Unishivaji Structural Mechanics - II May 2016 Solved Question Paper Q2 B)

A tapering chimney of hollow circular cross section is 30 m high. Its external diameter at the base is 2.4 m and at the top it is 1.6 m. The chimney is subjected to uniform wind pressure of 2.2 KN/m²of the projected area. If the weight of the chimney is 4000 KN and internal diameter at the base is 0.80 m, determine the maximum and minimum stress intensities at the base. Take k:1.

13 - Marks

>>Given:
Height of chimney = H = 30 m;
External diameter at the base = 2.4 m;
External diameter at the top = 1.6 m;
Maximum stress at base = σ_max = ?;
Minimum stress at base = σ_min = ?;
Wind pressure = P = 2.2 KN/m²;
Weight of the chimney = W = 4000 kN;
Internal diameter = 0.8 m;
K = 1
>>To find:
σ_max = ?
σ_min = ?

>>Solution:
Direct stress,
$\sigma _{0}=\frac{W}{A}$
Where,
W = Self weight = 4000 kN;
A = Base ares;
$A = \frac{\pi}{4} \left(D-d \right)^{2}= \frac{\pi}{4} \left(2.4^{2}-0.8^{2} \right)$
$A=4.0212 m^{2}$
$\sigma _{0}=\frac{W}{A}=\frac{4000}{4.0212}=994.71 kN/m^{2}$

Bending stress $\sigma _{b}=\frac{M}{Z}$
Where,
$M=Bending \: moment\: due \:to\: wind\: load = F \cdot e$
$M=Wind \: force \times eccentricity$
$M=\left(wind\: coefficient \times wind\: pressure \times projected\: area \right) \timex e$
$M=\left(k \times P \times \left( \frac{top+bottom}{2}\right)\right) \times Height/2$
$M=\left(1 \times 2.2 \times \left( \frac{1.6+2.4}{2}\right)\right) \times 30/2$
$M=66 kNm$

$Z=Section Modulus$
$Z= \frac{I}{y}$

$I= \frac{\pi}{64}\times \left(D^{4}-d^{4}\right)$
$I= \frac{\pi}{64}\times \left(2.4^{4}-0.8^{4}\right)$
$I= 1.6084 m^{4}$

$y= D/2=2.4/2= 1.2 m$

$Z= \frac{1.6084}{1.2}$
$Z= 1.3404 m^{3}$

$\therefore \sigma _{b}=\frac{66}{1.3404}$
$\sigma _{b}=49.2385 kN/m^{2}$

Now, $\sigma _{max}=\sigma _{0}+\sigma _{b}$
$\sigma _{max}=994.71+49.2385$
$\sigma _{max}=1043.9485 kN/m^{2}(C)$
$\sigma_{min} = \sigma_{0} - \sigma_{b}$
$\sigma _{max}=994.71-49.2385$
$\sigma_{min} = 945.4715 kN/m^{2}(C)$
>>Summery:
$\sigma_{max} = 1043.9485 kN/m^{2}(C)$
$\sigma_{min} = 945.4715 kN/m^{2}(C)$

Vinayak Hulwane

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Unishivaji Structural Mechanics - II May 2016 Solved Question Paper Q2 B)

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