Full-Scale Torsion, Shear, and Bending Tests of

Alan H. MattockProfessor of Civil EngineeringUniversity of WashingtonSeattle, Washington

A. N. WyssPartner

Balzari, Blaser, Schudel,Engineers and Planners

Berne, Switzerland

Full-Scale Torsion, Shear,and Bending Tests ofPrestressed I-Girders

Based on full-scale tests and a theoretical analysis,the authors propose a possible form of interactiondiagram for the design of eccentrically prestressedconcrete I-girders.

The primary objective of this studywas to determine the influence of

the moment to shear ratio on thetorsion-shear interaction relationshipfor diagonal tension cracking in I-section prestressed concrete girders.This relationship is of practical inter-est since the contribution of the con-crete to ultimate strength in bothshear and torsion in a beam with webreinforcement is related to the shearand torque at diagonal tension crack-ing. Also, serviceability considerationsmay require that diagonal tensioncracking be avoided under serviceloads.

Interaction between torsion and

shear occurs because both torsion andshear produce diagonal tension stress-es in concrete beams. These stressesare additive on one face of the beamand subtractive on the other. There-fore, when a beam is subject to bothtorsion and shear, diagonal tensioncracking will generally occur at atorque less than the diagonal tensioncracking torque in pure torsion, and ata shear less than that which wouldcause diagonal tension cracking ifshear and moment only acted on thebeam.

Two different types of diagonal ten-sion cracking can occur in a pre-stressed concrete beam subject to

22

shear and moment only. The so-called"shear-flexure cracking," in which di-agonal tension cracking occurs in aflexurally cracked region (this corre-sponds to v im; in Section 11.5.2 of ACI318-71 1 ); and "web cracking," inwhich diagonal tension cracking oc-curs in a region without flexure cracks(this corresponds to v, u in Section11.5.2 of ACI 318-71).

The shear at which diagonal tensioncracking occurs is very dependent onthe moment to shear ratio in the caseof "shear-flexure cracking," but is in-dependent of the moment to shearratio in the case of "web cracking."This study was designed to determinewhether the shear-torsion interactionrelationship is affected by the type ofdiagonal tension cracking which oc-curs when shear and moment only act.

Experimental Study

Test programTwo groups of five girders each

were tested. One girder (Al) whichwas subject to pure torsion, was com-mon to both series. All nine girdershad the same cross section and pre-stress distribution. The first group,consisting of Girders Al through A5,were tested at a moment to shear(M/V) ratio of 4 ft, so that under trans-verse loads only, diagonal tensioncracking would occur in a flexurallyuncracked region (i.e., "web crack-ing").

The second group, consisting ofGirders Al and A6 through A9, weretested with a moment to shear ratio of91/2 ft, so that under transverse loadsonly, diagonal tension cracking wouldoccur in a flexurally cracked region(i.e.,, "shear-flexure cracking"). Withineach series the ratio of torsion to shearwas different for each girder, rangingfrom infinity (pure torsion) to zero(transverse load only).

PCI JOURNAL/March-April 1978 23

14 Straight31 3 g strands

3 z+—t 2 4 No.

24 11444 5 4'-z

I ^ Iz2 -♦+i *+^ - -I— 6 No.32 -+^^r++ ++ 8 No.

"chamfer L7 at 12 I I`(All dimensionsI4 in inches.)

Fig. 1. Cross section of test girders.

Test specimensThe cross section of the 26-ft long

girders was that of the WashingtonState Department of Highways stan-dard "30-ft series" prestressed con-crete bridge girder. The cross-sectional dimensions and details ofthe reinforcement are shown in Fig. 1.Two-legged #4 bar stirrups were pro-vided at 2-in, centers in the end 6 in.of each girder, and at 8-in, centersover the remainder of the girder, ex-cept for that part of the girder whichwould fall within the test shear span(i.e., that part of the girder betweenthe applied load and the adjacentsupport, where cracking observationswere to be made). No web reinforce-ment was provided within the testshear span.

The girders were prestressed bypretensioned 3/s-in. seven wirestrands, each of which was initiallytensioned to 14 kips. The ultimatestrength of the strand was 259 ksi. Theelastic and creep shortening, andshrinkage of the girders, was mea-sured over a gage length of 100 in. atthe top and bottom of each girderweb. These deformations were taken

into account when calculating the ef-fective prestress force and eccentricityat the time of test (see Table 1).

The girders were fabricated by theAssociated Sand and Gravel Co. Inc.,of Everett, Washington. The concretehad a strength of about 5500 psi attransfer of prestress and reached astrength of about 7000 psi at 7 days,the time of test. It was made fromType III portland cement, sand, and aglacial outwash gravel of 3/a-in.maximum size. About 10 cylinderswere made at each casting and theywere cured alongside the test girders.The concrete strengths at the time oftest are summarized in Table 1.

Test arrangementsand procedures

The equipment used to apply thetorsional moment consisted of a tor-sion head resting on a spherical bear-ing, through which the torque wasapplied and which also supported oneend of' the test girder; and also a re-straint head which supported the op-posite end of the test girder and re-strained it against twisting. The re-straint head supplied a vertical reac-

24

tion only to the test girder, it did notrestrain longitudinal motion, nor rota-tion of the girder end due to bendingdeformations. The torsion head wasactuated through load cells by a pairof hydraulic rams which weresupplied with oil by a Pendomaticpumping and measuring unit. Fric-tional losses in the bearings were de-termined in separate calibration tests.

Two equal transverse loads wereapplied to the girders by a 2400-kiphydraulic testing machine, actingthrough a distribution beam. Each endof the distribution beam acted on thetest girder through a cylindrical bear-ing and a rocker arranged at right an-gles to one another and a horizontaltransversely sliding bearing. The axisof the cylindrical bearing was made

to coincide with the longitudinal axisof the test girder. Both surfaces of allbearings were coated with a 1/16-in.laver of Tetralan to reduce friction.

This arrangement of bearings wasused so that the test girder could twistand deform with a minimum of re-straint from the transverse loadingequipment. Rocker bearings were alsoprovided between the upper plattenof the testing machine and the top ofthe distribution beam. The arrange-ment of the equipment for test can beseen in Fig. 2.

The girders were tested on a span of22 ft. The transverse loads wereapplied 4 ft from the supports in thecase of Girders A2 through A5 and 9'/2

ft from the supports in the case of Gir-ders A6 through A9.

Table 1. Details of test girders. (Data relates to time of test.)

Girder M/V(ft)

T/V(approx.)

PrestressForce, F(kips)

Eccentricityes(in.)

Concrete Stress(psi)_______

fc(5)(psi)

ft(6)(psi)

Top Bottom

Al - on(l) 217.4 4.89 147 2066 7600 534

A2 4.0 0(2) 215.7 4.94 136 2060 7300 600

A3 4.0 2Tc/V (3) 215.2 4.95 136 2056 7160 536

A4 4.0 Tc /VCW 216.5 4.91 141 2061 7130 552

A5 4.0 Tc/2VcW 215.0 4.87 153 2040 7470 554

A6 9.5 0(2) 217.4 4.88 150 2063 8050 589

A7 9.5 2Tc/Vc^4) 222.2 5.04 121 2142 7720 584

A8 9.5 Tc/Vci 216.8 4.83 150 2049 7370 540

A9 9.5 Tc/2Vci 219.7 5.04 118 2117 6810 614

(1) Torsion only.(2) Transverse load only.(3) Tc = Diagonal tension cracking torque, pure torsion.

Vc _ "Web cracking" shear.(4) Vyi = "shear-flexure cracking" shear.(5) f = concrete compression strength at test (6xl2-in. cylinder).(6) ft = concrete splitting tensile strength at test (6x12-in. cylinder).


Fig. 2. Test rig showing prestressed girder viewed from the restraint end.

The angle of twist was measuredover the length of the shear spanusing an angular differential trans-former coupled to a strip chart recor-der. Deflections were measured usingindependently supported 0.001-in.dial gages. More complete details ofthe testing equipment and in-strumentation have been reportedelsewhere.2

The test girders were gripped in thetorsion test rig. The torsion head waslifted slightly and then reseated, toeliminate any twist that may havebeen imposed on the girder inadver-tently during the process of clampingthe ends. The transverse load bearingplates were carefully positioned andaligned. Dial gages were put in placeand the zero readings were thentaken. The distribution beam was nextplaced in position, and the testingmachine upper platten was broughtjust in contact.

The torsional and transverse loadswere increased incrementally until

the girder failed. At each load incre-ment, the torque was first increased,then the transverse load. The size ofthe load increments was varied duringthe test, generally reducing in mag-nitude as the load increased. Aftereach increment of load was applied,readings of all gages were taken andany new cracks were marked.

Behavior of girders under testGroup 1 (girders with M/V = 4

ft)—As expected, diagonal tensioncracking occurred in all the beams ofthis series before flexure cracking oc-curred in the shear span. Girder A3,subject to a high torsion to shear ratio,behaved in a similar manner to GirderAl, subject to pure torsion, except thatin Girder A3 the first diagonal tension.crack formed at mid-height of theweb, whereas in Girder Al the firstdiagonal tension crack formed on thetop of the girder. In both cases thecracks propagated in spiral fashionaround the girder, covering both faces

26

Girder A3 (west face) Girder A3 (east face)(a) High torsion to shear ratio.

Girder A4 (west face) Girder A4 (east face)(b) Intermediate torsion to shear ratio.

Girder A5 (west face) Girder A5 (east face)(c) Low torsion to shear ratio.

Fig. 3. Cracking behavior of typical girders (Girders A3, A4, and A5) after test.

of the web and penetrating deeplyinto the flanges. Failure occurred inthe skew bending mode after shear-compression failure of the flanges [seeFig. 3(a)] .

Girder A5, subject to a low torsion

to shear ratio, behaved in a similarmanner to Girder A2, subject to trans-verse loads only. The first diagonaltension crack penetrated completelythrough the web in both cases, slopingupward toward the loading plate on


Table 2. Torque and shear test data for Girders Al-A9.

Diagonal Tension Cracking Ultimate

Torque Shear Torque ShearGirder (in.-kips) (kips) (in.-kips) (kips)

Al 342.6 2.6 394.0 2.6

A2 0 81.3 0 101.3

A3 394.6 31.7 451.4 36.8

A4 311.8 51.6 393.6 72.1

A5 112.2 72.6 135.8 110.1

A6 0 41.8 0 57.6

A7 364.3 17.8 414.0 20.8

A8 301.8 29.6 334.7 34.1

A9 187.7 38.6 215.0 46.6

both faces and extending immediatelyfrom top to bottom of the web. Failurefinally occurred when the top flangefailed in shear compression after pen-etration of the diagonal tension cracksthrough about half its thickness. Fail-ure was abrupt. Flexure cracks oc-curred in the shear span between di-agonal tension cracking and failure[see Fig. 3(c)].

Girder A4 was subject to an inter-mediate ratio of torsion to shear andexhibited a transitional mode of be-havior. Cracking initially developedas in the girders with a low torsion toshear ratio. However, the final modeof failure was in the skew bendingmode. Failure was fairly abrupt. Noflexure cracks formed in the shearspan [see Fig. 3(b)].

Group 2 (girders with M/V = 91/2ft)-Flexure cracking occurred in theshear span before diagonal tensioncracking in Girders A6, A8, and A9.Flexure cracking did not occur in theshear span at any stage of loading inGirder A7.

Girder A7, subject to a high torsionto shear ratio, and Girder A8, subject

to an intermediate torsion to shearratio, behaved in a similar manner toGirder Al tested in pure torsion.However, in the case, of Girders A7and A8, the first diagonal tensioncrack initiated in the web rather thanon the top face, and cracks only oc-curred on the face of the web wherediagonal tension stresses due to shearand torsion were additive. The crackspropagated into the top and bottomflanges in spiral fashion. Failure oc-curred in the skew-bending modewhen the flanges failed in shear com-pression and crushing occurred on thepreviously uncracked face of the web.Failure was quite sudden in bothcases.

Girder A9, subject to low torsion toshear ratio, behaved in a similar man-ner to Girder A6, subject to transverseloads only. In both cases a diagonaltension crack formed over most of thedepth of the web, shortly after the oc-currence of a flexure crack in theshear span at about half the depth ofthe beam away from the loading point.This diagonal tension crack pene-trated completely through the web.

28

Under increasing load the crack prop-agated into the top and bottomflanges. Failure occurred by shearcompression in the flanges. Failurewas quite sudden, but not as abrupt asin the case of Girders A2 and A5.

The torques and shears at diagonaltension cracking and at ultimate areshown in Table 2.

Discussion ofTest Results

Interaction of torsion, shear,and bending at diagonaltension cracking

As a preliminary to constructing thetorsion-shear interaction diagramsfrom theoretical considerations, astudy was made of the maximum prin-cipal tensile stress at diagonal tensioncracking in those beams which did not

have flexure cracks in the shear spanat the time of diagonal tension crack-ing.

Initially, the shear stresses due totorsion and the transverse shearstresses were both assumed to be dis-tributed according to the elastictheory. The values of these stresses atthe critical locations are listed in Col-umns 2 and 4 of Table 3. In the case ofGirder Al subject to pure torsion, thecritical location is the center of the topflange. In the other girders, themaximum principal tensile stress wasfound to be at the neutral axis.

The principal tensile stresses, basedon the elastic theory distribution ofshear stresses, are listed in Column 6,and are repeated in non-dimensionalform in Column 8. In this column theyhave been divided by the square rootof the concrete compressive strength,(measured on 6 x 12-in. cylinders).

Table 3. Stresses at diagonal tension cracking in girders not previously cracked in flexure

GirderCritical Stresses at

Diagonal Tension Cracking Precom-pression

Maximum PrincipalTension Stress

t a t—(1)

2'e(2)

2(3)

ve(4)

6t'(5)

^tpfc fc

( ps i ) (psi) (psi) (psi) (psi) (psi)

A1 1 636 0 147 940 567 10.78 6.50

A2 0 0 902 1098 507 507 5.93 5.93

A3 1128 733 352 1096 1030 668 12.17 7.90

A4 892 580 572 1101 1013 727 13.39 8.62

A5 321 209 804 1096 704 604 8.15 6.98

A7 1043 677 197 1131 798 487 9.10 5.54

(1) Torsional shear stress according to elastic theory.(2) Torsional shear stress assuming fully plastic distribution of torsional

shear stresses.(3) Transverse shear stress according to elastic theory.(4) Principal tension stress based on total shear stress = (e + ye).

(5) Principal tension stress based on total shear stress = (2p + ve).

(6) Critical location is center of top face for Girder Al and at mid-height ofweb for all other girders.


It can be seen that the maximumprincipal tensile stress calculated inthis way varies from 5.93 in thecase of Girder A2 subject to transverseloads only, to 13.39 ,' in the case ofGirder A4 subject to combined loads.In the case of pure torsion, themaximum principal tension stress is10.78 J7,.

The values of 5.93 ,' in the caseof Girder A2 corresponds well withthe average "split cyclinder" tensilestrength of the concrete of 6.57which indicates a direct tensilestrength of about 5.5 V` to 6 VT,' . Itfollows that an elastic theory calcula-tion of maximum principal tensilestress at diagonal tension cracking dueto shear in an uncracked girder isrealistic. However, in the case of puretorsion and combined loading, themaximum principal tensile stressescalculated on the basis of the elastictheory are very different from theknown tensile strength of the con-crete.

Such a calculation is not thereforerealistic. This was recognized by Ny-lander3 for the case of reinforced con-crete beams and he suggested that afully plastic distribution of shearstress due to torsion be assumed.

The next step was therefore to re-calculate the maximum principal ten-sile stresses at diagonal tensioncracking on the basis of a fully plasticdistribution of torsional shear stressesand elastic theory distribution oftransverse shear stresses. The plastictheory torsional shear stresses arelisted in Column 3 of Table 3, and thecorresponding maximum principalstresses in Columns 7 and 9 of thesame table.

It can be seen that the calculatedmaximum principal tension stressesare much more consistent in mag-nitude and, except in the case of Gir-ders A3 and A4, correlate reasonablywell with the tensile strength of theconcrete. It appears, therefore, that

Nylander's 3 proposal for reinforcedconcrete (to combine the plastictheory torsional shear stress with theelastic theory transverse shear stress)also leads to reasonably realistic esti-mates of maximum principal tensilestress at diagonal tension cracking inprestressed concrete girders.

Torsion-shear interaction diagramswere therefore constructed for the twogroups of beams, based on the as-sumption that the torsional shearstress can be calculated on the basis ofa fully plastic distribution and thetransverse shear stress on the basis ofthe elastic theory distribution. Threepossible cases of cracking were con-sidered:

1. In the top face.2. In the web, without flexure

cracks (web cracking).3. In the web, after flexural crack-

ing (shear flexure cracking). .The calculations were based on the

following average values for the gir-ders tested.

Compressive strength:f,' = 7400 psi

Assumed tensile strength:f,' =6 c'= 516 psi

Precompression at top face:o-t=134psi

Precompression at bottom face:vb = 2070 psi

Precompression at neutral axis:o= 1102 psi

Case 1. Cracking in top face-In thiscase, only the torsional shear stressneed be combined with the precom-pression to obtain the principal tensilestress, since the transverse shearstress is zero at the top face. However,the bending moment coexisting withthe applied shear will cause addi-tional compressive stresses in the topface and this will increase the torquerequired to produce a principal tensilestress equal to the tensile strength ofthe concrete fz , and so cause cracking.

The additional compression caused

30

by the applied shear will diminish,moving from the transverse load to-ward the support. It was assumed thatthe location nearest the support atwhich a crack could initiate and prop-agate to the edges of the flange, is halfthe flange width out from the grippingplates of the restraint head, i.e., 14'/2in. from the center of support.

The effective compression in thetop face was therefore taken as:

= 134 1000V (14.5)

vt + St

(134 + 13.06V) psi (1)where V is the shear in kips and S t isthe section modulus with respect tothe top face in cubic inches.

The torsional shear stress T at diag-onal tension cracking is then given by:

vT – .ft 1+-

/

ftt

516 1+ 134+ 13.06V

=516516

= 516 1.26 + V/39.5 psi (2)

For the girder section used, the tor-sional shear stress at the diagonal ten-sion cracking torque Ter, assuming afully plastic distribution of shearstress, is given by:

T = 1.86 Tcr psi

where Tcr is in in.-kips. Hence:

Tcr = 277 1.26 + V/39.5 (3)

Case 2. Web cracking—In this casethe torsional shear stress r, must becombined with the transverse shearstress v, when calculating the princi-pal tensile stress corresponding to di-agonal tension cracking. Preliminarycalculations indicated that at diagonaltension cracking the maximum princi-pal tension stress occurred at theneutral axis, or that its value was veryclose to the value calculated at theneutral axis.

For simplicity therefore, it was as-sumed in calculating the interactionrelationship that the maximum princi-pal tensile stress would occur at theneutral axis. The longitudinal com-pressive stress to be combined withthe total shear stress is therefore con-stant and equal to o-, the precompres-sion at the neutral axis.

At diagonal tension cracking:

T +v = ft 1+t

=516 1+ 1102516

Therefore, T + v = 915.Now, -r = 1.86Tcr psi, according to

the plastic theory, Tcr is in in.-kips.Also, v = 11.07Vcr, according to the

elastic theory, Vcr is in kips.

Therefore:1.86Tcr + 11.07 Vcr = 915

and

Tcr + 5.95 Vcr = 492 (4)

or

Tcr + X= 1 (4a)492 82.6

Case 3. Shear-flexure cracking—Thehypothesis4 for shear-flexure diagonaltension cracking is that the diagonaltension crack will form at a shear 0.6bud f,' greater than the shear corre-sponding to the formation of a criticalflexure crack. The critical flexurecrack is considered to occur a distancefrom the section under considerationequal to half the effective depth of thesection, measured in the direction ofdecreasing moment.

The shear at shear-flexure crackingis then given by:

V,t = 0.6b^,d fJ + M Mcr + Vd

V 2(5)


where

b, = web widthd = effective depth of sectionMet. = net flexural cracking mo-

ment = S6(r + fpe — fd )Vd = dead load shearMAT = moment to shear ratio due

to applied loadsS b = section modulus with re-

spect to flexural tensionface

fPe = precompression at flexuraltension face

fd = flexural tension stress dueto dead load

f, = modulus of rupture of con-crete

Eq. (5) was Eq. (26-12) in ACI318-63. 5 Eq. (11-11) of ACI 318-71'was derived from Eq. (5) conserva-tively simplified by omitting "– d/2"from the denominator of the secondterm.

When developing the interactionrelationship for this case, it was as-sumed that the applied torque wouldinfluence the diagonal tension crack-ing shear both by its influence on thenet flexural cracking moment M cr andby reducing the first term of Eq. (5);0.6 b, dV , . This first term is as-sumed to be affected by coexistingtorque in the same way that Vcr is af-fected in Case 2, i.e., a linear interac-tion relationship is assumed betweenTer and OVj, the first term in the ex-pression for shear at shear-flexure di-agonal tension cracking:

Ter + 0 Ve{ — =1

492 0.6b,,, d f'

orTC1.

LVC1 = 0.6 b,u d Jf^ (1– 492)(6)

In Section 11.5.2 of ACI 318-71, themodulus of rupture of the concrete istaken as 1.5 times the direct tensilestrength, (6 f7TI4 f,' ). In these cal-

culations the modulus of rupture f,.was therefore also taken as 1.5 timesthe direct tensile strength, namely,9fj (i.e., for these girders, f,. = 774psi).

The net tension at the bottom facedue to moment and prestress, atflexural cracking is:

f = er — (fpe — J d )

Vf(a – –= S b (fre —.fd )

Therefore, for Group 2 girders:

_ 1000Vf (102)

f(2070 – 126) psi

1109

= 92V1 – 1944 psi (7)

where Vf is the applied load shear atflexural cracking in kips.

If the shear stress due to torsion is 7,

then at flexural cracking we have:

(^)22 + T2 + 2

That is, for the girders of Group 2:

774 = (46V f – 972) +

T2 + (46Vf – 972)2

Hence:Vf = 29.52 – T2/71,300 kips

= 29.52 – Tf/20,600 kipswhere Tf is the torque at flexuralcracking in in.-kips.

Hence:

V,j =0.6bw d f (i_ 492+

(29.52 – TT/20,600) + Vd(8)

If the torque is held constant as theshear increases, Tf = Ter and we maywrite:

32

V,=0.6b,,d -f (1- 492) +(29.52 - T /20,600) + Vd'

(9)

= 5.88 ( 1- 492 ) + (29.52 -

T^2 /20,600) + 0.3 (9A)

If the torsion and the shear are in-creasing in proportion to one anotherTf will be less than Tyr and Eq. (8)would have to be solved by iteration.Alternately, Eq. (9) may be used toobtain a reasonably conservative valueof Vci•

The calculated interaction relation-sh"ips are compared with the test datafrom Groups 1 and 2 in Figs. 4 and 5,respectively. [Eq. (9) was used forCase 31. For convenience the relation-ships are plotted in nondimensionalform, using as reference values thecalculated values of pure torsionalcracking torque T,, the "web cracking"shear V,,p in the case of Group 1, andthe "shear-flexure cracking" VCZ in thecase of Group 2. (Note that the coor-dinates of the data points are listed inTable 4.)

It can be seen that the calculatedinteraction relationships reflect the

Case I - D.T. cracking in top

AAIface of girder,(calc.)

A3

Test Results

OA4

0.6

^^xl •A5

0.2Group I Girders. M/V =Oft A2\

ri

Vcr /VVW(CalC.)

Fig. 4. Nondimensional interaction diagram for diagonal tensioncracking in Group 1 girders.

1.6

1.4

1.2

1.0

C)


S.-Case 2-S.D.T. cracking in webbefore flexural cracking

5- in shear span, (calc.)Test Results

U0C)^ 0.8

Case I -/

\5-S.

*A8D.T. cracking in topface of girder, (caic.)

Case 3 -D.T. cracking in webafter flexural crackingin shear span, (caic.)

Group 2 Girders, M/V = 9.5ft

0 0.2 0.4 0.6 0.8 I.0 1.2

Vcr/Vci(Caic.)Fig. 5. Nondimensional interaction diagram for diagonal tensioncracking in Group 2 girders.

1.4

1.2

0.6

0.4

0.2

trends of the data reasonably well, ex-cept that the values of cracking torqueand shear for Girders A3 and A4 ob-tained from the calculated relation-ships are rather conservative. Thecracking of Girder A7 was initiallyidentified as Case 2 (web) cracking.However, the first crack recorded wasat the web-top flange junction andcould have been triggered by an un-seen crack on the top rough trowelledface of the girder. (It was much easierto identify a fine crack on the faces ofthe girder which had been cast againstformwork, than on the top trowelledface.)

The torsion-shear ratios to whichGirders A3 and A4 were subjected, re-sulted in the torsional shear stressesand the transverse shear stressesbeing almost equal in absolute mag-nitude. Hence, on the web face inwhich these shear stresses were ofopposite sign, the resultant shearstresses were very small, so that no orvery small, principal tensile stresseswould exist in the concrete in thatface. The maximum principal tensilestresses would therefore extend onlyhalfway through the thickness of theweb.

In the cases of pure torsion and of

34

Table 4. Ratios of diagonal tensioncracking torque Tcr, and shear V 0, to cal-culated values* of T, (311 in.-kips), V,W

(82.6 kips), and V 0; (35.4 kips).'

Girder Tcr VcrV. Vci

Al 1.103 0.032 0.073A2 0 0.985 --A3 1.270 0.384 --A4 1.003 0.625 --A5 0.361 0.878 --A6 0 -- 1.180A7 1.172 -- 0.503A8 0.970 -- 0.836A9 0.603 -- 1.090

*Tc = Diagonal tension cracking torque inpure torsion, based on fully plastictorsional shear stress distribution.

VcW = Diagonal tension cracking shearin flexurally untracked beam basedon elastic theory distribution oftransverse shear stress.

U c^ = Diagonal tension cracking shear ina flexurally cracked beam.

shear without torsion, the principaltension stresses would be of equalmagnitude through the entire thick-ness of the web. It is possible thatcracking would occur at a lower valueof principal tension stress when theentire web thickness is stressed to thesame level, than when only half theweb thickness is subject to maximumstress and the other half is unstressed.The highly stressed half wouldperhaps be reinforced by the un-stressed half.

Alternatively, in the latter case itmay be that further internal redis-tribution of stress occurs, so that theprincipal tension stress calculatedwithout taking this into account, over-estimates the actual tension stress inthe concrete. This would parallel thesituation at flexural cracking, where

the ',modulus of rupture exceeds theactual tensile strength of the concrete,because in its calculation, redistribu-tion of the flexural tensile stresses isnot taken into account.

It has been shown2 that for the gird-ers of this study, if the principal ten-sile stress at web cracking is taken as:

r zv 1vt=2.5^f"[ 4-

r +v (10)

then the shears and torques at Case 2(web) cracking can be calculated al-most exactly. However, the generalapplicability of Eq. (10) is uncertain.A more generally applicable assump-tion might be that the principal tensilestress at web cracking would varylinearly between the concrete tensilestrength, when:

(r – v)/(T + v) =1

and the modulus of rupture of theconcrete when:

(T- v)/(r + v)I = 0

The case of(r–v)l(,r+v)I =1

corresponds to pure torsion (v = 0) orto shear and flexure only (T = 0).

The case of

(T–v)l(r+v) =0

corresponds to the shear stresses dueto torsion and to shear being equal inmagnitude. Further experimentalstudy would be needed to validatethis assumption.

Girder stiffness under combinedloading

The measured and calculated tor-sional and flexural stiffnesses arecompared in Table 5. The measuredtorsional stiffness is based on the twistmeasured over the length of the shearspan. The measured flexural stiffnessis based on the measured midspandeflection relative to the supports. Inboth cases the stiffnesses are those of


Table 5. Girder stiffnesses.

GirderTorsional Stiffness(10 3 kip. in. 2/deg.) Test

Cam.

Flexural Stiffness(106 kip. in. 2 ) Test

Calc.

Test Calc. Test Calc.

Al 71.8 67.5 1.06 -- -- --

A2-- -- -- 75.0 73.8 1.02

A3 75.2 65.6 1.14 77.3 73.2 1.06

A4 75.9 65.4 1.16 78.2 73.0 1.07

A5 68.4 67.0 1.02 76.2 74.7 1.03

A6 -- -- -- 84.3 77.6 1.09

A7 74.2 68.2 1.09 75.8 76.0 1.00

A8 69.3 66.5 1.04 76.0 74.2 1.02

A9 88.7 63.9 1.39 76.6 71.3 1.07

Average* = 1.12 Average = 1.05*Note that if Girder A9 is neglected the average becomes 1.08.

the uncracked girder. These are thestiffnesses of practical interest, sincethe girders would be uncracked underservice loads.

The calculated torsional andflexural stiffnesses are based on elas-tic theory. The modulus of elasticity Ewas assumed to be equal to:

5.52 f /7210 x 106 psi

Note that E was measured for con-crete having f' = 7120 psi and it wasassumed that E is proportional to fTfor the range of concrete strengths in-volved. The shear modulus G wasbased on E calculated as above and ona Poisson's ratio fir, of 0.16.

The measured and calculatedstiffnesses are in reasonably goodagreement, the measured values in gen-eral being slightly higher. The reasonfor the high value of measured tor-sional stiffness in the case of GirderA9 is not known. Instrument errormay be the cause, since the corre-sponding flexural stiffness is not un-usually high. The presence of shear

and flexure in the uncracked sectiondid not result in a reduction in tor-sional stiffness, as compared to thetorsional stiffness of the girders sub-ject to pure torsion. Also the flexuralstiffness of the uncracked section wasnot reduced by the presence of tor-sion.

Interaction Diagramsfor Design

The procedures described above forthe calculation of the interaction re-lationships could be used to developinteraction diagrams for standard gird-er sections and prestress distribu-tions, for use as design aids. The formthat such design aids might take isshown in Fig. 6.

Until other substantiating test dataare available, it would probably beprudent to base the line representingCase 2 (web) cracking on a constantlimiting principal tension stress equal

36

50030ft Series Gird_e_r

f 'c = 7500 psi

400n_Y

• 3000

rncVv

2000.N

top = 125 psiPrestress20 bottom = 2100 psi40 15 Simply supported span

12108 Continuous span

CC0O 100vD

M/V(ft) 151 19 8 7 6 5

i

0 20 40 60 80 .100

Diagonal tension cracking shear – kipsFig. 6. Possible form of interaction diagram for design.

to the tensile strength of the concrete.Two lines represent Case 1 cracking,i.e., cracking on the top face. The up-ward sloping line represents the situ-ation in a simply supported beamwhere flexural compression stresseson the top face increase the torsionalstrength.

In the case of a continuous beam, acrack could initiate at a point of con-traflexure where no flexural compres-sion stresses exist and hence therewould be no increase in torsionalstrength due to this effect. A family ofcurves represent Case 3 (shear-flexure) cracking, one curve corre-

sponding to each value of moment toshear ratio included on the diagram.Radiating lines corresponding to se-lected torsion to shear ratios enablethe torque and shear at diagonal ten-sion cracking to be interpolated forany combination of torsion to shearand moment to shear ratios.

Circular arc interaction diagram

The provisions for the design ofreinforced concrete members forcombined torsion and shear containedin ACI 318-71 1 are based on the use ofa circular arc, nondimensionaltorsion-shear interaction diagram. Its


validity for reinforced concrete hasbeen demonstrated in several separatestudies. o 9 It has also been sug-gested lo, ll that the circular arc in-teraction diagram could be used in thedesign of prestressed concrete mem-bers.

A comparison is made in Fig. 7 ofthe circular arc interaction diagramand the test data obtained in thisstudy. It can be seen that it is a rea-sonably conservative representation ofthe distribution of the test data, onlyone point falling below the curve. Ittherefore appears that it would be rea-sonable to use the circular arc interac-tion diagram in the design of eccentri-cally prestressed I-section girders,which, when subject to pure torsionwould crack on the top face.

Caution should be used in ex-

trapolating the above conclusion toother categories of prestressed con-crete members. If the distribution ofprestress and the girder section prop-erties are such that cracking first oc-curs in the web when the girder issubject to pure torsion, then in gen-eral Case 2 and Case 3 diagonal ten-sion cracking only will occur, and inthe limit (for small values of M/V),Case 2 cracking only would occur. Insuch a case the interaction diagramwould be more nearly a straight linethan a circular arc.

In the case of a box girder, the shearstresses would be uniform through thethickness of the web in which the ef-fects of shear and torsion are additive.In such a situation it is probable thatdiagonal tension cracking will occurwhen the principal tensile stress be-

1.2

I.0

0.8

Na,

0.6

0.4

0.2

0

^-M/V = 4ft0

^_o /_

M/V = 9.5 ft

(T 2 V 2) +CVO) = I +5%

–5%

0.2 0.4 0.6 0.8 1.0

V,,/V.(test)

Fig. 7. Comparison of test data with the circular arc torsion-shearinteraction diagram.

38

comes equal to the direct tensilestrength of the concrete, and not equalto the modulus of rupture as is prob-ably the case for a solid section (asdiscussed earlier). It is thereforeprobable that the shear-torsion in-teraction diagram for Case 2 diagonaltension cracking in a box girder is astraight line.

Ultimate strength

It can be seen in Table 2 that theultimate strength was in all casesgreater than the torsion and shear atdiagonal tension cracking. The ratio ofthe ultimate torque to the diagonaltension cracking torque was between1.14 and 1.26 for girders with M/V = 4ft, and between 1.11 and 1.15 for gird-ers with M/V = 9.5 ft. In line with thephilosophy of the ACI Code,' no ad-vantage should in practice be taken ofthe additional strength beyond diago-nal tension cracking of memberswithout web reinforcement.

Conclusions

Based on the tests reported here,the following conclusions are thoughtto be reasonable concerning the be-havior of I-section, eccentrically pre-stressed concrete girders subject tocombined loading.

1. Interaction diagrams which re-present with reasonable accuracy thetorque and shear at diagonal tensioncracking can be constructed as de-scribed in this paper. The diagramsrecognize the three possible ways inwhich diagonal tension cracking mayoccur:

a. In the top face of the girder.b. In the web of the girder in a re-

gion uncracked in flexure.c. In the web of the girder in a re-

gion containing flexural cracks.2. The circular arc torsion-shear in-

teraction diagram may be used in thedesign of eccentrically prestressedI-girders, which, when subject to puretorsion would crack in the top face.(Caution should be used in ex-trapolating this conclusion to othertypes of prestressed concrete mem-bers.)

3. The torsional and flexural stiff-nesses of an uncracked girder arenot reduced in combined loading, andmay be calculated with reasonable ac-curacy on the basis of the elastictheory.

Acknowledgments

This study was carried out in theStructural Research Laboratory of the Uni-versity of Washington. It was made possi-ble by the support of the Washington StateHighway Commission, Department ofState Highways in cooperation with theU.S. Department of Transportation, Fed-eral Highway Administration. The opin-ions, findings and conclusions are those ofthe authors and not necessarily those of theWashington State Highway Commission,Department of State Highways, or of theFederal Highway Administration.

References

ACI Committee 318, `Building CodeRequirements for Reinforced Concrete(ACI 318-71)," American Concrete In-stitute, 1971, 78 pp.Wyss, A. N., and Mattock, A. H., "AStudy of I-Section Prestressed Con-crete Girders Subject to Torsion, Shearand Bending," Structures and Me-chanics Report SM 71-1, Departmentof Civil Engineering, University ofWashington, June 1971.Nylander, H., "Vridning och vrid-ningsinspanning vid betongkonstruk-tioner," (Torsion and Torsional Re-

straint by Concrete Structures), Bulle-tin No. 3, Statens Kommitte forByggnadsforskning, Stockholm, 1945.


4. ACI Committee 318, "Commentary onBuilding Code Requirements forReinforced Concrete (ACI 318-63),"American Concrete Institute, Publica-tion SP 10, 1965, pp. 79-81.

5. ACI Committee 318, "Building CodeRequirements for Reinforced Concrete(ACI 318-63)," American Concrete In-stitute, June 1963.

6. Birkeland, C. J., Hamilton, M. E., andMattock, A. H., "Strength of Rein-forced Concrete Beams Without WebReinforcement in Combined Torsion,Shear and Bending," The Trend in En-gineering, University of Washington,V. 19, No. 4, pp. 8-12 and 29, October1967.

7. Ersoy, E., and Ferguson, P. M., "Con-crete Beams Subject to Combined Tor-sion and Shear—ExperimentalTrends," American Concrete Institute,SP-18, Torsion of Structural Concrete,1968.

8. Osburn, D. L., Mayoglou, B., andMattock, A. H., "Strength of Rein-forced Concrete Beams with WebReinforcement in Combined Torsion,Shear and Bending," ACI Journal, V.66, No. 1, January 1969.

9. Rajagopalan, K. S., "Combined Bend-ing, Shear and Torsion on Semicon-tinuous L-beams with Stirrups," PhDDissertation, University of Texas, May1969.

10. Gausel, E., "Spannbetonbalken mitI-formigem Querschnitt bei Gleich-zeitiger Einwirkung von Torsion,Querkraft and Moment," (Pre-stressed Concrete I-Beams in Com-bined Torsion, Shear and Bending),Norges Tekniske Hogskole, Trond-heim, Institutt for Betong, April 1968.

11. Zia, P., and McGee, W. D., "TorsionDesign of Prestressed Concrete," PCIJOURNAL, V. 19, No. 2, March-April,1974.

The Editors welcome discussions of paperspublished in the PCI JOURNAL. Thecomments must be confined to the scope ofthe paper under discussion. Please notethat discussion of papers appearing in thiscurrent issue must be received at PCIHeadquarters by Sept. 1, 1978.

D]

Appendix—Notation

Dimensions, section propertiesbw = width of web of beamd = effective depth of beames = eccentricity of prestress

force relative to centroid ofcross section

S b = section modulus with re-spect to flexural tension face

Material propertiesE = modulus of elasticity of con-

crete= concrete compressive

strength measured on6 x 12-in, cylinders

f, = modulus of rupture of con-crete

ft = split cylinder tensilestrength of concrete

fr = direct tensile strength ofconcrete

G = shear modulus of concreteµ = Poisson's ratio

Stress resultantsF = total prestress forceM = bending moment acting at a

sectionM c, = net flexural cracking mo-

mentT = torque applied at a sectionT, = diagonal tension cracking

torque in pure torsionTer = diagonal tension cracking

torque in combined loadingTf = torque at flexural crackingV = shear applied to a sectionV, = diagonal tension cracking

shear in a beam not subjectto torsion

VI, = diagonal tension crackingshear in a region of a beamcracked in flexure

OV,z = increment of shear between

due to moment and prestressat flexural cracking

.fa = flexural tension stress due todead load

fPe = compressive stress in con-crete due to prestress only,after all losses, at the ex-treme fiber of a section atwhich tension stresses arecaused by applied loads

V = shear stress due to shear, at

diagonal tension crackingVe = calculated shear stress ac-

cording to elastic theorycr = precompression stress in

concrete(Tt = principal tensile stress cal-

culated using elastic theorytorsional stress, Te

(Ti = principal tensile stress cal-culated using plastic theorytorsional stress, Tp

T = torsional shear stress at di-

agonal tension crackingTe = torsional shear stress calcu-

lated assuming elastic be-havior

Ty = torsional shear stress calcu-lated assuming a fully plasticdistribution of shear stress

critical flexural cracking anddiagonal tension cracking

Vcr = diagonal tension crackingshear in combined loading

= diagonal tension crackingshear in a region of a beamuncracked in flexure

Vd = shear due to dead loadVf = applied load shear at flexural

crackingStressesf = net tension at bottom face


Documents

Full-Scale Torsion, Shear, and Bending Tests of