Introduction:
In this question, we are asked to explain why the strain energy absorbed in a cantilever beam in bending under its own weight is k times greater than the strain energy absorbed in an identical simply supported beam. To answer this question, we need to understand the concepts of strain energy, cantilever beams, simply supported beams, and the effect of self-weight on the bending behavior of beams.

Strain Energy:
Strain energy is the energy stored within a material due to deformation caused by an applied load. In the case of beams, strain energy is primarily stored in the form of bending.

Cantilever Beam:
A cantilever beam is a beam that is supported at one end and is free to deform under applied loads at the other end. The fixed end provides support against rotation, while the free end is subjected to bending due to the applied load.

Simply Supported Beam:
A simply supported beam is a beam that is supported at both ends and is free to deform under applied loads. The supports at both ends allow the beam to rotate, resulting in bending under the applied load.

Effect of Self-Weight:
When a beam is subjected to its own weight, it experiences a distributed load along its length. This distributed load creates a bending moment in the beam, causing it to deform. The effect of self-weight is different for cantilever beams and simply supported beams.

Explanation:
When a cantilever beam is subjected to its own weight, the entire weight acts at the free end. This creates a greater bending moment compared to a simply supported beam, where the weight is distributed between the two supports. As a result, the cantilever beam experiences higher bending stresses and strains than the simply supported beam.

The strain energy in a beam is directly proportional to the square of the bending moment and inversely proportional to the flexural rigidity of the beam. Since the bending moment is higher in the cantilever beam due to the concentrated load at the free end, the strain energy absorbed in the cantilever beam will be greater than that in the simply supported beam.

Mathematically, the strain energy U in a beam can be expressed as:

U = (1/2) ∫(M^2/EI) dx

Where M is the bending moment, E is the modulus of elasticity, I is the moment of inertia, and dx is a differential length along the beam.

Since the bending moment is greater in the cantilever beam, the strain energy Uc in the cantilever beam can be expressed as:

Uc = (1/2) ∫(Mc^2/EI) dx

Similarly, the strain energy Us in the simply supported beam can be expressed as:

Us = (1/2) ∫(Ms^2/EI) dx

As the cantilever beam experiences a higher bending moment, the term Mc^2 will be greater than Ms^2, resulting in a greater strain energy Uc compared to Us.

Conclusion:
In conclusion, the strain energy absorbed in a cantilever beam in bending under its own weight is k times greater than the strain energy absorbed in an identical simply supported beam. This is due to the higher bending moment experienced by the cantilever beam, resulting in higher bending stresses and strains, and consequently, higher strain energy.