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Spring-stiffness model for flexible end-plate bare-steel joints in fire
Journal of Constructional Steel Research 61 (2005) 1672–1691 www.elsevier.com/locate/jcsr
Spring-stiffness model for flexible end-plate bare-steel joints in fire K.S. Al-Jabria,∗, I.W. Burgessb, R.J. Plankc a Department of Civil and Architectural Engineering, College of Engineering, Sultan Qaboos University,
P.O. Box 33, Al-Khod, Post Code 123, Oman
b Department of Civil and Structural Engineering, University of Sheffield, Sheffield, S10 2TN, UK c Department of Architectural Studies, University of Sheffield, Sheffield, S10 2TN, UK
Received 6 September 2004; accepted 3 May 2005
Abstract This paper describes a spring-stiffness model developed to predict the behaviour of flexible endplate bare-steel joints at elevated temperature. The joint components are considered as springs with predefined mechanical properties (i.e. stiffness and strength). They are also assumed to follow a trilinear force–displacement relationship. The elevated temperature joint’s response can be predicted by assembling the stiffnesses of the components which are assumed to degrade with increasing temperatures based on the recommendations presented in the design codes. Comparison of the results from the model with existing experimental data showed good agreement. Also, the predicted degradation of the joint’s stiffness and capacity compares well with the experimental results. The proposed model can be easily modified to describe the elevated temperature behaviour of other types of joint as well as joints under large rotations. © 2005 Elsevier Ltd. All rights reserved. Keywords: Bare-steel; Flexible end-plate; Connections; Joints; Elevated-temperature; Fire; Spring-stiffness model; Component model
∗ Corresponding author. Tel.: +968 2441 5335; fax: +968 2441 3416.
E-mail address: [email protected] (K.S. Al-Jabri). 0143-974X/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2005.05.003
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Nomenclature Joint’s overall response modulus of elasticity for steel at a given temperature E st Fn internal force at a given bolt row ultimate strength of the joint’s component fuc hn distance between bolt row n and the centre of rotation K cft final stiffness of the joint’s component at a given temperature K cst strain hardening stiffness of the joint’s component at a given temperature K ct elastic stiffness of the joint’s component at a given temperature K eqt equivalent single stiffness of all components in the tension zone at a given temperature the spring’s stiffness in the tension zone at a given temperature for the bolt K tt,n row under consideration M applied moment Scc joint rotational stiffness in the compression zone at a given temperature SCt global joint rotational stiffness of the bare-steel joint for a given temperature Stt joint rotational stiffness in the tension zone at a given temperature z distance from the centre of rotation to location of equivalent tension spring (the centre of the tension zone) φ rotation of the joint µs strain hardening coefficient for structural steel Column flange behaviour a half the column flange width = (Bc f /2) Bc f width of column flange D flexural rigidity of the plate Fcfpt force that causes yielding within the column flange, at a given temperature yield strength of the column flange at a given temperature fycft K cft out-of-plane stiffness of the column flange at a given temperature leff _c f effective length assuming the column flange to act as an equivalent T-stub m distance from bolt centre to 20% into the column root radius s distance from the centre-line of the column web to the centre of the bolt hole tc f thickness of column flange β dimensionless coefficient ν Poisson’s ratio for steel (≈0.3)
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Nomenclature Bolt behaviour As bolt shaft area E btt modulus of elasticity of bolts at a given temperature Fbtpt force that causes yielding within the bolts, at any given temperature f ybtt yield strength of bolts for a given temperature K btt stiffness of the bolt at a given temperature lbtt bolt elongation length Nbt number of bolts in tension at a given bolt row tbh thickness of the bolt head tbn thickness of the bolt nut tw thickness of the washer θb temperature of the bolt End-plate behaviour Dep depth of the end-plate dy assumed effective width for the end-plate segment = (Dep /n) Feppt force which causes yielding of the end-plate at a given temperature f yept yield strength of the end-plate at a given temperature K ept stiffness of the end-plate at a given temperature l AB assumed length of the end-plate element = (gauge length less diameter of bolt hole) n number of tension springs considered in the model tep thickness of the end-plate Column web behaviour beff assumed effective width of the column web beff -b effective buckling width of the column web dc depth of the web flange between fillets Dc depth of the column Fbcwt buckling force of the column web at a given temperature Fccwt force that causes crushing of the column web at a given temperature Fcwpt critical force of the column web at a given temperature f ywt column web yield strength at a given temperature k distance from the outer face of the flange to the web toe fillet K cwpt post buckled stiffness for the column web in the compression zone K cwt stiffness of the column web in the compression zone at a given temperature rc column root radius tcw thickness of the column web
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1. Introduction Flexible (or partial depth) end-plates are classified as ‘pinned’ joints with the end-plate partially welded to the beam web. They possess higher flexibility and larger rotational capacity than those joints, which are classified as ‘semi-rigid’. Flexible end-plate joints are widely used in the construction of braced multi-storey steel framed buildings due to the ease of fabrication and assemblage and speed of erection. Steel beams in multi-storey buildings are normally considered as simply supported with no transfer of moment to the columns. These joints have demonstrated significant performance in fire [1,2], albeit at large deformations, improving the survival time of the structure. In recent years, considerable research work has been conducted to understand the performance of joints at ambient temperature at both experimental and analytical modelling levels. Experimental tests were carried out on a wide variety of joints either in isolation or as part of complete steel-framed structures in order to understand their behaviour. Understanding of the behaviour of joints was further enhanced by developing analytical models which had the capability to predict the complete response of joints. Various forms of analysis and modelling methods have been suggested including simple curve-fitting techniques, simplified analytical methods and sophisticated finite element models for both bare-steel and composite joints. The European code for the design of steel structures (EC3: Part 1.8) [3] has adopted a simplified analytical procedure for the design of joints at ambient temperature. This method is based on dividing the joint into its basic components of known mechanical properties. By assembling the contributions of individual components which represent the joint as a set of rigid and deformable elements, the entire behaviour of the joint may be determined. This method is known as the springstiffness or component method. However, there is a paucity of elevated temperature component models due to the lack of experimental data that describes the joint’s behaviour. Recent experimental studies have been conducted on T-stubs [4,5] in order to define the elevated temperature mechanical characteristics of different components of the joint. Results from such studies will provide better understanding of the elevated temperature characteristics of individual components of joints, allowing the development of component models that have the capability of describing the joint response to an acceptable degree of accuracy. This consequently will lead to better understanding of the high performance of steel joints in fire and their effects on the behaviour of structural members with a fraction of testing costs and time. This paper describes a spring-stiffness model developed in an attempt to use the component method to predict the behaviour of flexible end-plate bare-steel joints at elevated temperature using the mechanical characteristics of the components that are available in the literature. In the model the joint’s components are treated as springs with predefined characteristics such as stiffness and strength. By assembling the characteristics of individual components, the joint’s response can be predicted with increasing temperatures. Only those parameters representing the stiffness and strength of the joint are degraded with increasing temperatures. Comparison of the results from the model with existing test data generated good results. Also, the predicted degradation of the joint’s stiffness and capacity compares well with the experimental results.
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Fig. 1. Behaviour of flexible end-plate joint.
2. Idealisation of flexible end-plate joint One of the main characteristics of some flexible end-plate joints is that the response has two stages (Fig. 1): stage one: the unobstructed rotation of the joint, and; stage two: the beam lower flange bears against the column with further rotation. Prior to contact, the joint is assumed to rotate about the lower edge of the end-plate, whilst after bearing the rotation is assumed to take place at the centre-line of the bottom flange resulting in an enhanced stiffness and capacity. Due to lack of elevated temperature experimental data for the second stage, only the first stage is accounted for in the model. However, a simple modification of the model is possible to accommodate this response once test data becomes available. The principle of spring-stiffness (component) models is based on dividing the joint into its basic elements as springs with defined mechanical characteristics (i.e. strength and stiffness). Components of the joint are simulated by individual springs with known stiffnesses at each bolt row which are assumed to follow a predefined force–displacement relationship. For simplicity, an equivalent single spring stiffness, K eqt , is used to represent the stiffness of all components in the tension zone in accordance with EC3: Part 1.8 [3] whilst the compression zone is defined by a separate spring, K cwt . The tension zone comprises bolt stiffness (K bt t ), end-plate stiffness (K ept ) and column flange stiffness (K cft ). An idealised representation of the flexible end-plate bare-steel joint is shown in Fig. 2. The global rotational stiffness, SCt , of the joint can be determined for any given temperature and moment based on the assembled stiffness, K eqt , of all components acting in the tension zone, and the compression zone (i.e. the column web stiffness). Therefore
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Fig. 2. General representation of the proposed model.
the overall stiffness of the joint may be expressed as: 1 1 1 = + SCt Stt Scc 1 1 1 + = 2 SCt (K eqt · z ) (K cwt · z 2 )
(1) (2)
where SCt : Stt : Scc : K eqt : z:
global rotational stiffness of the bare-steel joint for a given temperature; joint rotational stiffness in the tension zone at a given temperature; joint rotational stiffness in the compression zone at a given temperature; stiffness of the equivalent spring in the tension zone at a given temperature; distance from the centre of rotation to location of equivalent tension spring (the centre of the tension zone).
The rotation of the joint, at any given moment, M may be expressed as: φ=
M SCt
where φ:
rotation of the joint.
(3)
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2.1. Determination of the equivalent spring in the tension zone, K eqt The overall stiffness of the components in the tension zone at any bolt row at a given temperature may be expressed as: 1 1 1 1 = + + K tt,n K ept 2K cft Nbt · K btt
(4)
where K tt,n : K ept : K cft : K btt : Nbt :
spring’s stiffness in the tension zone at a given temperature, for the bolt row under consideration; stiffness of the end-plate at a given temperature; out-of-plane stiffness of the column flange at a given temperature; stiffness of the bolt at a given temperature; number of bolts in tension at a given bolt row.
EC3: Part 1.8 [3] permits the use of a single spring of equivalent stiffness to represent the stiffnesses of the springs in the tension zone where there is more than one bolt row in tension. This can be determined from the following expression: (K tt,n · h n ) n (5) K eqt = z where hn :
distance between bolt row n and the centre of rotation.
When considering the stiffness of the equivalent spring in the tension zone K eqt , it is necessary to determine the distance from the centre of rotation to the location of the equivalent tension spring. This is the lever arm, z. For bolted joints with only a single bolt row in tension, the lever arm, z, is taken as the distance from the centre of rotation to the centre-line of the bolt row in tension as illustrated in Fig. 3(a). However, for joints with more than one bolt row acting in tension (Fig. 3(b)), the lever arm, z, may be calculated by the following expression: (K tt,n h 2n ) n . (6) z= (K tt,n h n ) n
By knowing the joint’s rotation and bolt row stiffness, the internal force at a given bolt row, Fn , may be calculated as: Fn = φ · K tt,n · h n .
(7)
3. Behaviour of the joint’s components In order to determine the overall stiffness and capacity of the joint, the response of individual joint’s component must be defined. As stated earlier the joint’s components,
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(a) Single bolt row in tension.
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(b) More than one bolt row in tension.
Fig. 3. Determination of lever arm z.
which are assumed to act in the tension zone at each bolt row, consisted of column flange, end-plate and bolts whilst the column web is assumed to represent the joint’s component in the compression zone. The following sections describe the mechanical characteristics of each component. 3.1. Column flange behaviour In order to simulate the column flange in the tension zone, the model proposed by Jaramillo [6] was adopted. Jaramillo idealised the column flange as an infinitely long cantilever plate of breadth a and thickness tc f , subject to a transverse random point load P, acting at a distance s from the support as shown in Fig. 4. Hence, the out-of-plane stiffness of the column K cft in the tension zone may be given as follows: K cft =
πD βa 2
(8)
where D: β: a:
flexural rigidity of the plate; dimensionless coefficient; half the column flange width = (Bc f /2). The flexural rigidity of the plate, D, may be evaluated using the following expression: D=
E st tc3f 12(1 − ν 2 )
where E st :
modulus of elasticity for steel at a given temperature;
(9)
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(a) Column flange subject to point load.
(b) Infinitely long cantilever plate.
Fig. 4. Idealisation of column flange response in the tension zone.
Table 1 Values of coefficient β s/a β
0.25 0.0168
tc f :
column flange thickness;
ν:
Poisson’s ratio for steel (≈0.3).
0.50 0.0794
0.75 0.220
1.00 0.525
The dimensionless coefficient β is a function of the distance s which is the distance from the centre-line of the column web to the centre of the bolt hole, s (= gauge length/2), and half flange breadth, a. Representative values of β for different values of (s/a) are given in Table 1. The force Fcfpt that causes yielding within the column flange, at a given temperature, may be defined as: Fcfpt =
f ycft tc2f leff _c f m
(10)
where fycft :
yield strength of the column flange at a given temperature;
leff _c f : effective length assuming the column flange to act as an equivalent T-stub as presented in EC3: Part 1.8 [3]; m:
distance from bolt centre to 20% into the column root radius.
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3.2. Bolt behaviour The bolts are considered to be subjected to direct tensile force in isolation. By applying the principles of Hooke’s law, the elastic stiffness of the bolt K btt may be expressed as: K btt =
As · E btt lbtt
(11)
where As : E btt : lbtt :
bolt shaft area; modulus of elasticity of bolts at a given temperature; bolt elongation length.
The bolt elongation may be obtained from the following relationship based on Agerskov’s [7] recommendations: tbh + tbn (12) lbtt = tep + tc f + 2 · tw + 2 where tep : tw : tbh : tbn :
thickness of the end-plate; thickness of the washer; thickness of the bolt head; thickness of the bolt nut.
The force, Fbtpt , that causes yielding within the bolts, at any given temperature, may be defined as: Fbtpt = f ybtt · Nbt · As
(13)
where f ybtt :
yield strength of bolts for a given temperature.
3.3. End-plate behaviour When load is applied to the joint and the end-plate is pulled away from the column face, it is assumed that the element is subject to pure bending. A number of authors [8–10] have considered a flexible end-plate to act simply as a rigidly fixed beam subject to a point load. The end-plate is considered to act as a T-stub, where the flange of the T represents the endplate, and the stem simulates the beam web. The end-plate is restrained against rotation by the weld which connects the beam web, and at bolt locations. Fig. 5 shows the idealisation of the end-plate in the tension zone. From simple beam-deflection theory, it may be shown that the elastic end-plate stiffness, K ept may be expressed as: K ept = where
3 16E st d y tep
l 3AB
(14)
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Fig. 5. Idealisation of end-plate deformation.
dy : l AB : Dep : n:
assumed effective width for the end-plate segment = (Dep /n); assumed length of the end-plate element = (gauge length less diameter of bolt hole); depth of the end-plate; number of tension springs considered in the model.
The stiffness of the end-plate is assumed to be elastic until the formation of plastic hinges. The force, Feppt , which causes yielding of the end-plate at a given temperature may be expressed as: Feppt =
2 ·d 2 · fyept · tep y
l AB
(15)
where fyept :
yield strength of the end-plate at a given temperature.
It is worth remembering that the model is only applicable for the first stage of response (i.e. before the beam bears against the column), due to lack of elevated temperature data describing the second stage of response. However, a detailed description of the second stage response of the end-plate has been presented by Madas [10] for ambient temperature conditions.
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3.4. Column web behaviour The stiffness of the column web, K cwt , in the compression zone may be estimated by idealising it as a plate of dimensions dc × beff subject to a uniform compressive force from the beam over the whole depth of the end-plate. By assuming that the plate obeys Hooke’s law, the elastic stiffness of the column web may be expressed as: K cwt =
E st tcw beff dc
(16)
where tcw : beff : dc :
thickness of the column web; assumed effective width of the column web; depth of the column web between fillets.
Previous studies [11,12] on simple joints (mainly web cleats and fin-plates) have revealed that the depth of the compression zone in the column web can be as great as the depth of the end-plate. Therefore, the effective column web area resisting the compression force is substantial and overstressing of the column web is unlikely. However, for flexible end-plate joints, this may be true up to the point when the beam flange comes into contact with the column, but after this overstressing may occur in some cases. Therefore the effective width of the column flange in the compression zone may be determined from the following expression: beff = Dep + 5k
(17)
where k: k: rc :
distance from the outer face of the flange to the web toe fillet; tc f + rc ; column root radius.
When the column web is subject to a compression force, the stiffness of the column web is assumed to remain constant until one of two modes of failure occurs within the depth of the column web. These are compression (i.e. crushing) of the web close to the flange or buckling of the web over part or most of the depth of the member. The critical force, Fcwpt , for the column web may be determined as the lesser of the buckling and compression forces. The compression resistance is based on the assumption that the compression force applied is distributed across the depth of the end-plate and the column flange is assumed to act as a bearing plate. Therefore the force, Fccwt , that causes crushing of the column web at a given temperature may be given by the following expression: Fccwt = beff tcw f ywt where f ywt :
column web yield strength at a given temperature.
(18)
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When considering buckling of the column web, the effective buckling width of the column needs to be defined. The buckling force, Fbcwt , of the column web at a given temperature based on the empirical relationship suggested by Ahmed et al. [12] may be written as: 0.017 0.60 1.43 0.76 Fbcwt = 8.4 · beff · tcw · f ywt -b · Dc
(19)
where beff -b :
effective buckling width of the column web; 2 ); beff -b = (Dc2 + Dep
Dc :
depth of the column.
Therefore, the critical force, Fcwpt , of the column web may be determined as: Eq. (18) . Fcwpt = Min. of Eq. (19)
(20)
Subsequent to the onset of yielding, a reduced stiffness must be adopted. In order to estimate the post buckling stiffness of the column web, the technique proposed by Walker [13] has been used. Hence, the post buckled stiffness for the column web in the compression zone K cwpt may be expressed as: K cwpt =
K cwt . 2.45
(21)
4. Material properties and behaviour The yield and ultimate stresses of the components were taken as 545 and 412 MPa, respectively, while a value of 197 kN/mm2 was adopted for the Young’s modulus of structural steel. These values were selected based on tensile coupon tests conducted on structural steel specimens [14]. However, the bolts were assumed to have yield and ultimate stresses of 480 and 600 MPa, respectively. Geometrical properties of the steel sections were based on nominal values. The temperature profile across the joint depth was based on experimental observations [14]. At elevated temperature the entire joint response needs to be considered since large deformation and consequently rotations are possible. Strain hardening of the steel in the plastic zone may therefore have a significant influence. To model the entire joint’s response yet avoids complexities arising from complete nonlinear modelling, a trilinear force–displacement representation is proposed as shown Fig. 6. It is assumed that the joint components would experience an elastic response (represented by an elastic stiffness) until the onset of yielding as a result of yielding of one or more of the components. Then a reduced stiffness representing a strain hardening stiffness may be adopted based upon revised elemental stiffnesses. This procedure is repeated as more elements enter the strain hardening region.
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Fig. 6. Force–displacement behaviour of the joint’s components.
Following the onset of yielding, a strain hardening stiffness is adopted and may be expressed as: K cst = µs K ct
(22)
where K cst : K ct : µs :
strain hardening stiffness of the joint’s component at a given temperature; elastic stiffness of the joint’s component at a given temperature; strain hardening coefficient for structural steel.
The strain hardening coefficient is defined as the ratio of the strain hardening stiffness to the elastic stiffness and is usually based on existing data. Due to the lack of available test data, Atamiaz Sibai and Frey [15] recommended a value in the range 0.019–0.024 for mild steel at ambient temperature whilst Ren and Crisinel [16] adopted a value of 0.06 for modelling of double web cleat joints at ambient temperature. In the current model a strain hardening coefficient of 0.05 is adopted. The ultimate capacity of the component may be assessed in a similar manner to the plastic capacity as expressed in Eq. (22), replacing the yield strength of the component with the ultimate strength fuc . Following the onset of failure, the stiffness of the component approaches zero. The final stiffness, K cft , may be calculated using a reduced stiffness coefficient of 0.01 and expressed as: K cft = 0.01K ct .
(23)
5. Degradation of the joint’s characteristics at elevated temperature The degradation of stiffness and strength of the components was based on the degradation of structural steel at elevated according to EC3: Part 1.2 [17] as shown in Table 2.
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Table 2 Properties of structural steel at elevated temperature Steel temperature (◦ C)
200 100 200 300 400 500 600 700 800 900 1000 1100 1200
Reduction factors for yield stress, f y and Young’s modulus, E a , at temperature, θa k y,θ = f y,θ / f y k E,θ = E a,θ /E a 1.000 1.000 0.926 0.848 0.775 0.618 0.357 0.169 0.087 0.060 0.040 0.020 0.000
1.000 1.000 0.900 0.800 0.700 0.600 0.310 0.130 0.09 0.0675 0.0450 0.0225 0.000
However, the degradation of bolt stiffness and capacity is based on recommendations presented by Kirby [18] based on experimental tests using the following expressions: For θb ≤ 300 ◦ C SFR = 1.0.
(24a)
For θb < 300 ◦ C ≤ 680 ◦ C SFR = 1.0 − (θb − 300) × 2.128 × 10−3 .
(24b)
For θb < 680 ◦ C ≤ 1000 ◦ C SFR = 0.17 − (θb − 680) × 5.13 × 10−4 .
(24c)
where θb :
temperature of the bolt and SFR is strength retention factor of the bolt.
6. Validation of the proposed model In order to validate the spring-stiffness model under fire condition, a cruciform bolted flexible end-plate bare-steel joint tested by Al-Jabri et al. [14] was selected. This major axis joint configuration consists of two 356 × 171UB51 beams connected to a 254 × 254UC89 column by 8 mm thick flexible end-plates and eight M20 bolts. The joint detail is shown in Fig. 7. The elevated temperature tests were performed under anisothermal conditions. Two load cases were modelled at moments of 8 and 16 kN m which represent the behaviour of the joint in the first stage of rotation (until the beam flange comes into contact with the column flange).
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Fig. 7. Description of the modelled joint.
Before verifying the model against elevated temperature tests, initial verification was carried out at ambient temperature as shown in Fig. 8. This was based on relevant data from the joint fire tests as well as existing test data reported by Boreman et al. [19]. The proposed model predicts closely the initial stiffness of the joint but underestimates the response in the plastic zone. This may be attributable to the lack of experimental data in the strain hardening zone which represents a large proportion of the moment–rotation response. Also, the ultimate capacity of the joint was underestimated with a maximum difference of about 10 kN m. Fig. 8 also shows comparison between the model and the experimental ambient temperature response obtained from fire tests. The initial stiffness and strain hardening stiffness of the joint is closely represented, but it was not possible to compare the ultimate capacity because only the first stage of response (i.e. before the beam bears against the column) is accounted for in the model. Verification of the model at elevated temperature was based on the two joint fire tests. Fig. 9 compares the experimental degradation of stiffness and strength of the joint with those predicted by the model. Fig. 9(a) shows that the predicted stiffness compares well with that obtained experimentally for temperatures up to 400 ◦ C, beyond which the model gives a slight overestimate. The predicted and recorded degradation of joint strength are compared in Fig. 9(b), from which it may be seen that for temperatures between 500 and 600 ◦ C the model underestimates the strength. However, for higher temperatures there was close agreement between the experimental and predicted results. Unfortunately test data
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Fig. 8. Comparison of predicted ambient temperature response with experimental results.
Fig. 9. Degradation of joint’s stiffness and strength at elevated temperature.
was not available for temperatures below 500 ◦ C, due to the levels of loading applied during testing. Therefore, at this stage it is difficult to draw definite conclusions concerning the
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Fig. 10. Comparison of predicted elevated temperature response with fire tests.
degradation of joint capacity since only three test points were available, all at temperatures greater than 500 ◦ C. It is worth remembering that the degradation of the joint capacity was based on the first stage of behaviour. The elevated temperature component model for flexible end-plate joints was compared with two fire tests conducted at moments of 8 and 16 kN m. Fig. 10 compares the experimental temperature–rotation response for the two tests with those predicted by the model. These compare very closely for both tests. In both cases little rotation was measured for temperatures up to approximately 510 ◦ C, beyond which the joint gradually plastifies until failure due to excessive end-plate deformation. The proposed model predicts a similar response. Also the temperatures at which both test specimens failed was closely predicted by the model. 7. Conclusions A simplified spring-stiffness model was presented for modelling the elevated temperature response of flexible end-plate bare-steel joints. Joints were modelled by assembling the contributions of individual components. Tri-linear force–displacement modelling was adopted, with the separation of the joint into its main components allowing the use of any chosen temperature profile. The main parameters describing stiffness and capacity of the components were degraded with increasing temperatures. The predicted response of the joint was compared with the response observed from experimental tests. Results showed that there was good agreement between the predicted and the experimental responses. The rate of degradation of stiffness and strength of the
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joint was closely predicted by the model. In general the proposed spring-stiffness model was capable of predicting the joint response at both ambient and elevated temperatures to a reasonable accuracy. 8. Concluding remarks This research work is an initial attempt to model the behaviour of flexible end-plate joints in fire using a component model. The model was developed based on components characteristics of the joint as well as the elevated temperature behaviour of flexible end-plate joints that were presented in the literature. The proposed model needs further development to take into consideration the following features that can have significant influence on the joint’s behaviour at elevated temperature: • The applicability of the model to predict the joint behaviour at higher levels of moment than those presented; • Incorporation of the behaviour of second stage of response of flexible end-plate joints at elevated temperature. The model can easily be modified to describe the joint behaviour at large rotations. Further information may be found in the work conducted by Madas [10]. • The model was developed based on isolated joint tests, the effect of axial restrained on the joint behaviour needs to be addressed in the model since this can have significant influence on the behaviour of the structure in fire as observed from Cardington frame fire tests. Also, the joint behaviour during the cooling phase needs to be investigated.
References [1] Investigation of Broadgate phase 8 fire. In: Structural fire engineering. UK: Steel Construction Institute; 1991. [2] Robinson J. New thinking on multi-storey building fires: the Cardington fire test programme—initial conclusions. Fire Saf Eng 1997;4(6):25–6. [3] Eurocode 3. Design of steel structures, Part 1-8: general rules design of joints, prEN-1993-1-8. European Committee for Standardisation (CEN); 2003. [4] Spyrou S. Development of a component-based model of steel beam-to-column joints at elevated temperatures. Ph.D. thesis. University of Sheffield; 2002. [5] Block FM, Burgess IW, Davison JB. Numerical and analytical studies of joint component behaviour in fire. In: Third international workshop structures in fire. 2004. [6] Jaramillo TJ. Deflections and moments due to a concentrated load on a cantilever plate of infinite length. J Appl Mech ASME 1950;17(1):67–72. [7] Agerskov H. High strength bolted connections subject to prying. J Struct Div ASCE 1976;102(ST1):161–75. [8] Laurie Kennedy DJ, Hafez MA. A study of end-plate connections for steel beams. Can J Civ Eng 1984;11. [9] Owens GW, Moore DB. Steelwork connections: the robustness of simple connections. The Structural Engineer 1992;70(3):37–46. [10] Madas PJ. Advanced modelling of composite frames subject to earthquake loading. Ph.D. thesis. UK: University of London; 1993. [11] Xiao Y, Choo BS, Nethercot DA. Composite connections in steel and concrete. Part 2: moment capacity of end plate beam to column connections. J Constr Steel Res 1996;37(1):63–90. [12] Ahmed B, Li TQ, Nethercot DA. Design of fin plate and angle cleated connections. J Constr Steel Res 1997; 41(1):1–29.
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[13] Walker AC. The post buckling behaviour of simply-supported square plates. Aeronaut Q 1969;XX:203–22. [14] Al-Jabri KS, Burgess IW, Lennon T, Plank RJ. Moment–rotation–temperature curves for semi-rigid joints. J Constr Steel Res 2005;61:281–303. [15] Atamiaz Sibai W, Frey F. Numerical simulation of the behaviour up to collapse of two welded unstiffened one-side flange connections. In: Bjorhovde R et al., editors. Connections in steel structures: behaviour, strength and design. London: Elsevier Applied Science; 1988. p. 85–92. [16] Ren P, Crisinel M. Prediction method for moment–rotation behaviour of composite beam to steel column connection. In: Bjorhovde R et al., editors. Connections in steel structures III: behaviour, strength and design (Proc. of 3rd international workshop). 1995. p. 33–46. [17] Eurocode 3. Design of steel structures, Part 1-2: general rules structural fire design, prEN-1993-1-2. European Committee for Standardisation (CEN); 2003. [18] Kirby BR, Preston RR. High temperature properties of hot rolled structural steels for use in fire engineering design studies. Fire Saf J 1988;13:27–37. [19] Boreman J, Kirby PA, Davison JB. Cardington steel frame building: tests on nine additional isolated bare steel connections. Preliminary report to BRE. UK: University of Sheffield; 1995.
Spring-stiffness model for flexible end-plate bare-steel joints in fire K.S. Al-Jabria,∗, I.W. Burgessb, R.J. Plankc a Department of Civil and Architectural Engineering, College of Engineering, Sultan Qaboos University,
P.O. Box 33, Al-Khod, Post Code 123, Oman
b Department of Civil and Structural Engineering, University of Sheffield, Sheffield, S10 2TN, UK c Department of Architectural Studies, University of Sheffield, Sheffield, S10 2TN, UK
Received 6 September 2004; accepted 3 May 2005
Abstract This paper describes a spring-stiffness model developed to predict the behaviour of flexible endplate bare-steel joints at elevated temperature. The joint components are considered as springs with predefined mechanical properties (i.e. stiffness and strength). They are also assumed to follow a trilinear force–displacement relationship. The elevated temperature joint’s response can be predicted by assembling the stiffnesses of the components which are assumed to degrade with increasing temperatures based on the recommendations presented in the design codes. Comparison of the results from the model with existing experimental data showed good agreement. Also, the predicted degradation of the joint’s stiffness and capacity compares well with the experimental results. The proposed model can be easily modified to describe the elevated temperature behaviour of other types of joint as well as joints under large rotations. © 2005 Elsevier Ltd. All rights reserved. Keywords: Bare-steel; Flexible end-plate; Connections; Joints; Elevated-temperature; Fire; Spring-stiffness model; Component model
∗ Corresponding author. Tel.: +968 2441 5335; fax: +968 2441 3416.
E-mail address: [email protected] (K.S. Al-Jabri). 0143-974X/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2005.05.003
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Nomenclature Joint’s overall response modulus of elasticity for steel at a given temperature E st Fn internal force at a given bolt row ultimate strength of the joint’s component fuc hn distance between bolt row n and the centre of rotation K cft final stiffness of the joint’s component at a given temperature K cst strain hardening stiffness of the joint’s component at a given temperature K ct elastic stiffness of the joint’s component at a given temperature K eqt equivalent single stiffness of all components in the tension zone at a given temperature the spring’s stiffness in the tension zone at a given temperature for the bolt K tt,n row under consideration M applied moment Scc joint rotational stiffness in the compression zone at a given temperature SCt global joint rotational stiffness of the bare-steel joint for a given temperature Stt joint rotational stiffness in the tension zone at a given temperature z distance from the centre of rotation to location of equivalent tension spring (the centre of the tension zone) φ rotation of the joint µs strain hardening coefficient for structural steel Column flange behaviour a half the column flange width = (Bc f /2) Bc f width of column flange D flexural rigidity of the plate Fcfpt force that causes yielding within the column flange, at a given temperature yield strength of the column flange at a given temperature fycft K cft out-of-plane stiffness of the column flange at a given temperature leff _c f effective length assuming the column flange to act as an equivalent T-stub m distance from bolt centre to 20% into the column root radius s distance from the centre-line of the column web to the centre of the bolt hole tc f thickness of column flange β dimensionless coefficient ν Poisson’s ratio for steel (≈0.3)
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Nomenclature Bolt behaviour As bolt shaft area E btt modulus of elasticity of bolts at a given temperature Fbtpt force that causes yielding within the bolts, at any given temperature f ybtt yield strength of bolts for a given temperature K btt stiffness of the bolt at a given temperature lbtt bolt elongation length Nbt number of bolts in tension at a given bolt row tbh thickness of the bolt head tbn thickness of the bolt nut tw thickness of the washer θb temperature of the bolt End-plate behaviour Dep depth of the end-plate dy assumed effective width for the end-plate segment = (Dep /n) Feppt force which causes yielding of the end-plate at a given temperature f yept yield strength of the end-plate at a given temperature K ept stiffness of the end-plate at a given temperature l AB assumed length of the end-plate element = (gauge length less diameter of bolt hole) n number of tension springs considered in the model tep thickness of the end-plate Column web behaviour beff assumed effective width of the column web beff -b effective buckling width of the column web dc depth of the web flange between fillets Dc depth of the column Fbcwt buckling force of the column web at a given temperature Fccwt force that causes crushing of the column web at a given temperature Fcwpt critical force of the column web at a given temperature f ywt column web yield strength at a given temperature k distance from the outer face of the flange to the web toe fillet K cwpt post buckled stiffness for the column web in the compression zone K cwt stiffness of the column web in the compression zone at a given temperature rc column root radius tcw thickness of the column web
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1. Introduction Flexible (or partial depth) end-plates are classified as ‘pinned’ joints with the end-plate partially welded to the beam web. They possess higher flexibility and larger rotational capacity than those joints, which are classified as ‘semi-rigid’. Flexible end-plate joints are widely used in the construction of braced multi-storey steel framed buildings due to the ease of fabrication and assemblage and speed of erection. Steel beams in multi-storey buildings are normally considered as simply supported with no transfer of moment to the columns. These joints have demonstrated significant performance in fire [1,2], albeit at large deformations, improving the survival time of the structure. In recent years, considerable research work has been conducted to understand the performance of joints at ambient temperature at both experimental and analytical modelling levels. Experimental tests were carried out on a wide variety of joints either in isolation or as part of complete steel-framed structures in order to understand their behaviour. Understanding of the behaviour of joints was further enhanced by developing analytical models which had the capability to predict the complete response of joints. Various forms of analysis and modelling methods have been suggested including simple curve-fitting techniques, simplified analytical methods and sophisticated finite element models for both bare-steel and composite joints. The European code for the design of steel structures (EC3: Part 1.8) [3] has adopted a simplified analytical procedure for the design of joints at ambient temperature. This method is based on dividing the joint into its basic components of known mechanical properties. By assembling the contributions of individual components which represent the joint as a set of rigid and deformable elements, the entire behaviour of the joint may be determined. This method is known as the springstiffness or component method. However, there is a paucity of elevated temperature component models due to the lack of experimental data that describes the joint’s behaviour. Recent experimental studies have been conducted on T-stubs [4,5] in order to define the elevated temperature mechanical characteristics of different components of the joint. Results from such studies will provide better understanding of the elevated temperature characteristics of individual components of joints, allowing the development of component models that have the capability of describing the joint response to an acceptable degree of accuracy. This consequently will lead to better understanding of the high performance of steel joints in fire and their effects on the behaviour of structural members with a fraction of testing costs and time. This paper describes a spring-stiffness model developed in an attempt to use the component method to predict the behaviour of flexible end-plate bare-steel joints at elevated temperature using the mechanical characteristics of the components that are available in the literature. In the model the joint’s components are treated as springs with predefined characteristics such as stiffness and strength. By assembling the characteristics of individual components, the joint’s response can be predicted with increasing temperatures. Only those parameters representing the stiffness and strength of the joint are degraded with increasing temperatures. Comparison of the results from the model with existing test data generated good results. Also, the predicted degradation of the joint’s stiffness and capacity compares well with the experimental results.
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Fig. 1. Behaviour of flexible end-plate joint.
2. Idealisation of flexible end-plate joint One of the main characteristics of some flexible end-plate joints is that the response has two stages (Fig. 1): stage one: the unobstructed rotation of the joint, and; stage two: the beam lower flange bears against the column with further rotation. Prior to contact, the joint is assumed to rotate about the lower edge of the end-plate, whilst after bearing the rotation is assumed to take place at the centre-line of the bottom flange resulting in an enhanced stiffness and capacity. Due to lack of elevated temperature experimental data for the second stage, only the first stage is accounted for in the model. However, a simple modification of the model is possible to accommodate this response once test data becomes available. The principle of spring-stiffness (component) models is based on dividing the joint into its basic elements as springs with defined mechanical characteristics (i.e. strength and stiffness). Components of the joint are simulated by individual springs with known stiffnesses at each bolt row which are assumed to follow a predefined force–displacement relationship. For simplicity, an equivalent single spring stiffness, K eqt , is used to represent the stiffness of all components in the tension zone in accordance with EC3: Part 1.8 [3] whilst the compression zone is defined by a separate spring, K cwt . The tension zone comprises bolt stiffness (K bt t ), end-plate stiffness (K ept ) and column flange stiffness (K cft ). An idealised representation of the flexible end-plate bare-steel joint is shown in Fig. 2. The global rotational stiffness, SCt , of the joint can be determined for any given temperature and moment based on the assembled stiffness, K eqt , of all components acting in the tension zone, and the compression zone (i.e. the column web stiffness). Therefore
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Fig. 2. General representation of the proposed model.
the overall stiffness of the joint may be expressed as: 1 1 1 = + SCt Stt Scc 1 1 1 + = 2 SCt (K eqt · z ) (K cwt · z 2 )
(1) (2)
where SCt : Stt : Scc : K eqt : z:
global rotational stiffness of the bare-steel joint for a given temperature; joint rotational stiffness in the tension zone at a given temperature; joint rotational stiffness in the compression zone at a given temperature; stiffness of the equivalent spring in the tension zone at a given temperature; distance from the centre of rotation to location of equivalent tension spring (the centre of the tension zone).
The rotation of the joint, at any given moment, M may be expressed as: φ=
M SCt
where φ:
rotation of the joint.
(3)
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2.1. Determination of the equivalent spring in the tension zone, K eqt The overall stiffness of the components in the tension zone at any bolt row at a given temperature may be expressed as: 1 1 1 1 = + + K tt,n K ept 2K cft Nbt · K btt
(4)
where K tt,n : K ept : K cft : K btt : Nbt :
spring’s stiffness in the tension zone at a given temperature, for the bolt row under consideration; stiffness of the end-plate at a given temperature; out-of-plane stiffness of the column flange at a given temperature; stiffness of the bolt at a given temperature; number of bolts in tension at a given bolt row.
EC3: Part 1.8 [3] permits the use of a single spring of equivalent stiffness to represent the stiffnesses of the springs in the tension zone where there is more than one bolt row in tension. This can be determined from the following expression: (K tt,n · h n ) n (5) K eqt = z where hn :
distance between bolt row n and the centre of rotation.
When considering the stiffness of the equivalent spring in the tension zone K eqt , it is necessary to determine the distance from the centre of rotation to the location of the equivalent tension spring. This is the lever arm, z. For bolted joints with only a single bolt row in tension, the lever arm, z, is taken as the distance from the centre of rotation to the centre-line of the bolt row in tension as illustrated in Fig. 3(a). However, for joints with more than one bolt row acting in tension (Fig. 3(b)), the lever arm, z, may be calculated by the following expression: (K tt,n h 2n ) n . (6) z= (K tt,n h n ) n
By knowing the joint’s rotation and bolt row stiffness, the internal force at a given bolt row, Fn , may be calculated as: Fn = φ · K tt,n · h n .
(7)
3. Behaviour of the joint’s components In order to determine the overall stiffness and capacity of the joint, the response of individual joint’s component must be defined. As stated earlier the joint’s components,
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(a) Single bolt row in tension.
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(b) More than one bolt row in tension.
Fig. 3. Determination of lever arm z.
which are assumed to act in the tension zone at each bolt row, consisted of column flange, end-plate and bolts whilst the column web is assumed to represent the joint’s component in the compression zone. The following sections describe the mechanical characteristics of each component. 3.1. Column flange behaviour In order to simulate the column flange in the tension zone, the model proposed by Jaramillo [6] was adopted. Jaramillo idealised the column flange as an infinitely long cantilever plate of breadth a and thickness tc f , subject to a transverse random point load P, acting at a distance s from the support as shown in Fig. 4. Hence, the out-of-plane stiffness of the column K cft in the tension zone may be given as follows: K cft =
πD βa 2
(8)
where D: β: a:
flexural rigidity of the plate; dimensionless coefficient; half the column flange width = (Bc f /2). The flexural rigidity of the plate, D, may be evaluated using the following expression: D=
E st tc3f 12(1 − ν 2 )
where E st :
modulus of elasticity for steel at a given temperature;
(9)
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(a) Column flange subject to point load.
(b) Infinitely long cantilever plate.
Fig. 4. Idealisation of column flange response in the tension zone.
Table 1 Values of coefficient β s/a β
0.25 0.0168
tc f :
column flange thickness;
ν:
Poisson’s ratio for steel (≈0.3).
0.50 0.0794
0.75 0.220
1.00 0.525
The dimensionless coefficient β is a function of the distance s which is the distance from the centre-line of the column web to the centre of the bolt hole, s (= gauge length/2), and half flange breadth, a. Representative values of β for different values of (s/a) are given in Table 1. The force Fcfpt that causes yielding within the column flange, at a given temperature, may be defined as: Fcfpt =
f ycft tc2f leff _c f m
(10)
where fycft :
yield strength of the column flange at a given temperature;
leff _c f : effective length assuming the column flange to act as an equivalent T-stub as presented in EC3: Part 1.8 [3]; m:
distance from bolt centre to 20% into the column root radius.
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3.2. Bolt behaviour The bolts are considered to be subjected to direct tensile force in isolation. By applying the principles of Hooke’s law, the elastic stiffness of the bolt K btt may be expressed as: K btt =
As · E btt lbtt
(11)
where As : E btt : lbtt :
bolt shaft area; modulus of elasticity of bolts at a given temperature; bolt elongation length.
The bolt elongation may be obtained from the following relationship based on Agerskov’s [7] recommendations: tbh + tbn (12) lbtt = tep + tc f + 2 · tw + 2 where tep : tw : tbh : tbn :
thickness of the end-plate; thickness of the washer; thickness of the bolt head; thickness of the bolt nut.
The force, Fbtpt , that causes yielding within the bolts, at any given temperature, may be defined as: Fbtpt = f ybtt · Nbt · As
(13)
where f ybtt :
yield strength of bolts for a given temperature.
3.3. End-plate behaviour When load is applied to the joint and the end-plate is pulled away from the column face, it is assumed that the element is subject to pure bending. A number of authors [8–10] have considered a flexible end-plate to act simply as a rigidly fixed beam subject to a point load. The end-plate is considered to act as a T-stub, where the flange of the T represents the endplate, and the stem simulates the beam web. The end-plate is restrained against rotation by the weld which connects the beam web, and at bolt locations. Fig. 5 shows the idealisation of the end-plate in the tension zone. From simple beam-deflection theory, it may be shown that the elastic end-plate stiffness, K ept may be expressed as: K ept = where
3 16E st d y tep
l 3AB
(14)
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Fig. 5. Idealisation of end-plate deformation.
dy : l AB : Dep : n:
assumed effective width for the end-plate segment = (Dep /n); assumed length of the end-plate element = (gauge length less diameter of bolt hole); depth of the end-plate; number of tension springs considered in the model.
The stiffness of the end-plate is assumed to be elastic until the formation of plastic hinges. The force, Feppt , which causes yielding of the end-plate at a given temperature may be expressed as: Feppt =
2 ·d 2 · fyept · tep y
l AB
(15)
where fyept :
yield strength of the end-plate at a given temperature.
It is worth remembering that the model is only applicable for the first stage of response (i.e. before the beam bears against the column), due to lack of elevated temperature data describing the second stage of response. However, a detailed description of the second stage response of the end-plate has been presented by Madas [10] for ambient temperature conditions.
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3.4. Column web behaviour The stiffness of the column web, K cwt , in the compression zone may be estimated by idealising it as a plate of dimensions dc × beff subject to a uniform compressive force from the beam over the whole depth of the end-plate. By assuming that the plate obeys Hooke’s law, the elastic stiffness of the column web may be expressed as: K cwt =
E st tcw beff dc
(16)
where tcw : beff : dc :
thickness of the column web; assumed effective width of the column web; depth of the column web between fillets.
Previous studies [11,12] on simple joints (mainly web cleats and fin-plates) have revealed that the depth of the compression zone in the column web can be as great as the depth of the end-plate. Therefore, the effective column web area resisting the compression force is substantial and overstressing of the column web is unlikely. However, for flexible end-plate joints, this may be true up to the point when the beam flange comes into contact with the column, but after this overstressing may occur in some cases. Therefore the effective width of the column flange in the compression zone may be determined from the following expression: beff = Dep + 5k
(17)
where k: k: rc :
distance from the outer face of the flange to the web toe fillet; tc f + rc ; column root radius.
When the column web is subject to a compression force, the stiffness of the column web is assumed to remain constant until one of two modes of failure occurs within the depth of the column web. These are compression (i.e. crushing) of the web close to the flange or buckling of the web over part or most of the depth of the member. The critical force, Fcwpt , for the column web may be determined as the lesser of the buckling and compression forces. The compression resistance is based on the assumption that the compression force applied is distributed across the depth of the end-plate and the column flange is assumed to act as a bearing plate. Therefore the force, Fccwt , that causes crushing of the column web at a given temperature may be given by the following expression: Fccwt = beff tcw f ywt where f ywt :
column web yield strength at a given temperature.
(18)
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When considering buckling of the column web, the effective buckling width of the column needs to be defined. The buckling force, Fbcwt , of the column web at a given temperature based on the empirical relationship suggested by Ahmed et al. [12] may be written as: 0.017 0.60 1.43 0.76 Fbcwt = 8.4 · beff · tcw · f ywt -b · Dc
(19)
where beff -b :
effective buckling width of the column web; 2 ); beff -b = (Dc2 + Dep
Dc :
depth of the column.
Therefore, the critical force, Fcwpt , of the column web may be determined as: Eq. (18) . Fcwpt = Min. of Eq. (19)
(20)
Subsequent to the onset of yielding, a reduced stiffness must be adopted. In order to estimate the post buckling stiffness of the column web, the technique proposed by Walker [13] has been used. Hence, the post buckled stiffness for the column web in the compression zone K cwpt may be expressed as: K cwpt =
K cwt . 2.45
(21)
4. Material properties and behaviour The yield and ultimate stresses of the components were taken as 545 and 412 MPa, respectively, while a value of 197 kN/mm2 was adopted for the Young’s modulus of structural steel. These values were selected based on tensile coupon tests conducted on structural steel specimens [14]. However, the bolts were assumed to have yield and ultimate stresses of 480 and 600 MPa, respectively. Geometrical properties of the steel sections were based on nominal values. The temperature profile across the joint depth was based on experimental observations [14]. At elevated temperature the entire joint response needs to be considered since large deformation and consequently rotations are possible. Strain hardening of the steel in the plastic zone may therefore have a significant influence. To model the entire joint’s response yet avoids complexities arising from complete nonlinear modelling, a trilinear force–displacement representation is proposed as shown Fig. 6. It is assumed that the joint components would experience an elastic response (represented by an elastic stiffness) until the onset of yielding as a result of yielding of one or more of the components. Then a reduced stiffness representing a strain hardening stiffness may be adopted based upon revised elemental stiffnesses. This procedure is repeated as more elements enter the strain hardening region.
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Fig. 6. Force–displacement behaviour of the joint’s components.
Following the onset of yielding, a strain hardening stiffness is adopted and may be expressed as: K cst = µs K ct
(22)
where K cst : K ct : µs :
strain hardening stiffness of the joint’s component at a given temperature; elastic stiffness of the joint’s component at a given temperature; strain hardening coefficient for structural steel.
The strain hardening coefficient is defined as the ratio of the strain hardening stiffness to the elastic stiffness and is usually based on existing data. Due to the lack of available test data, Atamiaz Sibai and Frey [15] recommended a value in the range 0.019–0.024 for mild steel at ambient temperature whilst Ren and Crisinel [16] adopted a value of 0.06 for modelling of double web cleat joints at ambient temperature. In the current model a strain hardening coefficient of 0.05 is adopted. The ultimate capacity of the component may be assessed in a similar manner to the plastic capacity as expressed in Eq. (22), replacing the yield strength of the component with the ultimate strength fuc . Following the onset of failure, the stiffness of the component approaches zero. The final stiffness, K cft , may be calculated using a reduced stiffness coefficient of 0.01 and expressed as: K cft = 0.01K ct .
(23)
5. Degradation of the joint’s characteristics at elevated temperature The degradation of stiffness and strength of the components was based on the degradation of structural steel at elevated according to EC3: Part 1.2 [17] as shown in Table 2.
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Table 2 Properties of structural steel at elevated temperature Steel temperature (◦ C)
200 100 200 300 400 500 600 700 800 900 1000 1100 1200
Reduction factors for yield stress, f y and Young’s modulus, E a , at temperature, θa k y,θ = f y,θ / f y k E,θ = E a,θ /E a 1.000 1.000 0.926 0.848 0.775 0.618 0.357 0.169 0.087 0.060 0.040 0.020 0.000
1.000 1.000 0.900 0.800 0.700 0.600 0.310 0.130 0.09 0.0675 0.0450 0.0225 0.000
However, the degradation of bolt stiffness and capacity is based on recommendations presented by Kirby [18] based on experimental tests using the following expressions: For θb ≤ 300 ◦ C SFR = 1.0.
(24a)
For θb < 300 ◦ C ≤ 680 ◦ C SFR = 1.0 − (θb − 300) × 2.128 × 10−3 .
(24b)
For θb < 680 ◦ C ≤ 1000 ◦ C SFR = 0.17 − (θb − 680) × 5.13 × 10−4 .
(24c)
where θb :
temperature of the bolt and SFR is strength retention factor of the bolt.
6. Validation of the proposed model In order to validate the spring-stiffness model under fire condition, a cruciform bolted flexible end-plate bare-steel joint tested by Al-Jabri et al. [14] was selected. This major axis joint configuration consists of two 356 × 171UB51 beams connected to a 254 × 254UC89 column by 8 mm thick flexible end-plates and eight M20 bolts. The joint detail is shown in Fig. 7. The elevated temperature tests were performed under anisothermal conditions. Two load cases were modelled at moments of 8 and 16 kN m which represent the behaviour of the joint in the first stage of rotation (until the beam flange comes into contact with the column flange).
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Fig. 7. Description of the modelled joint.
Before verifying the model against elevated temperature tests, initial verification was carried out at ambient temperature as shown in Fig. 8. This was based on relevant data from the joint fire tests as well as existing test data reported by Boreman et al. [19]. The proposed model predicts closely the initial stiffness of the joint but underestimates the response in the plastic zone. This may be attributable to the lack of experimental data in the strain hardening zone which represents a large proportion of the moment–rotation response. Also, the ultimate capacity of the joint was underestimated with a maximum difference of about 10 kN m. Fig. 8 also shows comparison between the model and the experimental ambient temperature response obtained from fire tests. The initial stiffness and strain hardening stiffness of the joint is closely represented, but it was not possible to compare the ultimate capacity because only the first stage of response (i.e. before the beam bears against the column) is accounted for in the model. Verification of the model at elevated temperature was based on the two joint fire tests. Fig. 9 compares the experimental degradation of stiffness and strength of the joint with those predicted by the model. Fig. 9(a) shows that the predicted stiffness compares well with that obtained experimentally for temperatures up to 400 ◦ C, beyond which the model gives a slight overestimate. The predicted and recorded degradation of joint strength are compared in Fig. 9(b), from which it may be seen that for temperatures between 500 and 600 ◦ C the model underestimates the strength. However, for higher temperatures there was close agreement between the experimental and predicted results. Unfortunately test data
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Fig. 8. Comparison of predicted ambient temperature response with experimental results.
Fig. 9. Degradation of joint’s stiffness and strength at elevated temperature.
was not available for temperatures below 500 ◦ C, due to the levels of loading applied during testing. Therefore, at this stage it is difficult to draw definite conclusions concerning the
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Fig. 10. Comparison of predicted elevated temperature response with fire tests.
degradation of joint capacity since only three test points were available, all at temperatures greater than 500 ◦ C. It is worth remembering that the degradation of the joint capacity was based on the first stage of behaviour. The elevated temperature component model for flexible end-plate joints was compared with two fire tests conducted at moments of 8 and 16 kN m. Fig. 10 compares the experimental temperature–rotation response for the two tests with those predicted by the model. These compare very closely for both tests. In both cases little rotation was measured for temperatures up to approximately 510 ◦ C, beyond which the joint gradually plastifies until failure due to excessive end-plate deformation. The proposed model predicts a similar response. Also the temperatures at which both test specimens failed was closely predicted by the model. 7. Conclusions A simplified spring-stiffness model was presented for modelling the elevated temperature response of flexible end-plate bare-steel joints. Joints were modelled by assembling the contributions of individual components. Tri-linear force–displacement modelling was adopted, with the separation of the joint into its main components allowing the use of any chosen temperature profile. The main parameters describing stiffness and capacity of the components were degraded with increasing temperatures. The predicted response of the joint was compared with the response observed from experimental tests. Results showed that there was good agreement between the predicted and the experimental responses. The rate of degradation of stiffness and strength of the
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joint was closely predicted by the model. In general the proposed spring-stiffness model was capable of predicting the joint response at both ambient and elevated temperatures to a reasonable accuracy. 8. Concluding remarks This research work is an initial attempt to model the behaviour of flexible end-plate joints in fire using a component model. The model was developed based on components characteristics of the joint as well as the elevated temperature behaviour of flexible end-plate joints that were presented in the literature. The proposed model needs further development to take into consideration the following features that can have significant influence on the joint’s behaviour at elevated temperature: • The applicability of the model to predict the joint behaviour at higher levels of moment than those presented; • Incorporation of the behaviour of second stage of response of flexible end-plate joints at elevated temperature. The model can easily be modified to describe the joint behaviour at large rotations. Further information may be found in the work conducted by Madas [10]. • The model was developed based on isolated joint tests, the effect of axial restrained on the joint behaviour needs to be addressed in the model since this can have significant influence on the behaviour of the structure in fire as observed from Cardington frame fire tests. Also, the joint behaviour during the cooling phase needs to be investigated.
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