Geometric Stiffness Matrices Research Articles

Dynamic stability behavior of damped laminated beam subjected to uniformly distributed subtangential forces

Abstract The dynamic stability behavior of damped laminated beam with various boundary conditions subjected to the uniformly distributed subtangential forces is investigated using the finite element formulation. The formal engineering approach of the mechanics for the thin-walled laminated beam based on kinematic assumptions consistent with Vlasov beam theory is used. An extended Hamilton’s principle is employed to obtain the mass-, damping-, elastic stiffness-, geometric stiffness matrices, and the load correction stiffness matrix due to the subtangential forces, respectively. The method for the evaluation of critical values for divergence and flutter of the nonconservative systems is briefly introduced in case of considering and neglecting damping effects. Throughout numerical examples, the influence of various parameters on the dynamic stability behavior of the nonconservative laminated beam is newly investigated: (1) the variation of the divergence and flutter loads due to the nonconservativeness with respect to the fiber orientation, (2) the effect of boundary condition on the instability region of the divergence-flutter system, and (3) the influence of external and internal damping on the flutter load.

A New Method for Analysing Large Elasto-Plastic Deformations of a Thin Cosserat Shell

One of the best approaches for modelling the large deformation of shells is the Cosserat surface; however, the finite-element implementation of this model suffers from membrane and shear locking, especially for very thin shells. If the director vector is constrained to remain perpendicular to the mid-surface, during deformation, locking will be prevented. This constraint is in fact a limiting analysis of the Cosserat theory in which Kirchhoff's hypothesis is enforced. This has been considered for the first time. Simo's plastic approach is modified to implement the constrained director. This model includes both kinematic and isotropic hardening behaviours. A consistent elasto-plastic tangent modular matrix is extracted. Numerical solution is performed by interpolation of displacement on the whole domain, and a hierarchical finite-element scheme is developed. The principle of virtual work is used to obtain the weak form of the governing differential equations and the material and geometric stiffness matrices are derived through a linearization process. The validity and the accuracy of the method are illustrated by numerical examples.

GBT buckling analysis of thin-walled steel frames: A state-of-the-art report

This paper presents a state-of-the-art report on the use of Generalised Beam Theory (GBT) to assess the buckling behaviour of plane and space thin-walled steel frames. After a very brief overview of the main concepts and procedures involved in performing a GBT buckling analysis, one addresses the development and numerical implementation of a GBT-based beam finite element formulation that is able (i) to unveil local, distortional and global buckling modes, (ii) to handle arbitrary loadings (namely those causing non-uniform member internal force and moment diagrams) and (iii) to incorporate the presence of several frame joint configurations and arbitrary end and/or intermediate support conditions (including those associated with the modelling of bracing systems). In particular, one describes the procedures employed to establish the frame linear and geometric stiffness matrices – special attention is paid to the constraint conditions adopted to ensure the local displacement compatibility at the frame joints. The paper closes with the presentation and discussion of a number of numerical results that make it possible to illustrate the application and show the potential of the GBT-based approach to perform frame buckling analyses – they concern both plane and space frames. In order to validate and assess the numerical efficiency and accuracy of the GBT analyses and results (critical buckling loads and mode shapes), the frames are also rigorously analysed in the commercial code A nsys – both the members and joints are discretised by means of fine shell finite element meshes .

Nonlinear dynamic analysis of frames with plastic hinges at arbitrary locations

AbstractThis paper presents a method for nonlinear dynamic analysis of frames subjected to distributed loads, which is based on the semi‐rigid technique and moving node strategy. The plastic hinge is modelled as a pseudo‐semi‐rigid connection with nonlinear hysteretic moment–curvature characteristics at element ends. The stiffness matrix with material and geometric nonlinearities is expressed as a sum of products of the standard and geometric stiffness matrices with their corresponding correction matrices based on the plasticity‐factors developed from the section flexural stiffness at the plastic hinge locations. Each beam member is modelled by two elements. The moving node strategy is applied to the intermediate node to track the exact location of any intermediate plastic hinge that may be formed. Equilibrium iterations and geometry updating are carried out in every time step. Stiffness degradation is adopted to describe the deterioration of plastic hinges, and the effects of various parameters in the degradation model are evaluated. Examples are used to illustrate the applicability and excellent performance of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.

Stability analysis of frame structures by bubble-function method

In the stability analysis of frame structures, the results by conventional finite element method (FEM) in which one member is taken as one element are sometimes unavailable. This paper took a new basic function system with bubble functions as the shape function of a bar element to develop a bubble function finite element method (BFEM), in which the bending and the geometric stiffness matrices were derived from the principle of virtual work. Bubble functions are finite element modes that are located entirely within a single element and are zero on boundaries of the element, but are nonzero at the other points. BFEM is as concise as conventional bar FEM but has better accuracy, and is adaptable to the buckling analysis of all kinds of frame structures. The use of bubble functions significantly improves the convergence of finite element analysis, and efficiently reduces the computation cost for the buckling analysis of frame structures. Numerical results show that using bubble functions in finite element for the stability analysis of structures is very efficient, especially for high-rise and large-scale frame structures.

Nonlinear buckling analysis of hygrothermoelastic composite shell panels using finite element method

Hygrothermal stresses due to the change in environmental condition may induce buckling and dynamic instability in the composite shell structures. In the present investigation, the hygrothermoelastic buckling behavior of laminated composite shells are numerically simulated using geometrically nonlinear finite element method. The orthogonal curvilinear coordinate is used for modeling a general doubly curved deep or shallow shell surface. The geometrically nonlinear finite element formulation is based on general nonlinear strain–displacement relations in the orthogonal curvilinear coordinate system. The present theory can be applicable to thin and moderately thick shells. The mechanical linear and nonlinear stiffnesses, and the nonmechanical nonlinear geometric stiffness matrices and the hygrothermal load vector are presented. It is also observed that during the present numerical solution of nonlinear equilibrium equation, in order to construct the nonlinear stiffness matrices for the first load step, the initial deformation can be assumed as zero or any computer generated small random number or the properly scaled fundamental buckling mode shape. To verify the present formulations and finite element code, the present results are compared well with those available in the open literature. Parametric studies such as thickness ratio and shallowness ratio on buckling are performed for spherical, truncated conical and cylindrical composite shell panels. The buckling behavior and deflection shapes are characterized by multiple wrinkles along unreinforced direction at higher moisture concentrations or temperature rise.

Elastic beam finite element with an arbitrary number of transverse cracks

This paper formulates the finite element of a beam with an arbitrary number of transverse cracks. The derivations are based on a simplified computational model, where each crack is replaced by a corresponding linear rotational spring, connecting two adjacent elastic parts. The stiffness and geometrical stiffness matrices thus take into account the effect of flexural bending deformation caused by the presence of the cracks. The expressions for calculating the coefficients of stiffness and geometrical stiffness matrices, as well as the load vector of the element, are presented in closed forms. Since the corresponding interpolation functions were implemented in the derivations, transverse displacements within the finite element can also be obtained. Due to the fact that the number of parameters describing the cracked beam's structure is thus reduced to its minimum, it can be expected that this element could be efficiently implemented, not only in static and stability analysis, but also in inverse identification of cracks in beam-like structures.

Energy equations for elastic flexural–torsional buckling analysis of plane structures

Lateral–torsional buckling is a critical mode of failure of metal structures. When the values of the loadings on a member of a structure reach a limiting state, the member will experience out-of-plane bending and twisting. This type of failure occurs suddenly in members with a much greater in-plane bending stiffness than torsional or lateral bending stiffness. Slender members of a structural system may buckle laterally and twist before their in-plane capabilities can be reached. Energy equations are derived by considering the total potential energy of a beam-column element. The second variation of the total potential energy equal to zero indicates the transition from a stable state to an unstable state, which is the critical condition for buckling. Several energy equations are derived analytically by calculating the second variation of the total potential energy of a double symmetric thin wall beam-column element. In this article, in-plane deformations of the beam-column element are disregarded. Then energy equations are derived expressing in dimensional and non-dimensional forms. These energy equations will be implemented in a future article to derive elastic and geometric stiffness matrices for the beam-column element and calculate the lateral–torsional buckling of plane structures. Examples are provided to show the accuracy of the equations and applications.

Symmetric prismatic tensegrity structures. Part II: Symmetry-adapted formulations

This paper presents analytical formulations for the symmetry-adapted equilibrium, force density and geometrical stiffness matrices for prismatic tensegrity structures with dihedral symmetries. An analytical expression for the infinitesimal mechanisms is also derived.

Dynamic stability of shear-flexible beck’s columns based on Engesser’s and Haringx’s buckling theories

Based on Engesser’s and Haringx’s buckling theories for shear-flexible columns, dynamic stability behaviors of damped Beck’s columns subjected to sub-tangential follower forces are intensively investigated using fifth-order Hermitian beam elements. For this purpose, dimensionless equations of motion for the non-conservative systems are firstly derived based on the buckling approaches of Engesser and Haringx. FE procedures using the shear-flexible Hermitian interpolation functions are next presented by obtaining the mass matrix, the internal and external damping matrices, the elastic and the geometric stiffness matrices, and the load correction stiffness matrices due to sub-tangential follower forces, respectively. Evaluation schemes for flutter and divergence loads of non-conservative systems are described and the static buckling loads and natural frequencies of the shear-flexible beam-columns are compared through numerical examples using the two buckling theories. Finally, the influences of sub-tangentiality, shear deformation and Rayleigh damping on the dynamic stability of the non-conservative system are investigated and compared based on two buckling approaches.

Sub-tangentially loaded and damped Beck's columns on two-parameter elastic foundation

The dynamic stability of the damped Beck's column on two-parameter elastic foundation is investigated by using Hermitian beam elements. For this purpose, based on the extended Hamilton's principle, the dimensionless finite element (FE) formulation using the Hermitian interpolation function is presented. First, the mass matrix, the external and internal damping matrices, the elastic and the geometric stiffness matrices, Winkler and Pasternak foundation matrices, and the load correction stiffness matrix due to the sub-tangential follower force are obtained. Then, evaluation procedure for the flutter and divergence loads of the non-conservative system and the time history analysis using the Newmark- β method are shortly described. Finally, the influences of various parameters on the dynamic stability of non-conservative systems are newly addressed: (1) variation of the second flutter load due to sub-tangentiality, (2) influences of the external and the internal damping on flutter loads by analysis of complex natural frequencies, (3) the effect of the growth rate of motion in a finite time interval using time history analysis, and (4) fluctuation of divergence and flutter loads due to Winkler and Pasternak foundations.

An improved formulation of space stiffeners

This paper presents a displacement based finite element model for predicting the constraint torsion effect of stiffeners. In structural modelling, the plate/shell and the stiffeners are treated as separate elements where the displacement compatibility transformation between these two types of elements takes into account the constraint torsional warping effect in the stiffeners. The development is based on a general beam theory which includes flexural-torsion coupling, constrained torsion warping, and shear-centre location. The virtual work principle includes the second order terms of finite beam rotations. For finite element analysis, cubic Hermitian polynomials are used as shape functions of the straight space frame element with two nodes. Elastic stiffness and geometric stiffness matrices for an arbitrary cross-section are evaluated in a closed form, and load correction stiffness for eccentric stiffener loads are considered. To demonstrate the importance of torsion warping constraints and to illustrate the accuracy of this formulation, finite element solutions are presented and compared with available solutions.

Non-linear analysis of arbitrary quadrilateral plates by use of Kirchhoff-Love theory

Very large displacement but small strain of a very thin quadrilateral plate is studied using Kirchhoff-Love theory. The numerical investigation is based on the mapping of the quadrilateral plate onto the computational coordinates of a standard square and interpolation of the displacement field over the whole domain with no director assignment. The present investigation, which is basically a limiting analysis of the Cosserat’s theory, enforces the well known Kirchhoff’s hypothesis which denies the existence of shear strain in the direction of the plate’s thickness. After forming the decoupled nonlinear equations, the material and geometric tangential stiffness matrices are derived through a linearization process and different stages of the problem solution are presented. Finally through certain numerical examples and comparison of the results with some existing researches the validity and the accuracy of the present method are verified.

Stiffness Matrices for Linear and Buckling Analyses of Composite Beams with Partial Shear Connection

AbstractThis paper is concerned with linear and buckling analyses of composite beams with partial shear connection (partial composite beams) using the finite element method. Two elements derived from different types of shape functions are proposed in this study. The first element, referred to as exact, is based on the exact shape functions obtained by solving the differential equations governing the transverse displacement and the slip of the shear connector layer of a partial composite beam. The second element, referred to as approximate, is based on the conventional linear and cubic shape functions used in conventional axial and beam elements. By making use of these two types of shape functions, the elastic and geometric stiffness matrices can be derived explicitly from the strain energy and the load potential, respectively. Both types of elements can be used to carry out linear static and buckling analyses. As expected, the exact element is more accurate than the approximate element if the same discretization is adopted. However, the approximate element has the advantage of easy implementation since the expressions of its elastic and geometric stiffness matrices are very simple. Also, the solutions obtained from the approximate element converge very fast; with reasonable discretization, say 8 elements per member, very accurate solutions can be obtained.

Stability conditions of prestressed pin-jointed structures

Pin-jointed structures are first classified to trusses, tensile structures, and tensegrity structures in view of their respective stability properties. A sufficient condition for stability of an equilibrium state is derived for tensegrity structures. The condition is based on the bilinear forms of the linear and geometrical stiffness matrices considering the flexibility of members. The stability is defined by the positive definiteness of the tangent stiffness matrix, whereas the definition of prestress-stability is based on the geometrical stiffness matrix and the infinitesimal mechanisms. Numerical examples verify that the so-called super-stability condition might not be satisfied by a stable tensegrity structure, and that a prestress-stable structure can be unstable if the prestresses are moderately large.

Stability conditions for tensegrity structures

Stability conditions for tensegrity structures are derived based on positive definiteness of the tangent stiffness matrix, which is the sum of the linear and geometrical stiffness matrices. A necessary stability condition is presented by considering the affine motions that lie in the null-space of the geometrical stiffness matrix. The condition is demonstrated to be equivalent to that derived from the mathematical rigidity theory so as to resolve the discrepancy between the stability theories in the fields of engineering and mathematics. Furthermore, it is shown that the structure is guaranteed to be stable, if the structure satisfies the necessary stability condition and the geometrical stiffness matrix is positive semidefinite with the minimum rank deficiency for non-degeneracy.

Simplified buckling analysis of skeletal structures

A simplified approach is proposed for buckling analysis of skeletal structures, which employs a ‘rotational spring analogy’ for the formulation of the geometric stiffness matrix. The benefit of this analogy is that it offers an intuitive framework, which is based on the common notions of linear structural analysis. Assuming that the structural deflections prior to buckling are negligible, a linear eigenvalue problem, utilising the geometric and material stiffness matrices, can be easily formulated and solved for the critical buckling loads. This can be further simplified using an assumed mode, where the rotational spring analogy is shown to provide considerable computational benefits and significant insight into the buckling of various forms of skeletal structure. In this context, the use of different assumed modes can be conceived as a process of probing the structure to establish the most likely mode for buckling and the corresponding critical load. It is also shown that the approximation inherent in the assumed mode approach together with the discrete form of the rotational spring analogy can be significantly improved through modal combinations and increasing the number of elements, respectively, where convergence to the exact buckling solution is demonstrated. Several illustrative examples are provided in this paper, which highlight the simplicity of the proposed approach, its application using a linear structural analysis tool and its ability to shed significant light on important issues in buckling analysis of skeletal structures.

Buckling mode decomposition of single-branched open cross-section members via finite strip method: Application and examples

The objective of this paper is to provide implementation details of, and practical examples for, modal decomposition of the cross-section stability modes of thin-walled members by constraining a traditional finite strip method (FSM) solution. The theoretical development of the proposed method is provided in a companion to this paper [Ádány S, Schafer BW. Buckling mode decomposition of single-branched open cross-section members via finite strip method: derivation. Thin-walled Structures, submitted for publication, companion to this paper.] The constraint matrix, which is directly applied to the elastic and geometric stiffness matrices of a traditional FSM solution in order to constrain the deformations, is provided along with all formulae necessary in its construction. In addition, a completely worked out numerical example is provided to aid in implementing the constrained FSM solution. The authors implemented the constrained FSM in the open source program CUFSM. This modified version of CUFSM is then used to provide a series of numerical examples that illustrate (i) the advantages of performing modal decomposition, (ii) the importance of understanding and defining the deformation fields related to a desired mode, and (iii) the behavior of constrained FSM stability solutions compared with classical analytical solutions, GBT, and unconstrained FSM. Decomposition of the cross-section buckling classes related to global and distortional modes is demonstrated. Further, the impact of how to select the deformation fields and perform modal decomposition for cross-section stability modes within a class, e.g., for the traditional three global modes (weak-axis flexure, strong-axis flexure and flexural-torsional buckling), is explored and the impact of the deformation field definitions demonstrated. Comparisons of the constrained FSM solutions with other available solutions demonstrate the importance of properly determining when beam theory and plate theory should apply to the cross-section stability of thin-walled members.

Stability of tapered I-Beams under torsional moments

Previous investigations on the buckling of torsionally loaded bars indicate that the critical loads depend upon the mechanical device that generates the end torque. By representing the torsional moment on an I-section as the combination of St. Venant and warping torsions, this study presents a finite element model for buckling analysis of tapered I-beams subjected to torsional moments. Regarding a tapered I-beam as the formation of three tapered narrow beams (the top flange, web, and the bottom flange), one can directly express the total potential energy equation of the tapered I-beam by summing up all the potential energy equations of these narrow beams based on Yang et al.'s well-developed potential equation of a tapered solid beam. As a result, the linear elastic and geometric stiffness matrices of a torsionally loaded I-beam element with tapered cross-section can be established from the total potential energy derived herein. It is worthy mentioning that the second-order effects of warping moments and non-uniform torsions have been included in the present geometric stiffness matrix. By buckling analysis, the numerical examples indicate that the warping and St. Venant torsions may significantly affect the flexural buckling resistance of tapered I-beams against torsional loadings. Besides, the critical loads of the tapered I-beams with higher warping rigidity would be reduced by the instability effect of warping moments.

Very Large Deformation but Small Strain Analysis of Plates and Folded Plates by Finite Strip Method

In this paper a Finite strip method is developed to analyze very large deformations of thin plates and folded plates by use of the elastic Cosserat theory. The principle of virtual work is exploited to present the weak form of the governing differential equations. Through a linear mapping, a rectangular strip 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. The geometric and material tangential stiffness matrices are formed through linearization, and a step by step procedure is presented to complete the scheme. The validity and the accuracy of the method are illustrated through certain numerical examples and comparison of the results with other researches. The method is shown to be capable of handling numerical analysis of plates experiencing very large deformations.