Engineering Structures 94 (2015) 70–81
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Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
Exterior beam column joints – Shear strength model and design formula
Margherita Pauletta a,⇑, Daniele Di Luca b, Gaetano Russo b
a
b
Department of Civil Engineering and Architecture, University of Udine, Viale delle Scienze, 206, 33100 Udine, Italy
Department of Civil Engineering and Architecture, University of Udine, Viale delle Scienze, 208, 33100 Udine, Italy
a r t i c l e
i n f o
Article history:
Received 4 February 2014
Revised 17 March 2015
Accepted 18 March 2015
Keywords:
Beam–column joint
Shear strength
Strut-and-tie mechanism
Test data
Design formula
a b s t r a c t
A new model to determine the shear strength of exterior reinforced concrete (RC) beam–column joints
under seismic actions is proposed in this paper. An explicit formula that considers the shear strength contributions provided by the strut-and-tie mechanism due to two diagonal concrete struts, as well as the
horizontal hoops and the intermediate vertical bars of the column, is derived. The coefficients of each
contribution are calibrated using 61 test data sets available in the literature, most of them from cyclic
tests. This paper compares the shear strength predictions using the proposed expression, the model of
Hwang and Lee, and the model of Park and Mosalam, the last of which is valid for unreinforced joints
only. A design formula is also proposed and its predictions are compared to those of Eurocode 8 and
ACI Code.
Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction
The key point of the aseismic performance for beam–column
joints is to ensure and maintain the energy absorption capacity
of plastic hinges of adjoining members, usually the beams, avoiding any shear or anchorage failure in the joint core. By contrast,
due to the small value of shear span to depth ratio, the strength
of external beam–column joints can be governed by shear rather
than flexural strength.
Various codes and authors try to predict the real shear strength
of exterior beam–column joints under seismic loads [1–7] following different approaches. However, the obtained predictions are
often not accurate, mainly because of the several mechanisms
involved in the actual behavior.
This paper proposes a strut-and-tie model for determining
shear strength of exterior RC beam–column joints that represents
an evolution of the model proposed by Hwang and Lee [1]. A plane
frame joint is considered in the following for the sake of simplicity.
The proposed model is based on a softening approximate constitutive law for plain concrete and it considers diagonal compressed concrete strut, diagonal compression due to bond
resistance of beam longitudinal reinforcement, and resisting contributions of horizontal hoops and intermediate column bars
within the joint core. A single analytical expression is proposed
to evaluate the tensile stress trend in the longitudinal reinforcing
⇑ Corresponding author. Tel.: +39 0432 558065; fax: +39 0432 558052.
E-mail addresses: margherita.pauletta@uniud.it (M. Pauletta), diluca.daniele@
spes.uniud.it (D. Di Luca), gaetano.russo@uniud.it (G. Russo).
http://dx.doi.org/10.1016/j.engstruct.2015.03.040
0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.
bars of the beam when joint shear failure occurs. This allows one
to avoid a solution iterative procedure up to the ultimate load.
The coefficients of each contribution are calibrated on the basis
of 61 test data sets and results, which have been selected from
reports on exterior RC beam–column joints that failed due to shear
only mainly under reversed cyclic loads. The data also include 17
beam–column joints without transverse reinforcement.
Uniformity and accuracy of the model predictions are assessed
by comparing these predictions with those of the iterative procedure of Hwang and Lee [1] with the 61 experimental results.
The predictions regarding joints without transverse reinforcement are compared with predictions from the simplified strength
model proposed by Park and Mosalam [2]. A design formula is also
proposed, and its predictions are compared with those obtained
through the shear strength design formulas provided by
Eurocode 8 [3] and ACI Code 318-11 [4].
2. Research significance
The aim of the present study is to solve the problem of the
external RC beam–column joint shear strength prediction by
means of a single expression more accurate and consistent (uniform in the prediction) than existing formulas or time-consuming
computing procedures. The proposed expression highlights four
principal resistant contributions: the first two are based on a
mechanism consisting of two inclined concrete struts in the joint,
whose contributions are affected by the type of anchorage of the
beam longitudinal reinforcing bars into the joint region; the third
contribution is due to horizontal stirrups reinforcement; and the
M. Pauletta et al. / Engineering Structures 94 (2015) 70–81
last one is provided by the vertical intermediate column bars. A
design formula, whose predictions incorporate an adequate margin
of safety, is also proposed.
Asb f b jbd
L
Mb ¼ V b L
Vb ¼
L þ hc =2
¼
Vb
H
71
ð3Þ
ð4Þ
3. Model bases
V c1
For the case of a typical exterior RC beam–column joint subjected to seismic load, the shear and compression acting forces
are shown in Fig. 1. The horizontal joint shear force in the joint core
Vjh can be calculated as
where L is the length from beam inflection point to column face; hc
is the total height of column cross section; H is the height of the column, equal to the height between upper and lower column inflection points (Fig. 3); and jbd is the internal moment arm of the
beam cross section that can be calculated as follows
V jh ¼ T b V c1
ð1Þ
jdb ¼ hb where
T b ¼ Asb f b
ð2Þ
with Tb the tensile force in the beam longitudinal reinforcement, Asb
and fb the area and stress of this reinforcement, respectively, and Vc1
the column horizontal shear above the joint.
The shear on the beam Vb, the beam flexural moment Mb, and
the column shear above the joint (Fig. 2) are calculated as follows
xb
dsb
3
ð5Þ
ð6Þ
In Eq. (6) hb is the beam depth, dsb is the distance from the centroid
of the tensile beam reinforcement to the closest edge of the beam
cross section, and xb is the depth of the compression zone in the
beam cross section (Fig. 3). After having verified that concrete in
compression remains within the elastic range [6], the value of xb
can be obtained by imposing the equilibrium of beam internal
forces, which leads to
bb x2b
þ ðAsb þ A0sb Þnh;b xb ðAsb db þ A0sb d0sb Þnh;b A0sb d0sb ¼ 0
2
ð7Þ
where bb is the width of the beam cross section at the face of the
column, A0 sb is the area of the beam longitudinal compressive
reinforcement, db is the effective depth of the beam cross section,
d0 sb is the distance from the centroid of compressive beam reinforcement to the closest edge of the beam cross section, and nh,b is the
modular ratio given by
nh;b ¼
Esb
Ec
ð8Þ
with Esb the steel elastic modulus of the beam reinforcement, whose
value, if not provided by the experimental papers, can be assumed
equal to 200,000 [MPa] [4]; and Ec the concrete elastic modulus
0 0;5
Fig. 1. External actions on the exterior beam–column joint core.
whose value, if not provided, can be assumed equal to 4700ðf c Þ
[MPa] [4].
Substituting Eqs. (3) and (5) into Eq. (1), the horizontal shear
force in the joint core can be expressed by
ðL þ hc =2Þjdb
V jh ¼ Asb f b 1 HL
ð9Þ
The shear nominal strength of exterior RC beam–column joints Vn is
assumed equal to the sum of two independent resisting
contributions
V n ¼ V hc þ V hs
ð10Þ
where Vhc is the shear strength contribution provided by two diagonal struts, ST1 and ST2 in Fig. 2, and Vhs is the shear contribution
given by steel horizontal and vertical reinforcements.
It is assumed that failure is governed by the crushing of the
diagonal compressive strut ST1 in Fig. 2 stiffened by the steel horizontal and vertical reinforcement (Fig. 1). The formation of the
strut is highlighted by the appearance of inclined cracks in the
joint.
4. Strut and tie mechanism
Fig. 2. The two inclined struts in unreinforced exterior joints.
Similar to what is reported by Hwang and Lee [1], the proposed
strut and tie model is composed of diagonal (Vhc), horizontal and
vertical mechanisms (Vhs).
For joints without horizontal stirrups and intermediate vertical
bars in the joint region, the horizontal joint shear force can be
assumed to be resisted by the horizontal components of the forces
acting in the two inclined struts (Park and Mosalam [1]), which are
72
M. Pauletta et al. / Engineering Structures 94 (2015) 70–81
Fig. 3. (a) Forces acting on the exterior beam–column subassemblage, and (b) sections of beam and column.
considered parallel. Each contribution can be computed by considering the level of beam reinforcement tensile stress, which is
related to the bond resistance. Consequently, for these joints,
Vn = Vhc; hence, at the joint shear failure, it must occur that
Vjh = Vhc. In this condition, the shear forces in the two concrete
struts can be expressed as follows
V hc;ST1 ¼ aV hc
ð11Þ
V hc;ST2 ¼ ð1 aÞV hc
ð12Þ
where a is the fraction factor related to the bond deterioration of
the reinforcement [1]. If bond deterioration is widespread, joint failure can occur for bond failure and not for shear failure [8]. These
cases are excluded from this study.
For joints with horizontal stirrups and vertical intermediate
bars, by imposing Vn = Vjh and substituting Eq. (10) into Eq. (11),
the two strut contributions to horizontal joint shear resistance
can be expressed as follows
V hc;ST1 ¼ aðV jh V hs Þ
ð13Þ
V hc;ST2 ¼ ð1 aÞ ðV jh V hs Þ
ð14Þ
The strut ST1 is developed by the 90-degree hook of the beam
reinforcement, assuming that this anchorage can carry the joint
shear force without bond failure.
The angle of inclination hh of the first strut ST1 is defined as
(Fig. 2)
00
00
hh ¼ tan1 ðhb =hc Þ
ð15Þ
00
hb
where is the distance between top and bottom beam longitudinal
00
bars, and hc is the distance measured from the centroid of bar
extension at the free end of the 90-deg hooked bar to the centroid
of longitudinal column reinforcement in the opposite side.
The ST1 strut contribution to horizontal joint shear resistance
can be expressed as follows (Fig. 2)
V hc;ST1 ¼ Asb f b nb p/b
Z
lh
lðf b Þdx V hs
ð16Þ
0
where l(fb) is the bond stress distribution along the beam bar when
joint shear failure occurs, expressed as a function of the tensile
stress of the bar fb, which varies with the distance x from the
beam–column interface [9]; nb is the number of beam longitudinal
bars in tension, with diameter /b; and lh is the horizontal projection
of the strut ST2 (Fig. 2), which is given by
lh ¼ hc ac
ð17Þ
with ac the depth of the compression zone in the column. The ac
value can be approximated by [6]
ac ¼
0; 25 þ 0; 85
!
N
0 hc
Ag f c
ð18Þ
0
where N is the axial force in the column, f c is the cylindric compressive strength of concrete, and Ag is the gross area of the column
section.
The ST2 strut contribution is developed by the bond resistance
of the concrete surrounding the beam longitudinal reinforcement
[1]. In this study only 90-degree hooks and bars bent into the joint
region are considered. The contribution of ST2 strut to the joint
horizontal shear resistance can be expressed as follows
V hc;ST2 ¼ nb p/b
Z
lh
lðf b Þdx V c1 ð1 aÞV hs
ð19Þ
0
The tensile stress f b of the beam longitudinal bars present in Eq.
(16) and bond stress distribution lðf b Þ in Eq. (19) are unknown.
In order to determine these two terms without relying on iterative
procedures like those used by Park and Mosalam in their analytical
model [1], experimental results on 61 test specimens found in the
literature (Table 1) are used. The aim is to obtain a single analytical
expression giving the tensile stress trend f bi in the beam longitudinal bars when the joint shear failure occurs.
The analysis of the 61 test data sets shows that f bi decreases
with an increase in the mechanical percentage of beam tensile
reinforcement xb .
On the basis of the 61 test data, the function f bi ðxb Þ; interpolating the experimental values of f b , is
f bi ¼ ð0:630:21
Þf yb
b
ð20Þ
where
xb ¼
Asb f yb
0
b b hb f c
The form of function f bi is plotted in Fig. 4.
ð21Þ
Table 1
Mechanical and geometrical properties and reinforcement areas.
fc 0
(MPa)
Author references
and specimen
labels
fyb
(MPa)
fyh
(MPa)
fyv
(MPa)
Ec
(MPa)
Esb
(MPa)
Esh
(MPa)
Esv
(MPa)
bb
mm
hb
(mm)
xb
(mm)
Ag
(mm2)
ac
(mm)
hh (deg)
Asb
(mm2)
Ath
(mm2)
Atv
(mm2)
LL8c
LH8c
HL8c
HH8c
LL11c
LH11c
HL11c
HH11c
LL14c
LH14c
HH14c
56.6
56.6
56.6
56.6
74.5
74.5
74.5
74.5
92.5
92.5
92.5
457
457
443
443
457
457
443
443
457
457
443
447
447
447
447
447
447
447
447
447
447
447
463a
463a
457a
457
463a
463a
457a
457
463a
463a
457
39,673
39,673
39,673
39,673
45,251
45,251
45,251
45,251
49,369
49,369
49,369
180,052
180,052
171,778
171,778
180,052
180,052
171,778
171,778
180,052
180,052
171,778
187,721
187,721
187,721
187,721
187,721
187,721
187,721
187,721
187,721
187,721
187,721
176,956
176,956
180,052
180,052
176,956
176,956
180,052
180,052
176,956
176,956
180,052
318
318
318
318
318
318
318
318
318
318
318
508
508
508
508
508
508
508
508
508
508
508
125
125
134
134
120
120
128
128
117
117
125
126,451
126,451
126,451
126,451
126,451
126,451
126,451
126,451
126,451
126,451
126,451
101
101
110
110
98
98
108
108
95
95
101
58
58
57
57
59
59
59
59
59
58
57
2027
2027
2565
2565
2027
2027
2565
2565
2027
2027
2565
1161
1935
1161
1935
1161
1935
1161
1935
1161
1935
1935
776
776
1013
1013
776
776
1013
1013
776
776
1013
[16,17]
2b
4b
5b
6b
46.2
41.0
37.0
40.1
454
454
454
454
–
–
–
–
470
470
470
470
31,946
30,095
28,589
29,763
200,000
200,000
200,000
200,000
–
–
–
–
200,000
200,000
200,000
200,000
305
305
305
305
406
406
406
406
129
131
133
131
139,355
139,355
139,355
139,355
153
211
211
153
40
40
40
40
2565
2565
2565
2565
0
0
0
0
776
776
776
776
[18]
2c
3c
4c
67.3
64.7
67.3
455
455
455
455
455
455
455
455
455
38,570
37,811
38,570
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
300
259
259
480
439
439
114
109
117
115,845
89,832
89,832
98
92
89
58
60
60
1425
1251
1560
881
881
881
570
570
776
[19]
1B
3B
4B
5B
6Bc
33.6
40.9
44.6
24.4
39.8
337a
337a
337a
331
336a
437
437
437
437
437
490
490
490
414
490
27,245
30,064
31,403
23,196
29,656
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
259
259
259
300
300
480
480
439
480
480
149
145
134
155
140
89,832
89,832
89,832
115,845
115,845
90
90
89
122
104
68
68
66
62
62
2019
2019
2019
2328
2081
881
881
881
881
881
570
570
570
1013
570
[20]
B1
B2
B3
B4
30.0
30.0
30.0
30.0
1069
409
1069
1069
291
291
291
291
387
387
387
387
25,743
25,743
25,743
25,743
534,645
200,000
534,645
534,645
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
160
160
160
160
250
250
250
250
97
77
97
97
48,400
48,400
48,400
48,400
68
68
99
99
51
51
51
51
628
628
628
628
112
112
112
336
531
531
531
531
[21]
3
4
5
6
9
11
12
12
14
15
41.7
44.7
36.7
40.4
40.6
41.9
35.1
46.4
41.0
39.7
391
391
391
391
391
391
391
391
391
391
250
281
281
281
250
281
281
250
281
281
–
–
–
–
395
395
395
395
282
395
30,351
31,423
28,473
29,874
29,948
30,423
27,845
32,015
30,095
29,614
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
–
–
–
–
200,000
200,000
200,000
200,000
200,000
200,000
160
160
160
160
160
160
160
160
160
160
220
220
220
220
220
220
220
220
220
220
64
64
65
64
64
64
66
63
64
65
48,400
48,400
48,400
48,400
48,400
48,400
48,400
48,400
48,400
48,400
55
86
72
55
55
70
55
47
70
71
48
48
48
48
47
47
47
47
47
47
531
531
531
531
531
531
531
531
531
531
168
42
42
42
168
42
42
168
42
42
0
0
0
0
314
314
314
314
113
157
[22]
BS-Lb
BS-Ub
38.6
38.8
520
520
–
–
520
520
29,201
29,276
200,000
200,000
–
–
200,000
200,000
260
260
450
450
109
109
90,000
90,000
112
112
61
61
942
942
0
0
0
0
[23]
5
type2
6
type1
24.8
276
389
414
23,425
200,000
200,000
200,000
203
254
75
56,774
70
42
570
426
0
24.8
276
273
414
23,425
200,000
200,000
200,000
203
254
75
56,774
70
42
570
126
0
[24]
Ac
22.1
376a
374
366
25,207
178,275a
181,700a
169,050
255
460
137
125,400
102
53
1642
1330
760
[25]
1b
2b
3b
4b
5b
33.1
30.2
34.0
31.6
31.7
459
459
459
459
459
–
–
–
–
–
470
470
470
470
470
27,040
25,829
27,405
26,421
26,462
200,000
200,000
200,000
200,000
200,000
–
–
–
–
–
200,000
200,000
200,000
200,000
200,000
406
406
406
406
406
406
406
406
406
406
124
126
124
125
125
165,161
165,161
165,161
165,161
165,161
136
188
136
188
136
45
45
45
45
45
2565
2565
2565
2565
2565
0
0
0
0
0
0
0
0
0
0
73
(continued on next page)
M. Pauletta et al. / Engineering Structures 94 (2015) 70–81
[15]
As regards the bond stress lðf b Þ, on the bases of approximated
expressions available in the literature [10], an average value along
the joint portion lh (Fig. 2) is assumed herein
1257
1257
0
0
0
0
0
0
0
0
0
0
0
0
M. Pauletta et al. / Engineering Structures 94 (2015) 70–81
Atv
(mm2)
74
qffiffiffiffi
ð22Þ
where k is a factor to be determined on the basis of the experimental results.
Substituting Eq. (9) into Eq. (13) and, subsequently, Eqs. (13)
and (16) into Eq. (11), and using the approximate terms provided
by Eq. (20) and (22) for the tensile stress fb and the bond stress
along the beam bars lðf b Þ respectively, it follows that
c
a
P
P
Balanced average value of the mechanical property xeq ¼ ð i As;i xi Þ=ð i As;i Þ; xi is the mechanical property value of the i-th homogeneous steel reinforcement of area As,I.
Unreinforced joints.
Specimens satisfying Eurocode 8 and ACI Code 318-11 requirements.
qffiffiffiffi1
0
0
lh f c
2HL
4
@1 k
A 6 1:0
a¼
2HL ð2L þ hc Þjdb
/b f bi
b
1356
942
1885
2790
55
55
610
610
200,000
200,000
[27]
Unit 1c
Unit 2c
22.6
22.5
296
297a
326
326
296
296
22,344
22,294
200,000
200,000
200,000
200,000
356
356
165
186
208,849
208,849
135
173
452
452
452
452
452
982
982
1608
1608
1608
500
500
500
500
500
200,000
200,000
200,000
200,000
200,000
[26]
5b
5c
5d
5e
5f
54.0
54.0
54.0
56.0
54.0
485
485
515
515
515
480
480
480
480
480
485
485
485
485
485
34,538
34,538
34,538
35,172
34,538
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
200,000
250
250
250
250
250
113
113
134
133
134
90,000
90,000
90,000
90,000
90,000
64
64
64
64
64
0
0
0
0
0
452
91
106
75
90
106
0
982
982
982
982
982
982
64
64
64
64
64
64
92
110
75
92
111
75
2565
45
188
90,000
90,000
90,000
90,000
90,000
90,000
500
500
500
500
500
500
115
117
115
115
116
114
406
250
250
250
250
250
250
–
–
–
–
–
200,000
200,000
200,000
200,000
200,000
200,000
200,000
–
200,000
200,000
200,000
200,000
200,000
200,000
32,900
31,877
32,900
33,234
32,222
34,217
550
550
580
580
580
485
470
–
–
–
–
–
–
480
570
570
570
570
570
485
459
31.0
49.0
46.0
49.0
50.0
47.0
53.0
4bb
4cb
4db
4eb
4fb
5a
6b
Author references
and specimen
labels
Table 1 (continued)
[26]
26,168
200,000
200,000
406
126
165,161
hh (deg)
hb
(mm)
Esh
(MPa)
fc 0
(MPa)
fyb
(MPa)
fyh
(MPa)
fyv
(MPa)
Ec
(MPa)
Esb
(MPa)
Esv
(MPa)
bb
mm
xb
(mm)
Ag
(mm2)
ac
(mm)
Asb
(mm2)
Ath
(mm2)
lðf b Þ ¼ k f 0c
4.1. Shear strength contribution
ð23Þ
vhc,ST1
The horizontal shear capacity of strut ST1 (Vhc,ST1) is obtained
assuming that the depth of the strut is equal to the depth of the
flexural compression zone ac of the column (Fig. 2) given by Eq.
(16).
The width bj of the diagonal strut ST1 used for calculation is
assumed as the minimum value between bb and the width of the
column cross section bc.
The strut-and-tie mechanism leads to the following equilibrium
equation (Fig. 5b)
V hc;ST1 ¼ C c cos hh
ð24Þ
where Cc is the compression force in the inclined strut ST1 (Fig. 5b),
and hh (Fig. 5a) is the angle between the compression concrete strut
and the horizontal direction (Eq. (15)).
On the basis of the model of Hwang and Lee [1], the strut ST1
cross-sectional area is assumed equal to acj (Fig. 5b and c) and,
assuming that the principal axis of inertia coincides with the direction of the diagonal concrete strut, the maximum value of the compression force Cc is computed as
C c ¼ rd;max ac bj
ð25Þ
where rd,max is the maximum concrete compression stress (<0) in
the strut main direction; that is, the average in the cross section
of the strut in the presence of the transverse tensile strain er, and
it is given by [1]
rd;max ¼ f f 0c
ð26Þ
With
5:8
1
0:9
f ¼ qffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0
e
1
þ
400
1
þ
400er
r
fc
ð27Þ
Since Eq. (27) is verified if 5.8/(fc0 [MPa] 6 0.9), hence for
fc0 P 42 MPa, it follows that Eq. (26) can be written as
rd;max
8
0:9f 0c
ffi for f 0c < 42 MPa
>
< pffiffiffiffiffiffiffiffiffiffiffiffi
1þ400er
pffiffiffi0
¼
fc
>
: p5:8
ffiffiffiffiffiffiffiffiffiffiffiffi
ffi for f 0c P 42 MPa
ð28Þ
1þ400er
The mean shear strength within the joint core is given by
v hc;ST1 ¼
V hc;ST1
hc bj
ð29Þ
Eq. (29), using Eq. (24), gives
v hc;ST1;max ¼
C c cos hh
hc bj
ð30Þ
75
M. Pauletta et al. / Engineering Structures 94 (2015) 70–81
tension, fct. Therefore, the maximum value which can be assumed
for rt is always lower than the limiting value rt,lim = fct.
By imposing the limiting value of the concrete tensile stress,
rt = rt,lim = fct, the following limiting expression for rd, rd,lim, is
achieved
rd;lim ¼ rd;max jer ¼f ct =Ec
8
0
0:9f 0c
>
qffiffiffiffiffiffiffiffiffiffiffiffiffi
for f c < 42 MPa
>
>
f
>
1þ400 Ect
<
c
pffiffiffi0
¼
5:8
f
0
>
c
>
qffiffiffiffiffiffiffiffiffiffiffiffiffi for f c P 42 MPa
>
>
f
:
1þ400 ct
ð32Þ
Ec
To avoid a double expression for rd,lim, the approximation [11,12]
rd;lim ¼ vf 0c
ð33Þ
is used, where v is the non-dimensional interpolating function
"
0
f
v ¼ 0:74 c
105
3
#
0 0 2
fc
fc
þ 0:87
1:28 þ 0:22 105
105
ð34Þ
0
Fig. 4. Interpolating function fbi for tensile stress in the beam longitudinal bars.
and Eq. (30), using Eq. (25), gives
v hc;ST1 ¼
rd;max ac cos hh
hc
ð31Þ
The value of rd,max appearing in Eq. (30) is not known, because it is a
function of the unknown strain er.
To evaluate er, the concrete stress–strain curve in tension may
be taken as a straight line with constant slope up to the tensile
strength value, and within this range, the modulus of elasticity in
tension may be assumed to be the same as that in compression.
It follows that er can be expressed as er = rt/Ec, where rt is the
transverse concrete tensile stress acting within the joint at failure.
The strut concrete element disposed according to the principal
directions is subject to a two-dimensional tension–compression
(rt–rd) stress state. This stress state is unknown, because both
compressive and tensile stresses at failure, rd,max and rt, are
unknown. It is well known, however, that for the plain concrete
under consideration, the tensile strength under biaxial regime is
lower than the tensile strength of the concrete loaded in uniaxial
The v function is obtained by considering, for 10 6 fc 6 105 MPa,
the approximating functions corresponding to five fct expressions
and, in turn, the one leading to the lowest coefficient of variation
of Eqs. (32), (33) ratios [11,12]. The exact (Eq. (32)) and approximate (Eq. (33) with Eq. (34)) expressions for the limiting concrete
0
compression stress are plotted versus f c in Fig. 6. Consequently,
the approximating limiting value of vhc,ST1 (Eq. (31)), v hc;ST1;lim , is
given by
v hc;ST1;lim ¼
vf 0c ac cos hh
hc
ð35Þ
Because v hc;ST1;lim is obtained by approximating vhc,ST1, it follows that
v hc;ST1 ¼ a1 v hc;ST1;lim
ð36Þ
where a1 is a factor to be determined on the basis of experimental
results. Eq. (36) by means of Eq. (35) gives
v hc;ST1 ¼ a1 vf 0c ac cos hh
hc
ð37Þ
where hh is provided by Eq. (15).
4.2. Reinforcement mechanism
The joint can be horizontally reinforced by m layers of two-leg
closed stirrups, the i-th of which has cross-sectional area Ahi
Fig. 5. Diagonal strut ST1 mechanism: (a) beam and column longitudinal reinforcements and angle of inclination hh of strut ST1, (b) height ac of diagonal strut ST1, (c) beam
width bb and column width bc.
76
M. Pauletta et al. / Engineering Structures 94 (2015) 70–81
Cs due to the presence of steel reinforcements, the horizontal force
provided by the horizontal stirrups reinforcement q1Ahifyh and the
force provided by the intermediate column bars q2Avjfyv, with
Ah ¼
m
X
Ahi ¼ m Ahi
i¼1
ð38Þ
p
X
Av ¼
Av j ¼ p Av j
j¼1
where m equals the number of horizontal stirrups reinforcement
layers, and p equals the number of intermediate vertical bars within
the joint core (Fig. 7).
Therefore
V hs ¼ q1 Ah f yh þ q2 Av f yv = tan hh
Fig. 6. Limiting principal compression stress given by bilinear relation (rd;lim ), or
0
interpolating function (rd;lim ), versus concrete strength f c .
(i = 1, .., m) while the vertical reinforcement is provided by p intermediate column bars, the j-th of which has area Avj (j = 1, .., p).
Experimental tests on exterior beams provide evidence that the
maximum resistance is obtained after extensive crack formation
[13,14]. In this condition, as observed by Russo et al. [11,12] for
reinforced concrete deep joints and corbels, it can be assumed that
the stirrups near the mid-height of the joint core yield, but the far
ones are probably subjected to lower stresses than the stirrup
yielding strength fyh, while some stirrup layers could even be
ineffective in tension. Analogously, it is probable that, within the
joint, the intermediate column bars acting as vertical reinforcement attain their yielding strength fyv in the central region of the
shear span, while they are subjected to lower stresses in the outer
region. Consequently, it can be initially assumed that the mean
tensile stress in the horizontal stirrups is equal to q1fyh, with
q1 < 1.0, and the mean tensile stress in the vertical bars is equal
to q2fyv, with q2 < 1.0. It follows that the horizontal force carried
by the i-th horizontal stirrups layer is equal to q1Ahifyh, and the vertical force carried by the j-th intermediate vertical bar is equal to
q2Avjfyv (Fig. 7).
It is assumed herein that the compression force in the inclined
strut, in the presence or absence of steel reinforcement, respectively Ccs or Cc, acts on the same line in both cases. Because of this
collinearity, the scalar difference between Ccs and Cc simply equals
the contribution Cs (=Ccs Cc) provided by horizontal and vertical
reinforcements to the strut force.
The shear force Vhs, carried by the steel reinforcement alone,
must be equal to the vector sum (Fig. 7) of the compression force
ð39Þ
consequently, the exterior joint shear strength due to steel
reinforcement is given by
v hs ¼ c2 q1 Ah f yh þ q2 Av f yv = tan hh
V h;s
¼ c2 hc bj
hc b j
ð40Þ
where c2 (<1.0) is a factor to be determined on the basis of experimental results.
Obviously, for exterior joints without intermediate column bars,
the shear strength due to the steel reinforcement vhs is reduced
only to
v hs ¼ c2 q1 Ah f yh
hc bj
ð41Þ
4.3. Shear strength expression
The parametric expression for computing the nominal shear
strength of exterior RC beam–column joints is obtained from Eq.
(10) using Eqs. (11), (37) and (40)
vn ¼
a1
hc bj
a3 Av f yv
vf 0c bj ac cos hh
þ a2 Ah f yh þ
a
tan hh
ð42Þ
where a2 = (c2 q1)/a1; a3 = (c2 q2)/a1; while v and hh are respectively
provided by Eqs. (34) and (15).
Eq. (42) is a function with four unknown parameters k, a1, a2,
and a3, which are determined on the basis of the collected experimental results.
Sixty-one exterior joints have been selected from the following
12 investigations: Alameddine [15], Clyde et al. [16,17], Ehsani
et al. [18], Ehsani and Wight [19], Fujii and Morita [20], Kaku and
Asakusa [21], Kuang and Wong [22], Lee et al. [23], Megget [24],
Pantelides et al. [25], Parker and Bullman [26], and Paulay and
Scarpas [27]. In selecting these test data sets, the authors have considered only the exterior joints that failed due to shear and not due
to flexure or bearing modes. All the considered tests, excluding the
11 specimens of Parker and Bullman [26], were of cyclic type.
The original labels of the thus-selected exterior RC joints are
listed in Table 1, after the number that represents the reference
to the author.
On the basis of the test data collected and taking into account
the boundary conditions of the used interpolating function v (Eq.
(34)), a list of limitations to the parameters necessary to calculate
joint shear strength was defined and is reported below:
0
Fig. 7. Truss mechanism.
– Cylindric compressive strength of concrete: 22 MPa 6 f c 6
92 MPa.
– Angle of inclination hh of the joint strut ST1: 40 deg 6 hh 6
68 deg.
– Overall area of tensile principal reinforcement in the beam:
531 mm2 6 Asb 6 2790 mm2.
77
M. Pauletta et al. / Engineering Structures 94 (2015) 70–81
– Overall area of compressive principal reinforcement in the
beam: 396 mm2 6 A0sb 6 2790 mm2.
– Overall area of horizontal joint hoop reinforcement: Ath 6
1356 mm2.
– Overall area of vertical intermediate column bars: Atv 6
1257 mm2.
– Yield strength of beam tensile reinforcement: f yb 6 1069 MPa.
– Yield strength of joint hoop reinforcement: f yh 6 480 MPa.
– Yield strength of column bars: f yv 6 580 MPa.
The coefficients k in Eq. (23) and a2 and a3 in Eq. (42) are determined by minimizing the coefficient of variation (COV) which is
calculated as the ratio between the standard deviation (ST DEV)
and the average (AVG) of the ratios between the shear strength
measured in the experiments and the nominal shear strength from
Eq. (42) (Vjh,test/Vn) and using Eq. (15) for hh.
Moreover, the coefficient a1 is chosen with the purpose of making the AVG of these ratio equal to 1.0. The values k = 0.25,
a1 = 0.71, a2 = 0.79, and a3 = 0.52 have been found, consequently
Eqs. (23) and (42) become, respectively
qffiffiffiffi 1
0
0
lh f c
2HL
@1 A 6 1:0
a¼
/b f b
2HL ð2L þ hc Þjdb
0
Av f y v
vf c bj ac cos hh
V n ¼ 0; 71
þ 0:79Ah f yh þ 0:52
a
tan hh
ð43Þ
– The decrease of the contribution of the diagonal mechanisms,
which carries the largest share of the joint strength up to strut
ST1, for slopes up to approximately 65 degrees.
– The increase of the horizontal mechanism contribution, which
becomes the main resisting mechanism for strut ST1, for slopes
greater than approximately 65 degrees.
– The percentage of shear forces carried by the vertical mechanism is slightly decreasing, and it provides the smaller contribution, when its slope is greater than approximately
51 degrees.
5. Existing models
To evaluate the reliability of the proposed model with respect to
the existing ones, a comparison with the shear strength formulae
and procedures provided by Hwang and Lee [1] and Park and
Mosalam model [2] is performed.
5.1. Hwang and Lee
These authors have proposed an iterative procedure [1] that
originates from a strut-and-tie model, which satisfies equilibrium,
compatibility, and the constitutive laws of cracked reinforced
concrete.
5.2. Park and Mosalam
ð44Þ
For the 61 exterior joints tested, Eq. (44) provides a COV value of
0.157.
It can be observed that the coefficient a2 value, referring to the
horizontal reinforcement contribution, is approximately 30% larger
than the a3 value, which refers to the contribution of the column
intermediate bars. This means that the horizontal stirrups
reinforcement contribution to the ultimate shear strength of exterior joints is more effective than that of the vertical bars.
Due to the layout of the proposed strut-and-tie model, it is possible to plot the percentage of the joint shear forces resisted by the
different mechanisms, in relation to the slope of the diagonal strut
ST1, derived by geometry of the joint only (Fig. 8).
As shown in Fig. 8, increasing the inclination of the strut ST1,
one can observe:
These authors have proposed an analytical shear strength
model for unreinforced exterior beam–column joints based on
two inclined strut mechanisms in the joint region. From the evaluation of a large literature data set, the authors also proposed
the following simplified formula (Eq. (45)) for practical engineering
applications, instead of the iterative procedure required to solve
the analytical model [2]
qffiffiffiffi
cos hh
0
V n ¼ k cext f c bj hc
cosðp=4Þ
ð45Þ
where cext = 1.0 [MPa0.5]
SIj X 1
6 1:0
k ¼ 0:4 þ 0:6
X2 X1
As f y
h
qffiffiffiffi 1 0:85 b
SIj ¼
0
H
bj hc f c
cos hh
cosðp=4Þ
cos hh
X 2 ¼ cext
cosðp=4Þ
X 1 ¼ cST1
0 0;5
with cST1 ¼ 0:33ðf c Þ
ð46Þ
ð47Þ
ð48Þ
ð49Þ
[MPa0.5].
5.3. Model reliability
Fig. 8. Ratios of force distribution among mechanisms.
The shear strength Vn of the 61 considered exterior RC beam–
column joints, whose test results are listed in Table 2, has been
evaluated by means of the procedure provided by Hwang and
Lee [1] and the proposed formula (Eq. (44)). The corresponding
ratios between the experimental shear strength Vjh,test and calculated values Vn are plotted in Fig. 9, where the corresponding
AVG, COV and UP (number of Unsafe Predictions) values are also
reported.
For these 61 joints, the AVG and COV of Vjh,test/V ratios result
equal to 1.105 and 0.232, for the procedure of Hwang and Lee,
and 0.994 and 0.157, for Eq. (44), respectively. In comparison with
the tedious iterative procedure of Hwang and Lee, the simple proposed shear strength expression in Eq. (44) can be said to provide
78
M. Pauletta et al. / Engineering Structures 94 (2015) 70–81
Table 2
Forces, stresses and results.
Author references and
specimen labels
a
Vb
(kN)
Vc
(kN)
Nc
(kN)
fb
(MPa)
f bi
(MPa)
a
V hc
(kN)
V sh
(kN)
V sv
(kN)
V n;calc
(kN)
V jh;test
(kN)
V jh;test
V n;calc
V jh;test
V n;HL
V jh;test
V n;PM
[15]
LL8c
LH8c
HL8c
HH8c
LL11c
LH11c
HL11c
HH11c
LL14c
LH14c
HH14c
248
240
262
264
213
284
264
289
261
267
288
120
116
126
128
103
137
127
140
126
129
139
294
294
507
507
285
274
587
605
236
223
476
509
492
429
433
421
572
416
461
531
542
474
466
466
432
432
493
493
458
458
516
516
479
0.96
0.96
0.97
0.97
0.94
0.94
0.96
0.96
0.92
0.92
0.93
525
525
580
580
582
582
630
634
637
653
709
291
485
291
485
291
485
291
485
291
485
485
83
83
111
111
80
80
103
103
80
83
111
899
1093
982
1176
953
1146
1024
1222
1008
1221
1305
911
881
974
984
751
1023
940
1043
949
969
1076
1.01
0.81
0.99
0.84
0.79
0.89
0.92
0.85
0.94
0.79
0.82
1.28
1.12
1.27
1.16
0.99
1.20
1.15
1.13
1.19
1.06
1.11
–
–
–
–
–
–
–
–
–
–
–
[16,17]
2b
4b
5b
6b
267
276
267
262
156
161
156
153
644
1428
1289
559
471
488
472
463
400
390
382
388
0.95
1.00
1.00
0.96
968
1165
1069
862
0
0
0
0
162
162
162
162
1130
1327
1230
1023
1051
1090
1056
1035
0.93
0.82
0.86
1.01
1.05
0.88
0.93
1.11
1.04
1.12
1.15
1.08
[18]
2c
3c
4c
184
135
157
93
67
78
338
383
325
581
534
496
506
491
473
0.91
0.91
0.92
556
415
406
225
225
225
60
55
75
840
695
707
735
600
695
0.87
0.86
0.98
1.10
1.09
1.27
–
–
–
[19]
1B
3B
4B
5B
6Bc
140
173
165
166
153
109
135
129
96
89
178
222
222
357
304
314
387
412
232
228
311
324
324
287
329
1.00
1.00
0.96
1.00
0.99
174
206
244
255
339
216
216
216
216
216
42
42
46
82
55
431
464
506
554
610
524
646
703
444
385
1.21
1.39
1.39
0.80
0.63
0.89
1.63
0.88
1.02
0.76
–
–
–
–
–
[20]
B1
B2
B3
B4
58
52
64
66
43
38
47
49
98
98
343
343
489
432
534
550
761
356
761
761
1.00
0.90
1.00
1.00
123
137
180
180
18
18
18
55
62
62
62
62
202
216
260
296
264
233
288
297
1.30
1.08
1.11
1.00
1.40
1.24
1.10
1.04
–
–
–
–
[21]
3
4
5
6
9
11
12
13
14
15
47
52
48
46
52
50
45
46
49
50
31
34
31
30
33
33
29
30
32
33
0
360
160
0
0
160
0
–100
160
160
450
495
458
434
489
478
431
433
468
478
371
377
362
369
369
372
358
380
370
368
0.90
0.94
0.93
0.90
0.90
0.92
0.91
0.88
0.92
0.93
156
245
177
151
155
197
136
149
194
190
24
7
7
7
24
7
7
24
7
7
0
0
0
0
43
43
43
43
11
21
179
252
183
158
221
246
185
216
212
218
208
229
212
201
226
221
199
200
217
221
1.16
0.91
1.16
1.27
1.02
0.90
1.07
0.93
1.02
1.01
0.97
0.72
0.93
1.04
1.03
0.84
1.07
0.98
0.85
0.95
–
–
–
–
–
–
–
–
–
–
[22]
BS-Lb
BS-Ub
90
96
43
46
504
506
361
385
522
523
1.00
1.00
316
317
0
0
0
0
316
317
297
317
0.94
1.00
1.32
1.40
0.83
0.88
[23]
5
type2
6
type1
40
29
0
454
270
0.91
176
93
0
269
230
0.85
1.52
–
41
29
0
459
270
0.91
176
19
0
195
232
1.19
1.59
–
[24]
Ac
161
84
196
403
320
1.00
213
279
77
570
577
1.01
1.39
–
[25]
b
1
2b
3b
4b
5b
6b
195
190
188
211
170
192
102
100
98
111
89
101
547
1247
562
1305
524
1280
397
388
382
431
346
391
399
391
401
395
395
394
0.97
1.00
0.97
1.00
0.97
1.00
798
984
817
1025
767
1008
0
0
0
0
0
0
0
0
0
0
0
0
798
984
817
1025
767
1008
916
895
882
994
800
903
1.15
0.91
1.08
0.97
1.04
0.90
1.36
1.06
1.28
1.13
1.24
1.05
0.96
0.98
0.91
1.07
0.86
0.98
[26]
4bb
4cb
4db
4eb
4fb
5a
5b
138
170
150
160
183
213
236
69
85
75
80
92
107
118
300
570
0
300
600
0
300
308
380
335
357
409
475
526
594
586
594
596
588
531
533
1.00
1.00
1.00
1.00
1.00
1.00
1.00
271
308
220
273
316
232
284
0
0
0
0
0
122
122
0
0
0
0
0
0
0
271
308
220
273
316
354
406
233
288
254
270
310
360
398
0.86
0.93
1.15
0.99
0.98
1.02
0.98
1.28
1.37
1.75
1.50
1.47
0.62
0.62
0.65
0.83
0.71
0.75
0.88
–
–
[26]
5c
5d
5e
5f
242
226
295
322
121
113
148
161
600
0
300
600
539
313
408
446
533
504
508
504
1.00
1.00
1.00
1.00
333
235
289
333
122
122
122
122
0
0
0
0
455
356
411
455
408
391
509
556
0.90
1.10
1.24
1.22
0.58
0.67
0.79
0.79
–
–
–
–
[27]
Unit
1c
Unit
2c
157
112
250
372
294
0.89
432
248
96
776
590
0.76
0.94
–
220
157
705
375
272
0.91
534
172
96
803
888
1.11
1.25
–
P
P
Balanced average value of the mechanical property xeq ¼ ð i As;i xi Þ=ð i As;i Þ; xi is the mechanical property value of the i-th homogeneous steel reinforcement of area As,i.
b
Unreinforced joints.
c
Specimens satisfying Eurocode 8 and ACI Code 318-11 requirements.
M. Pauletta et al. / Engineering Structures 94 (2015) 70–81
79
Fig. 9. Calculated ultimate shear strength by means of (a) Hwang and Lee
procedure and (b) proposed basic expression (Eq. (44)).
more reliable results, because the COV obtained with this formula
is 30% lower than that obtained with the procedure of Hwang and
Lee (Fig. 9).
Among the 61 collected tests, 17 were performed on exterior
beam–column joints without transverse reinforcement. Their
results have been evaluated by means of the procedure of Hwang
and Lee (Fig. 10a), the simplified formula proposed by Park and
Mosalam (Eq. (45) and Fig. 10b) and the proposed one (Eq. (44),
and Fig. 10c).
For these 17 joints, the AVG and COV of Vjh,test/Vn ratios result
equal to1.245 and 0.181 for Hwang and Lee [1] procedure, 0.922
and 0.156 for the Park and Mosalam simplified expression, and
0.972 and 0.098 for Eq. (44), respectively.
Although the formula of Park and Mosalam [2] is specifically
proposed only for joints without transverse reinforcement, while
Eq. (44) is valid in general, the latter (with COV = 0.098) is more
consistent than the former (with COV = 0.156).
Fig. 10. Calculated ultimate shear strength of unreinforced joints by mean of (a)
Hwang and Lee procedure, (b) Park and Mosalam simplified formula, and (c)
proposed basic expression (Eq. (44)).
6. Design formula
The shear strength expression of the proposed model (Eq. (44))
can be said to yield reliable results, because the COV of experimentally measured and computed shear strength ratios is the lowest
for all of the computational methods considered in this investigation. However, it cannot be used for design, as the AVG of the
above-mentioned ratio is equal to one, and therefore does not
incorporate a factor of safety into the mean prediction. Because
80
M. Pauletta et al. / Engineering Structures 94 (2015) 70–81
the AVG value can be changed by multiplying Eq. (44) by a factor
without modifying the COV value, Eq. (44) is suitable to provide
a design formula for the joint shear strength. This multiplying factor is determined on statistical bases here, so that there is a 95-in100 probability that the predicted design shear strength increase is
lower than the experimental one. To this end only the data of 18
specimens, identified with mark in Tables 1 and 2, are used.
These specimens comply with both Eurocode 8 and ACI Code
requirements for beam–column joints. To obtain a safety factor
of 0.63, the proposed design formula derived from the corresponding characteristic expression is
0
Av f y v
vf c bj ac cos hh
V n;d ¼ 0:45
þ 0:79Ah f yh þ 0:52
a
tan hh
7. Conclusions
6.1. Eurocode 8 [3]
For exterior beam–column joints, the horizontal shear force acting on the concrete core of the joints is
ð51Þ
j jw
6.2. ACI Code 318-11 [4]
The shear capacity of exterior beam to column joints is set as a
function of only the compressive strength of the concrete. For normal-weight concrete and joints which are unconfined on two
opposite faces this capacity is given by (21.7.4)
Vn ¼
qffiffiffiffi
0
f c Aj
The shear strengths obtained by applying Eq. (50) to all the 61
exterior joints previously taken into account are plotted versus the
measured shear strengths in Fig. 11(c). Comparing these results
(AVG = 1.402 and COV = 0.128) with those obtained by means of
the Eurocode 8 (AVG = 1.697 and COV = 0.182) and ACI Code 31811 (AVG = 0.855 and COV = 0.165) design formulae (Fig. 11(b)), it
is evident that the proposed design formula (Eq. (50)) is more accurate and reliable.
ð50Þ
Eq. (50) provides AVG = 1.177.
To evaluate the reliability of the proposed design formula with
respect to the existing ones, the shear strength design formulae
provided by Eurocode 8 [3] and ACI 318-11 [4] are considered.
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
md
>
>
< 0; 8f cd 1 g bj hjc
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
V n ¼ min
Asw f yw
>
>
þ
f
þ
m
f
Þ
ðf
: ðbj hjw Þ
d
ctd
ctd
cd
bh
6.3. Comparison
ð52Þ
where Aj is the effective cross-sectional area within a joint computed from joint depth times effective joint width.
On the basis of a mechanical analysis of exterior RC beam–column joints shear strength and an experimental comparison with
61 specimens available in the literature, the following conclusions
can be drawn:
1. A consistent model for predicting the shear strength of exterior
RC beam–column joints subjected to seismic actions is obtained
by super-posing the shear strength contribution of the strutand-tie mechanism due to two diagonal concrete struts and
the shear strength contribution due to steel reinforcements.
2. The sum of the two concrete struts is the main resisting mechanism up to approximately a 65 degree angle of the ST1 strut,
which is developed by the 90 degree hook of the beam
reinforcement.
3. In exterior RC beam–column joints, horizontal stirrups
reinforcement is more effective in providing shear strength
than vertical bars reinforcement.
4. The proposed model for computing the exterior RC joints shear
strength leads to a single shear strength expression, which is
more reliable than the best existing iterative procedure and formula, because it provides the lowest COV value in the prediction of experimental results. It follows that the proposed
mechanical model, on which this expression is based, is consistent with the actual mechanical behavior.
5. On the basis of an experimental comparison with a reduced
sample of 18 specimens that comply with Eurocode 8 and ACI
Code requirements, an adequately conservative and reliable
Fig. 11. Calculated ultimate shear strength by means of (a) Eurocode 8, (b) ACI Code 318-11, and (c) proposed design formula (Eq. (50)).
M. Pauletta et al. / Engineering Structures 94 (2015) 70–81
design formula (Eq. (50)) is derived from the aforementioned
shear strength expression. In comparison to this, the Eurocode
8 expressions for computing the shear strength of exterior
joints (Eq. (51)) lead to very conservative shear strength predictions, while the ACI Code expression (Eq. (52)) often leads to
unconservative predictions.
Acknowledgements
The research has been partially funded by Italian Department of
Civil Protection (within the framework of Executive Projects
DPCReLUIS 2005–2008, 2010–2013 and 2014–2016), whose support is greatly appreciated.
References
[1] Hwang SJ, Lee HJ. Analytical model for predicting shear strengths of exterior
reinforced concrete beam–column joints for seismic resistance. ACI Struct J
1999;96(5):846–57.
[2] Park S, Mosalam KM. Analytical model for predicting shear strength of
unreinforced exterior beam–column joints. ACI Struct J 2012;109(2):149–59.
[3] UNI EN 1998-1. Eurocode 8: design of structures for earthquake resistance –
Part 1: General rules, seismic actions and rules for buildings. CEN, Comitè
European de Normalisation; 2004.
[4] ACI Committee 318. Building code requirements for structural concrete and
commentary. ACI 318M-11. Farmington Hills MI: American Concrete Institute;
2011.
[5] Ministero delle Infrastrutture. DM 14 gennaio 2008. Norme tecniche per le
costruzioni; 2008 [in Italian].
[6] Paulay T, Priestley MJN. Seismic design of reinforced concrete and masonry
buildings. New York: John Wiley and Sons; 1992.
[7] To NHT, Ingham JM, Sritharan S. Strut – and – tie computer modelling of
reinforced concrete bridge joint systems. J Earthq Eng 2003;7(3):463–93.
[8] Russo G, Pauletta M. Seismic behavior of exterior beam–column connections
with plain bars and effects of upgrade. ACI Struct J 2012;109(2):225–33.
[9] Russo G, Pauletta M, Mitri D. Solution for bond distribution in asymmetric R.C.
structural members. Eng Struct 2009;31(3):33–641. http://dx.doi.org/10.1016/
j.engstruct.2008.11.003.
81
[10] Russo G, Pauletta M. A simple method for evaluating the maximum slip of
anchorages. Mater Struct 2006;39:533–46. http://dx.doi.org/10.1617/s11527006-9092-1.
[11] Russo G, Venir R, Pauletta M, Somma G. Reinforced concrete corbels – shear
strength model and design formula. ACI Struct J 2006;103(1):3–10.
[12] Russo G, Venir R, Pauletta M. Reinforced concrete deep beams – shear strength
model and design formula. ACI Struct J 2005;102(3):429–37.
[13] Kriz LB, Raths CH. Connections in precast concrete structures – strength of
corbels. PCI J 1956;10(1):16–61.
[14] Mattock AH, Chen KC, Soongswang K. The behavior of reinforced concrete
corbels. PCI J 1976;21(2):52–77.
[15] Alameddine FF. Seismic design recommendation for high-strength concrete
beam-to-column connections. Doctoral Thesis 1990, University of Arizona,
USA; 1990.
[16] Clyde C, Pantelides CP, Reaveley LD. Performance-based evaluation of exterior
reinforced concrete buildings joints for seismic excitation. PEER Report 2000/
05. Berkeley: University of California; 2000.
[17] Clyde C, Pantelides CP, Reaveley LD. Performance-based evaluation of
reinforced concrete building exterior joints for seismic excitation. Earthq
Spectra, Earthq Eng Res Inst 2002;18(3):449–80.
[18] Ehsani MR, Moussa AE, Vallenilla CR. Comparison of inelastic behavior of
reinforced ordinary and high-strength concrete frames. ACI Struct J
1987;84(2):161–9.
[19] Ehsani MR, Wight JK. Exterior reinforced concrete beam-to-column
connections subjected to earthquake-type loading. ACI J 1985;82(4):492–9.
[20] Fujii S, Morita S. Comparison between interior and exterior RC beam–column
joint behavior. In: Jirsa JO, editor. Design of beam–column joints for seismic
resistance. Farmington Hills: American Concrete Institute; 1991. p. 145–65.
[21] Kaku T, Asakusa H. Ductility estimation of exterior beam–column
subassemblages in reinforced concrete frames. In: Jirsa JO, editor. Design of
beam–column joints for seismic resistance. Farmington Hills: American
Concrete Institute; 1991. p. 167–85.
[22] Kuang JS, Wong HF. Effects of beam bar anchorage on beam–column joint
behavior. Proc ICE – Struct Build 2006;159(SB2):115–24.
[23] Lee DLN, Wight JK, Hanson RD. RC beam–column joints under large load
reversals. J Struct Div ASCE 1977;103(ST12):2337–50.
[24] Megget LM. Cyclic behaviour of exterior reinforced concrete beam–column
joints. Bull NZ Natl Soc Earthq Eng 1974;7(1):27–47.
[25] Pantelides CP, Hansen CP, Nadauld J, Reaveley LD. Assessment of reinforced
concrete building exterior joints with substandard details. PEER Report 2002/
18. Berkeley: University of California; 2002.
[26] Parker DE, Bullman PJM. Shear strength within reinforced concrete beam–
column joints. Struct Eng 1997;75(4):53–7.
[27] Paulay T, Scarpas A. Behavior of exterior beam–column joints. Bull NZ Natl Soc
Earthq Eng 1981;14(3):131–44.