A New Design Equation for Predicting the Joint Shear Strength of Mono Tonic Ally Loaded Exterior Beam-column Joints

8/3/2019 A New Design Equation for Predicting the Joint Shear Strength of Mono Tonic Ally Loaded Exterior Beam-column Joints

1/13

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.


2/13

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.


3/13

1107P.G. Bakir, H.M. Boduroglu / Engineering Structures 24 (2002) 11051117

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)


4/13


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.


5/13


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


6/13


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.


7/13


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.


8/13


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.


9/13


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.


10/13


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:


11/13


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)


12/13


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-


13/13


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.

References

[1] Bertero VV, Shadh H. Algeria earthquake, October 10, 1980.

Oakland, CA: Earthquake Engineering Research Institute, 1983.

[2] Mitchell D. Structural damage due to the 1985 Mexico earth-

quake. In: Proceedings of the 5th Canadian Conference on Earth-

quake Engineering. Rotterdam: A.A. Balkema; 1987. p. 87111.

[3] Seismology Committee, Structural Engineers Association of Cali-

fornia. Recommended lateral force requirement and tentative

commentary, 1998.

[4] EERI Reconnaissance Team. Lome Prieta earthquake, October

17, 1989. Preliminary reconnaissance report. Oakland, CA: EERI.[5] Bakir PG, Boduroglu MH. Earthquake risk and hazard mitigation

in Turkey. Earthquake Spectra, accepted for publication.

[6] Sarsam KF, Phillips ME. The shear design of insitu reinforced

beam-column joints subjected to monotonic loading. Magazine

of Concrete Research 1985;37(130):1628.

[7] Vollum RL. Design and analysis of exterior beam column con-

nections. PhD thesis, Imperial College of Science Technology

and Medicine-University of London, 1998.

[8] Recommendations for design of beam-column joints for design

of beam column joints in monolithic reinforced concrete struc-

tures, 1991 edition. Reported by ACI-ASCE Committee 352,

ACI.

[9] Eurocode 8: design provisions for earthquake resistance of struc-

tures. London: BSI, 1995.

[10] Ortiz R. Strut and tie modelling of reinforced concrete shortbeams and beam column joints. PhD thesis, University of

Westminster, 1993.

[11] Kordina K. Bewehrungsfuhrung in Ecken und Rahmenendknoten,

Deutscher Ausschuss fur Stahlbeton, Heft 354, 1984.

[12] Scott RH. The effects of detailing on RC beam column connec-

tion behaviour. The Structural Engineer 1992;70(18):31824.

[13] Scott RH, Hamill SJ. Connection zone strain in reinforced con-

crete beam column connections, In: Proceedings of the 11th Inter-

national Conference on Experimental Mechanics, Oxford, UK,

1998. pp. 65-69.

[14] Taylor HPJ. The behaviour of in situ concrete beam column

joints. Technical report 42.492, Cement and Concrete Associ-

ation, May 1974.

[15] Parker DE, Bullman PJM. Shear strength within reinforced con-

crete beam-column joints. The Structural Engineer1997;75(4):537.

[16] Park R, Paulay T. Reinforced concrete structures. New York:

John Wiley and Sons, 1975.

[17] Paulay T. Critique of the special provisions for seismic design

of the building code requirements for reinforced concrete (ACI

318-83). ACI Journal 1986; 83(2): 274, 283.

[18] Pantazopoulou S, Bonacci J. Consideration of questions about

beam-column joints. ACI Structural Journal 1992;89(1):2737.

[19] Collins MP, Mitchell D. Prestressed concrete structures. Engle-

wood Cliffs, NJ: Prentice-Hall, 1991.

Documents

A New Design Equation for Predicting the Joint Shear Strength of Mono Tonic Ally Loaded Exterior Beam-column Joints