MODULE 03 Stage 2: Concrete Cracked – Elastic Stress Stage When the load is increased, the tensile stress in the bottom of the beam becomes equal to the modulus of rupture, the cracks then starts to develop. The moment at which these cracks begin to form is referred to as the cracking moment, π΄ππ This stage will continue as long as the compression stress in the top fibers is less than about one-half of the concrete’s compression strength, π ′ π, and as long as the steel stress is less than the yield stress. π = ππ When the bending moment is sufficiently large to cause the tensile stress in the extreme fibers to be greater than the modulus of rupture of concrete, it is assumed that all of the concrete on the tensile side if the beam is cracked and must be neglected in the flexure calculations. Modular Ratio, n ο· The ratio of the steel modulus to the concrete modulus π= ο· π¬π π¬π Modular ratio is used to convert the area of steel with an equivalent area of concrete Transformed-Area Method a. Singly Reinforced Beam 1. 2. 3. 4. Transform the steel section Locate neutral axis Compute moment of inertia Compute the bending stress b. Doubly Reinforced Beam Double reinforced beam is a beam that has compression steel as well as tensile steel. Compression steel is generally thought to be uneconomical, but occasionally its use is quite advantageous. Compression steel will permit the use of appreciable smaller beams than those that make use of tensile steel only. Reduced sizes can be very important where space or architectural requirements limit the sizes of the beams As a consequence if creep in the concrete, the stresses in the compression bars computed by the transformed area method are assumed to double as time goes by. The transformed area of the compression side equals the gross compression area of the concrete plus 2nA’s minus the area of the holes in the concrete (1A’s), which theoretically should not have been included in the concrete part Sample Problem 1 a. Compute the bending stress in the beam, in MPa, shown in the figure by using the transformedarea method, f’c = 20.7 MPa, n = 9 and M = 94,90 kN-m b. Determine the allowable resisting moment of the beam, in kN-m, if the allowable stress are fc = 9.30 MPa and fs = 137.90 MPa. ππ2 ) (3) 4 π(28)2 π΄π = ( ) (3) 4 π¨π = ππππ. ππ πππ π΄π = ( ππ¨π = ππ, πππ. ππ πππ π₯ π΄π ( ) = ππ΄π (435 − π₯) 2 π₯ 305(π₯) ( ) = (16,625.31)(435 − π₯) 2 π = πππ. ππ ππ πΌπ = πΌππ + πΌππ πΌππ = πβ3 3 (305)(169.98)3 3 π°ππ = πππ. πππ πππππππ πΌππ = πΌππ = π΄π2 πΌππ = 16,625.31 (265.02)2 π°ππ = π, πππ. πππ πππππππ π°π = π, πππ. πππ ππππ πππ Bending Stress ππ¦ πΌπ (94.90π₯106 π. ππ)(169.98ππ) ππ = 1,667.001π₯106 ππ4 ππ = π. πππ π΄ππ ππ = πππ¦ πΌπ (9)(94.90π₯106 π. ππ)(265.02ππ) ππ = 1,667.001π₯106 ππ4 ππ = πππ. ππ π΄π·π ππ = Allowable Resisting Moment ππ¦ πΌπ π π (169.98ππ) 9.30 π/ππ2 = 1,667.001π₯106 ππ4 π΄π = ππ. ππ ππ΅. π ππ = πππ¦ πΌπ (9)(π π )(265.02ππ) 137.91 π/ππ2 = 1,667.001π₯106 ππ4 π΄π = ππ. ππ ππ΅. π ππ = π¨ππππππππ πΉππππππππ π΄πππππ, π΄ = ππ. ππ ππ΅. π Sample Problem 2 a. Compute the bending stress, in psi, in the beam shown in th figure. Use n = 10 and M = 118 kipft π¨′π = π. ππ πππ π¨π = π. ππ πππ (2π − 1)π΄′π = 38.00 ππ2 ππ΄π = 40.00 ππ2 πππ’π‘πππ π΄π₯ππ , π₯ π₯ π΄π ( ) + (2π − 1)π΄′π (π₯ − 2.5) = ππ΄π (17.5 − π₯) 2 π₯ 14 (π₯) ( ) + 38(π₯ − 2.5) = 40(17.5 − π₯) 2 π = π. πππ ππ ππππππ‘ ππ πΌππππ‘ππ ′ +πΌ πΌπ = πΌππ + πΌππ ππ πΌππ = πβ3 (14)(6.454)3 = = 1254.60 ππ4 3 3 πΌππ ′ = π΄π2 = 38(3.954)2 = 594.11 ππ4 πΌππ = π΄π2 = 40 (11.046)2 = 4880.56 ππ4 π°π = ππππ. ππ πππ Bending Stress ππ¦ πΌπ 1000ππ 12ππ π = 118 πππ. ππ‘ π₯ π₯ = 1,416,000 ππ − ππ 1πππ 1ππ‘ ππ = (1416000)(6.454) 6729.27 ππ = π, πππ. ππ πππ ππ = 2πππ¦ πΌπ 2(10)(1416000)(3.954) 137.91 π/ππ2 = 6729.27 ππ′ = ππ, πππ. ππ πππ πππ¦ ππ = πΌπ (10)(1,416,000)(11.046ππ) ππ = 6729.27 ππ = ππ, πππ. ππ πππ ππ ′ = Exercise Problem 1. A reinforced concrete beam 300 mm wide has an effective depth of 600 mm. It is reinforced with 4-32 mm diameter bars for tension. f’c = 21 MPa and fy = 275 MPa. Find the moment capacity of the beam. 2. A 300 mm × 600 mm reinforced concrete beam section is reinforced with 4 - 28-mmdiameter tension steel at d = 536 mm and 2 - 28-mm-diameter compression steel at d' = 64 mm. The section is subjected to a bending moment of 150 kN·m. Use n = 9. a) Find the maximum stress in concrete. b) Determine the stress in the compression steel. c) Calculate the stress in the tension steel.
