Question

A beam is supported at its ends by supports which are 12 m apart. Since the load is concentrated at its center, there is a deflection of 3 cm at the center, and the deflected beam is in the shape of a parabola. How far from the center is the deflection 1 cm ?

Answer 1

Given: A beam is supported at its ends by supports which are 12 m apart. There is a deflection of 3 cm at the center, and the deflected beam is in the shape of a parabola. 

Need to find: How far from the center is the deflection 1 cm

Here EF are the ends of the beam and they are 12 m apart. 

IJ is the deflection of 3 cm at the center. We know, 

that the distance IF= 12/2 = 6 m = 600 cm and the deflection IJ = FH = 3 cm. 

So, the coordinate of the point F is (600, 3) 

Let, the equation of the parabola is: x² = 4ay 

F point is on the parabola. So, putting the coordinates of F in the equation we get, 

x² = 4ay 

⇒ 3600 = 4a x 3 

⇒ a = 300

Here KL denotes the deflection of 1 cm. 

So, at the point L the value of y-coordinate is (3 – 1) = 2 

So, by the equation, 

⇒ x² = 4ay = 4 x 300 x 2 = 2400 

⇒ x = 49 cm. 

So, the distance of the point of 1 cm deflection from the center is 49 cm.