Exact eigenvalue calculations for structures with rotationally periodic substructures

The theory presented enables rotationally periodic (i.e. cyclically symmetric) three-dimensional substructures to be included when using existing algorithms to ensure that no eigenvalues are missed when the stiffness matrix method of structural analysis is used, where the eigenvalues are the natural frequencies of undamped free vibration analyses or the critical load factors of buckling problems. A substructure can be connected in any required way to a parent structure which shares its rotational periodicity, or can be connected by nodes at each end of its axis of periodicity to any parent structure, i.e. the parent structure need not be periodic. The theory uses complex arithmetic, involves only one of the rotationally repeating portions of the substructure, allows nodes and members to coincide with the axis of rotational periodicity, permits efficient multi-level use of rotationally periodic substructures, and gives ‘exact’ eigenvalues if the member equations used are those obtained by solving appropriate differential equations. The competitiveness of the method is illustrated by approximate predictions of computation times and savings for two structures which contain rotationally periodic substructures.     

Reliable use of determinants to solve non-linear structural eigenvalue problems efficiently

A ‘multiple determinant parabolic interpolation method’ is described and evaluated, principally by using a plane frame test-bed program. It is intended primarily for solving the transcendental eigenvalue problems arising when the ‘exact’ member equations obtained by solving the governing differential equations of members are used to find the eigenvalues (i.e. critical buckling loads or undamped natural frequencies) of structures. The method has five stages which together ensure successful convergence on the required eigenvalues in all circumstances. Thus, whenever checks indicate its suitability, parabolic interpolation is used to obtain eigenvalues more rapidly than would the popular bisection alternative. The checks also ensure a wise choice of the determinant used by the interpolation. The determinants available are all usually zero at eigenvalues, and comprise the determinant of the overall stiffness matrix K _n and the determinants which result, with negligible extra computation, from effectively considering all except the last m (m=1, 2,…, n?1) freedoms to which K _n corresponds as internal substructure freedoms. Tests showed time savings compared to bisection of 31 per cent when finding non-coincident eigenvalues to relative accuracy ? = 10^?4, increasing to 62 per cent when ? = 10^?8. The tests also showed time savings of about 10 per cent compared with an earlier Newtonian approach. The method requires no derivatives and its use in the widely available space frame program BUNVIS-RG has demonstrated how easily it can replace bisection, which was used in the earlier program BUNVIS.     

Exact determinant for infinite order FEM representation of a Timoshenko beam–column via improved transcendental member stiffness matrices

Transcendental stiffness matrices for vibration (or buckling) have been derived from exact analytical solutions of the governing differential equations for many structural members without recourse to the discretization of conventional finite element methods (FEM). Their assembly into the overall dynamic structural stiffness matrix gives a transcendental eigenproblem, whose eigenvalues (natural frequencies or critical load factors) can be found with certainty using the Wittrick–Williams algorithm. A very recently discovered analytical property is the member stiffness determinant, which equals the FEM stiffness matrix determinant of a clamped ended member modelled by infinitely many elements, normalized by dividing by its value at zero frequency (or load factor). Curve following convergence methods for transcendental eigenproblems are greatly simplified by multiplying the transcendental overall stiffness matrix determinant by all the member stiffness determinants to remove its poles. In this paper, the transcendental stiffness matrix for a vibrating, axially loaded, Timoshenko member is expressed in a new, convenient notation, enabling the first ever derivation of its member stiffness determinant to be obtained. Further expressions are derived, also for the first time, for unloaded and for static, loaded Timoshenko members. These new expressions have the advantage that they readily reduce to corresponding expressions for Bernoulli–Euler members when shear rigidity and rotatory inertia are neglected. Additionally, the total equivalence of the normalized transcendental determinant with that of an infinite order FEM formulation aids understanding and evaluation of conventional FEM results. Examples are presented to illustrate the use of the member stiffness determinant. Copyright © 2004 John Wiley & Sons, Ltd.     

A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells

An exact free vibration analysis of doubly-curved laminated composite shallow shells has been carried out by combining the dynamic stiffness method (DSM) and a higher order shear deformation theory (HSDT). In essence, the HSDT has been exploited to develop first the dynamic stiffness (DS) element matrix and then the global DS matrix of composite cylindrical and spherical shallow shell structures by assembling the individual DS elements. As an essential prerequisite, Hamilton’s principle is used to derive the governing differential equations and the related natural boundary conditions. The equations are solved symbolically in an exact sense and the DS matrix is formulated by imposing the natural boundary conditions in algebraic form. The Wittrick–Williams algorithm is used as a solution technique to compute the eigenvalues of the overall DS matrix. The effect of several parameters such as boundary conditions, orthotropic ratio, length-to-thickness ratio, radius-to-length ratio and stacking sequence on the natural frequencies and mode shapes is investigated in details. Results are compared with those available in the literature. Finally some concluding remarks are drawn.     

A note on universal matrices for triangular finite elements for the quasi-harmonic equation

An algorithm to generate universal matrices for plane triangular finite elements for the general ‘quasi-harmonic’ equation is presented. For every member of the triangle family three numerical universal matrices are obtained which are independent of the size, shape and ‘material’ properties of the element. Of these, two are basic and the third can be generated from one of these two. The element ‘stiffness’ matrix is conveniently generated by manipulating these two basic matrices taking into account the size, shape and material properties of the element in a simple manner.     

基于多重多级动力子结构的Lanczos算法

提出利用多重多级子结构技术与Lanczos方法求解超大型复杂结构动力特性的子结构算法。该算法利用子结构周游树技术,分别对每个子结构进行Lanczos迭代,通过累加各个子结构的正交化系数组成全局三对角矩阵,最后求解得到整体结构的特征值。算法能够计算超大型结构特征值和特征向量,计算效率高;消耗计算机资源少,稳定性高。由于考虑了各子结构内部自由度对整体求解的贡献,算法精度得到显著提高,并与不作凝聚的单一整体结构分析具有相同的计算精度,计算结果不受复杂子结构划分方式的限制。数值算例验证了所提出算法的正确与有效性。     

A procedure for analysing space frames with partial warping restraint

Stiffness matrices for three-dimensional beam elements that include the effect of warping restraint on elastic torsional response have been derived by various investigators. Using one of the available stiffness matrices and assuming that the warping boundary conditions can be specified on a member-by-member basis, an elastic ‘warping’ support is introduced to represent conditions of partial warping restraint at the member ends. The concept of a ‘warping indicator’ is then introduced to facilitate use of warping springs. Following this, static condensation is used to eliminate the restrained warping degrees-of-freedom. The condensed stiffness matrices for the elements can then be assembled to yield a global stiffness matrix. In the global matrix, continuous warping degrees-of-freedom, that is, those internal to a member represented by several elements, are expressed in local co-ordinates. The remainder are expressed in global co-ordinates. In the force recovery phase, it is shown that an ‘indirect’ method yields most accurate results for the bimoment and warping torsion when the twist function is represented by a cubic polynomial. Solutions to examples of linear elastic analysis are compared with well-known analytical solutions to demonstrate the application of the method.     

Compact computation of natural frequencies and buckling loads for plane frames

Existing theory is assembled to give a method which needs only the core of a mini-computer to calculate the eigenvalues of large rigidly jointed plane frames with certainty, the eigenvalues being natural frequencies and critical load factors in free vibration and buckling problems, respectively. The method is illustrated by annotated listings of vibration and buckling programs, each involving under two hundred Fortran statements and with low number storage requirements (see Table I). The use of the programs as ‘black boxes’ is fully explained, with illustrative examples. The member theory used is the ‘exact’ classical Bernoulli-Euler uniform member theory. Possible applications include: evaluation of answers from approximate methods; calculation of critical loads for substitution in the modified Merchant-Rankine formula to estimate collapse loads of frames; and calculation of shifts in natural frequencies caused by structural damage, in connection with structural integrity monitoring of inaccessible structures.     

Skyline algorithms for multilevel substructure analysis

A matrix derivation of the method of partial decomposition for substructure equilibrium equations is given and algorithms and subroutines necessary for implementing a multi-level substructuring scheme are presented. The algorithms are based upon partial decomposition of skyline matrices by the Cholesky method. Transformation of co-ordinates and prediction of the skyline for higher level substructures are also discussed.     

Substructuring in linear and nonlinear analysis

The procedures for utilizing substructuring and static condensation in structural analysis are well known. However, until recently there have been no general-purpose structural mechanics computer systems that offer multi-level substructuring combined with a convenient method for defining the structural model. Now that such systems are available, engineers must decide when substructuring techniques are useful. Substructuring, with and without condensation, has proved to be highly efficient in the analysis of certain classes of structure. It can reduce computer costs by a factor of from 2 to 100. Yet, indiscriminate use of condensation may result in unnecessary and expensive computations. This paper examines the advantages and disadvantages of substructuring relative to data entry and computational efficiency. Guidelines are proposed for engineers to follow when using substructuring in the analysis of linear and nonlinear structures. The FINITE system is used to illustrate actual implementation of substructuring features in a general purpose finite element system.     

ACCURATE CALCULATION OF ELASTIC BUCKLING LOADS FOR SPACE FRAMES BUILT UP OF UNIFORM BEAMS WITH OPEN THIN-WALLED CROSS-SECTION

A so-called exact static stiffness matrix for a uniform beam element with open thin-walled cross-section carrying an axial compressive load is derived. This stiffness matrix is useful in an accurate calculation of bifurcation loads and corresponding buckling modes of space frames built up of such beam elements. One may also calculate displacements and sectional forces caused by external joint loads taking into account the second-order effect of the axial beam loads. The exact stiffness matrix is derived by use of the general solution to a set of three coupled differential equations. This means that no preselected shape functions need be introduced and that discretization errors are avoided. The differential equations model coupled Euler–Bernoulli bending in the two principal planes and Saint-Venant/Vlasov torsion and warping with respect to the shear centre axis. No cross-sectional symmetries are assumed. Numerical examples are given. One application will be to loaded pallet racks. The ‘effective length’ for a rack column is calculated.     

Symmetricability of pressure stiffness matrices for shells with loaded free edges

For the case of free edges which are loaded, follower forces remaining normal to the middle surface of a shell throughout the deformation history do not have a load potential. In finite element analysis, this results in an unsymmetric pressure stiffness matrix. Depending on the structure of the available computer program, implementation of an equation solver permitting solution of unsymmetric simultaneous systems of algebraic equations may be a tedious task. This explains the significance of the topic of symmetricability of pressure stiffness matrices, turning out to be of special importance in the case of static buckling under the assumption of a linear prebuckling path. At first, incremental equations for tracing the nonlinear load–displacement path are derived. Thereafter, the buckling condition is deduced. Then, it is demonstrated that symmetrization of the pressure stiffness matrix is admissible if the so-obtained ‘buckling pressure’ differs ‘very little’ from the ‘buckling pressure’ resulting from an alternative symmetric ‘buckling matrix’, as is shown to be the case for a simply supported cylindrical shell with a free upper edge, subjected to hydrostatic external pressure. The alternative symmetric ‘buckling matrix’ is a consequence of deleting the virtual work term, causing the unsymmetry of the pressure stiffness matrix, in the expression for the external virtual work. A mechanical interpretation of this virtual work term is given. It is shown to be equal to the difference of virtual work of the original pressure load and of a ‘substitute pressure-field’, of the form of a Fourier series of the former. This explains why, normally, the buckling coefficient resulting from the ‘substitute pressure-field’ represents a good approximation to the buckling coefficient stemming from the original pressure load.     

A higher-order triangular finite element for the solution of field problems in orthotropic media

This paper presents a triangular finite element for the solution of two-dimensional field problems in orthotropic media. The element has nine degrees of freedom, these being the potential and its two derivatives at each node. The ‘stiffness’ matrix is derived analytically so that no further integration is required when computations are performed using the element. The results obtained using the element are compared with the exact mathematical solution of both a temperature distribution and a torsion problem.     

A general finite element for beams or beam-columns with or without an elastic foundation

A general finite element is derived for beams or beam-columns with or without a continuous Winkler type elastic foundation. The need to discretize members into shorter elements for convergence towards an ‘exact’ solution is eliminated by employing in the derivation of the element exact shape functions obtained from the equation of the elastic line. Inter-nodal values of deflections, bending moments and shear forces are obtained using the exact shape functions and trigonometric series. The effect of heavy compressive or tensile axial forces on bending stiffness is treated as a linear problem by considering the axial force as a constant parameter affecting the stiffness. FORTRAN subroutines to compute the stiffness matrix, equivalent nodal forces, deflected shape, bending moments and shear forces are provided and verified by an example.     

A successive bounding method to find the exact eigenvalues of transcendental stiffness matrix formulations

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.     

A new finite element method for analysing symmetrically loaded thin shells of revolution

An essential feature of finite element methods of analysis for symmetrically loaded shells of revolution is the setting up of equations representing the response of a short ‘ring’ element to edge loading. In this paper the axisymmetric behaviour of a short elastic cylindrical shell element under edge loading is described in a new way by means of a matrix which is a combination of stiffness, flexibility and ‘neutral’ sub-matrices. The coefficients of the matrix are derived direct from the equations of the problem, which involves a trivial amount of work in comparison with conventional methods. The corresponding matrix for a short section of an arbitrary shell of revolution is set up with little additional effort, and its use is described for calculation of edge response coefficients for portions of spherical shells. Finally, the method is used to study by iteration the behaviour of a thin spherical shell of viscous material containing a rigid boss which is loaded radially inwards: changes in meridional profile are followed as deformation proceeds. Results are presented for both linear and non-linear viscous material.     

Finite elements based on energy orthogonal functions

It is shown how the convergence requirements for a finite element may be written as a set of linear constraints on the stiffness matrix. It is then attempted to construct a best possible stiffness matrix. The constraint equations restrict the way in which these stiffness terms may be chosen; however, there is normally still room for improving or optimizing an element. It is demonstrated how an element stiffness matrix may be found using rigid body, constant strain and higher order deformation modes. Further, it is shown how the constraint equations may be exploited in deriving an ‘energy orthogonality theorem’. This theorem opens the door to a whole new class of simple finite elements which automatically satisfy the convergence requirements. Examples of deriving plane stress and plate bending elements are given.     

p-version least-squares finite element formulation of Burgers' equation

A p-version least-squares finite element formulation for non-linear problems is presented and applied to the steady-state, one-dimensional Burgers' equation. The second-order equation is recast as a set of first-order equations which permit the use of C⁰ elements. The primary and auxiliary variables are approximated using equal-order p-version hierarchical approximation functions. The system of non-linear simultaneous algebraic equations resulting from the least-squares process is solved using Newton's method with a line search. The use of ‘exact’ and ‘reduced’ quadrature rules is investigated and the results are compared. The formulation is found to produce excellent results when the ‘exact’ integration rule is used. The combination of least-squares finite element formulation and p-version works extremely well for Burgers' equation and appears to have great potential in fluid dynamics problems.     

Eigensolution for large structural systems with substructures

A method for calculating natural frequencies and mode shapes of large structural systems with substructures and the subspace iteration is developed. The method uses only substructural stiffness matrices and the mass matrix for each finite element of the system. The mass matrix for the entire structure or any of its substructures need not be computed. However, efficiency of the method is improved when mass matrix for the entire structure is computed and saved in the computer core. No approximating assumptions are made. Thus, natural frequencies and mode shapes for the finite element model employed are the same with or without the substructuring algorithm. This is demonstrated by computing first ten natural frequencies and the corresponding mode shapes for an open truss helicopter tail-boom structure.     

Diagonalization procedure for scaled boundary finite element method in modeling semi‐infinite reservoir with uniform cross‐section

To improve the ability of the scaled boundary finite element method (SBFEM) in the dynamic analysis of dam–reservoir interaction problems in the time domain, a diagonalization procedure was proposed, in which the SBFEM was used to model the reservoir with uniform cross‐section. First, SBFEM formulations in the full matrix form in the frequency and time domains were outlined to describe the semi‐infinite reservoir. No sediments and the reservoir bottom absorption were considered. Second, a generalized eigenproblem consisting of coefficient matrices of the SBFEM was constructed and analyzed to obtain corresponding eigenvalues and eigenvectors. Finally, using these eigenvalues and eigenvectors to normalize the SBFEM formulations yielded diagonal SBFEM formulations. A diagonal dynamic stiffness matrix and a diagonal dynamic mass matrix were derived. An efficient method was presented to evaluate them. In this method, no Riccati equation and Lyapunov equations needed solving and no Schur decomposition was required, which resulted in great computational costs saving. The correctness and efficiency of the diagonalization procedure were verified by numerical examples in the frequency and time domains, but the diagonalization procedure is only applicable for the SBFEM formulation whose scaling center is located at infinity. Copyright © 2009 John Wiley & Sons, Ltd.