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A uniform beam of length ‘L’ is simply and symmetrically supported to a span ‘l’ find the ratio L/l so that the upward deflection at each end equals the downward deflection at mid span, due to a concentrated point load at mid span.
- a)3/5
- b)5/3
- c)7/5
- d)5/7
Correct answer is option 'B'. Can you explain this answer?
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A uniform beam of length ‘L’ is simply and symmetrically supported to...
Deflection at midpoint = deflection at free end
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A uniform beam of length ‘L’ is simply and symmetrically supported to...
Solution:
Given, a uniform beam of length ‘L’ is simply and symmetrically supported to a span ‘l’ and the upward deflection at each end equals the downward deflection at mid span, due to a concentrated point load at mid span.
Let us consider the deflection equation for a simply supported beam with a point load at mid span.
The deflection equation is given by,
δ = (Wx/48EI) * (3l^2 - 4x^2) …(1)
Where,
δ = deflection at a distance ‘x’ from one end
W = point load at mid span
l = span length
E = modulus of elasticity of the beam material
I = moment of inertia of the beam
Let us consider the deflection at the ends of the beam,
At x = 0, δ = (W/48EI) * (3l^2) = (Wl^2/16EI)
At x = L/2, δ = (W/48EI) * (4L^2 - 3l^2) = (W/16EI) * ((2L)^2 - (sqrt(3)l)^2)
Since, the upward deflection at each end equals the downward deflection at mid span,
(Wl^2/16EI) = (W/16EI) * ((2L)^2 - (sqrt(3)l)^2)
On simplifying, we get,
L/l = 5/3
Hence, the correct option is (B) 5/3.
Given, a uniform beam of length ‘L’ is simply and symmetrically supported to a span ‘l’ and the upward deflection at each end equals the downward deflection at mid span, due to a concentrated point load at mid span.
Let us consider the deflection equation for a simply supported beam with a point load at mid span.
The deflection equation is given by,
δ = (Wx/48EI) * (3l^2 - 4x^2) …(1)
Where,
δ = deflection at a distance ‘x’ from one end
W = point load at mid span
l = span length
E = modulus of elasticity of the beam material
I = moment of inertia of the beam
Let us consider the deflection at the ends of the beam,
At x = 0, δ = (W/48EI) * (3l^2) = (Wl^2/16EI)
At x = L/2, δ = (W/48EI) * (4L^2 - 3l^2) = (W/16EI) * ((2L)^2 - (sqrt(3)l)^2)
Since, the upward deflection at each end equals the downward deflection at mid span,
(Wl^2/16EI) = (W/16EI) * ((2L)^2 - (sqrt(3)l)^2)
On simplifying, we get,
L/l = 5/3
Hence, the correct option is (B) 5/3.
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A uniform beam of length ‘L’ is simply and symmetrically supported to a span ‘l’ find the ratio L/l so that the upward deflection at each end equals the downward deflection at mid span, due to a concentrated point load at mid span.a) 3/5b) 5/3c) 7/5d) 5/7Correct answer is option 'B'. Can you explain this answer?
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