Points of Contraflexure:
Assume M_{1}, M_{2}, . . . and M_{6} be the points of contraflexure, where bending moment changes sign. To determine the position of M_{2}, let a section XX at a distance x from the end C.
M_{ x} = - 15.2 x + 24
- 15.2x + 24 = 0
∴ x_{2} = 1.579 m
To determine the position of M_{3} and M_{4}, consider a section XX at a distance x from the end E.
M _{x} = 15.2x + 24 + 30 ( x - 3) - (( ½) ( x - 3) × 0.6 × ( x - 3) × (( x - 3)/3)
M _{x} = - 15.2 x + 24 + 30 x - 90 - 0.1 ( x - 3)^{3}
= 14.8 x - 66 - 0.1 ( x - 3)^{3}
= 14.8 x - 66 - 0.1 [ x^{3} - (3 × π^{2} × 3) + (3x × 9) - 33 ]
= 14.8x - 66 - 0.1 ( x^{3} - 9 x^{2 } + 27 x - 27)
=- 0.1x^{3} + 0.9 x^{2} + 12.1x - 63.3
On changing the sign and equating it to zero, we obtain
0.1x^{3} - 0.9 x^{2} - 12.1 x + 63.3 = 0
Solving out by trial and error, we obatin
x_{1} = 4.4814 m
x_{2} = 14.357 m
and x_{3} = - 9.84 m
As, the value of x_{3} is negative, it must be ignored.
To determine the position of M_{3}, assume a section XX at a distance x from the end E.
M x= 7.2 × x × (x /2) + 15.6
On equating it to zero, we obtain
- 3.6 x^{2} + 15.6 = 0
x = 2.082 m
The points of contraflexure are at distance of 1 m, 1.579 m, 4.4814 m & 14.357 m from the left end X & at distances of 1 m & 2.082 m from the right end F.