Statistical Evaluation of CPT and CPTu Based Methods for Prediction of Axial Bearing Capacity of Piles

Abstract

Piles are structural members made of steel, concrete, or wood installed into the ground to transfer superstructure loads to the soil. Nowadays, many structures are built on poor lands, and therefore piles have crucial roles in such structures. Performing in-situ tests such as cone penetration (CPT) and piezocone penetration tests (CPTu) have always been of great importance in designing piles. These tests have a brilliant consistency with reality, and as a result, the outcome data can be used in order to achieve reliable pile designing models and reduce uncertainty in this regard. In this paper, the capability of various CPT and CPTu based methods developed from 1961 to 2016 has been investigated using four statistical methods. Such CPT and CPTu based methods are adopted for direct prediction of axial bearing capacity of piles using CPT and CPTu field data. For this purpose, 61 sets of field data prepared from CPT and CPTu have been collected. The data sets were utilized in order to calculate the axial bearing capacity of piles (Q_E) through 25 different methods. In addition, the measured axial pile capacities (Q_M) have been collected, recorded and prepared from field static load tests, respectively. Then, four different statistical approaches have been applied to assess the accuracy of these methods. Finally, the most reliable and accurate methods are presented.

1 Introduction

Determination of accurate ultimate bearing capacity of piles has always been a challenge for geotechnical engineers in order to have a safe design. In this regard, estimating the axial bearing capacity of piles can be achieved via static analysis, in-situ testing methods, full-scale loading tests, and presumed values recommended by codes and handbooks (Obeta et al. 2018). The in situ tests which have been used for the estimation of the axial bearing capacity of piles are mostly CPT (and CPTu), SPT, and more recently DMT (Vukićević et al. 2018). The cone penetration test (CPT) is considered one of the most practical in-situ tests for the prediction of the ultimate bearing capacity of piles for the similarity between pile and cone (Amirmojahedi and Abu-Farsakh 2019). Furthermore, CPT is a powerful, fundamental, quick, dependable, and inexpensive test, and can provide continuous soundings of the subsurface soil (Eslami and Fellenius 1997). Site variability assessment, soil stratigraphy and liquefaction assessment are the other applications of CPT. The CPT measurements are comprised of the cone tip resistance (${q}_{c}$) and sleeve friction (${f}_{s}$). Soil identification, classification, and evaluation of soil properties such as strength and deformation characteristics can be carried out effectively by these measurements (Lunne et al. 2002). Piezocone penetration test (CPTu) is a more advanced CPT test for its additional capability of measuring pore water pressure. One of the most efficient applications of CPT data is the determination of pile capacity, due to the similarity between the cone and the pile. One of the major advantages of using CPT or CPTu data for pile designing is the attainment of undisturbed sampling and less need for subsequent standard laboratory testing (Eslami and Fellenius 1997). In the literature, there are some empirical and semi-empirical approaches for estimating the ultimate bearing capacity of piles directly based on CPT and CPTu data. These approaches have been developed to overcome the uncertainties in geotechnical engineering predictions. Empirical and semi-empirical methods are based on simplifying assumptions including soil stratigraphy, soil-pile interaction, and distribution of soil resistance along the pile. Therefore, they cannot be reliable in order to be directly useful for the foundation design (Eslami et al. 2014). In this research, 25 available CPT and CPTu-based direct methods are considered to determine the estimated axial pile capacity (${Q}_{E}$). In Appendix, Table 5 presents these methods in details.

The pile capacities predicted from CPT or CPTu data by 25 methods have been compared with measured values through static load tests. To compare ${Q}_{E}$ values with ${Q}_{M}$, four statistical criteria have been employed. These are the best-fitted line for ${Q}_{E}$ versus ${Q}_{M}$, the geometric average and standard deviation of the ratio ${Q}_{E}$/${Q}_{M}$, the cumulative probability for the ratio of ${Q}_{E}$/${Q}_{M}$, and the accuracy of 20 percent of the histogram and the lognormal distribution curve (Eslami et al. 2011). These comparisons will be presented below.

2 Collected Database from CPTu and Static Load Tests

A series of data has been collected, containing the results of 61 static loading tests of piles and CPT, and CPTu soundings performed close to the pile locations. For measuring the axial bearing capacity of piles, the static compression pile load tests have been performed which have acceptable accuracies. The main application of CPT and CPTu is to determine the soil classifications (Begemann 1963, 1965, 1969; Douglas 1981; Robertson et al. 1986; Robertson 1990; Jefferies and Davies 1991; Olsen 1995; Eslami and Fellenius 1997; Zhang and Tumay 1999; Hegazy and Mayne 2002). In these series of data, soils consist of clay, sand, and mixture of soils called mixed soil. For instance, a typical piezocone profile is shown in Fig. 1 and the soil profile is determined using the approach of Eslami and Fellenius (1997) by calculating corrected cone resistance (${q}_{E}$) and sleeve friction.

Fig. 1

A typical piezocone profile with soil classification from the chart given by Eslami and Fellenius (1997) and Yen et al. (1989)

Most piles have been installed by driving. These tests are collected from 26 various sites in 16 countries, the majority of which have been performed in the U.S.A. For most databases, the pore water pressure measurements are available. The piles are made of concrete or steel with circular, rectangular or H-shape cross sections, the majority of them is square concrete. Embedment lengths of the piles range from 8.2 m to 67 m, and 61% of embedment lengths are in the range of 10 m < L < 20 m. The pile diameters range from 235 to 1372 mm, and about 79% of the pile diameters are less than 450 mm. The axial ultimate bearing capacities of piles that have been determined from pile load tests are 290–7500 kN. Eighty percent of total capacities are calculated on the basis of Hansen (1970) criteria. In cases where the load obtained from this criterion exceeds the applied load in pile load test, the maximum load in the test was used. Figure 2 shows different percentages of pile shape and material, pile diameter, embedment length, soil type, and site location in 61 databases.

Fig. 2

Percentages of: (a). embedment length; (b) pile diameter; (c) pile shape and material, (d) soil type; (e) site location in 61 databases

Table 1 summarizes the main case recorded data, including pile specifications, pile loading test results, and soil profiles. These 61 databases are divided into three categories based on soil type as sand, clay and mixed soil.

Table 1 Case recorded data

3 Statistical Analysis

Briaud (1988), Eslami and Fellenius (1997), Long and Wysockey (1999), Abu-Farsakh and Titi (2004), Schneider et al. (2008), and Eslami et al. (2011) evaluated the accuracy of some of the CPT and CPTu-based methods in estimating the pile axial bearing capacity when both measured data on piles from real field tests and predicted data from empirical and semi-empirical methods were accessible. However, it is felt that it is necessary to perform a comprehensive investigation to rank different approaches according to their prediction accuracy and resulting in much more economic design. In this paper, an extensive range of CPT and CPTu-based methods from 1961 to 2016 have been investigated by four statistical methods. The aim is to carry out statistical analyses to rank CPT and CPTu-based methods comprehensively.

3.1 Equations of the Best-Fitted Line for ${{\varvec{Q}}}_{{\varvec{E}}}$ versus ${{\varvec{Q}}}_{{\varvec{M}}}$

The values of ${Q}_{E}$ from the CPT and CPTu-based methods are plotted against the ${Q}_{M}$ from pile loading test results in different soil types. The soil profiles reported are divided into sand, clay, and mixed soil. In order to compare the estimated and measured pile capacities for all piles in three soil types, three terms of “most,” “medium,” and “least” resembling “the most accurate,” “medium accurate,” and “the least accurate” are used, and the results are shown in Figs. 3, 4, and 5. The line with the angle of ${45}^{^\circ }$ which is named diagonal line, in each diagram, indicates a perfect consistency between estimated and measured pile capacities. For each CPT and CPTu-based methods, a regression analysis is conducted on 61 databases to obtain the best fitting line for ${Q}_{E}$/${Q}_{M}$. The relationship between ${Q}_{E}$ and ${Q}_{M}$, and corresponding coefficient (${R}^{2}$) are determined for each CPT and CPTu data in 25 CPT and CPTu-based methods. As can be seen in Figs. 3, 4, and 5, the method of the Niazi and Mayne (2016) method in sand, the SEU method (Cai et al. 2011, 2012) in clay and the Niazi and Mayne (2016) method in mixed soil have the best-fitted equation. Therefore, the method of Niazi and Mayne (2016) and the SEU method (Cai et al. 2011, 2012) in the sand, mixed soil and in clay are ranked first according to this criterion with ${R}_{1}$ = 1, while ${R}_{1}$ is the ranking based on this criterion.

Fig. 3

Values of $Q_{E}$ versus $Q_{M}$ for piles in sand

Fig. 4

Values of $Q_{E}$ versus $Q_{M}$ for piles in clay

Fig. 5

Values of $Q_{E}$ versus $Q_{M}$ for piles in mixed soil

Figures 3, 4, and 5 represent the most accurate methods for sand, clay, and mixed soil, respectively. As shown in Figs. 6, 7, and 8, the UniCone (Eslami and Fellenius 1997; Fellenius 2002), Cambridge (White and Bolton 2005), Lee and Salgado (1999), Tumay and Fakhroo (1981), Bogdanovic (1961), Eslami and Fellenius (1997) in the sand, Eslami and Fellenius (1997) in clay and UniCone (Eslami and Fellenius 1997; Fellenius 2002), Eslami and Fellenius (1997) in mixed soil with ${R}^{2}\ge 0.9$ have good agreement with the results of a real situation.

Fig. 6

Comparison of methods based on the best-fitted line criteria in sand

Fig. 7

Comparison of methods based on the best-fitted line criteria in clay

Fig. 8

Comparison of methods based on the best-fitted line criteria in mixed soils

3.2 Geometric Mean (μ) and Standard Deviation (σ) for ${{\varvec{Q}}}_{{\varvec{E}}}$/${{\varvec{Q}}}_{{\varvec{M}}}$

The geometric mean (μ) and standard deviation (σ) of ${Q}_{E}$/${Q}_{M}$ for each method are calculated and used as a second evaluation criterion. According to μ and σ values, various approaches have been ranked as summarized in column 7 of Tables 2, 3, and 4. The smaller the standard deviation, the better agreement can be determined between Q_E and Q_M. Consequently, the Eslami and Fellenius Method (Eslami and Fellenius 1997) ranks number one in sand, clay, and mixed soil. In this criterion, ${R}_{2}$ is the rank number based on the geometric mean (μ) and standard deviation (σ) of ${Q}_{E}$/${Q}_{M}\mathrm{ values}$.

Table 2 Summary results of the evaluation of bearing capacity of piles in sand

Table 3 Summary results of the evaluation of the bearing capacity of piles in clay

Table 4 Summary results of the evaluation of the bearing capacity of piles in mixed soil

3.3 Cumulative Probability for ${\mathbf{Q}}_{\mathbf{E}}$/${\mathbf{Q}}_{\mathbf{M}}$

Alsamman (1995) and Long and Wysockey (1999) suggest that the cumulative probability is a statistical criterion for evaluating the CPT and CPTu-based methods for estimating the ultimate bearing capacity of piles. For obtaining the cumulative probability, the following procedure should be taken into account:

• 1. Arrangement of ${Q}_{E}$/${Q}_{M}$ values for each method in ascending order (1, 2, 3,…, i,…,n).

• 2. Estimation of cumulative probability (P) from Eq. 1. (Long and Wysockey 1999):

  $$ p = {i \mathord{\left/ {\vphantom {i {\left( {n + 1} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {n + 1} \right)}} $$

  (1)

  where i denotes the order number given for the considered ratio and n is the pile number.

Fifty and ninety percent cumulative probabilities ($P_{50}$ and $P_{90}$) should be calculated and utilized to quantify the accuracy of various approaches estimating the ultimate pile capacity. For example, the ${P}_{50}$ value is used as a measure of the tendency of the method to over-predict or under-predict the measured pile capacity. The determination of over-prediction and under-prediction depends on the intended type of distribution. If the ${Q}_{E}$/${Q}_{M}$ values are close to 1, the estimated ultimate bearing capacity is almost equal to the measured value and the method has a precise prediction. The cumulative probability for $Q_{E}$/$Q_{M}$ values is shown in Fig. 9 for sand, clay, and mixed soil, respectively.

Fig. 9

Cumulative probability for $Q_{E} /Q_{M}$ computed from various methods

As shown in Fig. 9, according to the cumulative probability criterion, the method of Niazi and Mayne (2016) with ${P}_{50}$=1.007, 1.012 and ${P}_{90}$=1.080, 1.196 ranks number one in sand and mixed soil, respectively, and the Eslami and Fellenius Method (Eslami and Fellenius 1997) with ${P}_{50}$=1.009 and ${P}_{90}$=1.45 ranks number one in clay.

3.4 Twenty Percent accuracy from the Histogram and Lognormal Distribution for ${{\varvec{Q}}}_{{\varvec{E}}}$/${{\varvec{Q}}}_{{\varvec{M}}}$

The range of ${Q}_{E}$/${Q}_{M}$ value is from 0 to an unlimited upper value with an optimum value of unity. These result in a non-symmetric distribution of ${Q}_{E}$/${Q}_{M}$ values around the average value, not giving a uniform weight of under-prediction and over-prediction (Briaud 1988). In this respect, Briaud and Tucker (Briaud 1988) state that in order to assess the performance of the methods predicting the ultimate bearing capacity of piles, a log-normal distribution of ${Q}_{E}$/${Q}_{M}$ should be used. With this criterion, it is possible to determine the under-prediction and over-prediction of CPT and CPTu-based methods. The CPT and CPTu-based method under-predicts the measured capacity when ${Q}_{E}$/${Q}_{M}<1$ and over-predicts the measured capacity when ${Q}_{E}$/${Q}_{M}>1$. The lognormal distribution is defined as the distribution with the following density:

$$ f\left( x \right) = \frac{1}{{\sqrt {2\pi } \mathop \sigma \nolimits_{\ln } x}}\exp \left[ {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {2\mathop {\left( {\frac{{\ln \left( x \right) - \mathop \mu \nolimits_{\ln } }}{{\mathop \sigma \nolimits_{\ln } }}} \right)}\nolimits^{2} }}} \right. \kern-\nulldelimiterspace} {2\mathop {\left( {\frac{{\ln \left( x \right) - \mathop \mu \nolimits_{\ln } }}{{\mathop \sigma \nolimits_{\ln } }}} \right)}\nolimits^{2} }}} \right] $$

(2)

where $x = Q_{E} /Q_{M} ,$ $\mu_{ln}$ = mean of ln($Q_{E} /Q_{M}$), and $\sigma_{ln}$ = standard deviation of ln($Q_{E} /Q_{M}$).

In statistical analysis, the mean value (μ) and standard deviation value (σ) of $Q_{E} /Q_{M}$ are essential indicators of the accuracy and precision of the predicting method.

Due to the lack of information and accucy of tests, upper and lower limits are considered for $Q_{E} /Q_{M}$ values. The upper and lower limits mean that when more accumulation of data exists between upper and lower limits, lower distribution of the results obtained from the method exists, and thus, this method is better (Long and Wysockey 1999). For this purpose, the accuracy rate in this study is chosen 20%. Then, histograms and standard probability distribution graphs are drawn for $Q_{E} /Q_{M}$ values. When the area of two graphs is calculated as $0.8 \le Q_{E} /Q_{M} \le 1.2$, with approximation of data under the graph to 100%, the method will be more accurate. Figures 10, 11, and 12 show these criteria in sand, clay, and mixed soils for methods with the most, medium, and minimum accuracy. For other methods, such analyses are illustrated by bar charts in three types of soils in Figs. 13, 14, and 15.

Fig. 10

Twenty percent accuracy from the histogram and lognormal distribution for $Q_{E} /Q_{M}$ in sand

Fig. 11

Twenty percent accuracy from the histogram and lognormal distribution for $Q_{E} /Q_{M}$ in clay

Fig. 12

Twenty percent accuracy from the histogram and lognormal distribution for $Q_{E} /Q_{M}$ in mixed soil

Fig. 13

Comparison of methods based on the accuracy of 20 percent of the histogram and the lognormal distribution curve for the ratio of $Q_{E} /Q_{M}$ in sand

Fig. 14

Comparison of methods based on the accuracy of 20 percent of the histogram and the lognormal distribution curve for the ratio $Q_{E} /Q_{M}$ in clay

Fig. 15

Comparison of methods based on the accuracy of 20 percent of the histogram and the lognormal distribution curve for the ratio $Q_{E} /Q_{M}$ in mixed soils

The amounts of Log-Normal and Histogram for different methods have been illustrated in columns 11 and 12 of Tables 2, 3, and 4 in the sand, clay, and mixed soil, respectively. In the sand, clay, and mixed soil the Eslami and Fellenius Method (Eslami and Fellenius 1997), the UniCone method (Eslami and Fellenius 1997; Fellenius 2002), and the Niazi and Mayne (2016) method are the most accurate methods based on this criterion, respectively.

Figures 13, 14 and 15 show that the method presented by Niazi and Mayne (2016) in the sand and clay and the UniCone method (Eslami and Fellenius 1997; Fellenius 2002) in mixed soil also have a good agreement with measured capacity in a real situation.

4 Results of Statistical Analysis for Various Experimental Methods

In this study, four evaluation criteria based on statistical approaches have been taken into account for ranking the performance of a wide variety of CPT and CPTu-based methods for predicting the ultimate bearing capacity of piles. The overall performance of CPT and CPTu-based methods are then evaluated using the four different criteria via RI. For example, RI for the Eslami and Fellenius Method in the sand is determined from RI = ${R}_{1}$ + ${R}_{2}$ +${R}_{3}$  + ${R}_{4}$=1 + 1 + 1 + 1 = 4 in which ${R}_{1}$, ${R}_{2}$, ${R}_{3}$, and ${R}_{4}$ are rank numbers for the regression, geometric average and standard deviation, cumulative probability, accuracy of 20 percent of the histogram and the lognormal distribution, respectively. The RI values for all methods in sand, clay, and mixed soil are presented in column 14 of Tables 2, 3, and 4, respectively. The final overall rank for each CPT or CPTu method obtained based on RI values in sand, clay, and mixed soil has been presented in column 15 of Tables 2, 3, and 4.

According to the results of the current research, the Niazi and Mayne method (Niazi and Mayne 2016) in sand and mixed soil, and the Eslami and Fellenius Method (Eslami and Fellenius 1997) in clay rank first. These CPTu-based methods (Niazi and Mayne and Eslami and Fellenius Method) show the best efficiency according to the four evaluation criteria. The KTRI method (Takesue et al. 1998) in sand and mixed soil and the Penpile method (Clisby et al. 1978) in clay show the lowest performance amongst all methods. Therefore, the rank of the KTRI (Takesue et al. 1998) method is number 18 and 10 corresponding to sand and mixed soil. The rank of the Penpile method (Clisby et al. 1978) is 13 in clay. Generally, the CPT-based methods exhibit more dispersion than CPTu-based methods. The results of this investigation help to consider methods in terms of their accuracy for designing purposes. In addition, the results help to quantify the most accurate methods to have much better agreement with the real situation.

5 Summary and Conclusions

In spite of doing comprehensive research by numerous researchers, for example Abu-Farsakh (1999, 2004), Eslami et al. (2011), the topic is still very interesting and needs further research. Much research has been conducted in this area. For example, Abu-Farsakh et al. (1999) evaluated 8 CPT-based methods with 35 databases in all types of soils and square precast prestressed concrete piles (PPC) using four statistical evaluation criteria and the prediction of pile capacity was performed on 60 piles. Abu-Farsakh and Titi (2004) evaluated 8 CPT-based methods with 35 databases in all types of soil and square precast prestressed concrete piles (PPC) utilizing 4 statistical methods. In addition, Eslami et al. (2011) evaluated 5 CPT-based methods with 13 databases and with 5 statistical approaches.

To the best of our knowledge, no research has been carried out to cover 25 empirical correlations predicting axial pile capacity at the same time in three different soil types. Each of these criteria reveals the accuracy of these methods by comparing their results with the real results obtained from in-situ tests.

This paper evaluates the accuracy of CPT and CPTu- based methods predicting the ultimate capacity of driven piles in sand, clay, and mixed soil. Each CPT or CPTu test is performed near the location of piles tested in the same field. The measured and estimated ultimate axial capacities of piles are obtained from static load test and CPT and CPTu data, respectively. Four statistical analyses have been operated to rank CPT and CPTu-based methods in terms of their accuracy for estimating the axial bearing capacity of piles. Four methods are the best-fitted line for ${Q}_{E}$ versus ${Q}_{m}$, the geometric average and standard deviation for ${Q}_{E}$/${Q}_{m}$ ratio, the cumulative probability for ${Q}_{E}$/${Q}_{m}$ ratio and the accuracy of 20 percent of the histogram and the lognormal distribution curve for ${Q}_{E}$/${Q}_{m}$ ratio. Each of these criteria reveals the accuracy of these methods by comparing their results with the real results obtained from in-situ tests.

For each CPT and CPTu-based method, finally, the final rank of each criterion (RI) has been calculated. The main concluding remarks can be cited as:

• Statistical methods show that CPT and CPTu based methods have their drawbacks in predicting the axial pile capacity.

• Based on the assessment of 25 CPT and CPTu based methods with 61 databases, amongst 25 methods, the one proposed by Eslami and Fellenius (1997) is the most accurate in estimating the ultimate bearing capacity of piles driven into clay and Niazi and Mayne method (Niazi and Mayne 2016) has the most accurate prediction in sand and mixed soil.

• Three relative reliability values amongst the 25 presented methods belong respectively to Niazi and Mayne (2016), Eslami and Fellenius (1997), and Cambridge-05 method (White and Bolton 2005) with ranks 1, 2 and 3 in the sand.

• In clay, Eslami and Fellenius method (Eslami and Fellenius 1997) with rank number 1 has the most relative reliability. The UniCone (Eslami and Fellenius 1997; Fellenius 2002) and KTRI (Takesue et al. 1998) methods have the second and third relative reliability values, respectively.

• In mixed soil, Niazi and Mayne (Niazi and Mayne 2016) presented a method ranking one. Eslami and Fellenius method (Eslami and Fellenius 1997) and the UniCone method (Eslami and Fellenius 1997; Fellenius 2002) with rank number 2 and 3 have higher accuracy.

There are a number of reasons why some methods perform well in prediction of pile capacities while others do not. A chief reason is that some researchers have introduced methods ignoring the soil type, pile installation methods or pile material types in which approximation becomes a brilliant factor in such methods. If they had classified their methods in more details for pile installation, soil type and pile material, there would have been much more accurate methods. Unfortunately, some of the methods did not do these classifications. For instance, they presented their methods in all soil types or in all pile types and it is obvious that they did not consider some facts. For example, for pile-soil friction along the pile shaft, the soil type, pile material, and interaction features are of great importance, or steel and concrete piles have different α values in conventional α-method. These are because of the fact that these researchers seek simplicity in order not to confuse practicing engineers. However, nowadays for economical consideration in the pile design, it may be preferable to consider all contributing parameters in methods predicting the axial capacity of piles.

Moreover, these CPT-based methods are empirical fitting; sometimes their databases might not have any variations resulting in the accuracy reduction of the methods. The variations of databases have significant effect on all the empirical methods. In conclusion, the performance of CPT-based methods depends on classifying soil and pile types and the ranges of databases.

It is recommended that the results of this investigation be considered in practice which can assist geotechnical engineers to have more precise designs through the best methods in each soil type. Moreover, it is necessary to develop more CPT and CPTu based methods and qualify existing methods for better prediction of pile capacity. Both are beyond the scope of this paper and need further investigation.

Appendix: Direct CPT and CPTu Methods

For more details about these 25 methods refer to “Cone Penetration Test-Based Direct Methods for Evaluating Static Axial Capacity of Single Piles: A State-of-the-Art Review” (Niazi and Mayne 2013) (Table 5).

Table 5 Summary of direct CPT and CPTu based pile design methods