Table 1 List of notations

Each area of soil wedge in area A \(\mathrm {(m^2)}\)

\(A_{area}\)

Cohesion of the soil \(\mathrm {(N/m^2)}\)

c

Soil-tool adhesion \(\mathrm {(N/m^2)}\)

\(C_a\)

Soil-wheel adhesion \(\mathrm {(N/m^2)}\)

\(c_w\)

Diameter of the wheel \(\mathrm {(m)}\)

D

Total resistance force of the locked-wheel \(\mathrm {(N)}\)

\(F_{total}\)

Force against soil mass area B \(\mathrm {(N)}\)

\(F_1\)

Shear force beneath the soil wedge area A \(\mathrm {(N)}\)

\(F_2\)

Side friction force of the soil wedge area A \(\mathrm {(N)}\)

\(F_3\)

Side friction force of the side surface of the wheel \(\mathrm {(N)}\)

\(F_4\)

Friction force on the wheel surface \(\mathrm {(N)}\)

\(F_5\)

Total force acting on the plate \(\mathrm {(N)}\)

\(F_T\)

Horizontal component of the total force \(\mathrm {(N)}\)

\(F_{Th}\)

Earth gravity \(\mathrm {(m/s^2)}\)

g

Coefficient of earth pressure at rest \(\mathrm {(-)}\)

\(K_0\)

Soil modulus of deformation depend on c (\(\mathrm {N/m^{(n+1)}}\))

\(k_c\)

Internal friction angle modulus (\(\mathrm {N/m^{(n+2)}}\))

\(k_\phi \)

Tool length \(\mathrm {(m)}\)

l

Slip surface length \(\mathrm {(m)}\)

\(L_1\)

Base line length of the soil wedge in area A \(\mathrm {(m)}\)

\(l_s\)

Sinkage ratio \(\mathrm {(-)}\)

n

Terzahgi’s soil bearing capacity factors \(\mathrm {(-)}\)

\(N_\gamma , N_c, N_q, N_{ca}\)

Normal stress \({\mathrm {(N/m^2)})}\)

p

Surcharge on the soil surface \(\mathrm {(N/m^2)}\)

q

Surcharge loading \(\mathrm {(N)}\)

Q

Resultant force acting on slip surface \(\mathrm {(N)}\)

R

Wheel radius \(\mathrm {(m)}\)

r

Width of the plate or wheel \(\mathrm {(m)}\)

\(b, w_w\)

Weight of the plate \(\mathrm {(N)}\)

\(W_p\)

Weight of the soil wedge \(\mathrm {(N)}\)

\(W_s\), \(W_b\)

Weight of the wheel \(\mathrm {(N)}\)

\(W_h\)

Position toward the horizontal direction of the wheel\(\mathrm {(m)}\)

x

Sinkage \(\mathrm {(m)}\)

\( z_0, z\)

Rake angle \(\mathrm {(^\circ )}\)

\(\beta \)

Soil density \(\mathrm {(kg/m^3)}\)

\(\gamma \)

External friction angle \(\mathrm {(^\circ )}\)

\(\delta \)

Contact angle \(\mathrm {(^\circ )}\)

\(\theta \)

Static contact angle \(\mathrm {(^\circ )}\)

\(\theta _s\)

Shear plane failure angle \(\mathrm {(^\circ )}\)

\(\rho \)

Normal stress acting on wheel surface \(\mathrm {(N/m^2)}\)

\(\sigma \)

Horizontal normal stress \(\mathrm {(N/m^2)}\)

\(\sigma _h\)

Wheel weight per unit area \(\mathrm {(N/m^2)}\)

\(\sigma _b\)

Vertical normal stress \(\mathrm {(N/m^2)}\)

\(\sigma _v\)

Shear stress on the wheel surface \(\mathrm {(N/m)}\)

\(\tau \)

Internal friction angle \((^\circ )\)

\(\phi \)