- Email: [email protected]
Ultimate resistance of tubular double T-joints under axial brace loading
J. Construct. Steel Research 1,4 (1993) 205-228
Ultimate Resistance of Tubular Double T-Joints under Axial Brace Loading J. C. Paul, Y. Makino & Y. K u r o b a n e Kumamoto University, Faculty of Engineering Kurokami 2-39-1, Kumamoto 860, Japan (Received 12 December 1991; revised version received I0 March 1992; accepted 14 May 1992)
ABSTRACT The results of a series of tests on multiplanar double T-joints with circular hollow sections under compressive brace loading are reported. The objective of the tests is to investigate influences of the brace to chord diameter ratio ft, the transverse gap ratio ( and the out-of-plane angle c~ on the ultimate resistance of the joints. To interpret the results of the tests a simple plastic analysis of a closed ring model with collapse mechanisms corresponding to two observed failure types is attempted. In the range ~p= 60 ° through 120 ° multiplanar TT-joints can be treated in design as two separate T-joints. An ultimate resistance equation is proposed on the basis of present and past tests results.
NOTATION A
Be d d' D E Fr Fu gt
Cross sectional area chord Ring width Brace diameter Comparable brace diameter Chord diameter Modulus of elasticity Yield stress chord material Tensile strength chord material Transverse gap 205
J. Construct. Steel Research 0143-974X/93/$06-00 © 1993 Elsevier Science Publishers Ltd,
England. Printed in Malta
206
lb, Lbr
L Mi
Mp Ni
Np P Pc Pu
PU,V P; R t
T w
we wi
Ai
0i T
d. C. Paul, Y. Makino, Y. Kurobane
Brace length (specimen, analytical model) Chord length Bending moment yield hinge i Plastic moment Axial force yield hinge i Plastic load Load Collapse load Ultimate resistance TT-joint Vertical resultant ultimate resistance TT-joint Ultimate resistance T-joint with d= d' Chord radius Brace wall thickness Chord wall thickness Vertical displacement beam External work Internal work Chord length ratio = 2LID Location yield hinges Diameter ratio = d/D Local rotation yield hinge i Chord wall thinness ratio = D/2T Shortenings yield hinge i Error Transverse gap ratio =gt/D Rotation yield hinge i Dimensionless extension yield hinge i Out-of-plane angle Shear stress Rotation braces
INTRODUCTION Since studies in the ultimate behaviour of multiplanar tubular joints have started only recently, basis for the design of such joints is still insufficient. The American Welding Society Structural Welding Code (AWS D 1,1-88)1 is the only code that shows general design criteria applicable to many types of multiplanar joints, while the Cidect Design Guide for Circular Hollow Section Joints 2 proposes criteria applicable only to double T, X and K-joints for which data are available. 3-7 The AWS Code proposes the
Tubular double T-joints under axial brace loading
207
chord ovalizing parameter to account for multiplanar effects, which, however, proved to be contradictory to actual behaviour in certain cases: two compression braces suppress the chord ovalization giving a beneficial effect on the ultimate resistance according to the AWS prediction, while local deflection of the chord wall between the two compression braces gave adverse effect on the ultimate resistance according to Makino's double K-joint tests. 6'8 Clearly, there remain many questions to be studied with regard to the ultimate behaviour of multiplanar joints. Scola et al. 3"4 performed a series of tests on double T-joints. They showed that the transverse gap size is a primary factor influencing the resistance of double T-joints when compared with that of planar T-joints, which resembles the behaviour of double K-joints. 6 However, the transverse gap interrelates to the out-of-plane angle and the brace to chord diameter ratio. More experimental evidence is indispensible to interpret the combined effect of these geometrical variables on the ultimate resistance of double T-joints. Twelve double T-joints (called TT-joints hereafter) under compression were tested to fill the void of existing data and the results are discussed in this paper. An analysis of a closed ring model is attempted to identify the effect of the transverse gap. The theoretical model is calibrated using experimental results while parametric formulae based on curve fitting are provided for practical use.
EXPERIMENTAL
PROGRAM
Twelve TT-joint specimens with two identical braces were tested under axial compression (Fig. 1). The diameter ratio fl, the out-of-plane angle ~b and the transverse gap to chord diameter ratio ~ were varied. The
Fig. 1. Joint geometry.
208
J. C. Paul, Y. Makino, Y. Kurobane 0.0
TI-IIz~
v~-t 0.6
TI-I U
TT-IO z~
TT-6 D
T~-II
-~0,4 ii TI-I o
0.2
TT'I o TT-2 o
TT-II
TT-3 0 , . , . ~ , ~ , 0.L6 ' 0 . t ? ' 1 . 0.2 0.3 O,i 0,5 0.8
B =did Fig. 2. Test program.
specimen are divisible into three groups with ~b of 60 °, 90 ° and 120 °. Within each group the diameter ratio 13was varied in a range between 0-22 and 0.73. The variation of ~b and /3 determined a variation range of the transverse gap ratio ( resulting in values between 0.04 and 0.73. The three nondimensional geometrical variables for the 12 joints included in the test program are depicted in Fig. 2. The chord length ratio 0c and the chord thinness ratio ~ were kept constant throughout the study (~=9.81 and = 18.24), as these ratios were thought to give similar effects on both the TT and T-joint capacities. The specimens had no eccentricities.
Specimen details The measured dimensions for the specimens and the material properties of the chord are shown in Table 1. All chord tubes were taken from the same lot (D= 190.7mm, T=5.23 mm), while the brace diameter were varied between 42.8 and 139.7 mm. For the chords cold formed steel STK400 tubes were used with the specified minimum tensile strength of 402 MPa. Fillet welds and partial penetration butt welds were used and welding was carried out with illuminate electrodes type D4301 with the specified minimum tensile strength of 422 MPa. Weld sizes were conformed to the Japanese design recommendations. 9
Test set-up The test set-up is shown in Fig. 3. The end plates welded to the braces were bolted to two hinges, which were bolted to a stiffened spreader
Tubular double T-joints under axial brace Ioadin#
209
TABLE 1 Measured Dimensions and Properties of TT-Joint Specimens
TT- 1 TT-2 TT-3 TT-4 TT-5 TT-6 TT-7 TT-8 TT-9 TT-10 TT-11 TT-12
(o (de#.)
d (mm)
t (rnrn)
g, (ram)
60.6 60-3 60.3 90-2 90.1 90"0 89.9 120-2 120.0 120.1 120-4 120-2
42.78 60-60 89.14 42.78 60.60 89"14 114-31 42.78 60.60 89.14 114-31 139-68
3.45 3-93 4.03 3.45 3.93 4-03 4-24 3.45 3"93 4"03 4-24 4"31
55.9 38.0 7-1 101.2 85.1 56.2 27.2 139.6 126.5 101-5 75.1 42.5
D= 190-8 mm, T= 5.23 mm, L=850mm, Fy=384 MPa, Fu=444MPa, E = 2.06 x 105 MPa.
Fig. 3. Test set-up.
b e a m . T h i n e n d p l a t e s w e r e welded to the c h o r d e n d s a n d b o l t e d to a frame. T h e l o a d w a s a p p l i e d to the m i d d l e o f the b e a m b y a universal testing m a c h i n e giving axial forces in the b r a c e s t h r o u g h the hinges.
210
J. C. Paul, Y. Makino, Y. Kurobane
Measurements The load and deformations were measured and recorded using a high speed scanning amplifier and a micro computer which stored the data on diskettes. In this study the brace penetration on the chord surface was defined as the displacement of the brace ends relative to the centreline of the chord cross section in the direction of the brace axis. In order to calculate this brace penetration the following measurements were made (Fig. 4): 1. At two positions on each of the braces the horizontal displacements between the braces were measured by two linear differential transformers (LVDT) connected to sliding frames. 2. The vertical displacement of the upper sliding flame was measured at both sides of the braces by two LVDTs attached to a flame that is supported at the end plates at the ends of the chord. 3. At two cross sections of both the braces strains were recorded, in order to calculate the moments and axial forces in the braces. For the calculation of the brace penetration the braces were assumed to be elastic bars. Movements of one brace relative to the other were determined using measured values of displacements of the two sliding -Q.Pin
~
.
.
.
.
.
~in/fl.
r . . . . .
P _L ,_L
-~
Fig.4. Measurements.
p
Tubular double T-joints under axial brace loadino
211
systems, A1 and A2, as well as the measured strains at two cross sections on each brace. Vertical movements of the braces were determined by subtracting the elastic beam deflections of the chord from the overall deflection measurements. These displacements were combined to deduce the brace penetration relative to the chord centreline.
RESULTS All the joints except joint TT-1 failed by plastic failure of the chord wall. Two different types of plastic chord failure were clearly distinguishable and shown in Fig. 5. For failure type 1 the braces acted as one member, with no local deflections in the gap area (crotch). Failure mode 1 was observed for joints TT-2, TT-3 and TT-7. For failure type 2 significant local defections in the gap area were observed forming a fold in radial direction. Failure mode 2 was observed for joints TT-4 to TT-6 and joints TT-8 to TT-12. Joint TT-1 failed by brace buckling. However, significant local defection of .the chord wall in the gap area suggested the occurrence of failure type 2. The overall behaviour of TT-joints can be seen in load versus brace penetration curves shown in Fig. 6. The first peak in each curve is defined as the ultimate resistance of the joint. For the joints failing in type 2 mode, the load dropped quickly after an ultimate load and a minimum and second maximum were recorded. For the joints failing in type 1 mode, the load dropped much more gradually and no second maximum was recorded. Exceptions were joints TT-1 which failed by brace buckling and TT-7. After joint TT-7 reached the ultimate load an unsymmetrical failure pattern appeared, in which the brace penetration of one of the braces was significantly larger than the other, showing a rapid decline of the load. The values of the ultimate capacities for all the joints are given in Table 2(b).
Fig. 5. Failure types.
212
J. C. Paul, Y. Makino, Y. Kurobane
300
o Ollimell capacity
200
~lse
~:oo o
250
- - ~=n. 22 +. B=o. 32 :0. 4T -- ~ =a. aO -- ~ =O. 73
250
(b)
300
(a) ~=60"
', TI-7 200
o 150
TT-6
o T1-3 o
-~: IOa
,e-..TI-I Jlilt TT-I IT-2 T+-~
1 I0
TT-5
o
I00
....TT72
....... TI-4
fli[+ll till t r p l 2/5r|c+ buckling trio I till +
I 20
50
Joint TT-4 TT-5 TT-6
TT-7
I 30
--'
Brace penetration (mm)
I'0
;0
F i l l , r e lype lype Ivpe 2 tips Irpe I
30. . . . .
Brace penetration (mm)
(c) ¢:120" Jelpt TT-8 TT-g
250 I,,:~ .
,i
'~ 200 i
", T1-12 -+
TT-t2
type 2
TT-I~
" ,,,
;' ..... TT-11
T1-11
F a i l u r e type trpe type 2 type type 2
.
~," ° T+-~i-.......... ,, <+oo~ o T+:,
0
I0
20
--
30
Brace penetration (ram)
r I
40
Fig. 6. Load-brace penetration curves. (a] ~ = 60~'; (b) ~b= 90~; (c) ~b= 120'.
ANALYTICAL MODEL T o get a better insight into factors controlling the ultimate resistance of TT-joints ultimate loads of a closed ring having two different collapse m e c h a n i s m s are derived based on the observed failure types 1 and 2. The joint is replaced by a ring with an effective width Be s h o w n in Fig. 7 and
Tubular double T-joints under axial brace loading
213
TABLE 2(a)
Geometric Variables of TT-Joint Specimens
~b (deg)
fl
~,
~
:~
TT- I TT-2 TT-3 TT-4 TT-5 TT-6 TT-7 TT-8 TT-9 TT-10 TT- 11 TT- 12
60"6 60"3 60"3 90"2 90" 1 90"0 89"9 120"2 120"0 120-1 120.4 120.2
0-224 0"318 0.467 0.224 0-3 ! 8 0-467 0"599 0-224 0"318 0.467 0-599 0-732
18'24 18"24 18"24 ! 8.24 18-24 18"24 18"24 18"24 18-24 18-24 18"24 18.24
0"293 0"!99 0"037 0'530 0"446 0"294 0" 142 0.731 0"663 0-532 0'394 0"223
8"9 ! 8"91 8"9 ! 8'9 ! 8-9 I 8-9 I 8"9 ! 8-91 8"91 8"91 8'91 8'91
V- 1 V-2 V-3 V-4 V-5 V-6 V-7 V-8
90 90 90 90 60 120 90 120
0'405 0.406 0.646 0.222 0.406 0.406 0.407 0.769
22'35 ! 3'33 17.43 17-37 17"25 17'20 ! 7.44 ! 7.48
0"360 0"359 0"083 0"532 0"105 0'588 0"358 0" 170
4.8 I 4'8 I 5"29 4.44 4'8 ! 4'81 4"81 5"54
with the same geometrical and mechanical properties as the chord material. The braces are both fixed to a horizontal beam with two hinges, in accordance with the experimental test set-up. A concentrated vertical load 2P cos ~ is applied to the beam to give compressive forces P in both the braces. The force applied to the beam is in equilibrium with shear stresses z that act on two cross sections of the chord. For the shear stress z the following simple distribution is assumed: 2P cos T= - A
sin 0
(1) A = 2nRT
For failure type 2 yield hinges are assumed at the point D (0 = 0), point A (0=~), point B (0=),) and an unknown point C ( 0 = / ~ , ) , < / ~ < n ) . In the
214
J. C. Paul, Y. Makino, Y. Kurobane TABLE 2(b) Ultimate Capacities of TT-Joints (kN)
TT-Joint Test
Failure Ring type model I
TT-I TT-2 TT-3 TT-4 TT-5 TT-6 TT-7 TT-8 TT-9 TT-10 TT-11 TT-12
87.70 112.77 143.54 82.13 111.15 174.17 255"80 78.50 96.87 133-40 169.83 252.26
2 1 1 2 2 2 1 2 2 2 2 2
v-1 v-2 v-3 v-4 v-5 v-6 v-7 v-8
131.50 383.70 452.90 122.00 238.80 177.40 206.30 458.10
2 2 1 2 1 2 2 2
T-Joint
Ring model2
Equation (22)
83-46
79.03 105.40 165'21 78'61 104"30 164"80 248-91 74.85 91"88 129'80 178"70 252.43
70-97 88'81 117-69 85-65 109"88 148'70 182-93 70-43 88-52 117"13 141'90 169"82
74.75 89"72 124"45 74"75 89'72 124.45 166"06 74"55 89'48 124.12 165"63 217"85
145.45 433.54 449-20 132-28 227.74 186.65 222-51 424.13
122.87 358.14 293.47 129.10 162.61 162.54 200.54 267.57
115.82 344.80 284-84 126.06 180.04 179.84 176.85 355.60
119"96 145"48 95'97 115-39 165"06 244"62 79-85 92-29 122'20 169"15 274.58 130.18 402.57 485.85 135.01 217.47 149.55 203.55 454.67
A WS'
Kurobane ~1
yield hinges the b e n d i n g m o m e n t Mi and n o r m a l force Ni are a c t i n g ( i = A , B, C, D). T h e r e l a t i o n b e t w e e n M i / M p and Ni/Np are given b y a simplified i n t e r a c t i o n c u r v e for a r e c t a n g u l a r cross section ~° as s h o w n in Fig. 8. Assumed c o m b i n a t i o n s of Ni a n d Mi in the yield hinges A to D are s h o w n in this figure. T h e local r o t a t i o n s at these hinges are d e n o t e d b y flA, fl~, tiC, a n d tip while the s h o r t e n i n g s are d e n o t e d by AA, AB, Ac, and Ao. F i g u r e 8 also s h o w s the r e l a t i o n between the r o t a t i o n 0~ a n d the d i m e n sionless extension 2~. T h e following r e l a t i o n s hold:
0A=flA, 0B=flB, 0C= --tiC, 0D=flD
(2)
2i = -~-~Ai
(3)
Tubular double T-joints under axial brace loading
~ 2Pcos~)
E
~
iD A...'"'" .-'"'~
~ 2Pcosqb T~
E
Fig. 7. Ring model 2: TT-joint with failure type 2. A (;~, 0) o B (~,, e) r
t
'I
0.5
,
i
,
Ni
(I
C (~, 0) o 0 (~, 0) 0 Fig. 8. Interaction curve for plastic hinges, t°
215
d. C. Paul, Y. Makino, Y. Kurobane
216
with i= A, B, C, D, Np= BeFyT, Mp= BeFyT 2 0A 2A = ---~,
2B--
0B 2'
2C=-~,
2D=
0D 2
(4)
The relation between the shortening A~ and the local rotation fi are calculated from eqns (2) to (4):
Ai- fiT
(5)
8 with i = A, B, C, D. The vertical displacement of the beam is denoted by w; the rotation of the braces is denoted by ~0. Due to symmetry the horizontal displacements of points D and E are zero, while the vertical displacement of point E is equal to w. If Lbr is the brace length the following relations between w, the local rotations fi and shortenings Ai (i= A, B, C, D) yield: R [tic (1 - cos p) - fib (1 - cos y) - fA (1 -- COS at)] -- Ac cos p - AB cos y - - AA COS ~X- - 0 " 5 A D = 0
(6)
flc[R(cos ~b- cos #) + Lb, cos ~b] - fB [R(cos ~ - cos y) + Lb, COS q~] (7)
- Ac cos # - AB cos y = 0 f c JR(sin # - s i n 4~)- Lb, sin 4)] --fib JR(sin "y- sin q~)-Lbr sin 4']
(8)
+ Ac sin ~ + AB sin 7 ----w
/h = fc + ¢
(9)
flA ----0 " 5 f D - ~
(10)
Equations (6) to (10) result in the next matrix for which the local rotations
fi (i = A, B, C, D) and the rotation of the brace ~ are calculated: -1
0
fAI fBI
-1
0
0
1
1
o
o
A
B
C
D
0
0
0
E
F
0
0
0
0
G
H
0
0
w
-½
0
Tubular double T-joints under axial brace loading
217
with: A = - R ( 1 - b cos ~) B = - R ( 1 - b cos ~) C = R(1 - a cos #) O
T 16
~
- - - - -
E = - R ( c cos ( ~ - b cos y)
(11)
F = R(c cos (~ - a cos #) G = - R(b sin y - c sin 4)) H = R(a sin g - c sin ~b) T a=l+~-~,
T b=l-~-~,
Lbr c=l+--ff-
The internal work Wi d o n e in the yield hinges is equal to: W,= ,
o
M
M "~i
.
Ni
.
_
i~,4 P(MpO,+Np~.,)-Mp(2flA+2fla+2flcq-flo)
(12)
The external work Wc d o n e by the concentrated force 2P cos tk and the shear stresses z is equal to: We = 2P cos ~ w - - -
--2tic(1 - c o s ~-0"5/~ sin #)
+ 2fib (1 - cos ? - 0.57 sin 7) +
2BA( 1 - cos e - 0"5e sin a)
A A . sin ~ q--~-T A B . sin )~d--~-p Ac sin/~ 1 ) -I--~-~
(13)
The upper b o u n d of the collapse load Pc can be found by the next relation: W, = We
(14)
The value of/~ for which Pc is minimal can be obtained by: dPc ~ =0
(15)
218
J. C. Paul, Y. Makino, Y. Kurobane
For failure type 1 yield hinges are assumed at point A (0 = 0t), point C (0=n) and an unknown point B (0=~, 0~
J
We ~ 2Pc°t~mv
AC (,) d
i:::~.~:::..........,
:::++iii~.~::..........
i PI2 '
PI2
PI2i
~ PI2
|A B c
~c
(b) Fig. 9. Ring models. (a) Ring model 1: TF-joint failure with type 1; (b) T-joint. ~°
219
Tubular double T-joints under axial brace loading
where: W'e = ( 1 - ~ ) ( 1 + cos 0t)sin ~ - ( 1 - ~ ) ( 1
+ cos ~)sin ct
+ ( 4 (1 +cos ~)cos ~ + 2 ( 1 - ~ ) s i n ~ ) T 8R + ((1 --~)(1 + cos )')sin ~t+ (1 - ~ ) ( 1 - cos :0sin ~) ( ~ R ) z
(16)
Analogous to eqn (16) the collapse load Pc for the TT-joint is: T 1 +cos ~ +(1 - c o s ~)~--~ Pc = 2MpBe
R cos (p
W'c
(17)
The value of ~, for which Pc is minimal is obtained by: tgPc = c3~,
0
COMPARISON
(18) WITH EXPERIMENTAL
DATA
In Fig. 10 the nondimensional collapse load is shown as a function of the diameter ratio/~ and the transverse gap ratio ~. The width of the ring B= is taken equal to the radius R. The collapse loads governed by the two failure modes show trends opposite to each other in the response to variation of ~. For constant values of fl the collapse load Pc increases with the increase of for failure type 1 but decreases for failure type 2. This change in trend implies that the maximum collapse load is reached at a value of ~ where a change in failure type occurs. In Fig. 1 l(a) the boundary between the two failure modes for the ring model with Be = R is shown as a function of fl, ( and ~b. The value of ( for which a change in failure mode occurs decreases with the increase of fl while the corresponding value of ~b increases with an increase of ft. These trends found for the ring model with Bc = R agree with the experimental observations shown in Fig. l l(b). For joints with (+0.236/~>0.352 failure type 2 governs, while for joints with (+0"214/~<0.268 failure type 1 governs. Note that the tests performed by
220
J. C. Paul, Y. Makino, Y. Kurobane
t)=O. 22 t3=O, 32 15
=0.41
,'*~l
,~ =o. ¢7 t3=O. 8
V
I / /
i I
Flilur!
I"
tlpe
2
J
ure ,
0
0
tTpe I
I
0.2
i
I
0.4
0.6
~'=It/D
,
8
Fill. !0. Influence of { and/J on collapse load P<. (a)
o. 8 - -
~=80"
....
~=gO*
~:120"
0,6
'"'"".,. ~"0.4 ii a,.t,
ry
0.2
[
O,
[
i
I
I
L
1
0.2
0.3
0.4
0.5
0.8
0.7
,
O,B
B=d/D
Fig, !1. Influence of ~b,/~ and ( on failure type. (a) Closed ring analysis; (b) experiments.
Tubular double T-joints under axial brace loadin#
(b)
0. 0 TT2 TT-9
•
© ~=60"
•
0
TT-JO
H-4 v-,
Failure type 2
T~-5 i!
~ :90°
v-0
0,6
~"
221
TT-II
0,4 V-I V_-2 V-7
TT~I
TT-6°
TT-2 ~ L ~ , n °
0.2
Failvro type I
IT-12 '~ V-O
dnry ~
TT-3 o
0
O.
,
I
0.2
,
I
0,3
,
1
0,4
,
i
0.5
,
I
0,6
,
I
0.7
,
0.8
/3 =did Fig. ! I.--Contd.
Scola a'4 are included and denoted by V while the present tests are denoted by TT. Figure 12 shows the nondimensional ultimate capacities observed in the tests as function of fl and ( together with the predictions according to the closed ring analysis width Be fitted to the experimental data. Using the tests performed by Scola 3'4 and t h e present tests Be was calculated assuming the brace length as infinite. A multiple regression analysis was performed, in which the ring width Be was expressed in the following form:
Be= f(fl, ~, a)Re
(19)
The variable R, fl, ( and at signify the dimensions of the joints and e is the error term. As the model takes a multiplicative form it was transformed into an alternative model:
,20,
222
J. C. Paul, Y. Makino, Y. Kurobane
30 t
•
i' Y5
. ¥-I
i 'i ,, FIilVI~llPO 2 ' 0.2 ' 0,~ ' O,IO
:lilltl IVPOll
~-2 "
0.8
¢=gt/D (a)
0.2
0,4
0.16
O,S
¢=zt/O (b)
Fig. 12. Influence of ( and fl on ultimate resistance P, based on closed ring analysis. (a) Present test results; (b) Scola's test results. 3"4
Using a multipurpose solving programme the sum of squares of the errors lne was minimized. This analysis was performed for both models of the two observed collapse mechanisms each with corresponding data, leading to different expressions for Be. The following linear functions were selected as giving a satisfactory fit:
B¢, 1= R(8.43 - 6.37fl- 4.11 ~ - 0.233ct) Be, 2 = R ( - - 0"86-- 2"48fl + 6"48~-- 0.073cc)
(21)
B=,I and Be, 2 a r e the ring widths for the models based on the collapse mechanisms of type 1 and 2 respectively. In Table 2(b) the observed ultimate capacities are compared with predictions based on the closed ring analysis showing a good agreement between them. For failure type 1 and a constant value of fl the ultimate resistance predictions by the closed ring analysis show an increase in ultimate resistance with the increase of (. The opposite trend is observed for failure type 2. For constant values of fl the ultimate resistance decreases with the increase of (. However, for small values of fl combined with small values of ( this trend is contradicted, showing the limitations of the closed ring analysis using the effective width B=.
Tubular double T-joints under axial brace loading
223
TT-JOINT ULTIMATE RESISTANCE EQUATION The ring model seems to give good predictions for the ultimate resistance of TT-joints. However, for design simpler prediction equations are needed. To develop an equation for the ultimate resistance P. for TTjoints a model based on failure type 1 is adopted. This failure type was seen for values of ~+0.214/3 smaller than 0.268 and is also applicable to negative values of ( where the braces overlap each other. In the latter case the resultant of the brace forces in the vertical direction at the ultimate load of the TT-joint denoted by P.,v should be close to the ultimate resistance P'. of a planar T-joint with a large brace diameter d' (see Fig. 13). For values of (+0.236fl larger than 0.352 failure type 2 emanates with local defections of the chord wall between the braces. Therefore, when compared with T-joint resistance P~, a decrease of the resultant P~,v of the TT-joint with the increase of ( is assumed. For the prediction of P~ the formula of Kurobane et al. 11 is adopted, in which the decay of the P~,v/P'. ratio is corrected by an exponential function having ( and ~b as the independent variables. Based on these relations and performing a nonlinear regression analysis of experimental data reported by Scola et al. 3"4 as well as those reported herein a prediction equation for the ultimate resistance P. of multiplanar TT-joints is obtained as follows:
Pu--
P u, v
P ua( C~)e b(¢ )°'/ D
p' |
L rv, v : 2 " P v ' c o s ( ~ / 2 )
0
Fig. 13. AnalogybetweenTT-joints and T-joints with large//value.
J. C. Paul, Y. Makino, ¥. Kurobane
224
_ a( c~)eb~*)o,/°
/d,,~ 2-]/,D,~0.2 3 3
4.83 [ 1 + 4 " 9 4 ~ )
JL-~)
- .
(L)o,,s
Fy T 2
(22)
with: a(~b)= 1"4018 sin(b)- 0.1975 cos(~b) b(~b)= - 2.3255 sin(~b)+ 0.7700 cos(~b)
COV=0.068
p=4
n=16
The variables in this formula are shown in Fig. 1, while Fy is the yield stress of the chord material. The validity of eqn (22) is restricted to the following range: 4~<0c~<9, 0.2~
Figure 14 shows the predictions of the ultimate resistance by eqn (22) together with the data used in the analysis. Similar trends to those observed for the ring model are seen. 40
V,-3 V-8 .
30
---
_
TT-?TT-12
v,;s . . . . . . . . ""e'i't~ ~l . TT-II
A
v-2 . . . .
T Ir1~,
6 TT-5 9 .................... ~ V-4 TT-9 TT-I ...... * ..........
TT-3
TT-2
I0
TT-8
TT-4
B=~ ?' B=0.3' ~=0.'1 B=0.47 B:O. 8 I ,
I
0.2
,
I
I
0.4
0.8
,
0.8
~; =gt /D Fig. 14. Influence of ~ and fl on ultimate resistance Pu according to eqn (22).
Tubular double T-joints under axial brace loadin 0
225
COMPARISON WITH T-JOINTS The ultimate capacities of TT-joints are compared with the capacities of T-joint predicted by a formula proposed by Kurobane e t al. ~1 In Fig. 15 the TT-/T-joint ultimate resistance ratios are plotted against (, fl and 4). All the TT-joints demonstrate ultimate capacities larger than those predicted for T-joints. For joints with a constant value of the diameter ratio fl an increase of the resistance ratio with the decrease of ( is observed as long as failure mode 2 occurs. The resistance ratio increases with ft. A smaller increase or even a decrease in the resistance ratio with the decrease of the ( ratio is observed when the failure mode changes to failure mode 1. As the resistance ratio drops with the decrease of the ( ratio when failure type 1 occurs to a minimum of 0.5 in case of a 100% overlap (~b=0°), an optimum in the resistance ratio is expected to occur at the ( value or 4) value where a change of failure mode is expected. This is in accordance with the observations made based on the ring model.
COMPARISON
WITH AWS
The values of the ultimate resistance for TT-joints are compared with the lower bond (=allowable resistance times safety factor) predictions for Y-3
1.8
TT-7 m, x
*
o
",
1.4
•
~',
A...., ..~ 1.2 ~m
TT-6
x
V-5 V-8
TT-5
~:,,
,r-i"-~.: Q
....... ::~ .....
* T
TT-3
~
T'i-%'.
',,
o
6
TT-I I
i-.--
,--- 0.8
V-*4 ' ~ - 6
I [~=0. 22 ~=0. 32 ~:0. 41 B=0. 47 B:0. 8 I O
0,6
z~
,
---O
I
0.2
. . . . .
L
~ . . . . .
I
"41--
,
0.4
J
-. -'ll- -
I
0.6
,
0.8
¢=qlo Fig. 15. Comparison of TT-joint with T-joint.
226
d.C. Paul, Y. Makino, Y. Kurobane 1.8
• ~ ~:1207 1,6
V-3 l o
•
fT-12
m :z,,,, 1.4
",', ',, ", ',',,
a
z T
~
"1 TT-II
1.2
T't*.~
TT-IO
,::,
vi:i-,.y--t v;i;;::..i..~.5
• v-6
TT-9 ~-8 z~
"'%.TT-4 V-4
0.2
0.4
0.6
0,8
~=gt/D Fig. 16. Comparison of experiments with AWS predictions. TT-joints according to the AWS design recommendations. ~ In Fig. 16 the test/AWS resistance ratios are shown as a function of ~ or ~b. The AWS forms a lower bound for most of the data and only overpredicts the ultimate resistance for ~b= 9 0 ° and fl=0"22. That the influence of ( is not included in the AWS formulation is reflected in the increase of the underprediction with the decrease of ( when th is 90 ° and 120 °, as long as failure mode 2 governs. An opposite trend is seen for the joints with failure type 1 when ~b= 60 °. Joints V-3 and TT-7 sustain failure mode 1. Therefore the trend observed for ~b= 90 ° cannot be extrapolated for smaller values of where failure mode 1 is likely to occur.
CONCLUSIONS The failure type depends on the geometry and can be described by two of the three independent geometric variables fl, ( and q~. Failure type 1 with no local deflections in the gap area occurs for
Tubular double T-joints under axial brace loading
227
(+0.214fl<0.268, while failure type 2 with a fold forming between the braces occurs for ( + 0.236fl > 0.352. --The influence of ( on the ultimate resistance is dependant on the failure type. For constant values of fl the ultimate resistance increases with ( for failure type 1 and decreases for failure type 2. However, the trend for failure type I cannot be confirmed only looking at the test data. - - For the range of 60°4 ~b~ 120 ° the ultimate resistance of TT-joints is close to or larger than that of T-joints, and can therefore be treated as two separate T-joints in design. However, this is a conservative approach. -Within the given validity range the proposed eqn (22) predicts the ultimate resistance of TT-joints in a close range taking into account the multiplanar effects observed for TT-joints.
ACKNOWLEDGEM
ENTS
Acknowledgements are made to Nippon Steel Corporation for financial support and to Mr T. Ueno, M. Tsunoda and H. Matsuno, students of Kumamoto University, for assisting with the experiments.
REFERENCES 1. The Welding Society, Structural Welding Code-Steel, ANS/AWSDI.I-88. AWS, Miami, 1988. 2. Wardenier, J., Kurobane, Y., Packer, J. A., Dutta, D. & Yeomans, N., Design guide for circular hollow sections (CHS)joints under predominantly static loading, Comit6 International pour le D6veloppement et l'Etude de la Construction Tubutaire, Verlag TOV Rheinland, K61n, 1991. 3. Scola, S., Behavior of axially loaded tubular V-joints, MSc thesis. McGill University, Montreal, Canada, March 1989. 4. Scola, S., Redwood, R. G. & Mitri, H. S., Behavior of axially loaded tubular V-joints. J. Construct. Steel Res., 16 (1990) 89-109. 5. Paul, J. C., Van der Valk, C. A. C. & Wardenier, J., The static strength of multiplanar X-joints. Proc. 3rd Int. Symposium of Tubular Structures, Lappeenranta, September 1989, pp, 73-80. 6. Makino, Y., Kurobane, Y. & Ochi, K., Ultimate capacity of tubular double K-joints. Proc. 2nd lnt Conference on Weldin# of Tubular Structures, Boston, July 1984, pp. 45 I-8. 7. Van der Vegte, G. J., De Koning, C. H. M., Puthli, R. S. & Wardenier, J., Static behaviour of multiplanar welded joints in circular hollow sections. TNO-IBBC Report No, BI-90-106/63.5.3860/Stevin Report No. 25.6.90.13/A 1/
228
8. 9. 10. II.
J. C. Paul, Y. Making, Y. Kurobane
11.03, TNO Building and Construction Research/Stevin Laboratory Delft University of Technology, Delft, February 1991. Marshall, P. W., Design of Welded Tubular Connections--Basis and Use of the A WS Code Provisions, Elsevier Science Publishers, New York, 1991. Architectural Institute of Japan, Recommendations for the design and fabrication of tubular structures in steel, AIJ, Tokyo, 1990. Togo, T., Experimental study on the mechanical behavior of tubular joints, Doctoral dissertation, Osaka University, Osaka, 1967. Kurobane, Y., Making, Y. & Ochi, K., Ultimate resistance of unstiffened tubular joints. J. Struct. Enging, ASCE, 110(2) (February 1984) 385-400.
Ultimate Resistance of Tubular Double T-Joints under Axial Brace Loading J. C. Paul, Y. Makino & Y. K u r o b a n e Kumamoto University, Faculty of Engineering Kurokami 2-39-1, Kumamoto 860, Japan (Received 12 December 1991; revised version received I0 March 1992; accepted 14 May 1992)
ABSTRACT The results of a series of tests on multiplanar double T-joints with circular hollow sections under compressive brace loading are reported. The objective of the tests is to investigate influences of the brace to chord diameter ratio ft, the transverse gap ratio ( and the out-of-plane angle c~ on the ultimate resistance of the joints. To interpret the results of the tests a simple plastic analysis of a closed ring model with collapse mechanisms corresponding to two observed failure types is attempted. In the range ~p= 60 ° through 120 ° multiplanar TT-joints can be treated in design as two separate T-joints. An ultimate resistance equation is proposed on the basis of present and past tests results.
NOTATION A
Be d d' D E Fr Fu gt
Cross sectional area chord Ring width Brace diameter Comparable brace diameter Chord diameter Modulus of elasticity Yield stress chord material Tensile strength chord material Transverse gap 205
J. Construct. Steel Research 0143-974X/93/$06-00 © 1993 Elsevier Science Publishers Ltd,
England. Printed in Malta
206
lb, Lbr
L Mi
Mp Ni
Np P Pc Pu
PU,V P; R t
T w
we wi
Ai
0i T
d. C. Paul, Y. Makino, Y. Kurobane
Brace length (specimen, analytical model) Chord length Bending moment yield hinge i Plastic moment Axial force yield hinge i Plastic load Load Collapse load Ultimate resistance TT-joint Vertical resultant ultimate resistance TT-joint Ultimate resistance T-joint with d= d' Chord radius Brace wall thickness Chord wall thickness Vertical displacement beam External work Internal work Chord length ratio = 2LID Location yield hinges Diameter ratio = d/D Local rotation yield hinge i Chord wall thinness ratio = D/2T Shortenings yield hinge i Error Transverse gap ratio =gt/D Rotation yield hinge i Dimensionless extension yield hinge i Out-of-plane angle Shear stress Rotation braces
INTRODUCTION Since studies in the ultimate behaviour of multiplanar tubular joints have started only recently, basis for the design of such joints is still insufficient. The American Welding Society Structural Welding Code (AWS D 1,1-88)1 is the only code that shows general design criteria applicable to many types of multiplanar joints, while the Cidect Design Guide for Circular Hollow Section Joints 2 proposes criteria applicable only to double T, X and K-joints for which data are available. 3-7 The AWS Code proposes the
Tubular double T-joints under axial brace loading
207
chord ovalizing parameter to account for multiplanar effects, which, however, proved to be contradictory to actual behaviour in certain cases: two compression braces suppress the chord ovalization giving a beneficial effect on the ultimate resistance according to the AWS prediction, while local deflection of the chord wall between the two compression braces gave adverse effect on the ultimate resistance according to Makino's double K-joint tests. 6'8 Clearly, there remain many questions to be studied with regard to the ultimate behaviour of multiplanar joints. Scola et al. 3"4 performed a series of tests on double T-joints. They showed that the transverse gap size is a primary factor influencing the resistance of double T-joints when compared with that of planar T-joints, which resembles the behaviour of double K-joints. 6 However, the transverse gap interrelates to the out-of-plane angle and the brace to chord diameter ratio. More experimental evidence is indispensible to interpret the combined effect of these geometrical variables on the ultimate resistance of double T-joints. Twelve double T-joints (called TT-joints hereafter) under compression were tested to fill the void of existing data and the results are discussed in this paper. An analysis of a closed ring model is attempted to identify the effect of the transverse gap. The theoretical model is calibrated using experimental results while parametric formulae based on curve fitting are provided for practical use.
EXPERIMENTAL
PROGRAM
Twelve TT-joint specimens with two identical braces were tested under axial compression (Fig. 1). The diameter ratio fl, the out-of-plane angle ~b and the transverse gap to chord diameter ratio ~ were varied. The
Fig. 1. Joint geometry.
208
J. C. Paul, Y. Makino, Y. Kurobane 0.0
TI-IIz~
v~-t 0.6
TI-I U
TT-IO z~
TT-6 D
T~-II
-~0,4 ii TI-I o
0.2
TT'I o TT-2 o
TT-II
TT-3 0 , . , . ~ , ~ , 0.L6 ' 0 . t ? ' 1 . 0.2 0.3 O,i 0,5 0.8
B =did Fig. 2. Test program.
specimen are divisible into three groups with ~b of 60 °, 90 ° and 120 °. Within each group the diameter ratio 13was varied in a range between 0-22 and 0.73. The variation of ~b and /3 determined a variation range of the transverse gap ratio ( resulting in values between 0.04 and 0.73. The three nondimensional geometrical variables for the 12 joints included in the test program are depicted in Fig. 2. The chord length ratio 0c and the chord thinness ratio ~ were kept constant throughout the study (~=9.81 and = 18.24), as these ratios were thought to give similar effects on both the TT and T-joint capacities. The specimens had no eccentricities.
Specimen details The measured dimensions for the specimens and the material properties of the chord are shown in Table 1. All chord tubes were taken from the same lot (D= 190.7mm, T=5.23 mm), while the brace diameter were varied between 42.8 and 139.7 mm. For the chords cold formed steel STK400 tubes were used with the specified minimum tensile strength of 402 MPa. Fillet welds and partial penetration butt welds were used and welding was carried out with illuminate electrodes type D4301 with the specified minimum tensile strength of 422 MPa. Weld sizes were conformed to the Japanese design recommendations. 9
Test set-up The test set-up is shown in Fig. 3. The end plates welded to the braces were bolted to two hinges, which were bolted to a stiffened spreader
Tubular double T-joints under axial brace Ioadin#
209
TABLE 1 Measured Dimensions and Properties of TT-Joint Specimens
TT- 1 TT-2 TT-3 TT-4 TT-5 TT-6 TT-7 TT-8 TT-9 TT-10 TT-11 TT-12
(o (de#.)
d (mm)
t (rnrn)
g, (ram)
60.6 60-3 60.3 90-2 90.1 90"0 89.9 120-2 120.0 120.1 120-4 120-2
42.78 60-60 89.14 42.78 60.60 89"14 114-31 42.78 60.60 89.14 114-31 139-68
3.45 3-93 4.03 3.45 3.93 4-03 4-24 3.45 3"93 4"03 4-24 4"31
55.9 38.0 7-1 101.2 85.1 56.2 27.2 139.6 126.5 101-5 75.1 42.5
D= 190-8 mm, T= 5.23 mm, L=850mm, Fy=384 MPa, Fu=444MPa, E = 2.06 x 105 MPa.
Fig. 3. Test set-up.
b e a m . T h i n e n d p l a t e s w e r e welded to the c h o r d e n d s a n d b o l t e d to a frame. T h e l o a d w a s a p p l i e d to the m i d d l e o f the b e a m b y a universal testing m a c h i n e giving axial forces in the b r a c e s t h r o u g h the hinges.
210
J. C. Paul, Y. Makino, Y. Kurobane
Measurements The load and deformations were measured and recorded using a high speed scanning amplifier and a micro computer which stored the data on diskettes. In this study the brace penetration on the chord surface was defined as the displacement of the brace ends relative to the centreline of the chord cross section in the direction of the brace axis. In order to calculate this brace penetration the following measurements were made (Fig. 4): 1. At two positions on each of the braces the horizontal displacements between the braces were measured by two linear differential transformers (LVDT) connected to sliding frames. 2. The vertical displacement of the upper sliding flame was measured at both sides of the braces by two LVDTs attached to a flame that is supported at the end plates at the ends of the chord. 3. At two cross sections of both the braces strains were recorded, in order to calculate the moments and axial forces in the braces. For the calculation of the brace penetration the braces were assumed to be elastic bars. Movements of one brace relative to the other were determined using measured values of displacements of the two sliding -Q.Pin
~
.
.
.
.
.
~in/fl.
r . . . . .
P _L ,_L
-~
Fig.4. Measurements.
p
Tubular double T-joints under axial brace loadino
211
systems, A1 and A2, as well as the measured strains at two cross sections on each brace. Vertical movements of the braces were determined by subtracting the elastic beam deflections of the chord from the overall deflection measurements. These displacements were combined to deduce the brace penetration relative to the chord centreline.
RESULTS All the joints except joint TT-1 failed by plastic failure of the chord wall. Two different types of plastic chord failure were clearly distinguishable and shown in Fig. 5. For failure type 1 the braces acted as one member, with no local deflections in the gap area (crotch). Failure mode 1 was observed for joints TT-2, TT-3 and TT-7. For failure type 2 significant local defections in the gap area were observed forming a fold in radial direction. Failure mode 2 was observed for joints TT-4 to TT-6 and joints TT-8 to TT-12. Joint TT-1 failed by brace buckling. However, significant local defection of .the chord wall in the gap area suggested the occurrence of failure type 2. The overall behaviour of TT-joints can be seen in load versus brace penetration curves shown in Fig. 6. The first peak in each curve is defined as the ultimate resistance of the joint. For the joints failing in type 2 mode, the load dropped quickly after an ultimate load and a minimum and second maximum were recorded. For the joints failing in type 1 mode, the load dropped much more gradually and no second maximum was recorded. Exceptions were joints TT-1 which failed by brace buckling and TT-7. After joint TT-7 reached the ultimate load an unsymmetrical failure pattern appeared, in which the brace penetration of one of the braces was significantly larger than the other, showing a rapid decline of the load. The values of the ultimate capacities for all the joints are given in Table 2(b).
Fig. 5. Failure types.
212
J. C. Paul, Y. Makino, Y. Kurobane
300
o Ollimell capacity
200
~lse
~:oo o
250
- - ~=n. 22 +. B=o. 32 :0. 4T -- ~ =a. aO -- ~ =O. 73
250
(b)
300
(a) ~=60"
', TI-7 200
o 150
TT-6
o T1-3 o
-~: IOa
,e-..TI-I Jlilt TT-I IT-2 T+-~
1 I0
TT-5
o
I00
....TT72
....... TI-4
fli[+ll till t r p l 2/5r|c+ buckling trio I till +
I 20
50
Joint TT-4 TT-5 TT-6
TT-7
I 30
--'
Brace penetration (mm)
I'0
;0
F i l l , r e lype lype Ivpe 2 tips Irpe I
30. . . . .
Brace penetration (mm)
(c) ¢:120" Jelpt TT-8 TT-g
250 I,,:~ .
,i
'~ 200 i
", T1-12 -+
TT-t2
type 2
TT-I~
" ,,,
;' ..... TT-11
T1-11
F a i l u r e type trpe type 2 type type 2
.
~," ° T+-~i-.......... ,, <+oo~ o T+:,
0
I0
20
--
30
Brace penetration (ram)
r I
40
Fig. 6. Load-brace penetration curves. (a] ~ = 60~'; (b) ~b= 90~; (c) ~b= 120'.
ANALYTICAL MODEL T o get a better insight into factors controlling the ultimate resistance of TT-joints ultimate loads of a closed ring having two different collapse m e c h a n i s m s are derived based on the observed failure types 1 and 2. The joint is replaced by a ring with an effective width Be s h o w n in Fig. 7 and
Tubular double T-joints under axial brace loading
213
TABLE 2(a)
Geometric Variables of TT-Joint Specimens
~b (deg)
fl
~,
~
:~
TT- I TT-2 TT-3 TT-4 TT-5 TT-6 TT-7 TT-8 TT-9 TT-10 TT- 11 TT- 12
60"6 60"3 60"3 90"2 90" 1 90"0 89"9 120"2 120"0 120-1 120.4 120.2
0-224 0"318 0.467 0.224 0-3 ! 8 0-467 0"599 0-224 0"318 0.467 0-599 0-732
18'24 18"24 18"24 ! 8.24 18-24 18"24 18"24 18"24 18-24 18-24 18"24 18.24
0"293 0"!99 0"037 0'530 0"446 0"294 0" 142 0.731 0"663 0-532 0'394 0"223
8"9 ! 8"91 8"9 ! 8'9 ! 8-9 I 8-9 I 8"9 ! 8-91 8"91 8"91 8'91 8'91
V- 1 V-2 V-3 V-4 V-5 V-6 V-7 V-8
90 90 90 90 60 120 90 120
0'405 0.406 0.646 0.222 0.406 0.406 0.407 0.769
22'35 ! 3'33 17.43 17-37 17"25 17'20 ! 7.44 ! 7.48
0"360 0"359 0"083 0"532 0"105 0'588 0"358 0" 170
4.8 I 4'8 I 5"29 4.44 4'8 ! 4'81 4"81 5"54
with the same geometrical and mechanical properties as the chord material. The braces are both fixed to a horizontal beam with two hinges, in accordance with the experimental test set-up. A concentrated vertical load 2P cos ~ is applied to the beam to give compressive forces P in both the braces. The force applied to the beam is in equilibrium with shear stresses z that act on two cross sections of the chord. For the shear stress z the following simple distribution is assumed: 2P cos T= - A
sin 0
(1) A = 2nRT
For failure type 2 yield hinges are assumed at the point D (0 = 0), point A (0=~), point B (0=),) and an unknown point C ( 0 = / ~ , ) , < / ~ < n ) . In the
214
J. C. Paul, Y. Makino, Y. Kurobane TABLE 2(b) Ultimate Capacities of TT-Joints (kN)
TT-Joint Test
Failure Ring type model I
TT-I TT-2 TT-3 TT-4 TT-5 TT-6 TT-7 TT-8 TT-9 TT-10 TT-11 TT-12
87.70 112.77 143.54 82.13 111.15 174.17 255"80 78.50 96.87 133-40 169.83 252.26
2 1 1 2 2 2 1 2 2 2 2 2
v-1 v-2 v-3 v-4 v-5 v-6 v-7 v-8
131.50 383.70 452.90 122.00 238.80 177.40 206.30 458.10
2 2 1 2 1 2 2 2
T-Joint
Ring model2
Equation (22)
83-46
79.03 105.40 165'21 78'61 104"30 164"80 248-91 74.85 91"88 129'80 178"70 252.43
70-97 88'81 117-69 85-65 109"88 148'70 182-93 70-43 88-52 117"13 141'90 169"82
74.75 89"72 124"45 74"75 89'72 124.45 166"06 74"55 89'48 124.12 165"63 217"85
145.45 433.54 449-20 132-28 227.74 186.65 222-51 424.13
122.87 358.14 293.47 129.10 162.61 162.54 200.54 267.57
115.82 344.80 284-84 126.06 180.04 179.84 176.85 355.60
119"96 145"48 95'97 115-39 165"06 244"62 79-85 92-29 122'20 169"15 274.58 130.18 402.57 485.85 135.01 217.47 149.55 203.55 454.67
A WS'
Kurobane ~1
yield hinges the b e n d i n g m o m e n t Mi and n o r m a l force Ni are a c t i n g ( i = A , B, C, D). T h e r e l a t i o n b e t w e e n M i / M p and Ni/Np are given b y a simplified i n t e r a c t i o n c u r v e for a r e c t a n g u l a r cross section ~° as s h o w n in Fig. 8. Assumed c o m b i n a t i o n s of Ni a n d Mi in the yield hinges A to D are s h o w n in this figure. T h e local r o t a t i o n s at these hinges are d e n o t e d b y flA, fl~, tiC, a n d tip while the s h o r t e n i n g s are d e n o t e d by AA, AB, Ac, and Ao. F i g u r e 8 also s h o w s the r e l a t i o n between the r o t a t i o n 0~ a n d the d i m e n sionless extension 2~. T h e following r e l a t i o n s hold:
0A=flA, 0B=flB, 0C= --tiC, 0D=flD
(2)
2i = -~-~Ai
(3)
Tubular double T-joints under axial brace loading
~ 2Pcos~)
E
~
iD A...'"'" .-'"'~
~ 2Pcosqb T~
E
Fig. 7. Ring model 2: TT-joint with failure type 2. A (;~, 0) o B (~,, e) r
t
'I
0.5
,
i
,
Ni
(I
C (~, 0) o 0 (~, 0) 0 Fig. 8. Interaction curve for plastic hinges, t°
215
d. C. Paul, Y. Makino, Y. Kurobane
216
with i= A, B, C, D, Np= BeFyT, Mp= BeFyT 2 0A 2A = ---~,
2B--
0B 2'
2C=-~,
2D=
0D 2
(4)
The relation between the shortening A~ and the local rotation fi are calculated from eqns (2) to (4):
Ai- fiT
(5)
8 with i = A, B, C, D. The vertical displacement of the beam is denoted by w; the rotation of the braces is denoted by ~0. Due to symmetry the horizontal displacements of points D and E are zero, while the vertical displacement of point E is equal to w. If Lbr is the brace length the following relations between w, the local rotations fi and shortenings Ai (i= A, B, C, D) yield: R [tic (1 - cos p) - fib (1 - cos y) - fA (1 -- COS at)] -- Ac cos p - AB cos y - - AA COS ~X- - 0 " 5 A D = 0
(6)
flc[R(cos ~b- cos #) + Lb, cos ~b] - fB [R(cos ~ - cos y) + Lb, COS q~] (7)
- Ac cos # - AB cos y = 0 f c JR(sin # - s i n 4~)- Lb, sin 4)] --fib JR(sin "y- sin q~)-Lbr sin 4']
(8)
+ Ac sin ~ + AB sin 7 ----w
/h = fc + ¢
(9)
flA ----0 " 5 f D - ~
(10)
Equations (6) to (10) result in the next matrix for which the local rotations
fi (i = A, B, C, D) and the rotation of the brace ~ are calculated: -1
0
fAI fBI
-1
0
0
1
1
o
o
A
B
C
D
0
0
0
E
F
0
0
0
0
G
H
0
0
w
-½
0
Tubular double T-joints under axial brace loading
217
with: A = - R ( 1 - b cos ~) B = - R ( 1 - b cos ~) C = R(1 - a cos #) O
T 16
~
- - - - -
E = - R ( c cos ( ~ - b cos y)
(11)
F = R(c cos (~ - a cos #) G = - R(b sin y - c sin 4)) H = R(a sin g - c sin ~b) T a=l+~-~,
T b=l-~-~,
Lbr c=l+--ff-
The internal work Wi d o n e in the yield hinges is equal to: W,= ,
o
M
M "~i
.
Ni
.
_
i~,4 P(MpO,+Np~.,)-Mp(2flA+2fla+2flcq-flo)
(12)
The external work Wc d o n e by the concentrated force 2P cos tk and the shear stresses z is equal to: We = 2P cos ~ w - - -
--2tic(1 - c o s ~-0"5/~ sin #)
+ 2fib (1 - cos ? - 0.57 sin 7) +
2BA( 1 - cos e - 0"5e sin a)
A A . sin ~ q--~-T A B . sin )~d--~-p Ac sin/~ 1 ) -I--~-~
(13)
The upper b o u n d of the collapse load Pc can be found by the next relation: W, = We
(14)
The value of/~ for which Pc is minimal can be obtained by: dPc ~ =0
(15)
218
J. C. Paul, Y. Makino, Y. Kurobane
For failure type 1 yield hinges are assumed at point A (0 = 0t), point C (0=n) and an unknown point B (0=~, 0~
J
We ~ 2Pc°t~mv
AC (,) d
i:::~.~:::..........,
:::++iii~.~::..........
i PI2 '
PI2
PI2i
~ PI2
|A B c
~c
(b) Fig. 9. Ring models. (a) Ring model 1: TF-joint failure with type 1; (b) T-joint. ~°
219
Tubular double T-joints under axial brace loading
where: W'e = ( 1 - ~ ) ( 1 + cos 0t)sin ~ - ( 1 - ~ ) ( 1
+ cos ~)sin ct
+ ( 4 (1 +cos ~)cos ~ + 2 ( 1 - ~ ) s i n ~ ) T 8R + ((1 --~)(1 + cos )')sin ~t+ (1 - ~ ) ( 1 - cos :0sin ~) ( ~ R ) z
(16)
Analogous to eqn (16) the collapse load Pc for the TT-joint is: T 1 +cos ~ +(1 - c o s ~)~--~ Pc = 2MpBe
R cos (p
W'c
(17)
The value of ~, for which Pc is minimal is obtained by: tgPc = c3~,
0
COMPARISON
(18) WITH EXPERIMENTAL
DATA
In Fig. 10 the nondimensional collapse load is shown as a function of the diameter ratio/~ and the transverse gap ratio ~. The width of the ring B= is taken equal to the radius R. The collapse loads governed by the two failure modes show trends opposite to each other in the response to variation of ~. For constant values of fl the collapse load Pc increases with the increase of for failure type 1 but decreases for failure type 2. This change in trend implies that the maximum collapse load is reached at a value of ~ where a change in failure type occurs. In Fig. 1 l(a) the boundary between the two failure modes for the ring model with Be = R is shown as a function of fl, ( and ~b. The value of ( for which a change in failure mode occurs decreases with the increase of fl while the corresponding value of ~b increases with an increase of ft. These trends found for the ring model with Bc = R agree with the experimental observations shown in Fig. l l(b). For joints with (+0.236/~>0.352 failure type 2 governs, while for joints with (+0"214/~<0.268 failure type 1 governs. Note that the tests performed by
220
J. C. Paul, Y. Makino, Y. Kurobane
t)=O. 22 t3=O, 32 15
=0.41
,'*~l
,~ =o. ¢7 t3=O. 8
V
I / /
i I
Flilur!
I"
tlpe
2
J
ure ,
0
0
tTpe I
I
0.2
i
I
0.4
0.6
~'=It/D
,
8
Fill. !0. Influence of { and/J on collapse load P<. (a)
o. 8 - -
~=80"
....
~=gO*
~:120"
0,6
'"'"".,. ~"0.4 ii a,.t,
ry
0.2
[
O,
[
i
I
I
L
1
0.2
0.3
0.4
0.5
0.8
0.7
,
O,B
B=d/D
Fig, !1. Influence of ~b,/~ and ( on failure type. (a) Closed ring analysis; (b) experiments.
Tubular double T-joints under axial brace loadin#
(b)
0. 0 TT2 TT-9
•
© ~=60"
•
0
TT-JO
H-4 v-,
Failure type 2
T~-5 i!
~ :90°
v-0
0,6
~"
221
TT-II
0,4 V-I V_-2 V-7
TT~I
TT-6°
TT-2 ~ L ~ , n °
0.2
Failvro type I
IT-12 '~ V-O
dnry ~
TT-3 o
0
O.
,
I
0.2
,
I
0,3
,
1
0,4
,
i
0.5
,
I
0,6
,
I
0.7
,
0.8
/3 =did Fig. ! I.--Contd.
Scola a'4 are included and denoted by V while the present tests are denoted by TT. Figure 12 shows the nondimensional ultimate capacities observed in the tests as function of fl and ( together with the predictions according to the closed ring analysis width Be fitted to the experimental data. Using the tests performed by Scola 3'4 and t h e present tests Be was calculated assuming the brace length as infinite. A multiple regression analysis was performed, in which the ring width Be was expressed in the following form:
Be= f(fl, ~, a)Re
(19)
The variable R, fl, ( and at signify the dimensions of the joints and e is the error term. As the model takes a multiplicative form it was transformed into an alternative model:
,20,
222
J. C. Paul, Y. Makino, Y. Kurobane
30 t
•
i' Y5
. ¥-I
i 'i ,, FIilVI~llPO 2 ' 0.2 ' 0,~ ' O,IO
:lilltl IVPOll
~-2 "
0.8
¢=gt/D (a)
0.2
0,4
0.16
O,S
¢=zt/O (b)
Fig. 12. Influence of ( and fl on ultimate resistance P, based on closed ring analysis. (a) Present test results; (b) Scola's test results. 3"4
Using a multipurpose solving programme the sum of squares of the errors lne was minimized. This analysis was performed for both models of the two observed collapse mechanisms each with corresponding data, leading to different expressions for Be. The following linear functions were selected as giving a satisfactory fit:
B¢, 1= R(8.43 - 6.37fl- 4.11 ~ - 0.233ct) Be, 2 = R ( - - 0"86-- 2"48fl + 6"48~-- 0.073cc)
(21)
B=,I and Be, 2 a r e the ring widths for the models based on the collapse mechanisms of type 1 and 2 respectively. In Table 2(b) the observed ultimate capacities are compared with predictions based on the closed ring analysis showing a good agreement between them. For failure type 1 and a constant value of fl the ultimate resistance predictions by the closed ring analysis show an increase in ultimate resistance with the increase of (. The opposite trend is observed for failure type 2. For constant values of fl the ultimate resistance decreases with the increase of (. However, for small values of fl combined with small values of ( this trend is contradicted, showing the limitations of the closed ring analysis using the effective width B=.
Tubular double T-joints under axial brace loading
223
TT-JOINT ULTIMATE RESISTANCE EQUATION The ring model seems to give good predictions for the ultimate resistance of TT-joints. However, for design simpler prediction equations are needed. To develop an equation for the ultimate resistance P. for TTjoints a model based on failure type 1 is adopted. This failure type was seen for values of ~+0.214/3 smaller than 0.268 and is also applicable to negative values of ( where the braces overlap each other. In the latter case the resultant of the brace forces in the vertical direction at the ultimate load of the TT-joint denoted by P.,v should be close to the ultimate resistance P'. of a planar T-joint with a large brace diameter d' (see Fig. 13). For values of (+0.236fl larger than 0.352 failure type 2 emanates with local defections of the chord wall between the braces. Therefore, when compared with T-joint resistance P~, a decrease of the resultant P~,v of the TT-joint with the increase of ( is assumed. For the prediction of P~ the formula of Kurobane et al. 11 is adopted, in which the decay of the P~,v/P'. ratio is corrected by an exponential function having ( and ~b as the independent variables. Based on these relations and performing a nonlinear regression analysis of experimental data reported by Scola et al. 3"4 as well as those reported herein a prediction equation for the ultimate resistance P. of multiplanar TT-joints is obtained as follows:
Pu--
P u, v
P ua( C~)e b(¢ )°'/ D
p' |
L rv, v : 2 " P v ' c o s ( ~ / 2 )
0
Fig. 13. AnalogybetweenTT-joints and T-joints with large//value.
J. C. Paul, Y. Makino, ¥. Kurobane
224
_ a( c~)eb~*)o,/°
/d,,~ 2-]/,D,~0.2 3 3
4.83 [ 1 + 4 " 9 4 ~ )
JL-~)
- .
(L)o,,s
Fy T 2
(22)
with: a(~b)= 1"4018 sin(b)- 0.1975 cos(~b) b(~b)= - 2.3255 sin(~b)+ 0.7700 cos(~b)
COV=0.068
p=4
n=16
The variables in this formula are shown in Fig. 1, while Fy is the yield stress of the chord material. The validity of eqn (22) is restricted to the following range: 4~<0c~<9, 0.2~
Figure 14 shows the predictions of the ultimate resistance by eqn (22) together with the data used in the analysis. Similar trends to those observed for the ring model are seen. 40
V,-3 V-8 .
30
---
_
TT-?TT-12
v,;s . . . . . . . . ""e'i't~ ~l . TT-II
A
v-2 . . . .
T Ir1~,
6 TT-5 9 .................... ~ V-4 TT-9 TT-I ...... * ..........
TT-3
TT-2
I0
TT-8
TT-4
B=~ ?' B=0.3' ~=0.'1 B=0.47 B:O. 8 I ,
I
0.2
,
I
I
0.4
0.8
,
0.8
~; =gt /D Fig. 14. Influence of ~ and fl on ultimate resistance Pu according to eqn (22).
Tubular double T-joints under axial brace loadin 0
225
COMPARISON WITH T-JOINTS The ultimate capacities of TT-joints are compared with the capacities of T-joint predicted by a formula proposed by Kurobane e t al. ~1 In Fig. 15 the TT-/T-joint ultimate resistance ratios are plotted against (, fl and 4). All the TT-joints demonstrate ultimate capacities larger than those predicted for T-joints. For joints with a constant value of the diameter ratio fl an increase of the resistance ratio with the decrease of ( is observed as long as failure mode 2 occurs. The resistance ratio increases with ft. A smaller increase or even a decrease in the resistance ratio with the decrease of the ( ratio is observed when the failure mode changes to failure mode 1. As the resistance ratio drops with the decrease of the ( ratio when failure type 1 occurs to a minimum of 0.5 in case of a 100% overlap (~b=0°), an optimum in the resistance ratio is expected to occur at the ( value or 4) value where a change of failure mode is expected. This is in accordance with the observations made based on the ring model.
COMPARISON
WITH AWS
The values of the ultimate resistance for TT-joints are compared with the lower bond (=allowable resistance times safety factor) predictions for Y-3
1.8
TT-7 m, x
*
o
",
1.4
•
~',
A...., ..~ 1.2 ~m
TT-6
x
V-5 V-8
TT-5
~:,,
,r-i"-~.: Q
....... ::~ .....
* T
TT-3
~
T'i-%'.
',,
o
6
TT-I I
i-.--
,--- 0.8
V-*4 ' ~ - 6
I [~=0. 22 ~=0. 32 ~:0. 41 B=0. 47 B:0. 8 I O
0,6
z~
,
---O
I
0.2
. . . . .
L
~ . . . . .
I
"41--
,
0.4
J
-. -'ll- -
I
0.6
,
0.8
¢=qlo Fig. 15. Comparison of TT-joint with T-joint.
226
d.C. Paul, Y. Makino, Y. Kurobane 1.8
• ~ ~:1207 1,6
V-3 l o
•
fT-12
m :z,,,, 1.4
",', ',, ", ',',,
a
z T
~
"1 TT-II
1.2
T't*.~
TT-IO
,::,
vi:i-,.y--t v;i;;::..i..~.5
• v-6
TT-9 ~-8 z~
"'%.TT-4 V-4
0.2
0.4
0.6
0,8
~=gt/D Fig. 16. Comparison of experiments with AWS predictions. TT-joints according to the AWS design recommendations. ~ In Fig. 16 the test/AWS resistance ratios are shown as a function of ~ or ~b. The AWS forms a lower bound for most of the data and only overpredicts the ultimate resistance for ~b= 9 0 ° and fl=0"22. That the influence of ( is not included in the AWS formulation is reflected in the increase of the underprediction with the decrease of ( when th is 90 ° and 120 °, as long as failure mode 2 governs. An opposite trend is seen for the joints with failure type 1 when ~b= 60 °. Joints V-3 and TT-7 sustain failure mode 1. Therefore the trend observed for ~b= 90 ° cannot be extrapolated for smaller values of where failure mode 1 is likely to occur.
CONCLUSIONS The failure type depends on the geometry and can be described by two of the three independent geometric variables fl, ( and q~. Failure type 1 with no local deflections in the gap area occurs for
Tubular double T-joints under axial brace loading
227
(+0.214fl<0.268, while failure type 2 with a fold forming between the braces occurs for ( + 0.236fl > 0.352. --The influence of ( on the ultimate resistance is dependant on the failure type. For constant values of fl the ultimate resistance increases with ( for failure type 1 and decreases for failure type 2. However, the trend for failure type I cannot be confirmed only looking at the test data. - - For the range of 60°4 ~b~ 120 ° the ultimate resistance of TT-joints is close to or larger than that of T-joints, and can therefore be treated as two separate T-joints in design. However, this is a conservative approach. -Within the given validity range the proposed eqn (22) predicts the ultimate resistance of TT-joints in a close range taking into account the multiplanar effects observed for TT-joints.
ACKNOWLEDGEM
ENTS
Acknowledgements are made to Nippon Steel Corporation for financial support and to Mr T. Ueno, M. Tsunoda and H. Matsuno, students of Kumamoto University, for assisting with the experiments.
REFERENCES 1. The Welding Society, Structural Welding Code-Steel, ANS/AWSDI.I-88. AWS, Miami, 1988. 2. Wardenier, J., Kurobane, Y., Packer, J. A., Dutta, D. & Yeomans, N., Design guide for circular hollow sections (CHS)joints under predominantly static loading, Comit6 International pour le D6veloppement et l'Etude de la Construction Tubutaire, Verlag TOV Rheinland, K61n, 1991. 3. Scola, S., Behavior of axially loaded tubular V-joints, MSc thesis. McGill University, Montreal, Canada, March 1989. 4. Scola, S., Redwood, R. G. & Mitri, H. S., Behavior of axially loaded tubular V-joints. J. Construct. Steel Res., 16 (1990) 89-109. 5. Paul, J. C., Van der Valk, C. A. C. & Wardenier, J., The static strength of multiplanar X-joints. Proc. 3rd Int. Symposium of Tubular Structures, Lappeenranta, September 1989, pp, 73-80. 6. Makino, Y., Kurobane, Y. & Ochi, K., Ultimate capacity of tubular double K-joints. Proc. 2nd lnt Conference on Weldin# of Tubular Structures, Boston, July 1984, pp. 45 I-8. 7. Van der Vegte, G. J., De Koning, C. H. M., Puthli, R. S. & Wardenier, J., Static behaviour of multiplanar welded joints in circular hollow sections. TNO-IBBC Report No, BI-90-106/63.5.3860/Stevin Report No. 25.6.90.13/A 1/
228
8. 9. 10. II.
J. C. Paul, Y. Making, Y. Kurobane
11.03, TNO Building and Construction Research/Stevin Laboratory Delft University of Technology, Delft, February 1991. Marshall, P. W., Design of Welded Tubular Connections--Basis and Use of the A WS Code Provisions, Elsevier Science Publishers, New York, 1991. Architectural Institute of Japan, Recommendations for the design and fabrication of tubular structures in steel, AIJ, Tokyo, 1990. Togo, T., Experimental study on the mechanical behavior of tubular joints, Doctoral dissertation, Osaka University, Osaka, 1967. Kurobane, Y., Making, Y. & Ochi, K., Ultimate resistance of unstiffened tubular joints. J. Struct. Enging, ASCE, 110(2) (February 1984) 385-400.