Expression for internal Bending Moment

EXPRESSION FOR INTERNAL BENDING MOMENT

Let us consider a beam under the action of deforming forces. The beam bends into a circular arc as shown in Fig 1.8. Let AB be the neutral axis of the beam. Here the filaments above AB are elongated and the filaments below AB are compressed. The filament AB remains unchanged.

Let PQ be the arc chosen from the neutral axis. If R is the radius of curvature of the neutral axis and θ is the angle subtended by it at its centre of curvature 'C'.

Then we can write original length PQ = Rθ            ……...(1)

Let us consider a filament P'Q' at a distance 'x' from the neutral axis.

∴ We can write the extended length = P'Q' = (R+x) θ     ...... (2)

From eqn.(1) and eqn.(2), we have

Increase in its length = P'Q' ‒ PQ

(or) Increase in its length = (R+x) θ‒Rθ

Increase in its length = xθ            …..(3)

We know Linear Strain = Increase in length / Original length

(or)

Linear Strain = xθ / Rθ

Linear Strain = x / R          …….(4)

We know, The youngs modulus of the material Y = Stress / Linear strain

(or) Stress = Y × Linear strain          …….(5)

Substituting eqn.(4) in (5), we have

Stress = Yx / R

If δA is the area of cross section of the filament P'Q'. Then,

The tensile force on the area (δA) = Stress × Area

(i.e) Tensile Force = Yx/R . δA

We know that, Moment of force = Force × Perpendicular Distance

∴ Moment of the tensile force about the neutral axis AB (or) PQ = Yx/R . δA . x

PQ = (Y/R)δAx²

The moment of force acting on both the upper and lower halves of the neutral axis can be got by summing all the moments of tensile and compressive forces about the neutral axis.

The moment of all the forces about the neutral axis = (Y/R) Σx²δA

Here I_g = Σx²δA = AK² is called as the geometrical moment of inertia.

Where, A is the total area of the beam and

K is the radius of the Gyration.

Total moment of all the forces (or) Internal bending moment = YI_g/R         ………..(6)

SPECIAL CASES

(I) RECTANGULAR CROSS SECTION

If 'b' is the breadth and 'd' is the thickness of the beam, then

Area A = bd and K² = d²/12

I_g  = AK²

⇒ bd.d² / 12

⇒ bd³ / 12

Substituting the value of I_g in eqn.(6), we can write

Bending moment for a rectangular cross section = Ybd³ / 12R   ……..(7)

(II) CIRCULAR CROSS SECTION

For a circular cross section if 'r' is the radius, then Area A = πг²

and

 K² = r²/4

I_g  = AK²

I_g  = [ π r² × r² ] / 4

I_g  = πr⁴ / 4

Substituting the value of I_g, in eqn.(6), we can write

The Bending moment of a circular cross section = πYr⁴ / 4R  ………..(8)