Question

Two columns A and B are situated 5 m apart. The ratio of load transferred by column A to column B is 2. The ratio of vertical stress at a point 10 m below the ground surface at points vertically below B to that of A is ? (Use Boussinesq equations)

• A. 0.8
• B. 1.2
• C. 1.6
• D. 2.0

Answer 1

The correct option is A 0.8



Given, z = 10m, r = 5m

Vertically stress at point vertically below A,

${\sigma }_A=$ Vertical stress due to column A + Vertical stress due to column B

$=\frac{3\left(2Q\right)}{2\pi }.\frac{1}{(10{)}^2}+\frac{3Q}{2\pi }.\frac{1}{(10{)}^2}.{\left[\frac{1}{1+(5/10{)}^2}\right]}^{5/2}$

$=\frac{3Q}{2\pi }.\frac{1}{(10{)}^2}.[2+{\left[\frac{1}{1+(5/10{)}^2}\right]}^{5/2}]$

$=\frac{3Q}{2\pi }.\frac{1}{(10{)}^2}.\left[2.572\right]$

Similarly vertical stress at a point vertically below B

${\sigma }_B=\frac{3\left(2Q\right)}{2\pi }.\frac{1}{(10{)}^2}{\left[\frac{1}{1+(5/10{)}^2}\right]}^{5/2}+\frac{3Q}{2\pi }.\frac{1}{(10{)}^2}$

$=\frac{3Q}{2\pi }.\frac{1}{(10{)}^2}[\frac{2}{{[1+(5/10{)}^2]}^{5/2}}+1]$

$=\frac{3Q}{2\pi }.\frac{1}{(10{)}^2}\left[2.1449\right]$

$\frac{{\sigma }_B}{{\sigma }_A}=\frac{2.1449}{2.572}=0.834$