Chapter 8. Flexural Analysis of TBeams


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1 Chapter 8. Flexural Analysis of Ts 8.1. Reading Assignments Text Chapter 3.7; ACI 318, Section Occurrence and Configuration of Ts Common construction type. used in conjunction with either onway or twoway slabs. Sections consists of the flange and web or stem; the slab forms the beam flange, while the part of the beam projecting below the slab forms is what is called web or stem. (a) oneway slab (b) twoway slab 8.3. Concepts of the effective width, Code allowable values In reality the maximum compression stress in Tsection varies with distance from section Web. Real max, Longitudinal compression sress Simplified equivalent width, stress 156
2 157
3 Code allows the following maximum effective widths: Symmetrical ACI318, Section b 1) b span 4 2) b Flange on one side only (Spandrel ) ACI318, Section ) b clear distance between beams b 1) b span 12 2) b 6 3) b 1 2 clear distance to next web Isolated T ACI318, Section b 1) b 4 2) 2 158
4 8.4. Analysis of Ts  ( a > ) Consider the total section in two parts: 1) Flange overhangs and corresponding steel; 2) Stem and corresponding steel; b 0.85f c C c a T s =A s f y = + A sf Case I A s A sf Case II For equilibrium we have: Case I: A sf f y = 0.85f c (b ) (8.1) or A sf = 0.85f c (b ) f y (8.2) Case II: Solve for a : (A s A sf ) f y = 0.85f c a (8.3) a = (A s A sf ) f y 0.85f c (8.4) and nominal moment capacity will be: M n = A sf f y (d 2 ) + (A s A sf ) f y (d a 2 ) (8.5) 159
5 8.5. Balanced Condition for Ts See Commentary page 48 of ACI (old code). b Áu = c b a b 0.85f c C c h d A b s Á y dc T s =A s b f y From geometry: c b = Á u Á u + Á y d = 87, , f y d (8.6) 160
6 8.6. Example. Analysis of Ts in Bending: 40 =4 Á u = f c c C c 20.5 A s = 6.88 in 2 Á y dc T s =A s f y 10 Find the nominal moment capacity of the beam given above: f c = 2, 400 psi f y = 50, 000 psi Solution: Check to see if a Tbeam analysis is required: Assume a< a = A s f y 0.85f c b = = 4.22 in Since 4.22 in > 4.00 in, a Tbeam analysis is required. First find the reinforcement area to balance flanges (A sf =?) A sf = 0.85 f c (b b f w ) = (40 10) 4 = 4.90 in2 y 50 A s A sf = = 1.98 in 2 Solve for a 0.85f c a = (A s A sf )f y a = (A s A sf )f y = f c = 4.86 in > 4in o.k. Assumption is o.k. 161
7 c = a β 1 = = 5.72 c d = =.279 < Tensioncontrolled Find the nominal moment capacity of the beam: M n = A sf f y (d 2 ) + f y(a s A sf )(d a 2 ) M n = 4.9(in 2 ) 50(ksi) ( ) + 50(ksi) 1.98(in2) ( ) M n = = 6, 320 in k Note: This could have been done by statics with T s = A s f y C c = (b )( ) 0.85f c + a (0.85)f c 162
8 8.7. Example. Design of Ts in Bending Determination of Steel Area for a given Moment: A floor system consists of a 3 in. concrete slab supported by continuous T beams of 24 ft span, 47 in. on centers. Web dimensions, as determined by negativemoment requirements at the supports, are = 11 in. and d = 20 in. What tensile steel area is required at midspan to resist a moment of 6,400 inkips if f y = 60,000 psi and f c = 3,000 psi. b + A sf Case I A s A sf Case II Solution First determining the effective flange widtrom Section (8.3.1.) or ACI ) b span 4 = = 72 in 2) b 16 + = (16 3) + 11 = 59 in 3) b clear spacing between beams + = center to center spacing between beams = 47 in The centerline T beam spacing controls in this case, and b = 47 inches. Assumption: Assuming that stressblock depth equals to the flange thickness of 3 inches (beam behaves like a rectangular shape). A s = M u φf y (d a 2) = (20 3 2) = 6.40 in2 (8.7) 163
9 Solve for a : a = A sf y = f c b = 3.2 in > = 3.0 Assumption incorrect Therefore, the beam will act as a Tbeam and must be designed as a Tbeam. From Case I given above and Section (8.4.1.) we have A sf = 0.85f c (b ) f y = 0.85 (3ksi) (3in) (47 11) 60(ksi) = 4.58 in 2 (8.8) φm n1 = φa sf f y d ) = (60ksi) (20 3 2) = 4570 in kips 2 (8.9) φm n2 = M u φm n1 = = 1830 in kips (8.10) Find a value by iteration. Assume initial a = 3.5 inches A s A sf = Find an improve a value φm n2 φf y (d a 2) = ( ) = 1.86 in2 (8.11) a = (A s A sf ) f y = f c = 3.97 in (8.12) Iterate with the new a = 3.97 in. A s A sf = Find an improve a value φm n2 φf y (d a 2) = ( ) = 1.88 in2 (8.13) a = (A s A sf ) f y = f c = 4.02 in (8.14) A s A sf = φm n2 φf y (d a 2) = ( ) = 1.88 in2 (8.15) 164
10 Since there is no change between equations (8.13) and (8.15) we have arrived at the answer. Therefore, A s = A sf + (A s A sf ) = = 6.46 in 2 (8.16) Check with ACI requirements for maximum amount of steel (TensionControlled) c = a β 1 = = 4.73 (8.17) c d = =.237 < Tensioncontrolled Therefore, the Tbeam satisfies the ACI provisions for tension failure. Next steps will be to select the reinforcement and check all the spacing requirements and detail the beam. 165
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