This method, firstly introduced in (Randolph & Wroth, 1978), has been developed in order to explain the axial load transfer process between pile and soil. It is particularly useful in cases where the soil is non-homogenous, since the previously developed methods, the Poulos and Davis method amongst them, had great limitations in that aspect.
Initially, the shaft and base behaviours are studied separately. An imaginary horizontal plane AB at the depth of the pile base separates base and shaft, as represented in Figure 2.12(a). Thus, it is considered that above that plane the soil deforms due to the pile shaft only, and that below the plane the soil deforms due to the pile base only, as shown in Figure 2.12(b). The deformation above and below the plane is not compatible and that allows for interaction between the upper and lower layers of soil. This is a simplification which will obviously not provide the exact solution, but that has proved to be satisfactory.
(a) (b)
Figure 2.12: (a) Upper and lower soil layers; (b) Separate deformation patters of the upper and lower soil layers.
Adapted from (Randolph & Wroth, 1978), p.1469.
The soil is considered to be linear elastic. Thus, the effects of installation (residual stresses) are ignored. As explained before, it is also assumed that the parameters of the soil are not affected by the installation of the pile.
The deformation of the soil surrounding the pile is similar to shearing of concentric cylinders. The vertical equilibrium of an element of soil is given by eq. (2.16):
(2.16)
Where:
r: horizontal distance to the pile axis τ: shear stress
x: horizontal coordinate σz: vertical stress z: depth
Since the shear stress caused by pile loading is much greater than the vertical stress, the second summand is insignificant. Thus, the integration of the previous equation gives the shear stress in the soil surrounding the pile, according to eq. (2.17):
(2.17)
Where:
r0: radius of the pile shaft τ0: shear stress at the pile shaft
The shear strain of the soil is calculated through eq. (2.18):
(2.18)
As mentioned before, the displacement is considered to be mainly vertical (the radial component is negligible), so eqs. (2.16) to (2.18) may be rewritten as eq. (2.19):
∫ ∫
(2.19)
When including rm in the equation, an upper boundary of the radius of influence of the place, i.e. the distance past which shear stress becomes negligible, eq. (2.19) may be rewritten as eq. (2.20):
( ) (2.20)
The factor ζ is as a relation between the radius of influence of the pile and the radius of the pile shaft, as shown in eq. (2.21):
( ) (2.21)
The pile acts on the layer below as a rigid punch; its effect is more significant, i.e. the lower layer suffers more deformation, nearer the pile. The lower layer restrains the deformation of the upper layer.
The result is the generation of positive vertical stresses (σz>0, compression), which indicates that, by the equation of soil equilibrium, shear stresses will decrease more rapidly than linearly, contrary to what eq. (2.17) implies.
The consequence is that the magnitude of ∂/∂r(rτ) (the variation of shear stress with distance do the pile) decreases with depth along the pile and, consequently, so does the value of rm, as represented in Figure 2.13.
.
Figure 2.13: Hypothetical variation of the radius of influence of the pile, rm. Adapted from (Randolph & Wroth, 1978), p. 1471.
Rigid Piles in Homogeneous Soil
For rigid piles, the shaft settlement, ws, is independent of the depth, since there is no shortening (the pile is considered as non-compressible). Thus, the shear stress at the pile shaft, τ0, must also vary, increasing with depth, so that the shaft displacement given by eq. (2.20) is constant. Thus, the higher the shear stress, the smaller the distance at which its value is significant.
The variation of rm with depth is generally disregarded, and its value is taken, in the case of homogeneous soil, as an average, given by eq. (2.22):
(2.22)
Generally, ζ varies between 3 and 5.
The shear stress at the pile shaft may be considered as described in eq. (2.23):
(2.23)
Assuming both this and the radius of influence of the pile, rm, constant with depth, the settlement relative to the pile shaft is given by eq. (2.24):
(2.24)
Since the behaviour of the pile base resembles a rigid punch, the resulting displacement is obtained by eq. (2.25):
(2.25)
η is the factor of interaction between the upper and lower layers of soil.
The factor η refers to the stiffening effect of the soil above the loaded area. The value to attribute to this factor is subject to discussion, but it is generally accepted that to adopt the unity is adequate, according to (Randolph & Wroth, 1978).
In (Fleming, 1992), this factor is considered as the relation between the base and shaft diameters, as shown in eq. (2.26):
(2.26)
This factor is included in the following equations, but is considered equal to the unity whenever calculations are performed.
In a rigid pile, there is by definition no relative displacement within, and so eq. (2.27) is applicable:
(2.27)
Therefore, every point in the pile has the same settlement.
Besides, the total load is given by eq. (2.28):
(2.28)
For rigid piles in a homogeneous soil, the load settlement ratio may be calculated through eq. (2.29):
(2.29)
Also, the fraction of the load that is taken by the base may be calculated by eq. (2.30):
[ ]
(2.30)
Compressible Piles in Homogeneous Soil
Not always may the pile be considered as rigid. For piles with certain properties, relative displacement within the pile must be taken into consideration. Thus, for compressible piles, the shaft settlement varies with depth, since shear stress can no longer be assumed as constant along the pile, as in eq.
(2.31):
(2.31)
The pile is considered as elastic, and its compressive strain in depth may be expressed by terms of the transmitted load, as in eq. (2.32):
(2.32)
The soil-pile stiffness ratio, λ, is calculated by eq. (2.15).
It is relevant to point out that the Young’s modulus of the pile, Ep, is taken into account in the calculations relative to compressible piles only, since for rigid piles it is considered to be infinite.
The transmitted load is related to the variable shear stress on the shaft surface, as in eq. (2.33):
(2.33)
Differentiating and combining the last equations will result in the governing differential equation, as in eq. (2.34):
(2.34)
And its solution is described in eq. (2.35):
(2.35)
Taking the following measure of pile compressibility, described in eq. (2.36):
√
(2.36)
The constants A and B are found by using the boundary conditions at the base of the pile, and equation (2.35) may be rewritten as eq. (2.37):
([
] [
] ) (2.37)
The term (πr0λμ)-1 is very small (inferior to 0.02 in normal cases) and, through eq. (2.25), eq. (2.37) may be simplified to eq. (2.38):
( ) ( ) (2.38)
It is possible to particularize eq. (2.38), expressing the total settlement of the pile in terms of the settlement of the base, as in eq. (2.39):
(
) ( ) (2.39)
The mentioned simplification may also be used, as in eq. (2.40):
(2.40)
The integration of eq. (2.33) and substitution of the latest equations provides the expression of the total load supported by the pile in terms of depth, as in eq. (2.41):
(
( )
( )) (2.41)
Thus, the expression of the load settlement ratio is provided by eq. (2.42):
[
] [ ]
(2.42)
This allows for a comparison between the pile compressibility and the load deformation behaviour.
Also, the fraction of the load that is taken by the base may be calculated by eq. (2.43):
[
]
(2.43)
It is also useful to adapt the former equations to the case of a non-homogenous soil. Only vertical non- homogeneity is considered – the concept of radial inhomogeneity not being developed, given its lack of relevance for the present study.
As in the chapter referent to the Poulos and Davis method, two types of non-homogeneous soils may be considered: a layered soil (in which every layer has a constant shear modulus) and a soil in which the shear modulus varies linearly.
For the case of the layered soil, the shear strain γ distribution is unaltered (see eq. (2.18)), the shear stress τ0 distribution being obtained by multiplying by an appropriate shear modulus.
Once again, the second type of soil inhomogeneity, in which the shear modulus varies linearly, is more carefully analysed.
Rigid Piles in Non-Homogeneous Soil
The behaviour of a rigid pile in this type of soil is considered. The shear stress, which would be assumed constant in a homogenous soil, increases approximately linearly with depth in this case. The distance at which the shear stresses become negligible, rm, will also decrease.
For a pile in an infinite half space, rm may be calculated by eq. (2.44):
(2.44)
And, for a pile in a space where there is a rigid layer at the depth of 2.5L, which is more commonly used since it allows a comparison with finite element method programs’ results, rm may be calculated by eq. (2.45):
(2.45)
The inhomogeneity factor, ρ, is given by eq. (2.14).
Below the pile base, the shear modulus is considered to be constant. However, analyses using the finite element method have proved that the difference between this and the case where the shear modulus continues to increase is negligible (inferior to 5% in the total settlement value).
The value of shear stress may be written as in eq. (2.46):
(2.46)
And thus, the shaft settlement, ws, may be calculated through eq. (2.47):
( ) (2.47)
The total load taken by the pile shaft may be written as in eq. (2.48):
∫ ( ) (2.48)
Eq. (2.48) may be rewritten as eq. (2.49):
(2.49)
And the load settlement ratio may be written as in eq. (2.50):
(2.50)
The fraction of the load that is taken by the base may be calculated by eq. (2.51):
[
]
(2.51)
Compressible Piles in Non-Homogeneous Soil
For compressible piles, eq. (2.42), used for homogeneous soils, may be used for non-homogeneous soils when modified by the introduction of the inhomogeneity factor, as in eq. (2.52):
[
] [ ]
(2.52)
Also, the fraction of the load that is taken by the base may be calculated by eq. (2.53):
[
]
(2.53)
A relevant statement is that a pile with L/r0=100 could hardly be considered “rigid” – this would be the case of a pile with L=100m and d=2m, or L=8m and d=16cm. The authors of this method have not provided any means of distinction between rigid and compressible piles. According to (Fleming, 1992), eq. (2.54) can be used as a general rule to determine if a pile is may be considered as rigid:
√ (2.54)
In this case, for the conditions described, eq. (2.54) is calculated as shown in eq. (2.55):
√ (2.55)
In the charts that display results from the Randolph and Wroth method presented from now on, a black line represents this limit, and the charts are divided in “Rigid” and “Compressible” areas.
In Figure 2.14, results given by eq. (2.50) are plotted, for different values of the soil inhomogeneity factor, ρ. The factor of interaction between layers, η, is given the value of the unity, and so is the pile radius, r0, in every case. The value of Poisson’s coefficient, ν, is 0.3. The relation between the radius of influence of the pile and the radius of the pile shaft, ζ, is given by equation (2.21), using values of the radius of influence of the pile, rm, given by equations (2.22) and (2.45), for homogeneous (ρ=1) and inhomogeneous (ρ≠1) soils respectively; thus, the rigid layer is assumed to be at the distance of 2.5L from the surface. The Young’s modulus at the pile base, EL, is equal to 80ì103 kPa in every case – its value varying in the rest of the soil according to ρ. Below the pile, the Young’s modulus is constant and has the same value as at the pile base.
These conditions are the closest possible to the ones used to analyse the Poulos and Davis method (see Figure 2.10).
Figure 2.14: Load settlement ratio in terms of the pile slenderness ratio for rigid piles, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth method.
This chart shows that, for rigid piles, the load settlement ratio increases approximately linearly with the pile slenderness ratio. The equations that originate these curves show very clearly that the contribution of the pile base is constant, and that the shaft contribution varies linearly with the pile slenderness.
Besides, the higher the inhomogeneity factor, ρ, the higher the load settlement ratio. As expected, as ρ decreases, so does the slope of the line that represents the function.
The values that originate these curves are grouped in tables in Appendix B1.
0 20 40 60 80 100 120 140
0 10 20 30 40 50 60 70 80 90 100
Pt/(wtìr0ìGL)
L/r0
ρ=1 ρ=0,75 ρ=0,5
In Figure 2.15, results given by eqs. (2.42) and (2.52) are plotted, for different values of the soil inhomogeneity factor. The Young’s modulus of the pile is 30ì106 kPa. The conditions are similar to the ones described for Figure 2.14.
Figure 2.15: Load settlement ratio in terms of the pile slenderness ratio for compressible piles, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth Method.
This chart shows that, for compressible piles, the load settlement ratio increases non-linearly with the pile slenderness ratio. This suggests that the total settlement does not reduce infinitely as the pile slenderness increases, but instead that it stabilizes, indicating the existence of an asymptote. This did not happen for rigid piles, as in that case the contribution of the shaft for the load settlement ratio increased infinitely with the pile slenderness.
For compressible piles, the contribution of the base remains the same. However, with the introduction of the measure of the pile compressibility, àL, the proportion of the load taken by the shaft is reduced.
Thus, the load settlement ratio for a given value is lower if the pile is considered compressible than if it were considered rigid, and the difference increases with the pile slenderness (see fig. Figure 2.16).
This is an overall more realistic approach for high pile slenderness values.
Once again, the load settlement ratio also becomes higher as the inhomogeneity factor ρ approaches the unity.
The values that originate these curves are grouped in tables in Appendix B1.
0 20 40 60 80 100 120 140
0 10 20 30 40 50 60 70 80 90 100
Pt/(wtìr0ìGL)
L/r0
ρ=1 ρ=0,75 ρ=0,5
Figure 2.16 allows a clear comparison between curves relative to rigid and to compressible piles, as well as the mentioned division.
Figure 2.16: Load settlement ratio in terms of the pile slenderness ratio for rigid and compressible piles (λ=975), for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth method.
It is now evident how different is the behaviour of each type of pile, rigid or compressible. The domain of rigid piles is clearly very limited, according to the division criterion used.
This analysis would not be complete, however, if a soil-pile stiffness ratio sensitivity analysis was not performed, since the presented charts are not normalized by it. Figure 2.17 shows the load settlement ratio in terms of the pile slenderness ratio for three different values of soil inhomogeneity factor (ρ=1, ρ=0.75 and ρ=0.5) and for three different values of the soil-pile stiffness ratio (λ=3,000, λ=975 and λ=300). This sensitivity analysis is only applied to compressible piles, since the soil-pile stiffness factor is assumed to be infinite in rigid piles.
0 20 40 60 80 100 120 140
0 10 20 30 40 50 60 70 80 90 100
Pt/(wtìr0ìGL)
L/r0
ρ=1 compressible ρ=0,75 compressible ρ=0,5 compressible ρ=1 rigid
ρ=0,75 rigid ρ=0,5 rigid
Rigid Compressible
Figure 2.17: Load settlement ratio in terms of the pile slenderness ratio for rigid and compressible piles, for different inhomogeneity factors and soil-pile stiffness ratios, ν=0.3 and h=2.5L, according to the Randolph and
Wroth method.
This chart shows that the load settlement ratio is quite sensitive to the soil-pile stiffness factor, for any given value of the soil inhomogeneity factor. However, it should be borne in mind that, for a constant Young’s modulus of the pile of 30GPa, a soil-pile stiffness factor of 300 requires the shear modulus of the soil at the pile base to be equal to 100MPa, whereas a soil-pile stiffness factor of 3,000 requires the shear modulus of the soil at the pile base to be equal to 10MPa, which is quite a wide range. The curves relative to λ=3,000 are obviously closer to the rigid pile solution (see Figure 2.14).
Application of eq. (2.54) provides the boundary values between rigid and compressible piles, for the given values of soil-pile stiffness factor, shown in Table 2.1.
Table 2.1: Limit pile slenderness ratio between rigid and compressible piles, for different values of the soil-pile stiffness factor, according to (Fleming, 1992).
λ L/r0
300 8.66 975 15.61 3,000 27.39
This table shows that the boundary between rigid and compressible piles is also quite sensitive to the soil-pile stiffness factor (as is, in fact, clear in eq. (2.54)). This gives an idea of the utility of the rigid solution (up to which value of the pile slenderness ratio the rigid solution is applicable).
0 20 40 60 80 100 120 140
0 10 20 30 40 50 60 70 80 90 100
Pt/(wtìr0ìGL)
L/r0
ρ=1; λ=3000 ρ=0,75; λ=3000 ρ=0,5; λ=3000 ρ=1; λ=975 ρ=0,75; λ=975 ρ=0,5; λ=975 ρ=1; λ=300 ρ=0,75; λ=300 ρ=0,5; λ=300
In Figure 2.18, results given by eqs. (2.30) and (2.51) are plotted, for different values of the soil inhomogeneity factor. Once again, the conditions are the same as the ones described for Figure 2.14.
Figure 2.18: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for rigid piles, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth method.
This chart shows that, for rigid piles, the proportion of load that reaches the pile base decreases non- linearly with the slenderness ratio: its value reduces quickly for low values of L/r0, but tends to stabilize.
Besides, the values increase inversely with ρ; this is expectable, since the contribution of the base to the load settlement ratio is not affected by ρ and the contribution of the shaft is proportional to ρ (see eq. (2.50)), as it increases with the shear modulus along the pile (see eq. (2.24)).
The values that originate these curves are grouped in tables in Appendix B2.
0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40
0 10 20 30 40 50 60 70 80 90 100
Pb/Pt
L/r0
ρ=1 ρ=0,75 ρ=0,5
In Figure 2.19, results given by equations (2.43) and (2.53) are plotted, for different values of the soil inhomogeneity factor.
Figure 2.19: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for compressible piles, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth
method.
This chart shows that, in terms of the proportion of load taken by the pile base, piles regarded as compressible behave very similarly to the ones considered to be rigid. The shapes of the curves are almost identical, the only difference being that values relative to the compressible piles are slightly lower. This is due to the fact that, in this case, the calculation of the contribution of the shaft includes the variation of shear stress with depth, and its results are higher. Besides, the longer the pile, the more load is transferred to the shaft, resulting in less load reaching the pile base.
It is interesting that the difference between curves ρ=1 and ρ=0.75 is lower than the difference between curves ρ=0.75 and ρ=0.5.
The values that originate these curves are grouped in tables in Appendix B2.
0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40
0 10 20 30 40 50 60 70 80 90 100
Pb/Pt
L/r0
ρ=1 ρ=0,75 ρ=0,5
Figure 2.20 allows a clear comparison between curves relative to rigid and to compressible piles, as well as the mentioned division.
Figure 2.20: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for rigid and compressible piles, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and
Wroth method.
Once again, a soil-pile stiffness ratio sensitivity analysis is performed. Figure 2.21 shows the proportion of load transferred to the pile base in terms of the pile slenderness ratio for three different values of soil inhomogeneity factor (ρ=1, ρ=0.75 and ρ=0.5) and for three different values of the soil- pile stiffness ratio (λ=3,000, λ=975 and λ=300).
0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40
0 10 20 30 40 50 60 70 80 90 100
Pb/Pt
L/r0
ρ=1 compressible ρ=0,75 compressible ρ=0,5 compressible ρ=1 rigid
ρ=0,75 rigid ρ=1 rigid
Rigid Compressible