Flexture design strength

Question

The cross section of a beam is reinforced with 1.8 in^2 of steel. The depth of the neutral axis of the section at ultimate moment is 3.40 in. The width b (round to the nearest inch) of the beam most nearly:

Assume: ∅= 0.9; fc = 4.0 ksi; fy = 60 ksi

Tip: Cc = Ts

a) 10.0 in

b) 12.0 in

c) 14.0 in

d) 16.0 in

Answer 1

To determine the width b of the beam, we will use the given parameters and the principle that the compressive force in the concrete (Cc​) equals the tensile force in the steel (Ts​).

Given:

1. Area of steel reinforcement As=1.8in2
2. Depth of the neutral axis c=3.40 in
3. Strength reduction factor φ=0.9
4. Compressive strength of concrete fc′=4.0 ksi
5. Yield strength of steel fy=60 ksi

1. Calculate the tensile force in the steel (Ts):

Ts=As⋅fy=108kips

2. Calculate the compressive force in the concrete (Cc​):

The compressive force in the concrete is calculated using the Whitney stress block. The depth of the equivalent rectangular stress block is a

a=β1⋅c

Where β1​ is a factor that depends on the concrete strength. For fc′=4.0ksi:

β1=0.85

So,

a=0.85 .c =2.89in

The compressive force in the concrete (Cc) is:

Cc=0.85⋅fc′⋅b⋅a= 9.826kips/in⋅b

Since Cc=Ts

9.826 kips/in⋅b=10.99in

3. Round b to the nearest inch:

Rounding 10.99 to the nearest inch gives 11 inches.

However, the options given are 10, 12, 14, and 16 inches. Since the closest option to 11 inches is 12 inches, the correct answer is:

12.0 in