Civil Engineering - Strength of Materials - Discussion

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.
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).

Step 1: Calculate strain energy due to uniaxial loading.

Viz. U1 = {(f^2)xvol}/(2xE).

Step 2: Calculate strain energy due to pure bending.

Viz. U2 = 0.5xMxtheta.

= 0.5x[(fxI)/y]x[(Mxl)/(ExI)].

(from simple bending equation & moment area method resp.)

Solve it U2 = {(f^2)xvol}/(6xE).

i.e U2 = 0.3XU1.

:) :) :) :).

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

We know that,
FOR COMPRESSIOM OR TENSIOIN.

Strain energy(SE) = (1/2) * force * deformation = 1/2) * P * PL/AE = P^2L/2AE.

FOR BENDING.

SE = (P^2L^3)/6AE.

So bending (SE)= (L^2)/3 tension(SE).

Here, [SE] FOR COMPRESSION OR TENSION IS.

= 1/2 * force * deformation.

And what about [SE] for BENDING.

@Dheeraj Hindu.
Could you please explain this one.

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

U= (σ^2*v)/2E -----> (axial loading).

U=(σ^2*v)/6E -----> (bending).

So, answer is (1/3).

Due to avail stress =σ2/2E.vol.

Due bending = Σ2/6E.vol.

I do not understand this.

Please explain this clearly.