-conditions is demonstrated by numerical results. Such results are not generally available in the literature. Therefore, results obtained by partially restricting the present theory are compared with the existing literature wherever possible. The results indicate that the method is accurate and efficient.

Exact analytical expressions for coupled extensional-torsional dynamic stiffness matrix elements of a uniform structural member are derived from the basic governing differential equation of the member in free vibration. Application of the derived dynamic stiffness matrix is discussed with particular reference to an established algorithm. The theory developed is demonstrated by numerical results.

Two simplied methods for the buckling analysis of multi-spar wing boxes made from composite materials are presented. The spars divide such boxes into cells. Both methods, called the lower bound method and the repetitive analysis method, are computationally much quicker than analysing a complete wing box because they consider only one cell at a time. The lower bound method has only been briefly considered previously and the lower bounding property is guaranteed by theory. This method gives guaranteed conservatism when designing. The repetitive analysis method is very quick and the data preparation small. However it is not guaranteed to give lower bound answers and is thus not favoured by the authors except for structures that are nearly repetitive. Both methods are illustrated by using the computer program VICONOPT to analyse six variants of a laboratory test specimen of the type used when developing multi-spar wing boxes. The results indicate that the lower bound method usually achieves an accuracy well within 10%.

Existing theory and the associated computer program VICONOPT deal with infinitely wide plate assemblies given that boundary conditions on all sides of each panel form a rectangle. They also deal with cases when the four supports form a parallelogram so that the plate is a skew plate. This is true provided the panel is of finite width, i.e. isolated from any adjacent panels, which is the case commonly modelled in practice. It does not represent what happens in the real structure, however, where normally there is continuity with the adjacent panel. The present paper shows how the theory and the computer program VICONOPT can be modified so that skewed plate assemblies that are infinitely wide and repeat at transverse intervals can now be modelled exactly. The paper also shows that the theory can be used, if a small measure of approximation is accepted, to model this situation by analysing only one of the identical stiffeners with associated panel skin in the common situations where the panel has equally spaced, identical, longitudinal stiffeners between each adjacent pair of longitudinal lines of support. Illustrative results are given.

A general method for vibration response and wave propagation in a substructural chain-type structure is presented in this paper. The analysis is based on the finite element and symplectic matrix methods. The dynamic stiffness matrices of a typical substructure and an irregular substructure of the substructural chain-type system are obtained by finite element discretization. The symplectic matrix of a single typical substructure is derived from the condensed dynamic stiffness matrix. After determining the eigenvalues and the associated eigenvectors of the symplectic matrix, the state vector at any station of the substructural chain is expanded in terms of the eigenvectors. From the compatibility conditions between the irregular substructure and the typical substructures, the governing dynamic equations for wave propagation are derived. A numerical example for a periodically simply supported infinite uniform Timoshenko beam is given.

The wave propagation constants of periodic structures are computed using the exact dynamic stiffness matrix of a typical substructure. The approach used is to show that wave propagation and the natural vibration eigenproblem are similar to such an extent that methods used to find the natural frequencies of a structure can be applied to find its wave propagation constants. The Wittrick-Williams algorithm has been incorporated into a finite element program, JIGFEX, in conjunction with exact dynamic member stiffnesses, to ensure that no phase propagation eigenvalues are missed during computation. The accuracy of the present approach is then demonstrated by comparing the results that it gives to analytically determined wave propagation curves for a Timoshenko beam on periodic simple supports. Finally, phase propagation curves are given for a complex Timoshenko beam structure of a type that would be very difficult to analyze analytically.

The structural efficiency of a range of panels under uniaxial compression is investigated using the optimum buckling design program VICONOPT. The design uses very efficient VIPASA analysis to guard against all possible modes of failure, together with a tailored sizing strategy. The panels all have nine blades, zed or hat stiffeners and between three and fifteen design variables, covering traditional design using one size of stiffener and more sophisticated design with five sizes of stiffener. Results show that using two stiffener sizes or two stiffener types in alternate positions across the panel width can produce mass savings of up to 30% compared with traditional design. Convergence on an optimum normally occurs within six sizing cycles, but up to twelve sizing cycles are required for sophisticated designs when the initial configuration is poorly chosen. Computational efficiency and material strength constraints are also considered.

An analytical-numerical procedure is extended to investigate the local buckling and post-buckling behaviour of thin-walled laminated stiffeners under uniaxial compression. Two stiffener sections are considered: I and channel sections. The stiffener section is treated as a plate assembly. The analysis uses a deflection-type perturbation technique to determine the buckling load and the post-buckling equilibrium path, ensuring compatibility at the plate junctions, adding extra in-plane terms in the strain-displacement relations, and including shear stresses along the flange-web junctions. Meanwhile, the effects of geometrical imperfections are taken into consideration.

This review covers the many applications of the Wittrick-Williams algorithm, which ensures that no critical buckling loads, or natural frequencies of undamped free vibration, are missed even when using the ‘exact’ member equations obtained by solving the appropriate differential equations. The review includes: plane and space frames; prismatic assemblies of isotropic or anisotropic plates, including in-plane plate shear loads; exact multi-level substructuring; design; damping; efficient solution of rotationally or linearly repetitive structures; use of Lagrangian multipliers; programmable pocket calculator methods; program listings for small computers and; references to large computer programs.

Results are presented for the most heavily and lightly loaded of eight benchmark stiffened laminated wing panels defined from a Dornier wing by a GARTEUR (Group for Aeronautical Research and Technology in Europe) working party. These benchmark panels had three identical and equally spaced blade stiffeners. The results were chosen to help designers to understand many important aspects of the choice of design variables, and of the effects of changing the sophistication of modelling and theory used, for a wide range of wing panels. The percentage changes of (global) optimum mass are presented, along with the final values of the design variables. Some examples of mass histories and of (rejected) local optimum masses are also given. The principal design variables are skin and blade ply thicknesses and blade height. Additional factors considered include the effects of adding flanges to the blades whose plies either matched those of the blades or were allowed to vary independently, varying the number of stiffeners, allowing the stiffeners to differ from each other, varying stiffener spacing, varying some ply angles, including the stiffening effect of adjacent spars, including the effects of continuity with laterally adjacent panels, including through thickness shear deformation in the panel analysis and analysing the panel with its true skewed shape rather than approximating it as rectangular in plan.

An analytical-numerical method is presented for the postbuckling analysis of prismatic plate structures that may be idealized as an assemblage of laminated, rectangular, plate-strips and beams. The formulations are based on the classical laminated plate theory and include a number of refined features. The analysis uses a deflection-type perturbation technique to determine the buckling loads and the postbuckling equilibrium paths. A computer program has been developed and is used to examine the postbuckling behavior of perfect and imperfect blade stiffened panels, with and without longitudinal, transverse or orthogonal beam stiffeners.

This paper investigates a new family of fibre laminated plate assemblies. The assemblies covered are prismatic and consist of a series of thin, flat, rectangular plates that have a series of equally spaced voids, which run for the full length of the central layer of the plates and have constant rectangular cross-section. Optimization of an important sub-group of such plate assemblies, namely blade-stiffened panels, is studied parametrically with the objective of minimizing their mass subject to buckling constraint under pure longitudinal compression. Ply thicknesses, fibre orientation and height of the blade stiffeners are the design variables used when optimizing and the variables changed during the parametric study are the number of stiffeners, a nondimensional load parameter and the aspect ratio of the panels. The parametric study shows graphically the effects on the panel mass of the number of voids in the portion between stiffeners, the thickness of individual voids, the number of stiffeners and the effects of prescribing the values of the stiffener height or thickness. It is observed that a substantial, e.g. 16-20%, mass saving can be achieved by using voids in the way advocated. These percentages are very close to those obtained by using the same voids for an individual plate that is identical to the skin between blades and is simply supported.

Because of the extensive use of long‐span structures in modern engineering, much attention has been given to the extent to which ground motion phase‐lags affect the internal forces of such structures. In this paper, this problem is studied from the aspect of random seismic analysis, i.e. the random seismic responses of long‐span structures are explored with the phase‐lags of the ground joints of the structures taken into account. The earthquake is regarded as a stationary random process. Formulae for calculating the random responses of the structural displacements and internal forces are derived. Numerical examples are presented which illustrate some basic features of such random response, and also show that the ground motion phase‐lags have considerable effects on structural safety analysis.