Points of Contraflexure:
Assume M1, M2, . . . and M6 be the points of contraflexure, where bending moment changes sign. To determine the position of M2, let a section XX at a distance x from the end C.
M x = - 15.2 x + 24
- 15.2x + 24 = 0
∴ x2 = 1.579 m
To determine the position of M3 and M4, 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 [ x3 - (3 × π2 × 3) + (3x × 9) - 33 ]
= 14.8x - 66 - 0.1 ( x3 - 9 x2 + 27 x - 27)
=- 0.1x3 + 0.9 x2 + 12.1x - 63.3
On changing the sign and equating it to zero, we obtain
0.1x3 - 0.9 x2 - 12.1 x + 63.3 = 0
Solving out by trial and error, we obatin
x1 = 4.4814 m
x2 = 14.357 m
and x3 = - 9.84 m
As, the value of x3 is negative, it must be ignored.
To determine the position of M3, 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 x2 + 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.