J. Construct. Steel Res. Vol. 41, No. I, pp. 3160, 1997 © 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S01439"/4X(96)0~533 0143974X/97 $17.00 + 0.00 ELSEVIER
Prediction of Initial Stiffness and Available Rotation Capacity of Major Axis Composite Flush Endplate Connections B. A h m e d & D. A. Nethercot Department of Civil Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, UK (Received 10 May 1996; revised version received 5 November 1996; accepted 4 December 1996)
ABSTRACT Knowledge of the initial rotational stiffness of a connection is important for the global elastic analysis of frame structures. Based on a simple force transfer mechanism and consideration of the behaviour of individual components, a method has been developed to predict the initial stiffness of composite flush endplate connections. The approach is compatible with that proposed earlier to predict moment capacity for several types of composite connection. In order to apply plastic analysis to frame structures a knowledge of the available and the required rotation capacities is necessary. A simple technique to determine the available rotation capacity of composite flush endplates is described herein. Taken together, the two methods represent key steps in the development of an approach to predict the main measures of the behaviour of composite endplate connections. Used in conjunction with the moment capacity and failure mode prediction method developed earlier by the authors, they provide a complete representation of the design properties of composite flush endplate connections. © 1997 Elsevier Science Ltd.
1 INTRODUCTION Modern approaches to frame design, e.g. those contained in EC3 and EC4, emphasise the need for consistency between the type of connections employed and the approach used to determine the distribution of internal forces within the structure. A particular aspect of this is the recognition that many practical forms of beamtocolumn connection will exhibit moment capacities and 31
B. Ahmed, D. A. Nethercot
32
rotational stiffnesses intermediate between the two extremes of 'fullstrength and rigid' and 'pinned'. Design on the basis of partial strength and/or semirigid joint action has, therefore, been formally recognized. This brings with it the requirement to determine key properties of the connections for use in overall frame design. The quantities that are required are: moment capacity, rotational stiffness and rotation capacity. Considerable progress has been made in recent years in the development of methods to calculate the moment capacity of both bare steel and composite connections [110]. The most advanced techniques properly recognize the actual force transfer within the connection and provide design equations to cover all possible modes of failure. In contrast, the prediction of rotational stiffness and rotation capacity is much less advanced. Although some proposals have been made [1,2,4,914], both the accuracy and lack of generality of the existing techniques mean that there is considerable scope for improvement. The authors have recently undertaken extensive study of several aspects of the behaviour of composite endplate connections [1517]. By developing a detailed numerical model using the ABAQUS [18] software and validating this carefully against several aspects of detailed test histories, they have been able to examine several facets of the mechanics of the complex force transfers between the various components. This has led to the development of a comprehensive design approach to predict moment capacity that recognizes all possible modes of failure and that correctly allows for interaction between components [3,5]. This design model is used, herein, as the basis for the development of parallel methods to predict initial stiffness and rotation capacity. Validation of the predictions for both properties against all available test data demonstrate both the accuracy and consistency of the method.
2 REVIEW OF AVAILABLE METHODS TO CALCULATE INITIAL STIFFNESS From eight tests Aribert and Lachal [11] have proposed a simple form of equation for initial stiffness calculation. This is only suitable for flush endplate joints. The initial stiffness of the composite connection is defined as: D Sac = Sa "t
h
a
(1)
+
2E~4dD Nkhb where Sa is the stiffness of the corresponding steel connection; Er is the elastic
Prediction of initial stiffness and available rotation capacity
33
modulus of the reinforcement; Ar is the section area of the reinforcing bars; D is the distance from the reinforcement to the centre of the lower flange; N is the number of active shear studs; hb is the depth of the steel beam; h is the depth of the steel column; k is the secant stiffness of one shear connector; is the increase factor, taken ~2. The basis for eqn (1) is the assumption that total moment capacity is equal to the sum of the moment capacity of the steel connection plus the contribution from the reinforcement.
Mac=Ma qMr SacOac=&Oac]&Oac&c=Sa + Sr
(2)
where Mac is the composite connection moment capacity; 0ac is the rotation capacity of the composite connection; Sac is the stiffness of the composite connection. This simple summation approach gives no consideration to the actual distribution of internal forces, but in a composite connection the presence of the composite slab causes the beam to behave in a different way (i.e. usually more compression and less tension) than would be the case in a bare steel connection. Anderson and Najafi [1] have proposed a model that gives the relation between the moment and the rotation of a composite connection. The model assumed a rotation of the beam web about the bottom flange, but at the same time considered the slip of the studs. It assumes that the developed moment is due to rebar and bolt forces only and does not account for any possible compressive force that may develop in the beam web. The relationship is:
{ KvKsD D r
M= \ ~rr~~ss .a¢.KbD2b]~
(3)
where M is the moment and th is the rotation, subscripts r, s and b indicate rebar, shear stud and bolt, and K represents the stiffness of the associated member. D is the distance between the centreline of the beam flanges, Dr is the distance of the reinforcement centroid from the centreline of the bottom flange of the beam and Db is the distance of top bolt row from the centreline of the bottom flange of the beam, where
Kr
2E,A r Dc'
t4a)
Dc is the depth of column section. After the initial model verification, when the flexibility of the shear studs
34
B. Ahmed, D. A. Nethercot
was taken into account, it was proposed by the authors to increase the length of rebar up to the first row of studs. Ks is taken as 200 kN/mm; G Kb = Dbz.
(4b)
The stiffness C~ (in the range of 050 kN/m) can be calculated for bare steel endplate joints using EC3 or other prediction methods or it can be obtained from tests. The above formulae do not consider yield of the column web, or the column web stiffness. Also, more importantly, the influence of the actual number of studs present in the connection is not properly reflected. Ren and Crisinel [2] developed a relation between moment and rotation for composite flush endplate connections that may be used to predict the initial stiffness. The derivation of the formula again used the basic assumption that the moment capacity of a composite connection is the sum of the rebar capacity and the bare steel connection capacity. The method considers the deformation of the column web at beam bottom flange level due to the compression. The relationship is:
D2 M= 1
1
1$
(5)
E+E+ c where D is the distance of the rebar from the bottom flange centre; Kr 
ErAr 2r/(60 + 1.3ks)'
(6a)
r / i s taken as 0.35; k is 1 for pure tension, 0.5 for simple bending; s is the spacing of the reinforcement; 0.6Rr Ks = min
0.6R~'
(6b)
Au Au is the interface slip taken as 0.5 m m for 19 x 100 m m welded shear connectors; Rr and Rs are the resistances of the reinforced concrete slab and the
Predictionof initialstiffnessand availablerotationcapacity
35
shear connectors over the hogging moment length within the composite beam, respectively;
f EAwc L K~ = min EAb ;
(6c)
L
Awe = 2twc[2(tp + tee) + teo]
(7)
and Ab is the crosssectional area of the steel beam. It can be seen that while the Anderson and Najafi [1] and the Ren and Crisinel [2] methods deal with the stiffness of components to determine the initial stiffness of the composite connection, Aribert's method simply adds the stiffness of the steel beam to the contribution of the rebar and shear studs. The Anderson and Najafi method was not developed specifically to predict initial stiffness, its main objective was to trace the M~b curve.
3 PROPOSED METHOD TO CALCULATE THE INITIAL STIFFNESS OF COMPOSITE FLUSH ENDPLATE CONNECTIONS From the review it is apparent that the methods already proposed by Anderson and Najafi [1] and Ren and Crisinel [2] would be improved if they could be modified so that the stiffness of the individual components could be properly addressed. The problem with the Anderson and Najafi method is that it neglects the stiffness of the column web and also the stiffness of the shear studs is not properly represented as only the stiffness from one stud is used. In the Ping Ren method the assumption that the total moment M = Mc + F~Z is not reliable and since it uses rebar spacing to calculate the stiffness of the rebar, only connections with mesh reinforcement cannot be analysed. Improvements to the calculation of initial stiffness are suggested by including the factors that are presently neglected in the two methods. The following assumptions are made: (1) For initial stiffness the developed internal forces are low, so only the influence of the rebar, bolts and the column web at the level of beam bottom flange need be considered when calculating the initial stiffness. In other words compression in the beam web will not influence the connection initial stiffness.
36
B. Ahmed, D. A. Nethercot
(2) Only the top bolts will be in tension at the load level for which initial stiffness is determined. (3) Beam web deformation at this load level is linear. In the clause for the determination of the stiffnesses of basic components in the revised Annex J of EC3, [4] it is stated 0.4.4.3) that: "The following components need not be taken into account when calculating the rotational stiffness Sj: • beam flange and web in compression; • beam web in tension; • plate in tension or compression". Also in this same clause for rotational stiffness, the basic model (J.4.1) assumes that [J.4.1 (2)]: "In a bolted connection with more than one boltrow in tension, as a simplification the contribution of any boltrow may be neglected, provided that the contributions of all other boltrows closer to the centre of compression are also neglected. The number of boltrows retained need not necessarily be the same as for the determination of the design moment resistance". This supports the second assumption made herein. These assumptions can be verified by the test results presented in [19]. Table 1 shows the strain of different components at 45% of the ultimate moment capacity. The reason for checking against the level of moment is that the M~b curve can be treated as linear up to this limit. It is clear from the test results that the influence of the beam web is insignificant and may be neglected. It can also be seen that at this level of load (45% of ultimate) the bottom row of bolts is below the neutral axis, hence its main function is shear transfer only. Also results represented in [19] show that the web strain is linear at this load level, hence the web deformations must be linear. TABLE 1
Strain of Various Components as Observed in Tests at 45% of Ultimate Capacity Test
Rebar strain (Ix~)
CJS1 CJS2 CJS3 CJS4 CJS5 CJS6
1015 1095 632 780 1200 770
Top bolt strain ( ~ )
413 450 450 480 500 410
Beam web Bottom flange strain (l~E) strain (1~)
182 160 190 280 250 160
2000 920 670 780 750 900
Column web strain (IxE)
1105 953 550 1200 1000 650
Prediction of initial stiffness and available rotation capacity
37
3.1 Finite element analysis to identify the most important influences on connection stiffness ABAQUS [18], a general purpose finite element software, has been extensively used by the authors to numerically simulate the response of composite connections [15,20,21]. These studies have provided a vast source of information on detailed aspects of connection behaviour which, subsequently, led to the development of comprehensive design procedures for moment capacity [3,5]. They also contain valuable information regarding the initial stiffness of connections. These results, therefore, provide a valuable source from which the major parameters that influence connection initial stiffness can be identified. Reference [15] describes the overall connection behaviour for varying reinforcement areas, degrees of shear interaction, thickness of endplate, beam bottom flange and column web. It can be seen from the results that reinforcement area and degree of shear interaction have most influence. Column web thickness is less important, whilst other parameters are insignificant. Reference [17] explains the influence of the shear to moment ratio on connection behaviour. Results from the study show that for a given connection detail the initial stiffness is not influenced by the distance of the load from the column flange. Besides these, [19] describes three tests on identical connections and variable shear to moment ratios. The results confirm that the initial stiffness is independent of distance of load from the columnflange provided other things are constant. Hence, the load position need not be considered in the equation for initial stiffness. Reference [16] describes the effect of axial column loading on composite connection behaviour. It is clear from the results that the connection initial stiffness is independent of the level of column axial loading present. In addition, [22] describes tests (JY2P) using concrete encased columns, in which column axial loading was present, although the results were not presented for the full loading, the early part shows that column axial load does not influence connection initial stiffness. It was concluded by the author that "The influence of column load on the stiffness of the joint was small. This was demonstrated by the test result (Figure C. 18) of joint JY2P, which had a substantial amount of concrete encasement". This test actually failed due to local buckling of the beam bottom flange, which was followed by buckling of the steel beam. Thus, the presence of column loading need not be considered when determining initial connection stiffness.
3.2 Equation for initial stiffness Using the above assumptions the proposed spring model is shown in Fig. 1. The equilibrium condition gives:
38
B. Ahmed, D. A. Nethercot '
Ar
! a"b
i
.
h M
/
¢
Fig. 1.
A spring model for composite cruciform flush endplate connections.
Fr+Fb=FcFr=F
(8)
where F r is the tensile force in the reinforcement; Fs is the shear force transferred by the shear studs; Fb is the tensile force in the bolts; Fc is the compressive force transferred by the beam bottom flange. From the stiffness deformation relations: F r = g r A r Fb = g b A b Fs = g s A s
(9)
where K and A are the stiffnesses and displacements of the related components. From Fig. 1 the compatibility condition is: As+ Ar+ Ac Ab+Ac m
 ~
(10)
where Hb is the depth of beam section; db is the distance of the top bolt row from the bottom flange. Substituting the displacements from eqn (9) into eqn (10) the following equations are obtained: Fs
Fr
Fc
Fb
Fc=db~
where ~b is the rotation of the connection. Substituting values for F s and Fc from eqn (8) into eqn (11) gives:
(11)
Prediction of initial stiffness and available rotation capacity
39
F~~1 + ~1+ K 1) +KG H~+:°
(12a)
Kc+FB
(12b)
&4
+
=0.
Solving for F~ and Fb from eqns (12a) and (12b) gives:
F~=
(13a)
+~+
+
db Kr ~gss "l"
K~
Hbgc (13b)
F~=(~r +~1+ ~)(~ + ~)K~l ~
Since the interest is in initial stiffness, the internal tensile forces are low, so there is no compression force developed in the beam web and, hence, the moment can be calculated using the rebar force and bolt force in this case. The equilibrium condition gives: M = FrDr + Fbd b where M is the connection moment; Dr is the distance of the reinforcement from the bottom flange centreline. Substituting Fr and Fb from eqns (13a) and (13b) in the above equation and applying the condition M = Ki~b gives:
HbOr
gi=
1
+
1
2 1
+ db
1
+ Kss +
db(Hb +
(~r+ ~l+ 1)(K~ + K~) Kc1~
Or)gc (14)
where Ki is the initial stiffness of the connection. It can be seen that the Anderson and Najafi [1] equation is a special form of the above general equation corresponding to the assumption of an infinite stiffness for the column web.
B. Ahmed, D.A. Nethercot
40
3.3 Selection of key parameters An equation to represent the initial stiffness of the connection has been developed above. However, its successful use in practice depends on the accurate estimation of the individual terms, i.e. the component stiffnesses of the items present in the equation. Before the formula is applied, it is essential to decide on suitable values for these key parameters. These are the length of reinforcement to be considered to calculate the rebar stiffness, the stiffness of the shear studs and the stiffness of the bolts. To investigate this, a basic value is first selected from which the effect of variations of the parameters can be studied. The selection of the initial values is described below.
3.3.1 Length of reinforcement to calculate the rebar stiffness If the first stud is very near the columnflange, the distance to the next stud should be used. It was found by Benussi and Noe [23], and also confirmed by Anderson and Najafi [1], that if the length of rebar considered for calculating the rebar stiffness is equal to half the depth of column, the resulting model overestimates the stiffness of the connection due to overestimating the rebar stiffness. They concluded that this can be corrected by increasing the length of rebar considered for the elongation. For the basic setup the length selected is: lr =
(Dc/2 + 225) mm.
Hence
Kr 
A~Er Dc   + 225 2
(15)
where Dc is the depth of column web.
3.3.2 Stiffness of the shear studs This can be calculated for all the shear studs in the hogging moment region or if the connection has more studs than are required for full interaction the actual number of studs required for full interaction should be used. Assessment of pushout test results of welded studs by several researchers [2426] shows that the elastic stiffness of the shear stud can vary between 200 and 350 kN/mm at 45% of the stud load capacity. On the other hand the finite element analyses reported by Razaqpur and Nofal [27], which were verified against the tests of other researchers, used a constitutive law for the force
41
Prediction of initial stiffness and available rotation capacity
deformation response of the shear studs based on the empirical equations developed by other researchers. The stud stiffness resulting from the coefficient used by them represents a stiffness of 118 kN/mm. Also the load slip curve adopted for numerical analysis of composite beams by Mistakidis et al. [28] from test results of other researchers gives a stiffness of 110 kN/mm. It can be seen that the stiffness of the shear studs can be in a range between 110 and 350 kN/mm. As a reasonable estimate the value is taken as 200 kN/mm. Hence, stiffness of a single stud is: Ks1 = 200 kN/mm, but it is also acknowledged that the true stiffness of the shear connector tbr a specific test should provide better results, and the amount of slip that can occur in a connection can vary. If the average slip of the studs is A, the force transmitted by them is: Fs = 200 × Ns x A. Hence, from the stiffness deformation relationship: Ks = 200 x Ns
(16)
where Ns is the smaller of either the total number of studs present in the hogging moment region or the number of studs required for full interaction. 3.3.3 Stiffness o f the bolts It is taken as 155 kN/mm following the Anderson method [1].
Kb = 155 kN/mm.
(17)
It is accepted that there will be some variation according to the equation in [ 1], due to the variation of the distance of the top bolt from the bottom flange centreline. This is neglected as the relative contribution of the bolts to the connection stiffness is insignificant with respect to the other associated components as indicated in Tables 24. TABLE 2
Effect of Stiffness of the Shear Studs on the Prediction Method Ks (kN/mm )
Average Standard deviation
100
200
300
0.98 0.20
1.12 0.21
1.21 0.22
42
B. Ahmed, D. A. Nethercot TABLE 3 Effect of Stiffness of the Bolts on the Prediction Method
Kb (kNImm) Average Standard deviation
50
I00
1.00 0.19
1.06 0.20
155 1.12 0.21
200 1.19 0.21
TABLE 4 Effect of Rebar Length on the Prediction Method
lr (ram)
(Dr2 + 225)
Average Standard deviation
1.2 0.1
(Dr2 + p, + P2) 0.9 0.1
(Dr2 + p, + 2p2) 0.6 0.6
Note: p~ is the distance of the first stud from the column face. P2 is the pitch of the shear studs.
3.3.4 The stiffness of the column web It is calculated from the basic principles (AE/L) for a compression member only, using the depth of compression from EC3 and is: Kc = 2ECWD[tcw{tbf ~ + 2(tp + X/2ap)"4 5(tcf "at"/'c)}] without web stiffener
(18)
but when a column web stiffener is present its contribution towards resisting deformation must be included. Hence, the stiffness should be calculated as: 2Ecw Kc = ~[tcw{tbf + 2(tp + ~f2ap) + 5(tcf + re)} + twsbcf] with web stiffener (19) and in general E is constant for all members. References [1,1113,19,22,29,30] describe 32 flush endplate connection tests. These are selected to study the effect of the parameters. Calculations are performed to study the average and the standard deviations of the ratio of prediction to test initial stiffness ratio. Tables 24 show the results of the study. From the results obtained, the values that are selected are: lr: (Dc/2+pl +Pz)nun; K b" 155 kN/nun (this value has been used by other researchers [1]); Ks: 200N~ kN/mm.
Prediction of initial stiffness and available rotation capacity
43
3.4 Validation of the proposed equation In order to validate eqn (14), test results obtained from [1,1113,19,22,29,30] have been used. When the initial stiffness was not directly reported, the slope of the line connecting the origin to 45% of the ultimate moment has been taken as the initial stiffness. Table 5, which presents the comparison of test and prediction, indicates that except for a few cases a very good agreement is achieved. It can be seen that if the first four results are ignored the average is 0.99, with a standard deviation of 0.21 for the proposed method, whereas under the same conditions the Anderson and Najafi [1] method gives an average prediction of 0.64 with standard deviation of 0.16 and the Ren and Crisinel [2] method gives an average prediction of 1.41, with a standard deviation of 0.29. From these comparisons it is clear that the proposed method provides better predictions of the connection stiffness. From Table 5 it can be seen that tests conducted by Law [22] exhibit the largest discrepancies. These had a beam depth of 453.6 mm with connection initial stiffnesses of 50, 75 and 8 0 k N / m m (for JX1, JX2 and JC1, respectively), while the predicted values were 134 kN/mm for all, but it can also be seen that connections with similar beam depth and reinforcement level, for example JC2, Test 10 and S8FD (453.6, 449.8 and 449.8 m m depth of beam) gave an initial stiffness of 200, 154 and 141 in the test while the predicted values were 149, 115 and 115 kN/mm. These early tests by Law and Johnson had concrete encasement on the columns. While the other tests demonstrate that an increase of beam depth causes the initial stiffness to increase (with other things constant) the reason for reduced stiffness for these particular tests is not clear.
4 REVIEW OF AVAILABLE METHODS TO CALCULATE ROTATION CAPACITY The SCI [31] prepared a document in which the rotation capacity of the composite connection is calculated using the elongation of the rebar and with the assumption that the compression zone is located at the bottom flange of the steel beam. The ultimate reinforcement strain is assumed to be 2%.
l AD + Dr
JX1 JX2 JCI JC2 CJS 1 CJS2 CJS3 CJS4 CJS5 CJS6 SCJ3 SCJ4 SCJ5 SCJ6 SCJ7 Test I
Test
g~
Ks
540 540 540 540 348 348 348 348 348 348 115 573 573 573 691 520
4000 4000 4000 4000 2800 2800 2800 2000 800 2800 1400 1400 1400 800 1400 1400
( kN/mm ) (kNImm )
155.00 155.00 155.00 155.00 155.00 155.00 155.00 155.00 155.00 155.00 155.00 155.00 155.00 155.00 155.00 155.00
( kN/mm )
8739 8739 8739 8739 6340 6340 6340 6340 6340 6340 8873 8873 77,271 8873 7517 68,861
K¢ (kN/mm)
133.91 133.91 133.91 149.04 31.04 31.04 31.04 30.03 25.86 31.04 20.31 53.54 56.45 45.70 58.83 54.55
Ki (kN. mm/mrad)
50 75 80 200 31 28 42 33 36 32 29 49 61 65 50 63
Ki test (kN. mm/mrad) 2.68 1.79 1.67 0.75 1.01 1.12 0.74 0.90 0.71 0.98 0.70 1.08 0.93 0.71 1.19 0.86
PIT
69.16 69.16 69.16 75.36 21.55 21.55 21.55 21.55 21.55 21.55 23.88 29;68 29.68 29.68 29.77 30.78
Anderson and Najafi [11 metho~
TABLE 5 Comparison of Predicted and Test Connection Initial Stiffness
1.38 0.92 0.86 0.38 0.70 0.78 0.52 0.65 0.59 0.68 0.82 0.60 0.49 0.46 0.60 0.48
179.80 179.80 179.80 231.94 47.03 47.03 47.03 47,03 47.03 47.03 NA 87.94 87.94 84.80 86.79 86.38
3.60 2.40 2.25 1.16 1.53 1.69 1.13 1.41 1.29 1.48 NA 1.78 1.44 1.31 1.75 1.36
Anderson Ren and Ren and and Najafi Crisinel [2] Crisinel P/T P/T method ~
298 742 742 520 261 522 743 522 232 232 198 324 324 324 390 765
1400 155.00 68,861 1400 155.00 68,861 1400 155.00 68,861 1400 155.00 68,879 1400 155.00 68,861 1400 155.00 68,861 1400 155.00 68,861 1400 155.00 68,879 2789 155.00 87,000 2789 155.00 86,917 2789 155.00 85,081 3903 155.00 87,000 3903 155.00 87,000 3903 155.00 87,000 2901 155.00 100,071 4800 155.00 100,071 Average prediction/test Standard deviation
38.78 66.99 66.99 115.26 35.71 54.67 67.06 115.49 55.08 14.85 50.46 68.77 68.77 68.77 48.50 84.96
36 1.09 51 1.31 73 0.91 154 0.75 30 1.18 64 0.86 66 1.02 141 0.82 67 0.82 13 1.14 43 1.17 73 0.94 76 0.90 78 0.88 38 1.28 51 1.67 1.08 (0.99) 0.40 (0.21)
28.70 0.80 31.54 0.62 31.54 0.43 66.92 0.43 27.96 0.93 30.78 0.48 31.54 0.48 66.98 0.47 47.15 0.70 12.33 0.95 44.21 1.03 48.46 0.66 48.46 0.64 48.46 0.62 27.71 0.73 29.45 0.58 0.67 (0.64) 0.21 (0.16)
50.75 1.42 97.88 1.91 97.88 1.33 162.58 1.06 39.80 1.32 81.58 1.28 97.91 1.49 153.41 1.09 69.94 1.04 27.11 2.09 66.76 1.55 97.11 1.33 81.53 1.07 61.77 0.79 53.93 1.43 91.51 1.79 1.53 (1.41) 0.53 (0.29)
aNote: these were not calculated by the relevant authors. Using their proposed equations the present authors have calculated the values.
Test 3 Test 4 Test 7 Test 10 S4F S8F S12F S8FD A2 A3 A4 C1 C2 C3 SJB10 SJB14
p~
3"
B. Ahmed, D. A. Nethercot
46
where pl is the distance of the column flange to the first shear connector; P2 is the pitch of the shear connectors; n is the number of shear connectors needed to resist the longitudinal force; Dc is the depth of the column; D is the depth of the beam; Dr is the distance of the reinforcement to the top of the steel beam. Aribert and Lachal [ 11 ] proposed a method to calculate the rotation obtained from the condition of strain compatibility at the column face as: A
s
~b= ~ + hb
(20)
where A is the elongation for the rebar over half the depth of the column; s is the slip of the shear studs near the connection = aF/(Nk). Xiao et al. [12] have proposed a method similar to the SCI method, to calculate the available rotation capacity of composite connections. By calculating the extension of the rebar, the rotation capacity was defined as." ~bu= AD
L + Dr
(21)
where D is the depth of the beam; Dr is the distance of the upper flange of the beam to the centre of the reinforcement. The elongation of reinforcement AL consists of two parts: deformation of the reinforcement in the plastic zone (L1) and deformation of reinforcement in the remaining elastic zone (L2). It was concluded later by Xiao [13] that the elongation of the reinforcement in the elastic zone has no effect on the rotation capacity and should be neglected. The elongation in the plastic zone was calculated as:
where Pl is the distance of the column flange to the first shear stud; p is the pitch of the shear studs; e is the plastic strain of reinforcement, taken as 0.005. Xiao [13] used the location of the neutral axis to find the rotation capacity of the composite connections. The rotation capacity also considers the slip of the shear studs and was defined considering four different positions for the neutral axis, i.e. neutral axis in the concrete slab, top flange, between the top flange and the top bolt row and finally below the top bolt row. Among these possibilities the last one is the most likely and, hence, is described in detail. The rotation capacity for the last case is given by:
Prediction of initial stiffness and available rotation capacity
L1
~bu=
~
+
dc+Pl+ip+~p
L2
2
47
(22)
pl+ip+~p
where p is the bolt pitch; Pl is the distance of the top bolt row from the top flange; dc is the depth of the concrete slab; i is the number of tension bolts that yielded and is given by the integer part of the following expression: pytbfbbffs + pytbw Dpltbf +
i = integer part of
2Ft + pytbwP
~capacity of rebar. where py is the yield strength; Fs = m i n [ capacity of stud' Ft = (p~AtQt); p~ is allowable bolt tensile stress from BS5950; At is tensile area of bolt; Q~ is the prying force.
fs L2  N k where N is the number of shear studs; k is the secant stiffness of a shear stud, taken as 30 kN/mm. The main flaw of the above formula is that the depth considered for the calculation of rotation capacity becomes larger than the depth of beam and sometimes greater than the combined beam and slab depth, when compared with the test results. This is not acceptable, since it implies that the beam is rotating without having any physical contact with the column at the level of the bottom flange. It can be easily visualised from the equation that for a connection with two rows of bolts where the top bolt has yielded the considered depth can easily be larger than the beam depth. A typical example is test CJS1 of [19]. It is known from the test results that the top bolt row has fully yielded (180 kN force developed in the top row). For this connection dc= 90 mm, Pl = 49 ram, p = 157 mm, D = 257 turn. The calculated depth for the first part of eqn (7a), is 90 + 49 + 157 + 2/3 x 157 = 400 ram. This is impossible as it implies a physical separation between the beam and the column as shown in Fig. 2. 5 PROPOSED METHOD TO CALCULATE THE AVAILABLE ROTATION CAPACITY OF COMPOSITE FLUSH ENDPLATE CONNECTIONS While the determination of the initial stiffness assumes low internal forces, the determination of rotation capacity requires the opposite. The forces asso
48
B. Ahmed, D. A. Nethercot
I I
'
I
'll I I I
I i
I
I!II
I
/
/
I
I mm
Fig. 2. Rotation capacity model of Xiao.
ciated with the rotation capacity are the forces in the different components at the joint's ultimate capacity. Hence, to calculate the rotation capacity the possible ultimate forces are required, from which the elongation of the associated component may be determined.
5.1 Determination of the elongation of components and the rotation capacity A method has previously been described [3] to calculate the moment capacity of connections. The method included the determination of internal forces for all the components considering the effect of loading conditions and the necessary compatibility conditions at the beamtocolumn connection interface. The magnitude of the internal forces from this model are used herein to determine the deformation of the components. Whilst the model for initial stiffness dealt with the initial linear behaviour of the connection; the available rotation capacity model requires the plastic deformation of the components. The model is shown in Fig. 3. A process similar to that already described when selecting values for key parameters used to calculate the initial stiffness of the connection has been adopted. It is observed in the composite connection tests that the strain of the reinforcement can vary between 3000 and 15,000 ~E. Elongation of the rebar is calculated by considering plastic strain of the rebar as:
Prediction of initial stiffness and available rotation capacity Elongationof rebar A Slip of studs A s ,'/~ l ., ~ .s.~.., ........
49
r
t L.. . . . . .
Extenslon of boits
A
,,.._.. i
~,'
,z/~
,~bco,.,,.s,io.
i i
1i
(a)
(b)
(c)
(d)
(a) Connection before deformation (b) Connection after deformation (c) Deformation of components at connection face (d) Representation of deformations for rotation calculation Fig. 3. Beam to column connection available rotation model,
The slip of the stud at the interface of the slab and the beam top flange can be estimated as: Fr As  Ksr,
(24)
Note: gsr is determined with a secant stiffness of 50 kN/mm. This represents the average failure load to slip at failure observed in push out tests for shear studs, hence Ksr = 50Ns. Extension of the bolts can be calculated as: Ab = Fb
(25)
Kb
where Kb has already been defined through eqn (17). From this the rotation capacity can be calculated as: =
Ar
+
As
+
Drdc,bw Drdc,bwdc'
Ab Drdc,bwdb'
(26)
50
B. Ahmed, D. A. Nethercot TABLE 6
Effect of Secant Stiffness of the Shear Studs on the Rotation Capacity Prediction Method Ks (kN/mm)
Average Standard deviation
20
30
40
50
0.82 0.33
0.71 0.27
0.63 0.25
0.59 0.24
TABLE 7
Effect of Stiffness of the Bolts on the Rotation Capacity Prediction Method Kb (kN/mm)
Average Standard deviation
50
100
1.15 0.52
155
0.82 0.32
200
0.72 0.27
0.66 0.25
TABLE 8 Effect of Rebar Length on the Rotation Capacity Prediction Method
1,. (mm)
(Dd'2 + 225)
Average Standard deviation
(Dd'2 + Pl + P2)
0.59 0.24
(D/2 + pl + 2p2)
0.71 0.27
0.65 0.25
Note: p~ is the distance of the first stud from the column face. P2 is the pitch of the shear studs. TABLE 9
Effect of Strain of Rebar on the Rotation Capacity Prediction Method er ( tze)
Average Standard deviation
3000
0.62 0.24
5000
0.71 0.27
7000
0.79 0.30
10,000
0.92 0.36
w h e r e dc,bw is the depth o f c o m p r e s s i o n in the b e a m w e b to be calculated f r o m the equilibrium [3]; de' is the distance o f the top o f the b e a m f r o m the rebar; db' is the distance o f the bolts f r o m the rebar. T a b l e s 6  9 show the effect o f variations o f these c o m p o n e n t s on the ratio o f prediction to test rotation capacity. F r o m these the selected values are: Er =
10,000 p , e
Prediction of initial stiffness and available rotation capacity
51
lr = (De~2 + p l +p2) mm
gsr" 50Ns kN/mm Kb: 155 kN/mm where Ns is the smaller of the total number of studs present in the hogging moment region or the number of studs required for full interaction as described in section 3.3.2.
5.2 Validation of the proposed method Test results obtained from [1,1113,19,22,29,30] are used for the verification of the proposed method. Predictions and test values are compared in Table 10. In addition to the test results, comparisons are made against the other available prediction methods. It can be seen that the proposed method predicts the rotation capacity fairly well in many cases, but that in some cases it considerably underestimates the rotation capacity. The proposed model assumes full plasticity in the connection and, hence, does not account for the brittle type of failure which can occur in connections with low levels of reinforcement, especially with mesh reinforcement only such as SCJ3. Recommendations for connection design [7] generally advise against such practice. From Table 10 it can be seen that the other methods give higher standard deviations and unsafe predictions of rotation capacity. For example Xiao's method gives an average of 0.95 with a standard deviation of 0.39, SCI method gives an average of 1.66 with a standard deviation of 0.97, while the proposed method gives a standard deviation of 0.36 with an average of 0.93, making the prediction safer. If the brittle failure mode of the mesh reinforced connections is taken into consideration, i.e. the extension of the bolts and the slip of rebars is neglected, a much better comparison is obtained. For example, with this modification for SCJ3, rotation becomes 9.27 mrad, which makes the average 0.91 with a standard deviation of 0.29. The reason for the large variations observed in a few cases can be explained from Fig. 4. From the figure it is clear that the rotation is very sensitive to the moment at the plastic state of the connection, especially near the ultimate moment capacity. In this region a variation of moment of 510% can change the rotation by more than 100%.
6 THE OVERALL BEHAVIOUR It has been observed from the test results for endplate connections that the overall behaviour can be best described by a combination of straight lines and an elliptical arc. The first part of the curve can be assumed as linear elastic (up to 0.45Mu). Beyond this up to the point at which the connection attains
JXI JX2 JC1 JC2 CJS 1 CJS2 CJS3 CJS4 CJS5 CJS6 SCJ3 SCJ4 SCJ5 SCJ6 SCJ7
Test
5.30 5.54 4.94 7.26 8.42 8.17 6.89 8.24 9.15 8.08 3.59 6.84 8.11 5.31 6.87
30.6 30.6 30.6 27.0 36.4 36.4 36.4 36.4 36.4 36.4 8.4 36.4 36.4 36.4 43.3
(1)
Najafi and SCI [131 (mrad) a Anderson [11 (mrad) ~
2.09 2.09 2.09 2.09 2.12 2.12 2.12 2.12 2.12 2.12 1.17 5.83 5.83 5.83 7.00
Aribert and Lachal [111 (mrad) a
11.99 11.99 11.99 10.96 28.87 28.87 24.03 30.68 41.24 24.77 15.58 (9.27) 13.41 20.14 16.52 16.11
(3)
(2) 14.1 a 14.1" 14.P 14.1" 33.4 33.4 33.4 33.4 33.4 33.4 8.8 23.5 23.5 20.5 23.5
Proposed method (mrad)
Xiao [13] (mrad)
13 10 10 18 35 42 18 58 60 22 7.2 23.4 26 11.5 26.5
(4)
Test rotation (mrad)
TABLE 10 Comparison of Predicted and Test Rotation Capacities
2/4 1.09 1.41 1.41 0.77 1.04 0.87 2.02 0.63 0.61 1.65 1.22 1.00 0.90 1.78 0.89
1/4 2.36 3.06 3.06 1.50 1.04 0.87 2.02 0.63 0.61 1.65 1.16 1.55 1.40 3.16 1.63
0.92 1.20 1.20 0.61 0.82 0.69 1.34 0.53 0.69 1.13 2.16 (1.29) 0.57 0.77 1.44 0.61
3/4
Ratios o f prediction to test rotation
8.51 6.24 7.70 9.57 6.22 6.40 8.51 9.57 6.21 6.82 12.83 7.34 7.69 7.29 6.44 7.51 8.86
44.6 27.3 61.9 61.9 32.6 19.3 34.5 47.4 25.2 19.9 31.9 23.3 26.0 26.0 26.0 17.5 29.1
6.75 3.87 9.63 9.63 6.75 2.53 5.07 7.22 5.07 1.01 1.01 1.01 1.41 1.88 2.82 0.87 1.71
18.7 12.9 9.5 24.5 14.3 13.5 ~ 19.9 ~ 26.2 ~ 15.2 a 11.8" 16.3 a 11.8 ~ 14.0 ~ 15.5 ~ 21.8 ~ 16.8 20.5 Average SD
18.09 21.31 18.09 18.09 12.76 20.17 19.23 18.09 13.45 17.60 35.48 24.17 17.60 17.30 17.00 18.43 27.54
20 15.7 12 24 14 15 30 25 14 26 36 35 28 29 21 22 24
2.23 1.74 5.15 2.58 2.33 1.29 1.15 1.90 1.80 0.77 0.89 0.66 0.93 0.90 1.24 0.79 1.21 1.66 0.97
0.94 0.82 0.79 1.02 1.02 0.90 0.66 1.05 1.08 0.45 0.45 0.34 0.50 0.54 1.04 0.76 0.86 0.95 0.39
0.90 1.36 1.51 0.75 0.91 1.34 0.64 0.72 0.96 0.68 0.99 0.69 0.63 0.60 0.81 0.84 1.15 0.94 (0.91) 0.36 (0.29)
aNote: these were not calculated by the relevant authors. Using their proposed equations the present authors have calculated the values.
Test 1 Test 3 Test 4 Test 7 Test 10 S4F S8F S12F S8FD A2 A3 A4 C1 C2 C3 SJB10 SJB14
P~
?:
ta~
54
B. Ahmed, D. A. Nethercot
E
M Ig
Ma
0.45 M
U
h Ki v
0a
(0 u
Rotation
Fig. 4. Typical momentrotation curve showing the initial stiffness and the rotation capacity.
C
B
M
a
0"45Mu
A
R
0
¢u
¢
Fig. 5. Model for overall behaviour of flush endplate connection.
its ultimate moment capacity (0.45M:1.00Mu) the response can be represented by the arc of an ellipse (to keep the model mathematically simple and easy to use the continuity of the curve at point A is ignored, a more complex relation can be obtained by assuming an arc having both lines OA and BC tangent to it); the final part is plastic. The relationship can be defined as: M = Ki~b
0 ~ b <0"45Mu
(27)
Xi M=0.45Mu+b M = Mu
1
dp>c~u
0.45Mu 

Prediction of initial stiffness and available rotation capacity
55
200 180 160
S
140 ~ 120 •~ 100
~ so 6O 40 20
/
A °
0
•
CJSl, Prediction
o     CJS 1, Test
5
l0
15
20
25
30
35
40
Rotation (mrad) Fig. 6. Comparison o f test and predicted overall behaviour o f test C J S  l .
180
160 140 120
/
~ 8o ~ 6o 40 2O
•
CJS6, Prediction
or.. C.IS6,Test
0 0
5
10
15
20
25
Rotation (mrad) Fig. 7. Comparison o f test and predicted overall behaviour o f test CJS6.
where constants a and b are defined as: a = ~bu0.45~~"
(28)
b = 0.55Mu. The derivation of the overall behaviour is explained in Fig. 5. During the derivation of initial stiffness it was assumed that up to 0.45Mu the m o m e n t rotation curve is linear, hence the equation of line OA is the equation of the
56
B. Ahmed, D. A. Nethercot 3OO
.,..____."4"
//
250 200 ~ 150 lOO
"
SCJ5, Prediction
o
SCJ5, Test
50
0
5
10
15
20
25
Rotation (mrad) Fig. 8. Comparison of test and predicted overall b e h a v i o u r of test SCJ5. 180 160 140
/
120 loo
•
i t50,°
SCJ6, Prediction
o SCJ6, Test
40 20 0 0
5
10
15
20
25
Rotation (mrad) Fig, 9. Comparison of test and predicted overall b e h a v i o u r of test SCJ6.
line passing through the origin and having a slope equal to the initial stiffness. As soon as the connection attains its full moment capacity the m o m e n t rotation curve will have a zero or negative slope (unloading); this is approximated as a straight line parallel to the rotation axis. The relation in between is approximated as an elliptical arc. 'R', the centre of the ellipse is located by the intersection of lines AR and BR parallel to the rotation and moment axes, respectively. The successful tracing of a momentrotation curve depends on the successful determination of several items which are: moment capacity, initial stiffness and available rotation capacity. Comparisons of test and pre
57
Prediction of initial stiffness and available rotation capacity 300
250
f
2O0
•
150
TeJt 4, Prediction Test 4, Teat
100
50
0
5
10
15
20
25
R(~tat~n (mrad)
Fig. 10. Comparison of test and predicted overall behaviour of Test 4.
250
Y
200
,~ 150 •
SSF, Prediction
t3 SSF, T ~ t
50
0
5
10
15
20
25
Sotati~ (rand) Fig. 11. Comparison
of test and predicted
dicted overall behaviour are shown in SCJ5 [8], SCJ6 [8], Test 4 [29], S8F that the proposed method can predict connections with sufficient accuracy properties are known.
overall behaviour
of test S8F.
Figs 612 for CJS1 [19], CJS6 [19], [1] and SJB10 [30]. The results show the overall behaviour of the endplate if the geometry and exact material
58
B. Ahmed, D. A. Nethercot 250
21111
150 
$JB10, ~ l i c t i a a
0 SJBI0, T ~ t
50
0
5
10
15
20
25
R~atioa (tm'ad)
Fig. 12. Comparison of test and predicted overall behaviour of test SJB10. 7 CONCLUSIONS Starting from the basic mechanism of force transfer within the components of a composite connection, a method has been proposed herein to estimate the initial stiffness and available rotation capacity of major axis flush endplate composite connections. The method is fully compatible with the method proposed earlier for the prediction of moment capacity. Comparisons against test data have demonstrated that the method is capable of predicting the initial stiffness and rotation capacity of flush endplate connections with very good accuracy. A simplified model to represent the overall behaviour of a given connection detail has also been proposed.
ACKNOWLEDGEMENTS The work presented in this paper is a part of the research project that is now in progress for a higher degree sponsored by the Commonwealth Scholarship Commission.
REFERENCES 1. Anderson, D. and Najafi, A. A., Performance of composite connections: major axis end plate joints. Journal of Constructional Steel Research, 1994, 31(1), 3157. 2. Ren, P. and Crisinel, M., Prediction method for momentrotation behaviour of
Prediction of initial stiffness and available rotation capacity
3. 4. 5. 6.
7. 8. 9. 10. 11. 12.
13. 14. 15. 16. 17. 18. 19.
59
composite beam to steel column connections. In Connections in Steel Structures III." Behaviour, Strength and Design, ed. R. Bjorhovde, A. Colson and R. Zandonini. Proceedings of the 3rd International Workshop, Trento University, 2931 May 1995, pp. 3346. Ahmed, B. and Nethercot, D. A., Design of composite flush endplate connections. The Structural Engineer (in press). CEN/TC 250, New Revised Annex J of Eurocode 3: Part 1.1, CEN document N419E, 1994. Ahmed, B., Li, T. Q. and Nethercot, D. A., Design of composite tinplate and angle cleated connections. Journal of Constructional Steel Research, 1997, 41(1), 129. Li, T. Q., Nethercot, D. A. and Choo, B. S., Behaviour of flush endplate composite connections with unbalanced moment and variable shear/moment ratiosII. Prediction of moment capacity. Journal of Constructional Steel Research, 1996, 38(2), 165198. SCI, Moment connections in composite construction: interim guidance for endplate connections, Technical Report, SCI publication 143, 1995. Xiao, Y., Choo, B. S. and Nethercot, D. A., Composite connections in steel and concrete. I. Experimental behaviour of composite beamcolumn connections. Journal of Constructional Steel Research, 1994, 31, 330. Davison, J. B., Kirby, P. A. and Nethercot, D. A., Rotational stiffness characteristics of steel beamtocolumn connections. Journal of Constructional Steel Research, 1987, 8, 1754. Kishi, N. and Chen, W. F., Momentrotation relations of semirigid connections with angles. Journal of Structural Engineering, ASCE, 1990, 116(ST7), 18131834. Aribert, J. M. and Lachal, A., Experimental investigation of composite connections in global interpretation. Proceedings of COST C 1 conference on semirigid Joints, Strasbourg, France, 1992, pp. 158169. Xiao, Y., Nethercot, D. A. and Choo, B. S., Design of semirigid composite beamcolumn connections. Building the Future; Innovation in Design, Materials and Construction, 1st edn, ed. F. K. Garas et al. E & FN Spon, London, 1992, pp. 391406. Xiao, Y., Behaviour of composite connections is steel and concrete. Ph.D. thesis, Department of Civil Engineering, University of Nottingham, UK, 1994. Kukreti, A. R., Murray, T. M. and Abolmaali, A., Endplate connection momentrotation relationship. Journal of Constructional Steel Research, 1987, 8, 137157. Ahmed, B. and Nethercot, D. A., Numerical modelling of composite flush endplate connections. Journal of Singapore Structural Steel Society, 1995, 6(1), 87102. Ahmed, B. and Nethercot, D. A., Effect of column axial load on composite connection behaviour. Engineering Structures (in press). Ahmed, B. and Nethercot, D. A., Effect of high shear on the moment capacity of composite cruciform endplate connections. Journal of Constructional Steel Research, 1996, 40(2), 129163. ABAQUS User Manual, Version 5.4. Hibbitt, Karlsson & Sorensen, Inc., 1080 Main Street, Pawtucket, RI 028604847, USA, 1994. Li, T. Q., Nethercot, D. A. and Choo, B. S., Behaviour of flush endplate composite connections with unbalanced moment and variable shear/moment ratios
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20.
21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.
B. Ahmed, D. A. Nethercot
I. Experimental behaviour. Journal of Constructional Steel Research, 1996, 38(2), 125164. Ahmed, B., Li, T. Q. and Nethercot, D. A., Modelling composite connection response. In Connections in Steel Structures: Behaviour, Strength and Design, ed. R. Bjorhovde, A. Colson and R. Zandonini. Proceedings of the Third International Workshop, Trento, 2931 May 1995, pp. 259268. Ahmed, B. and Nethercot, D. A., Parallel use of largescale testing and numerical simulation. The role of large and full scale testing, held at City University, London, 13 July 1996. Law, C. L. C., Planar nosway frames with semirigid beamtocolumn joints. Ph.D. thesis, Department of Engineering, University of Warwick, 1991. Benussi, F. and Noe, S., On the modelling of semirigid connections. Journal of Construction Steel Research (submitted). Chapman, J. C., Experiments on composite beams. The Structural Engineer, 1964, 42(11), 369383. Lloyd, R. M. and Wright, H. D., Shear connection between composite slabs and steel beams. Journal of Constructional Steel Research, 1990, 15, 255285. Lain, D., Elliot, K. S. and Nethercot, D. A., Push off tests on studs with hollow cored floor slabs (in preparation). Razaqpur, A. G. and Nofal, M., A finite element for modelling the nonlinear behaviour of shear connectors in composite structures. Computers and Structures, 1989, 32(1), 169174. Mistakidis, E. S., Thomopolos, K. T., Avdelas, A. and Panagiotopoulos, P. D., Analysis of composite beams with shear connectors allowing for softening. Steel StructuresEurosteel '95, ed. Kounadis. Balkema, Rotterdam, 1995, pp. 7380. Najafi, A. and Anderson, D., Composite connections with structural steel endplate. Final report to SCI, Department of Engineering, University of Warwick, 1992. Benussi, F., Puhali, R. and Zandonini, R., Semirigid joints in steelconcrete composite frames. Construzioni Metalliche, 1989, 5, 128. SCI report, Partial strength moment resisting connections in composite frames. Document no. SCIRT275, Revision 0, April, 1992.