Q - Derive bending equation that is,; M/I = σ /y = E/R.
Sol.: With reference to the figure given to us, consider any two normal sections AB and CD of a beam at small distance δ L apart (that is, AC = BD = δ L). Let AB and CD intersect neutral layer at the points M and N respectively.
Let;
M = bending moment acting on beam
θ = Angle subtended at centre by the arc.
R = Radius of curvature of neutral layer M' N' .
At any distance 'y' from neutral layer MN, consider layer EF.
As shown in the figure the beam because of sagging bending moment. After bending, A' B', C' D' , M' N' and
E'F' represent final positions of AB, CD, MN and EF in that order.
When produced, A' B' and C' D' intersect each other at the O subtending an angle θ radian at point O, which is centre of curvature.
As L is quite small, arcs A' C' , M' N' , E' F' and B' D' can be taken as circular.
Now, strain in layer EF because of bending can be given by e = (E F - EF)/EF = (E F - MN)/MN
As MN is the neutral layer, MN = M' N'
Let; σ = stress set up in layer EF because of bending
E = Young's modulus of material of beam.
Equate the equation (i) and (ii);
Let; σ = stress set up in layer EF because of bending
E = Young's modulus of material of beam.
At distance 'y', let us consider an elementary strip of quite small thickness dy. We have already assumed that 'σ ' is bending stress in this strip.
Let dA = area of the elementary strip. Then, force developed in this strip = σ.dA.
Then the, elementary moment of resistance because of this elementary force can be
given by dM = f.dA.y
Total moment of resistance because of all such elementary forces can be given by
From the Equation (iii),
By putting this value of f in Equation (iv), we get
But
where I = Moment of inertia of whole area about neutral axis N-A.
Where;
M = Bending moment
I = Moment of Inertia about axis of bending that is; Ixx
y = Distance of the layer at which the bending stress is consider
(We take always the maximum value of y, that is, distance of extreme fiber from N.A.)
E = Modulus of elasticity of beam material.
R = Radius of curvature