Behavior and Design of Reinforced Concrete,

Steel, and Steel-Concrete Coupling Beams

Kent A. Harries

1

, M.EERI, Bingnian Gong

2

, and Bahram M. Shahrooz

3

Corresponding (first) author: Kent A. Harries

Mailing address: USC-Civil Engineering, 300 Main Street, Columbia, SC 29208

Phone: 803 777 0671

Fax: 803 777 0670

E-mail address: harries@engr.sc.edu

Submission date for review copies: February 7, 2000

Submission date for camera-ready copy: June 21, 2000

(KAH) Dept. of Civil and Environmental Engineering, University of South Carolina, Columbia, SC 29208

(BG) CBM Engineers, Inc., 1700 W. Loop So., Suite 830, Houston, TX 77027

(BMS) Dept. of Civil and Environmental Engineering, University of Cincinnati, Cincinnati, OH 45221-0071

Behavior and Design of Reinforced Concrete,

Steel, and Steel-Concrete Coupling Beams

Kent A. Harries

4

, M.EERI, Bingnian Gong

5

, and Bahram M. Shahrooz

6

The efficiency of coupled wall systems to resist lateral loads is well

known. In order for the desired behavior of the coupled wall system to be

attained, the coupling beams must be sufficiently strong and stiff. The

coupling beams, however, must also yield before the wall piers, behave in a

ductile manner, and exhibit significant energy absorbing characteristics. This

paper reviews the current state of the art for the design of conventional

reinforced concrete, diagonally reinforced concrete, steel and composite steel-

concrete coupling beams. Although not exhaustive, critical aspects of the

design of these systems are presented.

INTRODUCTION

An efficient structural system for seismic resistance can be achieved if openings in

structural walls can be arranged in a regular pattern as shown in Figure 1. In this manner, a

number of individual wall piers are coupled together to produce a system having large lateral

stiffness and strength. By coupling individual flexural walls, the lateral load resisting

behavior changes to one where overturning moments are resisted partially by an axial

compression-tension couple across the wall system rather than by the individual flexural

action of the walls. As a result, a significantly stiffer lateral load resisting system is achieved.

The beams that connect individual wall piers are referred to as coupling beams. Coupling

beams are analogous to and serve the same structural role as link beams in eccentrically

braced frames. Well-proportioned coupling beams above the second floor generally develop

plastic hinges simultaneously, and are subjected to similar end rotations over the height of the

structure (Aktan and Bertero 1981, Aristizabal-Ochoa 1982, Shiu et al. 1981). Hence,

dissipation of input energy can be distributed over the height of the building in the coupling

beams rather than concentrating it predominantly in the first-story wall piers (Aktan and

Bertero 1984, Fintel and Ghosh 1982, Park and Paulay 1975). For optimum performance, the

energy dissipating mechanism should involve formation of plastic hinges in most of the

coupling beams and at the base of each wall. This mechanism is similar to the strong

column-weak girder design philosophy for ductile moment resisting frames (Paulay 1971,

Park and Paulay 1975). In order for the desired behavior of the coupled wall system to be

attained, the coupling beams must be sufficiently strong and stiff. The coupling beams,

however, must also yield before the wall piers, behave in a ductile manner, and exhibit

significant energy absorbing characteristics.

(KAH) Dept. of Civil and Environmental Engineering, University of South Carolina, Columbia, SC 29208

(BG) CBM Engineers, Inc., 1700 W. Loop So., Suite 830, Houston, TX 77027

(BMS) Dept. of Civil and Environmental Engineering, University of Cincinnati, Cincinnati, OH 45221-0071

Coupling

Beams

(a) (b) (c)

Figure 1. Examples of coupled wall geometry.

The degree of coupling of a coupled wall system is defined as the ratio of the total

overturning moment resisted by the coupling action (i.e., the axial load couple formed in the

walls) to the total overturning moment. The coupling action between wall piers is developed

through shears in the coupling beams. The hysteretic characteristics of coupling beams,

therefore, may substantially affect the overall response of a coupled wall system particularly

for structures having a high degree of coupling (Aktan and Bertero 1981, Harries 1999,

Paulay 1986, Paulay and Taylor 1981, Paulay 1980). Coupling beams should be designed to

avoid over coupling, which causes the system to act as a single pierced wall with little frame

action. Similarly, light coupling should also be avoided as it causes the system to behave like

two isolated walls (Aktan and Bertero 1987, 1984, 1981, Aristizabal-Ochoa 1987, 1982,

Aristizabal-Ochoa et al. 1979, Lybas and Sozen 1977, Shiu et al. 1981, 1984).

Reinforced concrete coupling beams may have conventional longitudinal reinforcement,

diagonal reinforcement or a hybrid reinforcing steel arrangement. Coupling beams may also

be fabricated from rolled or built-up steel shapes, or have a composite steel-concrete design.

Additionally, recent investigations have proposed alternative hybrid designs.

REINFORCED CONCRETE COUPLING BEAMS

BEHAVIOR

It has been shown that the ductility capacity of a coupled wall system increases with

increasing degrees of coupling (Harries 1998). The degree of coupling is a function of the

relative stiffness and strength of the beams and walls. Saatcioglu et al. (1987) showed that

coupling beams must be capable of developing displacement ductilities in excess of 6 for

system ductilities of about 4 to be achieved.

Several researchers have investigated novel approaches for improving the ductility and

energy absorption capacity of reinforced concrete coupling beams. For span-to-depth ratios

less than about 2, a system using specially detailed diagonal reinforcement, developed by

Paulay (e.g., Paulay and Binney 1974), has been shown to significantly improve the reversed

cyclic loading response. Shiu et al. (1978) confirmed the improved behavior of diagonally

reinforced beams over conventionally reinforced beams. However, these tests also

demonstrated that for larger span to depth ratios (2.5 and 5) the diagonal reinforcement was

not as efficient due to its lower angle of inclination and hence its reduced contribution in

resisting shear.

The stiffness required of "efficient" coupling beams usually results in moment arm to

effective depth ratios (a/d ratio) of less than 2 (Aktan and Bertero 1981). Indeed, a/d ratios of

1 or less are common (Aktan and Bertero 1981, Bertero 1980, Bertero et al. 1974, Paulay

1977, Paulay and Binney 1974, Paulay 1971). As the a/d ratio decreases, arching action

contributes more significantly to shear strength than flexural behavior (Aktan and Bertero

1981, Paulay 1977, Paulay and Binney 1974, Paulay 1971). In conventionally reinforced

coupling beams, both top and bottom reinforcement may simultaneously undergo tension as a

result of diagonal cracking. Thus, it is recommended that the increased ductility due to

compression bars not be accounted for (Park and Paulay 1975, Paulay 1971). Coupling

beams require special detailing particularly if

'f5.0v

cu

(

'f6v

cu

in psi units) and

a/d<3.0.

Figure 2. Examples of reinforced concrete coupling beams (Harries et al. 1998).

Conventional reinforcement, i.e., longitudinal bars with transverse steel (see Figure 2a),

performs satisfactorily at nominal shear stresses of

'f25.0

c

(

'f3

c

) or less when the

flexural mode controls the failure mechanism (Aktan and Bertero 1981, Aristizabal-Ochoa

1987). Sliding shear at the face of the wall begins to affect the response of the beams in the

range of

'f3.0

c

(

'f5.3

c

) to

'f5.0

c

(

'f6

c

). By providing intermediate midheight

longitudinal bars, the hysteretic response is improved and strength deterioration due to shear

is delayed (Aktan and Bertero 1984, Scriber and Wight 1978, Tassios et al. 1996). Beams

with intermediate bars do not perform well when the shear stress is greater than

'f5.0

c

(

'f6

c

). Providing cranked diagonal reinforcement near the beam ends has been shown to

improve the hysteretic behavior by preventing sliding shear, and by spreading the hinging

regions away from the wall face (Paparoni 1972, Aristizabal-Ochoa 1987). This detail,

however, poses construction difficulties and results in extra cost.

8 - No 25

8 - No 25

2 - No 15

midside

7

0

0

m

m

400 mm

No 10 hoops

@ 90 mm

4 - No 30 in

No 10 hoops

@ 100 mm

4 - No 30 in

No 10 hoops

@ 100 mm

7

0

0

m

m

400 mm

No 10 hoops

@ 300 mm

(a) Conventionally reinforced coupling beam

(b) Diagonally reinforced coupling beam

3000 mm

1.3l =

1200 mm

d

1200 mm

1.3l = 1450 mm

d

wall steel

not shown

wall steel

not shown

Full-length diagonal reinforcement, as shown in Figure 2b, is a relatively simpler

detailing alternative, and has been found to maintain stable hysteretic behavior provided the

bars are adequately confined by spirals or hoops, and sufficient concrete cover is provided

(Aktan and Bertero 1981, Paulay and Binney 1974). This confinement provides resistance

against buckling to the individual bars and the bundled diagonal bar group. Such diagonal

reinforcement is perhaps the only successful solution for reducing the potential for sliding

shear, and enhancing the hysteretic characteristics of coupling beams having a/d ratios as

large as 3.33 (Aktan and Bertero 1981).

Considering that higher modes contribute to the response of coupling beams more than to

walls (Aktan and Bertero 1981, Mahin and Bertero 1975), these elements are subjected to

large inelastic end rotations and reversals. As a result, adequate anchorage of the coupling

beam reinforcement in the walls is necessary. The possibility of buckling of compression bars

at the beam-wall interface (where large cracks are anticipated) is also minimized by

providing effective confinement (Aktan and Bertero 1981, Barney et al. 1978, Paulay 1980,

Paulay and Santhakumar 1976, Paulay and Binney 1974). It is noted that anchorage and

confinement requirements often make these diagonally reinforced coupling beams very

difficult to construct due to congestion at the center of the beam and at the wall faces. (Recall

that there will often be concentrated vertical wall steel at these interfaces.) Figure 3 shows a

typical example of a diagonally reinforced coupling beam and its wall anchorage.

Figure 3. Diagonally reinforced concrete coupling beam (photo courtesy of Reinforcing Steel

Institute of Canada).

Table 1 summarizes the guidelines for the selection and detailing of reinforced concrete

coupling beams for seismic design given by the ACI 318-95 and ACI 318-99 Building Code

Requirements for Structural Concrete (ACI 1995, 1999), CSA A23.3-94 Design of Concrete

Structures (CSA 1994), and NZS 3101 Concrete Structures Standard (NZS 1995). When no

seismic design is required, coupling beams may be detailed as regular reinforced concrete

beams. It is noted, nonetheless, that these beams do typically exhibit very high shear to

moment ratios requiring special attention.

concentrated vertical

reinforcing at face

of wall

diagonal

primary

reinforcement

Table 1. Guidelines for selecting reinforced concrete coupling beams

Governing Limits

L/d shear stress, v

(psi)

Type of

Coupling

Beam

Design Criteria

ACI 318-95 Clause 21.6

2

L/d

y

f

n

?'

c

fv 167.0

2

L/d

y

f

n

?'

c

f

c

av

where 0.167 <

c

0.25

ACI 318-95 does not have provisions for diagonally

reinforced concrete members

ACI 318-99 Clause 21.6.7

4/

dL

y

f

n

?'

c

f

c

av

where 0.167 <

c

0.25

conventionally

reinforced

beams may be designed as flexural

members according to Clause 21.3,

however the commentary appears to

invoke Clause 21.6.4

4/

dL

'83.0

c

fv

diagonally or

conventionally

reinforced

see above or below

2/

dL

'83.0

'33.0

c

fv

c

fv

diagonally

reinforced

diagonal reinforcement to carry shear

(A

vd

f

y

V

n

/sin)

CSA A23.3-94 Clauses 21.3 and 21.5

4/

dL

')/(1.0

c

fdLv

conventionally

reinforced

i) beams capable of significant flexural

hinging (V

r

2M

p

/L)

ii) provide confinement and antibuckling

reinforcement

4/

dL

'

c

fv

diagonally

reinforced

i) diagonal reinforcement to carry shear

(

s

A

s

f

y

V

u

/sin)

ii) provide adequate confinement and

anchorage for diagonal bars

NZS 3101-1995 Clauses 12.4.6.2 and 9.4.4.2

')/(1.0

c

fdLv

conventionally

reinforced

i) beams capable of significant flexural

hinging (V

r

2M

p

/L)

ii) provide confinement and antibuckling

reinforcement

not

specified

'25.0

c

fv

some diagonal

reinforcement

required

i) diagonal reinforcement to carry shear

(

s

A

s

f

y

V

u

/sin)

ii) provide adequate confinement and

anchorage for diagonal bars

The ACI 318-95 seismic provisions do not specifically address details for coupling

beams. Clause 21.6.5.7 limits the nominal shear strength of a “horizontal wall segment” to

'

ccp

f0.83A (

'

ccp

f10A ). The accompanying commentary states that the coupling beam is

“in effect, a pier rotated through 90 deg.” Applying this interpretation allows the nominal

shear strength of a coupling beam to be determined from Clauses 21.6.5.2 or 21.6.5.3,

depending on the span-to-depth ratio of the beam. If this interpretation is applied, Clause

21.6.5.3 permits the shear stress in a very short coupling beam (L/d < 2.0) to be greater than

that in a longer beam. From the previous discussion of coupling beam behavior, it is clear

that this interpretation runs counter to observed beam behavior. ACI 318-95 does not address

diagonally reinforced concrete beams in any way.

ACI 318-99 has adopted specific provisions for coupling beams. For beams having span-

to-depth ratios, L/d, greater than 4.0, coupling beams are designed as flexural members

satisfying the requirements of Clause 21.3. It is noted, however, that Clause 21.6.4.5

(formerly 21.6.5.7 in 318-95) and the accompanying commentary continues to suggest that

“horizontal wall segments” may be treated as “pier[s] rotated through 90 deg.” as discussed

above. For coupling beams having span-to-depth ratios between 2 and 4, Clause 21.6.7.2

allows the use of diagonal or conventional reinforcement. For beams having span-to-depth

ratios less than 2 and a factored shear force exceeding

'

c

f0.33 (

'

c

f4 ), diagonal

reinforcement is required.

In Table 1, it is noted that the Canadian standard CSA A23.3 provides the clearest

direction as to the use of diagonal reinforcement: diagonal reinforcement requirements are

triggered by both span-to-depth ratios and shear stresses in the beam. The New Zealand

standard NZS 3101, on the other hand, combines these triggers into one: shear stress,

normalized by the span-to-depth ratio. NZS 3101 does, however, require diagonal

reinforcement if the shear stress exceeds an absolute threshold value of

'

c

f0.25 (

'

c

f3 ).

Furthermore, NZS 3101 Commentary Clause C9.4.4.2 appears to permit shear to be resisted

by a combination of diagonal and conventional reinforcement providing that the contribution

of the conventional reinforcement remains below the threshold value. For seismic design, the

New Zealand Standard does recommend, however, that 100% of the shear forces be carried

by diagonal reinforcement in any case where diagonal reinforcement is required. The design

and detailing requirements for diagonally reinforced concrete coupling beams are essentially

identical in both the Canadian and New Zealand standards.

ANALYSIS OF WALLS COUPLED WITH REINFORCED CONCRETE COUPLING

BEAMS

Geometric Modeling

A simple two-dimensional equivalent frame model suitable for elastic analysis of coupled

wall systems is illustrated in Figure 4. Such a model can be used to compute beam and wall

design forces. This model has several drawbacks; the most significant of which is that it does

not simulate the shift of the neutral axis in the wall piers beyond elastic limit. Other models

appropriate for inelastic analysis are available (Kunnath and Reinhorn 1994, Kabeyasawa et

al. 1983, Teshigawara et al. 1996, Milev and Kabeyasawa 1996, Vulcano and Bertero 1987).

Multiple pier walls may be similarly modeled or may be reduced to a simpler two-pier model

using further analytic tools (Stafford-Smith and Coull 1991).

Rigid

Elements

Frame

Elements

L

Figure 4. Two dimensional equivalent frame model of coupled wall system.

Modeling Beam and Wall Properties

Elastic analysis is sufficient to determine a theoretical degree of coupling for a coupled

wall structure and thus the distribution of internal forces in the walls and beams. Local

inelasticities at the beam-wall interfaces, redistribution of forces between coupling beams and

from the tension wall to the compression wall and strain hardening effects all contribute to

reducing the theoretical degrees of coupling. Reduced section properties, accounting for

cracking and loss of stiffness due to cycling can be used to determine a more appropriate

value for the theoretical degree of coupling. Table 2 shows the reduced stiffness values

suggested by the current ACI, CSA and NZS standards. ACI 318-99 recommends the use of

“any set of reasonable assumptions…[which are] consistent throughout the analysis” and

recommends using one half of the gross section properties for flexural members. It is

interesting to note that, using the reduction factors recommended by the New Zealand

Standard, one computes effectively stiffer coupling beams and more flexible wall piers,

resulting in a larger degree of coupling, and hence greater beam forces, than one computes

with the ACI or Canadian recommendations.

Table 2. Recommended reduced member stiffnesses

Member

ACI 318-99 CSA A23.3-94 NZS 3101-1995

1

compression wall in flexure 0.70EI (uncracked)

0.80EI 0.45EI

tension wall in flexure 0.35EI (cracked) 0.50EI 0.25EI

compression wall axial 1.00EA 1.00EA 0.80EA

tension wall axial 0.35EI (inferred) 0.50EA 0.50EA

average wall axial

2

not applicable 0.75EA

not applicable

conventionally reinforced

coupling beam

0.35EI

2

)/(31

20.0

Lh

EI

2

)/(81

40.0

Lh

EI

diagonally reinforced

coupling beam

not applicable

2

)/(31

40.0

Lh

EI

2

)/(7.27.1

40.0

Lh

EI

1. NZS 3101 has different recommendations for different limit states. The values corresponding to the most

critical limit state are given here.

2. CSA A23.3 suggests that an average axial wall stiffness is appropriate to simplify analyses.

Design

As shown in Table 1, depending on the shear span, coupling beams are designed

differently.

Flexural Failure Mode

Coupling beams are designed and detailed as standard beams if both L/d 4 and

db'f0.33V

wcu

(

db'f4V

wcu

). If the structure is located in high seismic regions (UBC

Zone 4, NEHRP Seismic Design Category D or E), provisions from ACI 318-99 Section 21.3

are used. ACI 318-99 Section 21.9 is followed for structures in moderate seismic regions

(UBC Zone 3, NEHRP Seismic Design Category C). It is noted that ACI 318-99 provisions

for minimum beam width (Clauses 21.3.1.3 and 21.3.1.4) may be waived if rational analysis

shows that lateral stability of the beam is adequate, or lateral stability is provided by

alternative means. Often the provision of an integral slab will be sufficient to provide lateral

stability.

The longitudinal flexural bars in the beam should be anchored adequately in the wall

piers. For high seismic regions (UBC Zone 4, NEHRP Seismic Design Category D or E),

development length requirements in ACI 318-99 Clause 21.5.4 must be met in addition to

those in ACI 318-99 Chapter 12. For moderate seismic regions (UBC Zone 3, NEHRP

Seismic Design Category C), development lengths for the longitudinal bars based on ACI

318-99 Chapter 12 are considered sufficient.

Shear Failure Mode

If L/d < 4 and

db'f0.33V

wcu

(

db'f4V

wcu

), two intersecting groups of diagonal

bars are to be used to reinforce coupling beams. Although it is preferable for each

intersecting group to contain the same number and size of reinforcing bar, this is often not

practical. Where this is the case, it is necessary to ensure that the amount of steel provided in

each diagonal is the same. If L/d < 4 and

db'f0.33V

wcu

(

db'f4V

wcu

), diagonal

reinforcement may also be used. Each group of diagonal bars must have a minimum of four

bars placed in a core with a minimum depth and width equal to the greater of b

w

/5 in the

plane of the diagonal bars or b

w

/2 in the plane of the beam (ACI 318-99 Clause 21.6.7.4 (a)).

The diagonal bars must be confined according to ACI 318-99 clauses 21.4.4.1 through

21.4.4.3. The required area of one leg of the diagonal reinforcement, A

vd

is (ACI 318-99

Clause 21.6.7.4):

dbffAV

wcyvdn

'83.0sin2 (1)

Although the diagonal reinforcement is intended to prevent sliding shear, the value of

coupling beam shear is limited to

dbf

wc

'83.0 (

dbf

wc

'10 ). Minimum horizontal and

vertical reinforcement requirements for deep beams (ACI 318-99 clauses 11.8.9 and 11.8.10)

must also be respected.

Considering the importance of adequate anchorage for the diagonal bars, it has been

recommended that the development length, calculated based on Chapter 21 of ACI 318-99,

inside the wall piers be increased by as much as 50%, if possible (Paulay 1986). Although

there are no provisions in the Standards for increasing the development length, it is

recommended that these bars be considered as “top cast” bars and that appropriate factors be

applied as shown in Figure 2 (Harries et al. 1998). Often geometric constraints within the

wall will require that these bars be provided with bent or hooked anchorages (see Figure 3).

As previously stated, this detail may be very difficult to construct.

STEEL OR STEEL-CONCRETE COMPOSITE COUPLING BEAMS

Structural steel members provide a viable alternative to reinforced concrete coupling

beams. Furthermore, a steel member may be encased with varying levels of longitudinal and

transverse reinforcement. The resulting steel-concrete coupling beams are referred to as

composite coupling beams herein. The advantages of steel or composite coupling beams

become particularly apparent in cases where height restrictions do not permit the use of deep

reinforced concrete beams, or where the required capacities and stiffness cannot be

developed economically by a concrete beam.

Similar to reinforced concrete coupling beams, energy dissipation characteristics of steel

or composite coupling beams play an important role in the overall performance. Depending

on the coupling beam length, steel coupling beams may be detailed to dissipate a major

portion of the input energy by flexure or by shear. For most coupling beams, however, it is

more advantageous to design the coupling beams as “shear critical,” or shear yielding

members since such members exhibit a more desirable mode of energy dissipation. Such a

choice is not possible for reinforced concrete members.

The coupling beam-wall connection depends on whether the wall boundary element is

reinforced conventionally or contains embedded structural steel columns. In the latter case,

the connection is effectively a joint between a steel beam and column, as shown in Figure 5.

Standard design methods for steel structures can be used for this connection (AISC 1995,

1997); however, proper measures must be taken to prevent failures similar to those observed

in steel moment-resisting frames during recent earthquakes. Furthermore, such a composite

wall system will not require significant amounts of boundary element reinforcement.

steel boundary column

shear studs

moment connection

face of wall

steel

coupling beam

holes in beam

wall steel not shown

Figure 5. Coupling beam connection to embedded steel column.

If the wall boundary element is reinforced with longitudinal and transverse reinforcing

bars, a typical connection involves embedding the coupling beam into the wall and

interfacing it with the boundary element, as shown in Figure 6. In addition to boundary

element reinforcing, embedded steel members may also be provided with welded vertical

reinforcing passing through their flanges or shear studs to resist horizontal loads (PCI 1992).

The vertical bars may be attached to the flanges through mechanical half couplers which are

welded to the flanges. These bars must have adequate development length, and must be large

enough to provide enough stiffness to the coupling beam. Shahrooz et al. (1992, 1993) have

found 25.4 mm diameter (#8) bars to be sufficient for this purpose. Such details result in

greater congestion in the boundary element. It has been shown that these additional welded

appendages are not necessary provided that the embedment region is sufficiently long and

that there is adequate boundary element reinforcing to control cracking at the beam-wall

interface (Harries 1995, Harries et al. 1997, Shahrooz et al. 1993). Harries et al. recommend

that two-thirds of the required vertical boundary element steel be located within a distance of

one half the embedment length from the face of the wall. Furthermore, the width of the

boundary element steel should not exceed 2.5 times the coupling beam flange width.

Satisfying these requirements will provide adequate control of the gap that opens at the beam

flanges upon cycling. With typical boundary element designs, these requirements are easily

met.

Figure 6. Steel coupling beam embedment details (Harries et al. 1998).

No. 10 hoops and hairpins

No. 10 hoops and hairpins

SECTION A-A

SECTION B-B

SECTION A-A

SECTION B-B

A

A

B

B

A

A

B

B

(a) flexure critical steel coupling beam (b) shear critical steel coupling beam

900

900

1000

14 - No. 35

No. 10 hoops @ 200

12 - No. 25

No. 10 hoops @ 150

No. 10 @ 150 vertical e.f.

No. 10 @ 100 horizontal e.f.

No. 10 @ 150 vertical e.f.

No. 10 @ 100 horizontal e.f.

As is illustrated in Figure 6a, lighter “flexure critical” coupling beams require more

boundary element steel in the wall since a greater proportion of the overturning moment is

carried by the wall. This may also result in the need to provide “barbells” at the toes of the

walls, further complicating construction.

It is not necessary, nor is it practical, to pass boundary element reinforcing through the

web of the embedded coupling beam. It is however necessary to provide good confinement in

the region of the embedded web. This may be accomplished using hairpins and cross ties

parallel to the web as shown in Figure 6 (section A-A). Additionally, vertical boundary

element reinforcement in this region may also be relied upon to provide significant

confinement to the embedded region (Harries et al. 1997).

STEEL-CONCRETE COMPOSITE COUPLING BEAMS

The selection of the embedded steel beam in a composite coupling beam typically

neglects the contribution of the encasing concrete. The contribution of the encasing concrete,

however, must be accounted for in subsequent stiffness and embedment length calculations.

The presence of the encasing concrete does permit the detailing requirements of the

embedded member to be relaxed. Gong and Shahrooz (1998) have shown that stiffeners are

not required in the case of a composite coupling beam. The surrounding concrete is sufficient

to provide adequate stability and thus ductility to the embedded steel beam.

ANALYSIS OF WALLS COUPLED WITH STEEL OR STEEL-CONCRETE COMPOSITE

COUPLING BEAMS

Stiffness

For steel coupling beams, standard methods are used to calculate the stiffness. The

stiffness of steel-concrete composite coupling beams needs to account for the increased

stiffness due to encasement. Stiffness based on gross transformed sections should be used to

calculate the upper bound values of demands in the walls, most notably wall axial force. The

cracked transformed section moment of inertia may be used when deflection limits or

coupling beam shear angles are checked. A previous study indicates that the additional

stiffness due to the presence of a composite floor slab is lost shortly after coupling beams

undergo small deformations (Shahrooz and Gong 1998). Until additional experimental data

become available, it is recommended to include the participation of the floor slab when the

stiffness of steel-concrete composite coupling beams are computed. Effective flange width

for T beams as specified in ACI 8.10.2 may be used for this purpose. Deflection limits should

be checked neglecting the contribution of the floor slab.

Geometric Modeling

Previous studies (Shahrooz et al. 1992, 1993, Gong et al. 1997, 1998, Harries et al. 1997)

suggest that steel or steel-concrete composite coupling beams are not fixed at the face of the

wall. As part of design calculations, the additional flexibility needs to be taken into account

to ensure that wall forces and lateral deflections are computed with reasonable accuracy.

Based on experimental data (Shahrooz et al. 1992, 1993, Gong et al. 1997, 1998), the

“effective fixed point” of steel or steel-concrete composite coupling beams may be taken as

approximately one-third of the embedment length from the face of the wall. Thus, the

effective clear span, L of the frame element representing the coupling beam in the model

shown in Figure 4 is:

eclear

LLL 6.0 (2)

The aforementioned method assumes that the walls have been modeled by column

elements located at the centroid of wall piers as shown in Figure 4. Finite element modeling

of structures have become more common in recent years where shell elements are used to

model the wall piers. These elements are typically connected to nodes along fictitious

column lines which are placed at the wall boundaries. One may attempt to account for the

additional flexibility of steel/steel-concrete composite coupling beams by shifting the column

lines, which define the walls, to a distance equal to 1/3 of the embedment length from the

face of the wall. However, this approach will alter the wall properties, and other methods are

necessary. A possible approach is to obtain equivalent values for the coupling beam axial

area and moment of inertia in order to incorporate the additional beam flexibility. Equations

to obtain the lower and upper bound values of the equivalent area and moment of inertia have

been derived (Tunc et al. 2000). Another alternative is to use an effective stiffness.

Approaching the same decrease in stiffness from another point of view, Harries et al.

(1998) propose the use of an effective stiffness of the steel coupling beam of 0.60kEI. The

factor k represents the reduction in flexural stiffness due to shear deformations:

1

2

12

1

w

GAL

EI

k

(3)

If the modeling technique used includes shear deformations, k = 1.

With this approach, the effective length of the beam in increased only to account for

spalling at the face of the wall:

cLL

clear

2 (4)

This approach does not require a preliminary design to determine the coupling beam

embedment length. The effective stiffness of 0.60kEI has been observed to agree very well

with experimental observations of beams having significantly different embedment details

(Harries 1995)

DESIGN

Steel Coupling Beams

Well established guidelines for shear links in eccentrically-braced frames (AISC 1997)

may be used to design and detail steel coupling beams. The expected coupling beam rotation

angle plays an important role in the required beam details such as the provision of stiffeners.

This angle is computed with reference to the collapse mechanism shown in Figure 7 which

corresponds to the expected behavior of coupled wall systems, i.e., plastic hinges at the base

of walls and at the ends of coupling beams. The value of

p

is taken as C

d

e

, where the

elastic interstory drift angle,

e

, is computed under code level lateral loads (e.g., ICBO 1994;

NEHRP 1997). Knowing the value of

p

, shear angle,

p

is calculated as:

L

L

wallp

p

(5)

Note that in this equation the additional flexibility of the steel or composite coupling

beams may be taken into account by increasing the length of coupling beam as indicated in

Equations 2 or 4. This approach provides a reasonable approximation even if shell elements

have been used to model wall piers.

Steel-Concrete Composite Coupling Beams

Previous research on steel-concrete composite coupling beams (Shahrooz et al. 1992,

1993, Gong et al. 1997, 1998) indicates that nominal encasement around steel beams

provides adequate resistance against flange and web buckling. Therefore, composite coupling

beams may be detailed without web stiffeners. Due to inadequate data regarding the influence

of encasement on local buckling, minimum flange and web thicknesses similar to steel

coupling beams should still be used.

Figure 7. Determination of coupling beam angle of rotation.

Coupling Beam Embedment

The coupling beam is embedded in the wall such that its capacity can be developed. A

number of methods may be used to calculate the necessary embedment length. Either method

can be used although the equations proposed by Marcakis and Mitchell (1980) generally

result in slightly longer embedment lengths.

Mattock and Gaafar (1982)

This model is based on mobilizing an internal moment arm between bearing forces C

f

and

C

b

as shown in Figure 8a. A parabolic distribution of bearing stresses is assumed for C

b

, and

C

f

is computed by a uniform stress equal to 0.85f'

c

. The bearing stresses are distributed over

the beam flange width, b

f

. Following these assumptions and calibrating against experimental

data, the required embedment length, L

e

, may be determined from:

e

ef

f

wall

cu

L

a

Lb

b

t

fV

88.0

22.058.0

'05.4

1

1

66.0

(6)

L

L

wa ll

p

p

p

p

=

L

L

w

a

l

l

plastic hinges

Marcakis and Mitchell (1980)

Using slightly different assumed stress distributions shown in Figure 8b, and assuming a

rigid-body motion of the embedded element, the following expression may be used for

determining the required embedment length, L

e

. This expression has also been calibrated

against experimental data.

)(

6.3

1

)(''85.0

cL

e

cLbf

V

e

ecc

u

(7)

Where “e” is the eccentricity from the midspan of the coupling beam to the center of the

effective embedment as shown in Figure 8b.

In either method, the value of the shear span, “a” is taken as one half of coupling beam

length if the inflection point is assumed to be at the midspan. It is suggested that the shear

span be increased to account for spalling of the walls as calculated in Equation 4.

The significant difference between these methods is that Marcakis and Mitchell account

for spreading of the compression forces within the wall. As a result, the width of the effective

compression zone, b’, may be greater than the width of the beam flange, b

f

.

Figure 8. Methods for determining embedment capacity.

cc L/2 = aL/2 = a

e

V

u

V

u

spalled cover

concrete

C

C

L

L

L - c

2

e

L - c

e

L

e

L

e

C

b

C

b

C

f

C

f

f

= 0.003

f

= 0.003

b

b

x

f

x

f

1/3(L -c)

e

x

b

x

b

0.85f

c

0.85f

c

f

c

b

<

2

.

5

b

f

b

f

C

b

3/4(L -c)

e

assumed

strain

distribution

assumed

stress

distribution

(a) Mattock and Gaafar (1982)

(b) Marcakis and Mitchell (1980)

spalled cover

concrete

load spreading

Calculation of Embedment Length

The embedment length, L

e

, required to develop the coupling beam shear, V

u

, may be

calculated by either of the two aforementioned methods.

For steel coupling beams, V

u

is taken as 1.5 times the plastic shear capacity of the steel

member, V

p

:

wfypu

tthFxVV )2(6.05.15.1 (8)

For “flexure critical” steel beams, the shear capacity may be taken as the shear

corresponding to the development of the plastic moment capacity:

L

M

L

M

V

r

s

r

u

2

5.1

2

35.1

(9)

The contribution of concrete encasement to the shear capacity of composite coupling

beams needs to be taken into account when the embedment length is calculated (Gong et al.,

1997 and 1998):

s

dfA

dbftthFVVV

ys

wcwfyRCpu

'167.0)2(6.056.1)(56.1 (10)

Additional “transfer bars” attached to the beam flanges (as discussed above) can

contribute to the capacity of the embedment. The embedment length computed by Mattock

and Gaafar (1982) or Marcakis and Mitchell (1980) can be modified to account for the

additional strength. To ensure that the calculated embedment length is sufficiently large to

avoid excessive inelastic damage in the connection region, it is recommended that the

contribution of “transfer bars” be neglected.

DESIGN EXAMPLES

Examples included in Appendix A illustrate the designs of coupling beams discussed in

this paper. Details of the beam and wall designs contained in Appendix A may be found in

Harries (1996) and Harries et al. (1998). These examples are intended for illustrative

purposes.

CONCLUSIONS

The efficiency of coupled wall systems to resist lateral loads is well known. In order for

the desired behavior of the coupled wall system to be attained, the coupling beams must be

sufficiently strong and stiff. The coupling beams, however, must also yield before the wall

piers, behave in a ductile manner and exhibit significant energy absorbing characteristics.

This paper reviews the current state of the art for the design of conventional reinforced

concrete, diagonally reinforced concrete, steel and composite steel-concrete coupling beams.

Although not exhaustive, critical aspects of the design of these systems are presented.

In the case of diagonally reinforced concrete coupling beams, only included in ACI

provisions for the first time in ACI 318-99, considerable additional guidance in design and

detailing may be found in following Canadian (CSA) or New Zealand Standards (NZS).

Steel coupling beams may be efficiently designed and detailed as link beams in

eccentrically braced frames. When encased in concrete, considerable stability is imparted to

the beam allowing simplified details to be used.

The proper design of any coupled system must ensure the desired hierarchy of hinging. In

addition it is critical that hinges in the coupling beam remain in their clear span and do not

propagate into the walls. As such proper design of the embedment region is critical. There are

a number of methods of determining the required embedment length and details of steel or

composite coupling beams. All of these methods are based on an assumed rigid body rotation

of the embedment region of the coupling beam.

APPENDIX A – COUPLING BEAM DESIGN EXAMPLES

Salient points of the design of four types of coupling beam are presented. The beams are

located in eighteen storey prototype structures located in Vancouver, Canada. The plan of

these prototype structures is shown in Figure A1. Seismic design forces were determined

using the 1995 National Building Code of Canada equivalent lateral force method. Base

shears were determined to be in the range of 0.036W to 0.041W depending on the prototype

structure. These same base shears would be found by applying the NEHRP equivalent lateral

force method using R=5.8 to 6.7. NEHRP recommends a value of R=6 for either prototype

structure.

Figure A1. Plan of prototype structures.

Each of the 3000 mm long prototype beams are designed as either conventionally

reinforced concrete coupling beams or “flexure critical” steel coupling beams. Similarly, the

1200 mm beams are designed as both diagonally reinforced concrete coupling beams and

“shear critical” steel beams.

CONVENTIONALLY REINFORCED CONCRETE COUPLING BEAMS

The final details of the 3000 mm long prototype conventionally reinforced concrete

coupling beam are shown in Figure 2a. The design forces on these beams were determined to

be 467 kN in shear and 700 kNm in flexure. The span-to-depth ratio, L/d, for these beams is

4.3 and the shear stress is approximately

'f42.0

c

, thus diagonal reinforcement is required.

The design progressed according to standard beam design methods of ACI 318-99 or CSA

A23.3-94.

Typical floor plan

9000

14 000

1200

64006400

500

Fully (heavily) coupled

base shear = 0.036W

Prototype cores

Note: wall thickness = 500 mm (storys 1 to 5)

= 400 mm (storys 6 to 18)

9000

14 000

3000

55005500

500

Partially (lightly) coupled

base shear = 0.041W

900 mm x 900 mm "barbells"

(storys 1 to 5)

800 mm x 800 mm "barbells"

(storys 6 to 18)

42 000

6000 (typ.)

3

0

0

0

0

6

0

0

0

(

t

y

p

.

)

550 x 550 column (typ.)

200 mm thick flat plate

N

5500

end bays

DIAGONALLY REINFORCED CONCRETE COUPLING BEAMS

The final details of the 1200 mm long prototype diagonally reinforced concrete coupling

beam are shown in Figure 2b. The design shear force for these beams was determined to be

770 kN. The span-to-depth ratio is 1.7 and the shear stress is approximately

'f25.0

c

, thus

conventional reinforcement is permitted. The design progressed according to standard beam

design methods of ACI 318-99 or CSA A23.3-94:

Determine required dimensions in diagonally reinforced beam

Assume diagonal bar bundles consisting of 2 layers of No. 30 bars. Bar bundles,

including ties are 100 mm deep and 200 mm wide (see Fig. 2b).

d = 700 - 40 - 100/2 = 610 mm d’ = 90 mm

tan = (h - 2d’)/L

u

= 700 - (2 x 90)/1200 therefore: = 24

o

Selection of main diagonal reinforcement

Diagonal reinforcement is determined from Equation A1:

2

2783

24sin40085.02

770000

sin2

mm

xxxf

V

A

ys

n

s

(A1)

therefore use 4 - No. 30 bars in each diagonal (A

s

= 2800 mm

2

)

Embedment and confinement of diagonal reinforcement

Recommended embedment length = 1.5L

d

= 1.5 x 1480 = 2220 mm

Seismic hoops are required over entire length of diagonal with spacing not exceeding:

6d

b

= 6 x 30 = 180 mm

24d

b

(hoops) = 24 x 10 = 240 mm

100 mm

Thus, provide No. 10 closed hoops at a spacing of 100 mm over the length of the

diagonals.

“FLEXURE CRITICAL” STEEL COUPLING BEAMS

Ideally, the selection of a steel coupling beam must provide adequate stiffness for the

desired degree of coupling, while limiting overstrength. Furthermore, the steel section must

be narrow enough to fit within the concentrated wall reinforcement. For longer, “flexure

critical” coupling beams, these criteria can often be satisfied with standard rolled shapes. For

shorter spans, however, built-up sections may be required.

Details of the “flexure critical” coupling beam are shown in Figure A2a. For the 3000

mm prototype steel coupling beam the design shear was determined to be 597 kN,

corresponding to a moment of 895 kNm. For these forces, a W610 x 140 (W24 x 94; inch-

pound units) section with a 915 mm embedment region was designed.

At this point, it is important to verify that the section will indeed behave as expected. In

this case, that it is “flexure critical.” The factored shear capacity is determined to be 1340 kN

while the shear corresponding to the unfactored probable moment capacity is:

kN

x

xxx

c

L

ZFx

c

L

M

yp

734

40

2

3000

25103005.12

2

5.12

2

2

(A2)

Thus the beam is “flexure critical.”

Required embedment length

The required embedment length is determined from the following relation, (based on

Marcakis and Mitchell, 1980), where the factored shear resistance of the embedment, V

u

, is

taken as 1.35M

r

/2L

s

= 1.35 x 727 / 0.9 = 1090 kN (see Figure 8b).

)(

6.3

1

)(''85.0

cL

e

cLbf

V

e

ecc

u

(A3)

Because of the presence of a “barbell” at the wall toe, b’ may be taken as 2.5b =

2.5 x 230 = 575 mm. Taking b’ = 575 mm, e = (3000 + L

e

+ 40) / 2, and c = 40 mm. Solving

Equation A3 yields: L

e

= 915 mm.

It is necessary to ensure that there is adequate concentrated boundary element

reinforcement, A

sc

, to develop V

e

:

2

3206

40085.0

1090000

mm

xf

V

A

ys

e

sc

(A4)

This requirement is essentially satisfied, as this steel represents a small portion of what is

required in the region of concentrated reinforcement, and there will typically be some

overstrength provided. Details of the embedment region of this prototype are shown in Figure

6a.

At the beam-wall interface, double-sided stiffeners are provided to simplify the formwork

and to aid in the force transfer between the steel beam and the concrete wall. Due to the high

moment to shear ratio and the need to ensure lateral stability, single-sided stiffeners are

located 335 mm from the face of the wall.

The required embedment length may also be determined using the equation proposed by

Mattock and Gaafar (1982):

e

ef

f

wall

cu

L

a

Lb

b

t

fV

88.0

22.058.0

'05.4

1

1

66.0

(A5)

If the bearing area width, t

wall

, is assumed to be equal to the entire 900 mm width of the

“barbell,” the required embedment length is found to be 725 mm. It is noted that if the width

of the effective bearing area is limited to 2.5b

f

(Marcakis and Mitchell 1980), the required

embedment length is increased to 870 mm.

Figure A2. Prototype steel coupling beams.

“SHEAR CRITICAL” STEEL COUPLING BEAM

For the prototype structure having the 1200 mm long steel coupling beam and a high

degree of coupling, the coupling beam was designed to yield in shear while remaining elastic

in flexure. A 600 mm deep built-up section having an embedment length of 1000 mm was

selected. The high shear demands on this beam necessitated the provision of single-sided

stiffeners spaced at 300 mm over the entire span, these were continued into the embedment to

provide stability in the event of excessive spalling. Figure A2b shows details for this “shear

critical” steel coupling beam. Details of the embedment region are shown in Figure 6b.

The design forces for this beam were 764 kN and 458 kNm for shear and moment,

respectively. The design criteria called for a depth of no more than 600 mm (similar to the

“flexure critical” and concrete beams). The resulting built-up section has dimensions shown

in Figure A2.

The required embedment length is determined from Equation A3 to be 1000 mm, similar

to that required for the longer “flexure critical” beam. Similarly, using Equation A5, the

required embedment length is determined to be 780 mm.

3000 mm

335 mm

915 mm

1200 mm

50 mm

1000 mm

300 mm typ.

(a) “flexure critical” prototype

(b) “shear critical” prototype

W610 x 140

built-up section

d = 617 mm

b = 230 mm

t = 22 mm

t = 13 mm

f

w

d = 600 mm

b = 225 mm

t = 25 mm

t = 9 mm

f

w

COMPOSITE COUPLING BEAMS

The previously designed steel coupling beams are assumed to be encased as shown in

Figure A3. The longitudinal reinforcement consists of four 10M (diameter = 11.3 mm) corner

bars and two 10M midside bars. The transverse reinforcement consists of 10M closed hoops

spaced at 390 mm. For simplicity, it is assumed that the steel beams are identical to those

used previously, i.e., the contribution of concrete is ignored when the steel coupling beams

are sized. This would often be the case for composite coupling beam design. The dimensions

of the steel beams are given in Figure A2.

Figure A3. Cross section of composite coupling beam.

The overall dimensions for both the 3000 mm and 1200 mm long coupling beams are h =

780 mm and b

w

= 390 mm. These dimensions were obtained based on assuming that the

longitudinal bars would be at about 25 mm from the corners of the flanges, and by providing

40 mm of cover to the transverse reinforcement. The concrete compressive strength is taken

as 35 MPa, and the yield strength of the longitudinal bars is taken as 420 MPa. The required

embedment length for the flexural and shear critical beam is calculated in the following.

Flexural Critical Beam

The shear capacity of the coupling beam is determined as the sum the contribution from

the steel beam and that of the of the concrete encasement (Equation 9):

s

dfA

dbftthFVVV

ys

wcwfyRCpu

'167.0)2(6.056.1)(56.1 (A6)

Using an effective depth, d = 780 – 81.5 = 698.5 mm, the ultimate shear capacity of the

coupling beam is found to be 2747 kN.

The required embedment length may be calculated using Equation A5 (Mattock and

Gaafar 1982). Similar to the previous example, t

wall

is taken as 900 mm. The value of b

f

is

taken as the flange width of the steel coupling beam. The value of “a” is taken as half of the

3000 mm span. By applying Equation A5, the required embedment length, L

e

, is found to be

1285 mm.

The required embedment length determined using the method proposed by Marcakis and

Mitchell (1980), given in Equation A3 is found to be 1690 mm. Similar to the previous

example, this assumes that b’ is taken as 2.5b = 2.5 (230) = 575 and e = 0.5L + c+ 0.5(L

e

-c) =

1520 + 0.5L

e

(see Figure 8).

760 mm

No. 10 corner

and midside bars

No. 10 closed hoops

or hairpins @ 390 mm

390 mm

It is suggested that since the beam is “flexure critical” the value of V

p

may be replaced

with the value of the shear required to achieve the ultimate flexural capacity of the beam,

shown above to be equal to 1.35 x 2M

r

/L

s

= 1.35 x 727 / 0.9 = 1090 kN. In this case, the

required shear capacity of the embedment becomes:

s

dfA

dbf

L

M

V

L

M

V

ys

wc

r

RC

s

r

u

'167.056.1

2

5.156.1

2

35.1

(A7)

For the “flexure critical” beam designed above, the required shear capacity of the

embedment region is V

u

= 1090 + 1.56(420) = 1745 kN. The require embedment length is

found to be 970 mm (Equation A5) or 1280 mm (Equation A3).

Shear Critical Beam

For the “shear critical” beam, the method is identical to that used for the “flexural

critical” composite beam although only Equation A6 is considered. The design shear force

for the embedment, V

u

is 2,026 kN. Based on Mattock and Gaafar’s model (Equation A5),

setting t

wall

= 500 mm, the necessary embedment length is 1025 mm. When using the model

proposed by Marcakis and Mitchell (Equation A3), the value of b’ is approximately 400 mm,

limited by the width of the confining steel (see Figure 8b), and e = 620 + 0.5L

e

. Based on this

method, the value of L

e

is 1350 mm.

The preceding calculations demonstrate the effect of including the capacity of the

concrete encasement. Although the embedments are longer, it has been shown that encased

steel coupling beams do not require the significant stiffener details (see Figure A2) in order

to ensure adequate ductility and stability.

DESIGN OF WALL PIERS

The design of the wall piers is then carried out for flexure using axial loads resulting from

full plastification of all coupling beams. Figure 6 gives some indication as to the resulting

boundary element reinforcement required for these prototype structures.

ACKNOWLEDGEMENTS

The authors would like to gratefully acknowledge the input and support of members of

American Concrete Institute (ACI) Committee 335 – Composite and Hybrid Structures and

Professor Denis Mitchell of McGill University. The design methods and recommendations

are based on several projects funded by the U.S. National Science Foundation (Grants No.

BCS-8922777 and BCS-9319838 with Dr. Shih Chi Liu as the program director) and the

Natural Science and Engineering Research Council of Canada. Any opinions, findings, and

conclusions or recommendations expressed in this paper are those of the authors and do not

necessarily reflect the views of the sponsors.

NOTATION

a = shear span of coupling beam, typically taken as L/2

A = cross sectional area of wall pier

A

s

= area of transverse reinforcement in concrete encasement

A

sc

= area of concentrated reinforcement in boundary element of wall pier

A

vd

= area of reinforcement in each group of diagonal bars

A

w

= area of web of steel beam

b

'

= effective width of the concrete compression block

b

f

= coupling beam flange width, also the bearing width of the embedment

b

w

= web width of concrete beam or concrete encasement

c = depth of concrete cover

C

d

= deflection amplification factor defined by NEHRP (BSSC 1997)

C

f

= resultant bearing (compressive) force in embedment

d = distance from the extreme compression fiber to centroid of longitudinal tension

reinforcement in concrete section

e = eccentricity from midspan of beam to center of embedment

E = Young’s modulus of coupling beam or wall pier

f

c

’ = specified concrete compressive strength

f

y

= yield strength of reinforcing steel

F

y

= yield strength of steel beam

G = shear modulus

h = overall depth of coupling beam

I = moment of inertia of coupling beam or wall pier

k = stiffness factor accounting for shear deformations in flexural members

L = effective length of coupling beam

L

clear

= length of coupling beam measured from face of wall piers

L

e

= embedment length of the steel coupling beams inside the wall piers

L

wall

= distance between centroidal axes of the wall piers

M

r

= moment capacity of steel beam

s = spacing of transverse reinforcement

t

f

= flange thickness of steel beam

t

w

= web thickness of steel beam

t

wall

= thickness of wall pier

v = shear stress

v

u

= factored shear stress

V

c

= factored shear force resisted by concrete

V

n

= nominal shear capacity of coupling beam

V

p

= plastic shear capacity of coupling beam

V

r

= factored shear capacity of coupling beam

V

RC

= shear force resisted by reinforced concrete encasement

V

u

= factored shear force on section

x

b

= length of compression block at back of embedment

x

f

= length of compression block at front of embedment

= angle of inclination between diagonal reinforcement and longitudinal reinforcement

c

= coefficient defining relative contribution of concrete strength to wall strength

or

1

= ratio of the average concrete compressive strength to the maximum stress

p

= shear angle

= shape factor

p

= plastic interstory drift angle

e

= elastic interstory drift angle computed under code level lateral loads

n

= ratio of area of distributed reinforcement parallel to plane of shear resistance to the

concrete area perpendicular to that plane

c

= material resistance factor for concrete = 0.60

s

= strength reduction factor for steel = 0.90

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