Civil, Structural & Geotechnical Engineering
Gibson-Anderson & Vesić Undrained Soil Cavity Expansion Limit Pressure Formula
Calculate undrained clay Rigidity Index I_r = G / s_u, Cylindrical Limit Pressure p_lim,cyl = p_0 + s_u * (1 + LN(I_r)) (pressuremeter / pile shaft), and Spherical Limit Pressure p_lim,sph = p_0 + (4/3) * s_u * (1 + LN(I_r)) (deep pile tip).
Why soil stiffness G / s_u governs Menard Pressuremeter limit pressure and deep pile tip bearing!
Rigid-plastic Prandtl bearing capacity ignores elastic compressibility outside the plastic zone — yet expanding a borehole pressuremeter or driving a deep pile tip must push surrounding clay radially outward against elastic shear modulus G! Derived by R. E. Gibson and W. F. Anderson (1961) and generalized by Aleksandar S. Vesić (1972, ASCE JSMFD), Undrained Tresca Cavity Expansion depends directly on the Soil Rigidity Index I_r = G / s_u (requiring I_r >= 1 so an outer elastic zone exists)!
Expanding a cylindrical borehole from finite radius to infinity reaches exact asymptotic limit pressure p_lim,cyl = p_0 + s_u * (1 + LN(G / s_u)), while expanding a spherical cavity (modeling cone penetration q_c and deep pile end-bearing) reaches p_lim,sph = p_0 + (4 / 3) * s_u * (1 + LN(G / s_u)), yielding Vesić’s compressible deep-foundation bearing factor N_c* = (4 / 3) * (1 + LN(I_r)) + 1!
| Cavity Expansion Quantity (p_0 = 120 kPa, s_u = 60 kPa, G = 5,400 kPa) | Formula | Evaluated Value |
|---|---|---|
| Undrained Rigidity Index I_r | G / s_u | 90.0 |
| Cylindrical Limit Pressure p_lim,cyl (Pressuremeter) | p_0 + s_u * (1 + LN(G / s_u)) | 449.99 kPa |
| Spherical Limit Pressure p_lim,sph (Deep Pile Tip) | p_0 + (4/3) * s_u * (1 + LN(G / s_u)) | 559.98 kPa |
Gibson-Anderson Cylindrical Cavity Limit Pressure p_lim,cyl (kPa)
Computes p_0 + s_u * (1 + LN(G / s_u)) for in-situ horizontal/mean total stress B2, undrained shear strength B3, and shear modulus B4 (with G >= s_u > 0).
=IF(OR(B2 < 0, B3 <= 0, B4 < B3), "Requires s_u > 0 and Shear Modulus G >= s_u", ROUND(B2 + B3 * (1 + LN(B4 / B3)), 2))Vesić Spherical Cavity Limit Pressure p_lim,sph (kPa)
Computes p_0 + (4 / 3) * s_u * (1 + LN(G / s_u)) for deep foundation point resistance and piezocone interpretation.
=IF(OR(B2 < 0, B3 <= 0, B4 < B3), "Requires s_u > 0 and Shear Modulus G >= s_u", ROUND(B2 + (4 / 3) * B3 * (1 + LN(B4 / B3)), 2))Try the example
Use the original values or change them to test your understanding.