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Answer :
The minimum amount of shear reinforcement needed for the beam can be calculated using the equation. The minimum amount of shear reinforcement needed is 0.003 square inches.
[tex]Vu = V_dead + V_live[/tex]
Where Vu is the ultimate shear force, V_dead is the shear force from dead load, and V_live is the shear force from live load. In this case, [tex]V_dead = 5 kips and V_live = 15 kips[/tex].
[tex]Vu = 5 kips + 15 kips[/tex]
[tex]Vu = 20 kips[/tex]
Next, we need to calculate the maximum shear force that can be resisted by the concrete alone. This can be done using the equation:
[tex]Vc = 2 √(f'c) bw d[/tex]
Where Vc is the shear force resisted by the concrete, f'c is the compressive strength of the concrete, bw is the width of the beam, and d is the effective depth of the beam.
Since only the minimum amount of shear reinforcement needed is required, we can use the minimum value of f'c = 3,000 psi. Let's assume the width of the beam (bw) is 12 inches and the effective depth (d) is 18 inches. Therefore,
[tex]Vc = 2 √(3,000 psi) (12 inches) (18 inches)[/tex]
[tex]Vc = 17,320 pounds[/tex]
Now, we can calculate the minimum amount of shear reinforcement (Asw) needed using the equation:
[tex]Asw = (Vu - Vc) / (fy * d)[/tex]
Where Asw is the area of shear reinforcement, fy is the yield strength of the steel, and d is the effective depth of the beam.
Given that fy for all steel is 60,000 psi, we can substitute the values:
[tex]Asw = (20,000 pounds - 17,320 pounds) / (60,000 psi * 18 inches)[/tex]
[tex]Asw = 0.003 square inches[/tex]
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