Bearing Capacity of Pile Foundations: Calculation Methods for Qp

Following the previous section that explained the general background and universal equations for the estimation of a single pile’s load-bearing capacity, we will continue with three specific methods for the calculation of the end point bearing capacity Q_p, where:

Q_p = A_p · q_p = A_p · (c’· N_c +q’·N_q)  (1)

 

Meyerhof’s Method

Meyerhof’s method distinguishes two cases for the estimation of Q_p, one for the case that the pile is buried in sand, and one for clay.

Sand:

In the case of sand we have no cohesion (c’=0), so equation (1) will simplify to:

Q_p = A_p · q_p = A_p · q’ · N_q  (2)

The values that equation (2) can produce are bound by a limiting point resistance q_l, which has the form:

q_l = 0.5 · p_a · N_q · tanφ’  (3)

where:

p_a is the atmospheric pressure (roughly 100 kPa)

φ’ is the soil’s effective friction angle at the pile tip

 

So, taking all the above into consideration we have:

Q_p = A_p · q_p = A_p · q’ · N_q ≤ A_p · q_l

Now, the values for factor N_q to be used are determined in the table shown below through interpolation based on Meyerhof’s theory.

Soil friction angle, φ (degrees)	Nq
20	12.4
21	13.8
22	15.5
23	17.9
24	21.4
25	26.0
26	29.5
27	34.0
28	39.7
29	46.5
30	56.7
31	68.2
32	81.0
33	96.0
34	115.0
35	143.0
36	168.0
37	194.0
38	231.0
39	276.0
40	346.0
41	420.0
42	525.0
43	650.0
44	780.0
45	930.0

 

Clay:

In the case of saturated clay and undrained conditions we have a friction angle of zero (φ=0), and the equation for the end point bearing capacity has the form:

Q_p = A_p · c_u · N_c = A_p · c_u · 9  (4)

where:

c_u is the clay’s undrained cohesion at the pile’s tip

 

Vesic’s Method

Similarly to the previous method, Vesic’s one also has two different forms for the cases of sand and clay.

Sand:

In the case of sand for Vesic’s method, end tip load-bearing capacity is expressed as:

Clay:

Again, in the case of saturated clay under undrained conditions we get:

Q_p = A_p · q_p = A_p · c_u · N_c  (9)

where:

the factor N_c = 4/3 · (lnI_r + 1) + π/2 +1  (10)

 

Finally, I_r is the rigidity index, for which:

I_r = E_s / (3 · c_u)  (11)

and E_s is the soil’s modulus of elasticity.

 

Coyle and Castello’s Method

This method’s correlations are the result of 24 large-scale field load tests of piles driven in sand. Hence, it is understood that the following correlation is applicable to piles present in similar conditions.

In this case, the end point bearing capacity is formulated as:

Q_p = q’ · N_q · A_p  (12)

where, once again:

q’ is the vertical effective stress at the pile tip

N_q is a bearing capacity factor taken by the following figure

Calculating Q_p from SPT and CPT results

Once again, several publications exist on how the end tip bearing capacity of a pile can be derived from in-situ test results, and we are going to present several of them here.

SPT:

According to Meyerhof, the unit ultimate point resistance q_p can be obtained from the following equation in the case of homogeneous granular soil:

q_p = 0.4 · p_a · N₆₀ · L/D ≤ 4 · p_a · N₆₀  (13)

where:

N₆₀ is the average SPT blow count at about 10D above and 3D below the pile end point

P_a is the atmospheric pressure

 

Briaud et al. in 1985 also suggested the following relationship for granular soils:

q_p = 19.7 · p_a · (N₆₀)^0.36  (14)

Finally, Poulos suggested in 1989 that the ultimate bearing capacity at the base of a pile is:

Q_p = A_p · C_b · N₆₀ = Q_bu  (15)

where:

N₆₀ is again the SPT blow count in the vicinity of the pile tip

C_b is a constant described in the following table

Pile type	Soil	Cb
Displacement (driven)	Sand	400-450
	Silt	350
	Glacial till	250
	Clay	75-100
Driven cast-in-situ	Cohesionless	150
Bored	Sand	100
	Clay	75-100

 

CPT:

As for CPT results, Meyerhof in 1956 suggested that in granular soil q_p is approximately equal to the cone penetration resistance, q_c.

In addition to this, it is suggested in Craig’s Soil Mechanics that the following relationship may be used for the unit ultimate point resistance:

C_cpt values have been suggested by Jardine et al. in 2005 and Lee and Salgado in 1999 according to the following table, for an average q_c value at over 1.5D above and below the pile’s end point.

Pile type	Soil	Ccpt
Driven (closed)	Sand	0.4
	Clay undrained	0.8
	Clay drained	1.3
Bored pile	Sand	0.2

References:

Das, B.M. (2007). Principles of Foundation Engineering (7th Edition). Global Engineering

https://skyciv.com/docs/skyciv-foundation/piles/various-equations-for-estimating-pile-capacity/

Craig, R.F. (2004) Craig’s Soil Mechanics. 7th Edition, Spon Press, London