Seismic Vibration Control of Retaining Walls Using a Compliant-Tuned Liquid Damper

Data Availability Statement

Acknowledgments

Notation

a

half of the width of the tank of the compliant TLD;

B

width of the base of the retaining wall;

b

length of the tank of the complaint TLD;

C_fr, C_da

coefficients related to wave breaking defined in Eqs. (10a) and (10b), respectively;

c

cohesion of the soil;

c_s, c_t

viscous damping coefficient of the structure and the elastomeric pad, respectively;

e

eccentricity;

F_SL

sloshing force;

${\hat{F}}_{SL}$

maximum sloshing force;

FS_s, FS₀

factor of safety in sliding and overturning, respectively;

g

acceleration due to gravity;

H

height of soil retained by the retaining wall;

h

undisturbed depth of liquid in the tank of the compliant TLD;

K_a

active earth pressure coefficient;

k

wave number;

k_s, k_t

stiffness of the structure and the elastomeric pad (combined assembly), respectively;

k_h

horizontal stiffness of a single elastomeric pad;

L

width of the tank of the compliant TLD;

M

sum of moments;

m_l

mass of liquid;

m₀

mass of the CTLD;

m_s

mass of the structure;

m_t

mass of the tank;

N

standard penetration test value;

n_p

number of elastomeric pads;

P_a

active earth pressure;

P₀

amplitude of excitation force used in sine sweep analysis;

q_max, q_min

maximum and minimum base pressures, respectively;

S

factor defining the surface contamination of the liquid in the compliant TLD;

T_h

term defined by Eq. (6c);

u

particle velocity at the free surface of the liquid in the compliant TLD;

V

sum of vertical load;

x, z

axes in the two-dimensional Cartesian coordinate system;

x_s, x_t

displacement of the structure and the tank, respectively;

x_tmax

maximum displacement of the tank;

${\dot{x}}_s,{\dot{x}}_t$

velocity of the structure and the tank, respectively;

${\ddot{x}}_s$, ${\ddot{x}}_t$

acceleration of the structure and the tank, respectively;

${\ddot{x}}_s^c$, ${\ddot{x}}_s^{uc}$

response (displacement or acceleration) of the controlled and uncontrolled structure, respectively;

${\ddot{x}}_g$

acceleration obtained from the earthquake ground motion database;

β

frequency tuning ratio;

γ

unit weight of the soil;

Δ

depth ratio;

δ

angle of friction between retaining wall and backfill soil;

ζ, ζ_b

damping ratio of the structure and elastomeric pad, respectively;

η

free surface elevation above the liquid surface;

η_n, η₀

free surface elevations in the right and left wall of the tank, respectively;

λ

term that accounts for the damping caused by boundary layer effects in the tank, as well as free surface contamination;

ν

kinematic viscosity of the liquid in the tank;

ρ

density of the liquid in the tank;

σ

term defined in Eq. (6a);

ϕ

angle of internal frictional of soil;

$\tilde{\varphi }$

mode shapes (eigenvector);

φ

term defined in Eq. (6b);

ω

frequency of the excitation force used in sine sweep analysis;

ω_s

fundamental frequency (or angular frequency) of vibration of the structure; and

ω_t

frequency of vibration of the tank of the compliant TLD.

References