A typical reinforced concrete floor system is shown in the sketches below.

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1 CE 433, Fall 2006 Flexure Anali for T- 1 / 7 Cat-in-place reinforced concrete tructure have monolithic lab to beam and beam to column connection. Monolithic come from the Greek word mono (one) and litho (tone). Two conequence of monolithic contruction are "T-haped" beam and reinforcing teel in the compreion zone (compreion reinforcement). A tpical reinforced concrete floor tem i hown in the ketche below. Exterior Span Interior Span Exterior Span A Span lab pan A L beam L beam L beam Plan View of Floor Stem L beam L beam L beam +'ve M teel -'ve M teel M Elevation View & Moment Diagram

2 CE 433, Fall 2006 Flexure Anali for T- 2 / 7 beam pacing /2 beam pacing /2 t lab max flange width h b w beam beam pacing Section A-A: Slab Elevation and X-Section We need four quantitie to decribe the dimenion of a T-beam: width of web b w flange width b f (derived from beam pacing, beam pan length, lab thickne) lab thickne t beam depth h The flange width (b f ) i not alwa equal to the beam pacing becaue the compreive tree in the flange are not evenl ditributed. Due to a phenomenon called "hear lag", the compreive tre in the flange decreae toward the flange end (ee figure below). For deign, the total compreive force in the flange i aumed to be ditributed uniforml over a flange width called the "effective" flange width. We will ue the mbol b f for effective flange width in deign; (actual flange width i not ued for deign). beam pacing beam pacing Total Compreive Force C Iometric view Actual Compreive Stre Ditribution Top view effective flange width b f effective flange width b f Total Compreive Force C Iometric view Top view Equivalent Compreive Stre Ditribution

3 CE 433, Fall 2006 Flexure Anali for T- 3 / 7 The compreion zone in a T-beam i divided into two region: the web, which we will analze jut a we did for a rectangular beam, and the flange overhang, of width b f b w. Important: The depth of the tre block in the flange overhang cannot be > the lab thickne, t. Or: a flange min(a web, t) b f ε f'c t a web C f t C w d b w ε f T Example. Analze two ection of a continuou beam: 1) at midpan of an interior pan, and 2) at the upport of an interior pan pan 27 ft w SD 10 pf w LL 40 pf f'c 5000 pi f 60,000 pi midpan reinforcement: 4 #6 bar in 2 laer at bottom A 1.76 in 2 Reinforcement at upport: 2 #6 bar in bottom #5 bar in top in one laer t_tab 5 in beam_pacing 12 ft h 16 in b_w in Midpan of interior pan: T-beam from preadheet: t 0.3 in Calc. M u w w unit_wt [ (b_w)(h) + (beam_pacing b_w)(t_lab)] w w (0.150kcf) [ (")(16") + (12' x 12"/' ")(5") ](1ft 2 /144in 2 ) 0.42 klf w SD (w SD )(beam pacing) (0.010kf)(12') klf w L (w L )(beam pacing) (0.040kf)(12') 0.40 klf

4 CE 433, Fall 2006 Flexure Anali for T- 4 / 7 w u 1.2 (w w + w SD ) w L w u 1.2 (0.42 klf klf) (0.40 klf) klf M u w u L 2 / 16 (ACI moment coefficient for +'ve M in interior pan, ACI.3) (1.922 klf) (27 ft) 2 / 16 M u 7.6 k-ft Calc. effective flange width, b_f: (ACI.10.2) b_f minimum of: pan / 4 16 t_lab + b_w beam pacing 27' (12"/') / 4 1" 16(5") + " " 12' x 12'/" 144" b_f 1" Calc. tre reultant a β 1 t 0.0 (0.3 in) in < 5 in t (therefore compreion block doe not extend below flange) C_w 0.5 f'c a b_w 0.5 (5000pi)(0.307")(") 10.4 k C_f 0.5 f'c a (b_f b_w) 0.5 (5000pi)(0.307")(1" ") 95.2 k T A f (1.76 in 2 )(60 ki) k (Aume teel ield) ΣF H : C T?, 10.4 k k k, OK Calc. train in bottom laer of teel (needed to check aumption that teel ield and to calculate φ) ε t dbottom laer 3" dbottom laer 16" [1.5" "] " " ε, 13.75" ε teel ha eilded ( ε > ε & φ 0.90 ( ε > 0.005) 0.002) Calc. φ M n Sum moment about neutral axi φ M n φ [C_w (_C_w) + C_f (_C_f) + T (_T_)] _C_w _t a/2 0.3" 0.307" / "

5 CE 433, Fall 2006 Flexure Anali for T- 5 / 7 _C_f _t a/2 0.3" 0.307" / " _T_ d _t 15.0" 6.02".9" 3" cover tirrup diam lowet laer where d 16 " [1.5" + + " + 1"] 12.", ue d 12.75"(round to nearet 1/4" ) φ M n 0.90 [ 10.4 k (0.23") k (0.23") k (12.37") ](1' / 12"), φ M n 99. k-ft > 7.6 k-ft M u, OK min. pace between laer (ACI 7.6.1) At Interior Side of Interior Support: Compreion Steel from preadheet: t 4.17" φ M n 99. k-ft The bottom of the web i in compreion, everthing above the neutral axi ( t from extreme compreion fiber) i cracked. The top teel i pread laterall acro the flange (therefore it i alwa in one laer). b f t ε T d b w d' ε '.003 t 0.5f'c a f C c C Calc. M u M u w u L 2 / 11 (ACI moment coefficient for -'ve M in interior pan, ACI.3) (1.922 klf) (27 ft) 2 / 11 M u 127 k-ft Calc. tre reultant a β 1 t 0.0 (4.17 in) 3.34 in C_c 0.5 f'c a b_w 0.5 (5000pi)(3.34")(") 114 k (compreive force in concrete)

6 CE 433, Fall 2006 Flexure Anali for T- 6 / 7 Strain in compreion teel ε'.003 ε d' ' " ε, 4.17" 2.25" ' ε where d' cover + diam_tirrup + ' diam_bar " " 2 (note: round d up to the nearet ¼ if necear) f' ε' x 29,000 ki x 29,000 ki 40.0 ki (< f 60 ki, o ue f' 40 ki) C_ A' x f' (0. in 2 ) (40 ki) 35.3 k T A f (2.4 in 2 )(60 ki) 14. k (Aume teel ield) ΣF H : C T?, C_c + C_ T?, 114 k + 35 k 149 k, OK Calc. train in tenion teel (needed to check aumption that teel ield and to calculate φ) ε t d 3" d 16" [1.5" "] ", ue d 13.75" " ε, 13.75" ε teel ha eilded ( ε > ε 0.002) & φ 0.90 ( ε > 0.005) Calc. φ M n Sum moment about neutral axi φ M n φ [C_c (_C_c) + C_ (_C_) + T (_T_)] _C_c _t a/2 4.17" 3.34" / " _C t d' 4.17" 2.25" 1.92" _T_ d _t 13.75" 4.17" 9.5" φ M n 0.90 [ 114 k (2.50") + 35 k (1.92") k (9.5") ](1' / 12"), φ M n 133 k-ft > 127 k-ft M u, OK φ M n 133 k-ft

7 CE 433, Fall 2006 Flexure Anali for T- 7 / 7

A typical reinforced concrete floor system is shown in the sketches below. Exterior Span Interior Span Exterior Span. Beam Span.

A typical reinforced concrete floor system is shown in the sketches below. Exterior Span Interior Span Exterior Span. Beam Span. CE 331, Fall 009 Analyi of Reforce Concrete 1 / 6 Typical Reforce Concrete Builg Cat place reforce concrete tructure have monolithic lab to beam an beam to column connection. Monolithic come from the Greek