Design of Reinforced and Prestressed Concrete …...Design of Reinforced and Prestressed Concrete Inverted T Beams for Bridge Structures Prefabricated concrete stringers with cast-in-place

S. A. MirzaProfessor of Civil EngineeringLakehead UniversityThunder Bay, OntarioCanada

R. W. FurlongDonald J. Douglass CentennialProfessor of Civil Engineering

The University of Texas at AustinAustin, Texas

Design of Reinforced andPrestressed ConcreteInverted T Beams for

Bridge Structures

P refabricated concrete stringers withcast-in-place slab are frequently

used to achieve economical and speedybridge construction schemes. Beamsconstructed in the form of an inverted Tpossess on each side of the web abracket or flange overhang that providesa convenient shelf or supporting surfacefor the precast stringers. Hence, cast-in-place, post-tensioned, prestressedconcrete (PC) and reinforced concrete(RC) inverted T beams are frequentlyused in bridges as bentcap girders as in-dicated in Fig. 1.

Inverted T beams can he simply sup-ported, cantilevered over simple sup-ports, or they can be constructed mono-lithically with columns or piers. Theyreduce overall floor depth by avoid-ing deep cross members beneath pre-

fabricated stringers, resulting in low-er abutments and shorter approachesfor the bridges.

Despite its frequent use for at leastthe past two decades, the inverted Tshape remains as one of the least inves-tigated cross sections. At present, noguidance for handling design problemsspecifically associated with the invertedT section is available in current NorthAmerican standards." Consequently,engineers have tended to rely on per-

,sonal judgment and discretion for de-sign of inverted T girders.

This paper summarizes design rec-ommendations that are based on obser-vations and analyses of cast-in-placenormal weight, PC and RC laboratorytest specimens reported in studies of in-verted T beams °-' The paper is directed

112

toward the design office audience.Hence, only the highlights and conclu-sions from the reported research asso-ciated with design aspects of the invert-ed T beam bridge bentcaps are present-ed here.

Further information and details arefully available elsewhere." It should bepointed out that these recommendationsare being successfully used by theTexas State Department of Highwaysand Public Transportation for design ofinverted T beams.

PROBLEMS ANDIMPLICATIONS

Stringer bearings on the top face ofthe flange of an inverted T beam imposevertical tensile forces (hanger tension)near the bottom of the web as indicatedin Fig. 2. Such forces are not ordinarilyencountered in conventional beams,where vertical forces are applied at thetop of the web. Furthermore, the lon-gitudinal and lateral bending of theflange of an inverted T beam produce avery complex stress distribution in theflange. Hence, the design of reinforce-ment for the web and for the flange of aninverted T section imposes specialproblems.

Loading conditions involving torsionon the inverted T section might createmore severe requirements for propor-tioning reinforcement. Torsion of in-verted T bentcap girders occurs withevery passage of design vehicles acrossthe bentcap. As traffic approaches thebentcap, stringer reactions cause twist-ing or torsion of the bentcap toward theapproaching load. The direction of twistreverses after the passage of traffic im-poses loads on stringers that react on theopposite flange overhang of the invertedT beam. Hence, the passage of traffictends to make the twist a reversingphenomenon.

The state of combined stress in an in-verted T beam cannot be obtained by

Synopsis

The structural behavior of rein-forced and prestressed concrete in-verted T Beams differs from that ofconventional top-loaded beams. Theloads that are introduced into the bot-tom rather than into the sides or thetop of the web of an inverted T beamimpose special problems, which arenot dealt with in existing structuralcodes.

This paper provides recommen-dations for proportioning cross sectiondimensions and reinforcement ofcast-in-place, post-tensioned andreinforced concrete inverted T beamsused in bridge structures. A designexample is included to elaborate theapplication of these recommen-dations.

superposition of simple stress cases.Concrete cracks at a nominal amount oftensile stress and the analytical de-scription of mechanisms and materialcharacteristics must change for eachsubsequent load stage. Hence, a generalanalytic solution for strength of an inv-erted T beam that is versatile enough forall possible load cases and simpleenough for design office applicationdoes not seem to he within reach at thepresent time. The authors have, there-fore, resorted to empiricism supportedby a rational interpretation of test re-sults in order to develop design criteriafor inverted T beams.

TEST PROGRAMLaboratory test specimens that formed

the basis of design recommendationspresented in this paper representedmodel bentcap girders at a scale of ap-proximately one-third the size of theprototype members used by the Texas

PCI JOURNALJJuly-August 1985 1 13

State Department of Highways andPublic Transportation. Strength of nor-mal weight concrete, reinforcing steeland prestressing strand used were typi-cal of those currently employed in theindustry. The prestressing strands werestraight and each strand was post-tensioned and grouted.

The principal variables for the speci-mens involved reinforcement details as-sociated with the web and the flange.The observations included service loadtwist reversals to simulate the passage oftraffic and subsequent loads that causedfailure in a region of the test specimens.Hence, the typical load sequences in-volved three phases of observations:

First, loads simulating service dead

load stringer reactions were applied andmaintained.

Second, service live load forces wereapplied and removed to produce severalcycles of simulated passage of traffic.

Third, loads were increased until fail-ure occurred in a region of the testspecimen.

Other details are reported else-where' - s and will not he repeated here.

RESULTS ANDRECOMMENDATIONS

Six modes of failure plus at least oneservice load condition should be con-sidered as a part of the design of in-

;t'1ie or Drilled Shat) A -.J

{a) ELEVATION

Fig. 1 _ Highway pier cap.

114

LONGITUDINAL

4'^ ^^^(o^CRTOR^PµG ^^Sle

^r, ter' ̂ il S`^ tFlp1ibY. OV

POSITIVE MOMENT REGION tEGATrVF MO EN7REGION

Fig. 2. Structural actions on inverted T beams.

verted T girders. The six modes of fail-ure involved the possibility of failuredue to:

(1) Flexure of the overall inverted Tbeam;

(2) Flexural shear acting on the over-ail inverted T beam;

(3) Torsional shear on the overallcross section;

(4) Hanger tension on web stirrups;(5) Flange punching shear at stringer

bearings; and(6) Bracket type shear friction in

flange at face of the web.The service load condition involves

the possible wide cracking at the inter-face of the web and the flange due topremature yielding of stirrups acting ashangers nearest the concentrated loads.Typical forces and stress types acting oninverted T beams are illustrated in Fig.2.

The overall strength of an inverted Tbeam must be adequate to support ulti-mate flexure, flexural shear and tor-sional shear forces and any possiblecombination of such forces. The localstrength of inverted T beam components

must he adequate to support forces thatare applied as concentrated loads onflange overhangs. Locally, the flangemust be deep enough to avoid punchingshear weakness, the transverse flangereinforcement must be strong enough tomaintain shear friction resistance at theface of the web, and the web stirrupsmust be sufficient to act as hangers totransmit flange loads into the web.

Service load conditions of deflectionand crack control may be more signifi-cant than strength requirements forsome components of design. Decisionsregarding the overall depth of web andthe distribution of tensile reinforce-ment, both for flexure and for stirrupsacting as hangers, may involve serviceload conditions of behavior.

The height of the web above the topface of the flange of an inverted T beamis determined by the required depth ofthe stringer to be supported on theflange. A minimum depth of the flangeitself can be derived from punchingshear requirements, but additionaldepth may be appropriate to provideenough flexural stiffness for the overall

PCI JOIJRNAUJuIy-August 1985 115

member. The width of the web can beselected for adequate strength in shearalone and in combined shear and tor-sion, or it may be determined by place-ment requirements of flexural rein-forcement.

The length of the flange overhang iscontrolled by the size of the bearingpads used to support stringers andshould not exceed the flange thickness.In addition to accommodating thebearing pad width, the flange overhanghas to provide for the edge distance, thesweep tolerance of the T beam, thelength tolerance of the stringers and theerection and placing tolerances.

The design of an inverted T beam canbe divided into three parts:

(1) Design of flange;(2) Design of web stirrups acting as

hangers to deliver flange forcesinto the web; and

(3) Overall design of the beam itself.Hence, the design recommendations

are presented in three parts as well.Note that these recommendations ap-ply to cast-in-place, post-tensioned con-crete and reinforced concrete invert-ed T beams employed in bridge con-struction.

Design of FlangeThe strength of the flange should be

adequate to sustain the punching shearaction of the stringer loads applied onthe flange. In addition, the flangeshould be able to resist the shear frictionforces at the face of the web caused bythe bracket action of the flange.

Punching shear in flange — Theflange should be deep enough to pre-vent punching shear failure. This can beachieved by satisfying the followingequation for effective flange depth (dr)from the top face of the flange to the topof the bottom layer of transverse rein-forcement in the flange:

4 0 V fc (B , + 2d) ds' Pu (la)

or

2 Pyi p

df 4 I* Ba ^'f^ –1

(1b)

in whichdl = effective depth of flange (see

Fig. 3 and definitions in Nota-tion section)

Bp =B+2B,,B = length of bearing pad along

edge of flangeB, = width of bearing pad perpen-

dicular to beam axisP,, = ultimate concentrated load

acting on one bearing pad= capacity reduction factor for

shear and equals 0.85Eq. (Ib) is a cumbersome expression.

Fortunately, the equation can be ap-plied readily as a graph with B p versus P,for various values ofd,. Such graphs forspecified concrete strength ff taken as4000 and 5000 psi (27.6 and 34.5 MPa)are shown in Figs. 3 and 4, respectively.

Similar graphs can be prepared forother concrete strengths. It may bepointed out that the most commonstrengths of concrete used for bentcapgirders are 4000 and 5000 psi (27.6 and34.5 MPa) for reinforced and post-tensioned prestressed concreteconstruction, respectively.

Eq. (lb) is based on a tensile strengthof concrete equal to 4 ^^ f,' acting on thesurface of a tnincated pyramid under abearing pad and is supported by testresults . 4 Stirrups that intersect a face ofthe truncated pyramid can help supportthe concentrated load if the anchorage ofthe stirrups can be developed above andbelow the face ofthe truncated pyramid.However, no such help from the stirrupswas included in Eq. (Ib) because thiswould require cumbersome checks ondesign and detailing of stirrups.

The surfaces of truncated pyramidsresisting ultimate punching shear underadjacent stringers should not overlap.This can be achieved by providingenough longitudinal and transverse

116

Bp = length of bearing plate plus twice Its width.

L.

dt

80C

Note:width (Bw) 1000 psi = 6.89 MPa

100 kip = 444 KNlength (B) 10 In. = 254 mm

700

600En0.

a 500tiR0J

rn 400c

300

200

100

IliUVJP4i1fliU!iiDIII .AIIiN4.iiliiluill1idiFiiiII

40 60 80 100 120 140

Bp (inches}

Fig. 3. Punching shear capacity of flange of an inverted T beam;f,' = 4000 psi (27.6 MPa).

distance between stringers. Hence, the b,,) in Fig. 5a] at least equals 2df +web width bu should be such that the B, where B,, is the width of the bearingcenter-to-center transverse distance pad perpendicular to the beam axis.between the two stringer reactions act- Furthermore, stringers along theing on opposite sides of the web [(2a + beam axis on each side of the web

PCI JOURNAUJuly-August 19B5 117

should be placed at a center-to-centerspacing (shown as S in Fig. 5b) that ex-ceeds 2df + B, where B is the length ofhearing pad along the edge of the flange.

For an end stringer, the distance fromthe edge of the bearing pad to the ton-

gitudinal end of the inverted T beam(shown as de in Fig. 5b) should be aminimum of df + B. This ensures thedevelopment of flange punchingstrength at end stringers at least as greatas that developed at interior stringers,

800

700

600

Q.

500a-tiro0

400

300

200

100

fc =5000 psipunching shear

/ f //

// / V

dt =

I6 i^^^es

20 40 60 80 100 120 140

Bp (inches)

Fig. 4. Punching shear capacity of flange of an inverted T beam;f,' = 5000 psi (34.5 MPa).

118

If the reaction from the end stringer isless than that from an interior stringer,an end distance smaller than d f + BWmay he provided and can be calculatedfomn the following expression:

` P" – (B +B^.+df )]de' [4c7 7 dr

(2)

in which 0 = 0.85. The distance deshould always be greater than zero inorder to accommodate the bearing padplaced near the end of the inverted Tbeam.

Bracket-type shear friction in flange— The effective depth of flange (df)from centroid of top layer of flangetransverse reinforcement to the bottomof flange shown in Fig. 6a and requiredto fulfill shear friction requirementsshould satisfy the following equation:

0.2 4i f, dr (B + 4a) -- P,, (3a)

or

d61',, (3b)

f 3 f (B + 4a)

in whichP,, = ultimate concentrated load

acting on bearing padB = length of bearing pad along

edge of flangea = distance from face of web to

center of bearing pad0.2fc = shear strength of concrete

resisting shear friction'•z0= 0.85

Use f, = 4000 psi for f,. _- 4000 psi(27.6 MPa) in computing dr from Eq.(3a). This upper limit onf,' is specified tolimit the shear strength of concrete,because Eq. (4) used later for computingA L f will become unconservativefor higher values offs s Since shear fric-tion seldom controls the flange depth ofinverted T beams, this limit on f' willnot affect most practical cases.

In Eq. (3b), the effective flange lengthresisting shear friction has been taken as

PU a e Pu

df

By,r

(a) Cross Section

Pu S Pu d e m

(b) Elevation

Fig. 5. Stringer spacings required forpunching shear.

B + 4a. In most cases, the stringerspacing along the beam axis will belarge enough to permit the full effectiveflange length. However, if the lon-gitudinal spacing of stringers is less thanB + 4a, the stringer spacing should beused in place of B + 4a in computing dffrom Eq. (3b).

For a stringer placed near the lon-gitudinal end of an inverted T beam, thestringer spacing for use in Eq. (3b)should be taken as twice the distancefrom the center of the bearing pad to theend of the inverted T beam or as thelongitudinal distances between two ad-jacent stringers, whichever is smaller.

The transverse reinforcement shouldbe placed perpendicular to the web nearthe top of the flange to resist flexuraltension and to ensure enough pressurefor sustaining shear friction force as in-dicated in Fig. 6a. The transverse rein-forcement required to satisfy shear fric-tion should be placed within a distance

PCI JOURNAUJuIy-August 1985 119

2a each side from the edge of a bearingpad as shown in Fig. 6b. The area ofcross section of such reinforcement (An!)should satisfy the following expres-sion:'•a

4U Arff, -_ P , (4a)

or

P.A°f 1.2f (4b)v

in which/+. = coefficient of sliding friction

taken as 1.4 for normal weightconcrete cast monolithically`•2

L = specified yield strength of rein-fo reeme nt

0= 0,85This reinforcement should be placed

in two or more layers in the top half ofthe flange thickness; the area of rein-forcement in the top layer should equal2Aaf/3 as indicated in Fig. 6b.

The flange transverse reinforcementplaced in the top layer also resistsflexural tension, which is caused by thecantilever action of the flange, andshould satisfy the following equation:

0.8 .6dfA, & - PY a (5a)

or

1.4P„a

AAf J v d(5b)fr

in whichA,f = cross section area of trans-

verse reinforcement re-quired to resist flexural ten-Sion

0.8d1 = effective distance betweencentroid of compression andcentroid of' tension (j df) forcalculating flexural rein-forcement in flange

=0.9The value of jd f in a deep cantilever is

expected to be smaller than that of anordinary depth, shallower beam. Thesuggested value of 0.8d f for bracket de-

sign is based on a finite elementanalysis .8 All transverse reinforcementwithin a longitudinal distance 2.5 a eachside of the bearing pad can be taken asAtf as shown in Fig. 6c.

Thus, the reinforcement placed in thetop layer should at least equal AYf or2A,.f13, whichever is greater, as indi-cated in Fig. 6a. Ifa r 0,4 di, A, will notcontrol the design. Anf and distributionof shear friction reinforcement shouldalways be checked through Eq. (4b).

The use of the term B + 5a as theeffective length of bracket for the dis-tribution of flexural steel in the top of abracket is based on a finite elementanalysis and is supported by testresults .8 The same holds for the use ofthe term B + 4 a as the effective lengthof bracket suggested for shear friction.

The suggested effective flange Iength(B + 4 a for A)f and B + 5a for A,f)

should not overlap for adjacentstringers. If the distance c between thecenter of a concentrated load and Ion-gitudinal end of the girder is less thanone-half the effective flange length il-lustrated in Fig. 6b or 6c, the effectiveflange length should be taken as 2 c oras the spacing of stringers, whichever issmaller.

The longitudinal forces due to suddenbraking of a vehicle are transmitted fromits wheels to the deck of the bridge. Themagnitude of this longitudinal forcedepends on the weight, the velocity, andthe braking time of the vehicle. TheAASHTO specifications' call for a lon-gitudinal force of 5 percent of the liveload in all lanes carrying traffic headedin the same direction. This longitudinalforce will add little stress to the deckand the stringers but may be importantfor the design of stringer bearings andthe supporting brackets of the invertedT beams.

Consequentl y, a 5 to 10 percent lon-gitudinal component of live loadstringer reactions could be included forthe design of flexural steel in the top ofthe bracket, The rigidity of the bridge

120

2 A„f/3 or Asf , whichever is greater

P uA !

df f̂l— i-- A,,f /3

(a) Cross Section

^^ cuC o

2A Vf /3 2A„f /3 TO

• • . • • •' : • • • • • • • . c7

Ayf /3 Av1/3

2a B 2a Applies at girder 2c

iB+4ra} ends If a < (a - 4a)/2<S

(b) Effective Flange Length for Shear Friction

PuvF C

T___ A 1 Asf'C7

2.Sa B Z.Sa__ ____

Applies at girder 2c ,.^(8+3e} ends If c < IB•Sa)/2

(c) Effective Flange Length for Flexure

Fig. 6. Effective flange length for design of bracket reinforcement.

deck tends to spread such loads amongall stringers, and the approximationsused in estimating the effective flangelength for flexure hardly justify the

superposition of tension due to lon-gitudinal component of live loadstringer reactions and flexure un-less stringers are spaced more closely

PCI JOURNALJJuIy-August 1985 121

Fig. 7. Anchorage of flange transverse bars withoutthe use of welding.

than 5 a plus the length of the bearingpad.

The AASHTO specifications' alsorequire that an additional longitudinalforce due to friction at expansion bear-ings be included in design. The mag-nitude of this frictional force dependson the type of bearing used. Unlessspecial provisions are made in the de-sign of stringer bearings to avoid thefrictional force, this force should becomputed and additional reinforcementshould be added to the flexural steelnear the top of the bracket to resist thisforce.

In place of an exact computation ofthe longitudinal forces due to friction,the ACI Code 2 provisions for bracketdesign may be used. These clauses re-quire the use of a longitudinal forcethat at least equals 20 percent of thestringer reaction due to dead plus liveload. These provisions also regard thelongitudinal force acting on a bracketas a live load even when this force re-sults from creep, shrinkage or tempera-ture change.

The anchorage of the flangetransverse bars may impose a problem,because the flange overhang is usuallytoo short to accommodate the develop-ment length of reinforcing bars of usualsizes employed in bridge construction.The detail shown in Fig. 7 is recom-mended wherever the reinforcementsize would permit the development ofyield strength in the flange transversebars.

However, it may be necessary insome cases to weld the ends of trans-verse bars to an anchor bar at the exte-rior face of the flange and perpendicu-lar to the transverse bars.

A welding detail used by the TexasState Department of Highways andPublic Transportation is illustrated inFig. 8. The development of flangetransverse bars can also be achieved byfurnishing a continuous steel angle (orplate) along the top corners of the flangeand connecting these bars to the angle(or plate). The welding should conformto the American Welding Society D1.4Code for reinforcing steel.'p

122

Use Detail A

Weld

Anchor Bar

Weld allthree bars

Detail A

Fig. 8. Anchorage of flange transverse bars with the useof welding.

Design of Web StirrupsActing as Hangers

Stirrups in the web of an inverted Tbeam act as hangers to deliver the con-centrated loads applied on the flangeinto the body of the web. The maximumhanger stresses at ultimate load existwhen both sides of the web are sub-jected to maximum live loads simul-taneously.

The longitudinal distance over whichhanger forces can he distributed,defined as the effective hanger distancein Fig. 9, is limited either by the shearcapacity of concrete in the flange oneach side of a bearing pad or by thelongitudinal center-to-center spacing ofstringers. Hence, to achieve safe deliv-

ery of flange loads into the web, the fol-lowing strength relationships [Eq. (6b)and Eq. (7b)] should be satisfied.

0 A fv[(B +2d,/sI[2P,, – 2 (2 ; hrdf )] (6a)

or

2P, 4 y. f^ br drAL

(6b)S f„ (B + 2 dr)

and

0A f„(SIs)r2P,, (7a)

or

s Of

2P,,(7b)

,,S

PCI JOURNAL/Jufy-August 1985 123

In the above equations:br = overall flange width (Fig. 9)dr = flange depth from top of

flange to center of bottom lon-gitudinal reinforcement as in-dicated in Fig. 9a

S = stringer spacing along beamaxis

Pu = ultimate concentrated loadacting on one bearing pad

A,,/s = cross section area of both legsof a web stirrup divided bythe spacing of stirrups

= 0.85For a stringer load placed near the

end of an inverted T beam, the distanceS in Eq. (7b) should be taken as twicethe distance from the center of thebearing pad to the longitudinal end ofthe inverted T beam or as the longitudi-nal distance between two adjacentstringers, whichever is smaller.

Note that Eq. (6b) controls the designof hangers in cases where the stringerspacing is large enough to permit a fail-ure mode in which flange strengthbecomes effective in resisting hangerforces as shown in Fig. 9a. For thesecases, the hanger reinforcement shouldbe provided for twice the stringer loadminus the shear strength of the flangeon each side of the stringer as indicatedby Eq. (fib).

If the stringers are too closely spaced,hanger failure will take place by sepa-ration of the flange from the web overthe entire loaded length of the beam.Hence, hangers should be designed forlull stringer loads as indicated by Eq.(7h). However, both equations shouldbe satisfied to insure the safe transferof hanger forces.

The premature yielding of hangersnearest the concentrated loads appliedto the flange and the size of cracks thatform at the junction of the web and theflange when these hangers reach highstress levels should be controlled at ser-vice load conditions. This can heachieved by satisfying the followingequation:

Avfa I(B + 3a)/s] i 2P, (Sa)

or

A„ = 3P, (8b)s f„(B+3a)

in whichP, = concentrated service load acting

on a bearing padft = hanger stress at service loads

limited to a maximum value of2f„13

In Eq. (8b) the effective distance overwhich hanger forces can be distributedat service loads has been taken as B +3a, If B + 3a exceeds the longitudinalstringer spacing, the stringer spacing Sshould he used in place of B + 3 a incomputing A„!s from Eq. (8b), For endstringers S should be taken the same asthat defined for Eq. (7b). Note that Eq.(8b) is based on test results and isdocumented in Ref. 5.

The Iargest value of A„/s from Eqs.(6b), (7b), and (8b) should be used. Onlyvertical stirrups anchored to develop bybond their tension yield forces aboveand below the top surface of the flangeshould be considered to carry hangerforces. Since most flanges have a depthinadequate for developing stirrup baryield forces, hangers should be closedacross the bottom of the inverted Tbeam as indicated in Fig. 9b.

It is not necessary to superimposeloads on stirrups acting as hangers andloads on stirrups acting as shear (andtorsion) reinforcement. This will be dis-cussed further in a subsequent section.

Overall Design of InvertedT Beam

The overall strength of an inverted Tbeam should be adequate to support ul-timate flexure, flexural shear, and tor-sional shear forces and any possiblecombination of such forces.

Design for maximum flexural momentand shear — The most likely design

124

Each Side of the WebEqual Maximum Force

d f df

Flange Top + B Hanger Yield Loads

df I ^L^ Flange45°

Effective Hanger Distance

(a) Elevation

b -If ^I

(b) Cross Section

Fig. 9. Hanger forces in response to flange loads shown on the beam elevationand details of hanger reinforcement shown in the beam cross section.

condition for bridge hentcap girders in-volves flexural moments and flexuralshear forces that are largest when tor-sion is absent, because traffic loadsstringers fully on both sides of the girderweb. Consequently, a logical proce-dure for overall design begins with theproportioning of the cross section andreinforcement solely on the basis ofmaximum flexural moment andmaximum flexural shear force.

Requirements of flexural reinforce-ment for the overall design of the beamare not altered by the location of the Tbeam flange. The ultimate strength andserviceability design requirements ofthe ACI Code2 and the AASHTO Speci-fications' are quite safe and adequate for

flexure. This applies to both post-tensioned prestressed concrete andreinforced concrete inverted T beams.

The ACI Code2 permits the designerto consider the maximum end shear asthat occurring at a distanced from theface of the support for nonprestressedmembers and at a distance h/2 from theface of the support for prestressed mem-bers, where d and h are, respectively,the effective depth and overall thicknessof the member. Of course, this is al-lowed only if no stringer load is placedbetween the face of the support and thecritical section at a distanced orh/2 fromthe face of the support.

While this criterion is reasonable forconventional beams, this is not appro-

PCI JOURNAL/July-August 1985 125

A q ecentroid of

flexural tension d

dreinforcement

bt bf

(a) Subjected to Negative (b) Subjected to PositiveBending Moment Bending Moment

Fig. 10. Effective area resisting shear force A, (cross-hatched).

priate for inverted T beams unless theterms df and h112 related to the flange aresubstituted in place of d and h/2, re-spectively. To simplify the shear calcu-lations, however, the critical section forthe maximum end shear in inverted Tbeams may be taken at the face of thesupport. This simplification will notcause a significant loss of accuracy inshear design of bridge benteap girders.

With a flange overhang to thicknessratio equal to or less than 1.0, the flangeof an inverted T beam is expected to bestiff enough to fully participate in reten-tion of the shear force. This seems par-ticularly valid for a cross section sub-jected to negative moment, creatingflexural compression in the flange. Thetests on inverted T beams support thishypothesis..

Hence, the flexural shear strength inabsence of torsion can be based onmodified versions of the ACI Code equ-ations:°For reinforced concrete:

V. f2^f^A,+At,.fv(^1] (9)L s

For prestressed concrete:

V. = 0 [n f, Ae + Aj, (g.)] (10)

In both equations, A r f„ dls s 8 ^1Tb^, d; n = 5 for M. IV„ d = 1, decreas-ing linearly to 2 as M,,IV d increasesfrom I to 5; A, = area of all concretebetween compression face and cen-troid of flexural tension (or prestress-ing) reinforcement as indicated in Fig.10; 4. = 0,85; andd is defined in Fig. 10.

The suggested approximation for napplies only to the beams with f, 46400 psi (44 MPa) and having an effec-tive prestressing force greater than orequal to 0.4 times the tensile strength offlexural reinforcement. Stirrups shouldbe designed to resist all applied ulti-mate shear force above that resistedby the concrete section illustrated inFig. 10. In addition, all the other shearprovisions of the Code'-2 should be sat-isfied.

The tests on inverted T beams haveshown that the hanger failure cracksoccur at the junction of the web and the

126

flange, whereas the flexural shear fail-ure cracks occur in the web above theflange .5 Consequently, the yielding ofstirrups acting as hanger reinforcementand as shear reinforcement takes placeat different locations in the stirrups. Thestirrups designed as hangers can then beused as part of the web reinforcementresisting flexural shear.

This can be further justified by thefollowing argument. The failure of aninverted T beam due to flexural shear isan overall failure, whereas the hangerfailure is a local failure of stirrups actingas hangers nearest an applied concen-trated load to transmit flange forces intothe web.

Hence, the hanger forces deliveredto the web are carried to the beamsupports by the web as flexural shearand it is not necessary to superimposeloads on stirrups acting as hangers andloads on stirrups acting as shear rein-forcement. However, the web rein-forcement in the stem of an inverted Tbeam should be proportioned on thebasis of hanger requirement or shearstrength requirement, whichever isgreater.

Design for combined effect of flexuralshear and torsion — When torsion is atmaximum, traffic loads stringers on onlyone side of the web and flexural shear isless than the maximum value. Hence,stirrups also serve as vertical reinforce-ment of the web that is subjected tocombined flexural and torsional shear.The local hanger forces need not besuperimposed on the web shear forcesfor designing stirrups as explained ear-lier, but the cross section should bedesigned for the more critical of the twoforces.

For the cross section to be adequatewhen the combined torsional andflexural shear acts, the following in-teraction expression should be sat-isfied:

(V.} 2 + ^To ) 2 -_ 1.0 (11)

in which V,, and T. are applied ultimateshear force and applied ultimate torque,respectively.

The flexural shear capacity V. can betaken from Eq. (9) or (10) and the ulti-mate pure torsion strength To for bothreinforced and prestressed concretemembers can be calculated from anadaptation of the pure torsion strengthequation recommended by the ACICode:'

To 0 [4 : A&zZ,, + f «` xI Yil

3 1 s J

184) ^' fe x2^ (12)3

in whichAt = area of one leg of a web stirrups = space of web stirrups

a, = [0.66 + 0.33 (y,/x i )l a 1.50 = 0.85

x, and iJ, – shorter and longer cen-ter-to-center dimension,respectively, of closed webstirrups

The contribution to torsion strength ofthe cross section of each componentrectangle should be computed sepa-rately using the smaller dimension x andthe larger dimension y for the rectangleunder consideration as indicated in Fig.11. Only closed rectangular web stirrupsshould be considered effective inresisting torsion, This assumption ofneglecting the contribution of flangetransverse reinforcement to torsionalstrength may lead to a slightly conser-vative design for the combined effect offlexural shear and torsion, but willgreatly simplify the calculations.

Although the ACI Code 2 permits avalue of 2.4 v'T for the torsional shearstrength of concrete, a value of 4 f, isused in Eq. (12). This value seems to bejustified for inverted T beams and isdocumented in Ref. 4.

Alternately, the combined effect offlexural shear and torsional shear on thecross section can be satisfied by usingthe following equations:

PCI JOURNAUJuly-August 1985 127

( a) (b)

Fig. 11. Component rectangles of inverted T section for torsion analysis[use larger of the two values of 1 x 2 y from (a) and (b) 1.

6 f' 1 x 2y (13)

and

2 T" – 1.33 ^' f^ E xzy]

s "(xi y1f

(14)

In which = 0 [1 –V/V ) and canbe easily determined from the plotshown in Fig. 12.

Eqs. (13) and (14) were derived bysolving Eq. (11) for T. and equating it toEq. (12). The expressions can be used inplace of Eqs. (11) and (12). Eq. (13) en-sures yielding of stirrups before crush-ing of concrete takes place. Hence, thecross section should be revised if Eq.(13) is not satisfied.

Eq. (14) determines A„/s, the area ofcross section of both legs of a closedrectangular web stirrup divided by thestirrup spacing, required for the com-bined effect of shear and torsion. Notethat only one set of equations for thecombined effect of flexural shear andtorsion needs to he checked: either Eqs.(13) and (14) or Eqs, (11) and (12).

If the area of transverse reinforcementin the web is controlled by the require-ments of maximum flexural shear, thereis apparently no need to check for lon-gitudinal reinforcement required for

torsion, The supplemental longitudinalreinforcement should be provided tohelp flexural reinforcement resist tor-sion for cases in which stirrup design iscontrolled by the combined effect of tor-sion and flexural shear.

If the area of web stirrups is increasedto satisfy Eq. (14) or Eq. (11), supple-mental longitudinal steel (A t ) with avolume at least equal to the volume ofextra web transverse reinforcement (A1)should be provided.

When the yield strength ofA, andA; isthe same:

sAi = 2(x,+yl )At =A;, (xl+y,)

and

A, (x1 + y,) (15)S

in which A,' Is = Ar,/s required for thecombined effect of flexural shear andtorsion minus A o ls required formaximum flexural shear acting alone.

The area of longitudinal reinforce-mentA, should be distributed among thefour comers of the web plus the fourcomers of the flange, and it should beadded to the flexural reinforcement bothfor prestressed and nonprestressed con-crete members. Note that all other rele-vant shear and torsion provisions of theCode”" should he satisfied.

128

0.850.830.78

0.68

0.51

0.37

0.26 --

00 0.2 0.4 0.6

VuV,

0.8 0.9 1.0

Fig. 12. Value of f3 for different ratios of VJVQ (linear interpolation of{3 may be conservatively used between plotted values of V0 /V ).

DESIGN PROCEDUREThe design procedure for inverted T

beam bentcap girders based on thecriteria proposed in this paper can besummarized as follows:

1. Compute the flange thickness farpunching shear requirements usingEq.(lb) (or Figs. 3 and 4) and select avalue that at least equals the flangeoverhang. Establish the web width, sothat the center-to-center transverse dis-tance between the two stringer reactionsacting on the opposite sides of the web(2a +b) at least equals twice the flangedepth plus the width of the bearing pad(2df+B,,).

Note that this web width must be ableto accommodate the longitudinal rein-forcement required for negative bend-ing moment. Check the requirementsfor minimum stringer spacings and enddistances controlled by punching shear.

The height of the web above the top ofthe flange is determined by the depth ofthe stringers to be supported on theflange. The overall depth of the beamcan then be computed and must providethe required flexural stiffness. Thenominal span-to-depth ratios for in-verted T beam bentcap girders appear tobe from 2 to 4 for cantilevered spans,and 4 to 8 for spans supported at eachend,'

2. Using Eq. (3b), compute the flangethickness required to resist shear fric-tion and revise the furnished value ifneeded. Eqs. (4b) and (5b), respectively,calculate the flange transverse rein-forcement required for shear friction(A„f) and that required for flexural ten-sion (A,, ,) in the flange. Place A,f or2A,r/3, whichever is greater, near the topof the flange and A Gf/3 in one or morelayers below the top layer within the tophalf of the flange thickness.

PCI JOURNAL/July-August 1985 129

21 84 84 84 84 84 21

(a) Elevation

1 1mil+ tl}^-

(#5 ©- #4 U) @6 for XY(#4©+ #4 U) @6 for YZ

ss64 2#6

a2#6

0

iIr_ - f1jE2#1

5#8

— #3@6# 5 L @6

(b) Cross Section Dimensions (c)Reinforcement Details

Fig. 13. Details of an inverted T beam designed in the example (all dimensions are ininches; 1 in. = 25.4 mm).

Include the effect of longitudinalforces at stringer bearings due to frictionin the design of top transverse rein-forcement in the flange unless provi-sions are made to avoid the frictionalforces at stringer bearings.

The flange transverse reinforcementin the top layer is also subjected to alongitudinal component of stringerreactions due to braking of live and im-pact loads. The superposition of flangetransverse reinforcement for tensiondue to this longitudinal force and thatfor tension due to flange flexure is not

needed unless the stringers are spacedcloser than B + 5a. Anchor the flangetransverse bars as in Fig. 7 or 8.

3. Provide stirrups in the web for themost critical effect from hanger tension,maximum flexural shear, and maximumtorque plus corresponding flexuralshear:

(a) Determine the area of stirrupreinforcement required to resist hangerforces as the largest value of A a,/.s ob-tained from Eqs. (6h), (7b), and (8b).Note that the maximum stresses due tohanger action act when both sides of the

130

web are subjected to maximum liveloads simultaneously.

(b) The maximum stresses due toflexural shear acting alone occur whentraffic loads the stringers fully on bothsides of the web. Determine the re-quired area of stirrups in terms of A„!sfrom Eq. (9) or (10). Note that the criti-cal section for the maximum end shearin an inverted T beam may be taken atthe face of the support.

(c) For maximum torsion to act on aninverted T beam, traffic loads thestringers on only one side of the weband the corresponding flexural shearwill be less than its maximum value.Again, the critical section near the endof the member may be taken at the faceof the support. Satisfy Eq. (13) to ensureyielding of stirrups prior to crushing ofconcrete under ultimate loads. Note thata larger cross section will be needed toresist combined shear and torsion if Eq.(13) is not satisfied. Calculate therequired area of stirrups (A,ls) using Eq.(14). The term j3 used in Eqs. (13) and(14)may be determined from Fig. 12.

The superposition of stirrup rein-forcement (A„Is) required for Cases (a),(b) and (c) is not needed. However, de-sign stirrups for the maximum value ofthe three effects at all critical sections.Provide either closed rectangular stir-rups or stirrups that are closed at leastacross the bottom of the beam as indicatedin Fig. 9b. Use only closed rectangularstirrups to resist forces due to torsion.

4. The most critical section for flexureoccurs at the face of the support and themaximum bending moment on an in-verted T beam acts when full Iive loadsare applied on both sides of the web.Determine the longitudinal reinforce-ment required to resist flexural tensionat all critical sections. If the web rein-forcement (A„/s) computed for maxi-mum flexural shear acting alone isgreater than that calculated for the com-bined effect of torsion and flexuralshear, no check on longitudinal rein-forcement for torsion is necessary.

Provide supplemental longitudinalreinforcement as per Eq. (15) to helpflexural reinforcement resist torsion incases where stirrup design is controlledby the combined effect of torsion andflexural shear. Distribute the supple-mental longitudinal reinforcementalong the perimeter of the cross section,particularly at the corners of the weband flange.

SUMMARYReinforced concrete and post-ten-

sioned prestressed concrete inverted Tbeams are frequently used for bridges.The structural behavior of inverted Tbeams differs from that of conventionaltop-loaded beams, because the loads areintroduced into the bottom rather thaninto the sides or the top of the web. Theapplication of loads near the bottom ofthe web in inverted T beams imposesspecial problems, which are not ad-dressed to in the current North Ameri-can structural codes.

This paper provides recommenda-tions for proportioning cross section di-mensions and reinforcement of cast-in-place normal weight concrete invertedT beams employed in bridge structures.These beams should be designed tohave adequate strength against possiblefailure due to flexure, flexural shear, tor-sion and any possible combination ofthese forces.

Reinforcement details for the flangesof inverted T beams should accommo-date flexure, shear friction, and punch-ing shear on the short cantilevered shelf.The transverse reinforcement in thewebs of inverted T beams should resisthanger tension forces caused by loadsapplied to the lower part of the web.The step-by-step procedure based onthe proposed criteria summarizes in theprevious section the design of invertedT beam bentcap girders. A designexample given in the Appendix elabo-rates upon the application of the pro-posed criteria.

PCI JOURNAUJuIy-August 1985 131

REFERENCES

1. Standard Specifications for HighwayBridges, 12th Edition, American Associ-ation of State Highway and Transporta-tion Officials, Washington, D.C., 1977,469 pp.

2. ACI Committee 318, "Building CodeRequirements for Reinforced Concrete(ACI 318-83):' American Concrete In-stitute, Detroit, Michigan, 1983, 111 pp.

3. Code for the Design of Concrete Struc-tures for Buildings (CAN3-A23. 3-M77),Canadian Standards Association, Rex-dale, Ontario, Canada, 1977, 131 pp.

4. Mirza, S. A., and Furlong, R. W.,"Strength Criteria for Concrete InvertedT-Girders," Journal of Structural En-gineering, ASCE, V. 109, No. 8, August1983, pp. 1836-1853.

5. Mirza, S. A., and Furlong, R. W., "Ser-viceability Behavior and FailureMechanisms of Concrete Inverted T-Beam Bridge Benteaps," ACI Journal,V. 80, No. 4, July-August 1983, pp. 294-304.

6. Mirza, S. A., "Concrete Inverted T-Beams in Combined Torsion, Shear, and

Flexure," PhD Dissertation, The Uni-versity of Texas at Austin, Austin, Texas,May 1974, 181 pp.

7. Furlong, R. W., and Mirza, S. A.,`Strength and Serviceability of InvertedT-Beam Bentcaps Subject to CombinedFlexure, Shear, and Torsion," ResearchReport No. 153-1F, Center for HighwayResearch, The University of Texas atAustin, Austin, Texas, August 1974, 78pp-

8. Furlong, R. W., Ferguson, P. M., and Ma,J. S., "Shear and Anchorage Study ofReinforcement in Inverted T-BeamBenteap Girders," Research Report No.113-4, Center for Highway Research,The University of Texas at Austin, Aus-tin, Texas, July 1971, 73 pp.

9. ACI Committee 318, "Commentary onBuilding Code Requirements for Rein-forced Concrete (ACI 318-83)," Ameri-can Concrete Institute, Detroit, Michi-gan, 1983, 155 pp.

10. Structural Welding Code—ReinforcingSteel (AWS D1.4-79), American WeldingSociety, New York, N.Y., 1979.

APPENDIX A - NOTATION

Ae = area of cross section of all con-crete between compressionface and centroid of flexuraltension (or prestressing) rein-forcement

Ai = total area of cross section ofsupplemental longitudinalreinforcement required to resisttorsion

A,, = area of cross section of trans-verse reinforcement required toresist flexural tension in aflange overhang

A, = area of cross section of one legof a web stirrup

A = area of cross section of both legs

of a web stirrupA,f = area of cross section of trans-

verse reinforcement required toresist shear friction in a flangeoverhang

Ar/s = A r/s required for combined ef-fect of flexural shear and torsionminus A r /s required formaximum flexural shear actingalone

a = distance from face of web tocenter of bearing pad takenperpendicular to beam axis

B = length of bearing pad alongedge of flange

Bp = B+2B,.B. = width of bearing pad perpen-

dicular to beam axishf = overall width of flange of in-

verted T beamb,, = width of webd = effective depth of inverted T

beam between compressionface and centroid of flexuraltension (or prestressing) steel

dr– distance from edge of hearing

132

pad to longitudinal end of in-verted T beam

d,, = effective depth of flange asdefined in Eqs. (1b), (3b), and(6b)

= specified strength of concrete= service load stress limited to a

maximum of 2f„13 in stirrupsacting as hangers

J,

specified yield strength ofreinforcement

h

= overall thickness of inverted Tbeam

h1

= overall thickness of flange ofinverted T beam

j df = effective distance betweencentroid of compression andcentroid of tension for cal-culating flexural reinforcementin flange overhang

M„ = applied ultimate bending mo-ment acting on overall crosssection of inverted T beam

n

= coefficient defined after Eq.(10)

P, = concentrated service load act-ing on one bearing pad

P„

= concentrated ultimate loadacting on one bearing pad

= stringer spacing along the beamaxis

s = spacing of web stirrupsTo = ultimate pure torsion strength

of overall cross section of in-verted T beam

T,, = applied ultimate torque actingon overall cross section of in-verted T beam

Vo = ultimate flexural shear strength(in absence of torsion) of overallcross section of inverted Tbeam

V. = applied ultimate flexural shearforce acting on overall crosssection of inverted T beam

x = smaller dimension of rectanglex, = shorter center-to-center dimen-

sion of closed web stirrupy = larger dimension of a rectangley, = Ionger center-to-center dimen-

sion of closed web stirrupat = [0.66 + 0.33( j1 Ix,)] _- 1.5p = 0y' 1— V. /V.)_

= coefficient of sliding frictiontaken as 1.4 for normal weightconcrete cast monolithically

= strength reduction factor takenas 0.9 for flexure and 0.85 forshear and torsion

APPENDIX B - DESIGN EXAMPLEConsider a reinforced concrete in-

verted T beam shown in Fig. 13a. Thebeam acts as a bentcap girder in abridge superstructure to support fourinterior and two exterior precast pre-stressed concrete stringers placed oneach side of the web. Each interiorstringer exerts a total force of 130,000and 221,000 lbs (580 and 985 kN) at ser-vice load and ultimate Ioad conditions,respectively.

The corresponding reactions fromeach of the exterior stringers are 90,000and 143,000 lbs (400 and 635 kN). Theultimate live load plus impact reac-tions included in the foregoing totalloads are 130,000 lbs (580 kN) for each

of the interior stringers and 65,000lbs (290 kN) for each of the exteriorstringers.

All stringers are placed on 20 x 15 in.(508 x 381 mm) bearing pads spaced at84 in. (2134 mm) on centers with an enddistance of 21 in. (533 mm) as indicatedin Fig. 13a. Assume that the frictionalforces produced at the stringer bearingsare negligible.

Specified concrete strength T,) andspecified yield strength of reinforce-ment (f) ) are 4000 and 60,000 psi (27.6and 414 MPa), respectively. A minimumclear cover of 2 in. (51 mm) is used forall reinforcement and the maximum ag-gregate size is 1% in. (38 mm).

PCI JOURNAL1July-August 1985 133

Flange Design for PunchingShear

Since B„ equals (20 + 2 x 15) = 50 in.,Pg = 221,000 Ibs, and f, = 4000 psi, therequired value of df is 13.4 in. from Eq.(lb) or Fig. 3. This gives a flange thick-ness (= 13.4 + 0.625 + 2.0) = 16.1 in.Use a flange thickness of 18 in. with dffurnished - 15.4 in.

These values are acceptable becausethe length of the flange overhang (_width of bearing pad plus 1 + 2 in. = 18in.) is not greater than the flange thick-ness.

The required minimum transversedistance between stringers on the oppo-site sides of the web is equal to 2 d, + B,,(= 2 x 15.4 + 15)=45.8 in. anda (= Y2 ofthe bearing pad width plus 2 in.) = 9.5in. Hence, the required web widthequals (45.8 - 2 x 9.5 =) 26.8 in. Use b,= 34 in.

This width will be required to ac-commodate the longitudinal reinforce-ment near the top of the web. Therequired minimum longitudinal spacingof stringers is (2d1 + B = 2 x 15.4 + 20_) 50.8 in., which is smaller than thespacing furnished.

At exterior stringers, P. is equal to143,000 lbs. Hence, the minimum enddistance required for punching shearand calculated from Eq. (2) is zero,which is less than the furnished value(= 21- 10) = 11 in.

Flange Design for Bracket ShearB + 4a equals (20 + 4 x 9.5 =) 58 in.,

which is less than S = 84 in. for interiorstringers. Hence, use an effective flangeIength of58 in. in computingdrfrom Eq.(3h). From Eq. (3b), the required dt iscalculated as 5.7 in., which is less thanthe actual df (= 18 - 2 - 0.625/2) = 15.6in. For end stringers, the effectiveflange length equals (2 x 21 =) 42 in. andthe required df = 5.1 in., which is againless than the actual d1.

From Eq. (4b), the requiredA„requals

3.1 and 2.0 in,2 for interior and exteriorstringers, respectively. The effectiveflange length associated with Eq. (4b) is58 in. for interior stringers and 42 in. forexterior stringers. This makes therequired Arf - 0.65 and 0.57 in.2/ft forthe interior and exterior stringers, re-spectively.

Note that two-thirds of this reinforce-ment (2A„r13 = 0.43 or 0.38 in.2/ft) is re-quired for the top layer.

From Eq. (5b), the required A,f is cal-culated to be 3.2 and 2.0 in . 2 for interiorand exterior stringers, respectively.Considering the effective flange lengthsassociated with Eq. (5b) (67.5 in. forinterior and 42 in. for exterior stringers),Agf is equal to 0.57 in.2/ft for both theinterior and exterior stringers.

The flange transverse reinforcementplaced in the top layer equals 2A„r/3 orA,f , whichever is greater. Use #5 bars at6 in. on centers in the top layer over theentire length of the beam. This providesa steel area of 0.62 in 21 ft, which satisfiesall the requirements for reinforcementin the top layer.

In order to develop the yield strengthof flange transverse reinforcement,these bars will be furnished in theshape of closed rectangular stirrups asshown in Figs. 7 and 13c. These barswill also provide support for longitudi-nal reinforcement in the flange.

The flange transverse steel requiredin other layers below the top layerequals (A„f13 =) 0.22 and 0.19 in.2 /ft forinterior and exterior stringers, respec-tively. Provide #3 bars at 6 in. on cen-ters in a layer that is placed 4 in. belowthe top layer over the entire length ofthe beam. This satisfies the require-ments on vertical spacing and area ofreinforcement for these bars.

The longitudinal components of thelive load stringer reactions were notincluded for the design of top rein-forcement in the flange. Since thestringer spacing exceeds B + 5 a, thereis sufficient reserve strength in theflange to resist such forces.

134

Similarly, the effect of longitudinalforces due to friction at stringer bearingswere not considered in the design sincethese forces are given as negligible.

Web Design for Hanger ActionSince bf equals 70 in., df (= 18 — 2

-- 0.625 — 0.5) — 14.9 in., P„ = 221,000lbs with S = 84 in. for interior string-ers, and B — 20 in., calculated A„/s forstrength requirements is 0.086 in. and0.103 in. from Eqs. (6b) and (7b), re-spectively.

P, equals 130,000 lbs and B + 3 a (= 20+ 3 x 9.5) = 48.5 in., which is less than S= 84 in. Hence, A,/s required for ser-viceability considerations and calcu-lated from Eq. (8h) is 0.134 in. Thelargest value ofAx /s obtained from Eqs.(6b), (7b), and (8b) is 0.134 in. and willbe used in design.

At exterior stringers, P. equals 143,000lbs, PB = 90,000 lbs, and S (=2x21)= 42in. The largest value of Ar ls obtainedfrom Eqs. (6h), (7b), and (8b) is again0.134 in. Hence, AV ls = 0.134 in. isrequired over the length XZ of the beamfor the design of hangers. The final se-lection of stirrups acting as hangers willbe delayed until the design for shearand that for shear plus torsion has beencompleted.

Web Design for MaximumFlexural Shear

The critical sections occur at X and Yas indicated in Fig. 13a. Since d equals(64 — 2 — 0.625 — 1.41 — 0.5 =) 59.5 in.,A, = 2670 in.', and the applied ultimateshear force (V„) at the face of the supportor Section X [ = 2 x (221,000 + 143,000)plus (57 ,000 for the self weight of thebeam)] = 785,000 lbs, the requiredvalue of A,; f„ d/s is calculated to be585,000 lbs from Eq. (9).

Because this value is less than 8 ^"Tb d (= 1,020,000 lbs), the stirrups canbe provided to resist the required force

and the cross section need not berevised. The required A„Is for shearforce at X is then calculated as 0.164in., which will control the design ofstirrups required for shear force indistance XY.

The applied ultimate shear force atSection Y is [2 x 143,000 plus (30,000 forthe self weight) = ] 316,000 lbs and thecalculated A z l.s from Eq. (9) is 0.01 in.This is lower than the minimum valuerequired. Hence, A„Is equals 50 b,,lf„(= 50 x 34/60,000) = 0.03 in., which willcontrol the design of stirrups for shearforce in distance YZ. The final selectionof stirrups will be delayed until thecheck for shear plus torsion has beendone.

Web Design for CombinedFlexural Shear and Torsion

The most critical section for torsiondesign occurs at X, where 7. x zy = 85,650in:”, x, = 29.4 in., y, = 59.4 in., and a t (=0.66 + 0.33 x (59.4129.4)] — 1.33.

The applied ultimate torque (T„) atSection X[= (130,000 + 65,000)x(9.5+34/2)] = 5,168,000 lb-in.

The applied ultimate shear force (V,,)at Section X I= 785,000 — (130,000 +65,000)) = 590,000 Ibs, and the flexuralshear capacity of the cross section (V0 ) atX will at least equal the appliedmaximum ultimate shear force actingalone which is 785,000 lbs.

From Fig. 12, 13 is calculated as 0.56which satisfies Eq. (13) and the crosssection need not be revised. The A,,/.srequired over distance XY for theeffect of flexural shear plus torsionis then computed from Eq. (14) as0.03 in.

Since the applied ultimate torque atcritical section Y is very small, onlyminimum reinforcement is required.Hence, A,Is equals 50 hl f, = 0.03 in.,which will control the design of stirrupsfor the combined effect of flexural shearand torsion in distance YZ.

PCI JOURNAL/July-August 1985 135

Action on Web Distance XY Distance YZ

(a) Hanger tension 0.134 in. 0.134 in. (controls)(b) Maximum flexural shear 0.164 in. (controls) 0.03 in.(c) Torsion plus flexural shear 0.03 in. 0.03 in.

Selection of Web StirrupsThe table above shows the sum-

mary ofA 1,ls required for the web underdifferent actions.

The stirrups required for hanger ac-tion or flexural shear alone will beclosed at least across the bottom of thebeam. Closed rectangular stirrups arerequired for resisting torsional forces.Use (1# 5 closed + 1# 4) stirrups at 6in. on centers in distance XY and (1# 4closed + 1# 4) stirrups at 6 in, on cen-ters in distances YZ as shown in Fig.13c. These stirrups will satisfy therequirements for all actions summarizedabove.

Longitudinal ReinforcementThe most critical section for flexure is

at the face of the support and themaximum bending moment occurswhen full live loads act on both sides ofthe web. Applied ultimate bendingmoment (M.) at X equals (2 x 221,000 x6.1 + 2 x 143,000 x 13.1 + 3800 x 14.82/2=) 6,859,000 lb-ft and d = 59.5 in.

This gives the required area of steel =27.2 in. 2 Use 18 #11 bars placed in twolayers near the top of the beam as indi-cated in Fig. 13c. Other longitudinalbars shown in Fig. 13c are required toprovide stiffness to the steel cage for

handling purposes and to resist the lon-gitudinal forces that occur when torsionacts.

A further check on longitudinal rein-forcement required for torsion is notnecessary since the design of web stir-rups is not controlled by torsion plusflexural shear. The longitudinal bars inthe top corners of the flange also act asanchor bars for flange transverse rein-forcement.

The intent of this example was merelyto elaborate the design requirementsspecifically associated with inverted Tbeams and recommended in the body ofthis paper. In addition to these require-ments, all code provisions,' especiallythose for spacing and development ofreinforcement, should be satisfied.Another area of major consideration isthe connection of the pier and the in-verted T beam.

Si Conversion Factors1ft= 12 in.=305 mm1 in.2 = 645 mm2I lb = 4.45 N1000 psi = 6.9 MPa1000 lb-ft = 12,000 lb-in,

= 1356 N-m

NOTE: Discussion of this paper is invited. Please submityour comments to PCI Headquarters by March 1, 1986.

136