Table 3 Radial, circumferential and... | Emerald Publishing

Table 3

Radial, circumferential and vertical displacements of the material of the half-space

	FILE half-space	TILE half-space
Vertical point load P_z	$U_{r} = \frac{P_{z}}{4 π G R} [\frac{r z}{R^{2}} - \frac{(1 - 2 μ) r}{R + z}]$	$U_{r} = \frac{P_{z}}{4 π G_{vh}} \sum_{i = 1}^{2} (- λ_{i}^{'} \frac{R_{i}^{*}}{r R_{i}})$
	U_θ = 0	U_θ = 0
	$U_{z} = \frac{P_{z}}{4 π G R} [\frac{z^{2}}{R^{2}} + 2 (1 - μ)]$	$U_{z} = \frac{P_{z}}{4 π G_{vh}} \sum_{i = 1}^{2} (\frac{- m_{i} λ_{i}^{'}}{R_{i}})$
	Boussinesq (1875)	Liao and Wang (1998), interpreted by Dean (2023)
Horizontal (+x) point load P_x	$U_{r} = \frac{P_{x} \cos θ}{4 π G R} [1 + \frac{r^{2}}{R^{2}} - \frac{(1 - 2 μ) z}{R + z}]$	$U_{r} = \frac{P_{x} \cos θ}{4 π G_{vh}} {(\sum_{i = 1}^{2} [- \frac{λ_{i}}{m_{i}} \frac{z_{i} R_{i}^{}}{r^{2} R_{i}}]) + \frac{2}{u_{3}} \frac{R_{3}^{}}{r^{2}}}$
	$U_{θ} = \frac{- P_{x} \cos θ}{4 π G R} [1 + \frac{(1 - 2 μ) R}{R + z}]$	$U_{θ} = \frac{P_{x} \cos θ}{4 π G_{vh}} {(\sum_{i = 1}^{2} [\frac{λ_{i}}{m_{i}} \frac{R_{i}^{}}{r^{2}}]) - \frac{2}{u_{3}} \frac{z_{3} R_{3}^{}}{r^{2} R_{3}}}$
	$U_{z} = \frac{P_{x} \cos θ}{4 π G R} [\frac{r z}{R^{2}} + \frac{(1 - 2 μ) r}{R + z}]$	$U_{z} = \frac{P_{x}}{4 π G_{vh}} \sum_{i = 1}^{2} (\frac{- m_{i} λ_{i}^{'}}{R_{i}})$
	Cerruti (1882–1885)	Liao and Wang (1998), interpreted by Dean (2023)
See also	Westergaard (1952), Davis and Selvadurai (1996), Podio-Guidugli and Favata (2014)	Gerrard and Wardle (1973), Anyaegbunam (2014). Note: $λ_{i}^{'} = λ_{3 - i}$ in formulae for vertical load