Buckling, flutter and vibration analyses of beams by integral equation formulations

This paper presents a model for the investigation of buckling, flutter and vibration analyses of beams using the integral equation formulation. A mathematical formulation based on Euler–Bernoulli beam theory is presented for beams with variable sections on elastic foundations and subjected to lateral excitation, conservative and non-conservative loads. Using the boundary element method and radial basis functions, the equation of motion is reduced to an algebraic system related to internal and boundary unknowns. Eigenvalue problems related to buckling and vibrations are formulated and numerically solved. Buckling loads, natural frequencies and associated eigenmodes are computed. The corresponding slope, bending and shear forces can be directly obtained. The load-frequency dependence is investigated for various elastic foundations and the divergence critical loads are evidenced. Under non-conservative loads, a dynamic stability analysis is illustrated numerically based on the coalescence of eigenfrequencies. The flutter load and instability regions with respect to the elastic concentrated and distributed foundations are identified. Using the eigenmodes, numerically computed, non-linear vibrations of beams are investigated based on one mode analysis. The presented model is quite general and the obtained numerical results are in agreement with available data.     

A simple method to impose rotations and concentrated moments on ANC beams

Recently introduced ANC beam elements furnish a simple formulation that allows to solve nonlinear problems of beams, including those with large displacements and strains, as well as complex nonlinear (inelastic) materials. The success and simplicity of these finite elements is mainly due to the fact that the only nodal degrees of freedom that they employ are displacements, and rotations are thus completely avoided. This in turn makes it very difficult to apply concentrated moments or to impose rotations at specific nodes of a finite element mesh. In this article, we present a simple enhancement to this beam formulation that allows to apply these two types of boundary conditions in a simple manner, making ANC beam elements more versatile for both multibody and structural applications.     

Inelastic distortional buckling of I-beams

A finite element method of analysis is developed for the inelastic distortional buckling of determinate, hot-rolled I-beams. The method permits an economic computer analysis to be made of inelastic distortional buckling of members under various conditions of loading, end support and restraint. Following studies of the accuracy and convergence of the method, its scope of application is demonstrated by studing the inelastic buckling of simply supported beams, beams on seats and beams with complete and continuous tension flange restraint.     

A study of finite element and mathematical programming models for inelastic analysis of concrete framed structures

This paper presents an investigation of three analytical models for inelastic analysis of reinforced concrete framed structures. The first model employs a plane stress reinforced concrete element formulation with reinforcement oriented in any direction. The second model is based on a simplified layered frame element. The third model is a mathematical programming model in which the inelastic analysis is formulated as a linear complementarity problem. Formulation and significant computational steps of the three models are explained briefly. The performance of the three models in predicting the bending moment distribution at ultimate load stages is discussed through application to a number of frames and the results are compared with the experimental investigation conducted on 14 frames. This study concludes that the mathematical programming model is a computationally efficient model for inelastic analysis and offers scope for practical application for limit-state design of concrete frames.     

A finite element method for elastic-plastic beams and columns at large deflections

A finite element scheme for the large-displacement analysis of elastic-plastic beams and columns is presented. The proposed method, which assures continuous deflections and continuous slopes at the element junctions, is shown to furnish fairly accurate results with a minimal number of elements. Comparisons are made with existing results for laterally loaded beams on elastic foundations and for elastic columns on bi-linear elastoplastic foundations. The effect of imperfections on the buckling of elastic-plastic columns is also investigated.     

Dynamic inelastic structural analysis by boundary element methods

Summary Boundary element methodologies for the determination of the response of inelastic two-and three-dimensional solids and structures as well as beams and flexural plates to dynamic loads are briefly presented and critically discussed. Elastoplastic and viscoplastic material behaviour in the framework of small deformation theories are considered. These methodologies can be separated into four main categories: those which employ the elastodynamic fundamental solution in their formulation, those which employ the elastostatic fundamental solution in their formulation, those which combine boundary and finite elements for the creation of an efficient hybrid scheme and those representing special boundary element techniques. The first category, in addition to the boundary discretization, requires a discretization of those parts of the interior domain expected to become inelastic, while the second category a discretization of the whole interior domain, unless the inertial domain integrals are transformed by the dual reciprocity technique into boundary ones, in which case only the inelastic parts of the domain have to be discretized. The third category employs finite elements for one part of the structure and boundary elements for its remaining part in an effort to combine the advantages of both methods. Finally, the fourth category includes special boundary element techniques for inelastic beams and plates and symmetric boundary element formulations. The discretized equations of motion in all the above methodologies are solved by efficient step-by-step time integration algorithms. Numerical examples involving two-and three-dimensional solids and structures and flexural plates are presented to illustrate all these methodologies and demonstrate their advantages. Finally, directions for future research in the area are suggested.     

Finite element analysis of two-dimensional electrochemical–mechanical response of ionic conducting polymer–metal composite beams

Attention has been focused on ionic conducting polymer–metal composites (IPMCs) as intelligent materials for artificial muscles and robotics for recent years. The two-dimensional finite element formulation based on Galerkin method is conducted for the basic field equations governing electrochemical response of IPMC beams with two pairs of electrodes upon applied electric field. The three-dimensional finite element analysis is also conducted for the deformation of IPMC beams due to water redistribution in the beams associated with the electrochemical response. Some numerical studies are carried out in order to show the validity of the present formulation.     

Finite element analysis of 3-D beam-columns on a nonlinear foundation

A numerically integrated finite element for the analysis of three-dimensional, nonlinear Winkler foundations is developed. Together with a linear beam-column element they are used to solve problems of beams, columns and beam-columns on linear and nonlinear foundations in two and three dimensions. The Newton-Raphson method is used in the solution. Excellent results are illustrated by comparisons with available solutions.     

Rigid body concept for geometric nonlinear analysis of 3D frames, plates and shells based on the updated Lagrangian formulation

In the nonlinear analysis of elastic structures, the displacement increments generated at each incremental step can be decomposed into two components as the rigid displacements and natural deformations. Based on the updated Lagrangian (UL) formulation, the geometric stiffness matrix [k_g] is derived for a 3D rigid beam element from the virtual work equation using a rigid displacement field. Further, by treating the three-node triangular plate element (TPE) as the composition of three rigid beams lying along the three sides, the [k_g] matrix for the TPE can be assembled from those of the rigid beams. The idea for the UL-type incremental-iterative nonlinear analysis is that if the rigid rotation effects are fully taken into account at each stage of analysis, then the remaining effects of natural deformations can be treated using the small-deformation linearized theory. The present approach is featured by the fact that the formulation is simple, the expressions are explicit, and all kinds of actions are considered in the stiffness matrices. The robustness of the procedure is demonstrated in the solution of several benchmark problems involving the postbuckling response.     

A discrete layer beam finite element for the dynamic analysis of composite sandwich beams with integral damping layers

A discrete layer finite element is presented for the dynamic analysis of laminated beams. The element uses C⁰ continuous linear and quadratic polynominals to interpolate the in-plane and transverse displacement field, respectively, and is free from the effects of shear locking. Modal frequencies and damping are estimated using both the modal strain energy method and the complex modulus method. A forced response version of the model is also presented. The model predictions are compared with experimental data for composite sandwich beams with integral damping layers. Four damping configurations are considered, a constrained layer treatment, a segmented constrained layer treatment and two internal treatments.     

Non-linear finite element analysis of composite beams with interlayer slips

This article presents a new non-linear finite element formulation for the analysis of two-layer composite plane beams with interlayer slips. The element is based on the corotational method. The main interest of this approach is that different linear elements can be automatically transformed to non-linear ones. To avoid curvature locking that may occur for low order element(s), a local linear formulation based on the exact stiffness matrix is used. Five numerical applications are presented in order to assess the performance of the formulation.     

Corotational mixed finite element formulation for thin-walled beams with generic cross-section

The corotational technique is adopted here for the analysis of three-dimensional beams. The technique exploits the technology that applies to a two-noded element, a coordinate system which continuously translates and rotates with the element. In this way, the rigid body motion is separated out from the deformational motion. In this paper, a mixed formulation are adopted for the derivation of the local element tangent stiffness matrix and nodal forces. The mixed finite element formulation is based on an incremental form of the two-field Hellinger–Reissner variational principle to permit elasto-plastic material behavior. The local beam kinematics is based on a low-order nonlinear strain expression using Bernoulli assumption. The present formulation captures both the Saint–Venant and warping torsional effects of thin-walled open cross-sections. Shape functions that satisfy the nonlinear local equilibrium equations are selected for the interpolation of the stress resultants. In particular, for the torsional forces and the twist rotation degree of freedom, a family of hyperbolic interpolation functions is adopted in lieu of conventional polynomials. Governing equations are expressed in a weak form, and the constitutive equations are enforced at each integration cross-section along the element length. A consistent state determination algorithm is proposed. This local element, together with the corotational framework, can be used to analyze the nonlinear buckling and postbuckling of thin-walled beams with generic cross-section. The present corotational mixed element solution is compared against the results obtained from a corotational displacement-based model having the same beam kinematics and corotational framework. The superiority of the mixed formulation is clearly demonstrated.     

Free vibrations of coupled walls by transfer matrices and finite element modelling of joints

A transfer matrix analysis of coupled shear walls is combined with finite element models for joint flexibility. Every joint is considered as a single rectangular element with a displacement model that is consistent with the kinematic constraints imposed by the adjacent beams. The derivation of the element stiffness is based on plane stress elasticity. The overall problem can be solved for the static and free vibrational response of coupled shear walls on flexible foundations. Assessment of its performance and comparison with simpler models for joint deformation is accomplished through the determination of natural frequencies and modes of vibration for a range of previously tested or analysed specimens. Having established the accuracy of the proposed high-order finite element joint model, the effect of joint flexibility on higher frequencies and modes of vibration is then assessed.     

Time-dependent creep and shrinkage analysis of composite beams curved in-plan

This paper develops a numerical formulation for the time-dependent creep and shrinkage analysis of steel–concrete composite beams that are curved in-plan under conditions of service load. The creep behaviour of the concrete is considered by using the viscoelastic Wiechert model, in which the aging effect of the concrete is taken into account. The curved composite beam model that is developed also accounts for the partial shear interaction at the deck-girder interface in the tangential (or longitudinal) direction, as well as in the radial (or horizontal) direction, due to the flexibility of the shear connectors. Models based on the developed formulation are validated by comparisons with sophisticated and computationally intensive ABAQUS shell element models, and with available results reported in the literature. The effects of initial curvature and partial interaction on the time-dependent behaviour of curved composite beams are also illustrated in the examples.     

Topology optimization of beam cross-section considering warping deformation

The beam cross-section optimization problems have been very important as beams are widely used as efficient load-carrying structural components. Most of the earlier investigations focus on the dimension and shape optimization or on the topology optimization along the axial direction. An important problem in beam section design is to find the location and direction of stiffeners, for the introduction of a stiffener in a closed beam section may result in a topologically different configuration from the original; the existing section shape optimization theory cannot be used. The purpose of this paper is to formulate a section topology optimization technique based on an anisotropic beam theory considering warping of sections and coupling among deformations. The formulation and corresponding solving method for the topology optimization of beam cross-sections are proposed. In formulating the topology optimization problem, the minimum averaged compliance of the beam is taken as objective, and the material density of every element is used as design variable. The schemes to determine the rigidity matrix of the cross-sections and the sensitivity analysis are presented. Several kinds of topologies of the cross-section under different load conditions are given, and the effect of load condition on the optimum topology is analyzed.     

Finite element method and limit analysis theory for soil mechanics problems

This paper deals with a finite element formulation of problems of limit loads in soil mechanics via limit analysis theory. After recalling the principal results of this theory, the authors describe a numerical formulation for both the static and kinematic approaches of the ultimate load. Thanks to linearization of the yield criterion, the finite element model leads to a linear programming problem. The efficiency of the two proposed computing procedures is demonstrated by their application to the problem of pulling out of foundations and slope stability.     

Free vibration analysis of beams on elastic foundation

A simple finite element method is developed and applied to treat the free vibration analysis of beams supported on elastic foundations. The entire analysis is programmed to run on a microcomputer and with few elements modelling the beam, gives quick and reliable results. Numerical examples pertaining to the free vibration of beams in some special situations are considered, such as a stepped beam on an elastic foundation, beam on a stepped elastic foundation and a continuous beam on an elastic foundation. Present results compare very well with those obtained from existing solutions, wherever possible.     

Finite element analysis for time-dependent inelastic material behavior

A representation of inelastic, time-dependent material behavior, due to Bodner and Partom, in which both elastic and inelastic deformations are considered to be present at all stages of loading and unloading, is shown to be well-suited to structural analyses using finite element modeling techniques and high speed digital computation methods. The formulation considerably simplifies the computational logic for non-monotonic and cyclic loading problems since no special unloading criteria or yield conditions are required. Examples demonstrating strain-rate sensitivity, work-hardening, and reversed loading behavior are given for problems in the small strain range. Experimental results for a titanium tensile specimen subjected to changes in crosshead velocity are compared with predictions based on a plane stress finite element model. Numerical analyses using axisymmetric, solid of revolution finite element models are presented for unstiffened and ring-stiffened cyclindrical shells subjected to time-dependent external pressure.     

Implicit dynamic three-dimensional finite element analysis of an inelastic biphasic mixture at finite strain: Part 1: Application to a simple geomaterial

Based on the mixture theory formulation for a fluid-saturated, inelastic, pressure-sensitive porous solid subjected to dynamic large strain deformation, a 3D finite element implementation with implicit time integration is presented. A recently published 2D implementation [Li et al. 2004, CMAME, v193, p3837–70] is extended to 3D, porosity-dependent permeability, and pressure-sensitive inelastic solid skeleton response at finite strain. The Clausius-Duhem inequality provides the form of the constitutive equations for the solid and fluid phases, as well as the dissipation function. A non-associative Drucker-Prager cap-plasticity model at finite strain is formulated based on a multiplicative decomposition of the deformation gradient, and numerically integrated semi-implicitly in the intermediate configuration to avoid questions of incremental objectivity. The elastic implementation is verified with available 1D analytical and 2D benchmark problems. New numerical solutions for 3D large strain dynamic behavior of saturated inelastic porous media are presented. The computational efficiency of the implemented formulation in achieving quadratic convergence is illustrated.     

Inelastic stability of laterally unsupported i-beams

A computer method to study the inelastic stability of laterally unsupported steel I-beams and based on a general non-linear theory is presented.Traditionally, the problem of flexural-torsional stability of beams is treated as a lateral buckling problem. Some of the draw-backs of these earlier studies are given below:The classical theory assumes that the deformations are small. In addition the deformation field is linearized. This theory is therefore valid only when the major axis flexural rigidity is much greater than its minor axis rigidity, so that deformations before the onset of lateral buckling are negligible.The lateral buckling theory is valid for straight beams, with loads applied rigorously in the plane of symmetry. Practical beams have initial imperfections and unavoidable load eccentricities. So the true behavior is better described by the stability phenomenon.For beams of intermediate length for which buckling occurs in the inelastic range, the tangent modulus theory is generally used. For ideally straight beams the tangent modulus theory provides an estimate for the collapse load which is slightly conservative. However, for practical beams with initial deformations, this need not be the case.In the majority of existing studies on inelastic lateral buckling, the differential equations for beams under uniform moment are used without modification for beams under moment gradient. In the later case the shear center line is inclined to the centroidal and geometrical axes. The differential equations for beams under uniform moment should therefore be modified by adding additional terms.The majority of the existing studies are limited to the behavior of isolated beams with simple end-conditions and so the beneficial effect of adjacent members on the beam collapse load cannot be studied accurately.A general non-linear theory to describe the spatial behavior of beams and that doesn't have the deficiencies mentioned above, is developed in the present paper.The paper also presents a computer method of solving these non-linear equations using the method of finite differences. Several numerical examples presented and comparison with the existing theoretical and experimental results show the applicability of the theory to a wide range of problems.