Importance of Lower-Bound Capacities in the Design of Deep Foundations

A convenient mathematical model to account for a lower-bound capacity is a mixed lognormal distribution (Fig. 2). For capacities greater than the lower bound, the distribution is a continuous probability density function that follows a lognormal distribution. Most reliability analyses for pile capacities have assumed lognormal distributions for the pile capacity based on the available database information (AASHTO 2004; McVay et al. 2000, 2002, 2003), and the model on Fig. 2 is consistent with this conventional approach. For capacities at the lower bound, there is a finite probability (that is, a probability mass function) that corresponds to the probability of being less than or equal to the lower bound in the nontruncated lognormal distribution. For driven piles in cohesive and cohesionless soils, mixed lognormal probability distributions provide a realistic model to the uncertainty in the capacity of piles, as observed in results of pile load tests (Najjar 2005).

When both the load and the capacity follow lognormal probability distributions, the probability of failure can be calculated analytically aswhere $\Phi \left(\right)=\text{standard}$ normal cumulative distribution function; ${\mathrm{FS}}_{\text{median}}=\text{median}$ factor of safety defined as the ratio of the median capacity $r_{\text{median}}$ , to the median load $s_{\text{median}}$ ; ${\delta }_R$ and ${\delta }_S=\text{coefficients}$ of variation of the capacity and load, respectively; and $\beta =\text{reliability}$ index.

For the case where a mixed lognormal distribution is used to model the capacity and assuming a lognormal distribution for the load, the reliability and the probability of failure are calculated based on the parameters presented in Fig. 3 aswhere $\mathrm{LB}=\text{lower}$ -bound capacity. Note that the median and COV for the lognormal distribution of capacity, $r_{\text{median}}$ and ${\delta }_R$ , respectively, correspond to the nontruncated distribution. A more general form of Eq. (2) is obtained by normalizing the load and the capacity by the median capacity, $r_{\text{median}}$ Hence, for given values of $\mathrm{LB}∕r_{\text{median}}$ , ${\delta }_R$ , and ${\delta }_S$ , the reliability can be expressed entirely as a function of the median factor of safety. Note that the factor of safety used in design is not necessarily equal to the median factor of safety. If conservative (versus “best guess”) values of the load and capacity are used in calculating the design factor of safety, then the median factor of safety will be greater than the design factor of safety. When $\mathrm{LB}∕r_{\text{median}}>0$ , an analytical solution for Eq. (3) is not available and it must be solved using numerical integration.

The probability of failure and the reliability of a pile foundation are expected to be affected by the lower-bound capacity, the median factor of safety, and the relative uncertainty in the load and capacity (Fig. 3). Curves showing the variation of the reliability index as a function of the ratio of the lower-bound to median capacity are shown on Fig. 4. The curves in Fig. 4 represent the case where the uncertainty in the capacity $({\delta }_R=0.4)$ is relatively large compared to the uncertainty in the load $({\delta }_S=0.2)$ . This example is representative of many geotechnical designs where the capacity is more uncertain than the load (McVay et al. 2000; Kulhawy and Phoon 2002; Phoon et al. 2003; AASHTO 2004).

The primary conclusion from Fig. 4 is that a lower-bound capacity can have a significant effect on the calculated reliability. For example, consider a typical case where the median factor of safety is 3.0. If the lower-bound capacity is anything greater than 0.6 times the median capacity, the probability of failure is reduced by more than an order of magnitude compared to the case where there is no lower bound (Fig. 4). The effect of the lower bound capacity on the reliability is influenced by the magnitude of the median factor of safety; as the median factor of safety increases, the lower bound is more significant in reducing the probability of failure.

To better illustrate the magnitude of the effect of the lower-bound capacity, the median factor of safety that is required to achieve different levels of reliability is plotted on Fig. 5 as a function of the ratio of the lower-bound to the median capacity. The required median factor of safety decreases as the lower-bound capacity approaches the median capacity. Results in Fig. 5 indicate that resistance factors in a LRFD should incorporate information about the lower-bound capacity if they are to provide a consistent level of reliability. The reliability index of a design that is based on conventional resistance factors may differ significantly from the reliability index obtained when the effect of the lower-bound capacity is accounted for in the design-checking equations.

In this section, the effect of varying the distributional form for capacity on the calculated reliability is investigated by considering a variety of bounded probability distributions. These include truncated normal and lognormal, uniform, and beta distributions, in addition to the mixed lognormal distribution used to generate Figs. 4 and 5. Mathematical expressions for the probability of failure and the probability density function for each distribution are presented in the Appendix. In all cases, the uncertainty in the load is modeled using a conventional lognormal distribution. The variation of the reliability index with the ratio of the lower-bound to median capacity for different capacity distributions is plotted in Fig. 6.

Results in Fig. 6 indicate that for a given ratio of lower-bound to median capacity, the probability distribution for capacity does not have a significant effect on the calculated reliability, especially as the median factor of safety increases. This result is important because it shows that the reliability of a design is significantly more affected by the ratio of the lower-bound to median capacity than it is by the type of probability distribution used to model the capacity. It is worth noting that reliability indices that are calculated using the mixed lognormal distribution tend to be smaller than those obtained using other capacity distributions, indicating that the effect of the lower-bound capacity on the reliability can be higher than that shown in Figs. 4 and 5.

Curves in Fig. 4 indicate that there is a threshold value for the ratio of the lower-bound to the median capacity above which the lower bound affects the reliability (the knee in the curves in Fig. 4). Below this threshold, the lower-bound capacity has essentially no effect on the reliability index. Above this threshold, the reliability index increases approximately linearly with increasing lower bound. The threshold decreases as the median factor of safety increases. A proposed approximation for this threshold is the “most probable failure point” for the capacity in the case where there is no lower bound, which is given by Ang and Tang (1984) for the case of a lognormal capacity and load distributions as

Curves in Fig. 4 also indicate that the shape of the curves for relatively large ratios of lower-bound to median capacities can be approximated analytically by assuming that the capacity is equal to the lower-bound value. For a load following a lognormal distribution, the probability of failure is bounded by the following analytical expression:

This bounding approximation is shown on Fig. 4. As the ratio of the lower-bound to median capacity approaches one, the curves approach the bounding approximation in Eq. (5). A noteworthy aspect of this approximation is that it does not depend on an assumed probability distribution for capacity.

Based on the observations presented above, a simple approximation is proposed to relate the reliability to the lower-bound capacity (Fig. 7). The model is defined by three points that can be evaluated without the need for numerical integration. First, the reliability index with no lower-bound capacity is calculated from Eq. (1). Then a horizontal line is drawn from zero to the most probable failure point defined by Eq. (4). Finally, the reliability index with $\mathrm{LB}∕r_{\text{median}}=1.0$ is calculated from Eq. (5) and a straight line is drawn from the reliability index with no lower-bound capacity at the threshold to this reliability index at $\mathrm{LB}∕r_{\text{median}}=1.0$ .

Bilinear approximations to the curves describing the variation of the reliability index with the lower-bound capacity are shown in Fig. 8. Reliability curves corresponding to a mixed lognormal capacity distribution are also shown in Fig. 8. The bilinear model provides a conservative but close approximation to the calculated reliability without the need for complicated computations.

In practice, the lower-bound capacity may not be known with certainty. For example, spatial variability in soil properties or inaccuracies in a proof-load test can lead to uncertainty in estimating a lower-bound capacity. In this section, a mathematical framework is presented for incorporating information about uncertainty in lower-bound capacities in reliability analyses.

If uncertainty in the lower-bound capacity exists, the probability of failure is calculated using the theorem of total probability aswhere $p_{f\mid \mathrm{LB}}=\text{probability}$ of failure given a lower-bound capacity LB; and $f_{\mathrm{LB}}\left(\mathrm{LB}\right)=\text{probability}$ density function of the lower-bound capacity (LB). To illustrate the effect of uncertainty in the lower-bound capacity on the reliability, the following case is considered: the load $\left(S\right)$ follows a lognormal distribution with a COV of 0.2 and the capacity $\left(R\right)$ follows a mixed lognormal distribution with a nontruncated COV of 0.4 and conditional on a lower-bound capacity (LB) that follows a lognormal distribution with a range of COV values.

The effect of uncertainty in the lower-bound capacity on the reliability index is shown in Fig. 9. Even when there is uncertainty in the lower-bound capacity, the presence of a lower-bound capacity increases the reliability compared to the conventional case where a lower-bound capacity is not included in the reliability analysis. Furthermore, the larger the ratio of the median lower-bound to the median capacity is, the greater the increase in reliability. When the lower bound is relatively close to the median capacity (i.e., the ratio of median lower bound to median capacity equal to 0.5 and 0.6 in Fig. 9), increasing uncertainty in the lower-bound value decreases the reliability. This result occurs because the possibility of smaller values for the lower bound tends to decrease the reliability. However, when the lower bound is relatively far from the median capacity (i.e., the ratio of the median lower-bound to median capacity equal to 0.4 in Fig. 9), increasing uncertainty in the lower bound actually causes a small increase in the reliability. This result occurs when the possibility becomes dominant where the lower bound is less than the threshold where the reliability is affected by it (i.e., to the left of the knees in the curves in Fig. 4).

For practical applications, uncertainties in lower-bound capacities are expected to be small relative to uncertainties about mean capacities. For example, a lower-bound capacity that is calculated based on lower-bound shear strength values and soil properties (Najjar 2005) is expected to be less sensitive to variability in the depositional environment, in situ soil structure, soil-structure interaction, and sampling and testing techniques. As a result, lower-bound capacities may be estimated with a higher degree of confidence compared to expected or mean capacities.

To illustrate and test this hypothesis, the effect of spatial variability in the undrained shear strength on the predicted (or mean) and lower-bound pile capacities is investigated for a number of offshore sites. Predicted capacities of piles in normally consolidated to slightly overconsolidated clay are generally obtained using undisturbed undrained shear strength properties while estimates of lower-bound pile capacities can be obtained using remolded undrained shear strength values using the model presented in Najjar (2005). Both vertical and horizontal variations in shear strength properties will affect the predicted and lower-bound capacities of piles.

To study the effect of vertical variations on the undisturbed and remolded shear strength, shear strength measurements obtained at two offshore sites in the Gulf of Mexico (Doyle 1999) are presented in Fig. 10. The data were obtained from unconsolidated, undrained triaxial shear tests on undisturbed and remolded samples. The following two observations can be made by comparing the shear strength profiles at the two sites: (1) the variability of the undisturbed shear strength with depth is much larger than that of the remolded shear strength, indicating that uncertainty in predicted pile capacity due to vertical variations in soil properties is expected to be larger than that of the lower-bound capacity; and (2) the remolded shear strength profile provides a lower bound to variations about the undisturbed profile.

To study the effect of horizontal variations on undisturbed and remolded shear strengths, strength measurements obtained from four different boreholes separated by about $6\mathrm{km}$ in an offshore site (Al-Awar 2002) are plotted in Fig. 11. The shear strength was measured using unconsolidated, undrained triaxial shear tests on undisturbed and remolded specimens. The variability in the measured strengths reflects both vertical and horizontal variations. In order to isolate the contribution of horizontal variations, the strength data were averaged vertically to different depths in the four borings. This vertically averaged strength is of interest in pile design since the side friction will be approximately proportional to it. The horizontal variations in the vertically averaged shear strength over varying depths are compared in Fig. 11 and in Table 1. The results indicate that horizontal variations in the remolded shear strength are significantly lower than those in the undisturbed strength and can be represented by a coefficient of variation of about 0.1–0.15. This small level of uncertainty in the lower-bound capacity will not appreciably reduce the impact of lower-bound capacities on the reliability, as indicated by Fig. 9.

The conventional design checking equation has the following general form:where $r_{\text{nominal}}=\text{nominal}$ capacity calculated using a design method; ${\phi }_R=\text{resistance}$ factor; $s_{\text{nominal}}=\text{nominal}$ load for design; and ${\gamma }_S=\text{load}$ factor. In order to incorporate the effect of a lower-bound capacity, this design checking equation is modified as follows:where the resistance factor, ${\phi }_{R\left(\mathrm{LB}\right)}=\text{function}$ of the lower-bound capacity. When there is no lower bound and when the load and the capacity follow conventional lognormal distributions, the reliability is calculated from Eq. (9) aswhere ${\mathrm{FS}}_{\text{median}}=\text{median}$ factor of safety; and ${\delta }_S$ and ${\delta }_R=\text{coefficients}$ of variation of the load and the capacity, respectively. Assuming ${\lambda }_R$ and ${\lambda }_S$ to be mean biases for the capacity and load respectively (mean value divided by the nominal value), the nominal values of the load and the capacity will thus be related to the median load and median capacity throughWhen there is no lower bound and the reliability is found from Eq. (9), the required value for the resistance factor can be expressed in terms of the target reliability index as follows:If the nominal values of the load and the capacity are assumed to be equal to the mean values (the methods for predicting the load and capacity are unbiased), the required value for the resistance factor can be expressed in terms of the target reliability index as follows:The required resistance factor can also be expressed in terms of the required median factor of safety asIf the methods for predicting the load and capacity are unbiased

For a nonzero lower-bound capacity, the required median factor of safety will decrease as the lower bound increases. Since the lower-bound capacity does not affect the other variables in Eqs. (13) - (14), the required resistance factor when there is a lower-bound capacity, ${\phi }_{R\left(\mathrm{LB}\right)}$ , can be expressed in terms of the conventional case as follows:where ${\phi }_R$ is obtained from Eqs. (13) - (14) and corresponds to the case where there is no lower-bound capacity; ${\mathrm{FS}}_{\text{median}(\mathrm{LB}=0)}=\text{median}$ factor of safety required for a target reliability index if there is no lower bound [obtained from Eq. (9)]; and ${\mathrm{FS}}_{\text{median}\left(\mathrm{LB}\right)}=\text{median}$ factor of safety required to achieve the target reliability index for a lower-bound value equal to LB. Therefore, the increase in the resistance factor due to an increase in the lower-bound capacity is inversely proportional to the corresponding decrease in the required value for the median factor of safety.

The variation of the ratio of the resistance factor incorporating a lower-bound capacity to the conventional resistance factor, ${\phi }_{R\left(\mathrm{LB}\right)}∕{\phi }_R$ , is shown as a function of the lower-bound capacity in Fig. 12 for different target values of the reliability index. For reasonable values of the ratio of the lower-bound to median capacity, 0.4–0.9, the effect of the lower bound on the required resistance factor is significant. Consider a design with a target reliability index of 3.0 and a COV value for the capacity of 0.5. If the lower-bound capacity is 0.7 times the median capacity, then the required resistance factor in the design-checking equation, Eq. (8), will be more than double the conventional resistance factor [Fig. 12(b)]. For example, a conventional resistance factor of 0.25 would be increased to 0.6 in accounting for the lower-bound capacity.

An alternative code format would be to have two design checking equationswhere the first design-checking equation is the conventional equation and the second equation includes a resistance factor, ${\phi }_{\mathrm{LB}}$ , that is applied directly to the lower-bound capacity. A design will be considered to be acceptable (specified level of reliability is achieved) if only one of the two design equations is satisfied. The motivation for this form of the design-checking equation is that the conventional approach is incorporated and does not need to be modified, whether or not there is a lower-bound capacity; the effect of a lower-bound capacity is reflected entirely in the second equation.

The required value for ${\phi }_{\mathrm{LB}}$ can be determined such that the target reliability is achieved by rearranging Eq. (13) so that it is expressed in terms of the lower-bound capacity rather than the nominal capacity

The nominal capacity, $r_{\text{nominal}}$ , is related to the median capacity, $r_{\text{median}}$ , through Eq. (10). As such, the lower-bound resistance factor given by Eq. (17) can be expressed in terms of the ratio of the lower-bound to median capacity asIf the methods for predicting the load and capacity are unbiased

A plot of ${\phi }_{\mathrm{LB}}$ versus the lower-bound capacity is shown in Fig. 13 for different target reliability indices. Approximations of ${\phi }_{\mathrm{LB}}$ as calculated using the simple bilinear reliability model that was proposed earlier are also plotted in Fig. 13 for comparison. The curves begin at values of the lower-bound capacity where the second design checking equation in Eq. (16) governs. One advantage of this approach with two design-checking equations [Eq. (16) versus Eq. (8)] is that ${\phi }_{\mathrm{LB}}$ is not very sensitive to either the magnitude of the lower-bound capacity or the target reliability index (Fig. 13). In fact, a conservative value of around 0.75 for ${\phi }_{\mathrm{LB}}$ could be used to cover a wide range of possibilities.

In this section, a case study presented by Goble (1996) is used to illustrate how the proposed LRFD checking equation that incorporates a lower-bound capacity can be used in the design of a bridge foundation. The foundation to be designed supports an isolated bridge column that is subjected to a critical factored axial load of $10\mathrm{MN}$ . The piles to be used in supporting the bridge column consist of closed-ended steel pipe piles with an outside diameter of $355\mathrm{mm}$ and a length of $25\mathrm{m}$ . A simplified schematic of the soil profile at the site is shown in Fig. 14.

The predicted nominal axial capacity of a single driven pile is $1.8\mathrm{MN}$ as calculated by the API method (API 1993). If the COV values for the capacity $\left({\delta }_R\right)$ and load $\left({\delta }_S\right)$ are 0.5 and 0.15, respectively, which are typical for driven piles in cohesionless soils and for bridge loading conditions, respectively, the mean biases ( ${\lambda }_R$ and ${\lambda }_S$ ) for the capacity and load are 1.0, and the target reliability index is 3.0, then the conventional resistance factor $\left({\phi }_R\right)$ is obtained from Eq. (12) and is equal to 0.23. The factored capacity per pile is calculated from Eq. (8): ${\phi }_R{r}_{\text{nominal}}=0.23\times 1.8\mathrm{MN}=0.42\mathrm{MN}$ . Since each pile provides a factored capacity of $0.42\mathrm{MN}$ and the total factored load is $10\mathrm{MN}$ , the total number of piles needed to support the column load is $10\mathrm{MN}∕0.42\mathrm{MN}=24$ piles based on the conventional approach assuming a group efficiency factor of 1.0.

To account for the effect of a lower-bound capacity in this design, a lower-bound capacity is calculated for each pile using the available soil profile (Fig. 14) and a lower-bound model that is consistent with driven piles in siliceous, cohesionless soils as presented by Najjar (2005). In this model, the lower-bound skin friction is calculated assuming that the lateral coefficient of earth pressure is replaced with the at-rest value and the soil-pile friction angle and end-bearing capacity factors are replaced with the values for one category less in density (e.g., the values for a “dense sand” are replaced with those for a “medium sand”). The skin friction is thus calculated as $k_{\mathrm{LB}}{\sigma }_v^{\prime }\tan {\delta }_{\mathrm{LB}}=0.55\times 145\mathrm{kN}∕{\mathrm{m}}^2\times \tan \left(20\right)=29\mathrm{kN}∕{\mathrm{m}}^2$ , where $k_{\mathrm{LB}}$ is equal to 0.55 (at rest value), and ${\delta }_{\mathrm{LB}}$ is equal to 20° (API value for loose sand and gravel assuming one category less in density). The skin friction in the soft sandy silty clay layer $(\mathrm{SPT}=6)$ in the top $6\mathrm{m}$ is neglected. The lower-bound capacity of the pile due to skin friction is thus equal to $0.61\mathrm{MN}$ . The lower-bound tip resistance can also be calculated assuming one category less in density as ${\sigma }_v^{\prime }{\mathrm{N}}_{q,\mathrm{LB}}=242.7\mathrm{kN}∕{\mathrm{m}}^2\times 12=\mathrm{2,870}\mathrm{kN}∕{\mathrm{m}}^2$ , resulting in a lower-bound end bearing capacity of $0.29\mathrm{MN}$ .

The calculated lower-bound capacity is thus equal to $0.9\mathrm{MN}$ , or 0.5 times the predicted capacity and 0.55 times the median capacity. From Fig. 12(b), the resistance factor accounting for the lower bound can be increased by 1.7 times the conventional value of 0.23: ${\phi }_R(\mathrm{LB}=0.5r_{\text{nominal}})=1.7\times 0.23=0.40$ . The corresponding factored capacity per pile is then found from Eq. (8): ${\phi }_R(\mathrm{LB}=0.5r_{\text{nominal}})r_{\text{nominal}}=0.40\times 1.8\mathrm{MN}=0.72\mathrm{MN}$ .

Alternatively, the lower-bound resistance factor can be found from Fig. 13(b) to be ${\phi }_{\mathrm{LB}}=0.81$ . According to Eq. (16), the target reliability will be achieved with the larger of the factored nominal capacity from the conventional equation, ${\phi }_R{r}_{\text{nominal}}=0.42\mathrm{MN}$ , and the factored capacity based on the lower-bound capacity, ${\phi }_{\mathrm{LB}}\mathrm{LB}=0.81\times 0.9\mathrm{MN}=0.72\mathrm{MN}$ . Therefore, the same factored capacity of $0.72\mathrm{MN}$ for design is obtained using either version of the design checking equation [Eq. (8) or (16)], and the required number of piles to support the column load is $10\mathrm{MN}∕0.72\mathrm{MN}=14$ . Incorporating the lower-bound capacity into this design has a significant effect on the level of conservatism required, reducing the required number of piles from 24 to 14 while still providing the target level of reliability for each foundation.

In this section, an illustrative example is provided whereby a lower-bound capacity is incorporated into a reliability-based design. The example is based on a case history for the design of a floating offshore facility for oil production. The target annual reliability index for the design of the foundations is set to 3.0. Due to the geometry of the structure and the metocean environment, the uncertainty in the load is relatively low $(\mathrm{COV}=0.2)$ . Conversely, the uncertainty in the capacity is relatively large $(\mathrm{COV}=0.4)$ because the site is located in a frontier area with very little available data.

From a conventional reliability analysis where the load and the capacity are assumed to follow lognormal distributions, a median factor of safety greater than 3.5 is needed to achieve the target reliability index of 3.0 (Fig. 15). For this example design, the median load is $10\mathrm{MN}$ and the required median capacity is $37\mathrm{MN}$ . The opportunity for optimization in this design is that full setup of the pile capacity is estimated to take about $1\text{year}$ . Therefore, oil production cannot begin until at least $1\text{year}$ after the piles are installed. If a lower-bound estimate of capacity based on pile installation data is incorporated into the reliability analysis, it may be possible to accelerate the time that oil production can begin.

Fig. 15 shows how a lower-bound capacity affects the reliability index. In this case, the estimated capacity from a re-tap analysis conducted several days after pile driving was $18\mathrm{MN}$ , or about 50% of the median capacity after full setup. The results in Fig. 15 show that a median factor of safety of 2.7 or a median capacity of $27\mathrm{MN}$ , is sufficient to provide an adequate level of reliability. The setup analysis indicates that this capacity will be achieved within $3\text{months}$ after installation. Therefore, the oil-production schedule can be moved up $9\text{months}$ by conducting a more realistic reliability assessment that incorporates information about the lower-bound capacity.

Mathematical expressions for the probability density functions and the probability of failure for truncated normal and lognormal distributions are presented in Eqs. (20) - (23), respectively

The uniform distribution is defined by a probability density function that has a constant value over a range defined by a lower-bound capacity LB, and an upper-bound capacity UB, and is expressed as

The expression for the probability of failure for the case where the capacity follows a uniform distribution is given by Eq. (25)

A beta distribution with a mean ${\mu }_R$ and a standard deviation ${\sigma }_R$ is defined over a range LB to UB and has the following probability density function:whereand $\Gamma \left(\right)=\text{gamma}$ function defined as $\Gamma \left(k\right)={\int }_0^{\infty }{x}^{k-1}{e}^{-x}dx$ for $k>0$ . Depending on the values and signs of the parameters $\alpha$ and $\beta$ , the beta distribution can assume different geometrical shapes.

The expression for the probability of failure for the case where the capacity follows a beta distribution is given by Eq. (27)