Table 1 Stiffnesses of circular footings...

Table 1

Stiffnesses of circular footings on FILE materials

Symbol hereina	Case ID	Description	Equationb	By; notes
K_vv	VS	Vertical displacement, smooth interface	$\frac{2 G B}{1 - μ}$	Boussinesq (1878); exact
K_vv	VR	Vertical displacement, rough interface	$\frac{2 G B}{1 - 2 μ} \ln (3 - 4 μ)$	Spence (1968); exact
K_hh	HR	Horizontal displacement, rough interface	$\frac{4 G B}{2 - μ}$	Gerrard and Harrison (1970); approximatec
			$\frac{16 G B (1 - μ)}{7 - 8 μ}$	Bycroft (1956), Poulos and Davis (1974); approximatec
			$\frac{G B}{2 - μ} (2 + \frac{5 π}{8})$	Gazetas and Tassoulas (1987); approximatec
K_mm	MS	Overturning displacement, smooth interface	$\frac{G B}{3 (1 - μ)}$	Borowicka (1943); exact
K_mm	MR	Overturning displacement, rough interface	(Graphical presentation)	Numerical results; Bell (1991)c
K_hm = K_mh	MR, HR	Rough interface	Up to 10% of direct stiffnesses (drained), 0 undrained	Numerical results; Bell (1991)c
K_tt	TR	Torsional displacement, rough interface	$\frac{2 G B}{3}$	Reissner and Sagoci (1944); exact

Symbol hereina	Case ID	Description	Equationb	By; notes
K_vv	VS	Vertical displacement, smooth interface	$\frac{2 G B}{1 - μ}$	Boussinesq (1878); exact
K_vv	VR	Vertical displacement, rough interface	$\frac{2 G B}{1 - 2 μ} \ln (3 - 4 μ)$	Spence (1968); exact
K_hh	HR	Horizontal displacement, rough interface	$\frac{4 G B}{2 - μ}$	Gerrard and Harrison (1970); approximatec
			$\frac{16 G B (1 - μ)}{7 - 8 μ}$	Bycroft (1956), Poulos and Davis (1974); approximatec
			$\frac{G B}{2 - μ} (2 + \frac{5 π}{8})$	Gazetas and Tassoulas (1987); approximatec
K_mm	MS	Overturning displacement, smooth interface	$\frac{G B}{3 (1 - μ)}$	Borowicka (1943); exact
K_mm	MR	Overturning displacement, rough interface	(Graphical presentation)	Numerical results; Bell (1991)c
K_hm = K_mh	MR, HR	Rough interface	Up to 10% of direct stiffnesses (drained), 0 undrained	Numerical results; Bell (1991)c
K_tt	TR	Torsional displacement, rough interface	$\frac{2 G B}{3}$	Reissner and Sagoci (1944); exact

Superscript u may be used for undrained conditions

All equations assume full contact between the footing base and soil. μ = 0.5 for undrained conditions

Improved formulae are developed in this paper; see Equations 71–74

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