Given: The strain energy absorbed in cantilever beam in bending under Owen weight is k times greater than the strain energy absorbed in an identical simply supported beam in bending under its own weight.

To find: The magnitude of k.

Solution:

Let us assume,

Length of the cantilever beam = L
Length of the simply supported beam = L
Load on cantilever beam = W
Load on simply supported beam = Weight of the beam = w

Strain energy in cantilever beam = Uc
Strain energy in simply supported beam = Us

We know that the strain energy in a beam in bending is given by,

U = (1/2) * M * y * θ

where,
M = Bending moment at a section
y = Distance from the neutral axis to the point where strain energy is to be calculated
θ = Angle of bending

For a cantilever beam, the maximum bending moment occurs at the fixed end and is given by,

M = W * L

For a simply supported beam, the maximum bending moment occurs at the mid-span and is given by,

M = w * L / 4

Therefore, the strain energy in cantilever beam is given by,

Uc = (1/2) * M * y * θ
= (1/2) * W * L * (h/2) * (θc)

where,
h = Height of the beam
θc = Angle of bending for cantilever beam

Similarly, the strain energy in simply supported beam is given by,

Us = (1/2) * M * y * θ
= (1/2) * w * (L/2) * (h/2) * (θs)

where,
θs = Angle of bending for simply supported beam

Given that Uc = k * Us

Therefore, (1/2) * W * L * (h/2) * (θc) = k * (1/2) * w * (L/2) * (h/2) * (θs)

Simplifying further,

W * L * (θc) = k * w * (L/2) * (θs)

W * 2 * (θc) = k * w * (θs)

θc = (k/2) * θs

From the theory of bending, we know that the angle of bending is inversely proportional to the moment of inertia of the beam. Therefore, we can write,

θc / θs = Ic / Is

where,
Ic = Moment of inertia of cantilever beam
Is = Moment of inertia of simply supported beam

Since the beams are identical, we can write,

Ic = Is = (1/12) * b * h^3

where,
b = Width of the beam
h = Height of the beam

Substituting the values, we get,

θc / θs = 1

Therefore, k/2 = 1

Hence, k = 2

Therefore, the magnitude of k is 2.

Answer: (A) 2.

B