Coupled Analytical Method for Braced Excavation Based on the Pasternak Foundation Model and Nonlinear p–y Curve Model

Data Availability Statement

Acknowledgments

Notation

A_L

gradient of P_u;

A_z

cross-sectional area of the support;

B

calculated width of the pile or wall;

b_a

calculated width of the retaining structure;

E_s

elastic modulus of the soil;

E_z

elastic modulus of the support material;

E_pI_p

flexural stiffness of the pile or wall;

G_c

shear stiffness of the soil;

[G_s]

shear layer stiffness of the soil in the embedded section;

[G_t]

shear layer stiffness of the soil in the critical section;

K

foundation reaction modulus;

K₀

initial foundation reaction modulus;

[K_s]

pile deformation matrix in the embedded section;

[K_t]

pile deformation matrix in the critical section;

k

number of iterations;

k′

foundation scale factor;

k_R

support stiffness coefficient;

l₀

support length;

M_si

bending moment of the pile in the embedded section;

M_ti

bending moment of the pile in the critical section;

N_g

gradient to correlate clay strength or sand density with P_u;

p(z)

soil pressure acting on the pile or wall;

P_u

horizontal ultimate resistance of the soil;

{P₁}

soil pressure vector behind the pile in the critical section;

{P₂}

solution complementary vectors of the pile in the critical section;

q_ur

uniaxial compressive strength;

q(x, z)

foundation reaction force;

Q_si

shear force of the pile in the embedded section;

Q_ti

shear force of the pile in the critical section;

{Q₁}

soil pressure vector behind the pile in the embedded section;

{Q₂}

solution complementary vectors of the pile in the embedded section;

S_u

average undrained shear strength;

s

horizontal support spacing;

T_i

support or anchor prestress;

t_p

shear thickness of the soil;

w(z)

foundation deformation or excavation deformation;

{w_s}

pile displacement vector in the embedded section;

{w_t}

pile displacement vector in the critical section;

{w_t}^(k)

pile displacements of the critical section at the kth iterations;

{w_s}^(k)

pile displacements of the embedded section at the kth iterations;

Z₁

excavation length in the critical section;

Z₂

excavation length in the embedded section;

Δp(z)

difference in soil pressure;

α₀

equivalent depth to include the force at the ground level;

α_R

support relaxation factor;

$\epsilon$

calculation accuracy;

γ_s

unit weight of soil;

φ_ti

rotation angle of the pile in the critical section;

φ_si

rotation angle of the pile in the embedded section;

λ

adjustment factor of the support; and

μ_s

Poisson's ratio of the soil.

References