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

Question

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).

Accepted Answer

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

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

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)	[latex]\left(\sqrt{6^{2}+15^{2}} ight) / 20=0.81[/latex]
(6, 0)	6/20 = 0.3	(10, 25)	[latex]\left(\sqrt{10^{2}+25^{2}} ight) / 20=1.35[/latex]
(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 [latex]Q=q B L=150 imes 12 imes 30=54000[/latex] kN acting at (0, 0) coordinate. [latex]Q / z^{2}=135[/latex].

Location	r/z	Boussinesq		Westergaard		[latex]\sigma_{B} / \sigma_{w}[/latex]
		[latex]I_{B}[/latex]	[latex]\sigma_{B}( k Pa )[/latex]	[latex]I_{W}[/latex]	[latex]\sigma_{w}( kPa )[/latex]
(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

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

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