ACI Structural Journal/January-February 2000 35

ACI Structural Journal, V. 97, No. 1, January-February 2000.Received June 18, 1998, and reviewed under Institute publication policies. Copy-

right © 2000, American Concrete Institute. All rights reserved, including the makingof copies unless permission is obtained from the copyright proprietors. Pertinent discus-sion will be published in the November-December 2000 ACI Structural Journal ifreceived by July 1, 2000.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

A softened strut-and-tie model has previously been developed fordetermining the shear strengths of exterior beam-column joints forseismic resistance. This existing model originates from the strut-and-tie concept and satisfies equilibrium, compatibility, and the constitu-tive laws of cracked reinforced concrete. This paper examines theapplicability of the previously proposed model to interior beam-column joints. The calculated shear capacities of 56 interior jointswere compared with the experimental results, and good agreementwas obtained.

Keywords: beams (supports); compressive strength; reinforced concrete;shear properties; strength.

INTRODUCTIONBeam-column joints are critical because they ensure conti-

nuity of a structure and transfer forces from one element toanother. The flow of forces within beam-column joints maybe interrupted if the shear strengths of the joints are notadequately provided. Understanding the strength behavior ofa beam-column joint under seismic actions and being able tomodel it analytically are important aims in the achievementof safe reinforced concrete structures.

A rational model for determining the shear strengths of exte-rior beam-column joints for seismic resistance has beenproposed in a companion paper.1 The proposed model, calledthe softened strut-and-tie model, is based on the concept ofstruts and ties and derived to satisfy equilibrium, compatibility,and the constitutive laws of cracked reinforced concrete.

This paper represents a continuation of the previouslymentioned research. The applicability of the proposedmodel1 to interior beam-column joints is explored. Also, theprecision of the analytical model is further gaged by theavailable experimental data.

RESEARCH SIGNIFICANCEThe current design provisions on the beam-column joints

of the ACI 318-95 Code2 are based empirically on results oftests. Consequently, they must be restricted to joints whoseproperties closely match those of the tested joints. This leadsto many design limitations, and little guidance is providedfor the design of joints that may not meet these limitations.

A good physical model is needed to predict the shearstrengths of joints under seismic attack. It becomes an imper-ative necessity to the seismically insufficient joints that typi-cally fall outside of the limited range of those considered inthe ACI 318-95 Code.2 This paper presents a rational modelthat is capable of predicting the shear strengths of interiorbeam-column joints for seismic resistance.

SOFTENED STRUT-AND-TIE MODELBefore introducing the analytical model, the forces around

and within a joint should be identified. Figure 1 shows theearthquake-induced forces acting on an interior joint. Thehorizontal joint shear force can be calculated as

(1)

where Vjh is the horizontal joint shear force; Tb1 is the tensileforce resulting from the steel of the beam at the right of thejoint; Cb2 is the compressive force resulting from thecompression zone of the beam at the left of the joint; and Vc1is the horizontal column shear above the joint. Actually thetensile force Tb1 is not necessarily coincided with thecompressive force Cb2, and Fig. 1 is only a simplifiedversion on this matter.

Considering the dimensions of beam and column tension-compression couples (Fig. 1), the intensity of the verticaljoint shear force Vjv can be approximated

(2)

Vjh Tb1 Cb2 Vc1–+=

Vjv

h′bh′c------- Vjh×≈

Title no. 97-S4

Analytical Model for Predicting Shear Strengths of Interior Reinforced Concrete Beam-Column Joints for Seismic Resistanceby Shyh-Jiann Hwang and Hung-Jen Lee

Fig. 1—External actions and internal shears at interior joint.

36 ACI Structural Journal/January-February 2000

where hb′ and hc′ are the internal lever arms in the beams andcolumns, respectively. The column axial load coupled withmoment will increase the vertical joint shear force Vjv anddecrease the internal lever arm in the column hc′ .

Macromodel Statically indeterminate strut-and-tie load paths are

proposed to model the force transferring within the joint.1

The proposed model consists of the diagonal, horizontal, andvertical mechanisms as shown in Fig. 2. The diagonal mech-anism (Fig. 2(a)) is a single diagonal compression strutwhose angle of inclination θ is defined as

(3)

where hb′′ and hc′′ are the distances between the extremelongitudinal reinforcement in the beams and columns,respectively. It is also assumed that the direction of the diag-onal concrete strut coincides with the direction of the prin-cipal compressive stress of the concrete.

The effective area of the diagonal strut Astr is defined as

Astr = as × bs (4)

where as is the depth of the diagonal strut, and bs is the widthof the diagonal strut, which can be taken as the effectivewidth of the joint as per the ACI 318-95 Code.2

According to the recommendation of Zhang and Jirsa,3 thedepth of a strut without a beam hinge occurring at the face ofthe column can be determined as

(5)

where ab and ac are the depths of the compression zones inthe beam and column, respectively. For joints where a beamhinge occurs at the face of the column, the spalling of thecompression zone in the beam is frequently observed. Sincethe crushing of concrete produces a small compression zonein the beam, the neglect of ab in computing as is assumed.Therefore, the depth of the strut can be estimated as3

as = ac (6)

Following the suggestion of Paulay and Priestley,4 the depthof the flexural compression zone of the elastic column can beapproximated by

(7)

where N is the axial force acting on the column; f ′c is thecompressive strength of a standard concrete cylinder; Ag isthe gross area of the column section; and hc is the thicknessof the column in the direction of loading.

The horizontal mechanism (Fig. 2(b)) is composed of onehorizontal tie and two flat struts. The joint hoops constitute thehorizontal tie. It is roughly assumed that the joint hoops withinthe center half of the core are considered fully effective whencomputing the cross area of the horizontal tie, and that the otherjoint core hoops are included at a rate of 50%. Figure 3 explainshow to determine the area of the horizontal tie.

The proposed vertical mechanism (Fig. 2(c)) includes onevertical tie and two steep struts. The vertical tie is made upof the intermediate column bars. The way to estimate thecross area of the vertical tie is also presented in Fig. 3.

The proposed model is a statically indeterminate system.The yielding of ties does not stop the development of the

θh′′bh′′c---------

1–

tan=

as ab2

a2c+=

ac 0.25 0.85+N

Ag fc′-----------

hc=

Fig. 2—Joint shear-resisting mechanisms.

ACI member Shyh-Jiann Hwang is Professor of Construction Engineering at theNational Taiwan University of Science and Technology. He received his MS and PhDfrom the University of California at Berkeley. His research interests include seismicbehavior of beam-column joints, shear strengths of reinforced concrete members, andbond and anchorage.

Hung-Jen Lee is a PhD candidate in construction engineering at the NationalTaiwan University of Science and Technology. His research interests include behaviorand design of RC beam-column joints.

ACI Structural Journal/January-February 2000 37

shear strength of the joint5 because the inherent diagonal strutis capable of transferring shear force alone. Failure is definedas the crushing of concrete in the compression strut adjacentto the nodal zone (Fig. 1). Therefore, the shear strength of thejoint is calculated as the concrete compressive stress on thenodal zone as it reaches its capacity. The boundary of thenodal zone coincides with the diagonal strut boundary, but theconcrete bearing force to be examined is the summation ofcompressions from the diagonal, flat, and steep struts (Fig. 2).

Equilibrium Figure 4 shows the proposed strut-and-tie model for an

interior beam-column joint. The horizontal joint shear to beresisted by the strut-and-tie model is found as

(8)

where D is the compression force in the diagonal strut; Fh isthe tension force in the horizontal tie; and Fv is the tensionforce in the vertical tie. Similarly, the vertical joint force isexpressed as (Fig. 4)

(9)

It is of interest to note that the ratio of Vjv/Vjh = tanθ isalways maintained for any combination of the selectedmechanisms.

There are three load-paths in the joint region, and theratios to divide the joint shear forces among the resistingmechanisms should be determined. If the intermediatecolumn bars do not exist or the yielding of the vertical tieoccurs, the horizontal shear (or increment) is then resistedonly by the diagonal and horizontal mechanisms as shown inFig. 5(a). According to Schäfer et al.6 and Jennewein and

Vjh D– θcos Fh Fv θcot+ +=

Vjv D– θsin Fh θtan Fv+ +=

Schäfer,7 the statically indeterminate tie-force Fh in thereduced mechanisms (Fig. 5(a)) is assumed to be

(10)

where γh is the fraction of horizontal shear transferred by thehorizontal tie in the absence of the vertical tie. Equation (10)is a linear interpolation of Fh between two borderline cases,namely, that the entire horizontal shear is carried by the indi-rect load-path (Fh = Vjh) for θ ≥ tan–1(2) and that the entirehorizontal shear is transferred by the direct compression strut

(Fh = 0) for θ ≤ tan–1(1/2).

The vertical shear (or increment) is resisted only by thediagonal and vertical mechanisms in the absence or yieldingof the horizontal tie. Figure 5(b) presents the fraction of thevertical shear assigned to the vertical tie6,7 in the previous case

(11)

where γv is the fraction of vertical shear carried by thevertical tie in the absence of the horizontal tie.

Fh γh Vjh×=

γh2 θ 1–tan

3--------------------- for 0 γh 1≤ ≤=

Fv γv Vjv×=

γv2 θ 1–cot

3----------------------- for 0 γv 1≤ ≤=

Fig. 3—Determination of areas of horizontal and vertical ties.

Fig. 4—Strut-and-tie model at maximum response.

38 ACI Structural Journal/January-February 2000

Based on the previous finding of Schäfer et al.6 and Jenne-wein and Schäfer,7 it is further assumed that the ratios of thehorizontal shear Vjh assigned among the three mechanismsare defined as

(12)

Also, the same fractions of the vertical shear Vjv are sharedamong the three mechanisms

(13)

where Rd, Rh, and Rv are the ratios of the joint shears resistedby the diagonal, horizontal, and vertical mechanisms,respectively. The values of these ratios are defined as

(14)

(15)

(16)

It is carefully scaled such that the sum of Rd, Rh, and Rvequals unity. To facilitate the calculation, Eq. (12) can berestated as

D θ: Fh:Fv θcotcos– Rd:Rh:Rv=

D θ: Fh θ:Fvtansin– Rd:Rh:Rv=

Rd

1 γh–( ) 1 γv–( )1 γhγv–

-------------------------------------=

Rh

γh 1 γv–( )1 γhγv–

-----------------------=

Rv

γv 1 γh–( )1 γhγv–

-----------------------=

(17)

(18)

(19)

To check whether the joint strength is being reached, thebearing pressure on the nodal zone (Fig. 2) where thecompressive forces from the diagonal, flat, and steep strutsmeet at a node (Fig. 4) should be estimated. Since theinclined joint shear is mainly transferred in the d-direction(Fig. 1), the maximum compressive stress σd,max acting onthe nodal zone is assumed to govern the failure. With somealgebraic efforts, the value of σd,max is given by

(20)

Constitutive lawsThe ascending branch of the softened stress-strain curve of

the cracked concrete, as proposed by Zhang and Hsu,8 is asfollows

(21)

(22)

where σd is the average principal stress of concrete in the d-direction; ζ is the softening coefficient; f ′c is the compressivestrength of a standard concrete cylinder in units of MPa; εdand εr are the average principal strains in the d- and r- direc-tions, respectively; and εo is the concrete cylinder straincorresponding to the cylinder strength f ′c that can be definedapproximately as9

(23)

20 ≤ fc′ ≤ 100 MPa

By recognizing Eq. (21), the shear strength of the joint isassumed to be reached whenever the compressive stress and

D1–θcos

------------Rd

Rd Rh Rv+ +( )----------------------------------- Vjh××=

Fh

Rh

Rd Rh Rv+ +( )----------------------------------- Vjh×=

Fv1

θcot-----------

Rv

Rd Rh Rv+ +( )----------------------------------- Vjh××=

σd max,1

Astr-------- D

θh′′b

2h′′c------------ 1–

tan– cos

h′′b2h′′c------------ 1–

tan cos

------------------------------------------------------Fh–

=

2h ′′bhc′′

------------ 1–

tan θ– cos

2hb′′hc′′

------------ 1–

tan sin

------------------------------------------------------Fv

–

σd ζ– fc′ 2ε– d

ζεo-------- ε– d

ζεo--------

2

– for ε– d

ζεo-------- 1≤=

ζ 5.8

f ′c

---------1

1 400εr+---------------------------

0.9

1 400εr+---------------------------≤=

εo 0.002 0.001 fc′ 20–

80-----------------

for+=

Fig. 5—Forces in struts and ties.

ACI Structural Journal/January-February 2000 39

strain of the concrete diagonal strut arrive at the followingsituations

σd = –ζ · fc′ (24)

(25)

If the stress-strain relationships of bare mild steel for thejoint hoops and the intermediate column bars are assumed tobe elastic-perfectly-plastic, then

(26)

(27)

where Es is the elastic modulus of the steel bars; and fs andεs are stress and strain in the mild steel, respectively. fsbecomes fh or fv, εs becomes εh or εv, and fy becomes fyh orfyv when applied to joint hoop reinforcement or intermediatecolumn bars, respectively.

Neglecting the tension stiffening effect due to concrete,the relationship between forces and strains of the tension tiescan be constructed.

(28)

(29)

where Ath and Atv are the areas of the horizontal and verticalties, respectively; and Fyh and Fyv are the yielding forces ofthe horizontal and vertical ties, respectively.

CompatibilityAccepting the predetermined angle of inclination of the

principal compressive stress θ, the principal tensile strain εrcan be related to the horizontal strain εh, the vertical strainεv, and the magnitude of the principal compressive strain εdbased on the two-dimensional compatibility condition10

(30)

This equality states that the sum of the normal strains in theperpendicular direction is a constant.

Equation (30) is used to estimate the value of the principaltensile strain εr, which is directly related to the extent of soft-ening of the concrete as per Eq. (22). The treatment of Eq.(30) is different from that which had been used in Reference1, and Eq. (30) is a simple but effective way for solutionprocedures.

Solution proceduresA set of solution procedures is proposed, as shown in the

flow charts in Fig. 6 and 7. The solution procedure is catego-rized into five types of analyses (Type E, YH, YV, YHV, andYVH) for varied yielding conditions of the ties. The case ofType E means that the concrete strut reaches its strengthwhile the horizontal and vertical ties remain in the elasticrange. The Type-YH analysis deals with the case that theyielding of the horizontal tie precedes the reaching of theconcrete strength but the vertical tie is still effective in

εd ζ– εo⋅=

fs Esεs for εs εy<=

fs fy for εs εy≥=

Fh AthEsεh Fyh≤=

Fv AtvEsεv Fyv≤=

εr εd+ εh εv+=

constraining the cracks. The Type-YV analysis treats the casethat the yielding of the vertical tie precedes the reaching of theconcrete strength, whereas the horizontal tie is still in the elasticrange. The scope of the Type YHV includes the case where theyielding of the horizontal tie occurs first, then the vertical tieyields, and finally the concrete strut arrives at its capacity. Theyielding sequence of the ties for Type YVH is in reverse. Moredetails of these analyses can be found in Reference 1.

EXPERIMENTAL VERIFICATIONThe proposed model was used to calculate the joint shear

strengths of 56 test specimens (Table 1) described in theliterature. Twenty of these joints were tested in the UnitedStates,12,15,18,21,23 ten in New Zealand,5,11,13,14,16,17 and 26in Japan.19,20,22,24-27 In selecting these test data, a number ofscreens were applied. They were as follows:

1. Specimens with floor slab, transverse beam, lightweightconcrete, or eccentricity between column and beam axeswere omitted; and

Fig. 6—Flow chart showing efficient algorithm.

ACI Structural Journal/January-February 200040

Table 1—Experimental verification

Authors Specimensfc′ ,

MPafyh,

MPafyv,

MPa N/Ag fc′Vjh,test

kNFailure mode θ degree

Astr ,

cm2Ath,

mm2Atv ,

mm2Calculation

typeVjh,calc ,

kNVjh,test/Vjh,calc

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)

Blakeley et al.11 Interior 48.5 297 289 0.03 1722 F1 54 1285 9088 4914 E 2669 0.65

Meinheit et al.12

I 26.2 409 457 0.39 1090 J3 44 980 516 1548 YH 1142 0.95II 41.8 409 449 0.25 1597 J3 45 804 516 3276 YH 1476 1.08III 26.6 409 402 0.39 1228 J3 46 973 516 6036 YH 1180 1.04IV 36.1 409 438 0.29 1454 J3 59 903 1032 1290 YH 1138 1.28V 35.9 409 449 0.04 1530 J3 45 593 516 3276 YH 971 1.58VI 36.8 409 449 0.47 1646 J2 45 979 516 3276 YH 1582 1.04VII 37.2 409 438 0.46 1468 J3 59 110 1032 1290 YH 1375 1.07XII 35.2 423 449 0.29 1948 J2 46 755 2000 3276 E 1315 1.48XIII 41.3 409 449 0.25 1557 J3 45 808 1290 3276 E 1562 1.00XIV 33.2 409 438 0.31 1539 J3 59 932 2580 1290 E 1309 1.18

Fenwick et al.13 Unit 1 42.9 275 280 0.00 521 J1 45 188 1533 402 E 391 1.33Unit 3 39.3 275 318 0.00 437 J1 47 188 1799 452 E 365 1.20

Birss14 BI 27.9 346 427 0.05 1217 J2 55 617 1520 1810 E 759 1.60B2 31.5 398 427 0.44 1213 J1 55 1302 398 1810 YH 1405 0.86

Viwathanatepaet al.15

BC3 31.1 448 490 0.36 793 F1 42 1038 633 1136 E 1444 0.55BC4 31.5 448 490 0.36 874 F1 42 1030 633 1136 E 1451 0.60

Beckingsale16B11 35.9 336 423 0.04 965 F1 55 596 3096 1548 E 977 0.99B12 34.6 336 422 0.04 982 F1 55 599 3096 1548 E 949 1.03B13 31.4 336 398 0.26 1015 F1 55 976 2580 1548 E 1328 0.76

Park et al.17 Interior 34.0 305 412 0.24 966 J2 49 559 1608 628 E 841 1.15

Park et al.5 Unit 1 41.3 320 473 0.10 1001 J1 50 415 2413 905 E 799 1.25

Durrani et al.18

X1 34.3 352 414 0.05 840 J2 51 388 881 1020 E 592 1.42X2 33.6 352 414 0.06 853 J1 51 389 881 1020 E 583 1.46X3 31.0 352 345 0.05 629 J1 51 386 881 568 E 529 1.19

Otani et al.19,20

J1 25.6 367 374 0.08 516 J1 45 284 128 762 YH 330 1.56J2 24.0 367 374 0.08 536 J1 45 288 256 762 E 328 1.63J3 24.0 367 374 0.08 576 J1 45 288 640 762 E 353 1.63J4 25.7 367 374 0.23 503 J1 45 400 128 762 YH 455 1.11J5 28.7 367 374 0.07 491 J1 45 277 128 0 YVH 308 1.60J6 28.7 367 374 0.07 336 F1 45 277 256 284 E 349 0.96

Otani et al.19,20C1 25.6 324 422 0.08 436 F1 45 284 128 762 YH 334 1.31C2 25.6 324 422 0.08 432 F1 45 284 512 762 E 364 1.19C3 25.6 324 422 0.08 410 F1 45 284 512 762 E 364 1.13

Abrams21 LIJ3 31.1 400 470 0.00 724 J1 33 392 284 0 YV 554 1.31LIJ4 34.3 400 470 0.00 789 F1 34 392 284 0 YV 598 1.32

Noguchi et al.22

No. 2 32.9 330 348 0.06 553 J1 45 271 64 762 YH 396 1.40No. 4 32.9 330 348 0.06 623 J1 45 271 64 762 YH 396 1.57No. 5 28.5 330 348 0.07 570 F1 45 278 640 762 E 398 1.43No. 7 28.5 330 348 0.07 605 F1 45 278 64 762 YH 356 1.70

Leon23BCJ2 30.3 414 448 0.00 358 J2 52 161 190 774 YH 205 1.75BCJ3 27.4 414 448 0.00 394 J1 45 194 190 774 E 252 1.56BCJ4 27.2 414 448 0.00 462 F1 40 226 190 516 E 323 1.43

Joh et al.24

B8-HH 25.6 1320 404 0.15 272 F1 50 342 400 762 E 380 0.72HL 27.4 1320 404 0.14 277 F1 50 335 400 762 E 394 0.70MH 28.1 377 404 0.14 275 F1 50 332 280 762 YH 387 0.71LH 26.9 377 404 0.15 274 F1 50 337 168 762 YH 365 0.75

MHUB 26.1 377 404 0.15 239 F2 50 340 280 0 YVH 371 0.64

Kitayama et al.25

A1 30.6 320 540 0.06 689 J3 46 355 168 1194 YH 486 1.42B1 24.5 235 351 0.08 570 J1 46 286 168 1194 YH 331 1.72B2 24.5 235 351 0.08 570 J2 46 286 168 1194 YH 331 1.72B3 24.5 235 371 0.08 515 J1 45 286 420 762 E 348 1.48

Fujii et al.26

A1 40.2 291 644 0.08 412 J3 51 223 112 762 YH 353 1.17A2 40.2 291 387 0.08 380 J3 51 223 112 762 YH 353 1.08A3 40.2 291 644 0.23 412 J3 51 269 112 762 YH 421 0.98A4 40.2 291 644 0.23 421 J3 51 269 336 762 YH 441 0.95

Total 56Average 48 Average 1.2

COV 0.11 COV 0.27

ACI Structural Journal/January-February 2000 41

Fig. 7—Algorithm for post-yielding cases.

2. Only specimens failing in a joint or a beam adjacent toa column were considered; specimens with a relocated beamhinge or those that had failed prematurely in a column wereomitted.

The specimens selected encompass a wide range ofmaterial properties, geometry, loading, loading sequence,and reinforcement detailing, as summarized in Table 1according to chronological order. The experimental jointstrengths (Vjh,test) in Table 1 were either reported in theliterature or derived using Eq. (1) based on the maximumvalue of the column or beam shears measured during thetest.

According to the seismic performance of the beam-column subassemblages, the failure modes of the selectedspecimens were classified into F1, J1, J2, and J3 groups (Fig. 8).The letter “F” designates beam flexural failure, and “J” indi-cates joint shear failure. The behavior of the subassemblage(Fig. 8) is judged by the H-Δ response. The quantity Hrepresents the equivalent horizontal load capacity of thesubassemblage, including the P-Δ effect, and it can beexpressed as28

(31)H H NΔLc-----+=

42 ACI Structural Journal/January-February 2000

where H is the horizontal load applied at the column end; Δis the horizontal displacement measured at the column end;and Lc is the hinge-to-hinge distance of the column.

The classification of F1 and J1 means that the jointstrength can reach its design value and that the ductility ofbeam-column subassemblages is up to 4 (Fig. 8). The designlevel of the joint shear strength was gaged by the ratio ofHmax/Hy to exceed 1.1 (Fig. 8), which is explained in Refer-ence 1. Yielding load Hy is defined as the equivalent hori-zontal load when yielding of the subassemblage occurred.Yielding of the subassemblage occurred when the yieldingmoment was exceeded in both beams at the column face.

The failure Mode J2 means that the yielding load Hyprecedes the joint shear failure, and the above sequence of J3is in reverse (Fig. 8). It is noted that, in the proposed model,the depth of the diagonal strut as of Specimens F1, J1, and J2was determined by Eq. (6), and that the as of Specimen J3was calculated by Eq. (5). For J3 specimens, the ab in Eq. (5)corresponds to the smaller value of the depths of thecompression zones in the elastic beams at the right and leftof the joint.

In Fig. 9 the experimentally determined shear strengthsfrom 56 joint tests are compared to the shear strengthspredicted by the method presented in this paper. The strengthratios that are defined as the ratio of the measured to thecalculated strength are listed in Table 1 to indicate the preci-sion of the proposed model. Figure 9 shows that satisfactoryresults were obtained for the comparison of measured andcomputed strengths. The average strength ratio is 1.20 andthe coefficient of variation is 27% (Table 1).

The accuracy of the proposed model under varied conditionsis further examined in Table 2. Table 2 displays a similartendency as obtained in the previous analyses for the shearstrengths of the exterior joints.1 For F1 specimens, the valueof Vjh,test was determined mainly by the beam flexuralstrength and not necessarily by the joint shear strength.Therefore, Table 2 shows a wider dispersion for F1 speci-mens (COV = 34%) and also the lowest strength ratio (mean= 1.0). The occurrence of more yielding mechanisms leads togreater damage accumulating within the joint. Because theproposed model does not include the effect of progressivedeterioration, the strength ratios of Type E are larger thanthose of Type YH for cases with joint failures (J1; Table 2).

Examination of existing experimental data indicated thatthe proposed model is also capable of predicting the shearstrengths of the interior beam-column joints. Table 2 showsthat the proposed model yields higher strength ratio for theinterior joint with the J1 failure mode (mean = 1.40) than thatof the exterior joint (mean = 1.07).1 The higher strength ratiofor the interior joint, approximately 1.3 (1.40/1.07) timeshigher, is attributed to the better end conditions of its diag-onal strut provided by the compression zones in beams andcolumns. Further investigation on selecting the depth ofdiagonal Strut as seems to be needed. It is interesting,however, to note that the predicting ratio 1.3 (1.40/1.07)between the interior to exterior joints is close to the shearstrength ratio 1.25 (15/12) recommended by the ACI 318-95Code2 for different conditions of joint confinement providedby the framing beams.

The proposed model suggests that the bond strengthbetween member longitudinal reinforcement and joint coreconcrete deteriorates under reversed cyclic loading. Thecomplete loss of the bond strength along the beam reinforce-ment, however, may impair the development of the verticalmechanism, since the horizontal force needed for the equi-

Fig. 8—Classification of failure modes of interior beam-column subassemblage.

Fig. 9—Correlation of experimental and predicted joint shearstrengths.

Table 2—Statistics of shear strength ratios between experimental and predicted values

Calculation type

Failure mode

F1 J1 J2 J3 Total

E 13*0.93†

91.42

41.41

21.09

281.17

0.32‡ 0.13 0.14 0.12 0.28

YH 41.12

61.37

31.50

111.14

241.24

0.42 0.24 0.27 0.18 0.26

YV 11.32

11.31

None None 21.31

— — 0.01

YVH None 11.60

None None 11.60

— —

Total 181.0

171.40

71.45

131.14

551.21

0.34 0.16 0.19 0.17 0.27*No. of specimens.†Average of Vjh,test/Vjh,calc.‡COV of Vjh,test/Vjh,calc.

ACI Structural Journal/January-February 2000 43

librium of vertical tie and steep strut (Fig. 4) is probablymissing. Specimen MHUB, tested by Joh et al.,24 was giventhe unbond treatment of the beam bars within the joint byusing vinyl chloride pipes. This specimen, categorized as theF2 failure mode (Table 1), failed by flexure of the beams butit did not reach the strain hardening range.24 Because thebeam bars had no bond within the joint, it is recommendedto remove the vertical mechanism from the shear resistanceby setting Atv = 0 in the analytical model (Table 1). Thedetailed calculation of the specimen MHUB is provided inthe Appendix.* A similar approach may be applied to theprecast beam-to-column joint with unbonded tendons.

The strength behavior of the beam-column joints underseismic actions is very complicated. The sensitivities of therelated parameters are still not very clear. The proposedmodel maintains consistency in its estimations from one situ-ation to another.

Therefore, the model proposed herein can be used as a toolto clarify the roles of different parameters. The effect ofcolumn axial load on the joint shear strength is briefly eval-uated in the following paragraphs.

In the proposed model, the column axial load provides thebeneficial effect on the joint shear strength because it increasesthe depth of the strut (Eq. (7)). Figure 10(a) shows the strengthratios predicted by the proposed model if the beneficial effectof the column axial load is removed. As shown in Fig. 10(a),the normalized tested results indicate that the increasingcolumn axial loads do increase the strength ratios.

By using Eq. (7) to include the effect of column axial load,Fig. 10(b) presents the strength ratios listed in Table 1. Itseems that the proposed model with Eq. (7) can reasonablyestimate the joint shear strength for the range of N/Agfc′ lessthan 0.1 (Fig. 10(b)). The proposed model overestimates thejoint shear strengths for the cases with higher axial loads(Fig. 10(b), N/Agfc′ ≥ 0.2).

The overestimations in Fig. 10(b) come from the adoptedassumption of the proposed model. The angle of inclinationθ of the diagonal strut is assumed to be oriented between theextreme longitudinal reinforcement in the columns forsimplicity, but this assumption is violated by the high axialloads in the columns. To direct the angle of inclination θwithin the centroids of the compression zones of the elasticcolumns (ac /3) is more realistic for the high axial loads in thecolumns, and this results in a steeper θ (Fig. 10(c)). Consid-ering the previously mentioned adjustments, the overestima-tions for shear strengths of the joints with N/Agfc′ greaterthan 0.35 are corrected (Fig. 10(c)).

In contrast to the beneficial effect of the column axial loadto increase the depth of the diagonal strut, the high axialcompression load in the column was reported to acceleratethe deterioration of the joint shear resisting mechanism.19,27

This detrimental effect of the column axial load can be readfrom Fig. 10(c), where the predicting strength ratios aredecreased with increased column axial load. With a suffi-cient amount of test data, the beneficial, as well as detri-mental, effects of the column axial loads on the joint shearstrength can be estimated with the aid of the proposed model.

CONCLUSIONSExamination of existing experimental data indicated that

the previously proposed model1 is also capable of predicting

*The Appendix is available in xerographic or similar form from ACI headquarters,where it will be kept permanently on file at a charge equal to the cost of reproductionplus handling at time of request.

the shear strengths of the interior beam-column joints forseismic resistance. The so-called softened strut-and-tiemodel is based on the strut-and-tie concept and derived tosatisfy equilibrium, compatibility, and the constitutive lawsof cracked reinforced concrete.

The proposed method was found to reproduce 56 testresults from the literature with reasonable accuracy. Theeffect of column axial load on the shear strength of the inte-rior joint was briefly studied. One illustrative example thatdescribes the computational procedures for the joint spec-imen with unbonded beam reinforcement is also provided.

ACKNOWLEDGMENTSThis research study was sponsored by the National Science Council of the

Republic of China under Project NSC 88-2211-E-011-011. The authorswould like to express their gratitude for the support.

NOTATIONab = depth of compression zone in beam adjacent to jointac = depth of compression zone in column adjacent to jointas = depth of diagonal strutAg = gross area of column sectionAstr = effective area of diagonal strutAth, Atv = areas of horizontal and vertical ties, respectivelybs = width of diagonal strutCb1,Cb2 = compressive forces resulting from compression zones of

beams at right and left of joint, respectivelyCc1, Cc2 = compressive forces resulting from compression zone of

columns above and below joint, respectivelyd = direction of diagonal concrete strut

= assumed direction of principal compressive stress of concreteD = compression force in diagonal strut (negative for compression)Es = elastic modulus of steel barf ′c = compressive strength of standard concrete cylinder

Fig. 10—Effect of column axial load on joint shear strength.

ACI Structural Journal/January-February 200044

fh, fv = steel stresses in h- and v-directions, respectivelyfs = average tensile stress of mild steel bars, taken as fh and fv in h-

and v- directions, respectivelyfy = yield strength of bare mild steelfyh, fyv = yield strength of bare mild steel of joint hoop reinforcement

and intermediate column bars, respectivelyFh, Fv = tension forces in horizontal and vertical ties, respectively

(positive for tension)Fyh, Fyv = yielding forces of horizontal and vertical ties, respectivelyh = direction of joint hoop reinforcementhb = beam depthh'b = internal lever arm in beamhb′′ = distance between extreme longitudinal reinforcement in beamshc = thickness of column in direction of loadingh'c = internal lever arm in columnhc′′ = distance between extreme longitudinal reinforcement in columnsH = horizontal load applied at column endH = equivalent horizontal load capacity of subassemblage

including the P-Δ effect= H + NΔ/Lc

Hmax = maximum value of HHy = load H at the yielding of both beams at column faceLc = hinge-to-hinge distance of columnN = axial force acting on column (positive for compression)r = direction perpendicular to d

= assumed direction of principal tensile stressRd,Rh,Rv = ratios of joint shears carried by diagonal, horizontal, and

vertical mechanisms, respectivelysh = spacing of joint hoopssv = spacing of intermediate column barsTb1, Tb2 = tensile forces resulting from steel of beams at right and left of

joint, respectivelyTc1, Tc2 = tensile forces resulting from tensile steel of columns above

and below joint, respectivelyv = direction of intermediate column barsVb1, Vb2 = vertical beam shears at right and left of joint, respectivelyVc1, Vc2 = horizontal column shears above and below the joint, respectivelyVjh, Vjv = horizontal and vertical joint shear forces, respectivelyΔ = horizontal displacement measured at column endΔy = horizontal displacement Δ at Hyγh = fraction of horizontal shear transferred by horizontal tie in

absence of vertical tieγv = fraction of vertical shear carried by vertical tie in absence of

horizontal tie= 0.002 + 0.001(fc′ – 20)/80 (fc′ in units of MPa)

εd, εr = average normal strains in d- and r- directions, respectively,(positive for tension)

= assumed principal strainsεh, εϖ = average normal strains in h- and v- directions, respectively,

(positive for tension)εo = strain at peak stress of standard concrete cylinder

= 0.002 + 0.001(fc′ – 20)/80 (fc′in units of MPa)εs = average tensile strain of mild steel bars, taken as εh and εv in

h- and v- directions, respectivelyεyh, εyv = yield strain of bare mild steel of horizontal and vertical rein-

forcement, respectivelyθ = angle of inclination of h-axis with respect to d-axisζ = softening coefficient of concrete in compressionσd, σr = average normal stresses in d- and r- directions, respectively,

(positive for tension)= assumed principal stresses

σd,max = maximum compressive stress exerting on nodal zone in d-direction

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