A concentrated load of 45000 Ib acts at foundation level at a depth of 6.56 ft below ground surface. Find the vertical stress along the axis of the load at a depth of 32.8 ft and at a radial distance of 16.4 ft at the same depth by (a) Boussinesq, and (b) Westergaard formulae for μ = 0. Neglect the

Question

A concentrated load of 45000 Ib acts at foundation level at a depth of 6.56 ft below ground surface. Find the vertical stress along the axis of the load at a depth of 32.8 ft and at a radial distance of 16.4 ft at the same depth by (a) Boussinesq, and (b) Westergaard formulae for μ = 0. Neglect the depth of the foundation.

Accepted Answer

(a) Boussinesq Eq. (6.la)

[latex]I_{B}= ext { Boussinesq stress coefficient }=\frac{3}{2 \pi} \frac{1}{\left[1+(r / z)^{2} ight]^{5 / 2}}[/latex] (6.1a)

[latex]\sigma_{z}=\frac{Q}{z^{2}} I_{B}, I_{B}=\frac{3}{2 \pi} \frac{1}{1+(r / z)^{2}}^{5 / 2}[/latex]

Substituting the known values, and simplifying

[latex]I_{B}=0.2733 ext { for } r / z=0.5[/latex]

[latex]\sigma_{z}=\frac{45000}{(32.8)^{2}} imes 0.2733=11.43 lb / ft ^{2}[/latex]

(b) Westergaard (Eq. 6.3)

[latex]\sigma_{z}=\frac{Q}{\pi z^{2}} \frac{1}{\left[1+2(r / z)^{2} ight]^{3 / 2}}=\frac{Q}{z^{2}} I_{w}[/latex] (6.3)

[latex]\sigma_{z}=\frac{Q}{z^{2}} I_{w}, I_{w}=\frac{1}{\pi}\left[\frac{1}{1+2(r / z)^{2}} ight]^{3 / 2}[/latex]

Substituting the known values and simplifying, we have,

[latex]I_{w}=0.1733 ext { for } r / z=0.5[/latex]

therefore,

[latex]\sigma_{z}=\frac{45000}{(32.8)^{2}} imes 0.1733=7.25 lb / ft ^{2}[/latex]

Question 6.2: A concentrated load of 45000 Ib acts at foundation level at ...

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