Question

a. The beam of the cross section shown has f'c = 3000 psi and fy = 60,000 psi. Calculate Mn. b. Same cross section as part (a) but the steel is changed to four No. 10 bars. Calculate Mn. Calculate the percent increase in Mn and As. c. Same cross section as part (a) but the depth d is increased to 28 in. Calculate Mn. Calculate the percent increase in Mn and d. d. Same cross section as part (a) but fc is increased to 4000 psi. Calculate Mn. Calculate the percent increase in Mn and f'c. e. For cases (a)-(d) verify that the tension steel yields. 16" 4-#9

Answer 1

b = GIVEN DATA Breadth of beam, 16 in Effective depth of beam, 24 in Reinforcement 4 No.9 (single layer) Strength of concrete, fa': 3000 psi Yield Strength of steel, f, = 60000 psi CALCULATIONS STEP-1: Computation of a, the depth of equivalent rectangular compressive stress block Area of steel, A = 3.98 in' Assuming that the stress in steel, f, -fy the depth a of the equivalent stress blcok is obtained as: 0.85f 3.98 x 60000 0.85 x 3000 x 16 5.85 in STEP-2: Check whether f = f, a = 0.244 Note: B = 0.85 i) Forf,' <= 4000 psi, B -0.85 ii) For 4000<fc' < 8000 psi, Bi = 1.05 -0.05%/1000 iii) Forf.' >= 8000 psi, p=0.65 ay Bl_87,000 87,000+ fi = 0.503 > 0.244 (= a/d] Therefore, fs = fy Check if the section is tension-controlled or not Note: i) For single layer of reinforcement, d = d ii) For double layer, d = d-clear cover-stirrup dia - half of bar dia Since the tension reinforcement is in single layer di = 24 in ald, = 0.244 al/d, = 0.375 B = 0.319 > 0.244 [= a/d] Therefore, the section is tension-controlled and = 0.90 Note: i) = 0.9 for tension-controlled sections ii) = 0.7 for compression-controlled sections iii) = 0.356 + 0.204/(a / d) for transition sections STEP-3: Computation of nominal moment capcaity of beam, M. Lever arm, jd = (d-a/2) = 21.075 in Nominal moment capacity, M = 41, (d-a/2) = 3.98 x 60000 X 21.075 = 5032710.0 in.lb = 5032710.0 / 12000 419.4 ft-kips

GIVEN DATA Breadth of beam, b = 16 in Effective depth of beam, d = 24 in Reinforcement 4 No. 10 (single layer) Strength of concrete, fc = 3000 psi Yield Strength of steel, fy = 60000 psi CALCULATIONS STEP-1: Computation of a the depth of equivalent rectangular compressive stress block Area of steel, A, = 4.91 in Assuming that the stress in steel, fs = f, the depth a of the equivalent stress blcok is obtained as: 0.855.6 = 4.91 x 60000 0.85 x 3000 x 16 = 7.22 in STEP-2: Check whether f = f 4 = 0.301 Note: B = 0.85 i) For fe' <= 4000 psi, Bi=0.85 ii) For 4000<fc' < 8000 psi, A = 1.05 -0.05f/1000 iii) For f' >= 8000 psi, p=0.65 87,000 87,000+ = 0.503 > 0.301 (= a/d] Therefore, fs = fy Check if the section is tension-controlled or not Note: i) For single layer of reinforcement, d. - d ii) For double layer, d, - d-clear cover-stirrup dia - half of bar dia Since the tension reinforcement is in single layer d. = 24 in a/d, = 0.301 ad/d, = 0.375 B. = 0.319 > 0.301 (= a/d] Therefore, the section is tension-controlled and = 0.90 Note: i) = 0.9 for tension-controlled sections ii) = 0.7 for compression-controlled sections iii) = 0.356 + 0.204/(a /(B d) for transition sections STEP-3: Computation of nominal moment capcaity of beam, M. Lever arm, jd = (d-a/2) = 20.390 in Nominal moment capacity, M = 4.8 (d-a/2) = 4.91 x 60000 X 20.390 = 6006894.0 in.lb = 6006894.0 / 12000 500.6 ft-kips

GIVEN DATA Breadth of beam, b = 16 in Effective depth of beam, d = 28 in Reinforcement 4 No.9 (single layer) Strength of concrete, fc = 3000 psi Yield Strength of steel, fy = 60000 psi CALCULATIONS STEP-1: Computation of a the depth of equivalent rectangular compressive stress block Area of steel, A, = 3.98 in Assuming that the stress in steel, fs = f, the depth a of the equivalent stress blcok is obtained as: = 0.8556 3.98 x 0.85 x 5.85 in 60000 3000 x 16 = STEP-2: Check whether f = f a = 0.209 B = 0.85 Note: i) For fe' <= 4000 psi, Bi=0.85 ii) For 4000<fc' < 8000 psi, Pi = 1.05 -0.05f/1000 iii) For f' >= 8000 psi, P=0.65 a = 87,000) 87,000+ S) = 0.503 > 0.209 = a/d] Therefore, fs = fy Check if the section is tension-controlled or not Note: i) For single layer of reinforcement, d. - d ii) For double layer, d, - d-clear cover-stirrup dia - half of bar dia Since the tension reinforcement is in single layer d = 28 in a/d, = 0.209 ad/d, = 0.375 B. = 0.319 > 0.209 [= a/d] Therefore, the section is tension-controlled and = 0.90 Note: i) 6 = 0.9 for tension-controlled sections ii) = 0.7 for compression-controlled sections iii) = 0.356 + 0.204/(a /(B d) for transition sections STEP-3: Computation of nominal moment capcaity of beam, M. Lever arm, jd = (d-a/2) = 25.075 in Nominal moment capacity, M = 4.1 (d-a/2) = 3.98 x 60000 X 25.075 = = 5987910.0 in.lb 5987910.0 / 12000 499.0 ft-kips

GIVEN DATA Breadth of beam, b = 16 in Effective depth of beam, d = 24 in Reinforcement 4 No.9 (single layer) Strength of concrete, fc = 4000 psi Yield Strength of steel, fy = 60000 psi CALCULATIONS STEP-1: Computation of a the depth of equivalent rectangular compressive stress block Area of steel, A, = 3.98 in Assuming that the stress in steel, fs = f, the depth a of the equivalent stress blcok is obtained as: 0.85fb = 3.98 x 60000 0.85 x 4000 x 16 = 4.39 in STEP-2: Check whether f = f a = 0.183 Note: B = 0.85 i) For fe' <= 4000 psi, Bi=0.85 ii) For 4000<fc' < 8000 psi, B = 1.05-0.05f/1000 iii) Forf' >= 8000 psi, P1=0.65 87,000 = B, 87,000+f,) = 0.503 > 0.183 = a/d] Therefore, fs = fy Check if the section is tension-controlled or not Note: i) For single layer of reinforcement, d = 0 ii) For double layer, d, - d-clear cover-stirrup dia - half of bar dia Since the tension reinforcement is in single layer d = 24 in ald, = 0.183 ad/d, = 0.375 p. = 0.319 > 0.183 (= a/d ] Therefore, the section is tension-controlled and = 0.90 Note: i) 6 = 0.9 for tension-controlled sections ii) = 0.7 for compression-controlled sections iii) = 0.356 + 0.204/(a /(B d) for transition sections STEP-3: Computation of nominal moment capcaity of beam, M. Lever arm, jd = (d-a/2) = 21.805 in Nominal moment capacity, M = 4.1, (d-a/2) = 3.98 x 60000 X 21.805 = 5207034.0 in.lb = 5207034.0 / 12000 433.9 ft-kips

Cases % increase in Mn abc Nominal moemnt capacity, Mn 419.4 500.6 499 433.9 19.4 19.0 3.46 23.4 % increase in As 16.7 % increase in d 33.3 % increase in fc'