Introduction
The Convergence-Confinement Method (CCM) is a widely used analytical approach in
tunnel engineering that helps in understanding the interaction between the
surrounding ground and the tunnel support system during excavation. This method
is particularly useful in the design and construction of tunnels in rock masses,
where the behavior of the ground and the effectiveness of the support systems
need to be carefully evaluated to ensure stability and safety.
In this series of articles, we are going to discuss these following concepts in CCM:
Stress-Displacement Development along The Radial Axis
%reset -f import numpy as np import matplotlib.pyplot as plt import matplotlib as mpl
def displacement_ratio(r_R,R,lamb,lambda_e,sigma_0,Kp,Kpsi,G,nu):
Rp_R = ((2/(Kp + 1))*((Kp - 1)*sigma_0 + sigma_c)/
((1 - lamb)*(Kp - 1)*sigma_0 + sigma_c))**(1/(Kp - 1))
F1 = -(1 - 2*nu) * (Kp + 1) / (Kp - 1)
F2 = 2*(1 + Kp*Kpsi - nu*(Kp + 1)*(Kpsi + 1))/(Kp - 1)/(Kp + Kpsi)
F3 = 2*(1 - nu)*(Kp + 1)/(Kp + Kpsi)
u_R_fullelastic = lamb*sigma_0/(2*G)/r_R
u_R_elastic = lambda_e*sigma_0/(2*G)*(Rp_R/r_R)*Rp_R
u_R_plastic = (lambda_e*sigma_0/(2*G))*(F1 + F2*((Rp_R/r_R)**(-Kp + 1)) +
F3*(Rp_R/r_R)**(Kpsi + 1))*r_R
if lamb < lambda_e: # Full Elastic Zone
return u_R_fullelastic
if r_R >= Rp_R: # Elastic Part of Elastic-Plastic Zone
return u_R_elastic
else: # Plastic Zone
return u_R_plastic
# Based on equation 4.
def sigma_rθ(r_R,R,lamb,lambda_e,sigma_0,sigma_c,Kp):
Rp_R = ((2/(Kp + 1))*((Kp - 1)*sigma_0 + sigma_c)/
((1 - lamb)*(Kp - 1)*sigma_0 + sigma_c))**(1/(Kp - 1))
Rp = Rp_R*R
r = r_R*R
sigma_r_plastic = ((sigma_c/(Kp - 1))*((r/R)**(Kp - 1) - 1) +
(1 - lamb)*sigma_0*(r/R)**(Kp - 1))
sigma_θ_plastic = ((sigma_c/(Kp - 1))*(Kp*(r/R)**(Kp - 1) - 1) +
Kp*(1 - lamb)*sigma_0*(r/R)**(Kp - 1))
sigma_r_elastic = (1 - lambda_e*(Rp**2)/(r**2))*sigma_0
sigma_θ_elastic = (1 + lambda_e*(Rp**2)/(r**2))*sigma_0
# Fully elastic
if lamb < lambda_e:
return (1-lamb*r_R**(-2))*sigma_0, (1+lamb*r_R**(-2))*sigma_0, Rp_R
# Not Fully Elastic
if r_R >= Rp_R: # elastic
return sigma_r_elastic, sigma_θ_elastic, Rp_R
else: # plastic
return sigma_r_plastic, sigma_θ_plastic, Rp_R
###############################################################################
# Soil Parameter
gamma = 20.0
ϕ = 30
ψ = 30
cohesion = 974 #kpa
G = 126164.0 #kpa
nu = 0.28
# Lateral Stress Coefficient
Kp = np.tan(np.pi/4 + np.deg2rad(ϕ)/2)**2
Kpsi = np.tan(np.pi/4 + np.deg2rad(ψ)/2)**2
# Mohr Coulomb Parameter
sigma_c = 2*cohesion * np.cos(np.deg2rad(ϕ)) / (1 - np.sin(np.deg2rad(ϕ)))
################################################################################
# CCM Calculation
N = 4 # stability Number
σ0 = N*sigma_c/2
λe = 1/(Kp + 1) * (Kp - 1 + sigma_c/σ0)
R = 5 # radius dinding [meter]
λs = np.linspace(0, 1, 10000)
r_Rs = np.linspace(1.0, 1.5, 5)
σ_r_collect = []
σ_θ_collect = []
for r_R in r_Rs:
u_R = np.array([
displacement_ratio(r_R=r_R,R=R,lamb=λ,lambda_e=λe,
sigma_0=σ0,Kp=Kp,Kpsi=Kpsi,G=G,nu=nu)
for λ in λs]) *2*G/σ0
σ_rθ = np.array([
sigma_rθ(r_R=r_R,R=R,lamb=λ,lambda_e=λe,sigma_0=σ0,sigma_c=sigma_c,Kp=Kp)
for λ in λs])/σ0
σ_r_collect.append(σ_rθ[:,0])
σ_θ_collect.append(σ_rθ[:,1])
u_Rs_at_λe = [displacement_ratio(
r_R=r_R,
R=R,
lamb=λe,
lambda_e=λe,
sigma_0=σ0,
Kp=Kp,
Kpsi=Kpsi,
G=G,
nu=nu
)*2*G/σ0 for r_R in r_Rs]
for index, (σ_r, σ_θ) in enumerate(zip(σ_r_collect, σ_θ_collect)):
cmap = mpl.colormaps["tab20b"].colors
ls = '-.' if index>0 else '-'
label_r = "Radial Stress ($σ_r$)" if index==0 else ""
label_θ = "Tangential Stress ($σ_θ$)" if index==0 else ""
plt.plot(u_R, σ_r, c=cmap[-index], ls=ls, label=label_r)
plt.plot(u_R, σ_θ, c=cmap[-index], ls=ls, label=label_θ)
for index, u_R_at_λe in enumerate(u_Rs_at_λe):
label = "λe" if index==0 else ""
plt.axvline(u_R_at_λe, label=label, c='k', ls='-.', lw='.7')
plt.xlabel(r"Displacement Ratio $\left(\frac{u}{R}\right)$")
plt.ylabel("Stresses (σ)")
plt.title("Stress - Displacement Curve at Wall ($r = R$)")
plt.xlim(0,1.2)
plt.ylim(0,2)
plt.grid()
plt.legend()
plt.show()