Large Displacement Behavior Research Articles

Snubbing Effects in the Pullout of a Fibrous Rod from a Laminate

In prior papers, a simple model of the constitutive behavior of a tow (fibrous rod or stitch) that bridges a mixed-mode delamination crack was introduced. The model yielded the relation between the bridging traction supplied by the bridging tow and the crack opening and sliding displacements. A feature of the model is that, when loading includes shear (mode II), a zone arises near the fracture plane where the tow laterally deflects through the surrounding material (e.g., a polymer composite laminate). This paper focuses on the case in which the reinforcement is a fibrous rod (finite length) and extends the model to find explicit solutions in the regime where the rod is pulled out of the laminate, including the regime of large crack displacements. The formulation of the model is generalized to include the possibility of enhanced values of friction acting in the domain of deflection. This is necessary to account for the strong hardening behavior of the pullout process that has been observed experimentally for rods. When loading is predominantly mode II, the pullout process is mechanically stable even after the whole rod has begun to displace. Encouraging agreement of the new model with experiments is found. For small crack displacements, the generalized model coincides with the prior model: the incorporation of two-valued friction appears necessary mainly to account for large-displacement behavior.

Realistic Modeling of Composite and Reinforced Concrete Floor Slabs under Extreme Loading. II: Verification and Application

This paper deals with the large displacement behavior of floor slab systems under extreme loading conditions. The analytical model, presented in the companion paper, is verified through comparisons against existing experimental results on both reinforced concrete and composite slabs, with flat or ribbed profiles. The new model incorporates a novel shell element and accounts for material and geometric nonlinearities under ambient as well as elevated temperatures. The verification studies examine the response of reinforced concrete slabs under unrestrained and restrained edge conditions. An assessment of the behavior of ribbed floor slabs is also undertaken, in addition to simulation of the structural response of a full-scale composite beam-slab floor system under a realistic compartment fire situation. The results show good correlation between the experimental findings and numerical predictions, and demonstrate the reliability and robustness of the proposed analytical model. Additionally, the studies and discussions presented in this investigation provide an insight into the key behavioral aspects of floor slabs under extreme conditions. In particular, the significance of compressive arching and tensile membrane actions, under various boundary conditions, is illustrated. Also, the importance of adopting a realistic representation of the composite slab geometry is highlighted. The proposed analytical model is of particular relevance to developments in performance-based structural fire design, where realistic assessment of the floor slab response is of paramount importance.

Beam element for large displacement analysis of elasto-plastic frames

The present paper develops a non-linear beam element for analysis of elastoplastic frames under large displacements. The finite element formulations are derived by using the co-rotational approach and expression of the virtual work. The Gauss quadrature is employed for numerically computing the element tangent stiffness matrix and internal force vector. A bilinear stress-strain relationship with isotropic hardening is adopted to update the stress. The arc-length technique based on the Newton-Raphson iterative method is employed to compute the equilibrium paths. A number of numerical examples is employed to assess the performance of the developed element. The effects of plastic action on the large displacement behavior of the structures as well as the expansion of plastic zones in the loading process are discussed.

Consistent coupling of beam and shell models for thermo‐elastic analysis

AbstractIn this paper, the finite element formulation of a transition element for consistent coupling between shell and beam finite element models of thin‐walled beam‐like structures in thermo‐elastic problems is presented. Thin‐walled beam‐like structures modelled only with beam elements cannot be used to study local stress concentrations or to provide local mechanical or thermal boundary conditions. For this purpose, the structure has to be modelled using shell elements. However, computations using shell elements are a lot more expensive as compared to beam elements. The finite element model can be more efficient when the shell elements are used only in regions where the local effects are to be studied or local boundary conditions have to be provided. The remaining part of the structure can be modelled with beam elements. To couple these two models (i.e. shell and beam models) at transitional cross‐sections, transition elements are derived here for thermo‐elastic problems. The formulation encloses large displacement and rotational behaviour, which is important in case of thin‐walled beam‐like structures. Copyright © 2004 John Wiley & Sons, Ltd.

Large displacement behaviour of a structural model with foundation uplift under impulsive and earthquake excitations

AbstractThis paper considers the dynamical behaviour of a structural model with foundation uplift. The equations of motion of the system considered are derived for large displacements thus allowing for the eventual overturning of the system. The transition conditions between successive phases of motion, derived in terms of the specific Lagrangian co‐ordinates used in the formulation of the equations of motion, present innovative aspects which resolve some previously inexplicable behaviour in the structural response reported in the literature. The dynamical behaviour of the model is considered under impulsive and long‐duration ground motions. The minimum horizontal acceleration impulses for the uplift and the overturning of the system are evaluated in analytical form. The sensitivity of the model to uplifting and to overturning under impulsive excitations is established as a function of few significant structural parameters. Numerical applications have been performed changing either the structural parameters or the loading parameter, in order to analyse several dynamical behaviours and also to validate the analytical results. For earthquake ground motions the results, reported in the form of response spectra, show that linearized models generally underestimate, sometimes significantly, the structural response. Copyright © 2003 John Wiley & Sons, Ltd.

Response of idealised composite beam–slab systems under fire conditions

This paper presents a numerical model for beam–slab floor systems in which a single compartment is subjected to fire. The system consists of a composite steel–concrete slab and a bare steel beam. Several idealisations are made in order to illustrate important behavioural patterns which occur under fire conditions. Nonlinear analyses are undertaken using an advanced yet computationally efficient computer program which accounts for the large displacement behaviour at elevated temperatures. The model is shown to capture the main parameters influencing the performance of the system under fire. In particular, the significance of axial restraint and thermal expansion on the overall deformation and capacity of the system is demonstrated. It is indicated that thermal expansion may have beneficial as well as detrimental consequences on the performance, depending on the particular structural configuration under consideration. The paper also closely examines the level of dependency of the response on the sequence of application of loading and elevated temperature as well as the assessment of the overall system response from consideration of the respective responses of individual components. It is shown that such concepts may be effectively employed in undertaking detailed studies for improved quantification of the fire resistance of beam–slab systems with a view to the development of more rational performance-based design procedures.

Buckling Analysis of Curved Beams by Finite-Element Discretization

Recently, an extensive theoretical investigation on the buckling and large-displacement behavior of thin-walled circular beams was reported in a two-paper series. Equilibrium equations governing the linear, bifurcation buckling, and large-displacement behaviors were derived using the principle of minimum total potential energy. This paper first presents the transformation process for finite-element stiffness relationships for a spatial curved beam element with a total of 14 degrees of freedom. It then presents numerical data demonstrating the applicability of the method for the lateral buckling of arches and the lateral-torsional buckling of horizontally curved beams. A numerical comparison between the present formulations and those presented by others is made, along with a comparison to results obtained from using three-dimensional finite-element models. Based on results from the lateral bifurcation buckling of horizontally curved beams, a regression equation is formulated representing the reduction in critical moment due to the simple addition of curvature. A comparison of results from using this regression equation to ultimate strength experimental test results of horizontally curved girders by others resulted in an unexpected excellent correlation.

A simple finite element model for cloth drape simulation

Uses Newmark’s method to carry out a time‐stepping finite element analysis to predict the behaviour of a cloth garment as it falls from an initial horizontal position to a final position draped around a human body form. Bases the finite element model on a simple beam element, in order to minimize the computational time. Accounts for large displacement behaviour by including the element geometric stiffness. Bases the body form on anthropomorphic data produced by a shadow scanner. Enlists a novel scheme to model the contact between the cloth and the underlying body form. Uses the finite element model to provide data for an animated display and finds that it produces sufficiently realistic results for the garment designer’s purposes.

A Simple Beam Element, Large Displacement Model for the Finite Element Simulation of Cloth Drape

The behaviour of cloth draped about a human body form is simulated by modelling a garment as a mesh of simple beam finite elements, with geometric stiffness to account for large displacement behaviour, and carrying out a dynamic, time-stepping analysis to predict its behaviour as it falls from an initial horizontal position to the final draped position. The body form is based on anthropomorphic data produced by a shadow scanner. A novel scheme is introduced to model the contact between the cloth and the underlying body form. The model is used to provide data for an animated display which is found to simulate the drape of the garment sufficiently realistically for the garment designer's purposes.

Inelastic Large Displacement Behavior and Buckling of Cooling Tower

Inelastic, large displacement behavior of reinforced concrete hyperbolic cooling towers is studied using a finite-element program developed on the Cray Y-MP supercomputer. The large displacement effects are formulated based on the Lagrangian approach in which the displacements are measured from the original configuration of the structure. The concrete cracking is modeled using the rotating smeared crack approach and the bending deformation is represented using the layering technique. The concrete constitutive behavior is represented by a biaxial inelastic material model. The tension stiffening of concrete is modeled using gradual linear unloading of the stress strain curve of concrete. Several nonlinear analyses of the Grand Gulf cooling tower are performed. Unlike the results of previous investigations that attribute the failure of the cooling tower to the yielding of the meridional reinforcement, the results of this investigation show that the failure occurs due to the circumferential buckling in the vicinity of the throat triggered by the reduction of the stiffness due to concrete cracking.

Behaviour and design of horizontally curved steel beams

This paper investigates the behaviour of I-beams curved in plan, and a proposed design method to estimate the ultimate strength of such beams subjected to gravity loading. First, a finite element method for analysing inelastic large-displacement behaviour of a horizontally curved I-beam is presented. The load—displacement and ultimate strength results generated by the finite element method are compared with the theoretical and experimental results published in the literature. The accuracy of the finite element analysis approach is thus established. An investigation is then followed to study the effects of residual stresses and radius of curvature-to-span length ratios on the ultimate strength behaviour of horizontally curved I-beams. An approximate equation is derived and proposed to evaluate the ultimate moment capacity of curved I-beams for several loading and boundary conditions. Based on the derived equation, standard charts are generated for the evaluation of ultimate strength of I-beams with different horizontal curvatures and span-lengths. Behaviour of curved beams with intermediate restraints is also studied both experimentally and analytically.

Thin‐Walled Curved Beams. I: Formulation of Nonlinear Equations

An extensive investigation on the buckling and large displacement behavior of thin‐walled circular beams has been conducted theoretically. Equilibrium equations governing the linear, the bifurcation buckling, and the large displacement behavior have been derived using the principle of minimum total potential energy. An explicit and clear approximation of the curvature effect is made in the derivation process. The paper concludes with a series of fundamental nonlinear equations that describe the elastic behavior of thin‐walled curved beams. A companion paper examines closed‐form solutions for arch‐buckling problems based on the formulations presented in this paper and demonstrates the rigor and the validity of the present formulation.

Prediction of the Fiber-Matrix Interface Failure due to Longitudinal Tensile Loading Using Finite Element Methods

ABSTRACTA 3D-unit cell for 0/90 laminated composites has been developed to predict the composite behavior under longitudinal tensile loading condition. 3D contact element has been used to model the fiber matrix interface. Two interface conditions, namely, infinitely strong and weakly bonded, are considered in the analysis. Both large displacement and plastic strain behavior for the matrix are considered to account for the geometric and material non-linearities. Investigations were carried out at three temperatures to compare the composite response obtained from mechanical tests at those temperatures. Stress-strain behavior and the local stress distributions at the fiber as well as at the matrix are presented, and their effects on the failure of the interface are discussed in the paper. The material under investigation was SiCf/Si3N4.

Finite Element Modeling of an Oriented Assembly of Continuous Fibers

A micromechanical model of the large displacement behavior of a sliver of oriented helical fibers has produced algorithms for predicting and updating all five independent mechanical constants. The incorporation of the micromechanics in a finite-element model of a uniform rope of initially untwisted fibrous material is described. The response of the whole model to combined external and torsional stresses such as occur during twist insertion is then described. A detailed analysis of the stresses and strains in the deformed model shows that fiber packing in the yarn is predicted to quickly become non-uniform. The inner regions become most densely packed, and fibers there are subject to the highest stress levels. Fibers further from the center largely avoid being stressed by moving toward the yarn axis.

Buckling, postbuckling, and nonlinear vibrations of imperfect plates

Formulations and computational procedures are presented for studying the geometrically nonlinear behavior, including buckling, postbuckling, and nonlinear vibrations of perfect and imperfect, isotropic and laminated thin plates. The finite-element method is used. The element used is a 48 degree-of-freedom thin flat plate rectangular element capable of modeling arbitrary imperfections. The incremental and total stiffness matrices for large displacement behavior are derived based on the total Lagrangian approach in conjunction with the Hamilton's principle. The geometric imperfections are treated by considering additional terms in the straindisplacement relations. Numerical results are presented for a variety of examples including: 1) postbuckling analysis of an isotropic, imperfect flat plate; 2) postbuckling analysis of a thin, cross-ply laminated imperfect plate; 3) free vibrations of an angle-ply laminated plate; 4) free-vibrations of an imperfect isotropic plate under the action of axial loads; 5) nonlinear free-vibrations of isotropic perfect flat plates; 6) nonlinear vibrations of axially loaded, isotropic perfect flat plates; and 7) nonlinear vibrations of an imperfect laminated plate. The results obtained in this study are compared with existing solutions and a good agreement is seen.

Material-dependent stability conditions in the buckling of nonlinear elastic bars

A thorough study of the critical and post-critical large displacement response of simple discrete and continuous systems made from a nonlinear elastic material, is presented. Simple material-dependent stability conditions are established, whose application does not require the solution of the intractable nonlinear differential equations of equilibrium. The predominent role of the nonlinear component of the curvature on the buckling mechanism of the foregoing systems, is revealed. Moreover, it is found that elastic systems which were considered as exhibiting post-buckling strength might be imperfection sensitive, if the effect of material nonlinearity is taken into account. An approximate but very efficient analytical approach leading to very reliable results is derived for establishing the large displacement behavior of a nonlinear elastic cantilever bar. The degree of accuracy of this approach is checked by comparing it with the “exact” numerical solution of Runge Kutta. The numerical results presented herein contribute to our understanding of this problem and at the same time show the efficiency, reliability and range of applicability of the proposed approach.

NONLINEAR ANALYSIS OF THIN-WALLED STRUCTURES BY A COUPLED FINITE ELEMENT METHOD

This paper presents a finite element method which enables to analyze a large displacement behavior of elasto-plastic thin-walled structures which fail by overall, local or interactive instability. An incremental equilibrium equation for a thin-walled beam-column is derived in a stiffness matrix form by using a moving element coordinate system and an incremental variational principle. So called multipoint constraints technique is used to connect plate elements with a beam element at the coupling nodal point in the cross section. Numerical results of the present method are compared with the exact solutions and experimental results. Validity and efficiency of the present method are confirmed.

Plastic straining, unstressing and branching in large displacement perturbation analysis

AbstractThe nonlinear system of equations and inequalities relating to the large displacement behaviour of elasto‐plastic structures can be transformed by the method of perturbations into an infinite sequence of recursive linear complementarity problems (LCP). Each such LCP can be solved by a simple variant of the simplex algorithm of linear programming. Special consideration is given to describing those steps of the algorithm that are needed for detecting and controlling the phenomena of plastic straining, unstressing and branching.

Improved finite element models for the large displacement bending and post buckling analysis of thin plates

Triangular finite element models are presented for the large displacement bending and postbuckling analysis of thin plates. The formulation is based upon a general variational theory which offers a wide scope for the selection of valid finite element trial functions. This makes it feasible to model the particular physical actions crucial to the successful calculation of large displacement behaviour. In this paper it is demonstrated by numerical examples that a notable improvement in performance is achieved, without a significant increase in computational effort, when simple inextensional bending deformations are included explicitly in the finite element trial functions.

A triangular plate finite element for large-displacement elastic-plastic analysis of automobile structural components

This paper presents a simple, nonlinear triangular plate finite element developed for analyzing elastic-plastic, large-displacement behavior of shell type automobile structural components. The element is derived from Lagrange's strain-displacement relations. By being a plate element, the interpolation functions for the in-plane and the out-of-plane displacements are uncoupled. The element employs linear functions for the in-plane displacements and a quadratic function for the out-of-plane displacement. The material of the element is assumed to follow the von Mises yield criterion and the Prandtl-Reuss flow rule. The incremental stiffness matrices are obtained from the principle of virtual work. Four examples are furnished to demonstrate the versatility and accuracy of the presented simple plate element in modeling structures of automobile applications.