Table 12. Depth variation functions for... | Emerald Publishing

Table 12.

Depth variation functions for Cowden till, Bothkennar clay and Dunkirk sand (GDSM) (calibrated for 2 ≤ L/D ≤ 6)

Soil reaction component	Parameter	Cowden till (Byrne et al., 2020a)	Bothkennar clay (AWG, 2018)	Dunkirk sand (GDSM) (Burd et al., 2020a)
Distributed lateral load, p	Ultimate displacement, ${\bar{v}}_{pu}$	241·4	173·8	146·1 − 92·11D_R
	Initial stiffness, k_p	$10 \cdot 60 - 1 \cdot 650 \frac{z}{D}$	$12 \cdot 05 - 1 \cdot 547 \frac{z}{D}$	$8 \cdot 731 - 0 \cdot 6982 D_{R} - 0 \cdot 9178 \frac{z}{D}$
	Curvature, n_p	$0 \cdot 9390 - 0 \cdot 03345 \frac{z}{D}$	$0 \cdot 7204 - 0 \cdot 002679 \frac{z}{D}$	0·917 + 0·06193D_R
	Ultimate reaction, ${\bar{p}}_{u}$	10·70 − 7·101e^{(−0·3085(z/D))}	7·743 − 3·945e^{((−0·8456)(z/D))}	$\begin{aligned} 0 \cdot 3667 + 25 \cdot 89 D_{R} \\ + (0 \cdot 3375 - 8 \cdot 900 D_{R}) \frac{z}{L} \end{aligned}$
Distributed moment, m	Ultimate rotation, ${\bar{ψ}}_{mu}$	${\bar{m}}_{u} / k_{m}$	${\bar{m}}_{u} / k_{m}$	${\bar{m}}_{u} / k_{m}$
	Initial stiffness, k_m	$1 \cdot 420 - 0 \cdot 09643 \frac{z}{D}$	$1 \cdot 698 - 0 \cdot 1576 \frac{z}{D}$	17·00
	Curvature, n_m	0·0	0·0	0·0
	Ultimate moment, ${\bar{m}}_{u}$	$0 \cdot 2899 - 0 \cdot 04775 \frac{z}{D}$	$0 \cdot 4862 - 0 \cdot 05674 \frac{z}{D}$	$0 \cdot 2605 - (0 \cdot 1989 - 0 \cdot 2019 D_{R}) \frac{z}{L}$
Base horizontal force, H_B	Ultimate displacement, ${\bar{v}}_{Hu}$	235·7	291·5	$\begin{aligned} 0 \cdot 5150 + 2 \cdot 883 D_{R} \\ + (0 \cdot 1695 - 0 \cdot 7018 D_{R}) \frac{L}{D} \end{aligned}$
	Initial stiffness, k_H	$2 \cdot 717 - 0 \cdot 3575 \frac{L}{D}$	$3 \cdot 008 - 0 \cdot 2701 \frac{L}{D}$	$\begin{aligned} 6 \cdot 505 - 2 \cdot 985 D_{R} \\ - (0 \cdot 007969 + 0 \cdot 4299 D_{R}) \frac{L}{D} \end{aligned}$
	Curvature, n_H	$0 \cdot 8793 - 0 \cdot 03150 \frac{L}{D}$	$0 \cdot 3113 + 0 \cdot 04263 \frac{L}{D}$	$\begin{aligned} 0 \cdot 09978 + 0 \cdot 7974 D_{R} \\ + (0 \cdot 004994 - 0 \cdot 07005 D_{R}) \frac{L}{D} \end{aligned}$
	Ultimate reaction, ${\bar{H}}_{Bu}$	$0 \cdot 4038 + 0 \cdot 04812 \frac{L}{D}$	$0 \cdot 5279 + 0 \cdot 06864 \frac{L}{D}$	$\begin{aligned} 0 \cdot 09952 + 0 \cdot 7996 D_{R} \\ + (0 \cdot 03988 - 0 \cdot 1606 D_{R}) \frac{L}{D} \end{aligned}$
Base moment, M_B	Ultimate rotation, ${\bar{ψ}}_{Mu}$	173·1	187·0	44·89
	Initial stiffness, k_M	$0 \cdot 2146 - 0 \cdot 002132 \frac{L}{D}$	$0 \cdot 3409 - 0 \cdot 01995 \frac{L}{D}$	0·3515
	Curvature, n_M	$1 \cdot 079 - 0 \cdot 1087 \frac{L}{D}$	$0 \cdot 6990 - 0 \cdot 1155 \frac{L}{D}$	0·300 + 0·4986D_R
	Ultimate moment, ${\bar{M}}_{Bu}$	$0 \cdot 8192 - 0 \cdot 08588 \frac{L}{D}$	$0 \cdot 8756 - 0 \cdot 09195 \frac{L}{D}$	$\begin{aligned} 0 \cdot 09981 + 0 \cdot 3710 D_{R} \\ + (0 \cdot 01998 - 0 \cdot 09041 D_{R}) \frac{L}{D} \end{aligned}$

Soil reaction component	Parameter	Cowden till (Byrne et al., 2020a)	Bothkennar clay (AWG, 2018)	Dunkirk sand (GDSM) (Burd et al., 2020a)
Distributed lateral load, p	Ultimate displacement, ${\bar{v}}_{pu}$	241·4	173·8	146·1 − 92·11D_R
	Initial stiffness, k_p	$10 \cdot 60 - 1 \cdot 650 \frac{z}{D}$	$12 \cdot 05 - 1 \cdot 547 \frac{z}{D}$	$8 \cdot 731 - 0 \cdot 6982 D_{R} - 0 \cdot 9178 \frac{z}{D}$
	Curvature, n_p	$0 \cdot 9390 - 0 \cdot 03345 \frac{z}{D}$	$0 \cdot 7204 - 0 \cdot 002679 \frac{z}{D}$	0·917 + 0·06193D_R
	Ultimate reaction, ${\bar{p}}_{u}$	10·70 − 7·101e^{(−0·3085(z/D))}	7·743 − 3·945e^{((−0·8456)(z/D))}	$\begin{aligned} 0 \cdot 3667 + 25 \cdot 89 D_{R} \\ + (0 \cdot 3375 - 8 \cdot 900 D_{R}) \frac{z}{L} \end{aligned}$
Distributed moment, m	Ultimate rotation, ${\bar{ψ}}_{mu}$	${\bar{m}}_{u} / k_{m}$	${\bar{m}}_{u} / k_{m}$	${\bar{m}}_{u} / k_{m}$
	Initial stiffness, k_m	$1 \cdot 420 - 0 \cdot 09643 \frac{z}{D}$	$1 \cdot 698 - 0 \cdot 1576 \frac{z}{D}$	17·00
	Curvature, n_m	0·0	0·0	0·0
	Ultimate moment, ${\bar{m}}_{u}$	$0 \cdot 2899 - 0 \cdot 04775 \frac{z}{D}$	$0 \cdot 4862 - 0 \cdot 05674 \frac{z}{D}$	$0 \cdot 2605 - (0 \cdot 1989 - 0 \cdot 2019 D_{R}) \frac{z}{L}$
Base horizontal force, H_B	Ultimate displacement, ${\bar{v}}_{Hu}$	235·7	291·5	$\begin{aligned} 0 \cdot 5150 + 2 \cdot 883 D_{R} \\ + (0 \cdot 1695 - 0 \cdot 7018 D_{R}) \frac{L}{D} \end{aligned}$
	Initial stiffness, k_H	$2 \cdot 717 - 0 \cdot 3575 \frac{L}{D}$	$3 \cdot 008 - 0 \cdot 2701 \frac{L}{D}$	$\begin{aligned} 6 \cdot 505 - 2 \cdot 985 D_{R} \\ - (0 \cdot 007969 + 0 \cdot 4299 D_{R}) \frac{L}{D} \end{aligned}$
	Curvature, n_H	$0 \cdot 8793 - 0 \cdot 03150 \frac{L}{D}$	$0 \cdot 3113 + 0 \cdot 04263 \frac{L}{D}$	$\begin{aligned} 0 \cdot 09978 + 0 \cdot 7974 D_{R} \\ + (0 \cdot 004994 - 0 \cdot 07005 D_{R}) \frac{L}{D} \end{aligned}$
	Ultimate reaction, ${\bar{H}}_{Bu}$	$0 \cdot 4038 + 0 \cdot 04812 \frac{L}{D}$	$0 \cdot 5279 + 0 \cdot 06864 \frac{L}{D}$	$\begin{aligned} 0 \cdot 09952 + 0 \cdot 7996 D_{R} \\ + (0 \cdot 03988 - 0 \cdot 1606 D_{R}) \frac{L}{D} \end{aligned}$
Base moment, M_B	Ultimate rotation, ${\bar{ψ}}_{Mu}$	173·1	187·0	44·89
	Initial stiffness, k_M	$0 \cdot 2146 - 0 \cdot 002132 \frac{L}{D}$	$0 \cdot 3409 - 0 \cdot 01995 \frac{L}{D}$	0·3515
	Curvature, n_M	$1 \cdot 079 - 0 \cdot 1087 \frac{L}{D}$	$0 \cdot 6990 - 0 \cdot 1155 \frac{L}{D}$	0·300 + 0·4986D_R
	Ultimate moment, ${\bar{M}}_{Bu}$	$0 \cdot 8192 - 0 \cdot 08588 \frac{L}{D}$	$0 \cdot 8756 - 0 \cdot 09195 \frac{L}{D}$	$\begin{aligned} 0 \cdot 09981 + 0 \cdot 3710 D_{R} \\ + (0 \cdot 01998 - 0 \cdot 09041 D_{R}) \frac{L}{D} \end{aligned}$

For the GDSM, relative density is expressed as a decimal in the calibration range 0·45 ≤ D_R ≤ 0·9.