Civil Engineering - Strength of Materials - Discussion

I do not understand this.

Please explain this clearly.

U= (σ^2*v)/2E.
Where V = area * length
= bL^2 (axial tension) ----> (1)

For bending, strain energy stored by the elemental cylinder is for the entire volume is:
U = Ʃ[ (M^2)/2E ] dx{hear integration from 0 to L}
U = [ (M^2)/2E ]*Ʃ1dx
U = [ (M^2)/2E ]*L.

Now M/I = E/R = σ/y.
So M = σ*I/y.
Putting I = bL^3/12,
(I/Y) = Z = BL^2/6.
U=(σ^2 z/E) =(σ^2*BL^2)/6E ---> (2)

Dividing (2) by (1),
U(bending)=1/3 U(axial).

Thanks @Abhishek Thakur.

For(tension and compression):

Due to the stress-strain curve the strain energy;

Resilience = 1/2 * stress * strain.
And proof resilience= 1/2* stress * strain * volume is;
Strain = stress/modulus of elasticity.
Then PR=stress^2*volume/2E.

And for bending moment due to pure axial loading.
M=Wx.

Strain energy due to bending moment = integration of M^2*xdx/2E= integration of (Wx)^2dx/2E (limit 0toL).

Put the value of Moment M here and on solving u will get;
W^2 l^3/6E.

On solving both the value you will get;
1/3.

U= (σ^2*v)/2E -----> (axial loading).

U=(σ^2*v)/6E -----> (bending).

So, answer is (1/3).

But the strain energy if load multiplied by displacement, Then in case of bending, how do we derive the formula.

Will you please explain it clearly?

Can't understand can you explain?

U= (σ^2*v)/2E -----> (axial loading)

U=(σ^2*v)/6E -----> (bending)

So ans is (1/3)

For bending
U=Ʃ[ (M^2)/2E ] dx{hear integration frm 0 to L}
U=[ (M^2)/2E ]*Ʃ1dx
U=[ (M^2)/2E ]*L
Now M/I = E/R=σ/y
So M=σ*I/y
On simplifying
U=(σ^2*v)/6E

Due to avail stress =σ2/2E.vol.

Due bending = Σ2/6E.vol.