Shell Finite Element Analyses Research Articles

Analysis of Deep Drawing Process for Stainless Steel Micro-Channel Array.

The stainless steel bipolar plate has received much attention due to the cost of graphite bipolar plates. Since the micro-channel of bipolar plates plays the role of fuel flow field, electric connector and fuel sealing, an investigation of the deep drawing process for stainless steel micro-channel arrays is reported in this work. The updated Lagrangian formulation, degenerated shell finite element analysis, and the r-minimum rule have been employed to study the relationship between punch load and stroke, distributions of stress and strain, thickness variations and depth variations of individual micro-channel sections. A micro-channel array is practically formed, with a width and depth of a single micro-channel of 0.75 mm and 0.5 mm, respectively. Fractures were usually observed in the fillet corner of the micro-channel bottom. According to the experimental results, more attention should be devoted to the fillet dimension design of punch and die. A larger die fillet can lead to better formability and a reduction of the punch load. In addition, the micro-channel thickness and the fillet radius have to be taken into consideration at the same time. Finally, the punch load estimated by the unmodified metal forming equation is higher than that of experiments.

CFS lipped channel columns affected by L-D-G interaction. Part II: Numerical simulations and design considerations

This work deals with the structural behaviour, ultimate strength and design of fixed-ended cold-formed steel (CFS) lipped channel columns experiencing various levels of local-distortional-global buckling mode interaction (more or less close critical local, distortional and global loads). First, a comparison between experimental results, obtained from three tests carried out at The University of Hong Kong (UHK), and the corresponding Abaqus shell numerical simulations are presented and discussed – the UHK test campaign is reported in Part I of this two-part paper. The comparison (i) involves equilibrium paths, deformed configurations, failure loads and collapse mechanisms, and (ii) provides enough evidence to ensure that the numerical model developed may be deemed validated. Then, this numerical model is employed to perform a parametric study, aimed at obtaining numerical failure load data concerning lipped channel columns experiencing different levels of local-distortional-global interaction – 368 fixed-ended columns, exhibiting different geometries (cross-section dimensions and lengths) and various yield stresses, thus covering a wide slenderness range. Finally, the present (Part II) paper also assesses the quality of the failure load estimates provided by various design approaches, based on the Direct Strength Method (DSM), for CFS columns affected by the triple coupling phenomenon under consideration. This assessment is based on the comparison with both (i) experimental failure loads, namely those reported in Part I and others collected from the literature, and (ii) numerical failure loads, obtained from Abaqus shell finite element analyses (SFEA) and either reported in this work or gathered from previous publications by the authors.

Local buckling behaviour of thin-walled members with curved cross-section parts

In this paper the buckling behaviour of thin-walled members with cross-sections with curved parts is investigated. Due to the curved parts, shell-like buckling is a potential mode of failure. The objective of the research is to understand whether shell-like buckling behaviour might be governing in practical cold-formed steel members. For this aim, numerical studies have been carried out, involving linear buckling analysis as well as nonlinear shell finite element analysis with imperfections, by considering a large number of various cross-sections. Standardized capacity prediction has also been completed, based on elastic critical loads by using the direct strength method, and the results have been compared to that of shell finite element analyses. Based on the results it is concluded that shell-like behaviour might be critical in certain cases, but only in case of unusual cross-section geometries. It was also found that the simple direct strength capacity prediction can reasonably be used in most of the cases.

Dynamic analysis of high-speed railway bridge decks using generalised beam theory

This paper presents the details and illustrates the application of a semi-analytical Generalised Beam Theory (GBT) formulation for the dynamic analysis of high-speed railway bridge decks. The technique, which takes into account the cross-section local/distortional and shear deformations, provides an original “doubly modal” representation of the dynamic response, which makes it possible to acquire deep insight into the mechanics of the deck structural behaviour. It is employed to investigate the dynamic behaviour of a real high-speed railway viaduct, comprising a series of identical 46m-span simply supported box girder pre-stressed concrete decks carrying two tracks – the augmented structural response associated with resonance due to the crossing of trains at high-speeds is captured. The results obtained provide evidence that local/distortional deformation can play a significant role on the dynamic response of such structures. Moreover, the GBT analyses, whose validity is confirmed through the comparison with values yielded by shell finite element analyses, are shown to provide accurate results while involving just a few degrees of freedom.

Constrained shell Finite Element Method for thin-walled members, Part 1: constraints for a single band of finite elements

In this paper a novel method for the analysis of thin-walled members is presented: the constrained finite element method. The method is basically a shell finite element analysis, but carefully defined constraints are applied which enforce the thin-walled member to deform in accordance with specific mechanical criteria, e.g., to force local, global or distortional deformations. The constrained finite element method is essentially similar to the constrained finite strip method, but the trigonometric longitudinal shape functions of the finite strip method are replaced by polynomial longitudinal shape functions, which – together with longitudinal discretization – transforms a finite strip into multiple finite elements. This change in longitudinal interpolation makes the method applicable for a wide range of practical problems not yet handled by other modal decomposition methods. The new shell finite element is briefly presented here, but the main focus of this paper is on how the constraining criteria can be applied for a thin-walled member. More specifically, in this paper a band of finite elements is discussed in detail, where ‘band’ is a segment of the member with multiple elements along the cross-section, but with one single finite element longitudinally. The possible base systems for the various deformation spaces are demonstrated here. Members built up from multiple bands are discussed and presented in a companion paper, where various numerical examples are also provided to illustrate the potential of the proposed constrained finite element method.

Finite element analysis of shells of revolution using triangular discretization elements with corrective Lagrange multipliers

A thin shell algorithm based on a special triangular finite element is described. The column of nodal variable parameters of this element additionally contains corrective Lagrange multipliers, which allows one to improve the conditions of compatibility between the elements in the framework of the variational formulation of the problem.

Behaviour and DSM design of stiffened lipped channel columns undergoing local-distortional interaction

The paper reports the results of a numerical (Abaqus shell finite element analysis) investigation on the influence of local-distortional (L-D) interaction on the ultimate strength and design of fixed-ended cold-formed steel lipped channel columns with web and flange intermediate stiffeners. The columns analysed exhibit various geometries and yield stresses, ensuring several combinations of ratios between the (i) local and distortional critical buckling stresses, and (ii) yield stress and the highest of the above two critical buckling stresses (local or distortional). The objectives of this work are (i) to acquire in-depth understanding on the mechanics underlying the L-D interaction in the columns under consideration, and also (ii) to provide a contribution towards the development of an efficient Direct Strength Method (DSM) approach to design these structural elements. The results presented and discussed concern the post-buckling behaviour, ultimate strength and failure mode of fixed-ended web-flange-stiffened lipped channel columns previously selected to undergo more or less severe L-D interaction. Special attention is paid to the comparison between the ultimate strength erosion, due to L-D interaction, occurring in the columns analysed in this work and the (i) “plain” cross-section (i.e., without intermediate stiffeners) and (ii) web-stiffened lipped columns investigated earlier by the authors. Then, available experimental ultimate strengths of web-flange-stiffened lipped channel columns failing in L-D interactive modes are employed, together with the numerical failure loads obtained in this work, to assess the quality of the estimates provided by the existing DSM design approaches. The paper closes with (i) the reliability assessment of the DSM-based L-D approaches to predict the numerical and experimental failure loads, and (ii) some considerations about the impact of the findings reported on the design of cold-formed steel columns undergoing L-D interaction.

Elastic buckling analysis of thin-walled structural members with rectangular holes using generalized beam theory

This paper presents a method to perform generalized beam theory buckling analysis on thin-walled structural members with holes. Generalized beam theory (GBT) is an ideal tool for analyzing thin-walled structures because it can directly compute buckling mode participation in an eigen-buckling analysis. The GBT extension to members with holes is made by treating a thin-walled structural member as an assembly of prismatic sub-members, and compatibility constraints on the GBT modal amplitudes are introduced to connect these sub-members. GBT shear modes with nonlinear warping deformation are included in both first order and buckling analyses to account for the nonlinear normal stress distribution in the vicinity of a hole. GBT buckling mode shapes are verified with shell finite element analysis (SFEA) in three examples that highlight the potential for quantitatively documenting buckling modes initiated by the presence of holes.

Exact matching at a joint of multiply-connected box beams under out-of-plane bending and torsion

Accurate analysis of thin-walled box beams meeting at a joint requires not only consideration of higher-order deformation degrees (such as warping and distortion) but also exact matching conditions at the joint. Especially when more than two box beams are connected at a joint, the deformation of the beam-joint system is so complicated that no one-dimensional beam analysis has yet predicted its structural behavior correctly. Since a beam theory incorporating higher-order deformations is available, the main difficulty is determining the exact matching conditions at the joint. In this paper, we derive the exact matching conditions for five field variables—bending deflection, bending/torsional rotations, warping, and distortion—of multiply connected box beams under out-of-plane bending and torsional loads. The derived relations are valid irrespective of the number of beams and angles. We introduced a new concept called “edge resultants” besides conventional (sectional) resultants, and demonstrated its effectiveness for exact derivation and physical interpretations of the derived equations. The accuracy and validity of the proposed theory are checked by comparing the predicted results with those of shell finite element analysis.

Numerical investigation and direct strength design of cold-formed steel lipped channel columns experiencing local–distortional–global interaction

This paper reports the results of a numerical investigation concerning the relevance and Direct Strength Method (DSM) prediction of the ultimate strength erosion caused by local-distortional-global (LDG) interaction in cold-formed steel fixed-ended lipped channel columns. The geometries of the columns analysed (cross-section dimensions and lengths) were carefully selected to ensure that the three competing critical buckling loads are not more than 20% apart, thus guaranteeing a fairly high level of LDG coupling, and ordered in all the various possible sequences. In order to cover a wide slenderness range, several yield stresses are considered, falling below, in-between and above the lowest and highest critical buckling stresses. After providing a brief description of the column selection procedure, which is based on buckling analyses performed with Generalised Beam Theory (GBT), the methodology adopted to identify the most detrimental initial geometrical imperfection shape (in the sense it minimises the column strength) is addressed – it employs Koiter's asymptotic method along with a Monte Carlo simulation. Then, columns containing those initial geometrical imperfections are compressed up to failure, by means of Abaqus shell finite element analyses (SFEA), making it possible to acquire in-depth knowledge on the behaviour of lipped channel columns undergoing LDG interaction and gather considerable failure load data. Finally, the last part of the paper is devoted to the DSM prediction of those failure loads and uses the obtained “data bank” to assess whether the available design approaches are able to handle adequately the ultimate strength erosion caused by the triple interaction phenomenon under investigation – if this is not the case, new design curves must be developed.

Finite Element Analysis of Funicular Shells with Rectangular plan ratio 1:0.7 under Uniformly Distributed Load using SAP2000

A thin shell is a Three-dimensional spatial structure made up of one or more curved surfaces whose thickness is small compared to their other dimensions. Shells belong to the class of stressed skin structures which, because of their geometry and small flexural rigidity of the skin, tend to carry loads primarily by direct stresses acting in their plane. The shells are subjected to pure membrane state of stress, under appropriate loading and boundary condition the resulting bending and twisting moments are either zero or small which may be neglected. In this study doubly curved thin shells are anal yzed using Finite Element software SAP 2000 with new version. Doubly curved shells which are in rectangular plan having 1mX0.7m are considered. The behavior of shells under uniformly distributed load varying from 1to5KN/m 2 is studied and compared with the slabs of same dimension and thickness .In this case study deflection curves, membrane stress and stress contour diagram are obtained. It is observed that with the increase in rise and thickness of funicular shell the deflection are reduced. The membrane st resses decreases with the increase in rise and thickness of concrete funicular shell. Keywords - Edge beam, Funicular shell, Rise, Thickness, Stress contour.

Shell element for constrained finite element analysis of thin-walled structural members

In this paper a novel shell finite element is introduced, specifically proposed for constrained shell finite element analysis. The proposed element is derived from the finite strips used in the semi-analytical finite strip method. The new finite element shares the most fundamental feature of the finite strips, namely: transverse and longitudinal directions are distinguished. Moreover, the new element keeps the transverse interpolation functions of finite strips, however, the longitudinal interpolation functions are changed from trigonometric functions (or function series) to classic polynomials. It is found that the proper selection of the polynomial longitudinal interpolation functions makes it possible to perform modal decomposition similarly as in the constrained finite strip method (cFSM). This requires an unusual combination of otherwise well-known shape functions. If the so-constructed shell finite elements are used to model a thin-walled member, (hence, with using discretization in both the transverse and the longitudinal directions,) modal decomposition can be done essentially identically as in cFSM, whilst the practical applicability of the method is significantly extended (e.g., various restraints, holes, certain cross-section changes can easily be handled). In this paper the focus is on the derivation of the novel shell finite element. Constraining capability is illustrated by some basic examples. Practical application of the novel element will be presented in subsequent papers.

Buckling behaviour of partially restrained cold-formed steel zed purlins subjected to transverse distributed uplift loading

This paper presents a numerical investigation into the buckling behaviour of cold-formed steel zed purlins when subjected to transverse distributed uplift loading. The study uses both linear and non-linear finite element methods (ABAQUS shell Finite Element analyses) to investigate the pre-buckling, buckling and post-buckling behaviour of zed-section purlins in purlin-sheeting systems. The numerical modelling strategy is carefully validated using the results of available experimental data. Influences of boundary conditions and restraints from sheeting on web shear buckling, local, distortional and lateral-torsional buckling behaviour, and the buckling interactions of the purlins are discussed. Furthermore, the strengths and failure modes of the purlins obtained from numerical results are compared with the design strengths predicted using the direct strength method (DSM) specified in the ‘North American Specification’ (AISI S100). The comparisons show that the ultimate strengths of buckling interactions are generally underestimated. Hence, the DSM is modified to cover the partially restrained zed-sections in purlin-sheeting systems.

On the influence of local-distortional interaction in the behaviour and design of cold-formed steel web-stiffened lipped channel columns

This work reports the results of a numerical (Abaqus shell finite element analysis) investigation on the influence of local–distortional (L–D) interaction in the ultimate strength and design of fixed-ended cold-formed steel web-stiffened lipped channel columns, hereafter termed “WSLC columns”. These results concern columns with various geometries and yield stresses, ensuring a wide variety of combined ratios between (i) the distortional and local critical buckling stresses, and (ii) the yield stress and the higher of the above two buckling stresses. The objectives of the work are two-fold: (i) acquire in-depth knowledge on the mechanics underlying the L–D interaction in the WSLC columns analysed, all selected to ensure that local buckling is triggered by the flanges, and also (ii) provide a first contribution towards the efficient Direct Strength Method (DSM) design of these columns. The results presented and discussed concern the (i) post-buckling behaviour (elastic and elastic–plastic), (ii) ultimate strength and (iii) failure mechanisms of WSLC columns undergoing L–D interaction. Special attention is paid to the comparison between the ultimate strength erosion, stemming from L–D interaction, exhibited by web-stiffened and plain lipped channel columns (the latter were studied earlier by the authors). The paper closes with some considerations about the impact of the findings reported in this work on the design of cold-formed steel columns experiencing different levels of L–D interaction.

Static and fatigue experimental tests on a full scale fuselage panel and FEM analyses

A fatigue test on a full scale panel with complex loading condition and geometry configuration has been carried out using a triaxial test machine. The demonstrator is made up of two skins which are linked by a transversal butt-joint, parallel to the stringer direction. A fatigue load was applied in the direction normal to the longitudinal joint, while a constant load was applied in the longitudinal joint direction. The test panel was instrumented with strain gages and previously quasi-static tests were conducted to ensure a proper load transferring to the panel. In order to support the tests, geometric nonlinear shell finite element analyses were conducted to predict strain and stress distributions. The demonstrator broke up after about 177000 cycles. Subsequently, a finite element analysis (FEA) was carried out in order to correlate failure events; due to the biaxial nature of the fatigue loads, Sines criterion was used. The analysis was performed taking into account the different materials by which the panel is composed. The numerical results show a good correlation with experimental data, successfully predicting failure locations on the panel.

Buckling and Vibration Analysis of Cold-Formed Steel CHS Members and Frames Using Generalized Beam Theory

This paper reports the results of an investigation on the use of Generalized Beam Theory (GBT) to assess the buckling and vibration behaviors of thin-walled members and frames built from cold-formed steel circular hollow section (CHS) profiles. Initially, the concepts and procedures involved in performing GBT buckling and vibration analyses are presented, paying particular attention to the derivation of the mass tensors that account for the influence of the inertia forces. Then, the formulation, numerical implementation and validation of a GBT-based beam finite element for isolated members are described. Next, the determination of the frame linear stiffness, geometric stiffness and mass matrices, which incorporate the influence of the frame joints, is addressed. Finally, in order to illustrate the application and capabilities of the proposed GBT finite element formulation, numerical results are presented and discussed — they concern the buckling and vibration behaviors of an "L-shaped" frame. For validation purposes, most GBT-based results are compared with values yielded by shell finite element analyses carried out in the code ANSYS.

Local–Distortional Interaction in Cold-formed Steel Columns: Mechanics, Testing, Numerical Simulation and Design

This paper provides an overview of the latest developments of the research activity carried out by the authors concerning the non-linear behaviour, ultimate strength and Direct Strength Method (DSM) design of cold-formed steel (plain and stiffened) columns experiencing local–distortional (L–D) interaction. Initially, numerical results obtained by means of Generalized Beam Theory (GBT) elastic post-buckling analyses are presented and their unique modal nature is used to illustrate and provide the main behavioural/mechanics features of this mode coupling phenomenon. Then, both experimental and numerical (obtained from Abaqus shell finite element analyses – SFEA) are presented and discussed, making it possible to provide clear evidence and characterize the column L-D interactive behaviour and failure. Besides reporting on two experimental test series performed by the authors, the paper also presents a critical review of other experimental results available in the literature concerning columns undergoing L–D interaction. The numerical results deal with columns exhibiting different cross-section shapes and fairly extensive parametric studies are carried out, in order to (i) assess the relevance of L–D interaction, as far as the ultimate strength erosion is concerned, and (ii) gather substantial ultimate strength data. Next, after showing the inadequacy of the currently codified DSM column design curves to predict L–D interactive failures, the paper presents and assesses the merits of DSM-based design approaches recently developed specifically to handle such failures.

Interlaminar stress recovery for arbitrarily curved shells

AbstractAn established postprocessing procedure for the determination of interlaminar shear and normal stresses based on the results of a conventional shell finite element analysis is generalized for arbitrarily curved laminated shells. The method relies on the three‐dimensional equilibrium conditions and an extended lamination theory. Underlying approximations are discussed and exemplary results are presented. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Cold-formed steel member bending stiffness prediction

The objective of this paper is to provide a simplified method for predicting the bending stiffness of thin-walled cold-formed steel members subject to elastic (or inelastic) local or distortional buckling. In thin-walled cold-formed steel members for serviceability and consideration of deformations (second-order effects) it is important to account for the loss of cross-section bending stiffness due to the use of slender elements and/or yielding. Although existing design specifications provide some guidance on how to predict the stiffness, limited information is available for cross-sections subject to distortional buckling or undergoing inelastic local and/or distortional buckling. Existing tests and shell finite element analysis on cold-formed steel beams failing in local or distortional buckling are used as the basis for the study conducted herein. Stiffness of the conducted tests and finite element models is compared with predictions based on the Effective Width Method and Direct Strength Method. Relationships between local and distortional cross-section slenderness and the observed and predicted secant stiffness up to the peak load are examined. New design expressions to predict cold-formed steel member bending stiffness depending on cross-section slenderness are presented. The new methods are proposed for adoption in cold-formed steel specifications.

Distortional failure and DSM design of cold-formed steel lipped channel beams under elevated temperatures

The aim of this paper is to report an ongoing shell finite element investigation on the distortional post-buckling behaviour, ultimate strength and DSM design of cold-formed steel single-span lipped channel beams under elevated temperatures due to fire conditions. The beam ultimate strengths are computed by means of a steady-state loading strategy that consists of applying an increasing major-axis uniform bending moment to a beam under a uniform and constant (elevated) temperature distribution, until failure is reached. The steel constitutive law at elevated temperatures is simulated by means of the stress–strain–temperature models prescribed in EC3-1.2 for cold-formed steel. The materially and geometrically non-linear response of the cold-formed lipped channel beams is determined through ANSYS shell finite element analyses that incorporate critical-mode (distortional) initial geometrical imperfections. Finally, since there are no specific rules to predict the bending strength of cold-formed steel beams failing in distortional modes at elevated temperatures, the above failure moments and temperatures are used to establish preliminary guidelines for the design of cold-formed steel lipped channel beams under fire conditions. The approach followed is based on the Direct Strength Method (DSM) and has been already employed by other researchers – the currently available design/strength equations/curves, developed for ambient temperature, are modified to account for the appropriate Young’s modulus and yield stress reductions due to the temperature increase. The results presented in the paper provide solid numerical evidence that the currently codified DSM distortional strength curve is not appropriate to predict beam failure moments at elevated temperatures. This is because it either (i) does not account for the material non-linearity due to the elevated temperature (low-to-moderate slenderness range) or (ii) is already inadequate to predict room temperature failure moments (unsafe estimates obtained in the high slenderness range).