Reinforcement strength and length requirement of each layer in a nailed slope subjected to seismic loading

ABSTRACT

Estimating support force and length requirement of each reinforcement layer based on limit equilibrium framework remains a statically indeterminate problem and consideration of seismic loading further increases its complexity. The present study addresses it by proposing a novel calculus-based methodology based upon the modified pseudo-dynamic approach to analyse the internal stability of nailed slopes. The effects of slope angle, shear strength parameters of soil, motion parameters, nail orientation, and surcharge were discussed. Though the deeper nail layers are observed to require more support forces, their active zone nail length requirement is lesser. This trend combined with the higher nail-soil bond strengths due to higher vertical stresses suggests that the bottommost nail layer need not be the longest. Both possible directions of initial vertical acceleration must be considered with and without surcharge for designs to arrive at the largest length requirement of each nail layer. The closed-form solutions presented herein to estimate layer-wise strength and length requirement would benefit nailed slope design practitioners.

KEYWORDS:

• Limit equilibrium methods
• dynamics
• soil nailing
• optimisation
• ground improvement

List of notations

$a_{h0},a_{v0}$	=	Amplitude of horizontal and vertical acceleration at the base
$c$	=	Cohesion intercept of soil
$F_c$	=	Resisting force due to cohesion
$F_q$	=	Force due to surcharge (q)
$F{S}_{PO},F{S}_{OS}$	=	Safety factor against pullout and overall stability
i, j	=	Numerical counters employed for depths and nails
$i_n$	=	Nail declination angle
H	=	Height of slope
$k_h$	=	$a_{h0}/g$
$k_v$	=	$a_{v0}/g$
$k_{s1},k_{s2},k_{p1},k_{p2}$	=	Wave numbers of S wave and P wave
$L_{A,max}^j$ , $L_{P,max}^j$	=	Active and passive zone nail length requirements of jth nail layer
$L_c$	=	Distance CD in Figure 1 normalised by dividing with H.
LMT	=	Locus of maximum tension
MPSD	=	Modified Pseudo-dynamic approach
PSD	=	Conventional Pseudo-dynamic approach
$Q_h,Q_v$	=	Horizontal and Vertical inertia force
$R$	=	Reaction by passive zone
$S_v,S_h$	=	Vertical and horizontal nail spacing
Teq	=	Total support force required for slope stability
$T_{eq,max}$	=	Maximum $T_{eq}$ corresponding to ${\alpha }_m$ and $t_m$
$T_p\mathrm{a}\mathrm{n}\mathrm{d}{T}_s$	=	Period of P wave and S wave.
$T_{jm}^{\ast }$	=	Dimensionless nail force required in jth nail corresponding to ${\alpha }_m$ and tm
$T_{j,max}^{\ast }$	=	Dimensionless maximum support force required by jth nail corresponding to ${\alpha }_{max}^j$ and $t_{max}^j$
$t$	=	Time elapsed after start of base vibrations
$V_S,V_P$	=	Velocity of shear and primary wave in the soil
W	=	Weight of trial failure wedge
$z$	=	Depth variable measured from the top of slope
$z_h^j$	=	Nail head depth for jth nail layer
$\alpha$	=	Inclination of failure surface with horizontal
${\alpha }_m$	=	${\alpha }_{\hspace{0.17em}}$ corresponding to $T_{eq,max}$
${\alpha }_{max}^j$	=	${\alpha }_{\hspace{0.17em}}$ corresponding to $T_{j,max}^{\hspace{0.17em}}$
$\beta$	=	Inclination of slope face with horizontal
$\gamma$	=	Unit weight of soil
${\lambda }_s$ , ${\lambda }_p$	=	S-wave and P-wave wavelengths.
${\xi }_s,{\xi }_p$	=	Damping ratios of soil in response to S-wave and P-wave
$\phi$	=	Angle of shearing resistance of soil
${\omega }_s,{\omega }_p$	=	Circular frequencies of S-wave and P-wave motion

Acknowledgments

The authors acknowledge the QIP initiative administered by AICTE, Government of India, New Delhi, and the Government of Maharashtra for their support in this research work.

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.