This method, firstly presented in (Poulos & Davis, 1968), allows a quick estimation of both the proportion of load which reaches the pile base and the total settlement of a pile in the conditions described in the last section.
It is necessary to determine the values of the stress acting on the pile (see Figure 2.1) that satisfy the condition of displacement compatibility. As previously stated, only vertical displacement are considered. In order to obtain the values of shear stress, τ, normal stress, σ, and displacement of the pile top, i.e. total displacement of the pile, wt, expressions that relate vertical displacement with unknown stresses must be determined, imposing the compatibility conditions and solving the resulting equations.
The vertical displacement of the soil adjacent to a pile element due to shear stress at the pile shaft is given by eq. (2.3):
(2.3)
Where:
d: diameter of the pile shaft
Is: vertical displacement factor for the pile element due to the shear stress at the pile shaft τ0: shear stress acting at the pile shaft
E: Young’s modulus of the soil
Considering all n pile elements, the resulting vertical displacement of the soil, wt, is provided by eq.
(2.4):
( ∑ ) (2.4)
Where:
Ib: vertical displacement factor for the pile element due to the normal stress at the pile base σ0: normal stress acting at the pile base
This expression is valid for piles with constant diameter. The mentioned factors are determined using the integration of the (Mindlin, 1936) equations for the displacement caused by a point load within a semi-infinite mass. If the presence of a rigid layer at a certain depth is to be accounted for, then the factor Is is to be altered accordingly.
For calculating the displacement of the pile elements, only axial compression of the pile is considered.
The vertical equilibrium of a cylindrical pile is provided by eq. (2.5):
(2.5)
Where:
σ: normal stress acting on the pile (average over the cross section) r0: radius of the pile shaft
Eq. (2.5) may be applied to the pile top, resulting in eq. (2.6):
(2.6)
Where:
Pt: total applied load
A: area of the pile cross section
Eq. (2.5) may also be applied to the pile base, resulting in eq. (2.7):
(2.7)
The displacement compatibility condition is satisfied by imposing the same displacement for the pile and the soil in each element (rigid interface).
The load settlement ratio is expressed in terms of a coefficient, I, as shown in eq. (2.8):
(2.8)
However, it has been mentioned that the shear modulus, G, is used instead of the Young’s modulus, E. Thus, eq. (2.8) may be rewritten as eq. (2.9):
(2.9)
The difference between the general shear modulus, G, and the shear modulus at the pile base, GL, is clarified as the soil inhomogeneity is taken into consideration, further in this chapter. The load settlement ratio is from now on represented as in eq. (2.9): Pt/(wtr0GL).
This coefficient I is obtained by multiplying other coefficients, as shown in eq. (2.10):
(2.10)
Where:
I0: settlement-influence factor for a rigid pile in a semi-infinite incompressible soil (ν=0.5) Rk: correction factor for pile compressibility
Rh: correction factor for finite depth of layer on a rigid base
Rν: correction factor for the Poisson’s ratio of the soil (when ν<0.5)
The values of I0, Rk, Rh and Rν are plotted in Figure 2.3, Figure 2.4, Figure 2.5 and Figure 2.6, respectively.
Figure 2.3: Settlement-influence factor for a rigid pile in a semi-infinite incompressible soil, I0, in terms of the pile slenderness ratio, L/d, and of the relation between the base and the shaft diameters, db/d. (Poulos & Davis,
1980), p.89.
= 1
L
Figure 2.4: Correction factor for the pile compressibility, Rk: in terms of the relation between the pile’s and the soil’s Young’s modulus, K, and of the pile slenderness ratio, L/d. (Poulos & Davis, 1980), p.89.
The correction factor for the pile compressibility, Rk, is function of the relation between the pile’s Young’s modulus, Ep, and the soil’s, E. This relation is represented by K, as shown in eq. (2.11):
(2.11)
The more relatively compressible the pile, the smaller the value of K.
Figure 2.5: Correction factor for the finite depth of the layer on a rigid base, Rh, in terms of the relation between the total depth of the soil layer and the length of the pile, h/L, and of the pile slenderness ratio, L/d. (Poulos &
Davis, 1980), p.89.
L L
Figure 2.6: Correction factor for the Poisson’s ratio of the soil, Rν, in terms of the soil’s Poisson’s coefficient, ν, and of the relation between the pile’s and the soil’s Young’s modulus, K. (Poulos & Davis, 1980), p.89.
Figure 2.6 confirms the previous statement that the Poisson’s ratio of the soil, ν, has no great influence in the total settlement of the pile, since the correction factor Rν varies between 0.8 and 1.0, for normal cases.
The proportion of load transferred to the pile base, Pb/Pt, for a floating pile may be calculated by eq.
(2.12), first presented in (Poulos, 1972):
(2.12)
Where:
β0: tip-load proportion for incompressible pile in uniform half-space (ν=0.5) CK: correction factor for the pile compressibility
Cν: correction factor for the Poisson’s ratio of the soil
The values of β0, CK and Cν are plotted in Figure 2.7, Figure 2.8 and Figure 2.9, respectively.
Figure 2.7: Tip-load proportion for incompressible pile in uniform half-space, β0, in terms of the pile slenderness ratio, L/d, and of the relation between the base and the shaft diameters, db/d. (Poulos & Davis, 1980), p.86.
L ν
Figure 2.8: Correction factor for pile compressibility, Ck, in terms of relation between the pile and the soil’s Young’s modulus, K, and of the pile slenderness ratio, L/d. (Poulos & Davis, 1980), p.86.
The pile’s compressibility has the effect of decreasing the load transferred to the tip. On the other hand, the load transferred to the tip tends to increase with the relative stiffness of this stratum, and this is more pronounced for slender piles.
Figure 2.9: Correction factor for pile compressibility, Cν, in terms of the soil’s Poisson’s coefficient, ν, and of the relation between the pile’s and the soil’s Young’s modulus, K. (Poulos & Davis, 1980), p.86.
The distance to a rigid layer, h, is not present nor has any influence on any term of eq. (2.12). In fact, the proportion of load which reaches the pile base is not greatly affected by it, when its value is higher than 2L, according to (Poulos & Davis, 1980). This must be taken into consideration when comparing results provided by different methods – if the depth h is not to be accounted for, then the limit of 2L must be respected.
There are obviously other factors that may have influence on the proportion of load that reaches the pile tip, such as the presence of a pile cap resting on the soil surface, or of enlarged bulbs along the pile, but they are beyond the scope of this study.
ν
L
A layered or a vertically non-homogeneous soil may also be analysed by using the equations in (Mindlin, 1936) for a uniform mass, if approximate values of Young’s modulus and Poisson’s ratio at various points along the pile are employed.
Thus, the stress distribution is assumed to be unaltered, as if the soil was homogeneous, but the soil displacement at a point adjacent to the pile is function of the soil’s Young’s modulus at that point. The result of the soil-displacement equation changes (see eqs. (2.3) and (2.4)), but not the pile- displacement’s one.
An average Young’s modulus may be calculated using eq. (2.13):
∑
(2.13)
Where:
Ei: Young’s modulus of layer i hi: thickness of layer i
n: number of layers/divisions of the soil
This may be used when the soil is divided into different layers but the Young’s modulus does not vary much. In those cases, the solution may be calculated with this new value of the Young’s modulus and is very close to the one provided by the finite element method (errors inferior to 15%), according to (Poulos & Davis, 1980). This approach is an approximation, but its solution is considered to be accurate enough for practical purposes. It must not be forgotten, however, that this is an approximation and that it does not provide an accurate solution of the load or settlement distribution along the pile. Only the total values are considered relevant. The variations in the Poisson’s ratio along the depth may be ignored, since, as discussed before, this parameter has little influence in the total settlement of the pile.
A relevant form of soil non-homogeneity is one in which the shear modulus varies linearly with depth.
A measure of this variation is the inhomogeneity factor, ρ; it is calculated through eq. (2.14):
(2.14)
In the extreme case of ρ=0.5, the shear modulus at the surface must be null – this is called a “Gibson soil”.
The factor of inhomogeneity enables a comparison between different types of soil, with more or less vertical inhomogeneity. Since eq. (2.13) is also applicable in this case, it is used in the calculation of ρ, through eq. (2.14), using the relation described in eq. (2.1).
In Figure 2.10, results given by eq. (2.9) are plotted, for different values of the soil inhomogeneity factor, ρ. The value of Poisson’s coefficient of the soil, ν, is 0.3. The radius of the pile, r0, is equal to the unity in every case, for simplification reasons. The length of the pile, L, varies between 4m and 100m. The rigid layer is assumed to be at a distance of 2.5L of the surface (h). 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.
This has very little influence in the overall results, since the charts are normalized for the soil rigidity;
thus, it only affects the value of K. K assumes the values of 375, 500 and 750 for ρ=1, ρ=0.75 and ρ=0.5, respectively. The reason why it is not given a constant value is that it is not possible, if the said values of soil inhomogeneity are to be tested and simultaneously the shear modulus at the pile base, GL, is to be the same in all cases, for K is a relation between the pile’s Young’s modulus and the average shear modulus along the shaft. Nevertheless, K has not great influence either over the load settlement ratio or over the proportion of load that is transferred to the pile base for normal values of Ep and GL. Besides, in (Randolph & Wroth, 1978) a similar relation is presented, the soil-pile stiffness ratio, λ, calculated through eq. (2.15):
(2.15)
Since both the Young’s modulus of the pile, Ep, and the shear modulus of the soil at the pile base, GL, are the same for every case, the soil-pile stiffness ratio is constant and λ=975.
Below the pile, the Young’s modulus is constant and has the same value as at the pile base. The Young’s modulus of the soil, E, used in its calculation is the one at the middle of the pile. The Young’s modulus of the pile is 30ì106 kPa.
The chart in Figure 2.10 was built according to non-linear functions created from the few exact points given by Figure 2.3 to Figure 2.6, since the intermediate values cannot be interpolated linearly.
Therefore, the load settlement ratio was calculated for each natural number of pile slenderness between 4 and 100. All those values are grouped in tables in Appendix B1.
Figure 2.10: Load settlement ratio in terms of the pile slenderness ratio, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Poulos and Davis method.
The chart shows that the load settlement ratio increases with the pile slenderness ratio. This is expected, since, when subject to the same conditions, longer piles settle less.
The shapes of the three curves are very similar. In fact, the inhomogeneity factor, ρ, influences the pile compressibility, K, and the expression of the load settlement ratio, eq. (2.9), only; the distance between the curves increases slightly with the slenderness ratio, L/r0.
The pile settlement ratio increases with the inhomogeneity factor, ρ. Since the same shear modulus at the pile base is considered for the three cases, the one with the smallest ρ is the one in which the shear modulus at the middle of the pile (G, in eq. (2.9)) is the smallest, i.e. the average Young’s modulus Eav is the lowest. Thus, according to eq. (2.9) and Figure 2.4 and Figure 2.6, the load settlement ratio will also be the lowest. It is expected that, the lower the ρ, the worst the results, since eq. (2.13) obviously provides a very gross approximation, the grosser the less homogeneous the soil.
Since there were few exact points to be extracted from the original charts, the error associated with these charts is considerable. The non-linearity of these functions is the cause of irregularity of the resulting curves.
In Figure 2.11, results given by eq. (2.12) are plotted, for different values of the soil inhomogeneity factor, ρ. The conditions of the pile and the soil are identical to the ones described for Figure 2.10.
Once again, it is built from non-linear functions created from the few exact points given by Figure 2.7 to Figure 2.9. Therefore, the proportion of load transferred to the pile tip was calculated for each natural number of pile slenderness between 4 and 100. All those values are grouped in tables in Appendix B2.
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 2.11: Proportion of the load taken by the pile base in terms of the pile slenderness ratio, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Poulos and Davis method.
This chart shows that the proportion of load transferred the pile base decreases non-linearly with the slenderness ratio: its value reduces quickly for low values of L/r0, but tends to stabilize. In longer piles, the shaft plays a more important role in the load transfer mechanism, given its dimension. Thus, less load reaches the pile base. In fact, some studies consider it to be null, for values of the pile slenderness ratio higher than a certain limit.
Besides, the values of the load settlement ratio increase inversely with ρ, although very slightly. In fact, the difference in the shear modulus of the soil distribution only affect Ck (Figure 2.8) and Cν
(Figure 2.9), and in this last case the change is negligible. However, it is natural that, in piles with a higher value of the relation K, more load is transferred to the tip.
Once again, it should not be forgotten that the information used to build this chart has come from Figure 2.7 to Figure 2.9, and so there is a significant error associated with it.
Although some of these parameters are taken as independent from each other (as the measure of pile compressibility, K, and the total depth of the soil layer, h, used in the calculation of the load settlement ratio), and other factors are not considered, this method is very convenient and adequate for practical purposes.
The Poulos and Davis method has proved to provide relatively good solutions, considering its simplicity. Its results for the load settlement ratio are usually slightly higher than the ones given by numerical methods, i.e. settlement values are lower, according to (Poulos & Davis, 1980). In Chapter 3, the pertinence of this statement is tested.
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