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Engineering Structures 24 (2002) 11051117
www.elsevier.com/locate/engstruct
A new design equation for predicting the joint shear strength ofmonotonically loaded exterior beam-column joints
P.G. Bakir ab,, H.M. Boduroglu a
a Istanbul Technical University, Civil Engineering Department, Maslak 80626, Istanbul, Turkeyb Postal address: Yazmaci Tahir sok, Derya apt. no 11/4, Catalcesme, Suadiye, Istanbul, Turkey
Received 11 June 2001; received in revised form 8 March 2002; accepted 8 March 2002
Abstract
In this study a new design equation for predicting the shear strength of monotonically loaded exterior beam column joints isproposed. For this purpose, the influence of several key variables on the behaviour of beam-column joints are inspected usingresults of parametric studies on an experimental database compiled from a large number of exterior joint tests. The design equationsuggested has three differences from the previously proposed equations. First, the equation proposed considers the influence ofbeam longitudinal reinforcement ratio, which was not taken into account in previously suggested design equations. Second, as theinfluence of this parameter is taken into account, a more realistic estimate of the influence of joint aspect ratio is obtained. Third,the influence of stirrups is considered differently for joints with low, medium and high amount of stirrup ratios, in a way, whichwas not considered in previously suggested equations. The results showed that the proposed design equation predicts the joint shearstrength of exterior beam column connections accurately with minimal standard deviation and is more reliable than the previouslysuggested equations. 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Reinforced concrete; Shear strength; Monotonically loaded exterior beam-column connections
1. Introduction
It is now generally believed that beam-column jointscan be critical regions in reinforced concrete framesunder severe seismic effects. Beam-column joint failureshave been observed in the 1980 El Asnam [1], 1985Mexico [2], 1986 San Salvador [3], 1989 Lome Prieta[4] and 1999 Kocaeli earthquakes [5]. During the pastforty years, significant amount of research has been car-ried out on seismic behaviour of beam-column joints allover the world. However, compared to cyclically loaded
joints, little information exists in literature for predictingthe shear strength of monotonically loaded exterior
joints.Remarkable differences exist in the design of joints
for seismic loading or monotonic loading. Parameterssuch as column axial load, concrete cylinder strength,stirrup ratio, stirrup index, joint aspect ratio, beam
Corresponding author. Fax: +90-216-386-9742.
E-mail address: [email protected] (P.G. Bakir).
0141-0296/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.
PII: S0 1 4 1 - 0 2 9 6 ( 0 2 ) 0 0 0 3 8 - X
reinforcement detailing, the ratio of beam longitudinalreinforcement etc influence the joint shear strength dif-ferently for interior or exterior and monotonically loadedor cyclically loaded joints. This investigation wasplanned with the objective of adding useful data to theunderstanding of the influences of the above-mentionedparameters on the joint shear strength of monotonicallyloaded exterior beam-column joints. Surprisingly, noneof the previously suggested design equations for mono-tonically loaded exterior joints consider the factors thatinfluence joint shear strength such as beam longitudinalreinforcement ratio, joint aspect ratio and concrete cylin-der strength together.
2. Previously suggested design equations for
exterior beam column joints
In this section, the existing empirical design equationsfor predicting the shear strength of the monotonicallyloaded exterior beam-column joints are reviewed.
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1106 P.G. Bakir, H.M. Boduroglu / Engineering Structures 24 (2002) 11051117
2.1. The design equation of Sarsam and Phillips
Sarsam and Phillips [6] have proposed the followingequation for the design of monotonically loaded exterior
beam column connections.
Vcol 5.08fcurc0.33dc
db1.33bcdc1 0.29
NuAg (1)
where Vcol is the column shear force at the column-joint
interface (N); fcu is the concrete cube strength (MPa); rcis the column longitudinal reinforcement ratio
rc Aso/bcdc (2)
where Aso is the area of the layer of steel furthest from
the maximum compression face in a column (mm2); Agis the gross cross-sectional area of the column at the joint
(mm2); Nu is the axial column load (N);dc is the effectivedepth of the layer of steel furthest away from the
maximum compression face in a column (mm); db is theeffective depth of beam tension reinforcement (mm); bcis the width of column section at the joint (mm).
The shear force resisted by the links is taken as:
Vsd 0.87Ajsfyv (3)
Vud Vcd Vsd (4)
where Ajs is the total area of horizontal link reinforce-
ment crossing the diagonal plane from corner to comer
of the joint between the beam compression and tensionreinforcement (mm2); fyv is the tensile strength of the
link reinforcement (MPa); Vsd
is the design link shear
force resistance (N); Vcd is the design shear force resist-
ance of concrete in a joint (N); Vud is the design ultimate
shear capacity of joint (N)
All the joint stirrups are considered to be effective in
increasing the joint shear capacity.
2.2. The design equation of Vollum
Vollum [7] proposed the following equation on the
design of exterior beam column connections:
Vc
0.642b1
0.555(2
hb
hc)beffhcfc (5)
Vj Vc (Asjefyabeffhcfc) (6)
a is taken as 0.2 for low, medium and high amount ofstirrups. b is 0.9 for U detail beam reinforcement and1 for L bent down. The joint shear strength should be
limited to:
Vj0.97beffhcfc1 0.555(2hbhc
) (7)1.33beffhcfc (8)
where Asje is the cross-sectional area of the joint stirrups
within the top five eighths of the beam depth below themain beam reinforcement (mm2); a is a coefficient thatdepends on factors including column load, concrete
strength, stirrup index, and joint aspect ratio, which isconservatively taken as 0.2 in Vollums equation. Vc is
the joint shear strength without stirrups (N); Vj is thetotal joint shear strength (N); hb is the section depth of
the beam (mm); hc is the section depth of the column
(mm); fc is the concrete cylinder strength (MPa); beff is
the average of the beam and column widths (mm); fy is
the yield strength of stirrups (MPa).
2.3. Present design guidelines
The ACI-ASCE Committee 352 [8], and EC8 [9] rec-
ommend the following design equations for the shearstrength of monotonically loaded joints.
Vjd 1.058fcbeffhc (ACI (9) ASCE Committee 352)
Vjd 0.525f2/3c beffhc (EC8 ductility class DCL) (10)
In order to investigate the reliability of the above
design equations, the authors carried out several para-
metric studies on monotonically loaded exterior beam-
column joints. The parametric studies are explained in
the next section followed by a comparison of the above
design equations with the parametric studies of authors.
3. Parametric studies on joint shear strength
The authors carried out a parametric investigation of
exterior beam-column joint behaviour based on 58 tests
conducted in Europe. The loading in all the tests was
monotonic. The authors realised that there might be
important interactions between variables that at firstwere supposed to act independently and this necessitated
viewing the entire population of results as a single para-
meter. For this purpose, monotonically loaded exteriorbeam column joints are investigated through test results
assembled and reconsidered as a unified whole. Examin-ing a large number of individual series of tests as a single
database has the advantage of observing which variables
have a significant influence on joint shear strength in alltests and which variables interact with each other. A newdesign equation is proposed based on the parametric
studies carried out on the experimental database. Table
1 shows the experimental database used in this study.
The database comprises of the tests of Ortiz [10], Kord-
ina [11], Scott [12], Scott & Hamill [13], Taylor [14],and Parker & Bullman [15]. The specimen forms
included in the database are shown in Fig. 1(a). Fig. 1(b)
shows the notations used in this study.
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Table1
Experimentaldatabase
Investigator
Specimen
Detail
H(mm)
L(mm)
hb/h
c
bc/bb
Beam
Columnrein.
fc(MPa)
Columnaxial
SI(MPa)0.5
Vjpredicted/Vjactual
Failure
rein.ratio
ratio
load
modes
Ortiz
BCJ1
Lbar
2000
1050
1.33
1.00
1.1
2.19
34
0
0
0.68
is
BCJ2
Lbar
2000
1100
1.33
1.00
1.1
2.19
38
0
0.16
0.77
is
BCJ3
Lbar
2000
1100
1.33
1.00
1.1
2.92
33
0
0
0.64
is
BCJ4
Lbar
2000
1100
1.33
1.00
1.1
3.65
34
0
0.33
0.78
is
BCJ5
Lbar
2000
1100
1.33
1.00
1.1
3.65
38
300
0
0.72
Is
BCJ6
Lbar
2000
1100
1.33
1.00
1.1
3.65
35
300
0
0.68
is
BCJ7
Lbar
2000
1100
1.33
1.00
1.1
3.65
35
300
0.76
0.68
b
Kordina
RE2
Lbar
3000
1000
2.00
1.00
0.9
2.41
25
240
0
0.66
is
RE3
Lbar
3000
1000
1.50
1.00
1.8
2.41
40
400
0.26
0.96
is
RE4
Lbar
3000
1000
1.50
1.00
1.2
2.41
32
51
0.19
0.83
is
RE6
Lbar
3000
1000
1.50
1.00
1.2
2.41
32
213
0.38
0.91
is
RE7
Lbar
3000
975
1.40
1.00
1.3
1.61
26
650
0.43
0.87
is
RE8
Ubar
3000
975
1.40
1.00
1.3
1.61
28
525
0.42
0.90
is
RE9
Ubar
3000
975
1.40
1.00
1.3
1.61
28
770
0.41
0.86
is
RE10
Ubar
3000
975
1.56
1.00
1.2
1.61
24
551
0.45
0.94
is
Taylor
P1/41/24
Lbar
1290
470
1.43
1.40
2.4
4.10
33
240
0.3
0.97
is
P2/41/24
Lbar
1290
470
1.43
1.40
2.4
4.10
29
240
0.3
0.94
is
P2/41/24A
Lbar
1290
470
1.43
1.40
2.4
4.10
47
240
0.26
0.92
is
A3/41/24A
Lbar
1290
470
1.43
1.40
2.4
4.10
27
240
0.3
0.88
is
D3/41/24
Lbar
1290
470
1.43
1.40
2.4
4.10
53
60
0.24
0.89
is
B3/41/24
Lbar
1290
470
1.43
1.40
2.4
4.10
22
240
0.75
0.92
is
C3/41/24BY
Ubar
1290
470
1.43
1.40
2.4
4.10
32
240
0.31
1.04
is
C3/41/13Y
Ubar
1290
470
1.43
1.40
1.4
4.10
28
240
0.33
0.95
is
Scott
C1AL
Lbar
1700
750
1.40
1.36
1.1
4.29
33
50
0.188
0.87
is
C4
Lbar
1700
750
1.40
1.36
2.1
4.29
41
275
0.203
0.89
is
C4A
Lbar
1700
750
1.40
1.36
2.1
4.29
44
275
0.196
0.86
is
C4AL
Lbar
1700
750
1.40
1.36
2.1
4.29
36
50
0.218
0.86
is
C7
Lbar
1700
750
2.00
1.36
1.4
4.29
35
275
0.22
0.90
is
C3L
Ubar
1700
750
1.40
1.36
2.1
4.29
35
50
0.22
1.03
is
C6
Ubar
1700
750
1.40
1.36
2.1
4.29
40
275
0.21
1.05
is
C6L
Ubar
1700
750
1.40
1.36
2.1
4.29
46
50
0.19
0.94
is
C9
Ubar
1700
750
2.00
1.36
1.4
4.29
36
275
0.22
0.93
is
(co
ntinuedonnextpage)
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1108 P.G. Bakir, H.M. Boduroglu / Engineering Structures 24 (2002) 11051117
Table1(continued)
Investigator
Specimen
Detail
H(mm)
L(mm)
hb/h
c
bc/bb
Beam
Columnrein.
fc(MPa)
Columnaxial
SI(MPa)0.5
Vjpredicted/Vjactual
Failure
rein.ratio
ratio
load
modes
Scott
&C4ALN0
Lbar
1700
750
1.40
1.36
2.1
4.29
42
50
0
0.88
p
Hamil
C4ALN1
Lbar
1700
750
1.40
1.36
2.1
4.29
46
50
0.229
0.85
js
C4ALN3
Lbar
1700
750
1.40
1.36
2.1
4.29
42
50
0.478
0.78
js
C4ALN5
Lbar
1700
750
1.40
1.36
2.1
4.29
50
50
0.718
0.85
js
C4ALH0
Lbar
1700
750
1.40
1.36
2.1
4.29
104
100
0
0.86
p
C6LN0
Ubar
1700
750
1.40
1.36
2.1
4.29
51
50
0
0.92
js
C6LN1
Ubar
1700
750
1.40
1.36
2.1
4.29
51
100
0.19
0.96
js
C4ALH1
Lbar
1700
750
1.40
1.36
2.1
4.29
95.2
100
0.159
0.93
b
C4ALH3
Lbar
1700
750
1.40
1.36
2.1
4.29
105.6
100
0.302
0.97
b
C4ALH5
Lbar
1700
750
1.40
1.36
2.1
4.29
98.4
100
0.469
1.00
b
C6LN3
Ubar
1700
750
1.40
1.36
2.1
4.29
49
50
0.44
0.92
js
C6LN5
Ubar
1700
750
1.40
1.36
2.1
4.29
37
50
0.765
0.74
js
C6LH0
Ubar
1700
750
1.40
1.3
2.1
4.29
101
100
0
0.72
js
C6LH1
Ubar
1700
750
1.40
1.36
2.1
4.29
102
100
0.153
0.98
js
C6LH3
Ubar
1700
750
1.40
1.36
2.1
4.29
97
100
0.472
0.93
js
Parker
4a
Lbar
2000
850
1.67
1.20
0.9
1.09
39
0
0
-
c
4b
Lbar
2000
850
1.67
1.20
0.9
1.09
39
300
0
1.05
js
4c
Lbar
2000
850
1.67
1.20
0.9
1.09
37
600
0
0.83
js
4d
Lbar
2000
850
1.67
1.20
0.9
4.38
39
0
0
0.97
js
4e
Lbar
2000
850
1.67
1.20
0.9
4.38
40
300
0
0.92
js
4f
Lbar
2000
850
1.67
1.20
0.9
4.38
38
600
0
0.78
js
5a
Lbar
2000
850
1.67
1.20
0.9
2.67
42
0
0.404
-
c
5b
Lbar
2000
850
1.67
1.20
0.9
2.67
43
300
0.4
1.08
js
5d
Lbar
2000
850
1.67
1.20
1.4
2.67
43
0
0.6
-
c
5e
Lbar
2000
850
1.67
1.20
1.4
2.67
45
300
0.589
-
c
5f
Lbar
2000
850
1.67
1.20
1.4
2.67
43
600
0.6
0.86
js
Average
0.88
Standarddevi-0.10
ation
Note:Ciscolumnfailure;bisbeamfailure;jsisjointshearfailure;pisconnection
zonereinforcementpulloutfailure.
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Fig. 1. (a) Typical specimen shape in the experimental database. (b)
Typlical elevation and notations used for exterior beam colum joints.
(c) The strut and truss mechanisms.
It is commonly accepted that beam column joints
resist shear by the strut and truss mechanisms as sug-
gested first by Paulay [16] as shown in Fig. 1(c). Thestrut mechanism accounts for the contribution of the con-
crete, whereas the truss mechanism represents the contri-
bution of stirrups to joint shear strength. In this study,
the resistance of the concrete to the joint shear will be
determined first and then the influence of the stirrupswill be added.
4. Influence of concrete cylinder strength
The relation between maximum joint shear stress atthe instant of joint failure and the uni-axial compressive
stress of concrete fc is shown in Fig. 2. The authors car-
ried out a regression analysis and found that concrete
Fig. 2. The influence of concrete cylinder strength on joint shear
strength.
cylinder strength is related to the joint shear strength by
the following equation.
nj0.9155fc (11)where f
c
is the uniaxial concrete cylinder strength with-
out any factors of safety and vj is in MPa.
C4ALHO and C4ALNO of Scott have been investi-
gated in order to find the possible influences of the inter-action of other parameters with the concrete cylinder
strength. These specimens were deliberately chosen so
as to eliminate other factors such as stirrups, columnaxial load and joint aspect ratio. Because they had no
stirrups, they had similar joint aspect ratios and they
belonged to the same group of tests.
The analysis confirmed the above relationshipbetween the concrete cylinder strength and the joint
shear strength. Hamill has also suggested that the joint
shear strength is proportional to the square root of theconcrete cylinder strength.
5. Determining the joint shear strength
The joint shear is calculated using the following pro-
cedure:
Mb P(L d1) (12)
where P is the failure load (N); L is the distance fromthe point of application of the load to the face of the
column (mm); d1 is the cover (mm).
A value is assumed for the strain in the beam tensile
reinforcement. The force in the beam tensile reinforce-
ment and the moment produced by it are calculated and
if this is equal to the moment calculated in Eq. (2), theprocedure is stopped. If not, the strain assumed is
increased in small increments up until, the moment is
equal to the moment given in Eq. (2). The reinforcement
was assumed to have an elastic modulus of 200 GPa.
The joint shear strength is calculated as below:
Vj TbVcol (13)
where Vj is the joint shear force (N); Tb is the tensile
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force in the beam longitudinal reinforcement (N); Vcol is
the shear force in the upper column (N).
The normalised joint shear strength is determined as:
nj Vj
beffhcfc(14)
where beff is the average of the beam and the column
width; fc is the concrete cylinder strength; hc is the height
of the column.The unit of v j is (MPa)
0.5.
6. Influence of column reinforcement ratio
The authors made a parametric study on the specimensin the experimental database which failed by joint shear,
which had low amount of stirrups and which weredetailed by L bars bent down detail. Parker and
Bullmann specimens were deliberately excluded fromthis parametric study because they have high joint aspect
ratios and low beam reinforcement ratio. Furthermore
due to low column longitudinal reinforcement ratio and
low column axial stress, some of them have failed due
to column failure. The Kordina specimens were also
excluded as they are provided by inclined reinforcement
in their joints which significantly increased their jointshear strength. The parametric study in Fig. 3 clearly
shows that joint shear strength is independent of the col-
umn longitudinal reinforcement ratio in joints that fail
by joint shear failure.
The authors also analysed the joints of Parker andBullman. The analysis of specimens 4a and 4d of Parker
and Bullman showed that these two specimens were
identical except for their column longitudinal reinforce-
ment ratios. The provision of 75% less column reinforce-ment than specimen 4d, did not only decrease the joint
shear strength of specimen 4a, but also caused the speci-men 4a to fail by column failure instead of joint shear
failure. It can be concluded that, the joints that have low
column longitudinal reinforcement ratios and column
axial stresses are more likely to fail by column hinging
Fig. 3. The influence of the column longitudinal reinforcement ratio
on the normalised joint shear strength.
and consequently have lower joint shear strengths with
respect to joints that fail by joint shear failure.
7. Influence of beam longitudinal reinforcement
ratio
The authors having investigated the specimens of
Parker and Bullman decided that the possible low
strength of Parker and Bullman specimens apart fromdetailing and high joint aspect ratio as well as low radius
of bend could be the very low beam longitudinal
reinforcement ratio. The relation between the beam
longitudinal reinforcement ratio and the normalised joint
shear strength is demonstrated in Fig. 4. Fig. 4 shows
that the ratio of the beam longitudinal reinforcement isrelated to the normalised joint shear strength by the fol-
lowing equation:
njAsbbbd0.4289
(15)
where Asb is the total area of beam reinforcement; bb is
the breadth of the beam; d is the depth of the beam.
In order to understand the possible influence of theratio of beam longitudinal reinforcement on the joint
shear strength, the authors investigated the specimens
C4AL and C1AL of Scott, which were nearly identicalexcept the beam reinforcement ratios they had. The joint
shear strength of C1AL that had a beam longitudinal
reinforcement ratio of 1.1 was nearly 30% lower than
C4AL that had a beam reinforcement ratio of 2.1. The
analysis of these two specimens showed that the beamlongitudinal reinforcement is related to the normalisedjoint shear strength by the above equation.
8. Influence of beam reinforcement detailing
The beam reinforcement detailing affects both the nor-
malised joint shear strength and the failure modes of
joints. The authors analysis of two identical specimensof Kordina with different beam reinforcement detailing
Fig. 4. The influence of the beam longitudinal reinforcement ratio on
the normalised joint shear strength.
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showed that the joint shear strengths of monotonically
loaded exterior beam column connections decrease by
15% if detailed by U bars rather than L bars. This is
shown in Fig. 5. Inspection of Scott and Hamill speci-
mens C4ALH3 (L bars bent down detail) and C6ALH3(U bars detail) which were identical except for their
beam reinforcement detailing shows that providing Lbars bent down detail reinforcement can change the fail-
ure modes of monotonically loaded exterior beam col-
umn connections from joint shear to beam failure.Further inspection of Kordina specimens with that of
Ortiz or Scott shows that the normalised joint shear
strengths of Kordina specimens are significantly higherthan that of other researchers specimens. The authorsare of the opinion that there are two important reasons
for this. First, the specimens of Kordina except for RE4are detailed by inclined reinforcement, which signifi-cantly increases the normalised joint shear strength.
Second, the concrete cylinder strengths of Kordinasspecimens have been underestimated because they are
based on the minimum rather than the average cube
strengths. The consequence of this is that the normalised
joint shear strengths may have been overestimated.
9. Influence of the vertical anchorage length and
the radius of bend
The analysis of BCJ3 of Ortiz showed that the joint
shear strength of BCJ3 was considerably increased com-pared to other specimens of Ortiz with low amount ofstirrups. The authors are of the opinion that this increase
is due to higher vertical anchorage length and higherradius of bend of beam reinforcement of Ortiz specimen
BCJ3. Thus, it can be concluded that there is evidence
from Ortiz tests that the shear strength of joints can be
increased if the vertical anchorage length is higher than
26db (db is the diameter of beam reinforcement) and the
radius of bend is higher than 8db.
Fig. 5. The normalised joint shear strength of identical specimens
with different beam reinforcement detailing.
10. Influence of joint aspect ratio
The authors have carried out a parametric study toinvestigate the influence of the joint aspect ratio on thenormalized joint shear strength. The specimens whichfail by failure modes other than joint shear failure, the
U bar specimens, the specimens with inclined bars, allof Parker and Bullmann specimens, all specimens with
medium and high amount of stirrups are excluded from
this study. In order to minimise the possible interaction
of parameters such as the concrete cylinder strength and
the beam longitudinal reinforcement ratio, the joint shear
strength is normalised by the square root of the concrete
cylinder strength and the 0.4289 power of the beamlongitudinal reinforcement ratio. Fig. 6 shows the
relation between the joint aspect ratio and the normalised
joint shear strength. It is evident that the joint aspect
ratio is related to the normalised shear strength as shown
in Eq. (16).
Vj
beffhcfchb
hc0.61 (16)
where hb is the cross-sectional height of the beam; hc is
the cross-sectional height of the column.In order to investigate the possible interactions of dif-
ferent parameters, the authors also inspected C7 and
C4AL of Scott, which were the only test data available
that investigates the joint aspect ratio. The notable differ-
ence between C7 and C4AL was that they had got differ-
ent beam reinforcement ratios as well as different jointaspect ratios. Analysis of the test data confirmed the
reliability of Eq. (16).
11. Influence of Joint Stirrups
Without the influence of transverse reinforcement, theauthors design equation takes the following form:
Vc
0.71bg100Asbbbd
0.4289
hbhc0.61
bc bb2
hcfc (17)
Fig. 6. The influence of the joint aspect ratio on the joint shear
strength normalised by the concrete sylinder strength and the beam
reinforcement ratio.
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where b 0.85 for joints detailed by U bars and b 1 for joints detailed by L bars. and g 1.37 for inclined
bars in the joint and g 1 for others.The constant 0.71 is a capacity reduction factor
determined empirically. The authors have plotted the
joint shear strengths of the specimens in the experi-
mental database and their stirrup ratios in Fig. 7. It isvery apparent from the figure that the relation betweenthe stirrup ratio and the normalised joint shear strength
is tri-linear. Up to stirrup ratios of 0.003, the joints are
named as joints with low amount of stirrups. Joints with
joint stirrup ratios between 0.003 and 0.0055 are named
as joints with medium amount of stirrups. All the stirrups
yield within this region. When the stirrup ratio is higher
than 0.0055, not all the stirrups yield as evident from
the analysis of specimen BCJ7 of Ortiz and C4ALN3 ofScott and Hamill and the contribution of stirrups to joint
shear strength substantially reduces under the yield
capacity of stirrups.
The stirrups that are considered effective in this study
are those that are situated above beam compressive
chord and below the top beam reinforcement. This is
because, the reported stirrup strains of Ortiz specimen
BCJ7 and BCJ4 both showed that the stirrup strains sub-
stantially reduce if positioned between the beam tensile
reinforcement and the beam compressive chord. Fig. 1
shows this.It is commonly assumed in the literature [16] that joint
shear strength is given by the addition of concrete resist-
ance to shear, symbolising the resistance of the concrete
strut mechanism and stirrup yield capacity, symbolising
the resistance of the truss mechanism:
Vj Vc Asjefy (18)
where Vc is the joint shear strength of the concrete
(without stirrups); Asje is the area of the stirrups; fy is
the stirrup yield strength.
As mentioned above, the analysis of tests showed that
when the joints have a stirrup ratio higher than 0.0055,
not all the stirrups yield within the joint. The authors are
of the opinion that Eq. (18) should be corrected as:
Fig. 7. The influence of stirrup ratio on the joint shear strength.
Vj
beffhcfc vc
aAsjefy
behcfc(19)
Therefore the final equation can be formulated as:
Vj
0.71bg
100
Asb
bbdb
0.4289
bc bb
2 hcfc
hbhc0.61
(20)
aAsjefy
where a 0.664 for joints with low amount of stirrups;a 0.6 for joints with medium amount of stirrups;a 0.37 for joints with high amount of stirrups
The stirrups not only affect the normalised joint shearstrength but also the failure modes of joints. The authorsinspection of joints C4ALNO (no stirrup in joint) and
C4ALN1 (single stirrup in joint) of Scott and Hamill has
shown that provision of a single stirrup in joints whichhave hc/db ratios less than 10 changes the failure mode
from connection zone reinforcement pull out to jointshear failure. Furthermore, it is apparent from the datab-
ase that anchorage failures are not anticipated in joints
which have medium or high amount of stirrups.
12. Influence of column axial stress
The authors have depicted the relation between the
normalised joint shear strength and the column axial
stress in Fig. 8. There is considerable scatter in the
experimental data. The results show that the columnaxial load certainly does not influence the joint shearstrength of monotonically loaded exterior beam column
connections. Vollum reaches a similar conclusion.
Nevertheless, the authors investigation of the speci-mens of Parker and Bullman showed that column axial
load influences the behaviour of joints by changing theirfailure modes. The authors comparison of Parker andBullmann specimen 4a with specimens 4b, 4c showed
that the former failed by column failure while the lattertwo specimens all failed by joint shear failure. The
Fig. 8. The influence of the column axial stress on the normalised
joint shear strength.
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experimental data indicates that high column axial
stresses and high column longitudinal reinforcement
ratios are necessary in order to avoid column failures.
13. Critique of previously suggested design
equations for exterior beam column joints
In this section, the previously suggested empirical
equations are discussed and compared with the designequation of authors. Because the influence of beamlongitudinal reinforcement ratio is not taken into
account, previous researchers estimates of the influenceof the joint aspect ratio on the joint shear strength are
flawed. None of the previously suggested design equa-tions take account of the influence of the stirrups realisti-cally. Using the authors new proposal, a more realisticestimate of joint shear strength was obtained.
13.1. Remark on Sarsam and Phillips equation
The authors as well as Vollums analysis showed thatthe column axial load does not influence the joint shearstrength. Contrary to the experimental evidence, Sarsam
and Phillips equation suggests that column axial loadinfluences the joint shear strength.
Furthermore, all stirrups are considered as effective in
Sarsam and Phillips equation while the tests of Ortiz
showed that the effective stirrups are those that are situ-
ated above the beam compressive chord and below the
beam reinforcement. Sarsam and Phillips equation pre-
dicts that all the transverse reinforcement yield. How-ever, inspection of BCJ7 of Ortiz demonstrates clearlythat not all the stirrups yield in some joints.
Sarsam and Phillips equation predicts that column
longitudinal reinforcement ratio affects the joint shear
strength. However, the parametric study of authors inFig. 6 showed that there is no apparent relationship
between the column longitudinal reinforcement ratio and
the normalised shear strength of joints that fail by joint
shear failure.
The equation predicts that there is a linear relationshipbetween the concrete cube strength and the normalised
joint shear strength. The authors parametric study inFig. 3 showed that the normalised joint shear strength is
proportional to the square root of the concrete cylinder
strength. The equation of Sarsam and Phillips overesti-
mates the influence of the concrete cylinder strength onthe normalised joint shear strength.
13.2. Remark on the design equation of Vollum
As mentioned in the above paragraphs, beamreinforcement ratio significantly influences the jointshear strength of exterior joints. Vollums equation neg-lects the influence of beam longitudinal reinforcement
Fig. 9. The influence of the beam reinforcement ratio on the predicted
normalised joint shear strength of the Vollum equation.
ratio on the joint shear strength. Because the effect ofbeam longitudinal reinforcement ratio is disregarded, the
equation incorrectly predicts a linear relationship
between the joint aspect ratio and the joint shear
strength. If the above parameters are taken into account,
joint shear strength is predicted as proportional with the
0.61 power of the joint aspect ratio as given by the sug-
gested equation in this paper.
Vollum has not given the basis of the upper limits
chosen. It is not clear what the basis of the constant 0.97is. Moreover, the author suggests that the other limit
1.33fcbeffhc is chosen because it is the highest shearstrength of the database used by the author. So the accu-
racy of the above constraint is limited by the experi-
mental database used. The authors are of the opinion that
more experiments are needed to propose upper limits for
joint shear strength.The authors plotted Vjpredicted/Vjactual values for both
Vollum and the authors equation against beamreinforcement ratio, joint aspect ratio, stirrup ratio and
the stirrup index in Fig. 916. The figures clearly showthat the authors model is an improvement on the designequation of Vollum. The suggested design equation is
more conservative than Vollum equation under varying
beam reinforcement ratio, joint aspect ratio, stirrup ratio
and the stirrup index.
Fig. 10. The influence of the beam reinforcement ratio on the pre-
dicted joint shear strength of authors equation.
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Fig. 11. The influence of the joint aspect ratio on the normalised joint
shear strength predicted by Vollum.
Fig. 12. The influence of the joint aspect ratio on the predicted shear
strength of the authors equation.
Fig. 13. The influence of the stirrup ratio on the predicted joint shear
strength of the Vollum equation.
Fig. 14. The influence of the stirrup ratio on the predicted joint shear
strength of the authors equation.
Fig. 15. The influence of the stirrup index on the predicted joint shear
strength of Vollums equation.
Fig. 16. The influence of the stirrup index on the predicted joint shear
strength of the authors equation/
14. Present design guidelines
Both the ACI-ASCE Committee 352 [8], and EC8 [9]
methods calculate the joint shear strength on the assump-tion that the tensile reinforcement yield and for bothequations factors of safety are included. The above
methods are considered to be inadequate by the authors
because they neglect the influence of the stirrups, beamlongitudinal reinforcement ratio, and joint aspect ratio
on the joint shear strength.
14.1. Comparison of the parametric studies with the
established principles of joint mechanics
In order to investigate the reliability of the parametric
studies in Fig. 29, the authors investigated the estab-lished equations on the basic mechanics of reinforced
concrete beam-column joints. This has been also pre-
viously discussed by Paulay [17] and by Bonacci & Pan-
tazopoulou for interior joints [18] who have also takeninto account the joint deformations. Both of the authors
use the average stresses for equilibrium as shown in Fig.
18. The typical loading system considered in analysis of
exterior beam-column joints is shown in Fig. 17. Fig. 18
depicts the equilibrium of vertical and horizontal forces.Fig. 18 shows that equilibrium of forces in the horizontal
direction require the average transverse compressive
stress in the joint sx defined as:
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Fig. 18. Stress equilibrium.
Fig. 17. Joint geometry.
sx Asb
beffhbfs
Asje
beffhbfw (21)
where fs is the average stress in the beam reinforcement;
fw is the average stress in the transverse reinforcement.
Consequently, the average normal concrete stress in
the y direction sy can be expressed as :
sy Ascol
beffhcfscol
N
beffhc(22)
where fscol is the average stress in the column reinforce-
ment; N is the column axial loadDefining the average joint shear stress in the joint as
tav, the maximum principal stress associated with thestress tensor is given as;
s sx tav 0
tav sy 0
0 0 sz (23)where sz is the confining stress provided by stirrups inthe z direction.
s3 I1s2 I2sI3 0 (24a)
In order to determine the principal stresses, Eq. (24a)
has to be solved; where I1 sx sy sz
I2 sxsy sysz sxszt2av (24b)
I3 sxsyszszt2av (24c)
The tensile stress in the concrete is negligible and
therefore s1 0, which consequently gives:
sy t
2
av
sx(25)
From the Mohrs circle,
tan2q 2tav
sxsy(26)
If Eq. (25) is substituted into Eq. (26), the following
quadratic equation ensues:
t2av 1tanq
tanqsxtavs2x 0 (27)which gives:
tav sx
tanq(28)
Using Eq. (25), we have:
sy tav
tanq(29)
Collins and Mitchell [19] suggest the following equation
for the maximum stress in concrete panels:
f2max fc
0.8 170e1fc (30)
The principal compressive stress is given by:
s2 2 e20.002
e20.002
2f2max (31)s2 is also given from Mohrs circle as:
s2 sx sy tavtanq 1tanq (32)
Thus the average joint shear stress can be expresses as:
tav s2
tanq 1tanq(33)
Eq. (30)(33) show very clearly that as the principaltensile strain increases, the average joint shear stress
decreases. Thus it is necessary to express the principaltensile strain in terms of the strains in the x and y direc-
tions in order to investigate the factors that influence the joint shear strength. From Mohrs circle, it is knownthat:
tan2q g
exey(34)
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From Mohrs circle, the principal tensile strain will be:
e1 (ex ey)
2(exey)2
2
g22 (35)
If Eq. (34) is substituted into Eq. (35) and appropriate
trigonometric transformations are carried out, Eq. (36)given by Bonacci and Pantazopoulou is obtained.
e1 exeytan2q1tan2q (36)
The next step will be to express the strains in the x
and y directions in terms of the stresses.
sx tavtanq Asbfs
beffhb
Asjefw
beffhb Asb
beffhbm (37)
Asje
beffhbfw
where
m fs/fw (38)
The strain in the x direction can therefore be
expressed as:
ex fw
Es
tavtanq
EsAsbmbeffhb
Asje
beffhb
(39)
The strain in the y direction can similarly be
expressed as:
ey fscol
Es tav
tanq
N
beffhc beffhcAscolEs
(40)
If Eq. (39) and (40) are substituted into Eq. (36):
e1 1
Es(1tan2q)tavtanq 1Asb
beffhbm
Asje
beffhb
(41)
1
Ascol
beffhc Ntan2q
beffhcAscol
beffhcIt is evident from the inspection of experiments that
cracks extend throughout the diagonal of the joint. So
the angle of principal stresses can be expressed as:
tanq hb
hc(42)
If Eq. (42) is substituted into Eq. (41),
e1 1
Es1hbhc2tavhbhc 1Asbmbeffhb Asjebeffhb (43)
1
Ascol
beffhc Nhbhc
2
Ascol The above equation shows that the principal tensile
strain is increased by the joint aspect ratio and column
longitudinal reinforcement ratio and the axial load on
the column whereas it is decreased by increasing beam
longitudinal reinforcement ratio and the stirrup ratio. The
shear stress in the joint is dependent on the principal
tensile strain as evident from Equations 30 and 33. It is
therefore evident from Eq. (28)(29) and (43) that the
joint shear strength increases as the beam longitudinalreinforcement ratio and the transverse reinforcement
ratio increases. Eq. (29) shows that the joint shear
strength increases as the column load and the column
longitudinal reinforcement increases but Eq. (43) shows
that as the longitudinal column reinforcement and the
column load increases, the principal tensile stresses
increase which consequently decreases the normalised
joint shear strength. Therefore the increase in the joint
shear strength due to Eq. (29) is offset by the increasein the principal tensile strain. The above conclusions are
totally in accordance with the predictions of theauthors equation.
15. Conclusions
The purpose of this investigation was to study the
effect of the parameters influencing the behaviour ofbeam to column connections and to determine if the
present design guidelines are unconservative. From the
analysis of the tests and results of the parametric studies,the following design recommendations can be made.
1. All the experimental evidence points to the fact that
the U bar details should not be used in monotonically
loaded exterior beam column joints. As mentioned in
the upper paragraphs, providing L bars bent downdetail beam reinforcement can change the failure
modes of monotonically loaded exterior beam-column
joints from joint shear to beam failure. Furthermore,
the joint shear strength of joints is increased by 15%
if detailed by L bars bent down detail beam reinforce-ment.
2. Experimental evidence shows that anchorage failures
are not anticipated in joints with stirrups in monoton-
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ically loaded exterior beam column joints. In joints
without stirrups on the other hand, hc/db values should
be larger than 10 in order to avoid anchorage failures.
3. Column axial load has no influence on ultimate shearcapacity of the joint but high column axial load and
high column longitudinal reinforcement ratios are
necessary in the joint to avoid column failures.4. Transverse reinforcement in the joint improves the
joint shear capacity but not in the same ratio as indi-
cated by the addition rule Vc+Vs. The authors plottedthe stirrup ratio against the joint shear strength. The
results showed that the diagram is trilinear. Based on
this parametric study, the authors classified the stir-rups into three in increasing the joint shear strength.
Up to stirrup ratios of 0.003, the joints are named as
joints with low amount of stirrups. Joints with joint
stirrup ratios between 0.003 and 0.0055 are named as
joints with medium amount of stirrups. All the stir-
rups yield within this region. When the stirrup ratiois higher than 0.0055, not all the stirrups yield as evi-
dent from the analysis of specimen BCJ7 of Ortiz and
C4ALN3 of Scott and Hamill and the contribution of
stirrups to joint shear strength substantially reduces
under the yield capacity of stirrups. Parametric studies
further carried out on the experimental database in
Table 1 showed that the addition rule Vc+Vs, shouldbe corrected as (Vc aVs)beffhcfc where is 0.664for low amount of stirrups, 0.6 for medium amount
of stirrups, 0.37 for high amount of stirrups.
5. Increasing the beam longitudinal reinforcement ratio
increases the joint shear strength. Because the influ-ence of beam longitudinal reinforcement ratio is taken
into account, the proposed equation predicts that the
joint shear strength is proportional to (hb/hc)0.61.
6. The present guidelines, design equations and code
recommendations for predicting the shear strength of
monotonically loaded exterior beam column joints are
unconservative. The suggested design equation gives
more conservative and reliable results for predicting
the joint shear strength under varying beam longitudi-
nal reinforcement ratio, joint aspect ratio, stirrup ratio
as well as stirrup index.
7. The analysis of BCJ3 of Ortiz shows that the joint
shear strength will further increase if the verticalanchorage length is higher than 26db and the radius
of bend of beam bars are higher than 8db. The authors
are of the opinion that these figures can be used aslower limits for the radius of bend and vertical
anchorage length.
8. The authors applied their equation on the experi-
mental database in Table 1. The results showed that
the average Vjpredicted Vjtest values for the authors equ-
ation applied on all the experiments in the experi-
mental database is 0.88 and the standard deviation is
0.1. The results show that the equation suggested
gives realistic and conservative estimates of the joint
shear strength.9. The extremely good results of the proposed design
equation on the experimental database confirmed andsupported the value of the parametric studies.
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