Response of Large Panel Precast Wall Systems: Analysis and Design

M. Reza Kianoush Ph.D., P.Eng.

Professor Department of Civil Engineering Ryerson Polytechnic University

Toronto, Ontario Canada

Mostafa Elmorsi Ph .D. Cand idate Department of Civil Engineering McMaster University Ham ilton, Ontario Canada

Andrew Scanlon, Ph.D., P.E.

90

Professor Department of Civil and

Environmental Engineering Pennsylvania State University University Park, Pennsylvania

A general design procedure proposed by Clough is applied to a precast coupled shear wall structure located in UBC Seismic Zones 2, 3 and 4. Both slender and deep coupling beams are considered. Inelastic static analyses using the finite element method are carried out to compare the results with Clough 's method. This study showed that large panel structures with deep coupling beams behave in a satisfactory manner in all seismic zones. In comparison, coupled walls with slender beams showed a less satisfactory response. The application of Clough 's method is illustrated with the aid of a fully worked design example.

L arge precast wall panel construction has been used extensively in Europe over the past 50 years. The use of a large panel system has several advantages in­

cluding high quality of the product and a short construc­tion time. This makes precast concrete buildings both eco­nomically and aesthetically competitive with other types of construction.

The excellent performance of the large panel system was observed in the 1977 Romanian earthquake and the 1988 Armenian earthquake as reported by Fintel .U A re­port by Ghosh3 on the 1995 earthquake in Kobe indicates that structures employing precast concrete shear walls performed very well. These structures , mostly mid-rise apartment buildings , suffered no damage except some minor cracking and spalling of concrete near the founda­tion . Despite this good performance, the North American building codes limit the use of these types of systems in high seismic zones.

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LITERATURE REVIEW The behavior of precast wall panels

has bee n s tudied a na ly ti ca ll y by Becker and Llorente; Schricker and Powell ,5 Pall and Marsh,6 and Pekau and Hum.' These studies have been limited to simple walls (i.e., without coupling beams) or coupled walls with vertica l jo ints onl y. Ki anoush and Scanl o n8

·9 s tudi ed th e in e las ti c re­

sponse of precast wall panels with coupling beams.

There have been some experimental tests of precast panel wall systems in the past. Smal l scale models have been tested by Oli va and Shahrooz 10 and Oliva et a!." The strength of wall panel joints has been experimentally studied by Soudki et a!. , "·14 Rizkalla et a!. , 15

Foerster et al. , 16 Hutchinson et a!. ," Mattock and Hawkins, 18 and Shiohara et al .19

Precast panel systems can be used in seismic areas provided that the connec­tions behave similarly to those used in monolithic cast-in-place concrete con­struction. Fig. 1 shows a "platform" connecti on detail that is co mmonly used in North America. For this type of detail , the monolithic des ign concept implies that the connections must be strong enough to form plastic hinges at the base of the wa ll. Beca use th e strength and stiffness of the horizontal connec ti ons are considerabl y lower than the wall panels, precast walls re­quire considerable detailing in order to meet the above requirement. This can be very difficult from an economic and construction perspective.

An alternati ve approach is to apply the provisions for cast-in-place con­crete structures to large panel systems, provided th at those prov isions are modified to meet the seismic strength requirements consistent with the avail­able ductility of precas t structures . The lateral force provisions of existing codes can be used if the inelastic de­fo rmation demands in the horizontal connections are accounted for.

The advantage of using coupling beams in precas t wall sys tems has been discussed by Kianoush and Scan­lon.9 The study showed that coupled wall s prov id e bette r stru c tural re­sponse compared to isolated walls. An optimum des ign approach can be a combination of energy diss ipation in

November-December 1996

Paper or Plastic Dam

Plank

Plastic Bearing Strip

.,_-+--Vertical Reinforcement

Fig. 1. Interior wa ll to floor platform connection detail.

LEGEN D:

• p lastic hinge

S length af coupling beam

H heig ht of wall panel

W width of wa ll pan el

Dp = plastic d isplacement at t op of bui lding

h ve rtical displacement

Op•W =-H-

Op 01=H

Fig. 2. Kinematic model of a one-story stru cture subjected to plastic d isplacement, Dp.

91

coupling beams with some controlled amount of inelastic action in the hori­zontal connections.

Clough20 has proposed a practica l design approach that specifically con­siders both strength and inelastic de­formati on of members. Fig. 2 shows the di splaced configuration of a one­story coupled wall structure with slen­de r co uplin g beams. Based o n Clough' s des ign method, and using the predicted inelastic displacement at the top of the structure, hinge rotations and vertical and horizontal displace­ments can be obtained by applying simple trigonometric relationships.

In th e c urre nt s tud y, Cl oug h 's method has been modified and applied to a ten-s tory precast coupled shear wall structure. Results of this study are presented in terms of di splaced config­uration, gap opening and its di stribu­tion over the length of the horizontal connection, and beam forces and duc­tilities over the height of the structure. These resul ts are based on the assump­ti on that the structure is located in Seismic Zones 2, 3 and 4 as specified by the UBC.2

'

In the current investigation, analyti­cal studies are also carried out to com­pare the results with Clough's method. Thi s is es tablished by co mpari ng Clough's results with those of the non­linear finite element method.

CLOUGH'S DESIGN METHODOLOGY

In thi s secti on, the basic assump­tions, design methodology, and appli­cations to coupled wall s wi th both slender and deep beams are discussed.

Basic Assumptions

It is assumed that the wall rotates about its corner. The gap opening is the only mode of deformation for the horizontal joints while s liding has been neglected. Previou s analytical studies by Kianoush and Scanlon8

·9 on

the behav ior of a ten- story precas t wall sys tem showed that fo r walls contai ning post-tensioning bars as ver­tical continuity, the mode of failure was primarily due to slip with some limited gap opening occurring at the hori zo ntal co nn ections. Th e North American "platform" type of connec-

92

I Detennine Natural Period I i

Perform Static Analysis

+ I Compute UBC Base Shear I Compute ESD I

ComputeR Value

R= ESI:! Rdesign

~esign < R J Select R Value for Design I. Rdesign = R for higher forces and for normal forces lower ductility I (Rdesign) I and ductility

Compute Base Shear for Design

V design = ESD Factored UBC

Compute Global Inelastic Displacement

D = Dy ( &:.:_! ) p 2

• [ Perform Kinematic Analysis J

Connector Performance Requirement I

Fig. 3. Simplified des ign proced ure for earthquake-resistant precast concrete buildings (Ciough20

) .

ti on was used in the in ves ti gati on. However, when mild reinforcing bars were used instead of post-tensioning as vertical continuity, the mode of fail­ure changed to rocking with a small amount of slip occurring mainly at lower story levels. Due to the effect of aggregate interl ock, interface shear transfer and dowel action of reinforce­ment, the amount of slip was reduced considerably; the maximum calculated amount of slip in thi s case was less than 1 mm (0.039 in.).

A study by Oli va et a!. 1' suggests th at a hori zontal connec ti on detail with adequate shear keys at the corner of the wall panels will circumvent a shear slip mechanism. This connection detail was used in their experimental investigation on the behav ior of pre­cast wall systems. An experimental

study by Soudki et a!. 14 also showed that the use of grouted shear keys sig­nificantly enhances the shear resis­tance of the horizontal joint in precast wall panels.

Design Methodo logy

The design methodology proposed by Clough for jointed precast concrete structures can be summarized by the simplified flowchart shown in Fig. 3. Major components of the methodol­ogy are described below.

As the initial step, the fundamental period of the structure should be eval­uated. The code-specified base shear using static analys is must be es tab­lished. This is achieved using the two load conditions shown in Fig. 4 and described as follows:

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Seismic Bose Shear

ATC Seismic Load

ESD

__ [~Elasti c Strength Demond)

.---Pi:-- Elastic Sys tem

Design Yie ld Strength of Elosto -Pios ti c System

Factored UBC Seismic Load (1 .4*UBC Load)

1:=::=::=::=::=::=::=::=::=::=::=::=::=::=::=::=::=::=::=::1

!================-;; =======~=======j 1---- ---- ---------::-::-::-::-::-::-::-::-::-:-::-::-::-::-::-::-::-::-::-1 - - ------------------

Displacement of Top of Building 1--------Dp -----~-1

Globa l Inelastic Displacement

Fig. 4. The two loading conditions and the equal-energy concept for estimating max imum se ism ic displacements of an elasto­plastic system (C iough20

) .

1. The usual load condition describ­ing the structure 's required ultimate capacity (Fy). The UBC" strength cri­terion is used to define the ultimate ca­pacity. Assuming that the behavior of the structure is elasto-plastic, the ulti­mate capacity is the same as the yield capacity.

2. An auxiliary load condition de­scribing the maximum seismic force that the structure would experience if it had infinite strength, referred to as the "Elastic Strength Demand" (ESD). Loads defining the ESD are derived from ATC-322 base shear formulas using an R value of 1.0.

A convenient approach to perform static analyses is to work initially with an equivalent static load case of "unit" magnitude . The "unit" base shear would be distributed over the struc­ture' s height according to UBC provi­sions. Based on the results of the static analysis, seismic loads can be deter­mined and the value of R, the response modification factor, can be established as follows:

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ESD R=----­

Factored UBC (1)

In selecting an appropriate R value, factors such as the observed system performance durin g earthquakes, damping, redundancy and other effects mu st also be considered. Using the calculated R value, if higher forces and reduced ductility are required, R design < R can be used. Otherwi se, R design = R is appropriate. The R design

value is used to calculate the new base shear for design as shown below:

V _ ESD design - R (2)

design.

The next step in the design process is to predict the global inelastic dis­placement of the structure. Thi s is achieved using the equal energy con­cept desc ribed by Newmark and HalF3 For structures with a fundamen­tal period in the range of 1/s to 112 sec­ond, Newmark and Hall have shown that peak internal strain energies of elastic and elasto-plastic SDOF sys-

terns subjected to seismic forces are approximately equal. This fact can be used to estimate the maximum inelas­tic di splacement of an elasto-plastic SDOF system.

Fig. 4 compares the force-displace­ment relation ship s for elastic and elasto-plastic SDOF systems. For ei­ther the elastic or elasto-plastic sys­tems , the area under the force -dis­placement curve is the strain energy absorbed by the system. The elastic system is under the force equal to the ESD. The elasto-plastic system is di s­placed by an amount Dep• the maxi­mum inelastic displacement the struc­ture wou ld experience under the de sign earthquake , ass uming its strength is less than the ESD. By the "equal energy" concept of Newmark and Hall, D ep is the value that makes a trapezoidal area equal to the triangular area of the elastic system under the displacement De.

As shown in Fig. 4, Dep consists of two components. One is the elastic di splacement, D , which is the dis-

93

placement the structure experiences before yield. This value is obtained by scaling upward or downward the re­sults of the unit load static analysis ac­cording to the code-specified base shear. The other component is the plastic displacement, DP, which can be easily calculated because the shaded areas are equal.

Using the predicted inelastic dis­placement at the top of the building, a kinematic analysis can now be per­formed to determine the correspond­ing deformations at individual joints.

In the kinematic analysis, inelastic displacements are significantly larger than the structure's elastic deforma­tions. Thus, motions of vertical ele­ments (such as walls, or similar stiff elements) can be approximated as rigid body rotations about their respec­tive foundations, with concentrated hinge lines or hinge points where they intersect the horizontal elements (such as floor and roof systems).

As was shown in Fig. 2, calculations of hinge rotations and vertical and horizontal displacements that corre- h, J spond to the lateral plastic displace-ment at the top of the structure can be made by applying simple trigonomet-ric relationships.

Applications to Coupled Walls With Slender Beams

Fig. 5 shows the deformed shape of a ten-story coupled shear wall building with slender beams connecting the two walls. It should be noted that base ro­tations and vertical and horizontal dis­placements are related by applying the approximation of rigid body rotations. This approximation is justified by the fact that the elastic deformations are insignificant when compared with the inelastic deformations.

Both the wall panels on the left and the right are treated as a rigid body and they experience the same amount of displacement and rotation. Plastic hinges are assumed to form at both ends of a beam element and rotation of the wall panel is assumed to occur about the panel edge at the bottom. The ductility at each floor is calcu­lated using a gap opening ratio of 40, 30, 20, and 10 percent at the base, first, second, and third floors, respec­tively (Clough20

).

94

LEGEND:

• = plastic hinge

h, vertical displacement at ground level

h2 vertical displacement at first level

h3 vertical displacement at second level

h4 vertical displacement at third level

Fig. 5. Kinematic analysis with four joint open ings.

Calculations of hinge rotations and vertical and horizontal displacement corresponding to the lateral plastic dis­placement at the top of the building can be made by applying simple trigonometric relationships. Appendix A(a) summarizes the general equa­tions and values of the rotation angles for this specific case. Details on the derivation of these equations are ex­plained by Yu.24

Calculation of Beam Ductility Demand - The ductility demand can be defined in terms of rotation angles. In Fig. 6, solid lines show the actual moment-rotation curve for an element, while dashed lines show the approxi­mate curve used for this study. When

a beam element is subject to yield mo­ment, My, at both ends, the ductility demand of a beam element can be de­fined in terms of rotation as:

where eep = maximum rotation ey = yield rotation ep = plastic rotation

Note that 8P for each beam element can be calculated using the equal energy concept described earlier. The yield ro­tation, 8y, is dependent on beam stiff­ness and yield moment. When a beam is subject to yield moment, My, and expe­riences a rotation angle of ey at both

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My

Mer

Moment

- ---;r--­

I I

I I

/1

Elcr 0y

Rotat ion Angle ( 0)

0ep

LEGEND:

0cr = crock rota t ion

0y approx imate yield ro tati on

0y actual yie ld rotation

0ep = maximum ine las t ic rotat ion

0p = plastic rotat ion

Mer = crock moment

My = yie ld moment

Fig. 6. Moment-rotation curve for an element.

ends, the yield rotation can be deter­mined using the following expression:

() =_!::_M (4) y 6EI y

y

where E = modulus of elasti city f y = moment of inertia L = length of beam In thi s study, f y was calculated on

the basis of the uncracked section, i.e., f y = 18• This leads to a calculated ()Y

value less then the actual beam rota­tion at yield e;, which could be calcu­lated using the cracked transformed moment of inertia. Further details of the application to slender couplin g beams are provided by Yu.24

November-December 1996

Applications to Coupled Walls With Deep Beams

Fig. 7 shows the deformed shape of a one-story coupled shear wall building with diagonal bars connecting the two walls. For clarity, only one diagonal bar is shown. This representation is based on the fact that the behavior of deep beams is mainly dependent on their di­agonal reinforcement when loaded be­yond the elastic range as described by Barney et al. / 5 Paulay and Binney / 6

and Paulay and Santhankumar.27

The ductility of a coupling beam is defined here as the ratio of the maxi­mum extension of the diagonal rein­forcement divided by the extension at the yield level. The max imum exten-

sion is evaluated geometrically by cal­culating the difference between the ex­tended diagonal length and its initial length by using the following relations:

L inirial = [(H - £)2 + SZ]112 (5)

l x = W + S- Esin()1 (6)

l y= Ecos()1 (7)

2x = Wcos()1 - Dp (8)

2y = h + Hcos()1 (9)

L deformed = [(2x- l x)2 + (2y- 1 y)2]

112

(1 0)

Extension = L deformed - L inirial

( 11 )

where lx. 2x = X coordin ates of th e two

nodes of the extended diago­nal member

l y, 2y = Y coo rdin ate s o f th e two nodes of the extended diago-nal member

1 ()1 = angle of base rotation

Once the extension is calculated for the member, the beam ducti lity can be obtained as fo llows:

where

Extension J.L=----

L1y ( 12)

(13)

L1y = yield extension of the diagonal reinforcement

f y = yield stress of the diagonal rein­forcement

L = total length of the diagonal rein­forcement

E =modulus of elas ticity of rein­forcement

Similar relationships can be deri ved for the case of a building taller than one story. For a ten-story building, the plastic rotations are expected to di s­tribu te among the lowest hori zontal floors rather than concentrating at the base only . The di stribution of gap openings is ass umed to be similar to those of slender beams.

Fig. 8 shows the di splacement pat­tern of a ten-story building. The duc­tility at each floor is calculated using the same di stribution of gap openings as above . The ex tended length of the diagonal bars at the first four levels is

95

Dp l I Diagonal Rebar

X

h

Fig. 7. Kinemati c analysis for a one-story build ing.

h J 1

LEGEND:

• = plastic hinge

h; vertical displacement at level i

0; angle of wall rotation at level i

Fig. 8. Kinematic analysi s for a ten-story build ing.

96

LEGEND:

• plastic hinge

S = length of coupling beam

H height of wall panel

W width of wall panel

Dp = plastic displacement at top of wal l

h vert ical displacement

d depth of coupling beam

E clear height of wall pone\ from bose to the bottom of beam reinforcement

01 = angle of base rotation

given in Appendix A(b). Because gap openings are assumed to concentrate only at the base level and the first three floor levels, the deformations of the diagonal bars at the other levels are found to be the same as that of the fourth floor.

APPLICATION OF CLOUGH'S

DESIGN METHOD TO A TEN-STORY COUPLED

WALL STRUCTURE This section provides a description

of the structure, the behavior of cou­pled walls with deep beams and a compari so n of co upl ed walls with slender beams vs. deep beams.

Description of Structure

Deta il s and dim e ns io ns of th e structure se lected for this study are shown in Fig. 9 . The structure is a large panel coupled shear wall with horizontal connections at each fl oor level. The coupling beams used are either deep beams of 1000 mm (39.4 in .) depth or slender beams of 700 mm (27 .6 in .) depth . It is assumed that the coupling beams are precast with wall panels. Thi s is the same structure that was used in previous studies by Ki anou sh and Scanlon.9

The structure is assumed to rest on a rigid base and the floor slabs are con­sidered infinitely rigid.

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Coupling Beom

(see Details A ond 8

Horizontal Connection

)

/ ~

""' / ~

~ -

-

-

'-------

~

f---

r----

r----

Wall Panel

32.5 m

-r 3 ,;,

_l

(a) Elevat ion

(b) Plan

Fig. 9. Detai ls of the coupled wall system se lected for the study.

The structure is designed for three different seismic zones with zonal ve­locity ratios of 0 .2, 0.3 and 0.4 as specified by the UBC. Details of the design method will be illustrated with the aid of a design example.

The fundamental period of the struc­ture was determined using the

November-December 1996

SAP-IV28 computer program and found to be 0.32 second. The effect of the horizontal joints and possible soften­ing that may occur as a result of a gap opening were considered by assigning the strength and stiffness of the joints to be 50 percent of wall panels (Back­ler and Baylick29

) . Because the funda-

Be om 0 II II II II I II II II II IEJ Wall Panel

Detail A - Slender Beam

Main reinforcement

5LliHiMia Detail B - Deep Beam

1.8 m (slender beams)

1.0 m (deep beams)

mental period of the structure is be­tween 1/s and 0.5 second, the equal en­ergy concept can be used. In this pe­riod range, the effects of higher modes of vibration are small and the SDOF approximation can be used, which will provide sufficient accuracy for practi­cal purposes.

97

(inch) Qnch) 0 0.04 0.08 0.12 0.16 0 0.4 0.8 1.2 1.6 2.0

10 10

9 9 .. -····

8 -- v=0.2 ............ v=0.3 8 -- v=0.2

············ v=0.3 7 --- - v=0.4 7 ---- v=0.4

~ 6

~ 5

Ci5 4

~ 6

~ 5

Ci5 4

3 3

2 2

1

0 0 0.5 1.5 2 2.5 3 3.5 4 4.5 5

Gap Opening (mm) 0 10.16 20.32 30.48 40.64

Horizontal Oellection (mm) 50.8

(a)

(kip) 0 40 80 120 160

10

9

8 --v-0.2 ' ............ v=0.3

7 ---- v=0.4

1 6 ... 5

f 4

3

1 l

I i

10

9

8

7

1 6 ... 5

f 4

3

--v=0.2 ............ v-0.3 - - -- v-0.4

2 -··' I .. -·· / ...... -······_····_· ______ ./ 2

1 ········· --- -- 1 0 0

0 100 200 300 400 500 600 700 800 900 Coupling Beam Forces (kN)

0 0.5

(c)

Fig. 10. Response of walls with deep beams.

Behavior of Coupled W alls With Deep Beams

Fig. lO(a) shows the distribution of gap opening over the height of the structure for the three seismic zones. The maximum amount of gap opening is 2.4, 3.6 and 4.7 mm (0.09, 0.14 and 0.19 in.) for v = 0.2, 0.3 and 0.4, re­spectively. In Clough's method, it is assumed that the wall will reach its ul­timate flexural capacity and rotate about its corner irrespective of the level of earthquake intensity. Choos­ing the wall corner to be the point of rotation appears to be conservative in terms of estimating the gap opening.

Fig. 1 O(b) shows the horizontal de­flection at each floor level. The maxi-

98

mum horizontal deflections are 33 and 45 mm (1.3 and 1.77 in.) and 58 mm (2.28 in. ) for v = 0.2, 0.3 and 0.4, re­spectively. Because the contribution of vertical continuity reinforcement to the overall stiffness of the structure was not considered, the displacements estimated using this method can be somewhat conservative.

Fig. 10(c) shows the internal forces in the coupling beams for the three seismic zones considered and Fig. I O(d) compares the beams' ductility demands over the height of the struc­ture . These results show that for all cases, the extensions in the coupling beams exceed the yield level. The amount of gap opening has a direct ef-

(b)

1 1.5 2 2.5 Beam Ductility Fac:tor

(d)

feet on the level of ductility in the coupling beams because the gap open­ing causes wall rotations and conse­quently creates extensions in the cou­pling beams. Therefore, it is important to determine the amount of gap open­ing in the horizontal joints as accu­rately as possible.

Further details on the behavior of coupled walls with deep beams have been reported by Elmorsi .30

Comparison of Coupled W alls With Slender Beams vs. Deep Beams

To study the behavior of coupled walls with slender coupling beams, it was assumed that the structure is lo-

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cated in Seismic Zone 3 as specified by the UBC, i.e., v = 0.3. The depth of coupling beams was assumed to be 700 mm (27 .6 in.) at every floor level and the yield moment of these beams was assumed to be 500 kN-m (368 .7 kip-ft). The behavior of cou­pled walls with slender beams is compared with those of deep beams as shown in Fig. 11.

Fig. 11 (a) shows the horizontal de­flection at each floor level. Coupled walls with slender beams show larger deflections. This is mainly due to the larger stiffness of deep coupling beams as compared to the slender beams. For the same reason, coupled walls with slender coupling beams show larger gap openings, as indicated in Fig. 11 (b).

Fig. ll(c) compares the beam duc­tility ratio of slender beams with deep beams. For slender beams, the ductil­ity ratio was determined based on ro­tations as described earlier. The very high ductility demands indicated for slender beams can be attributed, at least in part, to the calculation of yield rotation based on an uncracked sec­tion, which underestimates the rotation at onset of yielding.

MODIFICATIONS TO CLOUGH'S

DESIGN METHOD In Clough's design method, it is as­

sumed that wall rotation takes place about its corner. However, results from this study using the finite ele­ment method indicate that the wall ro­tation takes place approximately at a distance of one-third of the length of the wall panel from its comer. Studies by Becker and Llorrente, and Kianoush and Scanlon4

·9 have indi­

cated that wall rotation takes place at a distance of one-sixth to one-half away from the wall corner.

In thi s study, Clough ' s design ap­proach is examined by assuming that the wall rotation takes place at a dis­tance of one-third from its corner. Based on such an assumption, Fig. 12 shows the displacement pattern of a one-story coupled wall structure with diagonal bars connecting the two walls. This will be referred to as the modified Clough's method.

November-December 1996

(inch) 0 0.4 0.8 1.2 1.6 2.0

10 .·· 9 .-8

___ Deep Beams

··· ············· Slender Beams ...... ·· 7

1 6 ... ·· __J 5 -g4 .. ·

..·· 3

2 .. ··· .. ·"'

0 0 10 20 30 40 50 60

Horlzontal Deftectlon (mm)

(a)

(inch)

0 0.04 0.08 0.12 0.16 10

9

8 --- Deep Beams .. .. .. .... ....... Slender Beams

7

1 6 __J 5

g4 3

2

0.5 1.5 2 2.5 3 3.5 4 4.5 5 Gap Opening (mm)

(b)

10 .--.----------------------------------------,---,

9 !

87 --- Deep Beams l,',, ................ Slender Beams

!: I I 4 ~~

0 5 10 15 20 25 Beam Ductility Ratio

(c)

30

Fig. 11. Response of walls with slender beams and a comparison with deep beams.

ANALYTICAL INVESTIGATION

Analytical studies were performed to compare the results with Clough's design method. An inelastic model is used to model precast panel systems

with coupling beams under the ef­fects of static loading. The modeling of different components of the struc­ture is described as follows (see also Appendix B). • Wall Panels- Plane stress element

with linear elastic properties of

99

concrete and horizontal and vertical bars.

• Horizontal Connections - Concrete: inelastic springs with fi­

nite strength and stiffness in com­press io n and zero stre ng th a nd stiffness in tension to model a gap opening across the joint. The max­imum compress ive strength Cere) and the initial compress ive sti ff­ness (£0 ) for the horizontal con­nection is ass umed to be 50 per­ce nt of th e pa ne l stre ng th a nd stiffness .

- Vertical mild steel reinforcement: truss e lement with elasto-plastic properti es .

• Deep Coupling Beams - Inelasti c truss elements representing diagonal bars w ith yielding in tension and buckling in compression. Eight per­cent stra in hardening is ass ig ned based on experime ntal results by Barney et al. ,25 Paulay and Binney,26

and Paulay and Santhankumar.27

The computer program PC-ANSR3'

was used fo r the analys is. Thi s is a general purpose program for inelastic static and dynamic analysis. This pro­gram was originally developed at the Uni versity of California at Berke ley. For the current study, only an inelastic static a na lys is was co nducted . The only modi fication made to the pro­gram was the inclusion of a four-node rectangular plane stress element with eight degrees of freedom to represent the wall panels that are assumed to re­main linear elas ti c thro ug ho ut the analysis.

Selection of Parameters for the Ten-Story Coupled W all Structure

The finite element discretization and the material properties of the ten-story structure used for the inelastic static analysis are shown for a typical fl oor in Fig. 13. The amount of vertical con­tinui ty re inforcement selected is calcu­lated based o n the concept used by Clough,20 which assumes that the wall rotates about its corner and all the re­s isting mo ment is provided by the steel reinforcement and the dead load.

The amo unt of vertical re info rce­ment provided for the wall panels for the three seismic zones is also given in Fig. 13. It is assumed that the amount of reinforcement is similar throughout

100

h

Fig. 12. Deformed configuration of a one-story building based on Clough's mod ified method.

Vert ical W II P a one

Reinforcement r--- 1agona I R . f em orcemen / Joint Spring

Element

v; I/: v: I' v: I/: v: V:l/:: v: I/: ~ v: I/: v:: V: l/::

Vj 1/:: 1/: v: 1/:: 1/: [X V:: V:: IL 1/:: 1/:: 1/: 1/:: 1% ~ ILIL:IL IL: ~ iL: 1/:'i. 1/:'i. 1/:

r 3.2 5m 'i. 1/:'i. 'L'L

'i. v:v: iL: 'LV: 'L 'L 'L 'LV: v: I/: 10 v:v: Vj I/: v: v: I/: v: V:l/:: l I/: I/: 1/:: 10 I/: 1/:: lX V: l/: 1/: I/: 1/:: 1/:: ~ t;:..-::: IL:: V:: ~IL':;

3. 6m - --t-1 ,.:.;1 ·:.=8:.:..m:.-.rl ,.:.;1 m-"-l-,1""'. 8"-m---llr---- 3. 6m

Reinforcement

Concentrated Wall

Distributed

Beam Uniform

Wall Panels:

Diagonal Reinforcement:

Vertical Reinforcements:

Ratio % v=0.2 v=0 .3

~ 0.25% i.43%

I I 0.25% 0.62%

0.67% 1.00%

E=27,400 MPa (3,974 ksi) v=0.15

E=200,000 MPa (29,007 ksi) fy = 300 MPa ( 43.5 ksi)

E=200,000 MPa (29,007 ksi) f Y =300 MPo ( 43.5 ksi)

v: I/: ~ 10 V:::l/::

~ t::.-:::IL:: v:: 1/:V: 1/: 1/:'i. ~ 'L 'L 'i. v: '/ v:

(C:;

10 1/::1/:: v: 1/::1/:

v=0.4

2.86%

0.62%

1.40%

Connection Springs: K=6,400,000 N/mm (36 ,547 kip/in) fc = 15 MPa (2.2 ksi)

Fig. 13. Finite e lement discretization and properti es of the ten-story structure.

PCI JOURNAL

the height of the structure. This in­cludes concentrated reinforcement placed at the corner of the walls and distributed reinforcement placed uni­formly at other locations within the walls. In this study, deep coupling beams are used. The design of the di­agonally reinforced coupling beam was based on a statically determinant model.

COMPARISON OF ANALYTICAL RESULTS

WITH CLOUGH'S DESIGN METHOD

For the finite element analysis (FEA), the ten-story structure de­scribed above was analyzed using a static analysis. The same lateral loads used in Clough ' s method were applied to the structure at each floor level. These loads were determined based on the UBC equivalent static analysis. The results of the FEA are compared with those obtained from Clough's method. A comparison is also made with the results obtained from the modified Clough's method, which as­sumes that the wall rotation takes place at a distance of one-third from its corner.

Gap Openings

Fig. 14 shows the distribution of the gap opening over the height of the structure as predicted by Clough's method, the modified Clough's method and the FEA for three seismic zones. The maximum gap opening using the FEA is almost the same for all three cases. This is mainly due to the different amounts of reinforce­ment used in the structure in the three seismic zones.

The amount of gap openings deter­mined using Clough's method is greater than those obtained from the FEA for all cases. However, the re­sults obtained using the modified Clough ' s method are closer to the re­sults using the FEA. The difference gets larger by moving from low to high seismic zones. Fig. 15 shows the distribution of gap openings across the horizontal connection for the three seismic zones.

Results of the FEA indicate that ap­proximately two-thirds of the length of

November-December 1996

10

9

8

7

1 6 ...J 5 C!' .9 4 "' 3

2

0

10

9

8

7

1 6

~ 5

~ 4

3

2

1

0

10

9

8

7

1 6

~5 ~ 4

3

2

1

0

0 0.04

~''<- ...

· ~<~~ 0 0.5 1.5

0 0.04

0 0.5 1.5

0 0.04

(Inch)

0.08 0.12 0.16

--- Finte Element ooooo•--oo····oo PCI

- ---- Modified PCI

2 2.5 3 3.5 4 Gap Opening (mm)

(a) v = 0.2

(Inch)

0.08 0.12 0.16

--- Flnte Element 000000 0000 000000 PCI

----- Modified PCI

2 2.5 3 Gap Opening (mm)

3.5 4

(b) v = 0.3

(inch)

0.08 0.12 0.16

--- Finte Element 00 00000000000000 PCI

----- Modified PCI

4.5

4.5

\--·""'."'-·:.:.:·~------ .....

• ~ -<:::::::::::::::~-==:::::: - - .. 0 0.5 1.5 2 2.5 3

Gap Opening (mm)

(c) v = 0.4

3.5 4 4.5

Fig. 14. Distribution of gap openings over the height of the structure.

5

5

5

the horizontal connection experiences gap opening, which is consistent with that assumed for the modified Clough's method. Choosing the wall corner to be the point of rotation ap-

pears to be too conservative in terms of a gap opening.

The difference in the results be­tween the FEA and Clough's design method increases in higher seismic

101

(inch)

0 40 80 120 180 200 240 280 5~----,_-----r-----r-----T-----+-----4----~

0.18 4 ----- Flnte Element ··········· ····· PCI 0.12 - --- - ModlfHid PCI

0.08

0.04

········ 0

----=-----=----------..::~------04-----------------------~===-=-~------------~~~~

-1+-----~--~-----r----+-----~---+-----r--~

0 900 1800 2700 3800 4500 Horizontal Distance (mm)

5400 6300 7200

(a) v = 0.2

(inch)

0 40 80 120 180 200 240 280

5.-----,_-----r-----r-----T-----+-----4----~

0.18 4 ----- Flnte Element

la 0.12 ················ PCI - -- - - Modified PCI ·i 2 ----- ······· ·· ··-........... .

0 ----- · · ~ ...... ··--.. . 0.08

0.04 ··-... ··-...

~ ---- -...... . i-1 ~------

--o +---------------~~==~~----==~~~0

-1+-----~---+-----r----+-----~---+----~--~

0 900 1800 2700 3800 4500 Horizontal Distance (mm)

5400 8300 7200

(b) v = 0.3

(Inch)

0 40 80 120 180 200 240 280 5 ,-----,_-----r-----r-----T-----+-----4----~

0.18 ----- Flnte Element

---------....... ................ _

0.12

0.08

··········· ·· ··· PCI --- - - Modified PCI

p---____ -------- ·· ... . _

0.04

I -------- ........ ~~ o+-------------~======~----~==~~0

-1+-----~---+-----r----+-----~---+-----r--~

0 900 1800 2700 3800 4500 Horizontal Distance (mm)

(c) v = 0.4

5400 6300 7200

Fig. 15. Distribution of gap openings across the horizontal connection.

102

zones due to the fact that the elastic analysis, on which Clough's method is based, depends entirely on the gross concrete dimensions. Thus, when the structure is located in higher seismic zones and the amount of reinforce­ment is increased, Clough ' s force­displacement relationship remains unchanged.

Consequently, larger deflections and larger gap openings are expected for higher seismic zones irrespective of the amount of additional reinforce­ment used. However, in the FEA, as the reinforcement ratio is increased, the stiffness of the structure increases accordingly; this will have an effect on the deflections and gap openings. Results of the FEA showed that in all seismic zones, the maximum com­pressive stresses in the joint region never reached the joint crushing strength of concrete when the gap opening reached its maximum value.

Internal Forces and Ductilities of Coupling Beams

Fig. 16 shows the internal forces of the coupling beams as determined using the three different methods for three seismic zones. The forces calcu­lated using Clough 's method are higher than those determined using the FEA. Thi s is due to the fact that in Clough's method, the stiffnesses of the coupling beams are calculated using gross uncracked sections. This overestimates the stiffness of the beam and exaggerates the coupling force that the beams can transfer.

In the FEA, the stiffnesses of the coupling beams which are likely to de­crease drastically as concrete cracks, are considered by representing the deep beams as inelastic diagonal truss elements. Consequently, the beams' tendency to transfer shear forces de­creases and thus their internal forces decrease as the diagonal reinforcement is left alone to resist the coupling effect.

For v = 0.3 and 0.4, the bar forces fo r Clough's and the modified Cloug h 's methods are almost the same. This is because in Clough's method and in the modified Clough's method, the internal forces in the beams do not increase significantly beyond their yield level.

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0 10

9

8

7

~ 6

a- 5 iii 4

3

2

1

0 0

0 10

9

8

7

l 6 -J 5

i 4

3

2

1

0 0

0 10

9

8

7

l 6

~ 5 ~ 4

3

40 80

I I I I I I I I

/ /

I I I

(kip) 120 160

--- Finte Element ·········· ······ PCI ---- - Modified PCI

// .. ··· / :...-:.---: .. -···

100 200 300 400 500 600 700 Bar Forces (kN)

(a) v = 0.2

(kip) 40 80 120 160

I --- Finte Element I ················ PCI I ----- Modified PCI I

I I

) ---:::::.:::::::::::1.--·'

100 200 300 400 500 600 700 Bar Forces (kN)

(b) v = 0.3

(kip) 40 80 120 160

- -- Finte Element ················ PCI ---- - ModifiedPCI

800

800

I I I I I I I J 2 -----;·' - -::: ......... .

0 0 100 200 300 400 500 600 700 800

Bar Forces (kN)

(c) v = 0.4

Fig. 16. Internal beam forces over the height of the structure.

November-December 1996

900

900

900

DESIGN EXAMPLE The ten-story structure described

earlier is designed using Clough's method. Deep coupling beams of L = D = 1.0 m (39.4 in.) are used . The thickness of the wall panels is as­sumed to be 200 mm (7 .9 in.). It is assumed that the structure is located in UBC Seismic Zone 3 (Z = 0.3). Major components of the design are as follows:

1. Loads - The loads used include partition [0.5 kPa (10.5 psf)], mechan­ical [0.25 kPa (5.2 psf)], slab dead load [ 4.10 kPa (85. 7 psf)], built-up roof [0.3 kPa (6.3 psf)] and panel self weight of 34.0 kN/m (2.33 kips per ft).

2. Evaluation of the fundamental period - The fundamental period of the structure using the SAP-IV com­puter program as described previously was 0.32 second.

3. Elastic static analysis - A unit base shear of 1000 kN (224.8 kips) is distributed over the height of the structure in accordance with the UBC provisions. Tables la and lb show, re­spectivily, the distribution of the equivalent static loads at each floor level and the results of the elastic fi ­nite element analysis due to the distri­bution of these loads.

4. Calculation of Rdes ign value -The UBC base shear based on the equivalent static analysis is given by :

V= ZICW

R,v

where Z = 0.3, I = 1.0, W = 8407 kN (1890 kips), R w = 6.0 and C = 1.25S/T213

•

Using a soil profile factor of S = 1.5 and factored load of 1.4, C = 4.0 and V = 2358 kN (530 kips) .

The A TC-3 approach computes the base shear using the following expression:

V= C,.W

where C - 1·25A,S but C < 2·5Aa s - RT2/3 s R

where Av = 0.3, S = 1.5 and R = 1.0 (for Elastic Strength Demand). From the above expression, V = 6305 kN (1417 kips).

The Response Modification Factor, R, is expressed as:

103

Table 1 a. Distribution of unit 1000 kN (224.8 kips) base shear over height of structure.

Vx Story level kN (kips)

Roof 177 (39.8)

9 165 (37. 1)

8 146 (32.8)

7 128 (28.8)

6 110 (24.7)

5 91 (20.5)

4 ~ 73 (16.4)

3 55 (12.4)

2 : 7 (8.3)

I 18 (4.0)

ESD R=-----

Factored UBC

R = 6305 = 2.67 2358

Based on the above value of R, Rdes ign is assigned a value of 3.0 . Therefore, Vdesign = 2102 kN (472.6 kips).

5. Assignment of member strengths and prediction of global inelastic dis­placements - The inelastic displace­ment of the structure is determined using the "equal energy concept." The roof level horizontal deflection ac­cording to Figs. 4 and 17 is found to be 40 mm (1.57 in.). The amount of gap openings in the horizontal joints and the ductilities of the coupling beams are determined using Eqs. (5) to (13) and also the equations in Ap­pendix B(b). These values are shown in Fig. 10.

Based on the value of Vdesign , the design forces in the members are determined from the unit load static analysis described above. Based on these results, the amount of concen­trated reinforcement provided in the corners of the walls and distributed reinforcement provided along the wall sections are 1.43 percent and 0.62 percent of the gross concrete section, respectively.

The designs of the diagonally rein­forced coupling beams are based on the statically determinant model de­scribed by Paulay. 32 The maximum calculated beam reinforcement ratio is 1 percent of the gross concrete section.

104

Table 1 b. Results of elastic finite element analysis.

Maximum Maximum Base moment Base tension beam moment beam shear Top deflection kN-m (kip-ft) kN (kips) kN-m (kip-ft) kN (kips) mm (in.)

3550 (2617.8) 1700 (382.2) 218 (160.8) 243 (54.6) 4.76 (0.19)

Bose Shear (kN)

6305 -----

2102 r-

v Dp=40.02 mm

Displacement (mm)

~ Dy=10.01 mm - Dep=50.03 mm

LEGEND:

vdesign = Elastic Strength Demand = 6305 = 2102

Rdesign 3 kN

Dy = 2.1 02•4. 76 = 10.10 mm

Dp = Dy •(R2 -1)

= 40.02 mm 2

Dep= Dy + bp = 50.03 mm

Fig. 17. Selection of design base shear and prediction of global inelastic disp lacements .

CONCLUSIONS AND RECOMMENDATIONS

Based on the results of this study, the following conclusions and recom­mendations can be made:

1. This study showed that coupled walls with deep coupling beams provide satisfactory performance in all seismic zones. For the configuration used in the example structure, slender coupling beams showed large ductility demands that are beyond practical limits. How­ever, by using uncracked sections to estimate yield rotations, the ductility

demands have been overestimated. The ductility of deep coupling beams was found to be within acceptable limits.

2. Use of Clough's method to pre­dict the response of the structure was found to produce conservative results . The use of the modified Clough's method instead of Clough's design method by assuming that wall rotation takes place at one-third distance away from its corner improved the results.

3. Clough's method predicted large rotations in both walls, which conse­quently induced high internal forces and high ductilities in the coupling

PCI JOURNAL

beams. Taking the full gross moment of inertia of the uncracked coupling beams' sections in the elastic analysis overestimated the coupling effect and underestimated the walls ' contribution in resisting the overturning moments. The FEM showed that due to inelastic action in the coupling beams, the cou­pling effect was much less than that predicted by Clough's method.

4. In a precast wall system, it is most effective to use coupling beams with optimum strength to ensure that they reach beyond their yield level to dissi­pate energy. However, their level of ductility should be controlled to ensure

l. Fintel, M., "Performance of Precast Concrete Structures During Roma-nian Earthquake of March 4, 1977 ," PCI JOURNAL, V. 22, No. 2, March-April1977, pp. 10-15.

2. Fintel, M., "Performance of Build-ings With Shear Walls in Earth-quakes of the Last 30 Years," PCI JOURNAL, V. 40, No. 3, May-June 1995, pp. 62-80.

3. Ghosh , S. K., "Observations on the Performance of Structures in the Kobe Earthquake of January 17 , 1995," PCI JOURNAL, V. 40, No. 2, March-April1995, pp. 14-22.

4. Becker, J. M., and Llorente, C., "The Seismic Response of Simple Precast Concrete Panel Walls," Proceedings of U.S. National Conference on Earthquake Engineering, Stanford, CA, 1979.

5. Schricker, V., and Powell, G. H. , "In-elastic Seismic Analysis of Large Panel B uildings," Report No. UCB/EERC/80/38, College of Engi-neering, University of California, Berkeley, CA, 1980.

6. Pall, A. S., and Marsh, C., "Seismic Response of Large Panel Structures Using Limited-S lip Bolted Joints," Third Canadian Conference on Earth-quake Engineering, McGill Univer-sity, V. 2, Montreal, Quebec, Canada, 1979, pp. 899-916.

7. Pekau, 0 . A., and Hum, H., "Seismic Response of Friction Jointed Precast Panel Shear Walls," PCI JOURNAl::-, V. 36, No .2, March-Apri l 1991, pp. 56-71.

8. Kianoush, M. R., and Scanlon, A.,

November-December 1996

that these ductilities are not excessive. This can be achieved using an iterative design procedure. In addition, due to the presence of the horizontal joints, precast walls are designed for lower ductilities (i.e., lower Rdesign values) in comparison with monolithic cast-in­place concrete. The horizontal connec­tions should be of sufficient strength to resist seismic shaking. Adequate verti­cal reinforcement should be provided to ensure better structural response. The designer must also check that the vertical continuity bars are not rup­tured at the horizontal connection level at the base of the structure, where the

REFERENCES

"Analytical Modeling of Large Panel Coupled Walls for Seismic Loading," Canadian Journal of Civil Engineer-ing, V. 15, 1988, pp. 623-632.

9. Kianoush, M. R., and Scanlon, A. , "Behavior of Large Panel Precast Coupled Shear Wall Systems," PCI JOURNAL, V. 33, No. 5, September-October 1988, pp. 124-153.

10. Oliva, M. G., and Shahrooz, B. M., "Shaking Table Tests Panel Structures Using Limited-Slip Bolted Joints," Proceedings of the Eighth World Con-ference on Earthquake Engineering, v. 6, 1984, pp. 717-724.

11. Oliva, M. G., Clough, R. W ., and Malhas, F., "Seismic Behavior of Large Panel Precast Concrete Walls: Ana lysis and Expe riment," PCI JOURNAL, V. 34, No.5, September-October 1989, pp. 42-66.

12. Soudki , K. A., Rizkalla, S. H., and LeBlanc, B. , "Horizontal Connections for Precast Concrete Shear Walls Subjected to Cyclic Deformations, Part l: Mild Steel Connections ," PCI JOURNAL, V. 40, No. 4, July-August 1995, pp. 78-96.

13. Soudki , K. A., Rizkalla, S. H., and Diakiw, R. W ., "Horizontal Con-nections for Precast Concrete Shear Wall s Subjected to Cyclic Defor-mations , Part 2: Prestressed Con-nections," PCI JOURNAL, V. 40, No.5, September-October 1995, pp. 82-95.

14. Soudki , K. A. ; West, J . S., Rizkalla, S. H., and Blackett, B. , "Horizontal Connections for Precast Concrete Shear Wall Panels Under Cyclic

largest gap openings occur. 5. For structures with fundamental

periods in the range 1/s to 1 second, the equal energy concept as applied in this study to precast coupled walls appears to provide a conservative design ap­proach for seismic loading.

6. Based on this investigation, nu­merous research studies as well as the performance of large panel precast wall buildings in earthquakes, coupled walls with deep coupling beams can be used with safety and structural efficiency in all seismic zones. Design provisions for using such wall systems should be in­corporated in building codes.

Shear Loading," PCI JOURNAL , V. 41 , No. 3, May-June 1996, pp. 64-80. ,

15. Rizkalla, S. H., Serrette, R . L., Heuvel, J . S ., and Attiogbe, E. K., "Multiple Shear Key Connections for Precast Shear Wall Panel s," POI JOURNAL, V. 34, No. 2, March-April1989, pp. 104-120.

16. Foerster, H. R ., Rizkalla, S . H., and Heuvel, J. S. , "Behavior and De-sign of Shear Connections for Load-bearing Wall Panels," PCI JOUR-NAL, V. 34, No. 1, January-February 1989,pp. 102-119.

17. Hutchinson , R. L. , Rizkalla, S. H., Lau, M., and Heuvel, S., "Horizontal Post-Tensioned Connections for Pre-cast Concrete Loadbearing Shear Wall Panels," PCI JOURNAL, V. 36, No. 6, November-December 1991 , pp. 64-76.

18. Mattock, A. H., and Hawkins, N. M., "Shear Transfer in Reinforced Con-crete - Recent Research ," PCI JOURNAL, V. 17 , No.2 , March -April 1972, pp. 55-75.

19. Shiohara, H., Hosokawa, Y. , Naka-mura, T., and Aoyama, H. , "Direct Shear Transfer Mechanism at the In-terface of Reinforced Concrete Joints," Transactions, Japan Concrete Institute, V. 6, 1984, pp. 409-416.

20. Clough, D. P., "Design of Connec-tions for Precast Prestressed Concrete Buildings for the Effect of Earth-quakes," PCI Technical Report No. 5, Precast/Prestressed Concrete Institute, Chicago, IL, 1986.

21. Uniform Building Code, International

105

22 .

23 .

24.

25.

106

Conference of Building Offi c ia ls, and Corley, W. G ., "Earthquake Re- of California, Berkeley, CA, 1973. Whittier, CA, 1988. sis tant Structural Wall s - Tests of 29. B ac kler , A. P., a nd Baylik , M., Tentative Provisions fo r the Develop- Coupling Beams," Report to National "Loca l Behavior of Shear Transfer ment of Seismic Reg ula tions for Science Foundation, Portland Cement and Compression Transfer Joints -Buildings, Applied Technology Coun- Association, Skokie, !L, 1978, 58 pp. The Behavior of Large Panel Struc-cil, ATC Publication ATC-3-06, Na- 26 . Paul ay, T ., and Binney, J. R ., "Diag- tures," CIRIA Report 45 , Lo ndon , tiona! Bureau of Standards Special onally Reinforced Coupling Bea ms England, 1973. Publica tion 510, National Science of Shear Walls - Shear in Rein- 30. Elmorsi , M. , "Seismic Design of Pre-Foundation Publication 78-8, 1978. forced Concrete ," Special Publica- cast Wall s," M.S. Thesis, Department Newmark, N. M ., and H all , W . J ., tion 42, American Concrete Insti- of Civi l Engineering and Engineering "Procedure a nd Criteria for Earth- tute, Farmington Hill s, MI , 1974 , Mechanics, McMaster University , qu ake Resis tant Design," Building pp. 579-599 . H amilton , Ontario, Canada, 1994, Practice for Dis aste r Mitigation, 27. Paulay, T., and Santhankumar, A. R., 193 pp. Building Science Series 45 , National "Ducti le Behavior of Coupled Shear 31. Ma ison, B. F. , "PC-ANSR: A Com-Bureau of Standards , Washington , Wal ls," Journal of the Structural Di- puter Program for Non-Linear Struc-D.C. , 1973. vision, American Society of Civil En- tural Analysis ," University of Cali for-Yu , C., "Seismic Des ign of Precast gineers, V. 102, 1976, pp. 93-108. nia, Berkeley, CA, 1992. Coupled Walls ," M.S . Thesis, Depart- 28. Bathe, K. J ., Wil son, E. L. , and Peter- 32. Paulay, T ., "Simulated Seismic Load-ment of Civil Engineering, Pennsy l- son, F . E ., "A Structura l Ana lysis ing on Spandrel Beams," Journal of vania State University, 1992, 134 pp. Program for Static and Dynamic Re- the Structural Division, American So-B arney, G. B ., Shiu, K . N., Rabbat, spo nse of Linear Systems" SAPIV, ciety of Civil Engineers, Y. 97, Part B . G., Fiorato, A. E., Russell , H . G., Report No. EERC 73-11, University 3, 1971, pp. 2407-2419 .

DISCUSSION NOTE

The Editors w elcome discussion of reports and papers pub lished in the PCI JOURNAL. The comments must be conf ined to the scope of the art icle bei ng dis­cussed. Pl ease note that discussion of papers appear­ing in this issue must be received at PCI Headquarters by March 1, 1997.

PCI JOURNAL

APPENDIX A- EQUATIONS FOR ROTATIONS AND DISPLACEMENTS

(a) Equations for rotation angles (slender coupling beams):

Angle (radian) Equation

b --8 W I

DP -a-c-e

0.7H

0.1Hh1 a=--W

O.lHb c=--

W

Angle (radian)

(b) Equations for the extended length of the diagonal bars (deep coupling beams):

November-December 1996

Equation

107

108

APPENDIX B- LOAD-DEFORMATION RELATIONSHIPS

(a) Wall Panels Linear Elastic Plane Stress Element

Vertical Force

(c) Horizon tal Connections Inelastic Spring Element Zero Tension Model

Displacement

Force

Displacement --------~~--------~~--

(b) Vertical Reinforcement Inelastic Truss Element Yielding in Tension and in Compression

Force

Displacement --------~------~~~--

(d) Coupling Beoms Inelastic Truss Element Yielding in Tension and Buckling in Compression

APPENDIX C - NOTATION

Av = coefficient representing effective peak velocity related to acceleration

D e = elastic displacement D ep = maximum inelastic displacement D P = plastic displacement at top of building

d = depth of coupling beam E = modulus of elasticity f y = yield stress of diagonal reinforcement H = height of structure h = vertical displacement

h 1 = vertical displacement at bottom level h2 = vertical displacement at first floor h3 = vertical displacement at second level I = importance factor

ly = moment of inertia Me,.= cracking moment My = yield moment

R = response modification factor

S = soil profile coefficient S = length of coupling beam T = fundamental period V = base shear W = total gravity of structure W = width of one wall panel Z = seismic zone factor

L1y = yield extension of diagonal reinforcement f)ep = maximum rotation f)P = plastic rotation 8y = yield rotation e, = angle of base rotation J.1 = ductility factor

I X• 2x = X coordinates of two nodes of extended diagonal member

1 y, 2y = Y coordinates of two nodes of extended diagonal member

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