Question

If a three-hinged parabolic arch, (span l, rise h) is carrying a uniformly distributed load w/unit length over the entire span

• A. Shear force will be zero throughout
• B. Horizontal thrust is $\left(w{l}^2\right)/8h$
• C. Bending moment will be zero throughout
• D. All option are correct

Answer 1

The correct option is D All option are correct


1. Support reaction

$From\sum F_X=0,H_A=H_B=H$

$From\sum F_Y=0,V_A+V_B=w\times l$

$\sum M_A=0$

$w\times l\times \frac{l}{2}-V_B\times l=0$

$V_B=\frac{wl}{2},V_A=\frac{wl}{2}$

$\sum M_C=0$

$w\times \frac{l}{2}\times \frac{l}{4}-V_B\times \frac{l}{2}+H_B\times h=0$

$\frac{w{l}^2}{8}-\frac{w{l}^2}{4}+H_{B}h=0$

$H_B=\frac{w{l}^2}{8h}$

$H_A=H_B=H=\frac{w{l}^2}{8h}\text{(Remember)}$

2. Bending moment at any section xx

$BMatx-x=V_{A}x-H_{a}y-(w\times x)\times \frac{x}{2}$

$=\frac{wlx}{2}-\frac{w{l}^2}{8h}(\frac{4h}{l^2}\times x\times (l-x\left)\right)-w\times \frac{x^2}{2}$

$=\frac{wlx}{2}-\frac{w}{2}(xl-x^2)-\frac{w{x}^2}{2}$

$=\frac{wlx}{2}-\frac{wlx}{2}+\frac{w{x}^2}{2}-\frac{w{x}^2}{2}=0$

So, BM is zero (0) everywhere

3. Radial shear at x -x

Radial shear at x - x

$T=\left(\frac{w{l}^2}{8h}\right)sin\theta -(\frac{wl}{2}-wx)cos\theta ....\left(A\right)$

$y=\frac{4h}{l^2}\left(x\right)(l-x)$

$tan\theta =\frac{dy}{d\theta }=\frac{4h}{l^2}(l-2x)$

$Dividing\left(A\right)bycos\theta$

$\frac{T}{cos\theta }=\left(\frac{w{l}^2}{8h}\right)tan\theta -(\frac{wl}{2}-w_x)$

$\frac{w}{2}(l-2x)-\frac{wl}{2}+wx$

$=\frac{wl}{2}-wx-\frac{wl}{2}+wx$

$\frac{T}{cos\theta }=0$

So, Radial shear is zero everywhere in the arch.

4. Normal thrust at any section x -x

$N=\frac{w{l}^2}{8h}cos\theta +(\frac{wl}{2}-wx)sin\theta$

$\frac{N}{cos\theta }=(\frac{wl}{2}-wx)tan\theta +\frac{w{l}^2}{8h}$

$=(\frac{wl}{2}-wx)\frac{4h}{l^2}(l-2x)+\frac{w{l}^2}{8h}$

$N=\left[(\frac{wl}{2}-wx)\frac{4h}{l^2}\right(l-2x)+\frac{w{l}^2}{8h}]cos\theta$

Note :
1. If three hinged parabolic arch or a two hinged parabolic arch is subjected to udl throughout its length, BM & radial shear are zero.

2. Cross-section is subjected to only normal thrust every where.