FUNICULAR NATURE OF 3 HINGED PARABOLIC ARCH

FUNICULAR NATURE OF 3 HINGED PARABOLIC ARCH

whenever a three hinged parabolic arch is subjected to the uniformly distributed load throughout the entire span with same support level , then in such case the bending moment at any section of the arch is equal to zero. This is because the shape of bending moment diagram with respect to the geometry of the arch is same.This can be experienced only in case of flexible structure.In case of steel and Rcc structures this cannot be adapted due to its rigid nature.Apart from this ,if the rigid structure is designed by assuming the zero BM then the whole structure will experience moment under different load conditions and lead to failure of structure.

NUMERICAL

A three hinged parabolic arch of length L and rise h carries a uniformly distributed load of w/m span .Show that the there is zero bending moment at any section of the arch.

Step 1:

Applying vertical equilibrium condition for the arch

V _a + V _b = w (L)….. (1)

Since the arch is symmetric, the support reactions V _a = V _b = w (L) / 2

Taking moments about C

V _a (L/2) - H x h - w ((L) / 2) (L) / 4)) = 0

Solving the above equation

H = wL² / 8h

Step 2:

Consider the section X-X at the Horizontal distance x from support A and vertical distance y from the arch rib

M _x-x = V _a (x) - w(x) ²/2 –H (y)

W k t

M _x-x = V _a (x) - w(x) ²/2 –H (y)

M _x-x = [(w (L)/2(x)) - w(x) ²/2 – {(wL² / 8h) (4hx (L-x)/L²)}]

M _x-x = 0