torsion design of prestressed concrete journal...design provisions for torsion in reinforced...

TORSION DESIGN OFPRESTRESSED CONCRETEPaul ZiaProfessor and Associate Head

of Civil EngineeringNorth Carolina State UniversityRaleigh, North Carolina

W. Denis McGeeAssistant Professor of

Civil EngineeringWashington State UniversityPullman, Washington

Design provisions for torsion in reinforced concretemembers have been developed and included in the ACI318-71 Building Code. These provisions are notapplicable to prestressed concrete.Based on a thorough review of available research dataon torsion in prestressed concrete, this paper proposesan extension of the ACI design procedures to covertorsion in prestressed concrete. The basic approachconsists of determining the concrete contribution to theultimate strength of a member and then proportioningreinforcement for the remaining portion of therequired ultimate strength.Formulas for the nominal torsional stress and the basicequation for torsional strength of reinforced membersare presented, and followed by a discussion of thetorsion-shear interaction relation, taking into accountthe effect of the prestressing force. The requirements forboth web and longitudinal reinforcements for torsionare explained.The proposed procedures are illustrated by a designexample of a precast prestressed L-girder in a roofframing subjected to combined torsion, bending, andshear. It is believed that the proposed procedure isreasonably conservative and future refinements andsimplifications are possible when more completeresearch data become available.

46

The problem of torsion in concretestructures has received considerable at-tention of design engineers in recentyears. In 1971, the ACI Budding Code(ACI 318-71) 1 included for the firsttime explicit requirements of torsion de-sign for reinforced concrete developedon the basis of substantial research da-ta. However, the code provisions arenot applicable to prestressed concretemembers, for which research has beenin progress but design criteria have yetto be developed.

This paper presents a proposed de-sign procedure for torsion in prestressedconcrete, based on a thorough analysisof 394 test results available in the liter-ature. 2 The proposal is in close parallelwith the ACI Code procedure for rein-forced concrete.

The basic approach consists of deter-mining the concrete contribution to theultimate strength of a member and thenproportioning reinforcement for the re-maining portion of the required ulti-mate strength. It is believed that theproposed procedure is reasonably con-servative and future refinements andsimplifications are possible when morecomplete research data become avail-able.

In developing the proposed proce-dure, the authors have benefited fromhelpful discussions with several mem-bers of the ACI Committee 438 on Tor-sion. However, this proposal should notbe regarded as a committee-endorseddocument.

In the following discussion, the usualcapacity reduction factor 0 is not in-cluded for the sake of convenience.

Paul Zia

W. Denis McGee

torsional stress of a rectangular sectioncan be expressed by:

T.Tu— m, 2y (1)

TORSIONAL SHEAR STRESS

Rectangular section

A prestressed concrete member with-out web reinforcement fails, under tor-sion, in the form of skew bending. The

e wherex = shorter side of rectangular sec-

tiony = longer side of rectangular sec-

tiona = torsion coefficientIt will be recalled that Eq. (1) is of

the basic form for either the elastic the-

PCI Journal/March-April 1974 47

0.55

05 plastic1 coeff.

k i

6 0.45

00.40 ^• ._o !:V

0 035 ; j ' _ ,-AC I of 1/3

° 030 ; 0.75+x/y

elastic0.25 coeff.

0.20 position of ave.f

0.150 1 2 3 4 5aspect ratio y/x

Fig. 1. Variation of torsion coefficients with aspect ratio.

ory, the plastic theory or the skew wherebending theory of torsion. The coeffi- f,. = modulus of rupturecient a varies from 0.208 to 1/3 in the a- = average prestress on sectionelastic theory and from 1/a to 1/a in the The factor (0.85 f,.) is nearly equal toplastic theory. For simplicity, the ACI the tensile strength of concrete f,' andCode adopted a value of 1/3 in accor- may be taken as 6Vf j . Hence, corn-dance with the skew bending theory. 3bining Eqs. (1) and (2):

Hsu4 has shown that the torsionalcapacity of a prestressed section with- Tu = 6Vf^ ^1 + 10 a-/f0 — IXx2yout reinforcement, or the crackingtorque of a prestressed section with re- orinforcement is reached when: _ _ 3

Tu = (0.85 fr) 10 a- /f0' (2)a

x2y 6Vfc lrl + 10 a-/f0 ( )

48

Using the test results of 218 concen-trically or eccentrically prestressed rec-tangular beams, the value of a has beencomputed for each beam by Eq. (3)and these values are plotted against theaspect ratio y/x in Fig. 1. Also plottedfor comparison in Fig. 1 are the curvesfor the elastic and plastic torsion coeffi-cients as well as the value currentlyspecified by the ACI Code.

It can be seen that for sections of lowaspect ratio, the ACI 318-71 value of1/. could be quite unconservative.Since, as will be shown later, the con-

tribution of the concrete to the ultimatetorsional strength is a much larger por-tion of the cracking strength for pre-stressed members than for non-pre-stressed members, it is prudent and es-sential that the nominal torsional stressbe evaluated more accurately. There-fore, it is proposed that the coefficienta in Eq. (1) be taken as:

0.35« = 0.75 + x/y (4)

which represents a reasonably conserva-tive lower bound to the experimentaldata.

Table 1. Comparison of experimental cracking torque of prestressed flanged sec-tions with Eq. (5).

Inves-

tigator

Beam

No.

E(ax2y)

(in.3)

Tex

(in.-k)

Tth*

(in.-k)

Tex

Tth

Wyss, et. al. Al 294 253 234 1.08(I-beams)ḏ(5) A2 294 256 234 1.10

A3 294 300 233 1.29A4 294 316 233 1.35A5 294 329 233 1.42A6 294 313 233 1.35B1 294 203 185 1.10B2 294 216 185 1.17B3 294 222 188 1.19B4 294 222 188 1.19B5 294 209 188 1.12B6 294 200 188 1.06

Ziaḏ(6) 0.25T1 52.9ḏ(41.5)** 51.12 40.3 (31.6)** 1.27 (1.62)**(T-beams) 0.25T2 52.9ḏ(41.5) 51.98 40.3 (31.6) 1.29ḏ(1.64)

2.25T1 52.9ḏ(41.5) 35.20 39.9ḏ(31.3) 0.89ḏ(1.13)2.25T2 52.9ḏ(41.5) 32.92 39.9ḏ(31.3) 0.83ḏ(1.06)2.75T1 52.9ḏ(41.5) 33.96 39.8 (31.2) 0.88 (1.12)2.75T2 52.9ḏ(41.5) 33.16 39.9ḏ(31.3) 0.83 (1.06)

Zia (6) 0.75I1 57.1 (49.4) 28.0 39.4 (34.1) 0.71 (0.82)(I-beams) 0.75I2 57.1 (49.4) 35.0 39.4 (34.1) 0.89ḏ(1.03)

3.I1 57.1 (49.4) 30.0 39.7 (34.4) 0.75ḏ(0.87)3.I2 57.1 (49.4) 33.0 39.7ḏ(34.4) 0.83 (0.96)3.5I1 57.1 (49.4) 48.5 41.2ḏ(35.6) 1.18 (1.36)3.512 57.1 (49.4) 46.0 41.2ḏ(35.6) 1.11 (1.29)

*Computed from Equation (5).

**Neglecting outstanding flanges.

+Cracking torques for these members were not available; T wastaken as the elastic limit, which is less than the actuaTTXcracking load.


Table 2. Comparison of experimental cracking torque of prestressed box sectionswith theory.

InvestigatorBeam

No.

x, y

(in.)Core

Equiv.squarecore

4h/xTex

(in. -k)

T *th

(in.-k)

Tex

Tth

T 9 5" x 5" 5" x 5" 0.89 127.2 113.9 1.12

Johnston H-0-0-1 12 8.5" 7.5" x 7.5" 0.75 120 99.5 1.21and Zia (9)

H-0-3-1 12 8.5" 7.5" x 7.5" 0.75 120 73.7 1.63

H-0-6-1 12 8:5"c 7.5" x 7.5" 0.75 115 73.7 1.56

*Computed from Equation (1) multiplied by 4h/x.

Flanged sectionFor a flanged section, the same as-

sumption implied by ACI 318-71 willbe used. That is, the torsional strengthof a flanged section can be expressed asthe sum of the strengths of the individ-ual components. Hence:

Tu =^ 2y ^Jr)Eq. (5) is identical to the ACI 318-

71 Eq. (11-16) except for the modifi-cation of torsional coefficient a as ex-pressed by Eq. (4). The section maybe divided into component rectanglessuch that the quantity l.ax2y would bea maximum.

Comparison of Eq. (5) with test re-sults reported in the literature indicatesthat the method is reasonably safe asshown in Table 1. However, it shouldbe cautioned that only two investiga-tions5 ' 6 have been reported, consisting.of 24 test specimens, all being some-what stocky in cross section. The mar-gin of safety seems to reduce as thecross section becomes more stocky suchas the specimens tested by Zia. Forsuch disproportionate sections, it wouldseem logical to disregard the small out-standing flanges.

Box sectionBased on test results, ACI 318-71

specifies that for reinforced concretebox sections with a wall thickness h notless than x/4, where x is the overallwidth of the box section, the torsionalstrength of the box section may betaken as that of a comparable solid rec-tangular section with the same overalldimension.

If the wall thickness h is less thanx/4 but greater than x/10, the torsion-al strength of the box section is reducedand the strength reduction is accountedfor by a reduction factor 4h/x multi-plied to the denominator on the right-hand side of Eq. (1).

This design approach seems also ap-plicable to prestressed concrete box sec-tions. Although only four tests of pre-stressed concrete box sections underpure torsion have been reported in theliterature that can be used for compari-son, application of the ACI design rulefor these test results indicates a consid-erable margin of safety in all cases asshown in Table 2.

For box sections with a very thinwall, h approaching x/10, cautionshould be exercised against possible

50

wall buckling or crushing, particularlywhen high prestress is applied.

BASIC EQUATION FORTORSIONAL STRENGTH

The basic equation for torsionalstrength has been developed from thetest results of prestressed beams underpure torsion. Tests conducted by Hsu7show that the ultimate torsionalstrength can be expressed as the sum ofthe strength contributed by the con-crete and the strength contributed bythe web reinforcement, for both pre-stressed and non-prestressed beams

(see Fig. 2). The effect of prestress isto increase the contribution of the con-crete to the ultimate torsional strength,while the contribution of the reinforce-ment remains unchanged.

Referring to Fig. 3, it can be seenthat for non-prestressed beams the con-tribution by concrete To is only a por-tion of the torsional strength Tu5 of acorresponding plain concrete section.Likewise, for prestressed beams, thecontribution by concrete To is also aportion, but a greater portion, of thetorsional strength of a correspondingprestressed concrete section withoutweb reinforcement. Accordingly, the ba-sic strength equation for prestressed

600ҟ10"15"

500

400ҟ0 ^^j

300ҟ̂

200ҟ6\/``'

••e

o prestressedo nonprestressed

DO

00 100 200 300 400 500x^ y

I As fsy /s (ink)

Fig. 2. Tests on prestressed and nonprestressed reinforced concrete beams (afterHsu).


Tu^a

t^^Jn

lb 0

I—ҟ°i7̂

^^ Amin. reinf.

x i y1 As fsy /s

Fig. 3. Relation between ultimatetorque for prestressed and non pre-stressed reinforced, rectangular beams

(after Hsu).

beams can be expressed as:

Tu = To + Hx1y1Acf8y (6)

sand

fl = 0.66 m + 0.33 y1/x1where

To = strength contributed by con-crete

xl = shorter leg of a closed stirrupyl = long leg of a closed stirrupAt = cross-sectional area of one leg

of a stirrupf8y = yield strength of stirrups = spacing of stirrupm = volume ratio of longitudinal

reinforcement to web rein-forcement

If the yield strength of the longi-tudinal reinforcement is different from

that of the web reinforcement, then theparameter m should be replaced bymfj /f3i, in which fly is the yieldstrength of the longitudinal reinforce-ment. Hsu has recommended that:

0.7 mfl5/f8 1.5 (7)and

yl/xl 2.6 (8)In Fig. 3, the cracking (or ultimate)

torque of a plain concrete member canbe obtained by substituting 6Vf,,' forTu in Eq. (1). Thus:

T., = (ax2y)6 V/ff (9)The ordinate T° has been deter-

mined by Hsu experimentally as:

T0 =0.4(3xZy)6V

= 0.133(x2y)6Vf0'Hence:

To =T,,, /1 +10a-,/f0' — (T„P — To)= Tu5 [VI + 10 a-/f,, -

(1 — T0/T)]

_ (ax2y)6 Vi (V1 + 10 o-/f0' — k)(10)

wherek=1—T0/T,,5= 1-0.133/a (11)Thus, by combining Eqs. (6) and

(10), the basic strength equation be-comes:

1',, = (ax`y)6 Nlf (V1 + 10 a-/f,,' — k) +1y1A tf3v (12)

sLikewise, for a box section:

[(ax2y)6T,, = Vf0' X

(V1 +10a-/f0 - k) +

cXSYSAtfSY] (13)s

and, for a flanged section:

1',, = ( ax2y)6 Vf x(Vi + 10a-/f^ - k) +

1y1Atf8v (14)s

in which the value of k is determined

52

Table 3. Comparison of experimental ultimate torque of concentrically prestressedrectangular beams with Eq. (12).

Inves-

tigator

Seam

No.

Tex

(in.-k)

Tth*

(in.-k)

Tex

Tth

Chandler, C1 59.32 54.67 1.09

et. a1. 03 79.00 62.07 1.27

(10)C5C9

75.6085.10

66.9472.47

1.231.17

Cli 94.50 78.86 1.20C15 106.40 85.64 1.24C7 54.77 58.68 0.93C13 59.10 49.18 1.20C17 41.08 42.54 0.97C19 51.60 62.00 0.83SP1261 164.90 173.74 0.95SP1262 134.65 130.15 1.03SP1269 151.15 155.25 0.97SP1270 129.90 112.80 1.15SP1241 82.90 67.71 1.22SP1242 81.40 67.71 1.20SPC1243 70.95 58.51 1.21SPC1245 103.00 84.67 1.22SP1247 93.70 77.99 1.20SPC941 52.20 46.05 1.13SP1263 175.15 172.89 1.01SP1264 136.15 129.31 1.05SP1271 162.40 155.39 1.05SP1272 121.15 112.94 1.07SPI 152.15 118.55 1.28SPII 142.65 118.58 1.20SP1267 169.25 170.91 0.99

1

Mukherjee SP1 190.8 129.79 1.47Kempḏ(11) SP2 171.0 122.79 1.39

SP3 233.8 204.82 1.14SP4 198.6 164.45 1.21SP5 200.8 153.85 1.31SP6 153.0 122.99 1.24SP7 156.0 110.51 1.41SP8 186.0 190.83 0.97SP9 185.0 155.62 1.19SP10 178.0 143.42 1.24SP11 207.0 163.39 1.27SP12 205.0 153.93 1.33SP13 232.0 243.96 0.95SP14 232.0 200.34 1.16SP15 203.0 184.46 1.10

Jacobsen(16) 306 177.9 122.09 1.46

Inves-

tigator

Beam

No.

T,

(in.-k)

Tth*

(in.-k)

Teic

Tth

Mukherjee and 106 140.8 97.05 1.45Warwaruk (12) 206 146.4 121.24

Okada, et. al. RbIII-1I 47.0 25.05 1.88(13) RbIII-21 47.6 25.05 1.90

Superfeskyḏ(14) SP17 213.0 141.19 1.51SP18 246.0 229.24 1.07SP19 245.0 188.70 1.30SP20 215.0 169.94 1.274S1 63.5 35.63 1.78.452 77.5 58.68 1.32453 92.7 61.50 1.514S4 83.1 63.78 1.30455 95.8 68.91 1.394S6 72.0 42.30 1.70457 80.6 65.75 1.234S8 88.3 67.85 1.304S9 99.4 71.14 1.40'4S10 101.3 75.80 1.344511 49.0 40.00 1.234S12 88.7 63.90 1.39'4S13 90.7 66.02 1.374S14 89.8 68.64 1.31.4515 94.7 72.82 1.30

Ziaḏ(6) ORW1 50.64 40.25 1.260RW2 57.44 40.25 1.43

GangaRao 1-0 79.86 81.76 0.98and Zia (15) 2-0 111.68 89.50 1.25

2-C 136.20 89.07 1.533-0 104.70 104.29 1.004-0 120.00 102.38 1.17

Hsu (7) P1 382 318.76 1.20P2 440 384.67 1.14P3 500 468.25 1.07P4 534 563.47 0.95P5 424 349.29 1.21P6 496 414.17 1.20P7 530 504.05 1.05P8 546 687.99 0.79

*Computed from Equation (12)

on the basis of the largest componentrectangle where the web reinforcementis usually placed.

By comparison with available test re-sults, it has been shown that, with onlya few exceptions, the above three equa-tions are reasonably and consistently onthe safe side (see Tables 3 through 5).2It would appear, therefore, that theseequations are acceptable as the basis ofdesign.

TORSION-SHEAR INTERACTION

Tests2 have shown that the interac-tions between torsion and shear for pre-stressed beams without web reinforce-ment can be adequately represented bya circular curve. By following the samedevelopment adopted by ACI 318-71,it can be shown that the permissibleshear stress for torsion and shear are,respectively:

PCI Journal/Marcia-April 1974 53

Table 4. Comparison of experimental ultimate torque of eccentrically prestressedrectangular beams with Eq. (12).

Ines-

tigator

Beam

No.

Tex

(in.-k)

Tḏ*th

(in.-k)

Tex

Tth

Chandler, et. E2 52.82 54.90 0.96al.ḏ(10) E4 64.70 62.94 1.03

E6 70.20 66.94 1.05E10 79.70 72.47 1.10E12 92.20 78.86 1.17E16 107.82 85.64 1.26E8 50.44 58.68 0.86E14 61.90 49.18 1.26E18 41.35 42.54 0.97E20 53.10 62.00 0.87SPE1244 66.70 58.51 1.14SPE1246 97.00 84.67 1.15SPE942 44.70 46.05 0.97

Ganga Rao (15) 5-0 106.80 109.30 0.98and Zia 6-0 119.50 107.96 1.11

Jacobsenḏ(16) 326 154.2 143.50 1.07

Mukherjee and 126 147.5 99.50 1.48Warwaruk (12) 226 152.3 121.23 1.26

Ziaḏ(6) 2RW1 48.44 40.33 1.202RW2 52.24 40.09 1.302.5RW1 53.44 39.81 1.342.5RW2 52.44 39.81 1.32

McGee and W2e-4.5-LL 105 63.99 1.64Ziaḏ(2)

Woodhead and I1-1 105 70.89 1.48McMullenḏ(17) 111-4 125 87.50 1.43

*Computed from Equation (12).

To _ where

1 + (^ v5 1 21 2 (15) T0' = 6^/f' (^/1 + 10 a-/f0'-/f0'- k) (17)L v; = lesser of vj and v defined

h+ Tby ACI Eqs. (11-11) and 11-12), re-

v0' spectively.

1+ (,a r,)2 (16) Q - 2 (Tc, 1 (18)

Uu J

TC _

Va =

54

It is noted that r6' and v0' in Eqs.(15) and (16) are the permissible shearstresses for torsion and flexural shear,respectively, if torsion or shear is act-ing alone. The factor in the denomi-nator of these equations accounts forthe interaction between torsion andshear. For a discussion of Eq. (18), seeReference 2.

WEB REINFORCEMENTFOR TORSION

Recasting the basic equation for tor-sional strength discussed above and

taking into account the torsion-shearinteraction, we have:

T. = Y-ax2yr0 + SZxiyiAtfB„

s_ Y.ax2y-r^

Transposing terms:SLx1y1Aa f sy

'x2y(?u -r0)=S

orAt_ (ru-Tc)s laxly (19)

12xiyif"in which, for simplicity, the coefficientfZ may be taken as:

11= 0.66 + 0.33 y1/x1

Table 5. Comparison of experimental ultimate torque of flanged and box sectionswith Eq. (14) and Eq. (13), respectively.

Inves-

tigator

Beam

No.

Tex

(in. -k)

T*th

(in. -k)

Tex

Tth

Wyss,ḏet.ḏal. A2 318.0 194.65 1.63(5) A3 391.0 223.4 1.75

A4 430.0 255.9 1.68A5 477.0 313.7 1.52A6 493.0 385.9 1.28B2 219.0 137.2 1.60B3 296.0 163.4 1.81B4 320.0 195.3 1.64B5 544.0 252.9 2.15B6 582.0 325.3 1.79

Zia (6) 0.25TW1 50.20 34.42 1.46(T-beams) 0.25TW2 54.44 34.42 1.58

2.25TW1 41.00 33.79 1.212.25TW2 46.70 33.79 1.382.75TW1 46.84 33.59 1.402.75TW2 50.56 33.59 1.51

Zia (6) 0.75IW1 54.00 31.13 1.74

(I-beams) 0.751W2 50.20 31.13 1.613IW1 59.72 30.77 1.943IW2 57.76 30.69 1.883.5IW1 59.00 30.58 1.933.5IW2 62.10 30.58 2.00

Johnston H-0-3-1 210 109 1.93and Zia (9) H-0-6-1 176 74 2.38

*Computed from Equation (14) or Equation (13).


Thus, the web reinforcement is re-quired for the torsional shear stress inexcess of that carried by the concrete.The web reinforcement for torsion mustbe in the form of closed stirrups and itshould be added to the web reinforce-ment for flexural shear. Note that thelatter could be open stirrups.

MINIMUM AND MAXIMUMWEB REINFORCEMENT

To avoid brittle failure, a minimumamount of web reinforcement must beprovided to resist both torsion andshear. A series of tests 2 with prestressedrectangular beams, containing a mini-mum amount of web reinforcement asspecified by ACI 318-71 for shear, haveshown that the post-cracking ductilityof the beams was insufficient underhigh ratios of torsion and shear. In viewof the sudden and explosive nature ofthe torsional failure, and the lack ofsufficient research data, it is suggestedthat a beam should be reinforced for noless than its cracking torque.

At the other extreme, considerationsshould be given to the possible dangerof over-reinforcing a member such thata compressive failure of the concretemight occur before the reinforcementyields. To avoid this type of failure, anupper limit of web reinforcementshould be established by specifying amaximum permissible nominal torsionalstress. If the design stress exceeds thislimit, then a larger beam section wouldhave to be used.

Since the prestressing force exerts acompressive load on a member beforeit is subjected to torque, it seems logicalthat the upper limit on the nominal tor-sional stress should be a function of theprestress ratio o-/f'0 . Based on this rea-soning, it will be assumed that the max-imum torque a rectangular section cancarry is:

T(max) = 2y C /f,' 1/1 + 10 v/f(20)

in which C is an unknown coefficientyet to be determined. It is assumedthat C is a function of a-/f0'.

For the extreme case of o-/f,' = 0,i.e., reinforced concrete members with-out prestress, ACI 318-71 specifies 12Vf^ as the limiting stress. Thus:

T.(ma,) = 3 x2iy 12 Vfc = O x2y C Vfcor C = 4/a.

For a typical cross section with anaspect ratio of two:

_ 0.35 0.35 _a- 0.75-1-0.5 1.25-0.28

andC = 4/0.28 --x= 14

Hence, in Eq. (20), the coefficient Ccan be taken as 14 for o-/f0' = 0.

For members subjected to high val-ues of a-/f0', Zia° has shown that atransition from tension failure to com-pression failure would take place ata-/f0' 0.6. Since the tensile strengthof the concrete can be taken as 6V f 0' ,it is then reasonable to limit the maxi-

mum nominal torsional stress as 6Vf0'for a-/f0' = 0.6. Assuming a linear func-tion of a-/f0' between these two limit-ing values for C, one obtains:

C = 14 — 13.33 (o-/f0')Ɂ(21)In Fig. 4, test results are compared

with Eq. (21) and it is noted that, witha few exceptions, all the data represent-ing tension failure fall below the curveof Eq. (21) and the data for thosebeams retorted to have failed in com-pression fall above the curve. It wouldseem that the coefficient C as repre-sented by Eq. (21) is a very reasonablevalue to be used as the upper limit ofthe nominal shear stress for both rec-tangular and flanged sections.

Eq. (20) can be extended to the caseof combined torsion and shear, again

56

tens. comp.fail. fail.

20 Rect. sectionconc. p/s o 0eccen. p/s •

18 Flanged sec. 0 0

16

14C °

12 o°^:°

10

00 (9 ° •g o°o o0

•W 0 ° s ^oo

8 • o 0 0ep • •

G ° C =14 -13.33 a-/f'

00 0.2 0.4 0.6

0/ fcFig. 4. Comparison of Eq. (21) with tests results for upper limit of nominal torsion

stress.

based on the circular interaction rela- wheretion. Thus:

,^fcC'= C1/1+10a-/foC

^'u(mam) = 2 (22) C = 14 — 13.331 + [(c\ 1O Tull

The term 10 f^ in Eq. (23) is based\ 10/1/̀f' /Jon the ACI Code upper limit for flex-

)"(max) = (23) ural shear stress and is obtained by tak-1 + g a = V in Section 11.6.4 of ACI[()()]2 318-71.2


LONGITUDINAL TORSIONREINFORCEMENT

To resist the longitudinal componentof the diagonal tension induced by tor-sion, longitudinal reinforcement mustbe provided in addition to that re-quired by flexure. This longitudinal re-inforcement should be approximately ofequal volume as that of stirrups fortorsion in order that both will yield atfailure, i.e., m f l5/ f sy = 1. Therefore:

A1(xi + yi) fsv (24)

l = Sfly

A l is the total area of longitudinaltorsion reinforcement and must be dis-tributed around the inside perimeter ofthe closed stirrups.

The question of whether the pre-stressing steel can be considered as apart of longitudinal torsion reinforce-ment should be considered. The pre-stressing steel is provided primarily forresisting flexure and often is not effec-tively distributed around the perimeterof the closed stirrups.

Tests2 have shown that without addi-tional longitudinal reinforcement ofmild steel, prestressed beams would failabruptly under high torsion. In light ofthis observation, it is recommended thatuntil more research data become avail-able, only the prestressing steel in ex-cess of that required for flexure and lo-cated around the perimeter of closedstirrups should be considered as a partof the longitudinal torsion steel, havingan equivalent area of:

A (mild)eq — Aps(eff) fys(off)!!l1/in which APS(eff) is the excess area ofprestressing steel and f,, ( ,ff) is the ef-fective stress in prestressing steel.

OTHER LIMITATIONS

To ensure the development of ulti-mate torsional strength and to control

cracking and stiffness at service load,the maximum yield strength of rein-forcement should be limited to 60,000psi and the maximum spacing of stir-rups should not exceed (xl + yl)/4nor 12 in. If required, longitudinal tor-sion reinforcement of not less than aNo. 3 bar should be distributed aroundthe stirrup at no more than 12 in. apart.In any event, at least one No. 3 barshould be placed at each corner of thestirrup.

For reinforced concrete members,ACI 318-71 permits the torsional effectto be neglected if the torsional shearstress is less than 25 percent of thecracking stress. This same requirementwould seem justified for prestressedbeams. That is to say, the torsional ef-fect may be neglected in design if thetorsional stress r,,, is less than:

1 .5\/fo' V1+10cr/f0'

SUMMARY OF PROPOSEDDESIGN PROCEDURE

The design procedure proposed here-in may be summarized as follows:

1. Determine the nominal torsionalstress according to Eq. (5). The stressTu should not exceed that specified inEq. (22).

2. If T < 1.5 Alf' Vi + 10 cr/f0' ,torsion may be neglected.

3. If T,.> T. according to Eq. (15),determine the required web reinforce-ment from Eq. (19). Check the mini-mum web reinforcement requirement.

4. Determine the required longitudi-nal torsion reinforcement according toEq. (24). Check the minimum require-ment.

DESIGN EXAMPLE

To illustrate the above design proce-dure, let us consider the design of tor-sional reinforcement in a prestressed

58

Inverted tee ledger girder

c^ I a I I I _- Precast conc.

1 p r 8 wide dbl. tee roof columns (typ.)

i M

0LI

-d

O '"O

0 (Columns supportM floor below -

r L"- girder - (spandrel) industrial loading

30'-0° 30'-0° I 30'-0° 30'-0° 30'-0°O @

Fig. 5. Partial plan of precast concrete roof.

concrete L-girder in a roof framing sim-ilar to that given in the PCA DesignNotes18 as shown in Fig. 5. The designlive load plus insulation and roofing is40 psf. For this loading, a double tee8DT18A and a L-girder 18LB30 can bechosen according to the PCI DesignIlandbook.19

From an analysis for flexure, thirteeni -in. diameter 7-wire strands (270 ksi

grade) are required for the L-girderwith the eccentricity of prestress atmidspan being 7.3 in. and that at thesupport being 3.65 in. Assume that theloss of prestress is 22 percent; f'. =5000 psi; f'ci = 3500 psi; fn = fly = 40ksi. In the following calculations, theusual capacity reduction factor forshear and torsion (A = 0.85 will be ap-plied.

(a) Calculate Tu and V,,, for spandrelWDL = 30' X 0.062 = 1.85 kips per ft (double tees)

0.45 kips per ft (spandrel beam)2.30 kips per ft

WLj = 0.040 X 30' = 1.20 kips per ftAssume no continuity at columns.

DL = 1.4x2.30X30X2=48.3kips

LL = 1.7 x 1.20 X 30 x 2 = 30.6 kips

V,, at centerline of support = 78.9 kipsNeglecting restraints from double tees and walls.


DL = 1.4 X 1.85 x (9/12) x 15' = 29.1 ft-kipsLL = 1.7 X 1.20 x (9/12) X 15' = 22.9 ft-kips

Tu at centerline of support = 52.0 ft-kips (or 625 in.-kips)

(b) Calculate shear stress v,^

V,, at "d = 22.7 in." from support = 78.9 X 13/15 = 68.4 kips= V,/4b 4,d = 68,400/0.85 X 12 x 22.7 = 295 psi

(c) Calculate torsion stress with Eq. (5)

?'u = Tu/4(^.ax2y)For maximum J.ax2y, consider beam web and outstanding ledge as compo-nent rectangles.

_ 0.35.

= 0.304 (beam web)«I =075+12/30

_ 0.35 = 0.280 (ledge)a2- 0.75+6/12

.ax2y = 0.304 X 122 X 30 + 0.280 X 62 X 12=1313+121=1434in.3

T,, at "d" from face of support = 625 X 13/15 = 541 in.-kips= 541/0.85 X 1434 = 445 psi

Effective prestress = 0.78 X 189 = 147 ksio- = 13 x 0.153 x 147/432 = 0.677 ksi = 677 psiu/ f0' = 677/5000 = 0.135

1.5 V/0' /1 + 10 a/ f = 1.5 x 70.5 X 1.53 = 162 psi < 677 psiRequires torsion design.

(d) Check Tu(maz) and v,j („ a^) with Eqs. (22) and (23)

C=14- 13.33(cr/f0')= 14 - 13.33(0.135) = 12.2C'=C-V1+10cr/f0' = 12.2 x 1.53 = 18.7

C' V fcɁ_ 18.7V5000

(e) Calculate torsion stress and shear stress carried by concrete To and v0with Eqs. (15) and (16)

To = 6\/fo (V1 + 10 o-/f' - kweb)k,,,eb = 1 - 0.133/x1 = 1- 0 .133/0.304 = 0.563

= 6x/5000 (-/1 + 10(0. 135) - 0.563) = 412 psi

60

vo = lesser of v, ti and v,w from ACI Eqs. (11-11) and (11-12)VP= vaw =3.5 1/fo +0.3ff0 +bw

= 3•5 V5000 + 0.3(677) + 5930/12 x 22.7= 247 + 203 + 22 = 472 psi

1R () (412)=0.573

T^ 412

1 295\21 + ( • 445)1+ ( ĵ Tu)

2

412_ = 270 psi

x/2.34ye,

vo =

1+

_( T \\

472

—= 358 psi

\/L75

472

1 F0.573 445)

(f) Calculate web reinforcement for torsion with Eq. (19)x1 =12-2X1.5=9in.y1 =.30-2X1.5=27 in.11 =0.66+ 0.33 (yl/xl)

= 0.66 + 0.33 (27/9) = 1.65 > 1.5Use fZ =1.5

— (Tu — Tc)s XicYx2yAe iyif8v

A t — (445 — 270) x 1434 - 0.0173 sq in. per in.T 1.5x9x27x40,000

Maximum s = xl 4 yl

= 9+27 9 in. < 12 in. 4

A t =0.0173X9=0.155 sq. in.Use No. 4 closed stirrups at 9 in. on center

(g) Check minimum web reinforcement for torsionMinimum web reinforcement should be what is required to develop crackingtorque of section.Torsional stress atcracking:Ter = 6Vf" V1 + 10 v/fo

= 6i/5000 V1 + 10(0.135) = 650 psiTorsional stress carried by concrete under pure torsion is T0' = 412 psi.A t (650-412)x 1434—s 1.5 x 9 x 27 x 40,000 = 0.0234 sq in. per in.

At = 0.0234 X 9 = 0.21 sq in. — 0.20 sq in.Thus, No. 4 closed stirrups at 9 in. on center is adequate.


(h) Calculate web reinforcement for shear

v, = 295 psiv, = 358 psi > v,But v, > v,/2; so minimum web reinforcement is required.

50b,^,s =/50 x 12 x 9 40,000 = 0.135 sq in.Av = f8^

A,,/2 = 0.0675 sq in.

12"

tt 4 sti r. @ 9' o.c.with 44&

T

ҟj 4 tt 3 long. bars

JL1

c.g.c.N

c® 44 12".g.s.

18" 410'o.c.

Fig .6. Reinforcement details f or L-girder.

CONCLUSIONS

A procedure for design of torsion inprestressed concrete members has beenproposed, which follows the same ap-proach as ACI 318-71 for design of tor-sion in reinforced concrete members.Comparison2 with available test resultsof 132 beams under combined loadingas reported in the literature indicatesthat the proposed procedure is reason-ably conservative for rectangular beamsand quite conservative for flanged andbox beams. The application of the de-sign procedure has been illustrated bya design example.

When more complete research databecomes available, especially for pre-stressed flanged and box members, theprocedure can be further refined andsimplified. In the meantime, design ta-bles and curves can be easily preparedto aid the designers in reducing theamount of computation.

ACKNOWLEDGMENTS

This paper is based on the results ofone phase of a continuing torsion re-search program supported by the Na-tional Science Foundation Grant No,GK-1843X. This financial support isgratefully acknowledged.

REFERENCES

1. ACI Committee 318, "BuildingCode Requirements for ReinforcedConcrete (ACI 318-71)," Ameri-can Concrete Institute, Detroit,1971, 144 pp.

2. McGee, W. D. and Zia, P., "Pre-stressed Concrete Members underTorsion, Shear and Bending," Re-search Report, Department of CivilEngineering, North Carolina StateUniversity, Raleigh, N.C., July1973, 212 pp. (Also available as aPhD. dissertation by W. D. McGee,University Microfilm, Ann Arbor,Michigan).

3. ACI Committee 318, "Commentaryon Building Code Requirements forReinforced Concrete (ACI 318-71)," American Concrete Institute,Detroit, 1971, 96 pp.

4. Hsu, T. T. C., "Torsion of Structur-al Concrete—Uniformly PrestressedRectangular Members withoutWeb Reinforcement," PCI JOUR-NAL, Vol. 13, No. 2, April 1968,pp. 34-44.

5. Wyss, A. N., Garland, J. B., andMattock, A. H., "A Study of theBehavior of I-Section PrestressedConcrete Girders Subject to Tor-sion," Structures and MechanicsReport No. SM 69-1, University ofWashington, Seattle, Washington,1969.

6. Zia, P., "Torsional Strength of Pre-stressed Concrete Members," ACI


Journal, Vol. 32, No. 10, April1961, pp. 1337-1359.

7. Private communication to ACICommittee 438, Torsion, fromThomas T. C. Hsu, October 17,1967.

8. Swamy, N., "The Behavior and Ul-timate Strength of Prestressed Con-crete Hollow Beams Under Com-bined Bending and Torsion," Mag-azine of Concrete Research (Lon-don), Vol. 14, No. 40, March 1962,pp. 13-24.

9. Johnston, D. W., and Zia, P., "Hol-low Prestressed Concrete Beamsunder Combined Torsion, Bendingand Shear," Research Report, De-partment of Civil Engineering,North Carolina State University,Raleigh, N.C., December 1971, pp.178. (Also available as a PhD. dis-sertation by D. W. Johnston, Uni-versity Microfilm, Ann Arbor,Michigan)

10. Chander, H., Kemp, E. L., andWilhelm, W. J., "Behavior of Pre-stressed Concrete RectangularMembers Subjected to Pure Tor-sion," Civil Engineering StudiesReport No. 2007, West VirginiaUniversity, Morgantown, West Vir-ginia, 1970.

11. Mukherjee, P., and Kemp, E. L.,"Ultimate Torsional Strength ofPlain, Prestressed and ReinforcedConcrete Members of RectangularCross Section," Civil EngineeringStudies Report No. 2003, West Vir-ginia University, Morgantown,West Virginia, 1967.

12. Mukherjee, P., and Warwaruk, J.,"Prestressed Concrete Beams withWeb Reinforcement under Com-bined Loading," Structural Engi-neering Research Report No. 24,Department of Civil Engineering,University of Alberta, Edmonton,Alberta, Canada, 1970.

13. Okada, K., Nishibayashi, S., and

Abe, T., "Experimental Studies onthe Strength of Rectangular Rein-forced and Prestressed ConcreteBeams Under Combined Flexureand Torsion," Transactions, Japa-nese Society of Civil Engineers,No. 131, July 1966, pp. 39-51.

14. Superfesky, M. J., "Torsional Be-havior of Prestressed RectangularConcrete Beams," MS thesis, De-partment of Civil Engineering,West Virginia University, Morgan-town, West Virginia, 1968.

15. GangaRao, H. V. S., and Zia, P.,"Rectangular Prestressed Beams inTorsion and Bending," Journal,Structural Division, ASCE, Vol. 99,No. ST1, January 1973, pp. 183-198.

16. Jacobsen, E. B., "Torsion, Bendingand Shear in Rectangular Pre-stressed Concrete Beams," MS the-sis, Department of Civil Engineer-ing, University of Alberta, Edmon-ton, Alberta, Canada, 1970.

17. Woodhead, H. R. and McMullen,A. E., "A Study of Prestressed Con-crete under Combined Loading,"Research Report No. CE 72-43,University of Calgary, Calgary,Canada, 1972. See also PCI Jour-nal, Vol. 18, No. 5, September-Oc-tober 1973, pp. 85-100.

18. Notes on ACI 318-71 with DesignApplications, Portland Cement As-sociation, Skokie, Illinois, 1972, p.12-17.

19. PCI Design Handbook, PrestressedConcrete Institute, Chicago, Illi-nois, 1971, p. 3-26, and p. 3-47.

NOTATION

A l = area of longitudinal torsion re-inforcement

At = cross-sectional area of one leg ofstirrup

V,, = ultimate shear force

64

To = contribution by concrete to tor-sional strength of non-pre-stressed beam

To' = contribution by concrete to tor-sional strength of prestressedbeam

Tu = ultimate torque= torsional strength of plain con-

crete beam= concrete cylinder strength

fly = yield strength of longitudinal re-inforcement

f, = modulus of rupture of concretef $y = yield strength of stirruph = wall thickness of box sectionm = volume ratio of longitudinal re-

inforcement to web reinforce-ment

s = spacing of stirrupv, = permissible flexural shear stress

under combined torsion and

flexural shearv,' = permissible flexural shear stress

without torsion as defined byACI Eq. (11-11) or (11-12)

v, = nominal flexural shear stressx = shorter side of rectangular sec-

tionxl = shorter leg of closed stirrupy = longer side of rectangular sec-

tionyl = long leg of closed stirrupa = torsion constant, see Eq. (1)fl = reinforcement coefficient, see

Eq. (6)a- = average prestress on section,r, = permissible shear stress for tor-

sion under combined torsion andflexural shear

= permissible shear stress for tor-sion without flexural shear

,r,, = nominal torsional shear stress

Discussion of this paper is invited.Please forward your discussion to PCI Headquartersby August 1, 1974, to permit publication in theSeptember-October 1974 PCI JOURNAL


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