Depression of a cantilever Loaded at its ends

DEPRESSION OF A CANTILEVER ‒ LOADED AT ITS ENDS THEORY:

Let Let ‘l’ be the length of the cantilever OA fixed at 'O'. Let 'W' be the weight suspended (loaded) at the free end of the cantilever. Due to the load applied the cantilever moves to a new position OA' as shown in Fig.1.10.

Let us consider an element PQ of the beam of length dx, at a distance OP= x from the fixed end. Let 'C' be the centre of curvature of the element PQ and let 'R' be the radius of curvature.

Due to the load applied at the free end of the cantilever, an external couple is created between the load W at 'A' and the force of reaction at 'Q'. Here, the arm of the couple (Distance between the two equal and opposite forces) is (l‒x).

The external bending moment = W . (l‒x)                ………….(1)

We know that the internal bending moment = Yl_g / R                ………….(2)

We know that under equilibrium condition,

External bending moment = Internal bending moment.

Therefore, we can write Eqn. (1) = Eqn.(2)

(or) W(l‒x) = YI_g / R

(or) R = YI_g / W(l‒x)               ………….(3)

Two tangents are drawn at points P and Q, which meet the vertical line AA' at T and S respectively.

Let the smallest depression produced from T to S = dy

and Let the angle between the two tangents = dθ

Then we can write

The angle between CP and CQ is also dθ (i.e) ∠PCQ = dθ.

We can write the arc length PQ = Rdθ = dx

(or) dθ = dx/R       ………….(4)

Substituting eqn.(3) in eqn.(4), we have dθ = dx / [YI_g/W(l‒x)]

(or)       dθ = [ W(l‒x)   / YI_g  ] dx                  …………….(5)

From the ΔQA'S we can write sin dθ = dy / (l‒x)

If dθ is very small then we can write,

 dy = (l‒x) . dθ                 ...........(6)

Substituting eqn.(5) in eqn.(6) we have

 dy = W/YI_g  (l‒x)².dx                ...(7)

 ∴ Total depression at the free end of the cantilever can be derived by integrating the eqn(7) within the limits 0 to ‘l’.

Therefore

Depression of the cantilever at free end y = Wl³ / 3YI_g              …...(8)

SPECIAL CASES

(I) RECTANGULAR CROSS SECTION

If 'b' is the breadth and 'd' is the thickness of the beam then we know that

I_g = bd³ / 12

Substituting the value of I_g in eqn. (8), we can write

The depression produced at free end for a rectangular cross section

(II) CIRCULAR CROSS SECTION

If 'r' is the radius of the circular cross section, then

We know that

I_g = πr⁴ / 4

Substituting the value of I_g in eqn. (8), we can write

∴ Depression produced y = Wl³ / 3Y(πr⁴/4)

(or) y= Wl³ / 3πr⁴Y

Note: The angle between the tangents at the end of a cantilever can be got by integrating eqn. (5) within the limits 0 to ‘l’