CHAPTER REINFORCED CONCRETE Reinored Conrete Design A Fundmentl Approh - Fith Edition Fith Edition FLEXURE IN BEAMS A. J. Clrk Shool o Engineering Deprtment o Civil nd Environmentl Engineering 5 SPRING 004 B Dr. Ibrhim. Asskk ENCE 454 Design o Conrete Strutures Deprtment o Civil nd Environmentl Engineering Universit o Mrlnd, College Prk CHAPTER 5. FLEXURE IN BEAMS Slide No. Introdution Bending moment produes bending strins on bem, nd onsequentl ompressive nd tensile stresses. Under positive moment (s normll the se), ompressive stresses re produed in the top o the bem nd tensile stresses re produed in the om. Bending members must resist both ompressive nd tensile stresses.
CHAPTER 5. FLEXURE IN BEAMS Slide No. Introdution (ont d) Stresses in Bem P b w x Figure b h P R x x O V M R + τ da σ CHAPTER 5. FLEXURE IN BEAMS Slide No. Introdution (ont d) Sign Convention Figure V V M M L.H.F R.H.F V V () Positive Sher & Moment (b) Positive Sher (lokwise) M () Positive Moment (onve upwrd) M
CHAPTER 5. FLEXURE IN BEAMS Slide No. 4 Bems: Mehnis o Bending Review Introdution The most ommon tpe o struturl member is bem. In tul strutures bems n be ound in n ininite vriet o Sizes Shpes, nd Orienttions CHAPTER 5. FLEXURE IN BEAMS Slide No. 5 Bems: Mehnis o Bending Review Introdution Deinition A bem m be deined s member whose length is reltivel lrge in omprison with its thikness nd depth, nd whih is loded with trnsverse lods tht produe signiint bending eets s oppose to twisting or xil eets
CHAPTER 5. FLEXURE IN BEAMS Slide No. 6 Bems: Mehnis o Bending Review Pure Bending: Prismti members subjeted to equl nd opposite ouples ting in the sme longitudinl plne CHAPTER 5. FLEXURE IN BEAMS Slide No. 7 Bems: Mehnis o Bending Review Flexurl Norml Stress For lexurl loding nd linerl elsti tion, the neutrl xis psses through the entroid o the ross setion o the bem.
CHAPTER 5. FLEXURE IN BEAMS Slide No. 8 Bems: Mehnis o Bending Review The elsti lexurl ormul or norml stress is given b M b () I where b lulted bending stress t outer iber o the ross setion M the pplied moment distne rom the neutrl xis to the outside tension or ompression iber o the bem I moment o inerti o the ross setion bout neutrl xis CHAPTER 5. FLEXURE IN BEAMS Slide No. 9 Bems: Mehnis o Bending Review B rerrnging the lexure ormul, the mximum moment tht m be pplied to the bem ross setion, lled the resisting moment, M R, is given b M Fb I R () Where F b the llowble bending stress
CHAPTER 5. FLEXURE IN BEAMS Slide No. 0 Bems: Mehnis o Bending Review Exmple Determine the mximum lexurl stress produed b resisting moment M R o +5000 t-lb i the bem hs the ross setion shown in the igure. 6 6 CHAPTER 5. FLEXURE IN BEAMS Slide No. Bems: Mehnis o Bending Review Exmple (ont d) x First, we need to lote the neutrl xis rom the om edge: C 6 5 C ten ( )( 6) + ( + )( 6) 6 om Mx. Stress + 6 6 + M I b 7 4 5 mx
CHAPTER 5. FLEXURE IN BEAMS Slide No. Bems: Mehnis o Bending Review x Exmple (ont d) C 6 Find the moment o inerti I with respet to the x xis using prllel xis-theorem: 5 ( ) ( 6) 6 I + ( 6 )( ) + + 4 4 + 48 + 6 + 48 6 in ( 6)( ) ( 5) (5 ) Mx. Stress (om). ksi 6 CHAPTER 5. FLEXURE IN BEAMS Slide No. Bems: Mehnis o Bending Review Internl Couple Method The proedure o the lexure ormul is es nd strightorwrd or bem o known ross setion or whih the moment o inerti I n be ound. However, or reinored onrete bem, the use o the lexure ormul n be somewht omplited. The bem in this se is not homogeneous nd onrete does not behve elstill.
CHAPTER 5. FLEXURE IN BEAMS Slide No. 4 Bems: Mehnis o Bending Review Internl Couple Method (ont d) In this method, the ouple represents n internl resisting moment nd is omposed o ompressive ore C nd prllel internl tensile ore T s shown in Fig.. These two prllel ores C nd T re seprted b distne Z, lled the moment rm. (Fig. ) Beuse tht ll ores re in equilibrium, thereore, C must equl T. CHAPTER 5. FLEXURE IN BEAMS Slide No. 5 Bems: Mehnis o Bending Review Internl Couple Method (ont d) w P Neutrl xis Centroidl xis Z C x T da C d R Figure
CHAPTER 5. FLEXURE IN BEAMS Slide No. 6 Bems: Mehnis o Bending Review Internl Couple Method (ont d) The internl ouple method o determining bem stresses is more generl thn the lexure ormul beuse it n be pplied to homogeneous or non-homogeneous bems hving liner or nonliner stress distributions. For reinored onrete bem, it hs the dvntge o using the bsi resistne pttern tht is ound in bem. CHAPTER 5. FLEXURE IN BEAMS Slide No. 7 Bems: Mehnis o Bending Review Exmple Repet Exmple using the internl ouple method. x C 6 5 N.A Z C T
CHAPTER 5. FLEXURE IN BEAMS Slide No. 8 Bems: Mehnis o Bending Review Exmple (ont d) Beuse o the irregulr re or the tension zone, the tensile ore T will be broken up into omponents T, T, nd T. Likewise, the moment rm distne Z will be broken up into omponents Z, Z, nd Z, nd lulted or eh omponent tensile ore to the ompressive ore C s shown in Fig.. CHAPTER 5. FLEXURE IN BEAMS Slide No. 9 Bems: Mehnis o Bending Review Exmple (ont d) top C C 5 Z Z Z 6 Figure T mid T T
CHAPTER 5. FLEXURE IN BEAMS Slide No. 0 Exmple (ont d) C T vg T vg T vg vg C 6 re re re mid re top mid [()() 5 ] [()( ) ] [( )( 6) ] mid 5 top mid mid 5 4 [( )( 6) ] 6 6 mid top Z Z Z T mid C T T From similr tringles: mid mid CHAPTER 5. FLEXURE IN BEAMS Slide No. Exmple (ont d) C 6 5 top Z Z Z T mid C T T C T T + T + T 5 5 top top + 4 + 4 + 6 + 6 6 mid 5 5 top
CHAPTER 5. FLEXURE IN BEAMS Slide No. Exmple (ont d) Z Z Z C 6 () 5 + () () 5 4 in. 6 + in. 7 () 5 + + ( ) in. 5 top Z Z Z T mid C T T M ext M R 5000( ) ZT + ZT + ZT 60,000 Z T + Z T + Z T CHAPTER 5. FLEXURE IN BEAMS Slide No. Exmple (ont d) C 6 5 60,000 4 Thereore, 5 6 + 5 7 ( 4 ) + ( 4 ),.5 psi (Tension) The mximum Stress is ompressive stress : mx top top Z Z Z T mid 6 C T T (,.5),05.88 psi. ksi (Com)
CHAPTER 5. FLEXURE IN BEAMS Slide No. 4 Methods o Anlsis nd Design Elsti Design Elsti design is onsidered vlid or the homogeneous plin onrete bem s long s the tensile stress does not exeed the modulus o rupture r. Elsti design n lso be pplied to reinored onrete bem using the working stress design (WSD) or llowble stress design (ASD) pproh. CHAPTER 5. FLEXURE IN BEAMS Slide No. 5 Methods o Anlsis nd Design WSD or ASD Assumptions A plin setion beore bending remins plne ter bending. Stress is proportionl to strin (Hooke s Lw). Tensile stress or onrete is onsidered zero nd reinoring steel rries ll the tension. The bond between the onrete nd steel is peret, so no slip ours.
CHAPTER 5. FLEXURE IN BEAMS Slide No. 6 Methods o Anlsis nd Design Strength Design Method This method is the modern pproh or the nlsis nd design o reinored onrete. The ssumption re similr to those outlined or the WSD or ASD with one exeption: Compressive onrete stress is pproximtel proportionl to strin up to moderte lods. As the lod inreses, the pproximte proportionlit eses to exit, nd the stress digrm tkes shpe similr to the onrete stress-strin urve o the ollowing igure. CHAPTER 5. FLEXURE IN BEAMS Slide No. 7 Methods o Anlsis nd Design Conrete Compressive Strength Figure
CHAPTER 5. FLEXURE IN BEAMS Slide No. 8 Methods o Anlsis nd Design Comprison between the Two Methods WSD or ASD Working (servie) lods re used nd member is designed bsed on n llowble ompressive bending stress, normll 0.45 Compressive stress pttern is ssumed to vr linerl rom zero t the neutrl xis. Formul: m R n L i FS i ASD USD Servie lods re mpliied using prtil set tors. A member is design so tht its strength is redued b redution set tor. The strength t ilure is ommonl lled the ultimte strength m Formul: φr γ L n i i LRFD i CHAPTER 5. FLEXURE IN BEAMS Slide No. 9 Behvior Under Lod () At ver smll lods: Stresses Elsti nd Setion Unrked Reinored Conrete Bem b ε ( omp. ) ( omp. ) h d N.A. ε s ( tens. ) s ( tens. ) ε ( tens. ) Stresses re below modulus o rupture. ( tens. )
CHAPTER 5. FLEXURE IN BEAMS Slide No. 0 Behvior Under Lod () At moderte lods: Stresses Elsti nd Setion Crked Reinored Conrete Bem b ε ( omp. ) ( omp. ) h d N.A. ε s ( tens. ) s ( tens. ) Tensile stresses o onrete will be exeeded. Conrete will rk (hirline rk), nd steel brs will resist tensile stresses. This will our t pproximtel 0.5. CHAPTER 5. FLEXURE IN BEAMS Slide No. Behvior Under Lod Reinored Conrete Bem Formul The neutrl xis or onrete bem is ound b solving the qudrti eqution: b bx + na x na d b 0 s s () d d - x x C x E n E s n A s
CHAPTER 5. FLEXURE IN BEAMS Slide No. Behvior Under Lod () With urther lod inrese: Flexurl Strength ACI Approh Reinored Conrete Bem b ε ( omp. ) ( omp. ) h d N.A. ε s ( tens. ) s ( tens. ) Stress urve bove N.A. will be similr to the stress-strin urve o Fig.. Conrete hs rked, nd the proess is irreversible. Steel br hs ielded nd will not return to its originl length. CHAPTER 5. FLEXURE IN BEAMS Slide No. Strength Design Method Assumptions Bsi Assumption: A plne setion beore bending remins plne ter bending. Stresses nd strin re pproximtel proportionl up to moderte lods (onrete stress 0.5 ). When the lod is inresed, the vrition in the onrete stress is no longer liner. Tensile strength o onrete is negleted in the design o reinored onrete bems.
CHAPTER 5. FLEXURE IN BEAMS Slide No. 4 Strength Design Method Assumptions Bsi Assumption (ont d): 4. The mximum usble onrete ompressive strin t the extreme iber is ssumed equl to 0.00 (Fig. 4) 5. The steel is ssumed to be uniorml strined to the strin tht exists t the level o the entroid o the steel. Also i the strin in the steel ε s is less thn the ield strin o the steel ε, the stress in the steel is E s ε s. I ε s ε, the stress in steel will be equl to (Fig. 5) CHAPTER 5. FLEXURE IN BEAMS Slide No. 5 Strength Design Method Assumptions Bsi Assumption (ont d): 6. The bond between the steel nd onrete is peret nd no lip ours. Figure 4 0.00 Elsti region Figure 5 Stress ε Strin ε Strin Idelized Stress-Strin Curve
CHAPTER 5. FLEXURE IN BEAMS Slide No. 6 Flexurl Strength o Retngulr Bems Ultimte Moment Strength Cpit The ultimte moment or reinored onrete bem n be deined s the moment tht exists just prior to the ilure o the bem. In order to evlute this moment, we hve to exmine the strins, stresses, nd ores tht exist in the bem. The bem o Fig. 6 hs width o b, n eetive depth d, nd is reinored with steel re A s. CHAPTER 5. FLEXURE IN BEAMS Slide No. 7 Flexurl Strength o Retngulr Bems Ultimte Strength Flexurl Strength ACI Approh h b Figure 6 d ε ( 0.00 s limit) N.A. ε s ε Reinored Conrete Bem s s limit Z Strin Stress Fore C T
CHAPTER 5. FLEXURE IN BEAMS Slide No. 8 Flexurl Strength o Retngulr Bems Possible Vlues or Conrete Strins due to Loding (Modes o Filure). Conrete ompressive strin is less thn 0.00 in./in. when the mximum tensile steel unit equl its ield stress s limit.. Mximum ompressive onrete strin equls 0.00 in./in. nd the tensile steel unit stress is less thn its ield stress. CHAPTER 5. FLEXURE IN BEAMS Slide No. 9 Flexurl Strength o Retngulr Bems Nominl Moment Strength The ores C nd T, nd the distne Z seprted them onstitute n internl resisting ouple whose mximum vlue is termed nominl moment strength o the bending member. As limit, this nominl strength must be pble o resisting the tul design bending moment indued b the pplied lods.
CHAPTER 5. FLEXURE IN BEAMS Slide No. 40 Flexurl Strength o Retngulr Bems Nominl Moment Strength (ont d) The determintion o the moment strength is omplex beuse o The shpe o the ompressive stress digrm bove the neutrl xis Not onl is C diiult to evlute but lso its lotion reltive to the tensile steel is diiult to estblish s s limit Stress CHAPTER 5. FLEXURE IN BEAMS Slide No. 4 Flexurl Strength o Retngulr Bems How to Determine the Moment Strength o Reinored Conrete Bem? To determine the moment pit, it is neessr onl to know. The totl resultnt ompressive ore C in the onrete, nd. Its lotion rom the outer ompressive iber, rom whih the distne Z m be estblished.
CHAPTER 5. FLEXURE IN BEAMS Slide No. 4 Flexurl Strength o Retngulr Bems How to Determine the Moment Strength o Reinored Conrete Bem? (ont d) These two vlues m esil be estblished b repling the unknown omplex ompressive stress distribution b ititious (equivlent) one o simple geometril shpe (e.g., retngle). Provided tht the ititious distribution results in the sme totl C pplied t the sme lotion s in the tul distribution when it is t the point o ilure. CHAPTER 5. FLEXURE IN BEAMS Slide No. 4 The Equivlent Retngulr Blok An omplited untion n be repled with n equivlent or ititious one to mke the lultions simple nd will give the sme results. For purposes o simpliition nd prtil pplition, ititious but equivlent retngulr onrete stress distribution ws proposed.
CHAPTER 5. FLEXURE IN BEAMS Slide No. 44 The Equivlent Retngulr Blok This retngulr stress distribution ws proposed b Whine (94) nd subsequentl dopted b the ACI Code The ACI ode lso stipultes tht other ompressive stress distribution shpes m be used provided tht the re in greement with test results. Beuse o its simpliit, however, the retngulr shpe hs beome the more widel stress distribution (Fig. 7). CHAPTER 5. FLEXURE IN BEAMS Slide No. 45 The Equivlent Retngulr Blok Whitne s Retngulr Stress Distribution d s N.A Atul Compressive Stress Blok 0.85 s β Retngulr Equivlent Compressive Stress Blok Figure 7 C 0. 85 b Z jd d Internl Couple T A s s
CHAPTER 5. FLEXURE IN BEAMS Slide No. 46 The Equivlent Retngulr Blok Whitne s Retngulr Stress Distribution Aording to Fig. 7, the verge stress distribution is tken s Averge Stress 0.85 () It is ssumed to t over the upper re on the bem ross setion deined b the width b nd depth s shown in Fig. 8. CHAPTER 5. FLEXURE IN BEAMS Slide No. 47 The Equivlent Retngulr Blok Whitne s Retngulr Stress Distribution 0.85 b C 0. 85 b N.A. jd d () Figure 8 A s (b) T A s s
CHAPTER 5. FLEXURE IN BEAMS Slide No. 48 The Equivlent Retngulr Blok Whitne s Retngulr Stress Distribution The mgnitude o m determined b β Where distne rom the outer iber to the neutrl xis β tor dependent on onrete strength, nd is given b 0.85 β 05. 5 0 0.65-5 or 4,000 psi or 4,000 psi < 8,000 psi or > 8,000 psi () (4) CHAPTER 5. FLEXURE IN BEAMS Slide No. 49 The Equivlent Retngulr Blok Whitne s Retngulr Stress Distribution Using ll preeding ssumptions, the stress distribution digrm shown in Fig. 8 n be redrwn in Fig. 8b. Thereore, the ompressive ore C n be written s 0.85 b Tht is, the volume o the ompressive blok t or ner the ultimte when the tension steel hs ielded ε s > ε.
CHAPTER 5. FLEXURE IN BEAMS Slide No. 50 The Equivlent Retngulr Blok Whitne s Retngulr Stress Distribution The tensile ore T n be written s A s. Thus equilibrium suggests C T, or or 0.85 b A As 0. 85 b s (5) (6) CHAPTER 5. FLEXURE IN BEAMS Slide No. 5 The Equivlent Retngulr Blok Whitne s Retngulr Stress Distribution The moment o resistne o the setion, tht is, the nominl strength M n n be expressed s M A jd or M 0. 85 b ( ) ( ) jd n s n Using Whitne s retngulr blok, the lever rm is jd d (8) Hene, the nominl resisting moment beomes M n A s d or M n 0.85 b d (7) (9)
CHAPTER 5. FLEXURE IN BEAMS Slide No. 5 The Equivlent Retngulr Blok Whitne s Retngulr Stress Distribution I the reinorement rtio ρ A s /bd, Eq. 6 n be rewritten s ρd 0. 85 I r b/d, Eq. 9 beomes (0) M n ρd ρrd d ().7 CHAPTER 5. FLEXURE IN BEAMS Slide No. 5 The Equivlent Retngulr Blok Whitne s Retngulr Stress Distribution or M n [ ωr ( 0.59ω ) d ] () where ω ρ / `. Eq. sometimes expressed s M n Rbd () where ( 0. ) R ω 59ω (4)
CHAPTER 5. FLEXURE IN BEAMS Slide No. 54 Blned, Overreinored, nd Underreinored Bems Strin Distribution Figure 9 0.00 Elsti region Figure 0 Stress ε Strin ε Strin Idelized Stress-Strin Curve CHAPTER 5. FLEXURE IN BEAMS Slide No. 55 Blned, Overreinored, nd Underreinored Bems Strin Distribution 0.00 ε E Underreinored N.A. Blned N.A. Overreinored N.A. ε
CHAPTER 5. FLEXURE IN BEAMS Slide No. 56 Blned, Overreinored, nd Underreinored Bems Blned Condition: ε s ε nd ε 0.00 Overreinored Bem ε s < ε, nd ε 0.00. The bem will hve more steel thn required to rete the blned ondition. This is not preerble sine will use the onrete to rush suddenl beore tht steel rehes its ield point. Underreinored Bem ε s > ε, nd ε 0.00. The bem will hve less steel thn required to rete the blned ondition. This is preerble nd is ensured b the ACI Speiitions. CHAPTER 5. FLEXURE IN BEAMS Slide No. 57 Blned, Overreinored, nd Underreinored Bems Exmple Determine the nominl moment M n or bem o ross setion shown, where 4,000 psi. Assume A65 grde 60 steel tht hs ield strength o 60 ksi nd modulus o elstiit 9 0 6 psi. Is the bem underreinored, over-reinored, or blned? 5 in. 0 in. N.A. in.
CHAPTER 5. FLEXURE IN BEAMS Slide No. 58 Blned, Overreinored, nd Underreinored Bems Exmple (ont d) 0 #8 brs N.A ε s ε 0.85 jd d C 0. 85 b T A s CHAPTER 5. FLEXURE IN BEAMS Slide No. 59 Blned, Overreinored, nd Underreinored Bems b Exmple (ont d) Are or No.8 br 0.79 in ( 0.79).7 in Thereore, A s (see Tble) N.A. (Also see Tble 4.() Text) Assume tht or steel exists subjet to lter hek. A s 0.85 C 0. 85 b jd d T A s s C T 0.85 b A As.7 0.85 b 0.85 s ( 60) ( 4)( 0) 4.8 in. 5 in. 0 in. N.A. in.
CHAPTER 5. FLEXURE IN BEAMS Slide No. 60 Blned, Overreinored, nd Underreinored Bems Tble. ASTM Stndrd - English Reinoring Brs Br Designtion Dimeter in Are in Weight lb/t # [#0] 0.75 0. 0.76 #4 [#] 0.500 0.0 0.668 #5 [#6] 0.65 0..04 #6 [#9] 0.750 0.44.50 #7 [#] 0.875 0.60.044 #8 [#5].000 0.79.670 #9 [#9].8.00.400 #0 [#].70.7 4.0 # [#6].40.56 5. #4 [#4].69.5 7.650 #8 [#57].57 4.00.60 Note: Metri designtions re in brkets CHAPTER 5. FLEXURE IN BEAMS Slide No. 6 Blned, Overreinored, nd Underreinored Bems Exmple (ont d) Clultion o M n M n C d T d M 0.85 n b d As Bsed on steel : M n.7 ( 60),97.4 4.8,97.4 in.- kips 47.8 t - kips d 5 in. 0 in. N.A. 0.85 b N.A. A s in. C 0. 85 b jd d T A s s
CHAPTER 5. FLEXURE IN BEAMS Slide No. 6 Blned, Overreinored, nd Underreinored Bems Exmple (ont d) Chek i the steel rehes its ield point beore the onrete rehes its ultimte strin o 0.00: Reerring to the next igure (Fig. ), the neutrl xis n be loted s ollows: Using Eqs. nd 4 : β 0.85 β Thereore, 4.8 4.9 in. β 0.85 CHAPTER 5. FLEXURE IN BEAMS Slide No. 6 Blned, Overreinored, nd Underreinored Bems Exmple (ont d) 0 #8 brs N.A ε s 0.00 d 0.85 Figure Z d C 0. 85 b T A s
CHAPTER 5. FLEXURE IN BEAMS Slide No. 64 Blned, Overreinored, nd Underreinored Bems Exmple (ont d) B similr tringles in the strin digrm, the strin in steel when the onrete strin is 0.00 n be ound s ollows: 0.00 0.00 ε s d d 4.9 ε s 0.00 0.00 0.0in./in. 4.9 The strin t whih the steel ields is ε E s 60,000 9 0 6 0.0007 in./in. Sine ε s ( 0.0) > ε ( 0.0007), the bem is under-reinored ε s d