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Question

The maximum bending moment due to an isolated load in a three-hinged parabolic arch (span l, rise h) having one of its hinges at the crown, occurs on either side of the crown at a distance

A
L/(23)
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B
L/4
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C
h/4
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D
L/(32)
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Solution

The correct option is A L/(23)
Given :



Let us assume as isolated load W acting at distance x from end A.

Since there is no horizontal external force so horizontal reactions at end A and B will be HA=HB=H.

Taking moment about (A), we have

wxVB×2=0
VB=WxL

Taking bending moment at C:

We have (BM)C=0

VB×L2H×h=0
H=VBL2h=Wx2h

Now, (BM)D=VB(Lx)H×y

Since, y=4hxL2(Lx)(Equation of parabolic arch)

(BM)D=WxL(Lx)H×4hxL2(Lx)

(BM)D=Wx(Lx)LWx2h×4hxL2(Lx)

(BM)D=Wx(Lx)L2Wx2L2(Lx)

For absolute maximum bending moment
d(BM)Ddx=0

d(PWx(Lx)2)dxd(2Wx2L2(Lx))dx=0

WL(L2x)2WL2(2Lx3x2)=0

(L2x)2L(2Lx3x2)=0

6x26xL+L2=0

x=(6±2312)L
x = 0.78867L, 0.2113l




So, distance of maximum moments for given load condition either side of crown.

(L20.2113L) or (0.7886LL2)

= 0.2886L or 0.2886L from crown

L23=0.2886L

So, option (c) is correct.

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