A Numerical and Analytical Study on the Bearing Capacity of Two Neighboring Shallow Strip Foundations on Sand

Abstract

The bearing capacity of shallow foundations is affected by both the shear strength and the geometry of the problem. In this regard, the neighboring of two or more shallow foundations can affect their bearing capacity, which has been studied in this research. The finite element and the limiting equilibrium methods are applied to investigate the problem in detail. In traditional bearing capacity problem, the effect of a uniformly distributed surcharge at the level of the footing base or, at the ground level, which is governed by the second bearing capacity term has been well studied. The main emphasis of this research is on the effect of a finite area in the proximity of the footing, which is loaded uniformly. The main goal of this research is to estimate the bearing capacity of neighboring strip footings on sand and to find distances maximizing or minimizing the ultimate bearing pressure. A series of correction factors have been presented which represent the effect of neighboring of two footings and they can be used to find the reduced and/or increased bearing capacity.

Appendices

Appendix A

1.1 First Case

By using the equilibrium of forces in Fig. 9, formulas (13–19) are obtained:

$$2q_{\gamma } B + 2W_{\text{I}} + 2W_{\text{II}} + 2W_{\text{III}} = 2P_{P\gamma }$$

(13)

$$W_{\text{I}} = \frac{{\gamma B^{2} \tan \phi }}{4}$$

(14)

$$W_{\text{II}} = \frac{\gamma SB\tan \phi }{4}$$

(15)

$$W_{\text{III}} = \frac{{\gamma B^{2} {\text{e}}^{{\left( {\frac{3\pi }{4} - \frac{\phi }{2}} \right)^{2} \tan^{2} \phi }} \times \left( {\frac{3\pi }{4} - \frac{\phi }{2}} \right)}}{{8\cos^{2} \phi }}$$

(16)

$$P_{p\gamma } = \frac{1}{2}K_{p\gamma } \gamma h^{2} = \frac{1}{2}K_{p\gamma } \gamma \left( {\frac{B\tan \phi }{2}} \right)^{2} = \frac{1}{8}\gamma B^{2} \tan^{2} \left( {45 + \frac{\phi }{2}} \right)\tan^{2} \phi$$

(17)

Substituting Eqs. (14–17) and (S = Bα) into Eq. (13) results in Eq. (18):

$$\begin{aligned} q_{\gamma } & = \frac{1}{2}\gamma B\left[ {\frac{1}{4}\tan^{2} \left( {45 + \frac{\phi }{2}} \right)\tan^{2} \phi - \frac{\tan \phi }{2}} \right. \\ & \quad \left. { - \frac{{{\text{e}}^{{\left( {\frac{3\pi }{4} - \frac{\phi }{2}} \right)^{2} \tan^{2} \phi }} }}{{4\cos^{2} \phi }}\left( {\frac{3\pi }{4} - \frac{\phi }{2}} \right) - \frac{\alpha \tan \phi }{2}} \right] \\ \end{aligned}$$

(18)

In Eq. (19), the bearing capacity formula for two closely spaced footings is presented.

$$\begin{aligned} q_{\gamma } = \frac{1}{2}\gamma B[\frac{1}{4}\lambda tan^{2} \left( {45 + \frac{\phi }{2}} \right)tan^{2} \phi - \frac{\tan \phi }{2} \hfill \\ - \frac{{e^{{\left( {\frac{3\pi }{4} - \frac{\phi }{2}} \right)^{2} tan^{2} \phi }} }}{{4cos^{2} \phi }}\left( {\frac{3\pi }{4} - \frac{\phi }{2}} \right) - \frac{\alpha \tan \phi }{2}] \hfill \\ \end{aligned}$$

(19)

Fig. 9

A possible failure mechanism of two neighboring footings resting on sand (s/B ≥ 4)

Appendix B

2.1 Third Case

By using the equilibrium of forces in Fig. 10, formulas (20–24) are obtained:

$$q_{\gamma } B + W_{I} = 2P_{P\gamma }$$

(20)

$$W_{\text{I}} = \gamma B^{2} \tan \phi$$

(21)

$$P_{p\gamma } = \frac{1}{2}K_{p\gamma } \gamma h^{2} = \frac{1}{2}K_{p\gamma } \gamma \left( {\frac{B\tan \phi }{2}} \right)^{2} = \frac{1}{8}\gamma B^{2} \tan^{2} \left( {45 + \frac{\phi }{2}} \right)\tan^{2} \phi$$

(22)

Substituting Eqs. (21 and 22) and (S = Bα) into Eq. (20) results in Eq. (23):

$$q_{\gamma } = \frac{1}{2}\gamma B\left[ {\frac{1}{2}\tan^{2} \left( {45 + \frac{\phi }{2}} \right)\tan^{2} \phi - 2\tan \phi } \right]$$

(23)

In Eq. (24), the bearing capacity formula for two closely spaced footings is presented.

$$q_{\gamma } = \frac{1}{2}\gamma B\left[ {\frac{1}{2}\lambda \tan^{2} \left( {45 + \frac{\phi }{2}} \right)\tan^{2} \phi - 2\tan \phi } \right]$$

(24)

Fig. 10

A possible failure mechanism of two neighboring footings resting on sand (s/B = 0)