Static Stiffness Matrices Research Articles

Exact element static stiffness matrices of shear deformable thin-walled beam-columns

A simple but efficient method to obtain the exact static stiffness matrices is developed in order to perform the spatially coupled elastic and buckling analyses of shear deformable uniform beam-columns having non-symmetric thin-walled sections. First this numerical technique is accomplished via a generalized eigenvalue problem associated with 14 displacement parameters which produces both complex eigenvalues and multiple zero eigenvalues. Next polynomial expressions are assumed as trial solutions for displacement parameters and eigenmodes containing undetermined parameters equal to the number of zero eigenvalues are determined by invoking the identity condition to governing equations. And then the exact displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently exact static stiffness matrices are evaluated by applying member force–displacement relationships to these displacement functions. The lateral-torsional deflections and buckling loads of thin-walled beam-columns are evaluated and compared with analytic solutions and the results by straight beam element and ABAQUS’s shell element.

Exact element static stiffness matrices of shear deformable thin-walled beam-columns

A simple but efficient method to obtain the exact static stiffness matrices is developed in order to perform the spatially coupled elastic and buckling analyses of shear deformable uniform beam-columns having non-symmetric thin-walled sections. First this numerical technique is accomplished via a generalized eigenvalue problem associated with 14 displacement parameters which produces both complex eigenvalues and multiple zero eigenvalues. Next polynomial expressions are assumed as trial solutions for displacement parameters and eigenmodes containing undetermined parameters equal to the number of zero eigenvalues are determined by invoking the identity condition to governing equations. And then the exact displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently exact static stiffness matrices are evaluated by applying member force–displacement relationships to these displacement functions. The lateral-torsional deflections and buckling loads of thin-walled beam-columns are evaluated and compared with analytic solutions and the results by straight beam element and ABAQUS’s shell element.

Exact dynamic and static element stiffness matrices of nonsymmetric thin-walled beam-columns

An improved numerical method to exactly evaluate 14 × 14 dynamic and static element stiffness matrices is proposed for the spatial free vibration and stability analysis of nonsymmetric thin-walled straight beams subjected to eccentrically axial loads. Firstly equations of motion and force-deformation relations are rigorously derived from the total potential energy for a uniform beam element with nonsymmetric thin-walled cross-section. Next a system of linear algebraic equations with nonsymmetric matrices is constructed by introducing 14 displacement parameters and transforming the higher order simultaneous differential equation into the first order simultaneous equation. And then explicit expressions for displacement parameters are exactly evaluated by solving a generalized eigenproblem with complex eigenvalues. Finally exact element stiffness matrices are determined using force-deformation relations. Particularly straightforward application of the present method may not give the exact static stiffness because of existence of multiple zero eigenvalues in case of static buckling problems. Accordingly, a modified numerical method to resolve this difficulty is developed for two cases depending on the initial state of stress resultants. In order to demonstrate the validity and the accuracy of this method, the natural frequencies and buckling loads of nonsymmetric thin-walled beam-columns having bending-torsional deformation modes are evaluated and compared with analytical and F.E. solutions or results analyzed by ABAQUS’s shell element.

TOWARDS DEEP AND SIMPLE UNDERSTANDING OF THE TRANSCENDENTAL EIGENPROBLEM OF STRUCTURAL VIBRATIONS

When using exact methods for undamped free vibration problems the generalized linear eigenvalue problem ( K−ω 2 M) D=0 of approximate methods, e.g., finite elements, is replaced by the transcendental eigenvalue problem K (ω) D=0. Here ω is the circular frequency; D is the displacement amplitude vector; M and K are the mass and static stiffness matrices; and K (ω) is the dynamic stiffness matrix, with coefficients which include trigonometric and hyperbolic functions involving ω and mass because elements (for example, bars or beams) are analyzed exactly by solving their governing differential equations. The natural frequencies of this transcendental eigenvalue problem are generally found by the Wittrick–Williams algorithm which gives the number of natural frequencies below ω t , a trial value of ω, as ∑ J m + s{ K (ω t )} where s {} denotes the readily computed sign count property of K (ω) and the summation is over the clamped–clamped natural frequencies of all elements of the structure. Understanding the alternative solution forms of the transcendental eigenvalue problem is important both to accelerate convergence to natural frequencies, e.g., by plotting ∣ K (ω)∣, and to improve the mode calculations, which lack the complete reliability of natural frequencies obtained by using the Wittrick–Williams algorithm. The three solution forms are: ∣ K (ω)∣= 0; D= 0 with ∣ K (ω)∣ →∞; and ∣ K (ω)∣≠0 with D≠ 0. The literature covers the first two forms thoroughly but the third form has been almost totally ignored. Therefore, it is now examined thoroughly, principally by analytical studies of simple bar structures and also by confirmatory numerical results for a rigidly jointed plane frame. Although structures are unlikely to have exactly the properties giving this form, it needs to be understood, particularly because ill-conditioning can occur for structures approximating those for which it occurs.

Comparing Model Update Error Residuals and Effects on Model Predictive Accuracy

The effect of error residual choice on the predictive capability of an updated structural model is examined. First, analytical expressions are developed that relate errors in dynamically measured static flexibility matrices, stiffness matrices, and modal matrices. The analysis shows that flexibility, stiffness, and modal error residuals correspond to unique weightings on the error in the measured modal data. Next, the flexibility weighting indicator and the stiffness weighting indicator are defined to provide a simple means of ranking individual modes in terms of their potential contribution to errors in a measured static flexibility or stiffness matrix. Finally, expressions are developed for bounding model prediction errors when static flexibility, stiffness, or modal error residuals are used. The results show that the amount of uncertainty in predicted static displacements, static loads, and mode shapes and frequencies is directly related to the choice of error residual and is different in each case. These results are also illustrated using numerical simulations for a modified version of Kabe's problem, which includes model form errors.

Stability of thin-walled pultruded structural members by the finite element method

A finite element having seven degrees of freedom at each node is developed to study the stability problem of thin-walled fibre-reinforced plastic (FRP) structural members. The influence of the in plane shear strain on the stability of the members is taken into account. The shape functions for the rotation and unit length rotation induced by warping are derived. The static and geometric stiffness matrices of a general beam element are established on the basis of the developed shape functions. The bifurcation buckling problem of thin-walled pultruded open-sections subjected to various loading and boundary conditions is examined through a number of examples. It is shown that the influence of the shear strain on the buckling capacity of the FRP structural members is significant and must be taken into account in the design of such members.

Exact static and dynamic stiffness matrices for general variable cross section members

This paper concerns the formulation of a new finite-element method for the solution of beams with variable cross section. Using only one element, it is possible to derive the exact static and dynamic stiffness matrices (up to the accuracy of the computer) for any polynomial variation of axial, torsional, and bending stiffnesses along the beam. Examples are given for the accuracy and efficiency of the method.

A Derivation Procedure for the Dynamic Flexibility Matrix of a Triangular Bending Element

Of the two main finite element approaches, the displacement method has progressed more rapidly than the force method in the area of element representation. Many authors have contributed to the displacement method with both static and dynamic stiffness matrices for beam elements, plane stress elements and plate bending elements. The plane stress and bending elements being rectangular, triangular or quadrilateral in form. A dynamic stiffness matrix consists of a static stiffness matrix and a mass matrix. In the force method, .static flexibility matrices have been developed for beam elements and plane stress elements. However, static flexibility matrices for plate bending elements have only recently been published (Kaufman and Hall, Morley, Robinson, Przemieniecki). For the past few years, the author has been investigating the rank force method for structural vibration analysis. In ref. 6 a dynamic flexibility matrix is presented for a beam element. This matrix consists of a static flexibility matrix and an inverse mass matrix. Ref. 5 contains the derivation of a dynamic flexibility matrix for a rectangular plate element in bending, twisting and shear. The author, in collaboration with Petyt, demonstrated how a dynamic stiffness and flexibility matrix can be extended to include material damping.