Dinar Camotim

GBT buckling analysis of pultruded FRP lipped channel members

Abstract The main aspects concerning an orthotropic second order generalised beam theory (GBT), previously developed by the authors to enables the structural analysis of thin-walled members displaying special orthotropy, are briefly presented and discussed in the first part of the paper. Special attention is paid to the symbolic and numerical computational aspects related to the implementation of the GBT equations to perform member linear stability analyses. For this purpose, a finite element formulation is developed, validated and applied to solve the eigenvalue problem associated with the GBT system of differential equilibrium equations. Then, the second order orthotropic GBT is employed to investigate the buckling behaviour of pultruded fiber reinforced plastic lipped channel members, namely columns, beams and beam-columns. In particular, with the objective of illustrating the concepts and procedures involved in the performance of a GBT analysis, a detailed in-depth study of simply supported lipped channel columns is presented, in which the relevant (local and global) buckling modes are identified and the corresponding bifurcation stress values are determined. In addition, the results of an investigation concerning the influence of the applied stress distribution, cross-section geometry, material properties and end support conditions on the member buckling behaviour are presented, which required the completion of a number of parametric studies. In these investigations, careful consideration was given to the occurrence and characterisation of local-plate, distortional and mixed flexural–distortional buckling modes.

GBT-based Structural Analysis of Thin-walled members: Overview, Recent Progress and Future Developments

This paper provides an overview of the Generalised Beam Theory (GBT) fundamentals and reports on the novel formulations and applications recently developed at the TU Lisbon: the use of conventional GBT to derive analytical distortional buckling formulae and extensions to cover (i) the buckling behaviour of members with (i1) branched, closed and closed/branched cross-sections and (i2) made of orthotropic and elastic-plastic materials, and (ii) the vibration and post-buckling behaviours of elastic isotropic/orthotropic members. In order to illustrate the usefulness and potential of the new GBT formulations, a few numerical results are presented and briefly discussed. Finally, some (near) future developments are briefly mentioned.

Cold-Formed Steel Lipped Channel Columns Influenced by Local-Distortional Interaction: Strength and DSM Design

This paper deals with the ultimate strength and design of fixed-ended lipped channel columns experiencing local-distortional buckling mode interaction. First, the paper reports the results of an experimental investigation involving a set of 26 columns with several cross-section dimensions and yield stresses that were tested to determine their failure loads and also to provide experimental evidence of the occurrence of local-distortional mode interaction. These results consist of the column geometries, material properties, initial geometric imperfections, nonlinear equilibrium paths, and ultimate strength values. Then, after comparing the experimental column ultimate loads with the estimates provided by the current direct strength method (DSM) design curves against local and distortional failures, which clearly show that they lead to inaccurate and often very unsafe ultimate strength estimates, the paper presents and assesses the quality of DSM-based design procedures based on approaches providing nominal strengths against local-distortional and distortional-local interactive failures. Next, an in-depth comparison is made between all the experimental ultimate strength results available in the literature and their estimates provided by the preceding DSM design procedures. Finally, the paper closes with design considerations and recommendations, motivated by the conclusions drawn from this investigation.

Shear Deformable Generalized Beam Theory for the Analysis of Thin-Walled Composite Members

This paper presents the incorporation of shear deformation effects into a generalized beam theory (GBT) formulation developed to analyze the first-order (linear) and buckling behavior of composite thin-walled members made of laminated plates displaying arbitrary orthotropy, often designated as anisotropic laminates. Unlike other existing beam theories, the proposed GBT formulation incorporates in a unified fashion (1) elastic coupling effects, (2) warping effects, (3) cross-section in-plane deformation, and (4) shear deformation. The main concepts and procedures involved in the currently available GBT are adapted and/or modified to account for the specific aspects associated with shear deformation. In particular, the GBT equilibrium equations and boundary conditions are derived, and their terms are physically interpreted. A lipped channel section is considered to illustrate the performance of a GBT cross-section analysis, namely, the operations required to determine the (additional) set of shear deformation modes. Finally, to clarify the concepts involved in the proposed GBT formulation and illustrate its application and capabilities, two numerical examples are presented and discussed in detail: the first concerns the first-order and buckling behaviors of a lipped channel column exhibiting nonaligned orthotropy; and the second assesses the influence of shear deformation on the buckling behavior of lipped channel columns with cross-ply orthotropy.

GBT FORMULATION TO ANALYZE THE BUCKLING BEHAVIOR OF THIN-WALLED MEMBERS SUBJECTED TO NON-UNIFORM BENDING

In this paper, one investigates the local-plate, distortional and global buckling behavior of thin-walled steel beams subjected to non-uniform bending moment diagrams, i.e. under the presence of longitudinal stress gradients. One begins by deriving a novel formulation based on Generalized Beam Theory (GBT), which (i) can handle beams with arbitrary open cross-sections and (ii) incorporates all the effects stemming from the presence of longitudinally varying stress distributions. This formulation is numerically implemented by means of the finite element method: one (i) develops a GBT-based beam finite element, which accounts for the stiffness reduction associated to applied longitudinal stresses with linear, quadratic and cubic variation, as well as to the ensuing shear stresses, and (ii) addresses the derivation of the equilibrium equation system that needs to be solved in the context of a GBT buckling analysis. Then, in order to illustrate the application and capabilities of the proposed GBT-based formulation and finite element implementation, one presents and discusses numerical results concerning (i) rectangular plates under longitudinally varying stresses and pure shear, (ii) I-section cantilevers subjected to uniform major axis bending, tip point loads and uniformly distributed loads, and (iii) simply supported lipped channel beams subjected to uniform major axis bending, mid-span point loads and uniformly distributed loads — by taking full advantage of the GBT modal nature, one is able to acquire an in-depth understanding on the influence of the longitudinal stress gradients and shear stresses on the beam local and global buckling behavior. For validation purposes, the GBT results are compared with values either (i) yielded by shell finite element analyses, performed in the code ANSYS, or (ii) reported in the literature. Finally, the computational efficiency of the proposed GBT-based beam finite element is briefly assessed.

On the Use of Shell Finite Element Analysis to Assess the Local Buckling and Post-Buckling Behaviour of Cold-Formed Steel Thin-Walled Members

This paper deals with the use of shell finite element analyses to assess the (i) elastic bifurcation and (ii) elastic and elastic-plastic local-plate and distortional post-buckling behaviours of cold-formed steel thin-walled members (mostly columns, i.e., uniformly compressed members) all the geometrically and physically non-linear analyses are performed using the code ABAQUS and adopting 4-node isoparametric shell elements to discretise the members. First, one addresses several relevant issues concerning (i) the member discretisation (shell element type and mesh refinement), (ii) the simulation of the member end support conditions (a key aspect in numerical structural analysis), (iii) the modelling of the applied loading and material behaviour, (iv) the incorporation of member initial geometrical imperfections and residual stresses, (v) the assessment of buckling mode interaction effects and (v) the methods employed to solve either the eigenvalue problem or the system of non-linear algebraic equilibrium equations. Then, in order to illustrate the concepts and issues mentioned above and, at the same time, illustrate the power and versatility of the shell finite element analyses, one presents and thoroughly discusses a fairly large number of numerical results concerning the buckling and post-buckling behaviour of lipped channel (mostly), Zed-section and Rack-section cold-formed steel members some of the post-buckling analyses include interaction effects between local-plate and distortional buckling modes. These results consist of (i) buckling curves providing the variation of the critical stress with the member length (see Fig. 1(a)), (ii) elastic and elastic-plastic non-linear (post-buckling) equilibrium paths (see Figs. 1(b)-(c)), (iii) figures providing the evolution, along those equilibrium paths, of the elastic and elastic-plastic member deformed configurations, and (iv) figures showing the spread of plasticity along the members up to failure (see Fig. 1(d)) and conveying relevant information about the nature of their collapse mechanisms. Open image in new window Figure 1 Lipped channel simply supported columns: (a) elastic buckling, (b) elastic distortional post-buckling and (c) elastic-plastic distortional post-buckling results, and (d) distortional post-buckling plastic strain evolution.

Work-conjugacy between rotation-dependent moments and finite rotations

In this paper we investigate the work-conjugacy between rotation-dependent moments and finite rotation measures. The methodology adopted consists of writing all the relevant quantities in terms of the rotation vector, using expressions that remain exact in the finite rotation range. Through this procedure, we show how (i) to identify work-conjugate (rotation-dependent) moments and rotation measures, (ii) to derive a necessary condition for moment conservativeness and (iii) to obtain the general form of an isotropic conservative rotation-dependent moment. Several moment and finite rotation definitions that have been used in the past are investigated and, in particular, it is shown that the various existing definitions for the so-called semi-tangential moments are distinct in the finite rotation range and that not all of them are conservative. The tangent operator symmetry is discussed in the context of finite element analysis of conservative systems with rotational degrees of freedom, adopting either an additive or a multiplicative update.

Glass fibre reinforced polymer pultruded flexural members: assessment of existing design methods

Glass fibre reinforced polymer (GFRP) pultruded profiles are being increasingly used in bridge and building construction as an alternative to traditional materials because of their several favourable properties that include high strength, low self-weight, short installation times, low maintenance requirements and improved durability. In spite of these advantageous characteristics, there are some factors delaying the widespread use of GFRP pultruded profiles in civil infrastructure, one of which is the lack of widely accepted design codes. This paper presents the results of analytical, experimental and numerical investigations on the structural behaviour of GFRP pultruded profiles, the objective of which was to evaluate the relative accuracy of existing design methods. A survey of analytical formulae available for the design of GFRP pultruded flexural members at both service and ultimate limit states is first presented. Subsequently, results of a test programme carried out at Instituto Superior Técnico (IST) are briefly discussed—the experiments included material characterization tests and full-scale flexural tests on I-section simply supported beams and cantilevers. These tests allowed for the evaluation of the service behaviour of GFRP flexural members and some of their most relevant failure mechanisms and respective ultimate loads. Results from experimental tests are compared with those obtained from analytical formulae and numerical models in order to evaluate the relative accuracy of existing design methods.

AN ANALYTICAL STUDY ON THE LATERAL-TORSIONAL BUCKLING OF LINEARLY TAPERED CANTILEVER STRIP BEAMS

In this paper, the elastic lateral-torsional buckling of tapered cantilever strip beams acted by a tip load is investigated. Analytical solutions of the buckling problem are derived for cantilevers with a linearly varying depth. It is shown that the corresponding governing differential equation is a particular case of the general confluent equation, whose solution is given in terms of confluent hypergeometric functions. The buckling load corresponds to the lowest positive root of a 4×4 characteristic determinant arising from the introduction of the general solution of the differential equation into the boundary conditions. Finally, one discusses the optimization of the cantilever geometrical shape (for constant volume), as far as its resistance to lateral-torsional buckling is concerned.

GBT-BASED LOCAL AND GLOBAL VIBRATION ANALYSIS OF LOADED COMPOSITE OPEN-SECTION THIN-WALLED MEMBERS

This paper begins by presenting a Generalized Beam Theory (GBT) formulation for analyzing the vibration behavior of loaded composite thin-walled members, which accounts for the effects of (i) cross-section in-plane deformation, (ii) shear deformation, (iii) geometric and material coupling, (iv) primary, secondary and non-linear warping, and (v) rotary inertia. This formulation is then used to investigate the local and global vibration behavior of lipped channel columns and beams displaying cross-ply orthotropy, focusing on issues dealing with the variation of the fundamental frequency and vibration mode nature with the member length and applied stress level. For validation purposes, some GBT-based results are also compared with values obtained by means of 4-node shell finite element analyses using ABAQUS. Some relevant conclusions are drawn concerning the dependence of the member vibration mode shape (wave number) on the compression/bending level (applied-to-critical ratio).