Lateral Stability Of Long Prestressed Concrete Beams Part 2
SPECIAL REPORTLateral Stability ofLong PrestressedPart 2Concrete BeamsRobert F. Mast, P.E.ChairmanABAM EngineersA Member of the Berger GroupFederal Way, WashingtonLong-time PCI Professional Member Robert F. Mast is oneof the four co-founders of ABAM Engineers, headquarteredin Federal Way, Washington . The firm , which recentlycelebrated its 40th anniversary, is today a member of theBerger Group, a worldwide consulting organization. Duringhis professional career, Mr. Mast has been responsible forthe design of many important buildings, bridges and specialstructures. As a partner in the firm, he was involved in theinnovative use of prestressed concrete in the first bulb teebridge, the Walt Disney World and Seattle monorails,Disney World 's Space Mountain, the Indonesian LPGfloating vessel and many other noteworthy structures.Among his many engineering contributions, he was one ofthe original developers of the shear-friction principle, widelyused in the design of precast concrete connections. For hispaper on this topic, Mr. Mast won ASCE 's T.Y. Lin Award.He is also an authority on stability problems associated withthe handling and transportation of long prestressed concretemembers, which is the subject of this paper.70A theory for evaluating the lateral stability oflong prestressed concrete /-beams supported from below is developed. The sameprinciples are also applied to hanging beams,extending the work presented earlier in Part 1of this paper. The theory includes consideration of the post-cracking behavior ofprestressed concrete beams under lateralloads. Data from the testing of a full-sizedprestressed concrete beam are presented toverify the theory. A method of computingfactors of safety related to lateral stability ispresented. A numerical example is included,giving computations for the factors of safetyfor both hanging and bottom-supportedbeams. A computer program in BASIC isfurnished to facilitate these computations.art l of this paper' dealt with prestressed concretebeams hanging from lifting loops. The stability ofbeams hanging from loops is primarily a function ofthe elastic stiffness properties of the beam and may readilybe determined using the methods given in Part I .This Part 2 paper deals with the stability of beams whensupported on elastic supports below the beam. Such supports may be bearing pads or transportation equipment. Theunderstanding of the behavior of a beam supported on elastic supports was found to be different from and far morecomplex than that for a beam hanging from lifting loops.It was found that rollover of beams supported from belowis determined primarily by the properties of the supportPPCI JOURNAL
rather than the beam. Long prestressedC ATERALconcrete 1-beams of ordinary proportions (such as the PCI BT -72), whenjsupported from below, were usually 1.\I1\'I \'found to have sufficient lateral bend; \ing strength to withstand greater anC. G. OFI 1 eZ 0 s n8 eDEFLECiED:} ,gles of inclination than can be resistedBEAM, ,Iby the supports. \I yPart 1 stated that Part 2 would givemethods for determining the lateralROLL A X I S\ l w II, lstability of beams supported fromM K 9 18-ol 1 below, and that this requires evaluation of the rotational stiffness of sup- d J -fl-"-f s3uiuPERELEVAT ONIMOMENTNI \SPRING SUPPORTports and the post-cracking behavior8-o\ ANGLE1 \ 8-o\ ANGLEAT SPRINGof prestressed concrete 1-beams sub AT SPR lNG SuPPORTjected to lateral loads. The solution toCo 'i!'case eICase ysinethese questions proved more difficultthan the author had anticipated beFig. 1. Equilibrium of beam on elastic support.cause both the properties of the support and the post-cracking propertiesof the beam may be nonlinear. Analytical solutions to thethe center of gravity of the beam above the roll axis. The approblems were developed, and the results were verified by aplied overturning moment arm ca about the roll axis due tofull-scale test.the weight of the beam is:This Part 2 paper supplements Part 1. Eqs. (1) to (15) were(16)Ca Z COS 9 e; COS 9 y sin 9given in Part 1. Additional equations given in this Part 2 paperbegin with Eq. (16). Part 1 is still valid, with two exceptions:wherethe author recommends using Eq. (22) instead of Eqs. (1) and(2), and using Eq. (30) instead ofEqs. (14) and (15).e roll angle of major axis of beam with respect to verticali lateral deflection of center of gravity of curved arc ofdeflected beamBACKGROUNDe; initial eccentricity of center of gravity of beamClassic studies of lateral buckling of beams are based ony height of center of gravity of beam above roll axisthe assumption that the beams are rigidly restrained from roThe resisting moment arm c r is equal to the resisting motation at the supports. Buckling is caused by the middle partment of the spring supports divided by the weight W of theof the span twisting relative to the support, creating a sidebeam:ways deflection. This type of buckling is important in steel1-beams, which have low torsional stiffness.(17)The torsional stiffness of an 1-beam varies as the cube ofthe thickness of the web and flanges. Concrete 1-beams,wherewith relatively thick webs and flanges, are 100 to 1000times stiffer in torsion than steel 1-beams. As a result, lateralK 6 sum of rotational spring constants of supportsbuckling of the classic type is seldom critical in a concretea superelevation angle or tilt angle of supportsbeam. But, when the supports have roll flexibility, theThe rotational spring constant K 6 has units of moment dibeams may roll sideways, producing lateral bending of thevidedby rotation angle in radians. The rotational springbeam. This is the cause of most lateral stability problems inof an elastic support is found by applying a moconstantvolving long concrete 1-beams.ment and measuring the rotation. The quantity K 6 is equal toThe approach may be greatly simplified by assuming thethe moment divided by the rotation angle.beam to be rigid in torsion. For concrete 1-beams with websIt is convenient to let r K 8 /W. The quantity r has aand flanges 6 in. (150 mm) or more in thickness, the torsional stiffness of the beams will normally be much greaterphysical interpretation. It is the height at which the totalthan the roll stiffness of the supports. The assumption of torbeam weight W could be placed to cause neutral equilibriumsional rigidity for the beam transforms the problem from awith the spring for a given small angle 9 (see Fig. 2). Forneutral equilibrium, the overturning moment will just equalbuckling problem to a bending and equilibrium problem.the resisting moment when the member supporting W is displaced by a small angle. The quantity r may be called the raGENERAL SOLUTIONdius of stability.The equilibrium of a slender beam on elastic supports isThe equilibrium angle e may be found by equating Ca tocr. Since both ca and Cr may be nonlinear functions of e, theshown in Fig. 1. This figure is similar to Fig. C 1 of Part 1,but with a positive quantity y being used for the height ofsolution may be done graphically or by numerical iteration. F : -·.l-·-·-·-·-·-·-.I11January-February 1993I71
When a beam is hanging from liftingloops, the support spring constant K9 isnormally zero, and the resisting moment and resisting lever arm are provided by the weight of the beam itself.In this case, it is convenient to substitute the positive quantity Yr for the negative quantity -y, and move it from theapplied moment arm to the resistingmoment arm side of the equation.Also, z0 sin 8 may be substituted forz, where Z0 is the theoretical lateral deflection of the center of mass of thedeflected shape of the beam, with thefull dead weight applied laterally.Thus, for hanging beams:ca z0sin8cos8 e;cos8(18)r8FOR NEUTRALEQUILIBRIUM,wrr81 M er Ke Jr Ke/W,eLM 8lKelFig. 2. Definition of radius of stability r.When ca and cr are equated and thesmall angle approximations sin e e and cos e 1 aremade, the equations of Part 1 result. The equations for caand cr represent a more general solution to the stability ofhanging beams, as will be demonstrated later in this paper.In Part 1, linear elastic behavior of hanging beams was assumed, and thus the quantity z could be replaced by 20 sin 8,where z0 is the theoretical lateral deflection of the center ofgravity of the beam with the full dead weight applied laterally, using the gross lateral moment of inertia /g. The general solution requires that z be computed using cracked section stiffness, which varies with the roll angle e when eexceeds the tilt angle emax at which cracking begins.Once crackingoccurs, the neuFig. 3. Midspan compressive stresstral axis acquiresblock in a prestressed concretea large inclina1-beam, without tilt.tion with respectto the axes of thebeam, and the top flange cracks across about one-half of itswidth. This causes the centroid of the compressive force toshift laterally within the beam. For long I-beams, the prestress force (and the balancing compressive force) is typically 1000 kips (4500 kN) or more.The primary lateral load resistance is derived from thisBIAXIAL BENDING OFCRACKED PRESTRESSED I-BEAMSA common method of assessing the lateral bendingstrength of a long prestressed concrete beam is to limit thetensile stresses in the comer of the top flange to the modulusof rupture of the concrete, and to provide reinforcement inthe top flange. The author has found that this proceduregrossly underestimates the lateral bending strength of commonly used prestressed concrete I-beams, such as the PCIBT-72. These I-beams have the ability to resist lateral bending by a lateral shift in the centroid of the compressive forcewithin the beam.Fig. 3 illustrates the compressive stress block at midspanof a long prestressed concrete beam without tilt, where thereis no lateral shift of the compressive forces. In long (sayover 120 ft, or 36 m) beams, it is ordinarily not possible toplace the prestressing force low enough to fully compensatefor the dead weight of the beam. Even when supported afew feet from each end, there is usually some residual compression in the top flange.In very round numbers, the stresses shown in Fig. 3 mighttypically be 500 psi (3.5 MPa) in the top and 2500 psi (17.2MPa) in the bottom. Fig. 4 shows the compressive stress blockin the same beam when tilted 15 degrees from the vertical.72Fig. 4. Midspan compressive stress block in a prestressedconcrete 1-beam, with 15 degree tilt.PCI JOURNAL
Fig. 5. Overall view of test setup for 149ft (45.4 m) girder.Fig . 6. Test beam near maximum tilt of 32 degrees.large force acting on a lateral (minor axis) eccentricity ofseveral inches (or, say, 200 mm), as illustrated in Fig. 4. Ofcourse, reinforcement in the flanges also provides lateralmoment resistance, but its contribution is relatively smallcompared to the resisting moment provided by the prestressforce of 1000 kips (4500 kN) or more and the lateral eccentricity of the balancing compressive force.ANALYTICAL STUDIESThe analysis of a cracked concrete section subjected to biaxial bending is quite complex. The slope and depth of theneutral axis are both unknown, and both must be found bysuccessive approximations. Once the neutral axis slope andlocation are found, the net section in compression must beanalyzed as a section subjected to unsymmetrical bending,with principal axes inclined to the neutral axis and also inclined to the gross section major and minor axes. Furthermore, a complete solution to the problem involves the analysis of many sections along the length of the beam, so thatthe deflection and center of gravity of the deflected shapemay be found by numerical integration.The author wrote a computer program to solve this analytical problem. The program is an elastic analysis program.Tensile stresses are neglected in computing stresses and theneutral axis location , but a crude approximation of tenJanuary-February 1993sio n stiffeningwas attemptedin the deflection calculations.The author alsoha s a programthat computesnominal strengthin biaxial bending using a rectangular stressblock. This program does notcompute deflections.The results ofthe analysis ofseveral actualFig. 7. End view near maximum tilt oflarge I-beams in32 degrees.the 125 to 150 ft(38 to 46 m )span range indicated that the beams can tolerate tilt anglesof 25 to 30 degrees prior to failure . This result appeared tobe at odds with experience. The program was checked asthoroughly as the author was able, but it is difficult to provide a completely independent check to this complex program. It was decided, therefore, that a test would be necessary to verify the accuracy of the analytical program.TEST PROGRAMBecause the lateral bending of a tilted beam is a selfweight effect, it was decided that a full-scale test would benecessary. The Precast/Prestressed Concrete Institute, thePortland Cement Association, Concrete Technology Corporation, the University of Washington, and BERGER/ABAMEngineers jointly sponsored the test of a 149 ft (45.4 m)beam. The beam was available from a previous job and wastested at the Concrete Technology Corporation yard. Fig. 5shows an overall view of the test setup.The beam was supported on steel crad les and slowlytipped under controlled conditions. The supports were located 11 ft (3.3 m) in from each end, which is typical forshipping a beam of this length. Tilt and deflection readingswere taken at intervals during the test. Data were also collected from strain gauges located at two sections nearmidspan. A detailed description of the test results will bepublished as a separate paper.Three tests were made. In the first test, the beam wastipped to an angle of approximately 15 degrees and thentipped back to vertical. Although calculations indicated thatthe top flange would be cracked at the 15 degree angle (seeFig. 4), the beam showed no sign of any damage or permanent set after being brought back to vertical.The second test was performed with the beam braced witha king post bracing system commonly used in WashingtonState when transporting long beams. Little difference fromthe first test was observed at a tilt of approximately 15 degrees. The beam was then tipped back to vertical, and the73
bracing removed. Again, the beamshowed no evidence of having beentipped 15 degrees once it was broughtback to vertical.The third test was to destruction, without bracing. Figs. 6 and 7 show thebeam as it neared its maximum tilt angleof 32 degrees. Fig. 8 shows the comparison of predicted and actual test results.The failure tilt angle was almost exactlythat predicted, and the tilt angle vs.minor axis deflection curve has the sameshape as predicted, but with actual deflections being about 10 to 15 percentless than predicted. This is believed tobe due to tension stiffening effectswithin the cracked sections, which werenot fully accounted for.The test results also verified a fundamental assumption used in both Parts 1and 2 of this paper, that is, the torsionalflexibility of the beams may be neglected. At the failure roll angle of 32degrees, the twist at midspan relative tothe supports amounted to about 4 percent of the roll angle. At a 15 degreeroll angle, the twist was about 2 percentof the roll angle.The test results and the computerpredictions agree remarkably well. Thisgives the author confidence that thecomputer program can be used to predict the behavior of other beams withproperties different from the test beam.30Ulwwa: 25( ')w020wFOR fc' IO KSIt- "',::,«:- Qr!:!1{!;I'g1( ')z15 f-f---1 "'Ij0f'/ (;'/.'«:-'?"v*'(jIIII"'"' IIFOOT I-BEAMLATERAL LOAD TEST26 FEBRUARY 1991'?"zII149I'GJ"'10f-// , ::i/'vv"'/o;:: - ---- ,---- ------ ---- ------0510152025303540MIDSPAN DEFLECTION, INCHES(DEFLECTION RELATIVE TO SUPPORTS)Fig. 8. Predicted tilt and deflection behavior, and test results.6050"'z4010EFFECTIVE STIFFNESSOF CRACKED BEAMS030'44-The author's computer program wasQJ20I- used to analyze the effective lateral(minor axis) stiffness of nine differentlong 1-beams, as the stiffness de10creases due to cracking at various tiltangles. The beams included modifiedAASHTO beams, PCI bulb tees, and04010152035525300Washington and Oregon beam sec0.4 RADIANStions. All beams had a minor axis moTILT ANGLE 8, DEGREESment of inertia of at least 4 percent ofthe major axis moment of inertia, andFig. 9. Variation of effective lateral stiffness left with tilt angle.an average prestress level of at least1200 psi (8.3 MPa).Fig. 9 shows the results. All beams showed similar rela For tilt angles that produce top flange tensile stresses lesstionships between stiffness and tilt angle, and all beams hadthan 7.51!7:, use the gross 18 a predicted tilt angle at nominal strength of at least 25 de For tilt angles that produce top flange tensile stresses ingrees. The end points of the curves indicate the predictedexcess of 7. 5-y t:, use an effective stiffness:failure point.(20)A simplified relationship for the effective stiffness is proposed for long prestressed concrete 1-beams of ordinary pro Assume the maximum 8 at failure e'max to be 0.4 radiansportions, such as the PCI BT-72.(or 23 degrees).74PCI JOURNAL
Fig. 10 compares Eq. (20) to thenine beams analyzed. Although thetests show no sudden change in stiffness at first cracking, it is certainlytrue that deflections are less predictable after cracking. Therefore, theassumption of a sudden loss of stiffness after cracking is conservative.The length of the plateau prior tocracking will vary depending on thetop flange stresses.PART 1 REVISITEDI. 000.80Ol --- 0.60"- -Ieff lg/1 1 2.591 -GJH0.400.20The equations in Part 1 of "Lateral0 Stability of Long Prestressed Concrete4010152530355200Beams" were derived using the asTILT ANGLE 8, DEGREESsumption that the beam remained uncracked, and the maximum tilt angleFig. 10. fettvs. 9 relationship, and proposed equation.8max was defined as that at which thetensile stress in a corner of the topflange reached the modulus of rupture.Subsequent analyses, confirmed byDEFLECTION OF BEAMwtests, show that beams can sustainmuch larger tilt angles, but with reduced stiffness due to cracking.The tilt angle at failure may betaken as 0.4 radians, and the stiffnessCOMPONENT OFat angles between 8max and 0.4 radiansco (:;,:as i ne e i) cos 8-,/ /WEIGHT ABOUTmay be taken using Eq. (20). BecauseWEAK AX I S ---------- -0 ,nl::l,k ---20 is a function of stiffness, it is also a,TAKEMOMENTS\function of 8 for cracked beams. ThisCENTER OF GRAVITYW\dependency of 20 on 8 makes it moreOF CROSS SECT I ONe\CENTER OF MASS OFdifficult to solve for 8, compared toAT LIFTING POINTDEFLECTED SHAPEn\0OF THE BEAM - the solution for uncracked beams in'i:os' Part 1.This difficulty may be overcome byEQUILIBRIUM DIAGRAMEND VIEWplotting curves for applied overturningmoment arm and resisting momentarm, and solving for the intersectionFig. 11. Applied moment arm Ca and resisting moment arm c, for hanging beam.which represents equilibrium. Momentarms are used instead of applied andresisting moments. Moment arms are obtained by dividinging moment arm c, to applied moment arm ca:moments by the weight W. This eliminates W from all theequations, and produces a moment arm measured in inches(21)(or millimeters), which is easier to visualize than a bendingmoment measured in kip-in. (or kN-m).The initial roll angle 8; of a rigid beam is e; /y,. When 9 isFig. 11 shows how the applied and resisting moment armsequal to emax the tilt angle e at which cracking is expected,are defined. Taking moments about the center of gravity ofsubstituting 9; y, e; and simplifying, the factor of safetythe cross section at the lifting point, the applied moment armagainst cracking FS is found:ca is that due to the lateral deflection of the center of massof the deflected shape of the beam, and is given by Eq. (18).FS -----(22)The resisting moment arm c, is given by E
In Part 1, linear elastic behavior of hanging beams was as sumed, and thus the quantity z could be replaced by 2 0 sin 8, where z 0 is the theoretical lateral deflection of the center of gravity of the beam with the full dead weight applied later ally, using the gross lateral moment of inertia /g. The gen
Lecture 24 – Prestressed Concrete Prestressed concrete refers to concrete that has applied stresses induced into the member. Typically, wires or “tendons” are stretched and then blocked at the ends creating compressive stresses throughout the member’s entire cross-section. Most Prestressed concrete is precast in a plant.
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a story, is affected by vertical distribution of lateral loads, i.e., there is a unique displaced profile for each type of lateral load distribution. Conse quently, the lateral stiffness of a story is not a stationary property, but an apparent one that depends on lateral load distribution. In the analysis of
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