Higher-order non-linear analysis of steel structures. Part I : elastic second-order formulation

This paper presents a higher-order beam-column formulation that can capture the geometrically non-linear behaviour of steel framed structures which contain a multiplicity of slender members. Despite advances in computational frame software, analyses of large frames can still be problematic from a numerical standpoint and so the intent of the paper is to fulfil a need for versatile, reliable and efficient non-linear analysis of general steel framed structures with very many members. Following a comprehensive review of numerical frame analysis techniques, a fourth-order element is derived and implemented in an updated Lagrangian formulation, and it is able to predict flexural buckling, snap-through buckling and large displacement post-buckling behaviour of typical structures whose responses have been reported by independent researchers. The solutions are shown to be efficacious in terms of a balance of accuracy and computational expediency. The higher-order element forms a basis for augmenting the geometrically non-linear approach with material non-linearity through the refined plastic hinge methodology described in the companion paper.

Higher-order non-linear analysis of steel structures. Part II : refined plastic hinge formulation

In the companion paper, a fourth-order element formulation in an updated Lagrangian formulation was presented to handle geometric non-linearities. The formulation of the present paper extends this to include material non-linearity by proposing a refined plastic hinge approach to analyse large steel framed structures with many members, for which contemporary algorithms based on the plastic zone approach can be problematic computationally. This concept is an advancement of conventional plastic hinge approaches, as the refined plastic hinge technique allows for gradual yielding, being recognized as distributed plasticity across the element section, a condition of full plasticity, as well as including strain hardening. It is founded on interaction yield surfaces specified analytically in terms of force resultants, and achieves accurate and rapid convergence for large frames for which geometric and material non-linearity are significant. The solutions are shown to be efficacious in terms of a balance of accuracy and computational expediency. In addition to the numerical efficiency, the present versatile approach is able to capture different kinds of material and geometric non-linearities on general applications of steel structures, and thereby it offers an efficacious and accurate means of assessing non-linear behaviour of the structures for engineering practice.

Second-order elastic finite element analysis of steel structures using a single element per member

Finite element frame analysis programs targeted for design office application necessitate algorithms which can deliver reliable numerical convergence in a practical timeframe with comparable degrees of accuracy, and a highly desirable attribute is the use of a single element per member to reduce computational storage, as well as data preparation and the interpretation of the results. To this end, a higher-order finite element method including geometric non-linearity is addressed in the paper for the analysis of elastic frames for which a single element is used to model each member. The geometric non-linearity in the structure is handled using an updated Lagrangian formulation, which takes the effects of the large translations and rotations that occur at the joints into consideration by accumulating their nodal coordinates. Rigid body movements are eliminated from the local member load-displacement relationship for which the total secant stiffness is formulated for evaluating the large member deformations of an element. The influences of the axial force on the member stiffness and the changes in the member chord length are taken into account using a modified bowing function which is formulated in the total secant stiffness relationship, for which the coupling of the axial strain and flexural bowing is included. The accuracy and efficiency of the technique is verified by comparisons with a number of plane and spatial structures, whose structural response has been reported in independent studies.

Direct second-order elastic analysis for steel frame design

The traditional structural design procedure, especially for the large-scale and complex structures, is time consuming and inefficient. This is due primarily to the fact that the traditional design takes the second-order effects indirectly by virtue of design specifications for every member instead of system analysis for a whole structure. Consequently, the complicated and tedious design procedures are inevitably necessary to consider the second-order effects for the member level in design specification. They are twofold in general: 1) Flexural buckling due to P-d effect, i.e. effective length. 2) Sway effect due to P-D effect, i.e. magnification factor. In this study, a new system design concept based on the second-order elastic analysis is presented, in which the second-order effects are taken into account directly in the system analysis, and also to avoid the tedious member-by-member stability check. The plastic design on the basis of this integrated method of direct approach is ignored in this paper for simplicity and clarity, as the only emphasis is placed on the difference between the second-order elastic limit-state design and present system design approach. A practical design example, a 57m-span dome steel skylight structure, is used to demonstrate the efficiency and effectiveness of the proposed approach. This skylight structure is also designed by the traditional design approach BS5950-2000 for comparison on which the emphasis of aforementioned P-d and P-D effects is placed.

On the advanced analysis of steel frames allowing for flexural, local and lateral-torsional buckling

Detailed procedure for second-order analysis has been coded in the newest Eurocode 3 and the Hong Kong steel code (2005). The effective length method has been noted to be inapplicable to analysis of shallow domes of imperfect members exhibiting snap-through buckling, to portals with leaning columns and others. On the other hand, the advanced analysis is not limited to buckling design of these structures. This paper demonstrates its application to the design of a simple plane sway portal and a three diminsional non-sway steel building. The results by the advanced analysis and the first-order linear analysis are compared and the technique for practical second-order analysis steel structures is described. It is observed that the use of a straight element by itself cannot model the buckling resistance of columns governed by different buckling curves for hot-rolled and cold-formed sections of various shapes like I, H, hollow etc. Also the curvature of the conventional cubic Hermite element is not varied by the external axial force and thus it cannot simulate the response of a buckling column. Thus its use for second-order analysis is basically unacceptable. A technique for additional checking of beams undergoing lateral-torsional buckling is also suggested making the advanced analysis a complete design tool for conventional steel frames.

Performance-based structural fire design of steel frames using conventional computer software

Fire incident in buildings is common, so the fire safety design of the framed structure is imperative, especially for the unprotected or partly protected bare steel frames. However, software for structural fire analysis is not widely available. As a result, the performance-based structural fire design is urged on the basis of using user-friendly and conventional nonlinear computer analysis programs so that engineers do not need to acquire new structural analysis software for structural fire analysis and design. The tool is desired to have the capacity of simulating the different fire scenarios and associated detrimental effects efficiently, which includes second-order P-D and P-d effects and material yielding. Also the nonlinear behaviour of large-scale structure becomes complicated when under fire, and thus its simulation relies on an efficient and effective numerical analysis to cope with intricate nonlinear effects due to fire. To this end, the present fire study utilizes a second order elastic/plastic analysis software NIDA to predict structural behaviour of bare steel framed structures at elevated temperatures. This fire study considers thermal expansion and material degradation due to heating. Degradation of material strength with increasing temperature is included by a set of temperature-stress-strain curves according to BS5950 Part 8 mainly, which implicitly allows for creep deformation. This finite element stiffness formulation of beam-column elements is derived from the fifth-order PEP element which facilitates the computer modeling by one member per element. The Newton-Raphson method is used in the nonlinear solution procedure in order to trace the nonlinear equilibrium path at specified elevated temperatures. Several numerical and experimental verifications of framed structures are presented and compared against solutions in literature. The proposed method permits engineers to adopt the performance-based structural fire analysis and design using typical second-order nonlinear structural analysis software.

Bond performance of CFRP strengthened steel members subjected to axial compression

This paper focuses on the use of externally bonded Carbon Fiber Reinforced Polymer (CFRP) materials to strengthen steel plates subjected to compression. A fully slender steel section was selected in this test programme. CFRP strengthened steel plates and non strengthened plates were tested to fail under compressive load. The middle part of the strut was strengthened using CFRP sheet. The length of the strengthened zone was varied. Eight specimens were tested in this test programme. The test results showed a significant strength gain of 47% and delaying of lateral torsional buckling failure mode of strengthened members. This study confirms that there is great potential to increase the short term performance of CFRP strengthened steel structure under axial compression.

Effects of cold weather on durability of CFRP strengthened circular hollow steel members

Tubular members have become progressively more popular due to excellent structural properties, aesthetic appearance, corrosion and fire protection capability. However, a large number of such structures are found structurally deficient due to reduction of strength when they expose to severe environmental conditions such as marine environment, cold and hot weather. Hence strengthening and retrofitting of structural members are in high demands. In recent times Carbon Fibre Reinforced Polymers (CFRP) composites appears to be an excellent solution to enhance the load carrying capacity and serviceability of steel structures because of its superior physical and mechanical properties. However, the durability of such strengthening system under cold environmental condition has not yet been well documented to guide the engineers. This paper presents the findings of a study conducted to enhance the bond durability of CFRP strengthened steel tubular members by treating steel surface using epoxy based adhesion promoter under cold weather subjected to bending. The experimental program consisted of six number of CFRP strengthened specimens and one bare specimen. The sand blasted surface of the three specimens to be strengthened was pre-treated with MBrace primer and other three were remained untreated and then cured under ambient temperature and cold weather (3oC) for three and six months period of time. The beams were then loaded to failure under four point bending. The structural response of each specimen was predicted in terms of failure mode, failure load and mid-span deflection. The research findings show that the cold weather immersion had an adverse effect on durability of CFRP strengthened structures. Moreover, the epoxy based adhesion promoter was found to enhance the bond durability in elastic range.

Numerical modelling of load bearing light gauge steel frame wall systems exposed to realistic design fires

Fire safety has become an important part in structural design due to the ever increasing loss of properties and lives during fires. Conventionally the fire rating of load bearing wall systems made of Light gauge Steel Frames (LSF) is determined using fire tests based on the standard time-temperature curve in ISO834 [1]. However, modern commercial and residential buildings make use of thermoplastic materials, which mean considerably high fuel loads. Hence a detailed fire research study into the fire performance of LSF walls was undertaken using realistic design fire curves developed based on Eurocode parametric [2] and Barnett’s BFD [3] curves using both full scale fire tests and numerical studies. It included LSF walls without cavity insulation, and the recently developed externally insulated composite panel system. This paper presents the details of finite element models developed to simulate the full scale fire tests of LSF wall panels under realistic design fires. Finite element models of LSF walls exposed to realistic design fires were developed, and analysed under both transient and steady state fire conditions using the measured stud time-temperature curves. Transient state analyses were performed to simulate fire test conditions while steady state analyses were performed to obtain the load ratio versus time and failure temperature curves of LSF walls. Details of the developed finite element models and the results including the axial deformation and lateral deflection versus time curves, and the stud failure modes and times are presented in this paper. Comparison with fire test results demonstrate the ability of developed finite element models to predict the performance and fire resistance ratings of LSF walls under realistic design fires.

Vertical dynamic interaction of trains and rail steel bridges

Rail steel bridges are vulnerable to high impact forces due to the passage of trains; unfortunately the determination of these transient impact forces is not straightforward as these are affected by a large number of parameters, including the wagon design, the wheel-rail contact and the design parameters of the bridge deck and track, as well as the operational parameters – wheel load and speed. To determine these impact forces, a detailed rail train-track/bridge dynamic interaction model has been developed, which includes a comprehensive train model using multi-body dynamics approach and a flexible track/bridge model using Euler– Bernoulli beam theory. Single and multi-span bridges have been modelled to examine their dynamic characteristics. From the single span bridge, the train critical speed is determined; the minimum distance of two peak loadings is found to affect the train critical speed. The impact factor and the dynamic characteristics are discussed.

Fire performance of cold-formed steel wall panels and prediction of their fire resistance rating

Recent research at the Queensland University of Technology has investigated the structural and thermal behaviour of load bearing Light gauge Steel Frame (LSF) wall systems made of 1.15 mm G500 steel studs and varying plasterboard and insulation configurations (cavity and external insulation) using full scale fire tests. Suitable finite element models of LSF walls were then developed and validated by comparing with test results. In this study, the validated finite element models of LSF wall panels subject to standard fire conditions were used in a detailed parametric study to investigate the effects of important parameters such as steel grade and thickness, plasterboard screw spacing, plasterboard lateral restraint, insulation materials and load ratio on their performance under standard fire conditions. Suitable equations were proposed to predict the time–temperature profiles of LSF wall studs with eight different plasterboard-insulation configurations, and used in the finite element analyses. Finite element parametric studies produced extensive fire performance data for the LSF wall panels in the form of load ratio versus time and critical hot flange (failure) temperature curves for eight wall configurations. This data demonstrated the superior fire performance of externally insulated LSF wall panels made of different steel grades and thicknesses. It also led to the development of a set of equations to predict the important relationship between the load ratio and the critical hot flange temperature of LSF wall studs. Finally this paper proposes a simplified method to predict the fire resistance rating of LSF walls based on the two proposed set of equations for the load ratio–hot flange temperature and the time–temperature relationships.

Analysis of steel structures for first- and second-order element (displacement and force) solutions

Finite element method (FEM) relies on an approximate function to fit into a governing equation and minimizes the residual error in the integral sense in order to generate solutions for the boundary value problems (nodal solutions). Because of this FEM does not show simultaneous capacities for accurate displacement and force solutions at node and along an element, especially when under the element loads, which is of much ubiquity. If the displacement and force solutions are strictly confined to an element’s or member’s ends (nodal response), the structural safety along an element (member) is inevitably ignored, which can definitely hinder the design of a structure for both serviceability and ultimate limit states. Although the continuous element deflection and force solutions can be transformed into the discrete nodal solutions by mesh refinement of an element (member), this setback can also hinder the effective and efficient structural assessment as well as the whole-domain accuracy for structural safety of a structure. To this end, this paper presents an effective, robust, applicable and innovative approach to generate accurate nodal and element solutions in both fields of displacement and force, in which the salient and unique features embodies its versatility in applications for the structures to account for the accurate linear and second-order elastic displacement and force solutions along an element continuously as well as at its nodes. The significance of this paper is on shifting the nodal responses (robust global system analysis) into both nodal and element responses (sophisticated element formulation).

Non-linear refined finite element elastic analysis of steel frames with generalized transverse member loads

In the finite element modelling of steel frames, external loads usually act along the members rather than at the nodes only. Conventionally, when a member is subjected to these transverse loads, they are converted to nodal forces which act at the ends of the elements into which the member is discretised by either lumping or consistent nodal load approaches. For a contemporary geometrically non-linear analysis in which the axial force in the member is large, accurate solutions are achieved by discretising the member into many elements, which can produce unfavourable consequences on the efficacy of the method for analysing large steel frames. Herein, a numerical technique to include the transverse loading in the non-linear stiffness formulation for a single element is proposed, and which is able to predict the structural responses of steel frames involving the effects of first-order member loads as well as the second-order coupling effect between the transverse load and the axial force in the member. This allows for a minimal discretisation of a frame for second-order analysis. For those conventional analyses which do include transverse member loading, prescribed stiffness matrices must be used for the plethora of specific loading patterns encountered. This paper shows, however, that the principle of superposition can be applied to the equilibrium condition, so that the form of the stiffness matrix remains unchanged with only the magnitude of the loading being needed to be changed in the stiffness formulation. This novelty allows for a very useful generalised stiffness formulation for a single higher-order element with arbitrary transverse loading patterns to be formulated. The results are verified using analytical stability function studies, as well as with numerical results reported by independent researchers on several simple structural frames.

Design of light gauge steel roofing systems subject to high wind forces

Light gauge steel roofing systems made of thin profiled roof sheeting and battens are used commonly in residential, industrial and commercial buildings. Their critical design load combination is that due to wind uplift forces that occur during high wind events such as tropical cyclones and thunderstorms. However, premature local failures at their screw connections have been a concern for many decades since cyclone Tracy that devastated Darwin in 1974. Extensive research that followed cyclone Tracy on the pull-through and pull-out failures of roof sheeting to batten connections has significantly improved the safety of roof sheeting. However, this has made the batten to rafter/truss connection the weakest, and recent wind damage investigations have shown the failures of these connections and the resulting loss of entire roof structures. Therefore an experimental research program using both small scale and full scale air-box tests is currently under way to investigate the pull-through failures of thin-walled steel battens under high wind uplift forces. Tests have demonstrated that occurrence of pull-through failures in the bottom flanges of steel batttens and the need to develop simple test and design methods as a function of many critical parameters such as steel batten geometry, thickness and grade, screw fastener sizes and other fastening details. This paper presents the details of local failures that occur in light fauge roofing systems, a review of the current design and test methods for steel battens and associated short comings, and the test results obtained to date on pull-through failures of battens from small scale and full scale tests. Finally, it proposes the use of suitable small scale test methods that can be used by both researchers and manufacturers of such screw-fastened light gauge steel batten systems.

Effect of crack depth and specimen width on fracture toughness of a carbon steel in the ductile-brittle transition region

The effects of crack depth (a/W) and specimen width W on the fracture toughness and ductile±brittle transition have been investigated using three-point bend specimens. Finite element analysis is employed to obtain the stress-strain fields ahead of the crack tip. The results show that both normalized crack depth (a/W) and specimen width (W) affect the fracture toughness and ductile±brittle fracture transition. The measured crack tip opening displacement decreases and ductile±brittle transition occurs with increasing crack depth (a/W) from 0.1 to 0.2 and 0.3. At a fixed a/W (0.2 or 0.3), all specimens fail by cleavage prior to ductile tearing when specimen width W increases from 25 to 40 and 50 mm. The lower bound fracture toughness is not sensitive to crack depth and specimen width. Finite element analysis shows that the opening stress in the remaining ligament is elevated with increasing crack depth or specimen width due to the increase of in-plane constraint. The average local cleavage stress is dependent on both crack depth and specimen width but its lower bound value is not sensitive to constraint level. No fixed distance can be found from the cleavage initiation site to the crack tip and this distance increases gradually with decreasing inplane constraint.