TAMWAVE 3: Basic Results of Pile Capacity Analysis

With the soil properties and lateral loads finalised, we can proceed to look at the program’s static results. These are shown below. We will concentrate on cohesionless soils in this post; a sample case with cohesive results will come later.

Pile Data
Pile Designation	12 In. Square
Pile Material	Concrete
Penetration of Pile into the Soil, ft.	100
Basic “diameter” or size of the pile, ft.	1
Cross-sectional Area of the Pile, ft²	1.000
Pile Toe Area, ft²	1.000
Perimeter of the Pile, ft.	4.000
Soil Data
Type of Soil	SW
Specific Gravity of Solids	2.65
Void Ratio	0.51
Dry Unit Weight, pcf	109.5
Saturated Unit Weight, pcf	130.5
Soil Internal Friction Angle phi, degrees	32
Cohesion c, psf	0
SPT N₆₀, blows/foot	20
CPT q_c, psf	211,600
Distance of Water Table from Soil Surface, ft.	50
Penetration of Pile into Water Table, ft.	50
Active Earth Pressure Coefficient (K_min)	0.453
Frictional Angle Between Pile and Soil delta, degrees	27.9
Minimum Value for Beta	0.240
Pile Toe Results
Effective Stress at Pile Toe, ksf	8.880
N_q	22.8
Relative Density at Pile Toe, Percent	40
SPT (N₁)₆₀ at pile toe, blows/foot	10
Unit Toe Resistance q_p, ksf	202.7
Shear Modulus at Pile Toe, ksf	675.7
Toe Spring Constant Depth Factor	1.410
Toe Spring Constant, kips/ft	2,767.9
Pile Toe Quake, in.	0.879
Poisson’s Ratio at Pile Toe	0.310
Toe Damping, kips-sec/ft	13.2
Toe Smith-Type Damping Constant, sec/ft	0.065
Total Static Toe Resistance Q_p, kips	202.67
Pile Toe Plugged?	No
Final Results
Total Shaft Friction Q_s, kips	370.00
Ultimate Axial Capacity of Pile, kips	572.68
Pile Setup Factor	1.0
Total Pile Soil Resistance to Driving (SRD), kips	572.68

Pile Data

The pile data is pretty straightforward. Reproducing it here is an opportunity for you to confirm you’ve selected the correct pile.

Soil Data

Soil data affords the same opportunity for verification; however, it also shows the way the soil data is interpreted to generate the necessary parameters for shaft and toe resistance to load, both static and dynamic.

The first thing that is shown is assumed specific gravity and void ratio. TAMWAVE assumes cohesionless soils have a particle specific gravity of 2.65 and for cohesive soils 2.7. The void ratio is then computed using basic soil mechanics formulae. To do this it is necessary to know the unit weight. The typical properties tables show this in two ways. For cohesionless soils, the “moist” unit weight is shown, and for cohesive soils the saturated unit weight is shown. In both cases this is reduced to dry and saturated unit weights by assuming that S=50% for the cohesionless soils and S=100% for the cohesive ones. Thus, for cohesionless soils neither value will be the same as given in the typical properties.

The internal friction angle, cohesion and $N_{60}$ values are taken from the typical properties as modified (or not) by the user. The equivalent $q_c$ is also reported here, based on the Robertson and Campanella research as reported by Fellenius. As noted earlier, neither the $N_{60}$ values nor the $q_c$ values are actually used in the analysis.

Finally we get to the data necessary to compute the shaft friction. The methods used in TAMWAVE for ultimate shaft resistance are as follows:

Cohesionless soils: Randolph, Dolwin and Beck (1994), which are described here
Cohesive soils: Kolk and van der Velde (1996), which are described here

For cohesionless soils, it is necessary to compute the minimum/active earth pressure coefficient, which of course is strictly a function of $\phi$ . Discussion of $K_{act}$ brings us to the issue of computing $\beta$ . In this post $\beta$ was initially computed using the following formula

$\beta = K tan \phi$

However, as pointed out in the same place, both retaining wall practice and empirical pile capacity formulae show that the friction angle between the wall/pile shaft and the soil is not equal to the internal friction angle of the soil, and so this formula should really be written as

$\beta = K tan \delta$

This actually has a theoretical basis, and in fact is one of the knottiest problems in theoretical soil mechanics. We can consider this by considering the failure along the pile surface as a “direct shear” type of failure, where failure is induced along a predetermined surface. For the case where the principal stresses are normal and tangential to the surface (which is generally the case with driven piles) the failure surface predicted by Mohr’s circle and Mohr-Coulomb theory is not the same as the “predetermined” surface. The most acrimonious manifestation of this problem was with the shear failure of cellular cofferdams, which led to the dispute between Karl Terzaghi and Dmitri Krynine.

Although various studies have been made to determine friction on an empirical basis, probably the simplest solution, suggested by Šuklje (1969), is to compute the apparent friction angle by the formula

$\delta = tan^{-1} (sin \phi)$

Using this result and the active earth pressure coefficient, the minimum value for $\beta$ is readily computed.

Pile Toe Results

Now we get to the application of these parameters. The decision to not use equivalent CPT values has two immediate results. The first is that the unit toe resistance is most easily computed (for cohesionless soils) by the equation

$q_t = N_q \sigma'_{vo}$

Use of bearing capacity factors for toe resistance is both well embedded in literature and practice and well criticised in the same place. Additionally it is necessitated by the fact that the shaft friction is dependent upon $N_q$ , as discussed here.

So what value of $N_q$ should we adopt? As is all too common in geotechnical engineering, there has been a proliferation of values for this parameter. We experimented with several, including that of Vesic (1977). In addition to the usual theortical vs. empirical (and all the variations in between) divide, another factor is whether provision for soil elasticity is taken into consideration. Although methods such as Vesic (1977) do this, their main weakness is their tendency to be stiff, thus returning unrealistically high values of $N_q$ . Taking into account both theoretical methods and empirical ones such as Dennis and Olson, the first method tried for TAMWAVE was from Verruijt), namely:

$N_{\sigma} = K_p e^{\pi tan \phi}$

Note that we’re not at $N_q$ quite yet. For reasons explained by Vesic (1977), the pile toe unit resistance should be a function of $\frac {I_1}{3}$ . (An explanation of this quantity can be found here.) Thus,

$N_q = K_p e^{\pi tan \phi} \frac {I_1}{3 \sigma'_{vo}}$

If we use Jaky’s Equation for normally consolidated soils for the pile toe condition (we will definitely change this for the shaft,)

$\frac {I_1}{3 \sigma'_{vo}} = 1 - \frac {2}{3} sin \phi$

and so

$q_t = K_p e^{\pi tan \phi} \left( 1 - \frac {2}{3} sin \phi \right) \sigma'_{vo}$

This method yields conservative values of $N_q$ . It only takes into consideration the plasticity of the soil, and is additionally subject to the criticism of the entire concept of “bearing capacity” at the pile toe. Given this, a different (and more empirical) approach based on Randolph, Dolwin and Beck (1994) was chosen, which estimates the bearing capacity coefficient as follows:

$N_q=\frac{0.065S+25}{\left( \frac{\sigma'_{vo}}{p_{atm}} \right)^{0.25}}$

$S$ is the coefficient from the Hardin and Black (1968) method of estimating the shear modulus of the soil (see below.) Although this coefficient varies with silt content, for TAMWAVE we will assume $S = 315$ , thus this reduces to

$N_q=\frac{45.475}{\left( \frac{\sigma'_{vo}}{p_{atm}} \right)^{0.25}}$

If static capacity were our sole interest, we would be done with toe. But what about its response to movement? For both toe and shaft resistance, in both static and dynamic cases, we intend to use an elastic-purely plastic model. Assuming no preloading of the system, there are only two parameters we need to know: the ultimate/purely plastic resistance of the soil, and the deflection at which we reach that resistance. The spring constant can be computed by dividing the ultimate resistance by that deflection, or conversely we can determine that deflection by dividing the resistance by a known spring constant. It is the latter operation we will use in TAMWAVE, which leaves us to determine the spring constant of the toe and eventually along the shaft.

We will have occasion to return to this topic, but to determine spring constants we will use the model of Randolph and Simons (1985). For the toe this in turn is dependent upon Lysmer’s Analogue; both of these are discussed in detail in Warrington (1997). They are dependent upon determining values for the soil shear modulus $G$ . (They are also dependent upon the dry unit weight $\gamma$ and Poisson’s Ratio $\nu$ , but both of these parameters are known from basic soil properties and, indirectly, through Jaky’s Equation.) That in turn brings us to another “sticky wicket,” namely determining the shear (or for that matter the elastic) modulus of the soil. An interesting discussion of this topic can be found in Salgado, Loukidis, Abou-Jaoude and Zhang (2015). Assuming a hyperbolic type of soil deformation, there are two basic extremes to this parameter:

The small-strain (or tangent) value, the highest possible value.
The large-strain (or secant) value, the lowest possible value.

Based on their review of the literature, they conclude that the value for (2) can be 10-50% of (1). Although this problem is frought with uncertainties, it is hard to avoid the conclusion that this is a substantial spread and, for our purposes, raises as many questions as it answers. The “solution” to this problem is found in this post, where one attempts to define a ratio between (1) and (2) based on some consideration of anticipated deflections under load for a given application.

Based on some experimentation with the code and earlier considerations, we decided to use a ratio between the two of 0.15, i.e., the secant modulus used in elastic-purely plastic models is 15% of the tangent modulus from the hyperbolic model. We should emphasise that this is not “set in stone” but subject to variation. One of the advantages of a project such as TAMWAVE is the ability to alter parameters and see the results without affecting results on actual projects.

“Fixing” this ratio allows us to determine the shear modulus based on the tangent or small-strain value, and this can be computed by the method proposed in Hardin and Black (1968). There is little difference between the correlation for cohesionless and cohesive soils. There are many ways of expressing this; the one we used (for values of $G$ in psf) is as follows:

$G=S p_{atm}\frac{\left(3-e\right)^{2}}{1+e}\sqrt{\frac{\sigma'_{vo}}{p_{atm}}\frac{1+2K}{3}}$

As before, for TAMWAVE (and many applications) $S = 315$ . The same formula is used for the shaft friction, the main difference is that the lateral earth pressure coefficient is different thus the lateral/confining stresses are different. At the toe we use the result from Jaky’s Equation, which was explained in Verruijt’s method above.

Once this is computed, the pile toe stiffness is computed. The stiffness is increased by multiplying it by a depth factor (Salgado, Loukidis, Abou-Jaoude and Zhang (2015)

$D_f = 1+\left( 0.27- 0.12 ln \nu \right)\left\{ 1-e^{\left[ -0.83\left( \frac{D}{B} \right)^{0.83} \right]} \right\}$

Even at this, when compared to “conventional” toe quakes in dynamic analysis, the toe quake shown above seems rather large. We will leave this as it is for the static analysis and will return to this topic with the dynamic analysis.

Since we are computing stiffnesses for shaft and toe here, we will also do the same for damping. Traditionally wave equation programs have used “Smith damping,” but as we will see this will be modified for the wave equation analysis. To start let us redefine the “Smith type damping constant” as

$j = \frac {\mu}{R_u}$

In this case $\mu$ is the damping constant for the toe or shaft element in question, computed using the formulae given in Warrington (1997). $R_u$ is the ultimate resistance of the toe or shaft element in question. The toe damping constant that results in this case is somewhat lower than “standard” values; this will be discussed later.

Final Results

The final results are at the end of the table. The shaft friction computation will be discussed in the next post. The cohesive calculations have a provision for pile set-up using cavity expansion theory and this will be discussed later.

References

In addition to works already cited in this and the STADYN study, the following should be noted:

Hardin, B.O., and Black, W.L. (1968). “Vibration modulus of normally consolidated clay.” J. Soil Mech. Found. Div. 94, No. 2, 353-370.
Salgado, R., Loukidis, D., Abou-Jaoude, G., and Zhang, Y. (2015) “The role of soil stiffness non-linearity in 1D pile driving simulations.” Geotechnique 65, No. 3, 169-187. http://dx.doi.org/10.1680/geot.13.P.124
Vesic, A.S. (1977) Design of Pile Foundations. NCHRP Synthesis 42. Washington, DC: Transportation Research Board.