Collapse Load Analysis of Prestressed Concrete Structures

Transcription

1 Collapse Load Analyss of Prestressed Concrete Structures by K W Wong R F Warner Research Report No. R 6 July 998 ISBN

2 COLLAPSE LOAD ANALYSIS OF PRESTRESSED CONCRETE STRUCTURES by K W Wong R F Warner Department of Cvl and Envronmental Engneerng The Unversty of Adelade Research Report No. R 6 July 998

3 COLLAPSE LOAD ANALYSIS OF PRESTRESSED CONCRETE STRUCTURES K W Wong Research Fellow R F Warner Professor Department of Cvl and Envronmental Engneerng Unversty of Adelade ABSTRACT: Ths report descrbes a non-lnear method of analyss for prestressed concrete structures whch predcts behavour at all stages of loadng, from the ntal applcaton of prestress up to and beyond the collapse condton. The method uses a segmental lne element approach and s a modfcaton of an exstng method prevously developed for renforced concrete structures. Comparsons wth the results of tests of contnuous prestressed concrete beams show good correlaton. The method has been developed for the purpose of nvestgatng the collapse behavour of ndetermnate prestressed concrete structures and hence to develop desgn recommendatons on mnmum acceptable ductlty levels, moment redstrbuton and desgn safety coeffcents.

4 TABLE OF CONTENTS Secton Page ABSTRACT TABLE OF CONTENTS LIST OF FIGURES. INTRODUCTION. NON-LINEAR ANALYSIS OF PRESTRESSED CONCRETE STRUCTURES 3. PRESENT METHOD OF ANALYSIS 3 3. Modfcatons for Treatng Prestressed Concrete 4 3. Stage : Applcaton of Prestress and Self-weght Stage : Post-crackng Behavour up to Collapse 6 4. COMPARISON WITH TESTS 8 5. HYPERSTATIC REACTIONS AND SECONDARY MOMENTS DUE TO PRESTRESS 9 6. CONCLUDING REMARKS 9 7. REFERENCES APPENDIX A: STIFFNESS MATRIX FOR SEGMENTED ELEMENT APPENDIX B: FIXED END MOMENTS FOR A SEGMENTED ELEMENT 3 APPENDIX C: FORCE FROM PRESTRESSING CABLE 5

5 LIST OF FIGURES Page A Segmented element showng element end forces B Segmented element showng fxed end moments 3 C Prestressng effect at juncton of two segments 5 C Prestressng effect at left node of segmented element 6 C3 Prestressng effect at rght node of segmented element 7 A typcal segmented element 7 System of forces and moments actng at the junctures between segments 7 3 Stage analyss 8 4 Stage analyss 9 5 Detals of beams tested by Bshara and Brar (974) 6 Load deflecton plot for beam BC 7 Load deflecton plot for beam BC3 8 Load deflecton plot for beam BC4

6 . INTRODUCTION Ths report descrbes a non-lnear method of analyss for prestressed concrete structures at workng load and at collapse. The method of analyss was developed as a tool for nvestgatng nonductle behavour n prestressed concrete flexural structures and hence for the development of gudelnes on mnmum ductlty requrements for the desgn of prestressed concrete structures. The accuracy of the analytc procedure has been evaluated usng prevously publshed test data for contnuous prestressed beams. The predcted behavour was found to be n reasonably good agreement wth the test results. The analytc method s currently beng used n parametrc studes of the factors whch affect local secton ductlty and the modes of collapse of prestressed concrete contnuous beams. Of partcular nterest n these studes s collapse by premature fracture of the prestressng tendon. Parametrc studes are also beng carred out to evaluate global safety coeffcents for use n the non-lnear desgn of prestressed concrete structures.. NON-LINEAR ANALYSIS OF PRESTRESSED CONCRETE STRUCTURES Varous methods of analyss have been used for ndetermnate prestressed concrete structures. These nclude the methods developed by Kang and Scordels (98), Warner and Yeo (984); Kgoboko (987); Kgoboko, Wyche and Warner (988); and Campbell and Kodur (99). Kang and Scordels (98) descrbed a numercal procedure based on a fnte element formulaton for non-lnear analyss of plane prestressed concrete frames. The method uses a tangent stffness formulaton to calculate ncremental dsplacements. These dsplacements are added to the latest total dsplacements, and a secant stffness formulaton s then used to determne the resstng loads at the jonts based on the latest state of materals and geometry of the structure. From ths and the latest appled loads, the unbalanced jont forces are calculated for use n the next teratve cycle. Several such teratve cycles are requred to acheve convergence. Ths s usually known as the predctor-corrector method,

7 the predctor uses the tangent stffness formulaton, and the corrector uses the secant stffness formulaton. Ths non-lnear analyss s carred out by ncrementng load, and s not therefore sutable, wthout modfcaton, for tracng the collapse behavour of concrete structures up to and beyond collapse. Warner and Yeo (984) descrbed a lne element drect stffness approach to study partally prestressed concrete beams. In ths approach, a structure s modelled by dvdng members nto segments, each represented by a sngle lne element. A curvature control procedure, frst ntroduced by Warner (984), s used to trace the full-range behavour of the structure. The curvature ncrement procedure has been found to be superor to procedures based on ncrementng loads for tracng the full-range behavour of non-lnear structures. Moment curvature relatons for segments were pre-generated, and these relatons were used n the structural analyss. Kgoboko (987) used the same approach as Warner and Yeo n hs study of the collapse behavour of partally prestressed concrete structures but mproved the effcency by usng segmented elements to represent structural members. Segmented elements were frst used by Wong et al (987) n a study on the collapse behavour of renforced concrete structures. Wth ths approach, a beam can be modelled as a sngle segmented element. Ths reduces greatly the sze of the structural stffness matrx by reducng the number of nodes, whch n turn reduces both computer storage and program executon tme. A lnear-elastc analyss was carred out to determne the ntal effect of prestressng and self-weght. Ths approach was used to study the ductlty of prestressed concrete brdge grders (Kgoboko, Wyche and Warner, 988). Campbell and Kodur (99) also used pre-generated moment-curvature relatons and a curvature-ncrement soluton procedure for non-lnear analyss of prestressed concrete contnuous beams. Ths approach s smlar to that of Warner and Yeo, but dfferent from that of Kgoboko, as non-segmented elements were used to represent a member. Campbell and Kodur also used a lnear elastc analyss to determne the effect of prestressng.

8 3. PRESENT METHOD OF ANALYSIS The method of analyss descrbed here for prestressed concrete structures has been developed by adaptng an exstng procedure for renforced concrete structures. Ths method was chosen for adaptaton because t has been n use for many years and has proven to be numercally stable, robust and computatonally effcent. The method also can take full account of geometrc and materal non-lneartes. Although the method has been descrbed prevously (Wong et al, 987, 988), some of the key features wll be mentoned brefly here because of ther relevance to the analyss of prestressed concrete structures. Each member n the structural system s dscretsed nto a number of lne elements and each of these s further dscretsed nto a number of segments. The flexural behavour of each segment s determned at any stage of loadng on the structure by usng a secton analyss n whch a typcal cross-secton s represented as a large number of thn layers of concrete and renforcng steel. For a gven or assumed stran dstrbuton n the secton, stress-stran relatons for the component materals are used to determne stresses and hence forces n the varous layers. Summaton of the layer forces and ther moments gve the axal force and moment actng n the secton. In stuatons where loadng and unloadng occur n a secton, the stran hstory of each layer s used to determne the current stress and hence the stress resultants. In ths way, fully non-lnear materal behavour can be consdered, ncludng crackng of the concrete n tenson, tenson stffenng, compressve softenng of the concrete, and yeldng, stran hardenng or unloadng of renforcng steel. For the analyss of the structural system, a secant modulus method s adopted. The secant stffness of each segment n an element s determned on a tral bass and used to establsh a tral secant modulus for the element. The secant stffnesses of the segments n an element are used to form the element stffness matrx of the segmented element. The terms n the element stffness matrx are gven n Appendx A. The element stffnesses are then used to form the tral system stffness at the partcular stage of loadng beng consdered. 3

9 Loadng along the segmented element s ncluded by usng equvalent nodal forces based on fxed end moments. Equatons for fxed-end moments for a segmented element are gven n Appendx B. The structural analyss s carred out by defnng a unt pattern load. A key segment s chosen, and ncrements of curvature are appled to ths segment. The load factor λ s then determned for each ncremental curvature n the key segment untl collapse occurs Modfcatons for Treatng Prestressed Concrete In order to expand the exstng analyss method to treat prestressed concrete structures, two man modfcatons were requred. Frstly, a prelmnary analyss was added to take account of the ntroducton of the prestressng force nto the concrete-steel structure. In ths analyss the self-weght of the structure s consdered to be ntroduced at the same tme that the prestress s appled. Whle ths usually occurs n practce, t s also convenent to treat these processes smultaneously for computatonal reasons. Ths s dscussed further n Secton 3. below. The second man modfcaton made to the man structural analyss was to take account of the presence of the prestressng steel when determnng the stffness and secton stresses n local elements durng the postcrackng and overload stages of behavour. These two modfcatons are now consdered n turn. 3. Stage : Applcaton of Prestress and Self-weght In ths prelmnary analyss, the applcaton of the prestressng force to each concrete element s consdered. Intally, the case of post-tensoned elements whch contan a curved prestressng cable s consdered. As shown n Fg a, the curved prestressed cable apples a non-unformly dstrbuted force to the concrete element. To approxmate ths stuaton n the analyss, the cable s consdered to be straght wthn each segment, and to have a slght knk at each juncture between adjacent segments (Fg b). Statcally, the dstrbuted load actng on the concrete n Fg a s beng represented approxmately by a seres of

10 small nclned dscrete forces actng at the junctures between elements. The real cable at any pont s typcally eccentrc to the centrodal axs and nclned and curved. Ths produces a unformly dstrbuted force on the concrete segment under consderaton whch s perpendcular to the cable. The force appled at each juncture thus has a vertcal and a horzontal component. Furthermore, to allow for the eccentrcty of the real cable, a moment also acts at the juncture, as shown n Fg c. The system of vertcal and horzontal forces and moments actng at each juncture along the element s statcally equvalent to the contnuous force system beng appled to the element by the prestressed cable In most practcal stuatons, the slope of the cable s relatvely small and only the vertcal component s of sgnfcance. Expressons for vertcal forces from the prestressed cable at the junctures between segments and at the nodes of a segmented element are gven n Appendx C. Ths means that both the horzontal force and the moment (equal to the horzontal force tmes the cable eccentrcty at the node) are very small. Whle both the horzontal force and the moment can easly be allowed for, we gnore them n the followng analyss. If an analyss s undertaken of a practcal structure wth only the prestress actng, t s often found that crackng of the concrete s predcted. In the smple beam n Fg for example, the prestressng produces large tensle stresses n the upper fbres. Ths s because the cable s desgned to partally balance the stresses due to external load. It s convenent therefore to analyse the ntal state of the structure wth the effects of both the prestress and self-weght. If these effects are analysed separately, spurous non-lnear effects are ntroduced because of crackng. Because the behavour n the post crackng stage s sgnfcantly non-lnear, t s not possble to treat the two effects separately and superpose the results. The contnuous body forces due to self-weght are represented by a unformly dstrbuted force actng along the element. The structure s analysed for the smultaneous acton of the forces due to prestress and self-weght. A tral procedure s used whch usually requres only one or two teratons. In the frst cycle all concrete layers throughout the structure are assumed to be uncracked, and the secant modulus of elastcty of each layer of materal s set equal to the ntal tangent modulus for the materal. The stffness of each segment, and hence of each element and fnally the system, s determned and a lnear analyss s undertaken. 5

11 6 The Stage analyss s llustrated by the flow dagram shown n Fg 3. The steps n the analyss are:. Determne the equvalent loads actng on the system from the prestressng cable. These act at the junctures between segments and at the ends of the elements. The equvalent loads at the end of an element wth tendon anchorage nclude a horzontal prestressng force actng along the reference axs, and a moment actng on the secton at the reference axs f the tendon s anchored below or above the reference axs.. Determne the self-weght actng on the system. Ths s modelled by representng the self-weght of a horzontal element as an equvalent unformly dstrbuted load actng along t. 3. The total load pattern from equvalent loads and loads from selfweght s obtaned by supermposng the two loads descrbed above. 4. Dvde each ndvdual load (self-weght and equvalent prestress loads) by the magntude of the effectve prestressng cable force F prest to obtan the unt load pattern; ths unt load system s scaled by a load scalng factor λ. A load-ncrement procedure s used to analyse the system for progressvely ncreasng load untl the full load pattern wth λ equal to F prest s reached. Ten equal load ncrements are used, each of magntude F prest /., to reach the fnal state of λ = F prest. For each ncrement n load, the system under the latest load pattern s analysed takng nto consderaton materal non-lneartes. The use of a load-ncrement procedure for ths frst stage of analyss s acceptable as the system s assumed not to be near collapse at the end of ths stage of analyss. 3.3 Stage : Post-crackng Behavour up to Collapse In Stage, the system s subjected to dead and ncreasng lve load. The procedure, at present lmted to proportonal loadng, s llustrated by the flow dagram shown n Fg. 4.

12 The Stage loadng commences after the tendon has been bonded to the concrete. The stresses and strans of the concrete and non-prestressng steel at the end of Stage are stored as reference stresses and strans. The stress and stran n the prestressng tendon are obtaned from the effectve cable force and the cross-sectonal area of the tendon. Therefore, the tensle stran n the bonded tendon s much hgher than the tensle stran n the concrete surroundng t. The steps n the Stage analyss are: 7. Defne the unt load pattern from the dead and lve load. Ths unt load pattern s subjected to a load scalng factor λ, values of whch are determned for ncreasng curvatures n a chosen key segment.. At the begnnng of Stage loadng, the structural system s set to ts reference confguraton; that s, deflectons and curvatures are all zeros. All acton effects obtaned from the Stage analyss are relatve to ther reference values stored at the begnnng of stage. 3. A curvature ncrement procedure s used to trace the non-lnear Stage behavour. Ths s bascally the same as the procedure for renforced concrete structures descrbed n earler papers (Warner, 984; Wong et al, 988), but wth the renforcng steels (ncludng the tendon) and the concrete havng pre-strans and pre-stresses. Note that as the tendon s bonded to the concrete at the begnnng of stage, the tendon and concrete have the same Stage stran on loadng but dfferent total strans due to dfferent startng strans as a result of the prestressng and the subsequent groutng of the prestressng tendon. The man changes are n the secton analyss. The Stage curvature obtaned durng the system analyss s relatve to the reference curvature. Therefore, for a typcal materal layer, Stage stran s added to ts reference stran to gve the total stran. The total stran n turn gves the total stress based on the assumed materal stress-stran relaton, takng nto consderaton materal unloadng. After havng obtaned the total stress, the Stage stress s obtaned by subtractng the reference stress from the total stress. The flexural rgdty for the secton (and also the segment) s obtaned by dvdng the Stage bendng moment (calculated usng the Stage stresses) by the Stage curvature.

13 One smplfyng assumpton beng made s that the cable s straght wthn a segment, and the eccentrcty used s that at the md-segment. The segment s assumed to be unform wth a characterstc EI value equal to that of the end secton wth the larger bendng moment. Ths assumpton s conservatve n most stuatons COMPARISON WITH TESTS Three two-span contnuous prestressed beams tested by Bshara and Brar (974) were analysed usng the program SEGPCAN (acronymn for Segmental Prestressed Concrete Analyss) developed based on the procedures already descrbed. These are beams BC, BC3 and BC4. As the beams were symmetrcal about ther centre supports, they were each modelled as a propped cantlever (see Fg 5). Each strand had an ntal prestress of 6.38 kn. Effectve prestress was assumed to be 8% of the ntal prestress. The Young s Modulus of prestressng strand was assumed to be 94 MPa. The stress-stran relaton of the strand was assumed to be elastc-plastc, wth the yeld stress equal to the % offset stress of 7 MPa. The stress-stran relaton for non-prestressed renforcng bars was also assumed to be elastc plastc. The mean yeld strength assumed was 4 MPa, and the Young s Modulus was assumed to be MPa. The mean concrete strength f cm for beams BC, BC3 and BC4 were 4. MPa, 39. MPa and 39. MPa respectvely. The ntal modulus of concrete was assumed to be 55 f cm. The shape of the concrete stress-stran relaton for concrete n compresson was assumed to be that proposed by Warner (969) and for concrete n tenson, ncludng the effect of tenson stffenng, was assumed to be that proposed by Kenyon and Warner (993). The load versus md-span deflecton plots for beams BC, BC3 and BC4 are shown n Fgs 6, 7 and 8 respectvely. The results from the analyss agree reasonably well wth the test results. Two analyses were carred out on each beam, one ncludng the effect of tenson stffenng and the other wthout tenson stffenng. For these beams, the results obtaned from the analyses wthout tenson stffenng agree well wth test results. The results for the analyss whch ncluded the effect of tenson stffenng over-stffened the beams at low levels of loadng.

14 5. HYPERSTATIC REACTIONS AND SECONDARY MOMENTS DUE TO PRESTRESS In the lnear analyss of prestressed concrete adopted for desgn purposes, t s normal to analyse for the effects of prestress alone, and to determne from ths analyss the magntude of the hyperstatc reactons and secondary moments whch are produced by the deformatons n the structure. Such hyperstatc reactons and secondary moments are only meanngful f t can be assumed that the structure behaves n a lnear manner. Pror to crackng of the concrete, any materal non-lnearty n structural behavour s scarcely perceptble. However, n the present analyss the man attenton s on non-lnear behavour n the post-crackng workng load range and also on condtons at hgh overload. For ths reason t has been assumed, even n the prelmnary analyss of the applcaton of prestress, that the behavour s potentally non-lnear. In ths way, the analyss can proceed smoothly through each stage of loadng. A legtmate alternatve approach would be to assume lnear behavour pror to crackng and use the normal lnear analyss for ths ntal stage. The advantage of the lnear analyss would be that any hyperstatc effects could be evaluated. The dsadvantage s that an abrupt transton to nonlnear analyss has to be ntroduced, wth some slght msmatch of stresses and strans at the transton. The changeover would be when the frst crack appears n the structure CONCLUDING REMARKS The method of analyss descrbed n ths paper s more powerful than those methods prevously used whch rely on pre-generated momentcurvature relatons for the ndvdual elements. Generally t s smlar to the fnte element method developed by Scordels, but t has the advantage that the analyss can be undertaken n ether deformaton or load control, so that good numercal stablty s mantaned when condtons at collapse are nvestgated. It has the further advantage that t s a relatvely smple modfcaton of an exstng method orgnally developed for renforced concrete, whch has been well tested and appled to varous specal problems. Specal

15 analyses, for example allowng for non-proportonal loadng, can thus be carred out. The comparsons wth test data suggest that the method gves realstc results. 7. REFERENCES Bshara, A.G. and Brar, G.S.(974), Rotatonal Capacty of Prestressed Concrete Beams, Journal of the Structural Dvson, ASCE, Vol., No.ST9, September, pp Campbell, T.I. and Kodur, V.K.R.(99), Deformaton Controlled Nonlnear Analyss of Prestressed Concrete Contnuous Beams, PCI Journal, September-October, pp Kang, Y.J. and Scordels, A.C.(98), Nonlnear Analyss of Prestressed Concrete Frames, Journal of the Structural Dvson, ASCE, Vol.6, No.ST, February, pp Kenyon, J.M. and Warner, R.F.(993), Refned Analyss of Non-lnear Behavour of Concrete Structures, Cvl Engneerng Transactons, Insttuton of Engneers, Australa, Vol.CE35, no.3, August, pp.3-. Kgoboko, K.(987), Collapse Behavour of Non-ductle Partally Prestressed Concrete Brdge Grders, MEngSc Thess, Unversty of Adelade, November, 338 pp. Kgoboko, K., Wyche, P.J. and Warner, R.F.(988), Collapse Behavour and Ductlty Requrements n Partally Prestressed Concrete Brdge Grders, Research Report No. R79, Department of Cvl Engneerng, Unversty of Adelade, March, pp. Warner, R.F. and Yeo, M.F.(984), Ductlty Requrements for Partally Prestressed Concrete, Proceedngs of the NATO Advanced Research Workshop on Partally Prestressng, From Theory to Practce, Pars, France, June, pp Warner, R.F.(969), Baxal Moment Thrust Curvature Relatons, Journal of the Structural Dvson, ASCE, Vol 95,No.ST56, pp

16 Warner, R.F.(984), Computer Smulaton of the Collapse Behavour of Concrete Structures wth Lmted Ductlty, Proceedngs, Int. Conf. On Computer Aded Analyss and Desgn of Concrete Structure, Ed. Damjanc et al, Part II, Pne Rdge Press, Swansea, U.K., September, pp Wong, K.W., Yeo, M.F. and Warner, R.F.(987), Collapse Behavour of Renforced Concrete Frames, Research Report No.R78A, Department of Cvl Engneerng, Unversty of Adelade, August, 9pp. Wong, K.W., Yeo, M.F. and Warner, R.F.(988), Non-lnear Behavour of Renforced Concrete Frames, Cvl Engneerng Transactons, Instn Engrs Aust., Vol.CE3, No., July 988, pp

17 APPENDIX A: STIFFNESS MATRIX FOR SEGMENTED ELEMENT P y Segment P y M Px M x l L P x Fgure A: Segmented element showng element end forces The force deflecton relaton for a segmented element (Fgure A) s : Px Py M = Px P y M x y θ e x y θ [ K ] (A) The element stffness matrx [K e ] s: S S SYM ME TRY S S 3 33 S S S S S 3 L S LS S LS 3 3 S S LS LS S 33 (A) where: S = S( m) (A3) C S = CC3 C (A4)

18 S C = (A5) C C C C = (A6) S 3 33 CC3 C no. of seg. C = = EI l no. of seg. x C = = EI l no. of seg. x C = 3 = EI l (A7) (A8) (A9) L s the length of the element. S(m) s the axal stffness for the element. It s set to a large value for elements subjected manly to flexure. EI and l are the flexural stffness and length respectvely of the segment, and x s the dstance from the left end of the element to the centre of the segment. The dervaton of the matrx s gven by Wong, Yeo and Warner (988). APPENDIX B: FIXED END MOMENTS FOR A SEGMENTED ELEMENT Segment wth flexural stffness EI loadng M M y l L Fgure B: Segmented element showng fxed end moments

19 4 The fxed end moments M and M for the left and rght ends of a segmented element (Fgure B) subjected to loadng are: XT S M = R P ( L X ) T M = R S P (B) (B) X s the dstance from the left end of the element to the centrod of the /EI dagram of the entre element: X = no. of seg. = no. of seg. = y EI EI (B3) y s the dstance from the left end of the element to the centre of the segment, and: no. of seg. P = = EI l no. of seg. x R = = EI l no. of seg. M pp S = = EI l no. of seg. M pp x T = = EI l (B4) (B5) (B6) (B7) EI and l are the flexural stffness and length respectvely of the segment, and x s the dstance from the centrod of the /EI dagram to the centre of the segment. M pp s the bendng moment at the mddle of segment f the beam carryng the appled load were pn-supported at both ends.

20 5 APPENDIX C: FORCE FROM PRESTRESSING CABLE Ths appendx shows how the prestressng forces are determned for a segmented element. Case : At juncture between segments wthn the element: For small angles, θ and θ : y θ snθ = y θ snθ = y l seg. + l y seg. (C) (C) The vertcal pont force actng upward drectly above the juncton between the two segments: Fy = = = = F F F prest. prest. prest. l F seg. prest. l seg. ( snθ snθ ) ( θ θ ) ( y y y + y ) ( y + y y ) + + (C3) F prest. y - F prest. θ θ y y + l seg l seg Fgure C: Prestressng effect at juncton of two segments

21 6 Case : At the left end node of the segmented element: For small angle, y y θ snθ = (C4) l seg. The vertcal pont force (postve upward) actng on the left node (from the removed tendon) s: Fy = Fprest.snθ = Fprest. θ (C5) y = y F prest. lseg. Segment Node F prest. θ y y l seg Fgure C: Prestressng effect at left node of segmented element Case 3: At the rght end node of the segmented element For small angle, ynseg. + ynseg. θ snθ = l seg. (C6) The vertcal pont force (postve upward) actng on the rght node (from the removed tendon) s : Fy = Fprest.snθ = Fprest. θ (C7) y = nseg. + ynseg. F prest. lseg.

22 7 Last segment nseg y nseg. θ F prest. y nseg.+ l seg Node Fgure C3: Prestressng effect at rght node of segmented element (a) Element wth curved cable (b) Pece-wse lnear approxmaton wth knks at the junctures between segments Fgure : A typcal segmented element (a) Inclned local forces at (b) Vertcal and horzontal (c) resultants at junctures junctures between segments components on centrodal axs Fgure : System of forces and moments actng at the junctures between segments

23 8 INPUT Structural system: nodes, elements, renforcement detals ncludng cable profle and effectve prestressng force n the cable. Materals: mean propertes of concrete, mean propertes of renforcng steel, and mean propertes of prestressng steel. ) Determne equvalent element and nodal forces from prestressng cable. ) Determne element forces from self weght. 3) Supermposed equvalent forces from prestressng cable wth element forces from self weght to gve total pattern load. Obtan "unt" load pattern from total load pattern by scalng down usng the effectve prestressed force n the cable. λ = Effectve prestressed force / λ =. λ = λ + λ Lnear elastc analyss : For unt load pattern. Obtan curvatures κ_unt() Obtan tral curvature: κ_tral() = κ_unt().λ Secton analyses: Determne M_tral() for each κ_tral() Obtan new flexural stffnesses EI()=M_tral() / κ_tral() Yes Check convergence: Is (EI()-OLDEI()) / OLDEI() < tolerance? No OLDEI() = EI() Is λ = effectve prestressed force? No Yes Stage loadng Fgure 3: Stage analyss

24 9 Store the stresses and strans n all materal layers obtaned at the end of stage as reference values κ_target =. κ_target =κ_target + κ_target Lnear elastc analyss : For unt load pattern. Obtan curvatures κ_unt() Calculate the load scalng factor: λ = κ_target / κ_unt (key segment) Obtan tral curvature: κ_tral() =λ x κ_unt() Secton analyses relatve to the reference values (pre-stresses & pre-strans): () Determne M_tral() for each κ_tral() () Obtan new flexural stffnesses EI()=M_tral() / κ_tral() Yes Check convergence: Is (EI()-OLDEI()) / OLDEI() < tolerance? No OLDEI() = EI() Is κ_target less than a nomnated maxmum curvature value? Yes No End Fgure 4: Stage analyss

25 Load, P A 3.48 m A Beam BC Load, P B 3.48 m B Beam BC3 Load, P C 3.48 m C Beam BC4 7 mm 7 mm 7 mm 4 mm 6 mm 35 mm 33 mm 6 mm 35 mm 3 mm 6 mm 35 mm 4 mm 33 mm SECTION A-A SECTION B-B SECTION C-C 3/8 nch strand (area = 54.7 mm ) no. 3 rebar (area = 7.3 mm ) Note: Fxed support sectons are the same as md-span sectons except nverted. Fgure 5: Detals of beams tested by Bshara and Brar (974)

26 8 Load n kn 6 4 From analyss - wth tenson stffenng From analyss - no tenson stffenng From test Defln under load pont n mm Fgure 6: Load deflecton plot for beam BC 8 Load n kn 6 4 From analyss - wth tenson stffenng From analyss - no tenson stffenng From test Defln under load pont n mm Fgure 7: Load deflecton plot for beam BC3 Load n kn From analyss - wth tenson stffenng From analyss - no tenson stffenng From test Defln under load pont n mm Fgure 8: Load deflecton plot for beam BC4