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Question 6.3: A rectangular raft of size 30 x 12 m founded at a depth of 2...

A rectangular raft of size 30 x 12 m founded at a depth of 2.5 m below the ground surface is subjected to a uniform pressure of 150 kPa. Assume the center of the area is the origin of coordinates (0, 0). and the corners have coordinates (6, 15). Calculate stresses at a depth of 20 m below the foundation level by the methods of (a) Boussinesq, and (b) Westergaard at coordinates of (0, 0), (0, 15), (6, 0) (6, 15) and (10, 25). Also determine the ratios of the stresses as obtained by the two methods. Neglect the effect of foundation depth on the stresses (Fig. Ex. 6.3).

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Equations (a) Boussinesq: \sigma_{z}=\frac{Q}{z^{2}} I_{B}, I_{B}=\frac{0.48}{\left[1+(r / z)^{2}\right]^{5 / 2}}

 

(b) Westergaard: \sigma_{z}=\frac{Q}{z^{2}} I_{w}, I_{w}=\frac{0.32}{\left[1+2(r / z)^{2}\right]^{3 / 2}}

 

The ratios of r/z at the given locations for z = 20 m are as follows:

 

Location r/z Location r/z
(0, 0) 0 (6, 15) \left(\sqrt{6^{2}+15^{2}}\right) / 20=0.81
(6, 0) 6/20 = 0.3 (10, 25) \left(\sqrt{10^{2}+25^{2}}\right) / 20=1.35
(0, 15) 15/20 = 0.75

 

The stresses at the various locations at z = 20 m may be calculated by using the equations given above. The results are tabulated below for the given total load Q=q B L=150 \times 12 \times 30=54000 kN acting at (0, 0) coordinate. Q / z^{2}=135.

 

Location r/z Boussinesq Westergaard \sigma_{B} / \sigma_{w}
I_{B} \sigma_{B}( k Pa ) I_{W} \sigma_{w}( kPa )
(0,0) 0 0.48 65 0.32 43 1.51
(6,0) 0.3 0.39 53 0.25 34 1.56
(0, 15) 0.75 0.16 22 0.1 14 1.57
(6,15) 0.81 0.14 19 0.09 12 1.58
(10, 25) 1.35 0.036 5 0.03 4 1.25

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