Nail Forces and Locus of Maximum Tension of Nailed Cut Subjected to Seismic Excitations: A Calculus Approach with Log-Spiral Failure Surface

Data Availability Statement

Notation

a_h0, a_v0

amplitude of horizontal and vertical acceleration at the base of the slope;

c

soil’s cohesion intercept;

$F_c^H,F_c^V$

horizontal and vertical components of cohesive force $F_c^{}$;

F_q

force implied by live load surcharge (q);

FS_PO

factor of safety against pullout failure;

FS_OS

overall stability factor against rotational failure;

f_a

soil amplification factor;

G

shear modulus of the soil;

H

nailed cut depth;

i, j

counters used for depths and nails;

i_n

nail declination angle;

k_h

a_h0/g;

k_v

a_v0/g;

k_s1, k_s2, k_p1, k_p2

wave numbers related to S wave and P wave;

L_z

length of a horizontal differential strip;

M_c

total moment of cohesive forces about pole O;

Q_h, Q_v

horizontal and vertical inertia force;

q

live load surcharge intensity;

R

reaction by passive zone;

r, θ

polar coordinates of a point on log-spiral FS;

S_v

vertical nail spacing;

T_eq

total nail force demand;

T_eq,provided

total support force capacity of designed nail system;

$T_{j,m}^{*}$

dimensionless force demand of jth nail corresponding to θ_f,m and t_m;

$T_{j,max}^{*}$

dimensionless maximum force demand of jth nail corresponding to ${\theta }_{f,max}^j,t_{max}^j$;

t

time;

t_m

time corresponding to maximum $\sum T_j({\theta }_f,t)$;

$t_{max}^j$

time corresponding to $T_{j,max}^{*}$;

u_h, u_v

horizontal and vertical displacements;

V_s, V_p

velocity of S wave and P wave in the soil;

W

weight of failure wedge;

x_spiral, $x_{LMT}^j$

x-coordinate of a point on FS and LMT with Point B as an origin;

${\overline{x}}_W$, ${\overline{x}}_{F_q}$, ${\overline{x}}_{T_{eq}}$, ${\overline{y}}_{Q_h}$, ${\overline{y}}_{T_{eq}}$

lever arms of various forces about O;

z

depth variable measured from the top of a nailed cut;

z_jh

depth of jth nail’s head;

φ

soil’s angle of shearing resistance;

γ

soil’s unit weight;

η₁, η_s

viscosities of the soil;

λ

first Lame’s constant;

λ_s, λ_p

S-wave and P-wave wavelengths;

ρ

soil’s density;

ω_s, ω_p

S- and P-wave circular frequency of motion; and

ξ_s, ξ_p

damping ratios of soil in response to S wave and P wave.

References