Question

For the same cross-sectional area and for a given load, the ratio of depressions for the beam of a square cross-section and circular cross-section is ______.

विकल्प

• 3 : π

• π : 3

• 1 : π

• π : 1

Answer 1

For the same cross-sectional area and for a given load, the ratio of depressions for the beam of a square cross-section and circular cross-section is 3 : π.

Explanation:

δ = `("W"ℓ^3)/(3"YI")`, where W = load, ℓ = length of beam and I is geometrical moment of inertia for rectangular beam,

I = `"bd"^3/12` where b = breadth and d = depth 

For square beam b = d  ∴ I₁ = `"b"^4/12`

For a beam of circular cross-section, I₂ = `((pi"r"^4)/4)`

∴ δ₁ = `("W"ℓ^3xx12)/(3"Yb"^4)=(4"W"ℓ^3)/("Yb"^4)`  (for sq. cross-section)

and δ₂ = `("W"ℓ^3)/(3"Y"(pir^4//4))=(4"W"ℓ^3)/(3"Y"(pi"r"^4))`  (for circular cross-section)

Now `delta_1/delta_2=(3pi"r"^4)/"b"^4=(3pi"r"^4)/(pi"r"^2)^2=3/pi`

(∵ b² = πr² i.e., they have same cross-sectional area)