Calculation of the bearing capacity and settlements of shallow foundations, the methods implemented for the calculation of the bearing capacity are those of HANSEN, MEYERHOF, TERZAGHI while for the calculation of settlements the method uses the theory of elasticity (Timoshenko and Goodier (1951)).
The bearing capacity of a shallow foundation can be defined with reference to that maximum value of the load for which the failure condition is not reached at any point in the subsoil (Frolich’s method), or with reference to that value of the load, greater than the previous one, for which the failure phenomenon has extended to a large volume of soil (Prandtl’s method and subsequent ones)
The expressions used for the calculation of the bearing capacity are reported below.
HANSEN
For φ > 0
Nc = (Nq – 1)/tanφ
Nq =Exp(π·tanφ·tan(π/4+φ/2)2
Ng = 1.5·(Nq – 1)·tanφ
Qlimit = c’·Nc·sc·dc·Ic·gc·bc+?·D·Nq·Sq·dq·Iq·gq·bq +0.5·γ’·B’·Ng·sg·dg·Ig·gg·bg
For φ = 0
Nc = 5.14
Qlimit = 5.14·Cu·(1+scp+dcp-icp-bcp-gcp)+γ·D
Where:
sc, sq, sg shape factors;
dc, dq, dg shape factors;
Ic, Iq, Ig finclination factors.
gc, gq, gg ground inclination factors (foundation on slope)
bc, bq, bg base inclination factors (inclined base)
MEYERHOF
Nq =Exp(π·tanφ)·(tan(π/4 + φ/2))2
For φ>0
Nc =(Nq – 1)/tanφ
Per φ=0
Nc=5.14
Ng=(Nq – 1)·tan(1.4·φ)
TETA=0 (absence of load inclination)
Qlimit = c’·Nc·sc·dc + γ·D·Nq·sq·dq + 0.5·γ’·B’·Ng·sg·dg
TETA<>0 (presence of load inclination)
Qlimite = c’·Nc·sc·Ic + γ·D·Nq·dq·Iq + 0.5·γ’·B’·Ng·Ig·dg
where:
sc, sq, sg shape factors
dc, dq, dg depth factors
Ic, Iq, Ig load inclination factors.
TERZAGHI
a =Exp(((0.75·π)-φ/2))·tanφ)
Nq = a2/(2 ·cos(π/4)+φ/2)2
For φ> 0
Nc = (Nq-1)/tanφ
For φ= 0
Nc = 5.7
Ng = tanφ/2·(kpγ/(cos2φ-1))
Where kpγ is a coefficient elaborated by Terzaghi, provided in tabular form.
Qlimit = c’·Nc·sc + γ·D·Nq + 0.5·γ’·B’·Ng·sg
ELASTIC SETTLEMENTS
The settlements of a rectangular foundation of dimensions B·L placed on the surface of an elastic half-space can be calculated based on an equation based on the theory of elasticity (Timoshenko and Goodier (1951)).
ΔH=q0·B'(1-μ2)/Es)·(I1+(1-2μ)·I2/(1-μ))·IF
where:
q0 Contact pressure intensity
B’ Minimum dimension of the reacting area,
Es and μ Elastic parameters of the soil.
Ii Influence coefficients depending on: L’/B’, thickness of the layer H, Poisson’s ratio μ, embedment depth D
The coefficients I1 and I2 can be calculated using the equations provided by Steinbrenner (1934) (see Bowles), the influence coefficient IF is derived from Fox’s equations (1948).