Compute the factor of safety against a bearing capacity failure for the square spread footing shown in Figure 17.7 with the groundwater table at Position A.

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Accepted Answer

[latex]\sigma ^{\prime }_{D}=\gamma D - u=\left(121 lb/ft^{3}\right) \left(2 ft\right) - 0=242 lb/ft^{2}[/latex]

[latex]D=[/latex] 2 ft

[latex]D_{w}=[/latex] 7 ft

[latex]D+B=[/latex] 6 ft

[latex]D + B ≤ D_{w}[/latex], so groundwater case 3 applies - [latex]γ^{\prime }=γ[/latex]

Per Table 17.1

[latex]\Phi ^{\prime }[/latex] (deg)	[latex]N_{c}[/latex]	[latex]N_{q}[/latex]	[latex]N_{\gamma }[/latex]
0	5.7	1.0	0.0
1	6.0	1.1	0.1
2	6.3	1.2	0.1
3	6.6	1.3	0.2
4	7.0	1.5	0.3
5	7.3	1.6	0.4
6	7.7	1.8	0.5
7	8.2	2.0	0.6
8	8.6	2.2	0.7
9	9.1	2.4	0.9
10	9.6	2.7	1.0
11	10.2	3.0	1.2
12	10.8	3.3	1.4
13	11.4	3.6	1.6
14	12.1	4.0	1.9
15	12.9	4.4	2.2
16	13.7	4.9	2.5
17	14.6	5.5	2.9
18	15.5	6.0	3.3
19	16.6	6.7	3.8
20	17.7	7.4	4.4
21	18.9	8.3	5.1
22	20.3	9.2	5.9
23	21.7	10.2	6.8
24	23.4	11.4	7.9
25	25.1	12.7	9.2
26	27.1	14.2	10.7
27	29.2	15.9	12.5
28	31.6	17.8	14.6
29	34.2	20.0	17.1
30	37.2	22.5	20.1
31	40.4	25.3	23.7
32	44.0	28.5	28.0
33	48.1	32.2	33.3
34	52.6	36.5	39.6
35	57.8	41.4	47.3
36	63.5	47.2	56.7
37	70.1	53.8	68.1
38	77.5	61.5	82.3
39	86.0	70.6	99.8

[latex]N_{c}=40.4[/latex], [latex]N_{q}=25.3[/latex], [latex]N_{γ}= 23.7[/latex] when [latex]Φ^{\prime }=31°[/latex]

[latex]q_{ult}=1.3c^{\prime }N_{c} + σ^{\prime }_{D}N_{q} + 0.4γ^{\prime }BN_{γ}=0 + (242 lb/ft^{2})(25.3) + 0.4(121 lb/ft^{2})(4 ft)(23.7)=10,710 lb/ft^{2}[/latex]

[latex]q=\frac{P}{A} +\gamma _{c}D - u=\frac{76,000 lb}{\left(4 ft\right) ^{2}} +\left(150 lb/ft^{3}\right) \left(2 ft\right) - 0=5050 lb/ft^{2}[/latex]

Because we are computing the factor of safety for a given bearing pressure, we rewrite Equation 17.11 [latex]q_{a}=\frac{q_{ult}}{F} [/latex] in terms of [latex]q[/latex] instead of [latex]q_{a}[/latex]:

[latex]F=\frac{q_{ult}}{q} =\frac{10,710 lb/ft^{2}}{5,050 lb/ft^{2}} =[/latex] 2.1

Comments

For clean sands, a factor of safety of 2.1 would probably be marginally acceptable.

Question 17.1: Compute the factor of safety against a bearing capacity fail...

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