CHAPTER Reinored Conrete Design Fith Edition REINFORCED CONCRETE BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS A. J. Clark Shool o Engineering Department o Civil and Environmental Engineering Part I Conrete Design and Analsis 3a FALL 2002 B Dr. Irahim. Assakka ENCE 355 - Introdution to Strutural Design Department o Civil and Environmental Engineering Universit o Marland, College Park CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 1 Introdution to T-Beams Reinored onrete strutural sstems suh as loors, roos, deks, et., are almost monolithi, exept or preast sstems. Forms are uilt or eam sides the underside o slas, and the entire onstrution is poured at one, rom the ottom o the deepest eam to the top o the sla. 1
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 2 Introdution to T-Beams Floor-Column Sstems CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 3 Introdution to T-Beams Beam and Girder Sstem This sstem is omposed o sla on supporting reinored onrete eams and girder.. The eam and girder rameork is, in turn, supported olumns. In suh a sstem, the eams and girders are plaed monolithiall ith the sla. The tpial monolithi strutural sstem is shon in Fig. 1. 2
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 4 Introdution to T-Beams Beam and Girder Floor Sstem Sla Spandrel eam Beam Girder Column Figure 1 CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 5 Introdution to T-Beams Common Beam and Girder Laout Girder Girder Beam Column Beam Column Figure 2 3
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 6 Introdution to T-Beams Positive Bending Moment In the analsis and design o loor and roo sstems, it is ommon pratie to assume that the monolithiall plaed sla and supporting eam interat as a unit in resisting the positive ending moment. As shon in Fig. 3, the sla eomes the ompression lange, hile the supporting eam eomes the e or stem. CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 7 Introdution to T-Beams T-Beam as Part o a Floor Sstem Sla Eetive Flange Width h d Flange We or Stem Supporting Beam or Sla A s Figure 3 Beam Spaing 4
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 8 Introdution to T-Beams T-Beam The interating lange and e produe the ross setion having the tpial T- shape, thus the T-Beam gets its name. Negative Bending Moment It should e noted that hen the the T- Beam is sujeted to negative moment, the sla at the top o the stem (e) ill e in tension hile the ottom o the stem is in ompression. This usuall ours at interior support o ontinuous eam. CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 9 ACI Code Provisions or T-Beams 1. The eetive lange idth must not exeed a. One-ourth the span length. + 16h. Center-to-enter spaing o the eam The smallest o the three values ill ontrol 2. For eam having a lange on one side onl, the eetive overhanging lange idth must 5
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 10 ACI Code Provisions or T-Beams Not exeed one-telth o the span length o the eam, nor six times the sla thikness, nor one-hal o the lear distane to the next eam. 3. For isolated eam in hih the T-shape is used onl or the purpose o providing additional ompressive area, the lange thikness must not e less than one-hal o the idth o the e, and the total lange idth must not e more than our times the e idth. CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 11 T-Beam Versus Retangular Beam The dutilit requirements or T-eams are similar to those or retangular eams. The maximum steel ratio ρ shall not exeed 0.75ρ. Hoever, this steel ratio is not the same value as that taulated or retangular eams eause o the T-shaped ompressive area. 6
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 12 Formulas or Balaned T-Beam These ormulas an e used to ind A s. It ill e illustrated in Example 1: a N C 87,000 d + 87,000 β 1 0.85 [ h + ( a h )] (1) See Fig. 4 or deinitions o variales CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 13 Figure 4 ε 0.85 d h a N C N.A. N T ε s 7
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 14 Minimum Steel Ratio or T-Beams The T-eam is sujeted to positive moment: The steel area shall not e less than that given 3 200 As, min d d (2) Note that the irst expression ontrols i > 4440 psi ACI Code CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 15 Minimum Steel Ratio or T-Beams The T-eam is sujeted to negative moment: The steel area A s shall equal the smallest o the olloing expression: A s, min smallest o 6 d or 3 d (3) ACI Code 8
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 16 Notes on the Analsis o T-Beams Beause o the large ompressive in the lange o the T-eam, the moment strength is usuall limited the ielding o the tensile steel. Thereore, it sae to assume that the tensile steel ill ield eore the onrete reahes its ultimate strain. The ultimate tensile ore ma e ound rom N A (4) T s CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 17 Notes on the Analsis o T-Beams In analzing a T-eam, there might exist to onditions: 1. The stress lok ma e ompletel ithin the lange. 2. The stress lok ma over the lange and extend into the e. These to onditions ill result in hat are termed: a retangular T-eam and a true T-eam, respetivel. 9
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 18 Stress Blok Completel ithin the Flange (Retangular T-Beam) h ε a 0.85 N C N.A. d N T ε s CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 19 Stress Blok Cover Flange and Extends into We (True T-Beam) ε 0.85 d h a N C N.A. N T ε s 10
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 20 Example 1 The T-eam shon in the igure is part o a loor sstem. Determine the pratial moment strength φm n i 60,000 psi (A615 grade 60) and 3,000 psi. d 12 Beams 32 in. o.. 32 1 0 h 2 3 #9 (A s 3 in 2 ) CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 21 Example 1 (ont d) Sine the span length is not given, e determine the lange idth in terms o the lange thikness and eam spaing: + 16h Thereore, 10 + 16 ( 2) 42 in. Beam spaing 32 in.o.. Use 32 in.(smallest o the to) 11
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 22 Example 1 (ont d) Find N T assuming that the steel has ielded: NT As 3 ( 60) 180 kips I the lange alone is stressed to 0.85, then the total ompressive ore ould e N T ()()( 3 2 32) 163.2 kips 0.85 h 0.85 Sine 180 > 163, the eam should e analzed as true T-eam, and the stress lok ill extend into the e (Fig. 5) CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 23 Example 1 (ont d) d 12 32 h 2 ε a 0.85 Z N C N.A. 3 #9 (A s 3 in 2 ) N T 1 0 ε s Figure 5 12
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 24 Example 1 (ont d) The remaining ompression is thereore Remaining Compression N N T a h N C NT N a 0.85 0.85 NT N 0.85 C C + h ( a h ) T 180 163.2 + 2 2.66 in. 0.85 ()( 3 10) N C CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 25 Example 1 (ont d) Chek A s, min using Eq. 3 or Tale 1 From Tale1(also Tale A -5 Text) : A s,min 0.0033 ( )( 12) 0.40 in 2 2 ( A 3.0 in ) > ( A 0.4 in ) OK s d 0.0033 10 s,min In order to ind the internal ouple, e have to ind the ouple arm Z: A A 2 13
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 26 Tale 1. Design Constants (Tale A-5 Text) 3 Reommended Design Values 200 ( psi) ρ max 0.75 ρ ρ k (ksi) F 40,000 psi 3,000 0.0050 0.0278 0.0135 0.4828 4,000 0.0050 0.0372 0.0180 0.6438 5,000 0.0053 0.0436 0.0225 0.8047 6,000 0.0058 0.0490 0.0270 0.9657 F 50,000 psi 3,000 0.0040 0.0206 0.0108 0.4828 4,000 0.0040 0.0275 0.0144 0.6438 5,000 0.0042 0.0324 0.0180 0.8047 6,000 0.0046 0.0364 0.0216 0.9657 F 60,000 psi 3,000 0.0033 0.0161 0.0090 0.4828 4,000 0.0033 0.0214 0.0120 0.6438 5,000 0.0035 0.0252 0.0150 0.8047 6,000 0.0039 0.0283 0.0180 0.9657 F 75,000 psi 3,000 0.0027 0.0116 0.0072 0.4828 4,000 0.0027 0.0155 0.0096 0.6438 5,000 0.0028 0.0182 0.0120 0.8047 6,000 0.0031 0.0206 0.0144 0.9657 CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 27 Example 1 (ont d) 12 32 A 1 A 2 2 a 2.66 0.85 Z N C 10 N T Using a reerene axis at the top: A A [ 32( 2) ]() 1 + [ 10( 0.66) ]( 2 + 0.33) 32( 2) + 10( 0.66) 1.12 in 14
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 28 Example 1 (ont d) Z an e omputed as ollos: Z d 12 1.12 10.88 in. Thereore, ( ) 180 1.88 M n NT Z 163.2 t - kips 12 Thus the paratial moment is φm n ( ) 147 t - kips 0.9 163.2 CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 29 Example 1 (ont d) 12 Alternatel, the nominal moment an e ound as ollos: 32 0.85 N C A 2 1 a 2.66 A 2 N C Z Z M n Z NC + ZNC 1 12 10 N N N T C N + N N [( 12-1163.2 ) + ( 12 2 0.33)( 16.8) ] 163.1 t - kips C T C C, or N T 180 163.2 16.8 t - Kips 15
CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 30 Example 1 (ont d) Chek assumption or dutile ailure: From Eq. 1 N a C 87,000 87 d + 87,000 60 + 87 β1 0.85 0.85 0.85 ( 12) ( 7.1) 6.035 in. [ h + ( a h )] 7.10 in. () 3 [ 32() 2 + 10( 6.035 2) ] 266.09 kips N T CHAPTER 3a. R/C BEAMS: T-BEAMS AND DOUBLY REINFORCED BEAMS Slide No. 31 Example 1 (ont d) A A s s,max 0.75A 0.75 0.33 266.09 60 ( 4.44) 4.44 in 2 2 ( A 3.0 in ) < ( A 4.44 in ) OK s N T s s,max 2 16