Modeling of semi-rigid beam-to-column steel joints under extreme loading

Abstract

Joints play an important role in providing ductility for steel-composite structures subject to extreme loading conditions, such as blast, fire and impact. Due to sound energy dissipation capability and fabrication efficiency, semi-rigid joints have increasingly received attention during the last decade. This paper presents a component approach for modeling semi-rigid beam-to-column joints based on Eurocode3, where the post-elastic response, including component strain hardening and ultimate rotational capacity, is also considered. Failure criteria are defined based on the ultimate deformation capacity of components and bolt-rows. The model enables a direct integration of joint response into global frame models with the consideration of axial deformability, such that the interaction between bending moment and axial force within the joints can be realistically captured. In addition, elevated temperature can be considered in the joint model via the degradation of the component response. Through comparisons with available test data, the joint model is shown to have good accuracy, and the failure criteria are found to be reliable yet conservative. The strain hardening response of components is shown to have significant influence on the ultimate bending capacity of the joints, while neglecting it usually leads to a conservative prediction.

Keywords

semi-rigid joint / component method / ductility / elevated temperature

Introduction

Under extreme loading conditions, extensive inelastic deformations are expected, which are usually associated with excessive ductility demands concentrated in the joint regions near the affected structural members in typical steel-composite structures. Fire scenarios can further complicate the prediction of the response of such joints due to an increased uncertainty of their resistance and ductility. Traditional anti-fire treatments for joints consist of applying a fire protection coating or concrete encasement, where the joints can be designed for reduced or even ambient temperature. However, applying joint fire protection is costly, so there has been more interest in understanding the behavior of unprotected joints. Therefore, the ability of joints to accommodate excessive ductility demands without failure under both ambient and elevated temperature conditions becomes an essential factor for the mitigation of possible disastrous structural failure.

Commonly used approaches for investigating the large deformation and fire responses of semi-rigid joints include experimental tests [1-4], curve-fitting expressions of previous experimental results [5,6], detailed 3D finite element models [7-11], and the component method initially proposed by Eurocode3 [12]. Among the various available assessment methods, tests on joints are the most realistic one, but experimental data are only available for limited types of joints, and there are even discrepancies in the response between joints of the same type. Curve-fitting expressions can provide an efficient way for predicting the moment-rotation response of joints, and it can be easily included in frame models, but its application is limited to the specific joints that have been tested. Detailed finite element (FE) modeling offers an alternative reliable approach that can accurately predict the full response of joints; however, 3D joint FE models are computationally expensive and are difficult to incorporate directly into a global frame model due to their complexity.

The component method or the spring-assembly method was first introduced in Eurocode3 [12] as a simplified analytical approach for predicting the joint behavior under ambient conditions. The principle of this method is to subdivide a joint into several tension zones (T-stubs) and a compression zone, and each zone is further divided into basic components of known characteristics. The component method can possess sufficient accuracy if the properties (i.e., force-deformation relationship) of each component are represented accurately. In current research and design practice [12-14], the application of this method is mainly focused on obtaining a simplified moment-rotation relationship of major axis end-plate joints under ambient conditions, and applying this relationship for rotational springs embedded at the position of the joints in a global frame model, as shown in Fig. 1(a). In this frame model, the influence of the axial force on the joint response is not considered, which is unrealistic at large deflection, e.g., at catenary stage. As an alternative approach, a set of axial springs with predefined stiffness and resistance can be employed into the global frame model to represent individual bolts rows and compression zones, as illustrated in Fig. 1(b). Through this method, the axial force and moment resistances of the joints are coupled, and equilibrium of the joint internal forces is automatically satisfied via the compatibility conditions between the rigid links replicating the beam-end and column-face. In case of fire, temperature effects can be considered by degrading the properties of these springs. Furthermore, the spring approach is capable of simulating uneven temperature distributions across the joint, which can be implemented by degrading each spring independently based on the different temperatures at the locations of each bolt-row. In view of the advantages stated above, the component method as shown in Fig. 1(b) is considered for joint modeling in this paper, and the finite element program ADAPTIC [15] is employed to model the spring assemblies that replicate the joints.

In the following discussions, general procedures of the component method are presented first, where the properties of various joint components are discussed in detail. Considering the fact that Eurocode3 [12] mainly deals with the elastic response of the component, post-elastic component response (e.g., strain hardening and failure criteria) is further investigated. This is then followed by presenting a reliable strategy to consider the effect of elevated temperature on the joint model. Finally, available joint tests conducted by other researchers are used to compare with the current component joint model, where focus is given to the comparison of moment-rotation response and ductility supply under combined bending and normal actions.

General procedure of component method in EC3

Two assessment steps, i.e., individual component response and assembled joint response, are generally involved in the codified component method in order to establish the joint moment-rotation response. Considering a typical steel joint as shown in Fig. 2(a), the first step is to decompose the joint into several bolt-row/bolt-row group springs and contact springs as illustrated in Fig. 2(b), which can be further decomposed into a series of individual active components. The second step is to assemble the properties of these components and incorporate them into a rotational spring featuring the obtained moment-rotation characteristic, as shown in Figs. 2(c) and (d). The moment resistance and stiffness of the joint can be derived from the resistances and stiffness of its basic components to the applied rotation. The obtained moment-rotation relationship can be employed in further elastic-plastic global analysis on a frame model.

Step 1 – component response

The moment-rotation response is determined by the force-deformation characteristics of the components. Figure 3 illustrates a typical flush end-plate steel joint which can be decomposed into the individual components as noted [16]. Each component normally has two important elastic characteristics, which are the design resistance F_Rd and the stiffness coefficient k, where k is a general coefficient and the real stiffness is obtained by multiplying k with E (elastic modulus of the material, e.g., steel). One of the novel features of the component method is to model the deformations of the end-plate, the column flange, and the steel bolts by means of an equivalent T-stub with an effective width, as shown in Fig. 4.

Although the current application of the component method in design codes refers to specific joint configurations, e.g., end-plate joints, the component method can in principle be applied to practically any type of joint by decomposing the joint into the relevant components, provided that each component is comprehensively investigated. Due to different roles that components play in a certain equilibrium state, three component zones can be identified, namely, tension zone components, compressive zone components, and shear zone components. Table 1 lists the components that are normally considered in several widely used major-axis steel joint types, where the components considered for each joint type can be selected from a ‘component library’. For the web/flange cleat joint and the fin plate joint in the current table, the bottom beam flange is assumed not to directly contact the column face in the compressive zone due to the presence of gaps. However, when a more considerable rotation is considered, additional compressive zone components, e.g., beam web/flange in compression, should be activated to represent the gap closure. Moreover, when double-sided joints are considered, some components given in Table 1 may be deactivated, e.g., column web in shear under equal and opposite moments. The design resistance F_Rd and stiffness coefficient k expressions for all the components in end-plate joints and top/bottom flange cleat joints are specified in detail in Eurocode3 [12].

Step 2 – joint moment-rotation response

The design moment resistance M_j,Rd and the rotational stiffness S_j are two important parameters determining the moment-rotation characteristic of joints. In elastic global analysis, a linear relationship between the bending moment and the joint rotation is assumed, as illustrated in Figs. 5 (a) and (b). If the design bending moment M_j,Ed does not exceed 2/3 M_j,Rd, the initial rotational stiffness S_j,ini may be taken directly in the global analysis. As a simplified and conservative approach, the rotational stiffness S_j may be taken as S_j,ini/m in the analysis for all values of the design bending moment M_j,Ed, where m is the stiffness modification coefficient specified in Eurocode3 [12].When an elasto-plastic global analysis is conducted, the post-yield response can be considered in an idealised bilinear manner, as shown in Fig. 5(c).

The design moment resistance M_j,Rd of a bolted beam-to-column joint is calculated as

(1)M_{j,Rd}={\displaystyle \sum _r{h}_r{F}_{tr,Rd}},

where F_{tr,Rd} is the effective design tension resistance of bolt-row r, and is taken as the smallest component resistance in that bolt-row; h_r is the distance from bolt-row r to the center of compression, for bolted end-plate joints, the center of compression should be assumed to be in line with the center of the compression flange of the connected member, provided its compressive resistance exceeds the tensile resistance, and r is the bolt-row number starting from the furthest bolt-row from the center of compression. The design tension resistanceF_{tr,Rd} for each bolt-row should be determined in sequence from bolt-row 1 (furthest from the center of compression) toward nearer bolt-rows. For each bolt-row considered, the resistance F_{tr,Rd} should be calculated either individually or as a group with the bolt-rows further away from the center of compression, whichever is conservative. All other bolt-rows closer to the center of compression should be ignored.

Provided that the axial force applied on the connected member (e.g., beam) does not exceed 5% of the design resistance N_pl,Rd of its cross-section (this limit is not required in the current joint model), the rotational stiffness S_j of a typical major axis end-plate joint subject to a bending moment of M_j,Ed less than the design moment resistance M_j,Rd can be calculated from

(2)S_j={\displaystyle \frac{E_s{z}_{eq}^{\mathrm{2}}}{\mu \left({\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{1}}{k_{eq}}}\right)}},

where E_s is the elastic modulus of steel, k_{\mathrm{1}} is the stiffness coefficient of shear zone components, k_{\mathrm{2}} is the stiffness coefficient of compression zone components, k_{eq} is a single equivalent stiffness representing the characteristic of all bolt-rows, as given by

(3)k_{eq}={\displaystyle \frac{{\displaystyle \sum _r{k}_{eff,r}{h}_r}}{z_{eq}}},

where k_{eff,r} is the effective stiffness coefficient for each bolt-row calculated from all the active components in this bolt-row, z_{eq} is the equivalent lever arm given by

(4)z_{eq}={\displaystyle \frac{{\displaystyle \sum _r{k}_{eff,r}{h}_r^{\mathrm{2}}}}{{\displaystyle \sum _r{k}_{eff,r}{h}_r}}}.

Post-elastic response and ductility

With respect to the maximum rotation capacity of bolted joints, Eurocode3 [12] only presents several specific conditions where an adequate rotation capacity can be assumed for the joint in plastic global analysis, otherwise tests are recommended. Unlike normal loading conditions, larger deflections are inevitable when structures are subjected to extreme loading. The vulnerability to structural failure greatly depends on the capability of the joints to withstand large deformations. Predictions of joint ductility supply under fire require a clear definition of failure criteria that can explicitly account for the ductility supply of each individual joint component. In other words, the joint ductility capacity is closely associated with the deformation capacity of each active component. Although relevant research has been started on this front [17,18], available information with regard to the joint component ductility supply under fire conditions is still mainly empirical and insufficient.

In general, the behavior of each component can be represented by a force-deformation curve, which is usually idealized by multilinear response. For a typical bilinear force-deformation curve, the first segment represents the elastic performance with a stiffness of k_e before yielding, which is followed by a post-elastic curve with a stiffness k_pl = m₁k_e, where m₁ is the strain hardening coefficient. When a further reduced stiffnessm₂k_e needs to be considered beyond the ultimate strength f_u, a trilinear curve can be adopted. A piecewise linear force-deformation curve (e.g., bilinear curve) for each component is comprised of four key parameters, namely, elastic resistance F_Rd, elastic stiffness k, post-elastic stiffness m_ik, and maximum deformation capacity u_max, such that the overall joint behavior can be obtained from these properties, as depicted in Fig. 6. The calculations of the first two parameters (elastic resistance and elastic stiffness) for the joint components have already been comprehensively presented in Eurocode3 [12], but so far fewer investigations have been devoted to a systematic evaluation of the post-elastic response (post-elastic stiffness and maximum deformation capacity) for each individual component.

With respect to post-elastic stiffness of components, Eurocode3 [12] suggests a zero value as a conservative solution in elasto-plastic global analysis. In reality, however, the post-elastic stiffness may be considerable for some components, especially for highly ductile components; hence, the neglect of the post-elastic stiffness may lead to an underestimation of the bending resistance of joints. Table 2 summarizes the values of the component post-elastic stiffness studied by relevant researchers [19-24]. It can be seen that the predictions can vary considerably, which may be due to different joint configurations and material properties considered, as well as the diverse modeling methods. Despite the high level of uncertainty, it is found that most predictions of the component post-elastic stiffness fall between 1% and 5% for the bilinear response, except the predictions from [22], who gave much higher values of strain hardening coefficients.

Apart from the component post-elastic stiffness that contributes to the joint post-elastic behavior, the ultimate component deformation is another significant post-elastic parameter used for predicting joint failure. A component classification method with respect to ductility was first proposed by Kuhlmann et al. [25], where three main ductility classes were identified, namely, high ductility joint components, limited ductility joint components, and brittle joint components. For the components falling into the high ductility category, the force-deformation curve transfers from the linear elastic range into a second carrying mode allowing increasing resistance with deformation. The deformation capacity of the high ductility component was typically defined as infinite. The components that belong to the limited ductility class are characterized by the load-deformation curves with an elastic range and a subsequent limited post-elastic response followed by failure. For brittle components, little post-yield deformation is achieved, so failure is conservatively assumed at the point where yield strength is attained. The force-deformation responses (trilinear approximation) of the three component classes are illustrated in Fig. 7.

Based on this concept, a ductility index j = d_f/d_y for each component was further developed by Kuhlmann et al. [25] and Simões da Silva et al. [19], where d_f is the maximum deformation, and d_y is the elastic yielding deformation for each component. For the components with high ductility, the ductility index j is suggested as infinite or at least larger than 20. For limited ductility class components, a ductility index value of 3 to 20 is recommended. For brittle components, little post yield deformation is performed, so the ductility index j can be conservatively taken as 1.0. The elastic response and ductility index for typical joint components considered for end-plate joints and web cleat joints are provided in Table 3, where most of the listed ductility indices are based on [19] except the component ‘plates in bearing’ which was not initially provided. Fang et al. [26,27] proposed a ductility index of 15 based on the research of Sarraj [28], and this value is also adopted in this study.

Each component has now been categorized into its corresponding ductility classification, and the ductility index d_u/d_y for all the components is determined. Notwithstanding, as noted in Table 3, all the ‘high ductility’ components have an infinite deformation capacity in tension, which is unrealistic. In this study, the maximum deformations of 25 and 35 mm are employed as an additional failure criterion for all bolt-rows in end-plate joints and cleat/angle/fin plate joints, respectively, in accordance with relevant test data [29,30].

Influence of elevated temperature

Joints in fire exhibit significant reductions in strength and stiffness, and consequently the global stability and robustness of the structure subject to fire can be greatly reduced. Therefore, a method that can predict the actual response of joints subjected to fire has to be carefully implemented. Toward this aim, a ‘reduction factor approach’ is employed to estimate the elevated temperature behavior of joints, where the reduction factors for strength and stiffness (SRF) are determined. Based on three sets of bare-steel joint tests, Al-Jabri et al. [23] presented the observed strength and stiffness reduction factors of these joints, as shown in Fig. 8. According to the research of Al-Jabri et al., Ramli-sulong et al. [24] proposed new strength and stiffness reduction factors for the material of joints under fire, and employed them for a new joint element developed in ADAPTIC [15]. In this work, the component strength and stiffness reductions factors proposed by [24] are slight modified, as given in Fig. 8 and Table 4, which demonstrate closer trends compared with the results of [23].

There is no doubt that the fire degradation properties of an entire joint depend on the elevated temperature behavior of each individual joint component. Therefore, the reduction factors provided in Table 4 can be applied to individual components in the joint models which are simulated via spring assemblies. To address this in ADAPTIC, a newly developed spring element, based on piece wise linear degradation approximation of stiffness and strength with temperature, is employed for this study. Under elevated temperature, the spring element can feature a good fit to the reduction factors given in Fig. 8. It is noted that the current joint model only considers material degradations of joints, whereas the effect of thermal expansion within joints is considered to be negligible.

Summarizing all the above discussions, the key assumptions employed in the study for joint modeling are noted as follows:

1) The spring and rigid elements in ADAPTIC [15] are used to model the joint components and the rigid links interconnecting these components, respectively.

2) The force-deformation response of the components is represented by piecewise multilinear approximations, and the degradation properties of these components at elevated temperature are based on Table 4.

3) Failure occurs when the deformation capacity of either joint components or bolt-rows is exceeded, according to Table 3. This failure criterion is considered to be unchanged with temperature.

Validation of joint modeling

The joint modeling technique is validated through comparisons with selected ambient and fire tests. The tested joints were subjected to a combined tensile and bending action, which reflects a real load resisting mechanism of the joints under extreme loading conditions.

Flush end-plate joint

An experimental program on flush end-plate joints was conducted by Yu et al. [31]. The combined effect of axial force and bending moment was realized through a specially designed test rig as shown in Fig. 9. Three load angles (35°, 45°, 55°) and four levels of joint temperature (20°C, 450°C, 550°C, 650°C) were considered. The specimen was heated slowly until the desired temperature, and a gradually increased load was subsequently applied. The test results were given as the resultant force (force in the furnace bar) vs. rotation relationships under the four temperatures. All the tested joints were loaded until failure except the two ambient tests on joints with 35° and 45° load angles due to system malfunction.

The corresponding component/spring model is illustrated in Fig. 10, where altogether five spring series are considered to represent the three bolt-rows as well as the top and bottom beam flange-to-column contacts. For the three bolt-row spring series, the axial property in tension is contributed from four components, namely, column web in tension (cwt), column flange in bending (cfb), bolt in tension (bt), and end-plate in bending (epb). The “group effect” is considered for determining the effective widths of the upper two bolt-rows (T-stubs). The compressive characteristic for the five spring series are based on the resistance of column web in compression (cwc), while for the top and bottom outer spring series, the component of beam flange/web in compression (bfwc) is additionally considered. The effect of column web in shear (cws) is ignored in the current model due to relatively strong column and weak end-plate. Four post-elastic responses for each ductile component are considered, namely, no strain hardening (m = 0), bilinear response (m = 1%), bilinear response (m = 3%), and trilinear response (m₁ = 3%, m₂ = 1%). The material properties for steel beams are E = 176 kN/mm², f_y = 356 N/mm², and f_u = 502 N/mm². For the endplates the material properties are E = 166 kN/mm², f_y = 356 N/mm², and f_u = 502 N/mm². The properties for S355 column steel was not given in Yu et al. (2010), so typical values E = 195 kN/mm², f_y = 450 N/mm², and f_u = 560 N/mm² are employed.

The results obtained from the spring models are illustrated in Figs. 11 to 13 for the tests with the load angles of 35°, 45°, and 55°, respectively. Favorable comparisons are found between the spring model predictions and the test results within the elastic range, thus indicating a reliable elastic force-deformation response for each component listed in Table 3. However, the post-elastic joint behavior of the spring models can differ greatly with various values of the component post-elastic stiffness. It can be seen that under the ambient condition or a relatively low temperature (i.e., 450°C), the spring model with a 1% strain hardening coefficient for its ductile components has the best correlation with the tests, whereas under higher temperatures (i.e., 550°C and 650°C), the tests results are more consistent with the spring models employing components with no strain hardening. In some tests, the resistance even starts to decrease in the plastic range. This implies that low strain hardening coefficients may be assumed for joints under relatively high temperatures. With respect to the rotation capacity, the test results show that most of the tested joints achieved their maximum resistance at a rotation of around 2°, and they were able to maintain a moderate amount of resistance up to a rotation of 7°. According to the predictions from the component models, the failure rotations for most cases are approximately between 5° to 6°, which offer a reasonable prediction on the conservative side.

Double web cleat joint

The selected double web-cleat joint tests were undertaken by Yu et al. [4].A combined effect of axial force and bending moment was applied through the same test rig as introduced above. The geometric configuration of the joint specimen is shown in Fig. 14. Three load angles (35°, 45°, 55°) and four levels of joint temperature (20°C, 450°C, 550°C, 650°C) were considered. The specimens were heated slowly until the desired temperatures, and the load was then applied.

The component model for simulating the tests is shown in Fig. 15, where altogether five spring series are considered to represent the three bolt-rows as well as the top and bottom beam flange-to-column contact areas. For the three bolt-row spring series, the axial property in tension is contributed from six components, namely, column flange in bending (cfb), bolts in tension (bt), bolts in shear (bs), angle plate in bending (ab), angle plate in bearing (abr), and beam web in bearing (bwbr). For the three inner spring series in compression, bolts in shear (bs), angle plate in bearing (abr), and beam web in bearing (bwbr) are considered. The effective widths of the three bolt-rows (T-stubs) are taken as the smaller value of those considered individually or as part of a group. The compressive characteristic for the two outer spring series are based on the resistance of column web in compression (cwc) and beam web/flange in compression (bwfc). It should be noted that the responses of angle plate in bearing (abr) and beam web in bearing (bwbr) are different in tension and in compression. The ambient material properties for steel beams are E = 176 kN/mm², f_y = 356 N/mm², and f_u = 502 N/mm². For the angle plates the material properties are E = 134 kN/mm², f_y = 350 N/mm², and f_u = 455 N/mm². The properties for S355 column steel were not given in Ref. [4], so typical values E = 195 kN/mm², f_y = 450 N/mm², and f_u = 560 N/mm² are employed.

Since the bolt-hole diameter is 2 mm wider than the bolt diameter (20 mm), it is assumed in the model that the bolts are initially located at the center of the bolt-hole. This is simulated through additional gap-contact elements applied at the three bolt-rows. The friction effect is not considered in this model. Furthermore, two gap-contact elements are applied at the positions of beam top and bottom flanges to consider the 10 mm gap between the column flange face and the beam. For the post-elastic stiffness, bilinear curves are adopted, and three strain hardening coefficients (0%, 3%, 5%) are employed for the ductile components. The ADAPTIC results are compared with the test results as well as the results from the mechanical model developed by Yu et al. [4,14].

The results obtained from the component models are given in Figs. 16 to 18 for the tests with the load angles of 35°, 45°, and 55°, respectively. It is shown that the ADAPTIC predictions compare well with the test results, although for some cases the rotation capacity the joint is underestimated, particularly at ambient conditions. This is due to the limit of 35 mm maximum bolt-row tensile deformation capacity which can be conservative. The discrepancies of the results may also be attributed to the neglect of friction in the component model. In addition, it is assumed in the model that the load angle remains unchanged during the loading procedure, while during the test it was found that the load angle can be changed by±3°.

Conclusions

This paper has discussed the main component joint modeling techniques which can be employed in overall structural modeling frameworks. Emphasis is given to the current codified design procedure and its extension to different joint types (e.g., web cleat joint) under large deformations and elevated temperature conditions. Importantly, failure criteria of joints are proposed, which are essential to the investigation of the overall structural ductility supply which determines resistance to structural failure, e.g., progressive collapse. According to the predefined failure criteria and the elastic characteristics (i.e., initial stiffness and yield resistance) of the components provided in Eurocode3 and by other researchers, component joint models which are based on ADAPTIC spring and rigid link elements are developed. Available ambient and fire joint tests are selected to validate the component models, and conclusions are drawn as follows:

1) Favorable comparisons are found between the results obtained from the tests and the corresponding component models, particularly in terms of initial joint rotation stiffness under bending or combined bending-axial actions. This is essential for reliable predictions of joint performance under extreme loading, where significant axial forces may develop during the compressive arching or catenary stages. Furthermore, elevated temperature effects are considered via degradation of component properties, a strategy shown to offer good accuracy.

2) The post-elastic stiffness of ductile components has a significant influence on the elasto-plastic response of the considered joints. In general, assuming an elastic-perfectly plastic component force-deformation response (i.e., no strain hardening) can lead to an underestimation of the moment capacity. According to the findings of the joint tests considered for this study, a strain hardening coefficient of between 1% and 3% for ductile components typically yields favorable predictions for most cases.

3) With the ductility supply of joints considered in terms of rotation capacity, the joint multi-failure criteria defined in this study are found to be conservative in most cases, especially for joints under elevated temperature conditions where steel may be more ductile than that at ambient temperature. Toward a more clear understanding of the rotation capacities of various types of joint, further experimental investigations are needed.

References

[1] Lawson R M. Behaviour of steel beam-to-column connections in fire. Structural Engineer, 1990, 68(14): 263-271

[2] Leston-Jones L C, Burgess I W, Lennon T, Plank R J. Elevated-temperature moment-rotation tests on steelwork connections. Proceedings of the Institution of Civil Engineers. Structures and Buildings, 1997, 122(4): 410-419

[3] Wang Y C, Dai X H, Bailey C G. An experimental study of relative structural fire behaviour and robustness of different types of steel joint in restrained steel frames. Journal of Constructional Steel Research, 2011, 67(7): 1149-1163

[4] Yu H X, Burgess I W, Davison J B, Plank R J. Tying Capacity of Web Cleat Connections in Fire,Part 1: Test and Finite Element Simulation. Engineering Structures, 2009, 31(3): 651-663

[5] Al-Jabri K S, Burgess I W, Lennon T, Plank R J. Moment–rotation–temperature curves forsemi-rigid joints. Journal of Constructional Steel Research, 2005, 61(3): 281-303

[6] El-Rimawi J A, Burgess I W, Plank R J. The influence of connection stiffness on the behaviour of steel beams in fire. Journal of Constructional Steel Research, 1997, 43(1-3): 1-15

[7] Al-Jabri K S, Seibi A, Karrech A. Modelling of unstiffened flush end-plate bolted connections in fire. Journal of Constructional Steel Research, 2006, 62(1-2): 151-159

[8] Liu T C H. Finite element modelling of behaviour of steel beams andconnections in fire. Journal of Constructional Steel Research, 1996, 36(3): 181-199

[9] Qian Z H, Tan K H, Burgess I W. Numerical and analytical investigations of steel beam-to-column joints at elevated temperatures. Journal of Constructional Steel Research, 2009, 65(5): 1043-1054

[10] Dai X H, Wang Y C, Bailey C G. Numerical modelling of structural fire behaviour of restrained steel beam–column assemblies using typical joint types. Engineering Structures, 2010, 32(8): 2337-2351

[11] Kalogeropoulos A, Drosopoulos G A, Stavroulakis G E. Thermal–stress analysis of a three-dimensional end-plate steel joint. Construction & Building Materials, 2012, 29: 619-626

[12] EN 1993–1-8:2005. Eurocode 3: Design of Steel Structures – Part 1–8: Design of joints. European Committee for Standardization, Brussels, Belgium

[13] Block F M, Burgess I W, Davison J B, Plank R J. The development of a component-based connection element for endplate connections in fire. Fire Safety Journal, 2007, 42(6-7): 498-506

[14] Yu H X, Burgess I W, Davison J B, Plank R J. Tying capacity of web cleat connections in fire, Part 2: Development of component-based model. Engineering Structures, 2009, 31(3): 697-708

[15] Izzuddin B. Nonlinear dynamic analysis of framed structures. Dissertation for the Doctoral Degree. London: University of London, 1991

[16] Vlassis A G. Progressive collapse assessment of tall buildings. Dissertation for the Doctoral Degree. London: Imperial College London, 2007

[17] Spyrou S, Davison J B, Burgess I W, Plank R J. Experimental and analytical investigation of the ‘compression zone’ component within a steel joint at elevated temperatures. Journal of Constructional Steel Research, 2004, 60(6): 841-865

[18] Spyrou S, Davison J B, Burgess I W, Plank R J. Experimental and analytical investigation of the‘tension zone’ component within a steel joint at elevated temperatures. Journal of Constructional Steel Research, 2004, 60(6): 867-896

[19] Simões da Silva L, Santiago A, Vila R. Post-limit stiffness and ductility of end-plate beam-to-column steel joints. Computers & Structures, 2002, 80(5-6): 515-531

[20] Lima L R O, da Vellasco P C G, da Silva J G S, Borges L A C, da Silva L A P S. Post-limit stiffness prediction of semi-rigid joints using genetic algorithms. Latin American Journal of Solids and Structures, 2005, 2: 305-320

[21] Ren P, Crisinel M. Prediction method for moment–rotation behaviour of composite beam to steel column joint. Joints in steel structures III: Behaviour, strength and design. In: Proceedings of 3rd international workshop on connections in steel structures. Trento University, Italy, 1995, 33-46.

[22] Del Savio A A, Nethercot D A, Vellasco P C G S, Andrade S A L, Martha L F. Generalised component-based model for beam-to-column joints including axial versus moment interaction. Journal of Constructional Steel Research, 2009, 65(8-9): 1876-1895

[23] Al-Jabri K S, Burgess I W, Plank R J. Prediction of the degradation of Joint characteristics at elevated temperature. Journal of Constructional Steel Research, 2004, 60(3-5): 771-781

[24] Ramli-Sulong N H, Elghazouli A Y, Izzuddin B A. Behaviour and design of beam-to-column connections under fire conditions. Fire Safety Journal, 2007, 42(6-7): 437-451

[25] Kuhlmann U, Davison J B, Kattner M. Structural systems and rotation capacity. In: Proceeding of COST Conference on Control of the Semi-rigid Behaviour of Civil Engineering Structural Joints. Liege, Belgium, 1998, 167-176

[26] Fang C, Izzuddin B A, Elghazouli A Y, Nethercot D A. Robustness of steel-composite building structures subject to localised fire. Fire Safety Journal, 2011, 46(6): 348-363

[27] Fang C, Izzuddin B A, Obiala R, Elghazouli A Y, Nethercot D A. Robustness of multi-storey car parks under vehicle fire. Journal of Constructional Steel Research, 2012, 75: 72-84

[28] Sarraj M. The behaviour of steel fin plate joints in fire. Dissertation for the Doctoral Degree. Sheffield: The University of Sheffield, 2007

[29] Jarrett N D. Axial Tests on Beam/Column Joints. BRE Client Report CR 55/90, Building Research Establishment, Garston, Watford, UK, 1990

[30] Owens G W, Moore D B. The Robustness of Simple Joints. Structural Engineer, 1992, 70(3): 37-46

[31] Yu H X, Burgess I W, Davison J B, Plank R J. Experimental and numerical investigations of the behaviour of flush endplate connections at elevated temperatures. Journal of Structural Engineering, 2011, 137(1): 80-87

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg