Flexural comparison of the ACI and AASHTO LRFD structural concrete codes

Transcription

1 San Jse State University SJSU SchlarWrks Master's Theses Master's Theses and Graduate Research 28 Flexural cmparisn f the ACI and AASHTO LRFD structural cncrete cdes Nathan Jeffrey Drsey San Jse State University Fllw this and additinal wrks at: Part f the Structural Engineering Cmmns Recmmended Citatin Drsey, Nathan Jeffrey, "Flexural cmparisn f the ACI and AASHTO LRFD structural cncrete cdes" (28). Master's Theses This Thesis is brught t yu fr free and pen access by the Master's Theses and Graduate Research at SJSU SchlarWrks. It has been accepted fr inclusin in Master's Theses by an authrized administratr f SJSU SchlarWrks. Fr mre infrmatin, please cntact schlarwrks@sjsu.edu.

2 FLEXURAL MPARISON OF THE ACI AND AASHTO LRFD STRUCTURAL NCRETE DES A Thesis Presented t The Faculty f the Department f Civil Engineering San Jse State University In Partial Fulfillment f the Requirements fr the Degree Master f Science by Nathan Jeffrey Drsey May 28

3 UMI Number: Cpyright 28 by Drsey, Nathan Jeffrey All rights reserved. INFORMATION TO USERS The quality f this reprductin is dependent upn the quality f the cpy submitted. Brken r indistinct print, clred r pr quality illustratins and phtgraphs, print bleed-thrugh, substandard margins, and imprper alignment can adversely affect reprductin. In the unlikely event that the authr did nt send a cmplete manuscript and there are missing pages, these will be nted. Als, if unauthrized cpyright material had t be remved, a nte will indicate the deletin. UMI UMI Micrfrm Cpyright 28 by PrQuest LLC. All rights reserved. This micrfrm editin is prtected against unauthrized cpying under Title 17, United States Cde. PrQuest LLC 789 E. Eisenhwer Parkway PO Bx 1346 Ann Arbr, Ml

4 28 Nathan Jeffrey Drsey ALL RIGHTS RESERVED

5 APPROVED FOR THE DEPARTMENT OF CIVIL ENGINEERING j/jfl^l- Dr. Akthem Al-Manaseer Dr. Kurt M. McMullin 1 Mr. Daniel Merrick APPROVED FOR THE UNIVERSITY <? / Au0Lr~**~~ Vi/z/s'

6 ABSTRACT FLEXURAL MPARISON OF THE ACI AND AASHTO LRFD STRUCTURAL NCRETE DES by Nathan Jeffrey Drsey There are tw prevailing cdes utilized during the design f structural cncrete members in Nrth America, ACI and AASHTO LRFD. Each takes a unique apprach t achieve the same result, a safe wrking design f a structural cncrete sectin; hwever there are fundamental differences between the cdes regarding the calculatin f member prperties. The purpse f this paper was t investigate these differences between cdes encuntered during the calculatin f flexural strength r mment capacity f a sectin. This study fcused n the cde's influence during the classical apprach t analysis f a shallw reinfrced T-beam and the strut and tie mdel methd as applied t a series f deep beams with penings. Parametric studies were cnducted using sftware develped by the authr specifically fr this purpse. It was cncluded that bth cdes prvided similar and safe results regardless f the very different appraches t slutin taken by each.

7 ACKNOWLEDGEMENTS I wuld like t thank Dr. Akthem Al-Manaseer fr his patience and supprt during this prcess. Dr. Al-Manaseer gave me full academic freedm during the curse f this research and by ding s prvided me with a tremendus learning experience that went far beynd reinfrced cncrete. And I wuld especially like t thank my lvely wife Jaya fr always believing in me and fr making sure that I always believed in myself. v

8 Table f Cntents 1.0 INTRODUCTION General Scpe Objective Outline LITERATURE REVIEW Intrductin Shallw Beams Deep Beams THEORY OF FLEXURE Backgrund Thery f Flexure in Shallw Reinfrced Cncrete Sectins Thery f Flexure in Deep Beams ANALYTICAL PROCEDURE Shallw Beams Stress Blck Depth Factr^ ACI Limits f Reinfrcement Lcatin f Neutral Axis c and Depth f Cmpressive Blck a Strength Reductin Factr ^ AASHTO LRFD Limits f Reinfrcement Lcatin f Neutral Axis c, and Depth f Cmpressive Blck a Strength Reductin Factr <f> STM Intrductin Graphical Slutin Hand Calculatins Excel Spreadsheets ACI vs. AASHTO LRFD Strength Reductin Factr <j> Strut Effectiveness Factrs f3 s Nde Effectiveness Factrs f RESULTS and DISCUSSION Flexure A smin m&asmax, ACI versus ASHTO LRFD A s versus M u Lcatin f Neutral Axis c, and Depth f Cmpressive Blck a Strength Reductin Factr ^ STM Maximum Allwable Cncentrated Lad - STM versus FEA 99 vi

9 5.2.2 Maximum Allwable Cncentrated Lad withut $- STM versus FEA SUMMARY and NCLUSIONS Summary Cnclusin A sm in and A smax fr Flexure A s versus M u fr Flexure Depth f Neutral Axis c fr Flexure Strength Reductin Factr <f> fr Flexure Maximum Lad Capacityfr STM 105 Wrks Cited 106 Appendix A - Deep Beam STM Mdels 108 Appendix B - Deep Beam STM Truss Mdels 113 Appendix C - STM Truss Free Bdy Diagrams & Slutins 119 vii

10 List f Figures Figure 2-1 AASHTO LRFD Prcedure fr calculatin f s x 7 Figure 2-2 Summary f reinfrcement limits by 1999, 22 ACI cdes and AASHTO LRFD and Naaman recmmendatin 18 Figure 2-3 ACI and AASHTO LRFD limits f reinfrcement 22 Figure 3-1 Beam in flexure with saddling effect 38 Figure 3-2 Graphic representatin f frces within a reinfrced cncrete sectin 42 Figure 3-3 ACI Fig. RA.1.1 (a and b) - D-regins and discntinuities 45 Figure 3-4 ACI Fig. RA.1.5 (a and b) - Hydrstatic ndes 47 Figure 4-1 Sectin gemetry 49 Figure 4-2 IF lgic statement fr cmputatin f stress blck depth factr /?/ 50 Figure 4-3 ACI Fig. R Strength reductin factr 54 Figure 4-4 ACI calculatin f strength reductin factr ^ 55 Figure 4-5 Sample excel spreadsheet analysis 56 Figure 4-6 Sample excel spreadsheet shwing preliminary ACI calculatins 57 Figure 4-7 ACI excel prgram lgic flwchart 58 Figure 4-8 Determinatin f AASHTO LRSYi A smca 62 Figure 4-9 AASHTO LRFD and ACI limits f reinfrcement 63 Figure 4-10 Sample excel spreadsheet fr AASHTO LRFD analysis 64 Figure 4-11 AASHTO LRFD flwchart 65 Figure 4-12 Generic beam with and withut pening 67 Figure 4-13 Strut and tie mdeling prcedure 75 Figure 5-1 Graphic representatin f sectin gemetry 76 Figure 5-2 A smax anda smin as a functin f b 79 Figure 5-3 A smax anda smin as a functin f b w 80 Figure 5-4 A smax and A smi as a functin f d 81 Figure 5-5 A smax and A smin as a functin f tf 82 Figure 5-6 A s versus M u fr variable b 86 Figure 5-7 A s versus M u fr variable b w 87 Figure 5-8 A s versus M u fr variable d 88 Figure 5-9 A s versus M u fr variable tf 89 Figure 5-10 A s versus c fr variable b 94 Figure * versus c fr variable b w 95 Figure 5-12 A s versus c fr variable d 96 Figure 5-13 A s versus c fr variable tf 97 Figure 5-14 Generic beam with and withut pening 98 Figure 5-15 Predicted lad vs. maximum test lad cmparisns 102 Figure 5-16 Predicted lad withut ^ vs. maximum test lad cmparisns 102 viii

11 List f Tables Table 4-1 Values fr variable and fixed gemetry fr each case 49 Table 4-2 Cutut lcatins frm lwer left crner in inches 67 Table 4-3 ACI /AASHTO LRFD effective strength cefficients 74 Table 5-1 Rates f change fr ACI and AASHTO LRFD A smin and A smax 77 Table 5-2 Rati f AASHTO LRFD t ACI fr M u per unit A, 83 Table 5-3 Results fr lcatin f neutral axis 91 Table 5-4 Cutut lcatins frm lwer left crner in inches 98 Table 5-5 Predicted lad versus maximum test lad cmparisns 99 Table 5-6 Predicted lad withut ^versus maximum test lad cmparisns 101 ix

12 List f Variables a effective depth f the cmpressive blck a' distance frm center f cncentrated lad t edge f supprt A c ' area f cncrete n flexural cmpressin side f member A ps area f prestressed lngitudinal reinfrcement placed n the flexural tensin side f the sectin A s area f flexural reinfrcement A s area f nn-prestressed lngitudinal reinfrcement placed n the flexural tensin side f the sectin A s area f cnventinal nn-prestressed tensile reinfrcing steel A s ' area f cnventinal nn-prestressed cmpressive reinfrcing steel A v crss sectinal area f prvided stirrup reinfrcement within distance s b sectin width b v effective web width b w web width c distance frm extreme cmpressive fiber t neutral axis d distance frm the extreme cmpressive fiber t the centrid f the flexural reinfrcement D verall member depth d c strut width d e depth frm the extreme cmpressive fiber t the centrid f the tensile frce in the nn-prestressed reinfrcement, i.e. reinfrcing bars als called the effective depth f main flexural reinfrcement d effective shear depth, can be taken as 0.9d dp depth frm the extreme cmpressive fiber t the centrid f the tensile frce in the prestressed reinfrcement d s depth frm the extreme cmpressive fiber t the centrid f the tensile frce in the nn-prestressed reinfrcement, i.e. reinfrcing bars d v effective shear depth, taken as 0.9d E mdulus f elasticity f the material, cmmnly referred t as Yung's Mdulus E c elastic mdulus f cncrete E p mdulus f elasticity f prestressed reinfrcement E s mdulus f elasticity f nn-prestressed reinfrcement f* c effective cncrete strength f' c cmpressive cncrete strength f ps stress in the prestressing steel at nminal bending resistance f pu tensile strength f prestressing steel fsy yield strength f main lngitudinal reinfrcing steel f y yield strength f cnventinal nn-prestressed reinfrcing steel kz rati f the in situ strength t the cylinder strength M u factred mment; taken as a psitive quantity nt less than Vud n efficiency factr factred axial frce; psitive if tensile, negative if cmpressive N u x

13 s stirrup spacing tf flange thickness T u factred trsin V shear frce V c shear strength cntributin frm cncrete V s shear strength cntributin frm reinfrcing steel V u factred shear w nde width ver which shear frce is applied z distance between nde centers s strain Si majr principal strain, nrmal t strut 2 minr principal strain, parallel t strut Ex strain in the hrizntal directin Q equivalent strut width ver which ties cntribute 9 angle f strut t hrizntal 6 angle f strut t lngitudinal axis 6 a stress xi

14 1 1.0 INTRODUCTION 1.1 General The purpse f this thesis is t cmpare and cntrast the ACI and the AASHTO LRFD 3 rd Ed. cde prvisins and design philsphies. 1.2 Scpe The crux f this thesis is the cmparisn f the tw prevailing cncrete design cdes regarding the design and detailing f cncrete beams in pure flexure with n ther lading present. The discussin n shallw beams used a series f twelve flanged beams as its fcus while the deep beam discussin fcused n a series f five deep beams and the Strut-and-Tie Mdel (STM) methd. T accmplish this a series f Excel spreadsheets were created t ensure the accuracy and cnsistency f all calculatins perfrmed; hwever as prgramming is nt the pint f this research several simplifying assumptins were made t reduce the time required t create, vet and utilize these tls. Bth sectins fcus slely n analytical results; n labratry experiments were carried ut. Cmparisns were based n but nt limited t maximum predicted allwable lad. 1.3 Objective The bjective f this thesis is t clarify the differences between the tw prevailing cncrete design cdes, ACI and AASHTO LRFD 3 rd Ed. and categrize them as

15 2 majr, minr, r insignificant. Fr simplicity when the AASHTO LRFD 3 r Ed. is referenced, the 3 rd Ed. shall be mitted. In cases where ther editins are referenced, the editin under discussin shall be nted. An additinal cmparisn will be made between the results prduced by the tw cdes when using the methd f STM and the actual experimental test lads used during experimental wrk carried ut by Ha in 22. A cmprehensive literature review prviding cverage f examples illustrating additinal differences fund between the ACI 318 and AASHTO LRFD cdes beynd pure flexure and deep beams is included. 1.4 Outline Chapter 2 cntains a literature review f relevant academic and industry articles regarding a specific structural member type, a specific cncrete design cde r a cmparative study f bth cdes. Chapter 3 prvides a brief intrductin and review f the design philsphy f bth flexural analysis fr shallw reinfrced cncrete beams and the STM methd f design fr reinfrced cncrete deep beams. Chapter 4 details the analytical prcedure as described by each cde and what methds were implemented by the sftware tls develped. Chapter 5 cntains the results btained frm this study. Chapter 6 prvides the cncluding remarks n the findings f this study.

16 3 2.0 LITERATURE REVIEW 2.1 Intrductin The flexural resistance r mment capacity f a structural member is a fundamental part f the verall analysis required when designing r evaluating an assembly f structural cncrete sectins. This flexural resistance is nly ne f a myriad fferees that needs t be cnsidered when designing r evaluating a structural member. Other frces due t direct lading r reactins such as axial, trsinal and shear frces must nt be verlked. Of the many cmparisns between the American Cncrete Institute (ACI) 318 Building Cde Requirements fr Structural Cncrete and the American Assciatin f State Highway Transprt Officials Lad and Resistance Factr Design (AASHTO LRJFD) Bridge Design Specificatins fr nn-prestressed cncrete members reviewed nly ne article directly addressed differences when analyzing r designing a structure with respect t the flexural resistance. Hwever many f the cnclusins presented by the reviewed articles fllwed the same general trend. Articles detailing differences in methd and results when using the strut-and-tie mdel (STM) methd were rather numerus and articles reviewed r discussed in this paper date back as far as 1996, lder articles n these subjects were available but were nt cnsidered fr review due t the significant revisins made t the cdes since the time f publicatin.

17 4 2.2 Shallw Beams Gupta and Cllins (537-47) perfrmed a study that primarily fcused n the questin f the safety f using traditinal shear design prcedures based n the failure f the Sleipner ffshre platfrm n 23 August, T aid in their determinatin f the safety f these traditinal methds f analysis and design 24 reinfrced cncrete elements having cncrete cmpressive strengths ranging frm 4280 t 12,6 psi were laded under a variety f shear and cmpressive axial lad cmbinatins. The results frm these tests were used t evaluate the design prvisins f the ACI and AASHTO LRFD "Bridge Design Specificatins and Cmmentary " 2" Ed. The ACI cde prvisins made a general assumptin that the shear stress f a member V, culd be defined as the sum f the shear lad at which diagnal cracks frm V c, and the prvided shear capacity f any stirrups present using the traditinal 45 degree truss equatin, V s. V =K+V S (2-1) The traditinal 45 degree truss equatin is defined as V S =^IA (2-2) s The cde als allwed fr a simplified cnservative calculatin t be used fr V c althugh the detailed apprach prvided fr mre accurate and less cnservative results. Bth equatins are shwn belw. The simplified and cnservative apprach is detailed in

18 5 Equatin 2.3 while the detailed apprach is shwn in Equatin 2.4. Bth equatins require the use f English units. V.' = N rxd (2-3) 1.9^ ^-^V* (2-4) N u in Equatin 2-3 represents the lad due t axial cmpressin and M m in Equatin 2-4 represents the mdified mment as defined by the relatinship shwn in Equatin 2-5. 'Ah-d^ M m =M u -N u 8 (2-5) Regardless f the methd utilized fr calculatin f the shear lad at which diagnal cracks frm the ACI cde placed a restrictin n the maximum value fr V c as defined by V c <3.54J r Xdfi + N 54. (2-6) These detailed equatins fr V c were derived by ACI-ASCE Cmmittee 326 in 1962 and were based n the principal stress as fund at the lcatin f the diagnal tensin cracking. The AASHTO LRFD shear design prcedure did nt use the general assumptins utilized by the ACI prvisins, rather it relied n the mre invlved mdified cmpressin field thery (MCFT) which in turn uses relatinships between equilibrium,

19 6 cmpatibility and stress and strain t predict the shear capacity f cracked cncrete sectin/elements. In additin t the bvius difference between the backgrunds f the tw cde prvisins nte that the AASHTO LRFD used SI units f MPa whereas the ACI prvided slutins in English units f psi. Hence the AASHTO LRFD expressin fr shear resistance f a sectin, V, was mre invlved and incrprated several different sub-equatins. AASHTO LRFD defined shear resistance as V n = Jf\b v d v + -^d v ct0 (MPa) (2-7) s The variables were defined as: A v = crss sectinal area f prvided stirrup reinfrcement within distance s b v = effective web width d v = effective shear depth, taken as 0.9d f' c = cmpressive strength f cncrete f y = yield stress f steel reinfrcement s = stirrup spacing Values fr /?and 6 were derived frm calculating the stresses transmitted acrss diagnal cracked cncrete sectins which cntained n less than the minimum required

20 7 transverse reinfrcement fr crack cntrl. This minimum amunt f transverse reinfrcement A v>mi, was defined and calculated by AASHTO LRFD as < mi = 0.083V77^ (MPa) J y (2-8) The values fr /? and 9 were dependant n the shear stress, v, and the lngitudinal mid-depth strain f the sectin, s x. Figure 2-1 details the idealized sectin used by AASHTO LRFD in the calculatin f s x fr this prcedure. A's A< \ Flanged Cmpressin \ \ / Sectin» %\ M d. N, V -*_ / f \ Ac A<i Flanged Cmpressin Idealized Sectin Sectin Flanged Frces, Web Frces and Sectin Frces d v -^ +0.57V -0.5F,ct^ Vcid ^ 0.5N+Q.5Vci6 Lngitudinal Strains Figure 2-1 AASHTO LRFD Prcedure fr calculatin f e x Mathematically the lngitudinal mid-depth strain f this sectin is defined by the relatinship shwn in Equatin 2-9. = e,s c (2-9) Where the terms s, represented the lngitudinal tensile strain in the flexural tensin flange and e c represented the lngitudinal cmpressive strain in the flexural cmpressive flange.

21 8 Mathematically s t and s c were calculated as shwn in Equatins 2-10 and ^* 0.5N+0.5F ct<9 7 U U i =_l (2. 10 ) ^± + Q.5N -0.5V, cte 7 U U = Ji (2.ii) EA + EA' The newly intrduced variables were defined as: A c ' = area f cncrete n flexural cmpressin side f member E c = elastic mdulus f cncrete E s - elastic mdulus f steel And Kwas defined by the relatinship shwn in Equatin v = ^~ (2-12) The result f the relatinships defined abve was that fr an increase in axial cmpressin the variables N u, s x and # decreased while /? wuld increase. This cntributed t an increased shear capacity fr any given sectin. In rder t btain empirical data t crrbrate the analytical predictins frm each cde's prcedure a series f 24 specimens were built and tested in the University f Trnt's shell element testing apparatus. Specimens were designed t represent the

22 9 sectins f a structural wall that was simultaneusly subjected t high axial cmpressin and high ut f plane shear lads. Specimen length L, verall sectin depth h, ttal percentage f lngitudinal reinfrcement p x, and ttal percentage f transverse reinfrcement acrss the specimen width p y were nt varied during the curse f investigatin. The parameters that were variable were the cmpressin t shear rati N/V, the cncrete cmpressive strengthf' c, specimen width b and the shear reinfrcement prvided r :. Lads were induced by five series f six jacks that applied pressure nt steel transfer beams lcated n the tp and bttm faces f the specimens. A cmpressive stress f up t 94 psi was applied alng with equal but ppsite mments applied t each end f the specimen bending it in duble curvature. In each test case the lads frm axial, shear and mments were increased prprtinally until the specimens reached failure. Tangential defrmatin was used as the means f quantifying the ttal defrmatin f each specimen. Strains frm the flexural tensin side f the member e xt and the flexural cmpressin side f the member s xc, were measured frm lcatins directly adjacent t the steel end plates. Strain fr the shear reinfrcement was measured directly frm the T- headed bars used t prvide shear reinfrcement and reinfrcement acrss the member width.

23 10 Failures were classified as shear S, r flexural F, based n the bserved strains in the Ext and s xc directins at the ends f the specimen when cmpared t the magnitude f the applied mment at the specimen ends ME and the shear strains y x:. Of the 24 specimens tested 18 were classified as failing in shear while the remaining 6 were classified as flexural failures. In the six flexural failures it was determined that the lngitudinal reinfrcement yielded and the applied mment ME, apprximately equaled r exceeded the predicted mment at failure M 0. This experiment demnstrated that when the ACI cnservative methd fr calculating V c was utilized the shear strength and mde f failure fr a reinfrced specimen laded under high axial cmpressin can be cnsistently predicted; whereas when using the mre detailed methd f calculating V c was used, the mde f failure can be brittle shear and ccur at lads significantly less than thse predicted, n average 68% less. It was demnstrated that it was pssible t prperly design a sectin using the detailed methd described by this prcedure and yet nly prvide a factr f safety as lw as The AASHTO LRFD prcedure fr calculating V c prvided far mre accurate and cnsistent results fr shear failure and the upper limit f shear capacity as induced by an increase in axial cmpressin was crrectly predicted.

24 11 Tw majr recmmendatins arse frm these results: The detailed expressin fr V c be remved frm the ACI 318 cde The term fr axial cmpressin NJA S in Equatins 2-3 and 2-6 nt be taken greater than 30 psi Rahal and Cllins (277-82) undertk research t prvide an evaluatin f the design prvisins fr cmbined shear and trsin lad cases as described by bth the ACI and the AASHTO LRFD Bridge Design Specificatins, 2 nd Editin. When Rahal and Cllins' reprt was published the current versin f each cde cntained trsinal design prvisins that were similar ther than the methd used t determine the angle 6. Rahal and Cllins nted that the AASHTO LRFD prvisins had been extensively checked fr shear cases whereas the ACI prvisins had been extensively checked fr pure trsin as well as cmbined trsin and bending. It was cncluded that there was a lack f data available that directly crrelated the trsinal prvisins f each cde. Hence Rahal and Cllins ran fur large scale experiments t cmpare the results f the calculated trsin-shear interactin diagrams btained frm bth the AASHTO LRFD and ACI prvisins. The beams tested were slid, rectangular sectins 340mm wide and 640mm deep reinfrced with nn-prestressed lngitudinal bars. The ACI cde described the basic truss equatin relating the prvided hp reinfrcement t trsinal strength as fllws:

25 12 T = 2A 0 - ^ ct 9 (SI Units) (2-13) s Where A 0 represented the area enclsed by the shear flw path and was permitted t be taken as 0.85*A h, and where A 0 h represented the area enclsed by the utermst transverse trsinal reinfrcement. f yv was the yield strength f the hp reinfrcement and A t represented the crss sectinal area f ne leg f the transverse reinfrcement. 0 represented the angle f inclinatin f the cmpressive diagnals and was nted t have rather ambivalent instructins t its suggested versus its analyzed values. As described in ACI the angle 6shall nt be less than 30 degrees but it is suggested that <9= 45 degrees fr nn-prestressed members and 6= 37.5 degrees fr members that are prestressed. A similar truss equatin, see Equatin 2-14, related the trsinal strength t the quantity f lngitudinal reinfrcement prvided. A f T =2A 0 -!+*- tan 3 (SI Units) (2-14) Ph Rahal and Cllins bserved that when cmparing equatins 2-13 and 2-14 the equivalent trsinal strengths culd be btained by using less hp reinfrcement but increasing the lngitudinal reinfrcement.

26 13 When these tw equatins were set equal t each ther the required area f transverse reinfrcement culd be slved fr as shwn by Equatin A f A l=^- Ph^ct 2 0 (SI Units) (2-15) s f y t The ACI equatin fr a nn-prestressed sectin that described the relatinship between transverse reinfrcement and shear strength is shwn in Equatin K = K + K= 0A66j]\b w d + -^d (MPa) (2-16) The variables were defined as: A s = area f flexural reinfrcement b w - web width d = distance frm the extreme cmpressive fiber t the centrid f the flexural reinfrcement f' c = cmpressive cncrete strength f y = yield strength f reinfrcing steel V c - shear strength cntributin frm cncrete V s = shear strength cntributin frm reinfrcing steel Additinally the ACI cde required that the nminal shear stress fr slid sectins be limited t avid cncrete crush prir t reinfrcement yield. Equatin 2-17 describes this limitatin.

27 14 v^2 f b Jj -^r < 0.83SK (SI Units) (2-17) The AASHTO LRFD cde used the same basic truss equatin as the ACI , Equatin 2-13, t relate the area f ne leg f the transverse reinfrcement^,, t the required trsinal strength T. Hwever the AASHTO LRFD relatinship between the minimum required shear strength V and the required area f transverse reinfrcement A v was nticeably different frm that shwn fr ACI in Equatin V = V c + V s = Jf~ c b v d v + < ct 9 (SI Units) (2-18) s The variable b v represented the web width, hwever d v was defined as the effective shear depth which can be taken as 0.9*d. Fr sectins cntaining stirrups the values fr J3 and # depended n the nminal shear stress v, and the mid-depth lngitudinal strain e x. Rahal and Cllins nted that when /?was set equal t 2.22 the AASHTO LRFD value matched the ACI value fr V c exactly. Als when examining nnprestressed sectins s x culd be taken as 1.x10" which prvided a value f 36 degrees fr 0, and when the AASHTO LRFD value fr 6 equaled 36 degrees the results fr V s were 24% higher than thse btained frm the ACI prcedure.

28 15 Equatin 2-19 details the nminal shear stress v, fr a slid sectin as calculated under the AASHTO LRFD prvisins fr a slid sectin under cmbined shear and trsin which was required t be n greater than 0.25/' c t avid failures due t cncrete crushing. PA j 2 (T A 2 \ A h J <0.25/' c (MPa) (2-19) The lngitudinal stress s x, at the mid-plane culd be taken as l.ooxlo' 3 r it culd be calculated using the relatinship shwn in Equatin M.!L N ct 6JV + 0-Wu "0.7/aA v x = -1 ' 2 f J (2-20) 2(E s A s + E p A ps ) Variables in Equatin 2-20 were defined as: A s = area f nn-prestressed lngitudinal reinfrcement placed n the flexural tensin side f the sectin A ps - area f prestressed lngitudinal reinfrcement placed n the flexural tensin side f the sectin d v = effective shear depth, can be taken as 0.9d E s = mdulus f elasticity f nn-prestressed reinfrcement E p = mdulus f elasticity f prestressed reinfrcement f pu = tensile strength f prestressing steel M u = factred mment; taken as a psitive quantity nt less than Vud v

29 16 N u = factred axial frce; psitive if tensile, negative if cmpressive T u = factred trsin V u = factred shear Rahal and Cllins nted that fr prestressed sectins s x wuld ften apprach zer. In these cases the value f 6 wuld vary frm 22 t 30 degrees depending n the level f shear stress present. The AASHTO LRFD prvisins required that the tensile capacity f the lngitudinal reinfrcement n the flexural tensin side f the sectin be n less than the frce T E, t prevent premature failure f the lngitudinal reinfrcement. TE was calculated as shwn in Equatin fir V / / W C T - V WJMM (2-21) M. <i> \\<t> ) \ <t> 1A j The test variable in the series was the trsin t shear rati which varied frm zer t 1.216m. Specimen failures were attributed t excessive yielding f the clsed stirrups as well as spalling and crushing f the cncrete in the test regin. Rahal and Cllins determined that use f the ACI prvisins prduced very cnservative results when the maximum value f 45 degrees fr the angle f the cmpressin diagnals 6, was used. Cnversely when the minimum value fr #f 30

30 17 degrees was used the ACI prvisins demnstrated less cnsistent results and prvided failure lads fr high trsin-t-shear ratis much higher than thse bserved. Results btained frm use f the AASHTO LRFD prvisins prvided a cnsistent value f apprximately 36 degrees fr 6. Use f this value prvided results f a cnsistent and reasnable nature that clsely replicated the bserved crack patterns. Naaman (209-18) investigated the differences between the ACI and AASHTO LRFD cdes regarding sectins that were classified as being between tensin cntrlled and cmpressin cntrlled, i.e. in the transitin zne. Naaman fund and described several examples where the ACI prvisins regarding the limits f reinfrcement fr flexural members lead t unintended errneus results that brught the validity f the prvisin int questin. These flaws were nt directly crrelated t the crrespnding AASHTO LRFD prvisins but they did reflect similar results fr ther ACI prvisins as described by several ther publicatins where slutins appeared cnservative but in reality did nt prvide adequate factrs f safety. Naaman nted that the changes made frm the ACI t the ACI cdes relcated the limits fr tensin and cmpressin cntrlled sectins and added the transitin regin between the tw; the flaw lie in this definitin fr these reginal bundaries. The varius regins fr reinfrcement limits and the definitins and frm different cdes are shwn in Figure 2-2.

31 18 a) LU Q O O O < > Ptmm 0.75ft,, A 1 i i 1 i Under-reinfrced r i A C l fr bending -»- <>fr cmpressin (n transitin) ^ factr b) LU Q O O < CM O O CM +tr bending Transitin _J_ -(t fr cmpressin ft factr c C) Q LU Q. OH +fr bending Minimum I t ft factr Transitin -ft fr cmpressin Tensin cntrlled m I < Transitin ^ I ^ cmpressin cntrlled " (under-reinfrced) " (ver-reinfrced) -L O Minimum AASHTO I LRFD Under-reinfrced (t fr bending 0.42 ± -m- j fr cmpressin (n transitin) c Figure 2-2 Summary f reinfrcement limits by 1999, 22 ACI cdes and AASHTO LRFD and Naaman recmmendatin

32 19 The ACI cde used the rati f c/d s where c represented the depth f the cmpressin blck and d s represented the depth frm the extreme cmpressive fiber t the centrid f the tensile frce in the nn-prestressed reinfrcing bars. In the 22 editin f the ACI 318 cde the rati was changed t c/d t where d t was defined as the depth frm the extreme cmpressive fiber t the centrid f the extreme layer f the nn-prestressed reinfrcement. This allwed fr values f if) fund using the ACI t be different frm thse btained using the AASHTO LRFD fr identical sectins because the definitin f d t and its crrespnding s t was defined as the distance t the centrid f the extreme tensile reinfrcement nly. This did nt take int accunt tensile resistance prvided by a multi-layered arrangement and implied that a sectin which cntained mre than ne layer f reinfrcement r a sectin having any cmbinatin f plain reinfrced, partially prestressed r fully prestressed steel wuld be cntrlled by the extreme layer f reinfrcement exclusively. While the ACI cde ffered this smewhat cnflicting definitin fr the limits f reinfrcement between editins the crrespnding AASHTO LRFD prvisin detailed that fr all cases the maximum reinfrcement was bund by the relatinship detailed in Equatin < 0.42 (2-22) d e

33 20 Where c again represented the depth f the cmpressin blck and d e was calculated as the weighted sum assuming yield f the steel reinfrcement prvided. Reference Equatin 2-23 d e = AJpAp + Afyds (2-23) A ps f ps + Asfy The variables used were defined as: A s = area f nn-prestressed reinfrcement A ps = area f prestressed reinfrcement f ps = stress in the prestressing steel at nminal bending resistance f y = yield strength f cnventinal reinfrcing steel dp - depth frm the extreme cmpressive fiber t the centrid f the tensile frce in the prestressed reinfrcement d = depth frm the extreme cmpressive fiber t the centrid f the tensile frce in the nn-prestressed reinfrcement, i.e. reinfrcing bars Naaman nted that when the quantity d e was used t calculate the limits f reinfrcement fr a sectin the results were guaranteed t be the same, independent f whether the sectin was plain reinfrced, partially prestressed r fully prestressed. This cnsistency was attributed t the fact that the equatin guaranteed simultaneus equilibrium f frces as well as strain cmpatibility in any case. It als made the type f reinfrcement present irrelevant since the tensile frce T must equal the cmpressin frce C fr all cases.

34 21 Mre specifically Naaman described that because the assumed failure strain in the cncrete e cu, was taken as a cnstant in equatins 2-24 and 2-27 there was a direct relatinship between c/d e and the tensile strain in the cncrete at the centrid f the tensile frce s te that was unique. (2-24) d e =d s fr A ps =0 (2-25) d e =d p fr A s =0 (2-26) d-c va j (2-27) The use f d e was prpsed by Naaman t replace d t when determining the rati c/d t t avid errneus reinfrcement level classificatin f a sectin. It shuld be nted that the ACI prvisins did include sme but nt all f the recmmendatins prpsed by Naaman. The relatinship used t define the regins was changed t use the quantity d e rather than the less accurate d t, and the upper limit f the transitin regin remained 0.60 cnsistent with Naaman's recmmendatin; hwever the lwer limit f the transitin regin remained set at rather than the 0.44 that Naaman had prpsed. Reference Figure 2-3 t cmpare the new limits f the ACI t thse f the AASHTO LRFD.

35 i I 1 ' 4. ~4 1 J ' ^ Factr L f> fr bending I. ^ fr cmpressin ACI Minimum Trans.tin I ^ Tensin cntrlled i Transitin Cmpressin cntrlled ( (under-reinfrced) p " f (ver-reinfrced).. ~ Minimum AASHTO l 0-42,_J ^L _ UWD,. Under-reinfrced unaer-reinrrceq ^frcinpres$jl](ngtransjtid) I (j> fr bending ' J * Over-reinfrced Figure 2-3 ACI and AASHTO LRFD limits f reinfrcement Rahal and Al-Shaleh (872-78) cnducted a study t examine the differing requirements fr minimum transverse reinfrcement as specified by the ACI Cde, Canadian Standards Assciatin (CSA) A23.3 and AASHTO LRFD Specificatins 2 nd Ed. The bserved cracking patterns, crack widths at the estimated service lad and at the pst-cracking reserve strength were used t evaluate the perfrmance f each specimen. Their study fcused n high-strength cncrete (HSC) based n its increased use in cnstructin. HSC sectins require larger amunts f transverse reinfrcement due t their behavir f cracking at much higher shear stresses than cnventinally reinfrced cncrete sectins.

36 23 The ACI used detailed equatins that accunted fr the cntributin frm the cncrete V c, n the effect f the lngitudinal steel as well as the stress resultants fr bending mment and axial lad. The AASHTO LRFD Specificatins accunted fr the influence f lngitudinal reinfrcement, flexural, trsinal and axial lading in the calculatin f the strain indicatr s x, which has effects n bth cncrete and steel reinfrcement cntributins V c and V s respectively. The CSA A23.3, the ACI , and the AASHTO LRFD were nt united in their apprached t lngitudinal reinfrcement prvisins and were nted t have differed significantly. Regardless f the difference in the apprach t prvided lngitudinal reinfrcement nne f the afrementined cdes accunted fr the influence f the lngitudinal reinfrcement; addressing this influence frm lngitudinal reinfrcement was the bjective f Rahal and Al-Shaleh's test prgram. Eleven fur pint lad shear tests were perfrmed n 65 MPa (95 psi) beams that had minimal transverse reinfrcement and tw levels f lngitudinal reinfrcement. Beam dimensins fr all specimens were 2 mm (7.87 in.) wide, 370 mm (14.57 in.) deep and 2750 mm (108 in.) lng with a shear span f 9 mm. The average cncrete cylinder strength^ was 75% f the cncrete cube strength f cu and split tensile strength/^ was apprximately 0.74 Jf~.

37 24 Their study prduced the fllwing cnclusins: Behavir f members cntaining large amunts f lngitudinal steel was far superir when cmpared t members that cntained very little r n lngitudinal steel The ACI and CSA A23.3 prvided adequate perfrmance fr members cntaining large amunts f lngitudinal reinfrcement N evidence was fund in perfrmance between beams designed with the maximum stirrup spacing as defined by each cde Shear capacity equatins in the ACI and AASHTO LRFD 2 nd Ed were cnservative 2.3 Deep Beams Brwn, Sankvich, Bayrak and Jirsa (348-55) cmpleted a study f the behavir and efficiency factrs assigned t bttle shaped struts when used in calculatins as described by the methd f STM. Histrically these efficiency factrs were assigned based n gd practice rather than actual results frm experimentatin. A fundamental difference fund between the tw prvisins was in the calculatin f the strength f a strut based n specified cncrete strength as determined by a cylinder test/' c, and the strut efficiency factr fi s. ACI defined^,, as shwn in Equatin f cu =0.S5/3J< c (2-28)

38 25 The strut efficiency factr J3 S, is based n the type f strut under cnsideratin and the amunt f transverse reinfrcement present. When ACI is used in cases f a bttle shaped strut that was crssed by adequate transverse reinfrcement as defined by ACI Equatin A-4 (Equatin 2-29) the strut will cntrl the strut-nde interface in all cases ther than CTT, i.e. an interface nde having ne strut and tw ties. Y^sinr,< 0.3 (2-29) bs i The AASHTO LRFD utilized the MCFT t define f cu and therefre the definitin is much mre invlved than that described by the ACI methd. Equatin 2-30 repeats the AASHTO LRFD Equatin / = Ls. < 0.85/' c (2-30) " s, where e l =e, + (e, + 0.2) ct 2 a s (2-31) a s was defined as the smallest angle between the cmpressive struts and the adjining tie. It was nted that many engineers have difficulty chsing an apprpriate tensile cncrete strain t be used during design and have therefre expressed reservatins abut using the MCFT based ASSHTO LRFD prvisins.

39 26 T mre directly measure the effect these tw appraches had n mdeling struts 26 cncrete panels measuring 36 x 36 x 6 in. (914 x 914 x 152 mm) were laded using a 12 x 6 x 2 in. (305 x 152 x 51 mm) steel bearing plate. One test used a different panel thickness and bearing plate t bserve and examine the effect f specimen gemetry n the efficiency factr. Each specimen had a unique amunt and placement f reinfrcing steel and in each islated strut test the same mde f failure was bserved. Failure was first indicated by a vertical crack which wuld frm in the center f each panel. That crack then prpagated frm panel midheight t the lading pints but wuld nt intersect them, rather it wuld change directin. Failure was described as crushing and spalling f the cncrete near but nt adjacent t the lading plate. The same failure mde was bserved in every test regardless f the bundary cnditins present. Of the 26 specimens tested the efficiency factrs presented in ACI prvided a safe estimate f the islated strut capacity. Of these 25 specimens it was nted that the results were cnservative but erratic when cmpared t the test data. When the average value fr the experimental efficiency factr was divided by the predicted ACI efficiency the result was Of the 26 test specimens 20 f the AASHTO LRFD determined efficiency factrs yielded results fr the islated struts that were less cnservative but mre cnsistent with

40 27 the test data. All f the AASHTO LRFD data was gverned by the limitatin placed n the maximum strut strength f 0.85/ c as described by AASHTO LRFD Equatin (Equatin 2-30). Fster and Malik (569-77) reviewed a cmprehensive set f test data n deep beams and crbels and cmpared them t the prpsed efficiency factrs. The STM mdel that was mst extensively investigated was the plastic truss mdel where all truss members enter the ndal znes at 90. The plastic truss mdel has tw pssible failure mdes; cncrete crush in the struts and yielding f the ties. A third failure mde was prpsed by in 1998 by Fster where splitting r bursting f the strut shuld be cnsidered. Due t the well knwn behaviral and material prperties f reinfrcing steel tensin failures can be predicted with a high degree f cnfidence and therefre this mde is nt discussed. T simplify their discussin regarding deep beams and crbels Fster and Malik standardized the nmenclature used in their study. They defined the clear span a, as the distance frm the centers f the strut ndes and they als split up the in situ strength factr ks and the strut efficiency factr v; histrically these tw strength factrs were cmbined int a single parameter. All relevant equatins were then recalculated t incrprate this split between variables.

41 28 The fundamental plastic truss mdel equatins that relate material prperties, gemetry and strength f a member in equilibrium are shwn belw. Material: T = Af s sj sy C = f. c bd c fc=k,yf\ (2-32) (2-33) (2-34) Gemetry: a = a + - w (2-35) = d + Q 2 z m0 = a w m (2-36) (2-37) n = d-jd 2-2aw<2(D-d) (SI) (2-38) Strength: d=- w sin^ (2-39) V = min a (2-40) The variables were defined as: a = shear span a' = distance frm center f cncentrated lad t edge f supprt

42 A s = area f tensile reinfrcement b = sectin width d = effective depth f main flexural reinfrcement d c = strut width D = verall member depth f' c = cmpressive cncrete strength f c effective cncrete strength fay = yield strength f main lngitudinal reinfrcing steel ks = rati f the in situ strength t the cylinder strength V= shear frce w = nde width ver which shear frce is applied z = distance between nde centers v= efficiency factr 6 - angle f strut t lngitudinal axis Q - equivalent strut width ver which ties cntribute Using these relatinships Fster and Malik calculated the efficiency factr shwn in Equatin V V ~Kf' c bw

43 30 This efficiency factr was used t reduce predicted member capacity t cmpensate fr the fact that cncrete is nt a perfectly plastic but rather a brittle material. In 1986 the MCFT was prpsed t describe and define that cncrete is nt perfectly plastic and the crrespnding lss f strut capacity due t transverse tensin fields. Fster and Malik als investigated the cntrversy and debate surrunding the assignment and values used fr efficiency factrs in this same reprt. A 1986 study suggested v= 0.6, whereas anther undertaken in 1997 suggested v= 0.85 and placed greater emphasis n the selectin f an apprpriate truss mdel. Mdels created in 1987, 1990 and 1997 als used efficiency factrs that were functins f strut r nde lcatin and the degree f disturbance these struts r ndes experienced. The greater this disturbance the lwer the efficiency factr assigned. ASCE-ACI Cmmittee 445 ffered a cmprehensive review f this wrk. In 1978 Nielsen intrduced an efficiency factr t calibrate the cncrete plasticity mdels they had develped fr members in shear. In 1998 Chen revised these factrs fr deep beams and prpsed the fllwing relatinship v. 0.6(1-0.25Z))(1/? + 2)(2-0.4-) v = P= &- fr If. a/d<0.25 /?<0.02 Z)<1.0 (SI) (2-42) Where p was defined as the reinfrcement rati fr main lngitudinal steel

44 31 In 1986 Batchelr and Campbell prpsed a reductin f the effective cmpressive strength based n their thery that the diagnal struts are in a state f biaxial tensin which in turn reduced the strength f the web cncrete. Based n their parametric study Bachelr and Campbell prpsed the fllwing equatin be used t define the efficiency factr. In ( v d\ V b) = ^ d ) (SI) (2-44) Warwick and Fster investigated the effects f cncrete strength n the efficiency factr using a range f 20 t loompa and prpsed and v = 1.25 ^-0.72[ (f) + A {f) - Xf ra /d- 2 (SI) (2_45) v = *- fr a A>2 (SI) 5 /d (2-46) Equatins 2-44 and 2-45 were develped in parametric studies using the cncrete cmpressive strength f' c, and the quantities f hrizntal and vertical reinfrcement as the variable parameters and by cmparing a series f experimental data with nn-linear finite element analyses. Frm this Warwick and Fster cncluded that cncrete strength and the rati f the shear span t member depth were the main cntributrs that affected the efficiency factr.

45 32 Based n panel testing perfrmed by Vecchi and Cllins (1986), Cllins and Mitchell prpsed Equatin where Lv = (2-47) *, Variables were defined as: e-sx + tezzhl (2-48) 1 tan 2 # / = majr principal strain, nrmal t strut 2 = minr principal strain, parallel t strut Sx = strain in the hrizntal directin 6 - angle f strut t hrizntal Fster and Gilbert demnstrated in 1996 that the relatinship defined by Cllins and Mitchell culd be mdified as a functin f/' c and the rati a Yd. In ding s the strut angle was apprximated as tan9~ d/a'; mre precisely tan6~ z/a. This prduced the result shwn in Equatin k 3 v = l - (SI) (2-49) J c " 470 A z) Fster and Gilbert further simplified this equatin, termed the mdified Cllins and Mitchell relatinship, based n the insensitivity f vtf' c t prduce Equatin 2-50.

46 33 V = T-77 (SI) (2-50) (- u Fr special situatins where a/z = 0, i.e. si = 0 the MCFT infers that v= 1 and thus the relatinship is further mdified t yield Equatin v = l - j- (2-51) (a\ \z) It must be nted that when Equatin 2-51 is used ks = where MacGregr prpsed that the efficiency factr be defined by Equatin 2-52 k 3 v = v l v 2 (2-52) v 2 = HL (MPa) (2-53) The variable vi in Equatin 2-52 was defined as a factr dependant n the ptential f damage t the strut(s) under cnsideratin. Vecchi and Cllins later revised their efficiency factr based n newly available data f the time t prduce Equatin K c K f (2-54) where

47 34 /- \0.80 k = 0.35 ^ V c 2 J (2-55) and k f =0.1*25Jf' e >1.0 (M/ty (2-56) Using an assumed value fr 2 f-0.25 and the relatinship shwn in Equatin 2-57 k-g 2 ), = X tan 2 # (2-57) The fllwing is btained fr k c ( K = a A 0.80 (2-58) Fr cases where a/z = 0 the MCFT implied an efficiency factr value f v- 1 and kf= 1.0 and substituting Equatin 2-58 int Equatin 2-54 with these special case values prduced Equatin v = c 7 (2-59) Based n the develpment f a reinfrced cncrete cracked membrane mdel fr plane stress elements the efficiency factr define by Equatin 2-60 was adpted 1 v = ( *,)/'/ (2-60)

48 35 When the majr principal strain si, is treated as a functin f bth s x and l/tan 2 6 then Equatin 2-60 takes the frm f Equatin K = 1 c l+ c 2 (a/zy\f</i where Ci and C2 are empirically derived cnstants l -, jt-<1 (2-61) Frm their review f sixteen previus studies, 135 specimens that had been determined t have failed in cmpressin were analyzed and efficiency factrs assigned based n cncrete strengthf' c, multi-parameter mdel predictins and MCFT. Equatins 2-51, 2-59 and 2-61 were cmpared against the experimental data and mdels prpsed by the AS36 mdel with cutff and mdels prpsed by Batchelr and Campbell, Chen, MacGregr and Warwick and Fster. Tw parameters were used t define the efficiency factrs,/' c and a/z and the fllwing bservatins were made: Pr crrelatin with high degrees f variability existed between experimental data and efficiency factr mdels based slely n cncrete strengthf' c Multi-parameter efficiency factr mdels als exhibited pr crrelatin with high degrees f variability between experimental data and predicted behavir MCFT mdels prvided cefficients f variatin frm when cmpared t the experimental data

49 36 Bundary cnditins play are a significant factr in btaining data that is reliable and has a lw degree f scatter Strut angle was the mst significant factr and mdels based n the shear span t depth rati a/z, prvided the best predictins fr efficiency factrs Mst f the reviewed articles addressed tpics ther than the same tw specific areas f pure flexure and deep beams cvered by this thesis; this was attributable t the paucity f relevant available literature. Hwever their inclusin was justified by the wide array f examples cntained within these articles. The examples illustrated that there has always been divisin between the cdes n apprach t analysis and design as well as the actual frmulae t be utilized when designing r analyzing a reinfrced cncrete sectin, member r structure.

50 THEORY OF FLEXURE 3.1 Backgrund In the classical apprach t slving fr the flexural strength f a sectin generally there are tw assumptins required in rder t prvide a simplified methd t the slutin f the required calculatins. These tw assumptins frm the backbne f the elastic case fr flexural thery that states that nrmal stresses within a beam due t bending vary linearly with the distance frm the neutral axis. The assumptins are: 1) Plane sectins remain plane. 2) Hke's Law can be applied t the individual fibers within the beam sectin. There are tw cmpnents t the first assumptin as discussed in 6.4 f Mechanics f Materials: The first cmpnent is based n rigrus mathematical slutins frm the thery f elasticity that demnstrate sme warpage des actually ccur alng plane sectins and that this warpage is greatest when shear is applied alng with a mment. Hwever adjining planes are als similarly warped and therefre the distance between any tw pints n adjining sectins fr all practical purpses remains cnstant whether r nt warpage is cnsidered. Flexural thery is based upn the relative distances between sectins and because it has been prven that warpage des nt vilate

51 38 this relatinship between plane sectins and the assumptin that plane sectins remain plane remains valid. The secnd part f this assumptin is that when a beam is subjected t pure bending the psitive strains n the utermst tensile surface are accmpanied by negative transverse strains, this curvature is classified as anticlastic curvature. Likewise the negative strains alng the utermst cmpressive surface are accmpanied by psitive transverse strains; this type f curvature is classified as anti-synclastic curvature. See Figure 3-1. The classical apprach t slutins f reinfrced cncrete sectins ignre this behavir. Remain Straight Figure 3-1 Beam in flexure with saddling effect

52 39 The secnd assumptin, based n Hke's law, Equatin 3-1, states that individual fiber strains can be used t calculate individual fiber stresses and visa-versa. e = - (3-1) E Where the variables are defined as fllws: s = strain E = mdulus f elasticity f the material, cmmnly referred t as Yung's Mdulus a= stress 3.2 Thery f Flexure in Shallw Reinfrced Cncrete Sectins In the study f reinfrced cncrete additinal assumptins are made in series with the first tw presented; assumptins 4, 6 and 7 are generally attributed t Charles S. Whitney but they were btained frm "Design f Reinfrced Cncrete ACI Cde Editin" fr this thesis. 3) The strain in the reinfrcing steel is the same as the surrunding cncrete prir t cracking f the cncrete r yielding f the steel - this is a cntinuatin f the secnd assumptin 4) The tensile strength f cncrete is negligible and assumed t be zer 5) The stress-strain curve f the steel is elastically perfectly plastic 6) The ttal frce in the cmpressin zne can be apprximated by a unifrm stress blck with magnitude equivalent t 0.85/' c multiplied by a depth f a 7) The maximum allwable strain f cncrete is 0.3

53 40 The general arguments that frm the basis f Hke's Law remain as valid fr reinfrced cncrete sectins as they d fr istrpic hmgenus material sectins when subjected t small strains. In general the tensile strength f cncrete is arund 10% f the cmpressive strength. When laded the tensin zne f a cncrete sectin will begin t crack under very light lads destrying the cntinuity f the sectin and any tensile reinfrcement will be frced t carry the tensile lad in its entirety. The assumptin that the stress-strain curve f steel is elastically perfectly plastic implies that the ultimate strength f steel is equivalent t its yield strength. In effect, this results in an underestimatin f the verall ultimate strength f a given sectin due t the reinfrcing steel but it prduces a mre predictable mde f member failure. The stress distributin in the cmpressive regin des nt maintain a linear relatinship with respect t distance frm the neutral axis due t the nature f the cnstituent materials used t manufacture cncrete. Rather the stress distributin is in the frm f a parabla as shwn in Figure 3-2 c. Whitney develped an equivalent rectangular stress blck that prvides results f equal accuracy fr the cmpressive strength f a cncrete sectin that avids the rigrus mathematical calculatins required t cmpute the area f a parabla. This blck has

54 41 depth, a, and an average cmpressive strength equivalent t 0.85/' c. The value f 0.85 was derived frm extensive labratry testing f cre test results f cncrete in structures where the cncrete was a minimum f 28 days ld. T calculate the depth f the cmpressin blck the distance frm the extreme cmpressive fiber c, is multiplied by a mdificatin cefficient Pi yielding the result shwn in Equatin 3-2. a = p x c (3-2) The cefficient /?/ varies as summarized in Equatin far f'c<40psi A /'c-40j 40 </'c< 80 mz 10 J 0.65 fr fc> &0psi (3-3) The ACI has adpted a strain f 0.3 in the extreme cmpressive fiber as the assumed maximum allwable strain, r limit strain, fr a cncrete sectin. Cmpared with results determined frm extensive empirical data this value represents the lwer bund f the limit strain. Using these seven basic assumptins a series f equatins can be derived that prvide quantitative values fr the equivalent, cunteracting frce required fr a beam t remain in equilibrium while under the applicatin f an external frce. This can be

55 42 explained by visualizing the tw frces as vectrs, equal in magnitude and ppsite in directin, acting at any tw lcatins alng the centerline f the crss sectin. Summing the frces in the axial directin T = C, i.e. the tensile frces prvided by the reinfrcement present in a sectin are ppsite and equal t the sum f the cmpressive frce prvided by the cncrete; reference Equatins 3-4 thrugh 3-8. T = A,f y (3-4) C = 0.85/' c /?,c6 (3-5) where a = (5 x c :.c-al[5 x T = C=zA t f y =0.Z5f c ab (3-6) (3-7) (3-8) C=0.85f c ab a -P lc T=Af y (a) (b) (c) (d) Figure 3-2 Graphic representatin f frces within a reinfrced cncrete sectin

56 43 Variables are defined as: a = effective depth f the cmpressive blck A s = area f nn-prestressed steel f y = yield strength f nn-prestressed steel f' c = cmpressive strength f the cncrete b = the width f the sectin c = distance frm extreme cmpressive fiber t neutral axis Using the relatinships shw in Equatins 3-4 thrugh 3-8 and slving fr a yields the result shwn in Equatin 3-9. A f a = '-^ (3-9) 0.85 f c b Slving fr the internal resisting cuple between the tensile and cmpressive frces yields the nminal strength r mment resistance f the sectin M, shwn in Equatin M n -AJ^d-^j = 0.85/>Z>(V ) (3-10) 3.3 Thery f Flexure in Deep Beams Deep beams are a cmmn structural member fr which an accurate slutin cannt be reached using the afrementined techniques. Deep beams are structural members defined by the relatinship between beam width b w, beam depth h and clear

57 44 span /. Per ACI , deep beams are members laded n ne face and supprted n the ppsite face s that cmpressin struts can develp between the lads and the supprts and have either: a) clear spans /, equal t r less than fur time the verall member depth; r, b) regins with cncentrated lads within twice the member depth frm the face f the supprt Deep beams r deep beam regins begin t shw crack prpagatin at lads in the range f x hp u t x lip u, where P u represents the cncentrated lad, and plane sectins are n lnger assumed t remain plane. This invalidates the first and mst fundamental assumptin discussed in this paper fr the slutin f reinfrced sectins. Therefre ther mre advanced methds must be emplyed. One f these is the methd f Strutand-Tie Mdel (STM) an inherently cnservative methd fr slving the frces within these member's sectins. The verarching mtivatin f STM is weighted tward cnservatism in the slutins it prvides and t transfer as much f the applied lading as pssible int cmpressin using the fewest number f members. The fundatins f STM are attributed t the truss methd wrk dne by Ritter in 1899 develped as a means t explain the dwel actin f stirrup reinfrcement. Hwever it was nt until the 1980s that the truss methd transfrmed int STM and was used t find slutins fr the discntinuus regins within deep beams.

58 The methd f strut and tie mdels is nt exclusively limited t deep beams, it als has valid applicatins in crbels, dapped-end beams and the discntinuus regins, r D-regins, within shear spans. Mathematically a D-regin is defined as a regin lcated within a distance equal t the member depth h, frm the beam/supprt interface r a regin lcated a distance h frm each side f a cncentrated lad; reference Figure 3-3 fr examples. KH 4)1*, h (HE h, h. h \>>t Ay? 'TO -4 ~A~ JJ ttmmtmt O (tf 2h 1)1' lt)l" (a) Gemetric discntinuities fw Lading and gemetric discntinuities Figure 3-3 ACI Fig. RA.1.1 (a and b) - D-regins and discntinuities (Reprinted with permissin frm the American Cncrete Institute) A D-regin within a beam is defined as an inter-beam span where traditinal mment and shear strength thery n lnger applies as based n the assumptin that plane sectins remain plane, and lads are nt reacted by beam actin but rather they are reacted primarily by arch actin, as such these D-regins can be islated and viewed as a

59 46 deep beam. In these regins the rati f shear defrmatins t flexural defrmatins can n lnger be cnsidered negligible. At its cre STM disregards kinematic restraints, nly gives an estimatin f member strength, and it cnfrms t the lwer bundary f thery f plasticity, i.e. nly equilibrium and yield cnditins need be satisfied. The slutins fr the capacity f members fund using the applicatin f this lwer bund thery are estimates that will prvide member capacities which will be less than r equal t the lad required t fail the member; thus the inherently cnservative nature f STM slutins. The tw greatest similarities between the ACI and AASHTO LRFD cde prvisins are the basic set up f the slutin, and hw the truss members are chsen. After the glbal frces have been determined fr a particular member, a truss mdel is chsen t represent the flw f frces within that member. The gal fr develpment f any truss mdel is t use the lwest number f members required t satisfy equilibrium and safely transmit the frces int the supprts. After a truss mdel has been develped the frces within that truss are analyzed by using the methd f jints, the methd f sectins, a cmbinatin f bth r using a CAD prgram. Members in cmpressin are designated struts and members in tensin are designated ties. The intersectin f any tw r mre members is designated a nde, as shwn in Figure 3-4.

60 47 T-c,r (a) CCC Nde (bjcctnde Figure 3-4 ACI Fig. RA.1.5 (a and b) - Hydrstatic ndes (Reprinted with permissin frm the American Cncrete Institute) There are few similarities that exist between the ACI and AASHTO LRFD cdes with respect t STM efficiency factrs. The basic premise fr the implementatin f the methd using either cde is the same but the way that these frces are reslved within the individual strut and ndal members is the area f greatest divergence between the tw cdes. 4.2 prvides a cmprehensive explanatin f the differences in the STM between the ACI and the AASHTO LRFD 2 nd Ed. and 5.2 discusses the differences in the results btained frm each cde.

61 ANALYTICAL PROCEDURE 4.1 Shallw Beams In rder t simplify the prgramming phase f the prject, an analytical apprach was used rather than a design apprach. Ding s reduced the required number f IF lgic statements, shrtened the verall length f the EXCEL prgram, simplified the prgram used and fcused the prject n the analysis f results rather than prgramming. All spreadsheets used in the shallw beam analysis were vetted via direct cmparisn against example prblems 7.4 and 7.5 frm the Ntes n ACI Building cde Requirements fr Structural Cncrete. Results frm the spreadsheets matched thse in the example prblems exactly. Each series f analytical calculatins were perfrmed using sectins having similar gemetric and identical material prperties. The yield strength f the reinfrcing steely, was set at 60ksi and the crushing strength f the cncrete f' C) was set at 4ksi. Ten different gemetries were analyzed in a series f fur different calculatins with each calculatin using three different values fr the fllwing variables; flange width b, web width b w, flange depth tf, and the depth f reinfrcement d. Ten arbitrary sectins with the fllwing gemetries, graphically described in Figure 4-1, were chsen fr analysis.

62 49 ik -» b» tf * i d 1 b w * -fig w Figure 4-1 Sectin gemetry f h "! The values used fr each variable's analytical case are listed belw in Table 4-1. Table 4-1 Values fr variable and fixed gemetry fr each case VARIABLE VALUES 36in. b 42in. 54in. 14in. b w 18in. 21 in. 24in. d 30in. 36in. 2in. t f 6in. 10in. FIXED GEOMETRY t f 4in. b w 14in. d 24in. b 36in. tf 4in. d 24in. b 36in. b w 14in. tf 4in. b 36in. b w 14in. d 24in. Fr each case the assumed area f prvided reinfrcing steel was varied frm zer t an arbitrary value f 20 square inches. This was accmplished in incremental steps f 0.50 square inches. Hwever nly data within and inclusive f the bundaries determined by A smi and A smax wuld be cnsidered fr final analysis.

63 50 Because each cde specifies a unique analytical prcedure, the fcal crux f this paper, separate prgramming appraches were required fr bth the ACI and the AASHTO LRFD in rder t reach slutins. Each prcedure is described in detail in the fllwing sectins Stress Blck Depth Factr fit The sle quantity that was independent f the cde prvisin used and culd be calculated simultaneusly was the stress blck depth factr, /?/. Used in cmputatin f the lcatin f the neutral axis the methd fr cmputing the value f /?/, is identical fr bth ACI and AASHTO LRFD cdes and was cmputed using a simple nested IF statement as described in Figure 4-2 belw. 5 Yes f c < 4.0fc/ P l = 0.85 N 4.0 < f\ < 8Msi Yes A=0.85-(0.05*(/' c -4)) N f\ > 8.0*5/ > Yes» p x = 0.65 Figure 4-2 IF lgic statement fr cmputatin f stress blck depth factr /?/

64 ACI Limits f Reinfrcement As defined by ACI the minimum area f reinfrcement A smin, is the maximum value f the fllwing tw equatins; bth f which are functins f sectin gemetry, the yield strength f reinfrcement and the crushing strength f cncrete. A,. = Maximum] s mm V 3V/' C 10 Ai ->y Kd (( 2 v/,10 hd j (4-2) ACI defines the maximum area f reinfrcement A smax, as a fractin f the balanced area f steel fr a sectin. A, = A.. smax. sbal Where the balanced area f reinfrcement A S M, is cmputed as shwn in Equatin 4-4. (4-3) Asbai is als a functin f sectin gemetry, yield strength f reinfrcement f y, stress blck depth factr ft, and the crushing strength f cncrete f' c. 4 =0.85 ((6-0/ AM) (4-4) i f, \ J y J

65 Lcatin f Neutral Axis c and Depth f Cmpressive Blck a The ACI utilizes a variable called the reinfrcement index zu, based n the rati f reinfrcement and cmputed as shwn in Equatin 4-5. A f f m = - ^ = p^>- (4-5) bdf F c f c crwas used t evaluate the preliminary value fr the depth f the cmpressive sectin, dented in this paper as a' fr clarity, as shwn in Equatin 4-6. a'=1.18fflr/ (4-6) This preliminary value a' was used in the calculatin f the preliminary depth f the neutral axis which has been dented in this paper fr clarity as c' shwn in Equatin 4-7. a' c'= (4-7) A Bth f these preliminary values were used nly fr the evaluatin f sectin behavir via cmparisn t the flange depth tf, i.e. the determinatin as t whether the sectin was in rectangular actin r flanged actin and therefre if the web carried any cmpressive lad. These preliminary results fr a' and c' determined the methd used t calculate the wrking value fr the depth f the cmpressive sectin a, and the wrking value fr

66 53 the depth f the neutral axis c, by means f IF lgic statements. The grayed ut cells n the left hand side f the flw chart in Figure 4-6 ffers a cmprehensive view f this prcedure. After a value fr the depth f the cmpressive sectin a, had been btained and the depth f the neutral axis c, had been established, an IF statement was used t determine whether the sectin behaved under T-sectin r rectangular actin, i.e. if the depth f the cmpressive sectin was greater than r equal t the flange depth a < t/. This in turn prvided the apprpriate frmula t slve fr the factred mment capacity fr the sectin. See Figure 4-6.

67 Strength Reductin Factr (j> The depth f the neutral axis was als used in the determinatin f the strength reductin factr <f> based n ACI reprduced belw in Figure <p = (, - 0.2) (2QQ Other <p = (,- 0.2) (22S) Cmpressin! cntrlled e, = 0.2 = 0.6 Transitin, = 0.5 i = Tensin cntrlled Figure 4-3 ACI Fig. R Strength reductin factr (Reprinted with permissin frm the American Cncrete Institute) Anther nested IF statement was used within the Excel prgram t determine this value as shwn in Figure 4-4. The mment capacity culd then be cmputed using the given values fr A, sectin gemetry, f y,f c, and the calculated values fr c, /?/, and if required the web and flange mment capacities, M nl and M n2 respectively. A sample Excel wrksheet is shwn in Figure 4-5.

68 55 Figure 4-6 shws the preliminary calculatins which were cntained in hidden cells, these cells are represented by lightly shaded clumns. This prcedure is illustrated by the flwchart in Figure 4-7. Figure 4-4 ACI calculatin f strength reductin factr ^

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