PROCEEDINGS PAPER

Prestressed Concrete Anchoragesfor Large Tainter Gates

by Keith O. O'Donnell*

INTRODUCTION

The radial or tainter gate is themost common type of spillway crestgate used on large dam projects.Because of the simplicity, lightweight, and low hoist requirementsof a tainter gate, economy has beenthe basic reason for the trend to-ward its increased usage. However,this type of gate offers additionaladvantages over other types, suchas permitting a very favorabledesign from the standpoint of hy-draulics, ease of service, and main-tenance.

During the past decade, the Corpsof Engineers has designed taintergates for approximately 70 projectsincluding over 600 gates. Most ofthese gates have been constructedor are presently under construction.Prior to that time, it was somewhatunusual to design tainter gates inexcess of 40 ft. in height. It is nownot unusual to design 50-ft. widetainter gates subjected to a hydrau-lic head of 60 ft. or more. For gatesof this size, the direct thrust fromhydraulic loading delivered to eachspillway pier is approximately sixmillion pounds. This load is trans-mitted from the trunnion yokes to

*Structural EngineerOffice, Chief of EngineersWashington, D.C.

52

the supporting trunnion anchoragegirder and thence into the pier.While the discussion in this paper isconfined primarily to the anchor-ages, a photograph of a tainter gateshowing its principal parts is in-cluded as Fig. I. The principalstructural components of the gateare the skin plate assembly, thehorizontal girders, the end frames,and the trunnion assembly.

EVOLUTION OF ANCHORAGES

While the size of tainter gateshas continuously increased over thepast ten or fifteen years, anchoragesand anchorage girders have changedto provide more efficient and moreeconomical supports. Although therehas been quite a variation in typesof anchorages used, the more gen-eral types will be presented beforediscussing the design of prestressedconcrete anchorages.

Prior to the advent of the largertainter gates, the anchorage systemused, more than any other type,consisted of a steel girder, usuallybox type, cantilevered at each faceof the pier to receive the trunnionloads. An example of this type isshown in Fig. 2. The loads from thetrunnion girder are transmitted intothe pier by an anchorage consistingof two standard rolled beams orbuilt-up sections welded to the trun-

PCI Journal

Fig. 1—Tainter Gate

BUTT WELDS FACE OF PIEROF PIN

SLOT DOWNSTREAM FACE OF PIER eFLANGE OF CROSS GIRDERFOR TIE BEAM FLANGES

PLAN

UNBONDED OUTLINE OF PIEROF PIN

Nf BEAM ._,•. ;e e' °•'. `^:"' K

'°'` APPR. CONSTR. JOINTe

FOR ERECTION ONLYTIE BEAM-ROLLEDOR BUILT-UP SECTION

SIDE ELEVATION

Fig. 2—Horizontal Tension Tie Anchorage

June 1965 53

nion girder and to an embeddedgirder or grillage at the upstreamend. These tie beams are placedwith at least eight inches of con-crete cover at the pier faces. Theembedded girder is designed to re-ceive the entire load from the tiebeams in bearing. To allow freemovement or deformation of the tiebeams and prevent tension in thepier concrete, the beams are un-bonded or mechanically isolatedfrom the concrete for the entirelength by 1/2 inch of cork mastic.

For an intermediate pier with onegate loaded and the adjacent gateunloaded, the anchorage frameformed by the trunnion girder and

the tie beams is subjected to severeracking. In addition to the directtension in the tie beam nearest theloaded gate, there are bendingstresses induced into the frame. Thistype of anchorage has short verticalbeams embedded at the inner facesof the tie beams to transmit boththe side thrust from the inclinedend frames and the vertical compo-nent of trunnion reactions into thepier. Sliding at the downstreamends of the tie beams is allowed byuse of bronze plates.

Another type of anchorage thathad widespread use on the smallerto the moderately sized gates is onein which the trunnion loads are

E OF PIN

e ^°°,° STE E L PLATES BEAMt ;TOP AND BOTTOM_._ _

PLAN€ OF PIN

^_ __^__ __ _ BACK OFPIER°:OF PIN

OFTRU NN ION

TAINTER GATE vv '}^, . ° Std'/O^,P Q^p1^' If

SPILLWAY CREST,°° ;°. S1E^^ W BEAM I I

• °8 ^ °° .:APPROXIMATE CONSTRUCTION JOINT STEEL PLATE .-

*' ANCHOR BOLTS ANCHOR BOLTS 1

SIDE ELEVATION

Fig. 3—Inclined Tension Tie Anchorage

54 PCI Journal

SYMMETRICAL ABOUT ( PIER

ii

l

FACE OF PIER

PLAN

II

IIp\\ \\ \

j,.:.... A^ • ° ate.a • IPRESTRESSING STEEL,

• a-:a

a

^p .p a ' 4 ELEVATION•a

Fig. 4—Verticle Cantilevered Anchorage

transmitted to the concrete belowthe pier rather than to the pier it-self. This is shown in Fig. 3. In thistype, the anchor ties are inclined atan angle with the horizontal largeenough to place the embeddedgirder well below the level of thespillway crest. Columns are pro-vided under the trunnion girder totake the vertical components of thetrunnion loads. The cork mastic isalso used in this case to isolate theanchorage members from the con-crete except at the embeddedgirder and the column footings.While this anchorage requires great-er amount of structural steel than

the previous one, the reduction inthe amount of pier reinforcing steelsomewhat compensates for this.

While the above anchorages arequite satisfactory structurally forsmaller gates, they are not suitablefor large gates. As the size of taintergates is increased, the problem ofproviding an anchorage system be-comes more complex. It is generallyeconomical to utilize the higherstrength structural steels in the de-sign of larger tainter gates. How-ever, their use is not feasible for theanchorages because of the increasein elastic deformation due to thehigher stresses, and because of in-

June 1965 55

crease in length of embedment re-quired. Resulting elongations of thetie beams would be excessive. Thenatural solution is to prestress theanchorage system into the pier. Pre-stressing is preferred since it willeliminate the relative movementbetween the pier concrete and theanchorage structure. A prestressedanchorage system appears most es-sential for large gates spanning be-tween relatively narrow spillwaypiers.

Two types of prestressed anchor-age systems that have receivedgeneral usage consist of steel trun-nion girders and post-tensioninginto the piers. The system shown inFig. 4 is one in which the trunnionloads are taken by separate verticalcantilever girders. While this typeof anchorage is not consideredeconomical, there is a constructionadvantage in that each anchoragecan be aligned independently with-out affecting adjacent gates. A moreconventional type consists of awelded structural steel box girder,similar to those used without pre-stressing, cantilevered out on eachside of the pier to receive thetrunnion loads. The first designs inthe Corps of Engineers to incorpo-rate post-tensioned anchorages werefor Markland and Greenup projectson the Ohio River and were pre-pared by the Seattle District.

For very large tainter gates inwhich the total thrust anchored intothe pier amounts to more than, say6,000,000 pounds, the amount ofstructural steel involved in a trun-nion girder becomes excessive. In1961 a prestressed concrete anchor-age was conceived and developedfor the spillway tainter gates onLower Monumental Lock and Damon the Snake River. The design ofLower Monumental spillway gateswas also prepared by the Seattle

District. The gates on this projectare 50 ft. wide by 61 ft. high. Thegirders are of high strength con-crete, cast-in-place, post-tensioned,square in cross-section, and abutthe downstream face of the pier.Prestressing into the pier is similarto that previously used for structuralsteel girders.

MAJOR DESIGN CONSIDERATIONS

The prestressed concrete anchor-age consists of two systems of post-tensioned tendons; one for prestress-ing the girder itself, and one forprestressing into the pier. For dis-cussion in this paper, the system ofprestressing of the girder will bereferred to as the "transverse" pre-stressing and the prestressing intothe pier as "longitudinal." Thisanchorage system is depicted inFig. 5.

A compressible filler is installedover the central portion of the verti-cal contact surface between the pierand the girder. This gives a moreefficient arrangement by providinga larger moment couple for the pre-stressing forces and reducing thebending in the girder. After pre-stressing is completed, the prestress-ing tendons are grouted and con-crete caps are placed over theexposed ends of both the transverseand longitudinal tendons.

There are usually two major de-sign conditions for the intermediatepiers: one in which both gates areclosed, and the other condition inwhich one gate is closed with theadjacent gate open. The latter con-dition governs the design of thelongitudinal prestressing.

The longitudinal prestressing ele-ments are arranged in two groups,each group being placed as near thepier face as practicable, allowingabout 9 inches of cover for the out-side rows of tendons so that con-

56 PCI Journal

ADD GROUT CAP AFTER PRESTRESSINGIS COMPLETE. (TRANSVERSEPRESTRESSING STEEL NOT SHOWN) t PIER

TRUNNIONv YOKE NOT SHOWN THIS VIEW : GROATCAP ANCHOR PLATE

H___-- i:';'ANCHOR PLATE

ELEVATION DOWNSTREAM ELEVATION

TRANSVERSE PRESTRESSINGSTEEL

LONGITUDINAL.'•°,, • =1/2" COMPRESSIBLE°FILLER GROUT COVERPRESTRESSING STEEL , ° `•, QPIER; ;°^,• _ NOTE: CONVENTIONAL REINFORCING°y

=I+ STEEL IN GIRDER AND PIER NOT

Yt SHOWN.TRUNNION YOKE

PLAN GROUT COVER\

Fig. 5—Prestressed Concrete Anchorage

ventional pier reinforcing steel maybe placed outside the prestressingsteel. The amount of prestressingsteel is determined assuming theunbalanced condition of one gateloaded with the adjacent gate un-loaded.

While the bearing stress distribu-tion is not exactly known, longitu-dinal prestressing that will providean average residual bearing stressbetween the girder and the pieron the loaded side of about 20 per-cent of the final prestress will gen-erally be satisfactory. However, itmay be necessary to investigate theinterface pressures to assure that notension exists under unbalancedloads.

The girder is considered a rigidmember with a straight-line varia-tion in the stress distribution. How-ever, for the girder design, the uni-form stress distribution assumptionwill be conservative.

Another consideration in the de-sign of the longitudinal prestressing

into the pier is to unload the pre-stressing load at the embedded endgradually rather than abruptly on asingle vertical plane. This is ac-complished by staggering the cutoffsof tendons and will alleviate theobjectionable concentration of largequantities of reinforcing steel thatwould otherwise be required to con-trol vertical tension cracks in thepier concrete. Where practicable,the gate trunnion assembly is posi-tioned so that the vertical compo-nent of thrust is negligible with thegate closed and full pool. Theoreti-cally, this places the trunnion atabout one-third the height of thegate above the sill which permitsplacing the pier prestressing ele-ments horizontally.

A zone of high strength concreteis incorporated into the pier for theentire length and height of the pre-stressing elements. Placing the pre-stressing elements horizontally is aconstruction convenience in thatfewer lifts of the high strength con-

June 1965 57

crete are required and also highsteel support frames for the down-stream portion of the prestressingelements are eliminated.

The design of the post-tensioningelements into the pier can be ac-complished without regard to thetrunnion girder design requirement.After the longitudinal prestress hasbeen determined, all external loadson the trunnion girder are known.Design of the girder actually in-volves two steps: (1) choice of thegeometry of the girder and (2)the analysis of the trunnion girderto determine the unit stresses andestablish the amount and arrange-ment of transverse prestressing.

The facility of selecting a mem-ber of proper section comes withexperience. Usually a concrete of5,000 psi strength is required exceptfor larger gates which may requirehigher strength concrete. The sizeof the girder required is determinedby the magnitude of the bending,shear (diagonal tension) and tor-sion. Generally, the minimum widthof girder (vertical height) will beestablished by the space required toaccommodate the longitudinal pre-stress tendons. Maximum torsionusually will occur in the girder whenthe gate is partially raised and thepool is at maximum level. When itis practicable to position the trun-nion yoke on the girder to minimizethe effects of torsion under themaximum loading condition, thecombined shear, including torsion,will not generally be critical forother loading conditions. The flex-ural stresses in most cases will beconsiderably below those permittedby specifications.

The ordinary beam theory orstraight-line distribution of flexuralstresses does not consider the effectsof the normal pressures from the

external loads and reactions at thetop and bottom edges of the beam.For deep beams', however, theseeffects are substantial and cause anon-linear distribution of bendingstresses and the shear stress distribu-tion is not considered parabolic.Also, it has been shown in ordinaryshear investigation that even inshallow beams the maximum princi-paI tensile stress is exaggerated,and that the critical section for in-clined tension is not at the reactionbut some distance from the support.The reason for this is that, near thereaction, the values of the verticalcompressive stresses are large. Ananalysis by the finite element meth-od in plane stress analysis indicatesthat the effects of the trunnion loadreduce the principal tension sub-stantially.

For the reasons expressed in theabove paragraph, it is assumed thatthe critical section for principaltension is near the upstream edgeof the girder. This appears to be thelogical area for any cracks to de-velop. With this assumption, theprincipal stress obtained from com-bining average shearing stress withone normal stress is considered con-servative, i.e., neglecting any com-pressive stress caused by the trun-nion reaction.

Limiting values for principal ten-sile stress vary considerably. Basedupon working load, the limiting val-ues of principal tension, as given inT. Y. Lin's Design of PrestressedConcrete Structures2 range fromabout 0.015 f to 0.033 f , for beamswithout web reinforcement. Consid-ering ultimate load, the range isfrom 0.045 f, to 0.08 f, without webreinforcement, and to about 0.11 fwith web reinforcement. The codesfor prestressed concrete in theUnited States make no reference to

58 PCI Journal

limiting values of working load prin-cipal tension and generally the sheardesign is based on ultimate strength.A limiting principal tensile stress of2 under working load is consid-ered reasonable when no web rein-forcement is provided. This is quitecomparable to the British StandardCode of Practice. When web rein-forcement is provided, a limitingvalue of 3i is conservative. Thismay serve as a guide; however, thedesign should in the end be investi-gated for ultimate strength and re-inforcement provided accordingly.

For the small to moderate sizegates, it is practicable to limit thecomputed principal tension to a val-ue for which theoretically no webreinforcement is necessary. However,for the larger gates, the principaltension, as computed often governsthe design, and it may not be prac-ticable to limit the principal tensionto that in which no shear reinforce-ment is required. While this methodof determining principal tension isadmittedly conservative, it is con-sidered reasonable for design untilmore is known on this subject. Inany case, it is considered prudent toprovide a nominal amount of webreinforcement and this may be com-puted by rule of thumb. The mini-mum percent of web reinforcementdesirable is about 0.003 bs, in whichb is the width of girder and s thespacing of web reinforcement.

The ultimate strength of girdersshould be investigated both forflexure and shear. The ultimate loadon the girder, based upon 1.8 timesthe normal trunnion working load,should not exceed the ultimateflexural strength produced by thefollowing formula given in the ACICode:

Mu = 0 [A3 tau d (1-0.59 q)]

The principal tension in a pre-stressed concrete member may in-crease quite rapidly compared to theunit shearing stress for an increasein the external loading. For this rea-son, it becomes necessary to inves-tigate the principal tension underultimate shear load conditions. Thisis accomplished by computing theunit shear stress resulting from aprincipal tension of 4V which isconsidered the diagonal crackingstress. The difference between theshear value required for ultimatestrength and that for the shear crack-ing load should be taken care of byproviding web reinforcement. Thisis computed by the formula:

Av= T.-OV^)s0dt,

The areas of concrete beneath theprestressing loads in both the girderand the pier end anchorage zonesor end blocks are subject to tensilestresses. The stress distribution israther complicated in the end an-chorage zone of the pier with twogeneral areas of tension: (1) thecenter of the section which is termedthe "bursting zone", (2) on thesides and end surface called the"tensile spalling zone". Guidancefor this type of behavior is dis-cussed in Guyon's Prestressed Con-crete. While there remains consider-able latitude for the designer'sjudgment, it is considered that gen-erally conventional grid reinforce-ment placed as near the surface aspracticable to take about 4 percentof the total prestressing force will beadequate in the spalling zone. Toprevent spalling of the concrete atthe corners of the pier, the outerlayer of reinforcement is welded toangles embedded along the verticaledges of the pier. Tensile stresses inthe bursting zone are evaluated byGuyon and the ratio of the bursting

June 1965 59

tension to the prestressing force isshown to be a maximum of about0.18 for the typical pier load ar-rangement. Reinforcement for thesestresses should be provided for adistance in from the downstreampier face of approximately one-halfthe width of the pier. The theoreti-cal stress pattern for these zones isincluded in the above reference.

The actual final load on the pre-stressing steel should not exceed 60percent of the minimum ultimatestrength of the steel. The initialtension on the steel, immediatelyafter seating of the anchorages,should not exceed 70 percent of theultimate strength of the steel. Thecomputed losses in steel stress dueto elastic shortening of the concrete,shrinkage, creep, and plastic flowwill be less for gate anchorage sys-tems in massive piers than withnormal beam or slab prestressing.Therefore, the initial prestress willgenerally be somewhat below 70percent of the ultimate strength toassure that the final stress does notexceed 60 percent.

To minimize integral action be-tween the girder and the pier, thegirder is prestressed prior to longi-tudinal prestressing.

ANALYSIS OF DESIGN

The steps in design procedure are:1. Determine magnitude of the

longitudinal prestressing (into thepier) required for the differentloading conditions which will com-plete the applied loads for thegirder.

2. Determine the shear and mo-ment diagrams for the appropriateloading conditions.

3. Assume a girder cross sectionand compute its properties.

4. Determine the magnitude andarrangement of prestressing force toprovide compressive stresses to off-

60

set tensile stresses caused by bend-ing moments and to limit theprincipal tensile stresses.

5. Investigate stresses at both thecritical section for moment and thecritical section for shear with twocombinations of conditions, namely,final prestress plus full design loadand initial prestress plus dead loadonly.

6. Compute ultimate strength ofgirder to see if it meets the require-ments.

7. Design shear steel by ultimatestrength method.

8. Determine end block require-ments.

Design of the longitudinal pre-stressing is based upon an unbal-anced loading condition of one gateclosed with the adjacent gate open,see Fig. 6. To maintain a residualbearing stress of 20 percent of thefinal prestress, the prestress afterlosses is

F=TXAX1.25and the number of tendons requiredin each prestress group is

FN=

Itswhere f t8 is the final prestress pertendon (maximum value of 60 per-cent of the ultimate strength). Theprestress force is then corrected forthe actual number of prestressingtendons selected. The loading condi-tions for the girder are now knownand the shear and moment diagramsmay be computed as indicated inFig. 6. Based on a trial section, thegirder can now be analyzed. Thewidth (vertical height) of the girderis somewhat governed by the re-quirement of the longitudinal pre-stress force.

The girder prestressing arrange-ment should be such that the down-stream fibers of the girder under an

PC'I Journal

PRESTRESS FORCE

fflllUUthddJ _____ _________

H'no- N d

v ^x 11

xW

3W

°`u

W oct,

aN

a. 0 owe

eel w x >

INTERFACE BEARINGRTRUNNION THRUST STRESESS

IT B/2 % (HORIZ.^i 1 COMPONENT)F

1. A

SHEAR

MOMENT

Fig. 6—Load, Shear, and Moment in Girder

unloaded condition will not be intension due to prestressing anddead load. It is suggested that aminimum compression stress of 0 to100 psi be maintained for the un-loaded condition to insure compres-sion across the entire section evenwhen the pool is below the levelof the gates. Fig. 7 indicates thenormal stress distribution on thesection of maximum shear due toprestressing and moment at thatsection. At the section of maximummoment, no tension should result inthe upstream fibers under maximumworking load condition. At the sec-tion of maximum shear, a fairly uni-form distribution of normal stresses

across the section usually resultsfrom bending and prestressing whenthe prestressing is based on principalstress limitations.

For the conditions in which tor-sion is a consideration, the torsionalshear stress should be combinedwith the direct shear stress. Whereit is required that torsional stress becalculated, Bach's method may beused as presented in AdvancedMechanics of Materials4 by Seelyand Smith:

_9MtSb 2 b d2

Sd = d Sbb

June 1965 61

100 psi f2 f4= f 2 + 100

HORIZ.3 e U t+— DEPTHO PJLL

d

fl f2 f3= f 1 —f2

(a) (b) (c)

Fig. 7—Girder Stress Diagram

The principal tension may be ex-pressed by:

ft=f2\/3— (f I2 +v2

V `

v bdwith a limitation of f t = 2 i/7 whenno design web reinforcement is pro-vided and f t = 3V7 with web rein-forcement. The above equation issolved for f 3 which is then expressedby:

P Pec Mcf3 = A — I--- I

where e<d/6and finally the required girder pre-stress force, P. is determined.

The above computations theoreti-cally represent the condition at thesection of maximum shear stress. Itis now necessary to investigate thestress conditions at the section ofmaximum moment as indicated instep 5 of the design procedure toinsure that allowable stresses are not

exceeded.The ultimate flexural strength is

determined by the formula for M.as previously stated. However, theratio of prestressing steel used should

be such that p f8;` does not exceedfel

0.30. If this factor exceeds 0.3, thenthe M. is computed by a differentformula and is given in the ACICode (Formula 26-8). The load fac-tor of 1.8, as stated earlier in thispaper, is considered adequate forthis type of prestressed memberwhere the working loads are quitedefinite. The shear steel, A, is thendetermined based upon ultimateshear strength. Maximum shear rein-forcement should be provided aspreviously discussed.

RESULTS OF MODEL TESTING

Because of its configuration, thetrunnion girder is beyond the rangeof the usual beam or girder design.Since the trunnion girder is canti-levered at each side of the pier and

62 PCI Journal

the span is short, the behavior issimilar to that of a deep beam.Stress distribution is quite complexas a result of transverse prestressingof the girder, longitudinal prestress-ing into the pier, and gate loadingswhich may be either symmetricalor unsymmetrical. The interactionbetween the girder and pier andthe compression in the beam be-tween the yoke and the longitudinaldownstream anchor plate furthercomplicate the pattern of stress dis-tribution and is quite indeterminate.

The applicability of the assump-tions for conventional beam flexureand shear to this type of structuralmember was questioned. As a re-sult, structural model tests wereperformed to determine the struc-tural strength and behavior, verifythe original design, and to provideinformation for future designs. It isbeyond the scope of this paper todiscuss the model tests in detail,and only a brief resume of the re-sults is included.

The test program was performedunder the direction of Dr. Arthur R.Anderson of Anderson, Birkeland,Anderson & Mast of Tacoma for

the Seattle District of the Corps ofEngineers. The model tests includednine beams which were tested tofailure. The prototype girders forLower Monumental tainter gate an-chorages are 8 ft. by 8 ft. in crosssection by 23 ft. 2 in. long. A specialfacility for testing one-fourth scalemodels of the trunnion girders wasconstructed with a block of concreteto simulate the spillway pier. Thegirder models were fastened to thepier by post-tensioned high-tensilestrength steel rods. The girdermodels were cast-in-place concreteof 6,000 psi strength. A cross headwas built into the pier to jackagainst for applying thrust (trun-nion load) to the girder. The testfacility is shown in Fig. 8.

Various prestress combinationswere employed in the model testing.The width of girder for the modelalso was varied to determine theeffect of girder width on strain dis-tribution in the concrete. Twenty-four SR-4 type A-12 electric straingages were attached to the uppersurface of each model. Model No. 1was cast on the pier which had thesurfaces of both the vertical face

Fig. 8—Model and Test Facility

June 1965 63

and the horizontal face sandblasted.While this model was able to resistup to 2.2 times the design load, itwas not possible to interpret thestrains measured because of the un-known conditions resulting fromshear action between the model andthe pier. The other models testedwere cast by preventing bond onthe horizontal surface between thegirder and pier with the idea thatthe model was free to deflect underhorizontal loads and that a stressdistribution conforming to that of abeam subjected to bending couldbe achieved.

The model tests revealed a verycomplex behavior of the prestressedconcrete girder due to the interac-tion of the trunnion girder and thepier. As a result of this interaction,the strains measured were not al-ways linear with applied load whichmade the interpretation of the datadifficult, Before the trunnion loadsare applied, the interface betweenthe girder and pier is under highcompression from the longitudinalprestress, and as the load approachesthe prestress value, the pressure onthis face approaches zero. Thechange in local concentrated stressalong the upstream face of thegirder is rather severe and a verysteep stress gradient results. Despitethe limitations of the strain meas-urements in the girder model, theyare instructive in presenting thebehavior under load of a structure ofthis type. The torsional effect fromvertical loads on the trunnion wasnot serious. The crack patterns runparallel to the hoop reinforcementnear the zone of maximum moment,demonstrating that this reinforce-ment contributes very little to theultimate shear strength of the gird-ers. This was verified by one modelin which nearly all of the web rein-forcement was omitted.

The model tests confirm that theLower Monumental girder design isadequate and that the prestress inthe girder could be reduced some-what and yet resist cracking at 1.5design loads. In the model tests,when a load 1.5 times the designload was applied, the transverseprestressing was reduced until crack-ing occurred. At this point, the ten-sion in the concrete on the upstreamface was approximately 1,000 psi,computed by the straight-line meth-od. With this reduced prestress forceapplied for the design loads, a ten-sion of 500 psi will exist based upona straight-line assumption.

Figs. 9 and 10 represent the resultsof a computer program performedby the Seattle District for the mod-els. These theoretical analyses ofprincipal stresses are based on thefinite difference method and the fi-nite element method. Fig. 11 gives acomparison of flexural stresses basedupon theoretical analyses andstraight-line beam theory.

SUMMARY

Since the idea of prestressed an-chorage systems was introduced, ithas been possible to utilize largertainter gates, in which the trunnionIoads on a single pier exceed sixmillion pounds, with far less em-bedded anchorage metal than pre-viously required and without asubstantial increase in pier size. Theuse of the prestressed concrete trun-nion girders has proven to beeconomical. For the anchorage sys-tem of the Lower Monumentalgates, the bid price indicates abouta 40 percent saving over a preced-ing project having gates of similarsize but utilizing structural steeltrunnion girders. Studies and bidson subsequent projects, includingsmaller gates, also reflect a sub-stantial saving though not generally

64 PCI Journal

srCD

I-

2" 11 s/4 113/4 ' 2"

184 k I21" 21 184k

+653 I I x+653F+980PSI+980PS1

PSI I.HOIE^ I PSI

F=310 k ►^5 y o n i"^' ^0 i 0 ^c ♦ $ F=310k

E=3.25"t q5 45 E=3.25

X

GIRDER ,, so ^►oR

l

so \o

^o

o'^i s yR?SO lzso I ^000

+100PSI +100PSI

+1302PSI I +1302PSI1 9" 10'/2 , 10'Y2^ 19"

280 kLY69'/s" 280kMAXIMUM PRINCIPAL STRESS IMINIMUM PRINCIPAL STRESS

Fig. 9—Stress Countours•Finite Difference MethodC:,Ut

2" 11 3/4" 11 3/4" 2"

0184 K+ o 184K

00

+653PSI 100 +653PSI+980 PSI /,,_1E +980 PSI

1000 ^{100 2$o

600

j / SOp ^`c y^i X00 R tiho -ter ^R SOO R ptc-F 310 t ^R ^R ^R >^ SCR y^ i R ^R R ^1R?SO ^IR F=310k

E=3.25" _^ pp x x dc) E=3.25"

GIRDER yR ^ ^R sow ^r ; } }^o „ 100 x ¢ GIRDER

'^R1000 1^ + +^ I 4 ' y 11R 750 DSO 0

?SO 1500 1000100 PSI 100 PSI

+1302 PSI I I M+1302PSI

I9" 10 1/2" 10 1/2"

280 K 1=69.1/2" 280K

MAXIMUM PRINCIPAL STRESS MINIMUM PRINCIPAL STRESS

rn

C)

0

Fig. 10—Stress Countours-Finite Element Method

35 psi 29 psi

t!'? - 109 psi

O

// COMPRESSION

— TENSION

/ / • FINITE DIFFERENCE METHOD

i 0 FINITE ELEMENT METHOD

/ x STRAIGHT LINE METHOD

FINITE /DIFFERENCE I /̂— STRAIGHT LINE

METHOD / METHOD

^ ^ 3OLL

FINITE ELEMENT METHOD

+1069/ pfd

+ 634 psi

+ 780 psi COMPRESSION TENSION

^--TRANS. t GIRDER

Fig. 1 1—Comparison of Flexural Stress

June 1965 67

of this degree. The prestressed con-crete anchorage is proposed forseveral Corps of Engineers' projects.

Comparison of the results ofstructural model tests, as well astheoretical analyses by the finite ele-ment method and finite differencemethod confirm that the straight-line theory for these structural mem-bers is reasonable. While the modeltests demonstrate that a fairly highflexural tensile stress, say about 400psi, appears reasonable under work-ing load conditions and would alsoprovide adequate ultimate strength,it is not recommended that tensionbe allowed in the design. This is inaccordance with the codes. While it

is recognized that the maximumcomputed principal tension will besomewhat greater than actuallyexists, conservative design assump-tions for this type of structural mem-ber are recommended.

REFERENCES1. Chow, L., Conway, H. D., and Winter,

George, "Stress in Deep Beams", Trans-actions, ASCE Vol. 118, 1953.

2. Lin, T.Y., Design of Prestressed Con-crete Structures, John Wiley and Sons,Inc., New York.

3. Guyon, Y., Prestressed Concrete, Con-tractors Record, Ltd., London and JohnWiley and Sons, Inc., New York.

4. Seely, F. B. and Smith, J. 0., AdvancedMechanics of Materials, John Wileyand Sons, Inc., New York.

Presented at the Tenth Annual Convention of the Pre-stressed Concrete Institute, Washington, D.C. September 1964

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