Geometric Stiffness Matrices Research Articles

Nonlinear static analysis of arbitrary quadrilateral plates in very large deflections

Through a linear mapping, an arbitrary quadrilateral plate is transformed into a standard square computational domain in which the deformation and director fields are developed together with the general forms of the uncoupled nonlinear equations. By proper interpolation of displacement and rotation fields on the whole domain, such that the boundary conditions are satisfied, a mathematical model based on the elastic Cosserat theory, is developed to analyze very large deformations of thin plates in nonlinear static loading. The principle of virtual work is exploited to present the weak form of the governing differential equations. The geometric and material tangential stiffness matrices are formed through linearization, and a step by step procedure is presented to complete the method. The validity and the accuracy of the method are illustrated through certain numerical examples and comparison of the results with other researches.

A refined finite element formulation for flexural and torsional buckling of beam–columns with finite rotations

This paper describes a consistent formulation of a tangent stiffness matrix for the geometrically nonlinear analysis of the space beam–column elements allowing for axial–flexural, lateral–torsional and axial–torsional buckling. In the proposed formulation, three deformation matrices are derived for moderately large rotations in practical three-dimensional space frames subjected to axial force and moments. These matrices are functions of the element deformations and include the coupling among axial, lateral and torsional deformations. The proposed matrices are used together with linear and geometric stiffness matrices for beam elements to study the large deflection behavior of space frames which comprise members with negligible warping effects. Numerical examples show that the proposed element formulation is accurate and efficient in predicting the nonlinear behavior of space frames even when only a few elements are used to model a member.

Geometric stiffness of membranes using symbolic algebra

This paper is concerned with derivation of the geometric stiffness matrix for membrane shells which are represented by constant stress triangular finite elements. Symbolic algebra is used to calculate the gradient of the member nodal force vector of each element when the stresses are kept fixed. This gradient defines the geometric stiffness matrix of the element in global coordinates. The present approach follows the earlier works associated with trusses, plane frames and space frames. It has the advantage of explicitness in derivation, while showing clear physical insight. For the case of small rotations, all the mathematical manipulations can be handled by hand. However, for the case of finite rotations, one must have recourse to symbolic algebra programs. The geometric stiffness matrices derived were implanted into an existing nonlinear membrane analysis program that was used to study two examples from the available literature.

COMPUTATIONAL TOOLS FOR SEMI-ANALYTICAL FINITE ELEMENT STABILITY ANALYSIS OF THIN-WALLED STRUCTURES

Computational tools implementing the semi-analytical finite elements for the linear and stability analysis of thin-walled beams are considered. The proposed method presents two-stage discretisation, where the thin-walled cross section is approximated by the semi-analytical finite elements at the first stage, while conventional longitudinal finite element discretisation is performed at the second stage. The developed computational tools for the processor allow assembling of the global linear and geometric stiffness matrices and present them in a form as it is used in conventional finite element software. The proposed computational tools for pre- and post-processors enable to perform communication between the user and convention finite elements software. Solution examples illustrate the use software tools and the quality of the results.

Free vibration and stability analysis of stiffened sandwich plates

Description is given of the development of a B-spline finite strip method for predicting the natural frequencies of vibration and the buckling stresses of rectangular sandwich panels. Each panel comprises a flat sandwich main plate which is reinforced by a number of longitudinal compact plate stiffeners attached to the faceplates. The core of the main plate is represented as a three-dimensional solid whilst the faceplates and the stiffening plates may in general be composite laminates which are represented, in turn, as either shear-deformable plates or classically thin plates. Details are given of the development of the elastic stiffness, geometric stiffness and consistent mass matrices of the finite strip models. The nature of the structural equations and the sophisticated procedure for their solution are discussed. Both vibrational and buckling applications of the developed analysis capability are presented and these include situations in which the corresponding mode shapes are of an overall nature and, for some of the buckling problems, of a highly localised nature.

A new proportioning method for member sections of single layer reticulated domes subjected to uniform and non-uniform loads

This paper presents a new proportioning method for member sections of single layer reticulated domes based on linear buckling stress obtained from linear buckling analysis using geometric stiffness matrices. This method includes elasto–plastic behaviors and instability of the members and the global shell buckling. To confirm the validity of this method, firstly roller supported single layer reticulated domes subjected to uniform load are designed by the method and then it is discussed how the domes satisfy the required load conditions, and lastly it is verified by the inelastic buckling analysis. Also the buckling load and strength of single layer reticulated domes under non-uniform loads are presented with emphasis on how to apply the buckling axial stress expressed by the generalized slenderness ratio of members.

MODELING WITH INCREASED EFFICIENCY AND VERSATILITY FOR FLEXURAL-TORSIONAL BUCKLING OF UNSYMMETRICAL THIN-WALLED STRUCTURES

Aiming at the performance-enhancement in coarse mesh modeling, we utilize a number of closed form solutions of a class of torsionally loaded thin-walled bars to formulate a two-noded element for spatial buckling analysis. The key in this relates to the use of the "exact" solution for the displacement fields (as oppose to the more conventional finite element approach where polynomial/Lagrangian-type interpolation is employed). That is, in addition to the well known "exact" solution for the coupled flexure/transverse-shear problem, we utilize a new "exact" solution for the more difficult case of coupled system of differential equations governing a torsionally loaded thin-walled beam using the higher-order theories of non-uniform twist/bi-moment with coupled warping-shear deformations. For the linear analysis, convergence and accuracy study indicated that the proposed model to be rapidly convergent, stable and computationally efficient; i.e. one element is sufficient to exactly represent an end loaded part of the beam. Such model has been extended to account for nonlinear analysis, in particular, the flexural torsional buckling of thin-walled structures. To this end, the effect of finite rotations in space is accounted for as per the modern theories of spatial buckling, resulting in second-order accurate geometric stiffness matrices. Compared with the classical theory of thin-walled structures, the present approach is more general in that all significant modes of stretching, bending, shear (due to both flexure and torsional/warping), torsion, and warping are accounted for. The inclusion of non-uniform torsion is accomplished through adoption of the principle sectorial area. This requires incorporation of a warping degree of freedom in addition to the conventional six degrees of freedom at each node. The element is derived for general cross sections including the Wagner-effect contributions. The model's properties and performance, particularly with regard to the resulting (significant) improvements in mesh accuracy, are assessed in a fairly complete set of numerical simulations.

Push-over analysis for performance-based seismic design

The paper presents a simple computer-based push-over analysis technique for performance-based design of building frameworks subject to earthquake loading. The technique is based on the conventional displacement method of elastic analysis. Through the use of a ‘plasticity-factor’ that measures the degree of plastification, the standard elastic and geometric stiffness matrices for frame elements (beams, columns, etc.) are progressively modified to account for nonlinear elastic–plastic behavior under constant gravity loads and incrementally increasing lateral loads. The behavior model accounts for material inelasticity due to both single and combined stress states, and provides the ability to monitor the progressive plastification of frame elements and structural systems under increasing intensity of earthquake ground motion. The proposed analysis technique is illustrated for two building framework examples.

Incrementally small-deformation theory for nonlinear analysis of structural frames

An incrementally small-deformation theory that is physically self-explainable is presented for the large-displacement nonlinear analysis of structural frames. Strictly based on the assumption of small strains, small rotations, and small displacements within each incremental step, the elastic and geometric stiffness matrices for the beam element are derived from the force–displacement relations. Due consideration is taken of the 3D rotational behavior of nodal moments. The geometric stiffness matrix derived for the element is asymmetric. However, by enforcing all the joints to remain in equilibrium in the deformed configuration, the antisymmetric parts of the geometric stiffness matrices cancel out, resulting in a symmetric stiffness matrix for the structure. Also described is the procedure for updating the element forces and geometry in an incremental-iterative analysis. The present approach in its entire set is demonstrated to be robust and efficient for solving the nonlinear, postbuckling response of structural frames.

Stability and Second-Order Analyses of Frames with Semirigid Connections under Distributed Axial Loads

The stiffness and geometric stiffness matrices and load vector of a prismatic beam-column with semirigid connections under any distributed axial load are presented. The derived matrices can be used in the stability and second-order analyses of 2D and 3D framed structures with sidesway uninhibited, partially inhibited, or totally inhibited in any type of construction. The effects of diagonal truss bracing and connection flexural stiffness on the lateral stability of frames are discussed. Four examples are included that show the simplicity and effectiveness of the proposed method in the analysis of 2D and 3D framed structures with semirigid connections subjected to concentrated and distributed axial loads. It is shown that the stability and second-order behavior of framed structures depend on (1) geometry and column layout; (2) size and type of bracing against sidesway; (3) type and degree of restraints at the column supports; (4) type of connection (i.e., degree of restraint) and the relative flexural stif...

Alternative Approach to Modify a Discrete Kirchhoff Triangular Element

In this paper, the existing modified discrete Kirchhoff triangular bending element is further modified, using the least-squares method, to enable the analysis of arbitrary thin plate with different material and geometric properties. In the vibration and buckling analyses, the combined mass and combined geometric stiffness matrices are employed to improve the calculations of natural frequency and buckling load. A comparison between the proposed element and some existing elements shows that the former is a high precision element for thin plate bending, vibration, and buckling analyses.

Overall and local buckling of sandwich plates with laminated faceplates, Part I: Analysis

After discussion of related past work, a description is given of a B-spline finite strip method (FSM) for predicting the buckling stresses of rectangular sandwich plates. The core is represented as a three-dimensional solid in which the in-plane displacements vary quadratically through the thickness whilst the out-of-plane displacement varies linearly. The faceplates may in general be composite laminates which are represented, in turn, as either shear-deformable plates or classically-thin plates. The displacement field of a finite strip requires the representation of 12 (for shear-deformable faceplates) or eight (for thin faceplates) fundamental quantities over the strip plan area, with each quantity expressed as a summation of a series of products of longitudinal B-spline functions and crosswise shape functions. Full details are given of the development of elastic stiffness and geometric stiffness matrices. The nature of the solution of the buckling problem is described and this incorporates multi-level sub-structuring, through the use of superstrips. This allows the efficient prediction of buckling stresses for both overall modes and highly-localised, wrinkling-type modes, but description of applications of the developed capability is deferred to the companion Part II paper.

Customizing the mass and geometric stiffness of plane beam elements by Fourier methods

Teaches by example the application of finite element templates in constructing high performance elements. The example discusses the improvement of the mass and geometric stiffness matrices of a Bernoulli‐Euler plane beam. This process interweaves classical techniques (Fourier analysis and weighted orthogonal polynomials) with newer tools (finite element templates and computer algebra systems). Templates are parameterized algebraic forms that uniquely characterize an element population by a “genetic signature” defined by the set of free parameters. Specific elements are obtained by assigning numeric values to the parameters. This freedom of choice can be used to design “custom” elements. For this example weighted orthogonal polynomials are used to construct templates for the beam material stiffness, geometric stiffness and mass matrices. Fourier analysis carried out through symbolic computation searches for template signatures of mass and geometric stiffness that deliver matrices with desirable properties when used in conjunction with the well‐known Hermitian beam material stiffness. For mass‐stiffness combinations, three objectives are noted: high accuracy for vibration analysis, wide separation of acoustic and optical branches, and low sensitivity to mesh distortion and boundary conditions. Only the first objective is examined in detail.

Double Nonlinear Analysis of the Loading Capacity of Drilling Derrick Steel Structures Containing Defects and Defacements

This paper presents a method of analyzing the loading capacity of drilling derrick steel structures containing defects and defacements in keeping with the current demand in the oil industry for evaluating safety and the loading capacity of derricks during their service lives. Adopting the three dimensional beam element as the basic element for FEM analysis, the problem of large deformation analysis is treated using the incremental tangent stiffness matrix method incorporating both the geometric and deformation stiffness matrices of the element, and the material nonlinearity is simplified as a concentrated inelasticity approach. Thus allowing for here two types of non-linearity analysis of the loading capacity of a perfect derrick structure has been studied. Applying a new Referential Uniform Reduction Method, special defects and defacements could then be introduced into the mathematical model of the structure, and a group of load-displacement curves representing different states of imperfections are obtained. The ultimate strength of the derrick in service can be determined finally by comparing the computed curves and the full-scale displacement test results. Two examples of different types of derrick structure are analyzed.

Analysis of self-weight induced collapse of long-deep beams

A numerical procedure is developed in this paper for analysing the collapse of long-deep thin-wall beam-type structures caused by distributed loads like their self-weights. It is based on the first-order shear deformation plate theory, and the critical load is determined by a linear buckling eigenvalue problem. A two-step analysis is involved in the procedure of which the first is to determine the pre-buckling in-plane stress distribution and the second is to solve an eigenvalue problem, which is formed with the structural stiffness and geometric stiffness matrices of which the latter is constructed with the obtained in-plane stress distribution. As an example, the load factor of the self-weight induced lateral–torsional buckling of a long-deep tee beam in lifting is calculated using the procedure.

An improved discrete Kirchhoff triangular element for bending, vibration and buckling analyses

In this paper, an improved triangular discrete Kirchhoff thin plate element IMDKT is introduced for bending, vibration and buckling analysis. In the case of bending analysis, new boundary displacements coupled with a correction factor are introduced in the proposed element for improving the accuracy. As for vibration and buckling analyses, the combined mass and combined geometric stiffness matrices are employed to improve the calculations of natural frequency and buckling load, respectively. Several numerical examples have been used to illustrate the versatility and potential accuracy of the present methods. A comparison between the proposed and some existing elements shows that the former is superior to the latter for thin plate bending, vibration and buckling analyses.

REFINED HIGHER ORDER FINITE ELEMENT MODELS FOR THERMAL BUCKLING OF LAMINATED COMPOSITE AND SANDWICH PLATES

Two refined higher order theories, one that neglects and the other that takes into account the effect of transverse normal deformation, are used to develop two discrete finite element models for the thermal buckling analysis of composite laminates and sandwiches. The two models, one with nine degrees of freedom per node and the other with eleven degrees of freedom, are based on a nine-node Lagrangian isoparametric element. The geometric stiffness matrices are developed by taking into consideration the effects of the higher order terms on the initial in-plane and transverse shear stresses. The accuracy of the present formulations is first evaluated by analyzing sample problems for which analytical three-dimensional solutions exist in the literature. Numerical results are presented for the first time for sandwich plates, demonstrating the importance and accuracy of the higher order theory in comparison to first-order theory. Some new results are also given for sandwich plates with angle-ply composite face sheets, showing the effects of various boundary conditions and of variations in geometric and lamina tion parameters on critical temperature.

Improved formulation for spatial stability and free vibration of thin-walled tapered beams and space frames

A consistent finite element formulation is presented for the free vibration and spatial stability analysis of thin-walled tapered beams and space frames. The kinetic and potential energies are derived by applying the extended virtual work principle, introducing displacement parameters defined at the arbitrarily chosen axis and including second order terms of finite semitangential rotations. As a result, the energy functional corresponding to the semitangential rotation and moment is obtained, in which the elastic strain energy terms are coupled due to axial–flexural–torsional coupling effects but the potential energy due to initial stress resultants has a relatively simple expression. For finite element analysis, cubic polynomials are utilized as the shape functions of the two-noded Hermitian space frame element. Mass, elastic stiffness, and geometric stiffness matrices for the unsymmetric thin-walled cross-section are precisely evaluated, and load-correction stiffness matrices for off-axis concentrated and distributed loadings are considered. In order to illustrate the accuracy and practical usefulness of this formulation, finite element solutions for the free vibration and lateral–torsional buckling problems of thin-walled tapered beam-columns and space frames are presented and compared with available solutions.

Finite Element-Based Analytic Shape Sensitivities of Local and Global Airframe Buckling Constraints

An examination of available shell finite elements suitable for buckling analysis of thin walled airframe structures leads to the selection of a simple, accurate, design-oriented element, which is, then, used with slight modifications to obtain explicit, closed form equations for the stiffness and geometric stiffness matrices. In turn, these equations are used to derive explicit expressions for the analytic sensitivities of stiffness and geometric stiffness matrices with respect to shell shape design variables. With analytic shape sensitivities of structural matrices and corresponding buckling eigenvalues at hand, the resulting new computer capability makes it possible to construct buckling constraint approximations for Approximation-Concepts based structural synthesis, as well as to examine sources of numerical noise which might appear when parametric studies or finite difference sensitivities are carried out using existing FE codes. The simplicity of the shell elements used and the elimination of the need to carry out numerical integration, lead to computational savings, especially when repetitive analyses have to be carried out during shape design optimization of typical airframes.The new capability is aimed at capturing both local and global modes of buckling failure with the same FE model. Subcomponent interaction during buckling can, thus, be taken into account during shape optimization of wing and fuselage structures. Numerical tests involving plates, thin walled channel sections and a complete wing box of a typical fighter airplane demonstrate the effectiveness and accuracy of the new design-oriented capability.

ANALYTIC SENSITIVITIES IN COMPOSITE WING-SHAPE SYNTHESIS PART ONE.: PANEL BUCKLING CONSTRAINTS

Equivalent plate modeling techniques based on Ritz analysis with simple polynomials proved to be efficient tools for structural modeling of wings in the preliminary design stage. This paper describes a formulation of wing equivalent plate modeling in which it is simple to obtain analytic, explicit expressions for terms of the stiffness and geometric stiffness matrices without the need to perform numerical integration. The explicit expressions for the terms of the stiffness and geometric stiffness matrices are used for the buckling analysis of trapezoidal fiber composite wing skin panels. The formulation based on Ritz analysis using simple polynomials leads to explicit (axpressions for the analytic sensitivities of the stiffness and geometric stiffness matrices with respect to layer thickness, fiber orientations, and panel shape. Integration of the resulting buckling analysis based on the equivalent plate approach is discussed. To assess the new capability, test cases were chosen to address the convergence rate with increased polynomial order, the accuracy of analysis results for panels of general shape and the integration of wing box structural analysis with panel buckling analysis for fast structural analysis of airplane wings.