An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Calculate the modulus of rupture: f r = 7.5 5000 ≈ 0.53 ksi.
Calculate the cracking moment: M cr = 9.3 0.53 × 12550 ≈ 715.66 kips-in.
Determine the service moment strength based on allowable stresses: M s ≈ 2482.02 kips-in and M c ≈ 419.81 kips-in, so the service moment strength is 419.81 kips-in.
Calculate the nominal moment strength and design moment strength: M n ≈ 4666.67 kips-in and ϕ M n ≈ 4200.00 kips-in.
Explanation
Problem Setup and Given Information We are given a singly reinforced concrete beam and asked to find the cracking moment, service moment strength, nominal moment strength, and design moment strength. We have the following information:
Beam width, b = 16 in
Beam height, h = 20 in
Effective depth, d = 18.5 in
Reinforcement: Six No. 8 bars
Concrete compressive strength, f c ′ = 5 ksi
Steel yield strength, f y = 60 ksi
Neutral axis location: 10.7 in from the top
Moment of inertia, I = 12 , 550 in 4
Modulus of rupture, f r = 7.5 f c ′ psi
Calculate Modulus of Rupture and Steel Area First, we need to calculate the modulus of rupture, f r , in psi and ksi. We have f r = 7.5 f c ′ = 7.5 5000 ≈ 530.33 psi = 0.53033 ksi Next, we calculate the area of the steel reinforcement, A s . Since we have six No. 8 bars, and each No. 8 bar has an area of 0.79 in 2 , we have A s = 6 × 0.79 = 4.74 in 2
Calculate Cracking Moment Now, we can calculate the cracking moment, M cr , using the formula M cr = y t f r I where y t is the distance from the neutral axis to the tension face. We have y t = h − 10.7 = 20 − 10.7 = 9.3 in. Therefore, M cr = 9.3 in 0.53033 ksi × 12 , 550 in 4 ≈ 715.66 kips-in
Calculate Service Moment Strength Next, we calculate the service moment strength. The allowable stress in the concrete is 0.45 f c ′ = 0.45 × 5 = 2.25 ksi, and the allowable stress in the steel is 0.5 f y = 0.5 × 60 = 30 ksi.
To find the service moment strength, we first assume the steel yields. We calculate 'a' based on steel yielding: a = 0.85 f c ′ b A s f s = 0.85 × 5 × 16 4.74 × 30 ≈ 2.09 in Then, the moment capacity based on steel is: M s = A s f s ( d − 2 a ) = 4.74 × 30 × ( 18.5 − 2 2.09 ) ≈ 2482.02 kips-in
Now, we consider the concrete. We need to find the depth to the neutral axis, 'c', assuming the concrete reaches allowable stress. Using similar triangles: c f c , a ll o w ab l e = d − c f s , a ll o w ab l e 2.25 ( 18.5 − c ) = 30 c 41.625 − 2.25 c = 30 c c = 32.25 41.625 ≈ 1.29 in
The moment capacity based on concrete is: M c = 2 1 f c b c ( d − 3 c ) = 2 1 × 2.25 × 16 × 1.29 × ( 18.5 − 3 1.29 ) ≈ 419.81 kips-in
The service moment strength is the smaller of M s and M c , which is 419.81 kips-in.
Calculate Nominal and Design Moment Strength Finally, we calculate the nominal moment strength, M n , and the design moment strength, ϕ M n . First, we calculate 'a': a = 0.85 f c ′ b A s f y = 0.85 × 5 × 16 4.74 × 60 ≈ 4.18 in Then, the nominal moment strength is: M n = A s f y ( d − 2 a ) = 4.74 × 60 × ( 18.5 − 2 4.18 ) ≈ 4666.67 kips-in
To find the design moment strength, we need to determine the strain in the steel, ϵ t . We have c = 0.85 a = 0.85 4.18 ≈ 4.92 in. Then, ϵ t = 0.003 × c d − c = 0.003 × 4.92 18.5 − 4.92 ≈ 0.0083 Since 0.005"> ϵ t > 0.005 , the section is tension-controlled, and ϕ = 0.9 . Therefore, the design moment strength is: ϕ M n = 0.9 × 4666.67 ≈ 4200.00 kips-in
Final Answer The bending moment that can cause tension cracks is approximately 715.66 kips-in. The service moment strength is approximately 419.81 kips-in. The nominal moment strength is approximately 4666.67 kips-in, and the design moment strength is approximately 4200.00 kips-in.
Examples
Understanding the bending capacity of reinforced concrete beams is crucial in structural engineering. For example, when designing a bridge, engineers need to ensure that the concrete beams can withstand the bending moments caused by the weight of vehicles and the bridge itself. By calculating the cracking moment, service moment strength, nominal moment strength, and design moment strength, engineers can select appropriate beam dimensions and reinforcement to ensure the bridge's safety and longevity. This ensures structures can safely handle applied loads, preventing failures and ensuring public safety.
