# Improved Wave Equation Analysis for Piles in Soil-Based Intermediate Geomaterials with LRFD Recommendations and Economic Impact Assessment

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## Abstract

**:**

## 1. Introduction

_{u}) of any fine-grained geomaterials is equal to or greater than 129 kPa, then it would be classified as a fine-grained soil-based IGM (FG-IGM). Furthermore, when an SPT N-value is equal to or greater than 58 blows/0.3 m, coarse-grained soil-based IGM (CG-IGM) is classified for pile resistance predictions [24].

_{s}) and unit end bearing (q

_{b}) in S-IGM were developed for different subgroups of CG-IGM and FG-IGM to reduce the uncertainties associated with the prediction of pile resistances in S-IGMs [7,24]. For instance, CG-IGM is further differentiated based on pile types (steel H piles and steel pipe piles), and FG-IGM is further divided into clay-based IGM and silt-based IGM based on grain size. This paper presents a comprehensive wave equation analysis of piles driven in S-IGM using GRLWEAP14 software version 2010–2014 [25] by incorporating classification criteria and proposed static analysis methods for S-IGMs developed by Masud et al. [7,24].

## 2. Existing Studies on Wave Equation Analysis of Pile Driving

_{s}and Q

_{t}) and damping factors (J

_{s}and J

_{t}). The quake and damping parameters can be predicted from the results of Static Load Tests (SLT) and dynamic load testing utilizing the PDA with a subsequent signal-matching analysis using the CAPWAP. Several combinations of quake and damping factors can be obtained to match results from load tests and WEAP [30]. Several past studies have been performed to determine the dynamic parameters summarized in Table 1. Liang and Sheng [31] provided theoretical expressions for predicting the dynamic parameters of soil in terms of pile penetration velocities, accelerations, pile sizes, and soil properties. McVay and Kuo [32] expressed the dynamic soil parameters regarding the SPT N-value and hammer energy (Er). It should be noted that these recommended dynamic parameters were developed for piles driven in soils. However, the dynamic parameters for piles driven in S-IGMs have yet to be investigated.

_{t}values of D/120 for S-IGMs [35]. Since J

_{s}for IGMs were unavailable, the authors assumed J

_{s}values of 0.16 s/m for non-cohesive geomaterials, 0.65 s/m for cohesive geomaterials, and 0.33 s/m for silt-like geomaterials. Q

_{s}= 2.5 mm and J

_{t}= 0.5 s/m were used independent of the geomaterial and pile types. The analysis yielded a mean resistance bias (ratio of CAPWAP measured pile resistance to the WEAP predicted pile resistance) of 1.0 but a relatively high Coefficient of Variation (COV) of resistance biases of 0.30. However, this study was limited to only one state and used no new static analysis methods. Similarly, Islam et al. [10] used 46 test pile data from Kansas, Montana, Iowa, and Wyoming to back-calculate the dynamic parameters for piles driven in shale. However, no similar studies have been conducted to develop a WEAP procedure for driven piles in S-IGMs.

_{s}or q

_{b}to the S-IGMs. The q

_{s}and q

_{b}need to be manually input into WEAP; however, this creates another challenge as reliable static analysis methods for predicting these unit pile resistances in S-IGMs are unavailable in the default WEAP method. S-IGMs often exhibit SPT N-values greater than the maximum allowable input of 60 in the SA method. In those cases, other geomaterial properties, such as the friction angle (ϕ) or unconfined compressive strength (q

_{u}), can be used to describe the geomaterial, but these properties are usually unavailable. In addition, WEAP limits the maximum q

_{s}and q

_{b}assigned by the SA method to the geomaterials (Table 2), and for a past study performed for the Wyoming Department of Transportation (WYDOT), the catalog of geomaterial properties and CAPWAP-measured resistances prepared by Adhikari [35] suggested that the CAPWAP-measured q

_{s}and q

_{b}of the cohesive soil-based IGMs ranged from 52 kPa to 172 kPa and 3687 kPa to 10,677 kPa, respectively. These average q

_{s}and q

_{b}were 78 kPa and 6608 kPa, respectively, which are higher than the maximum q

_{s}and q

_{b}recommended for clay limited by the SA method (Table 2). Applying maximum q

_{s}and q

_{b}values could lead to the underprediction of pile resistances by WEAP for piles driven in S-IGMs. Therefore, a user-defined input of the unit resistances for soil-based IGMs in WEAP is recommended.

## 3. Pile Load Test Data

_{u}, ϕ, and q

_{u}are obtained from the respective boring logs and geotechnical reports. Among 34 test piles, 25 are steel H-piles and 9 are steel pipe piles. The total pile penetration ranges from 5.9 m to 41.6 m. The stroke height ranges from 1.68 to 3.05 m. These 34 test piles are used to conduct the bearing graph analyses, improve WEAP input methods based on the developed new static analysis methods for S-IGMs by Masud et al. [7,24], and recommend new damping parameters for both CG-IGM and FG-IGM for three different Subsurface conditions.

## 4. Four WEAP Procedures for Bearing Graph Analyses

#### 4.1. WEAP SAD Method

_{u}). Since the WEAP SA method only allows input of N ≤ 60 blows/0.3 m, it is difficult to define the S-IGMs that generally have N > 60 blows/0.3 m. An alternative procedure is thus proposed to define the S-IGMs in the default SA method for WEAP analysis as follows:

- N ≤ 60: N and γ are used to define the geomaterial.
- N > 60 for coarse-grained soil-based IGM (CG-IGM): ϕ and γ are used to define the geomaterial.
- N > 60 for fine-grained soil-based IGM (FG-IGM): q
_{u}and γ are used to define the geomaterial.

#### 4.2. WEAP UWD Method

#### 4.3. WEAP SAR Method

#### 4.4. WEAP UWR Method

_{s}and q

_{b}for S-IGMs. Hence, the new static analysis methods (Table 6) developed by Masud et al. [7,24], authors from the University of Wyoming “UW”, are adopted to predict the q

_{s}and q

_{b}for the S-IGMs that are manually input into WEAP. The new percentage shaft resistance obtained from the drivability analysis is used in the bearing graph analysis. The recommended “R” dynamic parameters determined from the back-calculation procedure discussed in the next section are manually input into the bearing graph analysis through the “pile segment” and “soil segment damping/quake” options. For all WEAP methods, bearing graph analysis is conducted to predict the ultimate pile resistance using the field-reported hammer blow count and stoke height for each test pile shown in Table 5.

## 5. Back-Calculation Procedure for Dynamic Parameters

_{s}have the most effect on the bearing graph.

_{s}value has little effect on the bearing graph, and thus, the default Q

_{s}value of 2.5 mm recommended in WEAP is adopted in this study for S-IGMs. Since S-IGMs are more rigid and stiffer than soil, it is reasonable to assume the WEAP-recommended Q

_{t}value of D/120 (hard and stiff soils) for the S-IGMs. For the damping factors, a J

_{s}-to-J

_{t}ratio of one is assumed to create a determinate problem as pile resistance from WEAP depends on the relative effect of both damping factors. The bearing graph is generated by changing J

_{s}and J

_{t}for the S-IGM layers while considering the WEAP-recommended values for the overburdened soil layers. The predicted resistance from WEAP is determined using the stoke height and blow count at the End of Driving (EOD) given in pile driving reports. The best back-calculated damping factors are determined by matching the predicted ultimate resistance from WEAP with the resistance determined from CAPWAP at EOD until the difference is less than 0.1%. A sample calculation is illustrated using the HP 310×79 test pile (pile ID-26 from Table 4) at the Abutment 1 location of the Pine Bluff-Beech Street bridge project in Wyoming. The CAPWAP resistance of the test pile at EOD is 1272 kN at this particular location. The pile length, penetration, hammer, stroke height, and blow counts at the EOD are 15.3 m, 13.6 m, Delmeg D 16–32, 2.07 m, and 62 b/0.3 m, respectively. Pile resistance from WEAP methods depends on quake and damping values, creating an indeterminate problem. To eliminate this indeterminacy, the shaft quake Q

_{s}value of 2.5 mm, toe quake Q

_{t}value of 2.58 mm, and J

_{s}-to-J

_{t}ratio of 1 are chosen for the bearing graph analysis. After several trials and errors, the damping factor J

_{s}= J

_{t}= 0.5 s/m was finally selected for the bearing graph analysis, and the WEAP predicted a resistance of 1270.9 kips. The resistance difference between WEAP and CAPWAP at EOD is 0.09% in this test pile. Based on this methodology, all the back-calculated damping parameters for 34 test piles are determined.

#### 5.1. Back-Calculation Results

#### 5.2. Subsurface Condition I (Fine-Grained Soil-Based IGM as Bearing Layer)

_{t}of 0.50 s/m for cohesive soils, but lower than the J

_{s}of 0.65 s/m for cohesive soils. The higher average back-calculated J

_{t}and lower average back-calculated J

_{s}indicate that the current WEAP-recommended J

_{t}and J

_{s}for cohesive soils will underpredict the shaft resistance and overpredict the end bearing of FG-IGMs. A relationship shown in Figure 2a is developed for the back-calculated damping factor based on the s

_{u}(kPa) of FG-IGM at the pile tip and slenderness ratio (i.e., the ratio of pile penetration L (m) to pile dimension D

_{p}(mm)). The fitted linear regression function for predicting ${\widehat{J}}_{t}$ (s/m) and ${\widehat{J}}_{s}$ (s/m) is given by Equation (1) with a relatively high coefficient of determination (R

^{2}) of 0.74. Equation (1) infers that those piles with a deeper bearing FG-IGM layer will demand higher damping factors, but the damping factors will decrease with increasing pile size. Equation (1) covers 6.1 m ≥ L ≤ 41.61 m, 129 kPa ≥ s

_{u}≤ 672 kPa, and a wide range of pile sizes of HP 310 mm to 360 mm and 406 mm to 510 mm of open-ended pipe piles (OEP). Equation (1) is applicable for damping factor ranges from 0.1 s/m to 1.23 s/m.

#### 5.3. Subsurface Condition II (Coarse-Grained Soil-Based IGM as Bearing Layer)

_{t}, but higher than the J

_{s}recommended in WEAP for non-cohesive soils. This comparison indicates that the current WEAP recommendations will underpredict the end bearing and overpredict the shaft resistance in CG-IGMs. Figure 2b shows a non-linear relationship between the back-calculated damping and the combined term of pile penetration length, L (m), and corrected N, (N

_{1})

_{60}(blows/0.3 m) below the pile tip. The nonlinear trend can be best described by the exponential relationship given by Equation (2)

_{1})

_{60}of CG-IGMs below the pile tip. Because of limited data points with 8.5 m ≥ L ≤ 14.8 m, 55 blows/0.3 m ≥ (N

_{1})

_{60}≤ 151 blows/0.3 m. The pile types used in the development of the prediction equation in Equation (2) include HP 310×79, HP 360×108, and 406-mm open-ended pipe piles (OEP). Future studies are recommended to improve the prediction equation for damping factors by including additional data with longer pile lengths and variable pile types and sizes.

#### 5.4. Subsurface Condition III (Either Soil or Soil-Based IGM as Bearing Layer)

## 6. Determination of Pile Resistances from Bearing Graph Analysis

## 7. Validation of Improved WEAP Methods

_{u}values ≥ 129 kPa following the criteria conducted by Masud et al. [6,22]. Bearing graph analyses are performed using the reported stoke heights and blow counts at the EOD for 22 independent test pile data. The predicted ultimate pile resistances from four different WEAP methods are compared with those from CAPWAP in Figure 5. The mean and COV of resistance biases for the four WEAP methods are included in Figure 5 for comparative study. The highest COV of 0.25 is obtained from the WEAP SAD and WEAP UWD methods. Having the mean resistance bias of 1.04, the pile resistances are underpredicted by the WEAP SAD and WEAP UWD, on average, by only approximately 4%. However, the WEAP UWR and WEAP SAR improve the pile resistance predictions by reducing the underprediction of pile resistances by around 5% (from 1.04 to 0.99) and the COV by 20% (from 0.25 to 0.20)

## 8. LRFD Resistance Factors

_{T}) of 2.33 and 3.0 for a redundant pile group and a non-redundant pile group, respectively [39]. The calibration of resistance factors using FOSM, FORM, and MCS required the mean, standard deviation, and COV of resistance biases. The details of the calibration procedures are discussed in the literature [39,40].

_{T}= 2.33. Compared with φ = 0.50 recommended in the AASHTO [38] for soils, the calibrated φ values using FOSM at β

_{T}= 2.33 for piles in S-IGMs are 40%, 40%, 44%, and 50% higher for WEAP SAD, WEAP UWD, WEAP SAR, and WEAP UWR, respectively. The obtained resistance factors are also higher than the φ = 0.40 recommended by Paikowsky et al. [40]. These higher calibrated φ values are attributed to mean bias closer to one and lower COV values. In this study, φ values based on FORM and MCS, on average, are 17%, 17%, 22%, and 24% higher than that from FOSM for WEAP SAD, WEAP UWD, WEAP SAR, and WEAP UWR, respectively, for β

_{T}= 2.33. Among all the different methods, WEAP UWR has the highest resistance and efficiency factors. The values of φ for the WEAP UWR are, on average, 12% higher than those for the default WEAP SAD methods for β

_{T}= 2.33. In addition, the efficiency factor of the WEAP UWR, on average, is approximately 20% higher than the WEAP SAD for β

_{T}= 2.33. Compared with the default WEAP SAD, the selected proposed WEAP UWR method, on average, reduces the underprediction of pile resistances by 6% and improves the reliability with a 43% reduction in the coefficient of variation (COV) for β

_{T}= 2.33.

## 9. Economic Impact Assessment

_{C}) is determined.

_{D}) for the default WEAP SAD using φ = 0.5 [38]. The WEAP SAD resistances are obtained from the bearing graph based on the stoke height and blow count at the EOD condition.

_{P}) for the WEAP UWR method using φ = 0.75 based on FOSM at β

_{T}= 2.33.

_{C}from Step 2, φR

_{D}from Step 3, or φR

_{P}from Step 4.

## 10. Summary and Conclusions

- Quake values of 2.5 mm and D/120 for Qs and Q
_{t}, respectively, are adequate for the S-IGMs. Smith damping factors are found to depend on the pile driving and subsurface conditions. Single values of Smith damping factors are inadequate for different pile and driving conditions. Hence, new Smith damping factors are proposed for three different subsurface conditions. - Using 34 training pile test data and 22 independent test pile data, it is found that WEAP UWR is the most efficient as it provides a mean resistance bias of 1.02 closer to 1 and the lowest COV of 0.18.
- A φ value of 0.75 for WEAP UWR calibrated based on FOSM at β
_{T}= 2.33 for piles driven into S-IGMs is higher than the value of φ of 0.5 recommended in AASHTO [38] for piles in soils. Compared with the default WEAP SAD, the selected proposed WEAP UWR method, on average, reduces the underprediction of pile resistances by 6% and improves the reliability with a 43% reduction in the coefficient of variation (COV) for β_{T}= 2.33. - The economic impact assessment reveals that the average differences in steel weight per unit load for the WEAP SAD and WEAP UWR are −0.54 kg/kN and 0.06 kg/kN, respectively. Compared with the WEAP SAD method, the WEAP UWR method seems to be more efficient as the average difference in steel weight per unit load is closer to zero, which will reduce construction challenges.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Relationship between back-calculated damping factor and the product of undrained shear strength and slenderness ratio for FG-IGMs and (

**b**) relationship between damping factor and the ratio of pile penetration to corrected SPT N-value for CG-IGMs.

**Figure 4.**Comparison of measured pile resistances from CAPWAP and predicted pile resistance from WEAP SAD, WEAP UWD, WEAP SAR, and WEAP UWR for the 34-training pile dataset.

**Figure 5.**Comparison of measured pile resistances from CAPWAP and predicted pile resistances from WEAP SAD, WEAP UWD, WEAP SAR, and WEAP UWR for the 22 independent pile data.

References | Dynamic Parameter | |||
---|---|---|---|---|

Q_{s} | Q_{t} | J_{s} | J_{t} | |

Smith [27] | 2.5 mm | 2.5 mm | 0.16 s/m | 0.49 s/m |

Coyle and Gibson [33] | NA | NA | NA | $\frac{1}{{V}^{N}}\left(\frac{{P}_{d}}{{P}_{s}}-1\right)$ |

Coyle et al. [18] | 2.5 mm | 2.5 mm | 0.66 s/m (clay) 0.16 s/m (sand) 0.33 s/m (silt) | 0.03 s/m (clay) 0.49 s/m (sand and silt) |

Hannigan et.al. [34] | 2.5 mm | D/120 (very dense and hard soil) D/60 (soft soils) | 0.66 s/m (cohesive soil) 0.16 s/m (non-cohesive soil) | 0.49 s/m for all soils |

Liang and Sheng [31] | $\frac{{f}_{s}{r}_{0}}{\overline{G}}\mathrm{l}\mathrm{n}\left(\frac{{r}_{m}}{{r}_{0}}\right)$ | $\frac{1+\upsilon}{2E}{p}_{y}\left(\frac{D}{2}\right)$ | 1/3 of toe damping | $\frac{\rho}{3{f}_{t}}\left(2{r}_{0}\frac{\dot{{v}_{p}}}{{v}_{p}}+3{v}_{p}\right)$ |

McVay and Kuo [32] | ${e}^{\left(\frac{B}{A\frac{{E}_{r}}{{N}_{s}}-1}\right)}$ | ${e}^{\left(\frac{B}{A\frac{{E}_{r}}{{N}_{t}}\frac{D}{12}-1}\right)}$ | ${e}^{\left(\frac{B}{A\frac{E}{{N}_{s}}-1}\right)}$ | ${e}^{\left(\frac{B}{A\frac{E}{{N}_{t}}\frac{D}{12}-1}\right)}$ |

_{s}= Shaft quake; Q

_{t}= Toe quake; J

_{s}= Shaft damping; J

_{t}= Toe damping; NA = Not Available; V = Velocity of soil deformation; P

_{d}= Peak dynamic load; P

_{s}= Peak static load; N = Exponential power modification (0.2 for sand and 0.18 for clay); f

_{s}= Unit shaft resistance; r

_{0}= Radius of pile; r

_{m}= Radius of influence zone; G = Average shear modulus in the influence zone; $\upsilon $ = Soil Poisson’s ratio; E = Elastic modulus of soil; p

_{y}= Yielding stress on cavity; D = Pile diameter (mm); $\rho $ = Soil density; f

_{t}= Unit toe resistance; $\dot{{v}_{p}}$ = Pile penetration acceleration; ${v}_{p}$ = Pile penetration velocity; A, B = Regression constants [27]; N

_{s}= Weighted average skin SPT N-value; N

_{t}= Weighted average toe SPT N-value; and E

_{r}= Rated energy of SPT hammer.

**Table 2.**Maximum unit shaft resistance and unit end bearing used in the SA method [35].

Soil Type | Unit Shaft Resistance, q_{s} (kPa) | Unit End Bearing, q_{b} (kPa) |
---|---|---|

Sand and Gravel | 250 | 12,000 |

Clay | 75 | 3240 |

Silts | 75 (non-cohesive); 250 (cohesive) | 6000 |

**Table 3.**WEAP recommended quake and damping values [35].

Dynamic Parameter | Geomaterial Type | Pile Type | WEAP Recommended Value |
---|---|---|---|

J_{s} | Non-cohesive Soils | All Pile Types | 0.16 s/m |

Cohesive soils | 0.66 s/m | ||

J_{t} | All soil types | 0.5 s/m | |

Q_{s} | All soil types | All Types | 2.5 mm |

Q_{t} | All soil types, soft rock | Non- Displacement piles | 2.5 mm |

Very dense or hard soils | Displacement piles | D/120 | |

Soils that are not very dense or hard | Displacement piles | D/60 | |

Hard rock | All Types | 1 mm |

_{s}= Shaft damping; J

_{t}= Toe damping; Q

_{s}= Shaft quake; Q

_{s}= Toe quake; D = Pile diameter or width.

State | Project | Location | ID | Pile Type | L (m) | Bearing IGM Layer | EOD Information | |||
---|---|---|---|---|---|---|---|---|---|---|

S (m) | B | Q_{m} (kN) | Hammer Types | |||||||

MT | M FK Po. CK | P-4 at B-1 * | 1 * | OEP 406 mm | 14.4 | FG-IGM | 2.44 | 23 | 1416 | ICE I-30 |

Cottonwood Cr | P-3 at B-1 * | 2 * | OEP 508 mm | 41.6 | FG-IGM | 2.90 | 313 | 2888 | ICE I-36 | |

Capitol Interchange | P-47 at B-4 ^{#} | 3 | CEP 406 mm | 9.8 | CG-IGM | 2.68 | 37 | 2895 | ICE I-36 | |

P-8 at B-5 ^{#} | 4 | CEP 406 mm | 14.8 | CG-IGM | 2.68 | 32 | 2253 | ICE I-36 | ||

P-8 at B-1 ^{#} | 5 | CEP 406 mm | 8.8 | CG-IGM | 2.65 | 48 | 2148 | ICE I-30 | ||

P-2 at B-1 ^{#} | 6 | CEP 406 mm | 8.5 | CG-IGM | 3.05 | 60 | 2973 | ICE I-30 | ||

P-11 at B-5 ^{#} | 7 | CEP 406 mm | 14.7 | CG-IGM | 2.90 | 74 | 2442 | ICE I-30 | ||

P-1 at B-1 ^{#} | 8 | CEP 406 mm | 9.0 | CG-IGM | 2.87 | 30 | 2323 | ICE I-36 | ||

P-38 at B-4 ^{#} | 9 | CEP 406 mm | 8.4 | CG-IGM | 2.68 | 44 | 2538 | ICE I-30 | ||

ID | US-95 WR B | P-10 at Pi-1 * | 10 * | HP 360×174 | 12.7 | FG-IGM | 2.87 | 590 | 4217 | ICE I-30 |

SH-55 SR Bridge | P-3 at Pi-1 * | 11 * | HP 360×174 | 17.1 | FG-IGM | 2.50 | 1158 | 4168 | Del D 36-32 | |

P-1 at Pi-4 * | 12 * | HP 360×174 | 6.1 | FG-IGM | 2.44 | 333 | 4191 | Del D 36-32 | ||

P-1 at Pi-5 * | 13 * | HP 360×174 | 10.4 | FG-IGM | 2.32 | 144 | 3522 | Del D 36-32 | ||

P-10 at A-2 * | 14 * | HP 360×174 | 10.1 | FG-IGM | 2.35 | 105 | 4178 | Del D 36-32 | ||

P-4 at A-1 * | 15 | HP 360×174 | 17.2 | FG-IGM | 1.95 | 420 | 4455 | Del D 36-32 | ||

P-1 at Pi-2 ^ | 16 | HP 360×174 | 14.6 | FG-IGM | 2.41 | 71 | 3403 | Del D 36-32 | ||

P-2 at Pi-3 ^ | 17 | HP 360×174 | 10.7 | CG-IGM | 2.35 | 72 | 3190 | Del D 36-32 | ||

SH-51 SR Bridge | P-1 at A-1 ^ | 18 | HP 360×174 | 21.0 | FG-IGM | 2.56 | 73 | 2825 | ICE I-30 V2 | |

P at A-2 ^ | 19 | HP 360×174 | 21.0 | FG-IGM | 2.74 | 52 | 2578 | ICE I-30 V2 | ||

SH-28 LR B | P-2 at A-1 ^ | 20 | HP 360×132 | 12.5 | CG-IGM | 2.29 | 16 | 1721 | Pileco 30-32 | |

SH-52 UPPR B | P-4 at A-1 ^ | 21 | HP 360×174 | 13.1 | CG-IGM | 2.59 | 31 | 2234 | ICE I-30 V2 | |

ND | Memorial Bridge | P-1 at Pi-10(N) ^ | 22 | HP 360×152 | 27.4 | CG-IGM | 2.59 | 40 | 3044 | Del D-36 |

P-1 at Pi-10(S) ^ | 23 | HP 360×152 | 29.6 | CG-IGM | 2.59 | 49 | 3015 | Del D-36 | ||

WY | PB-Muddy Creek | P-1 at A-2 * | 24 * | HP 310×79 | 16.3 | FG-IGM | 2.26 | 109 | 1695 | Del D16-32 |

P-1 at B-2 * | 25 * | HP 310×79 | 10.8 | FG-IGM | 2.41 | 108 | 2006 | Del D16-32 | ||

PB-Beech Street | P-1 at A-2 * | 26 * | HP 310×79 | 13.6 | FG-IGM | 2.07 | 62 | 1272 | Del D16-32 | |

P-3 at A-2 * | 27 * | HP 310×79 | 14.1 | FG-IGM | 2.32 | 82 | 1357 | Del D16-32 | ||

PBME (BS) | BS P-2 at A-2 * | 28 * | HP 310×79 | 12.5 | FG-IGM | 1.77 | 35 | 1477 | APE D 30-32 | |

Hunter Creek | P-3 at A-1 ^{#} | 29 | HP 310×79 | 5.9 | CG-IGM | 1.68 | 850 | 1090 | MKT DE 40 | |

P-2 at A-1 ^ | 30 | HP 310×79 | 11.0 | CG-IGM | 1.92 | 63 | 1010 | MKT DE 40 | ||

Elk Fork Creek | P-5 at A-2 ^ | 31 | HP 360×108 | 12.2 | CG-IGM | 2.50 | 49 | 1802 | ICE 42S | |

Clark’s Fork | P-1 at A-2 ^{#} | 32 | HP 360×108 | 13.7 | CG-IGM | 2.13 | 119 | 1957 | Del. D 19-42 | |

PBME (Parson Street) | PS P-4 at A-1 * | 33 * | HP 310×79 | 22.3 | FG-IGM | 1.92 | 32 | 1481 | APE D 30-32 | |

PS P-3 at A-2 * | 34 * | HP 310×79 | 21.3 | FG-IGM | 2.04 | 35 | 1446 | APE D 30-32 |

_{m}= CAPWAP measured pile resistance at EOD; HP = H pile; CEP = Closed end pipe pile; OEP = Open ended pipe pile; * = Subsurface Condition I;

^{#}= Subsurface Condition II; ^ = Subsurface Condition III.

Method | Geomaterial Input for q_{s} and q_{b} | Quake | Damping |
---|---|---|---|

WEAP-SAD (Default) | SPT N-based (SA) Procedure | ${Q}_{s}$ = 2.5 mm; ${Q}_{t}$ = D/60 or D/120 | ${J}_{s}$ = 0.16 s/m (Coarse); ${J}_{s}$ = 0.66 s/m (Fine); ${J}_{t}$ = 0.50 s/m |

WEAP-UWD | Proposed Static Analysis Methods | ${Q}_{s}$ = 2.5 mm; ${Q}_{t}$ = D/120 | ${J}_{s}$ = 0.16 s/m (Coarse); ${J}_{s}$ = 0.66 s/m (Fine); ${J}_{t}$ = 0.50 s/m |

WEAP-UWR | Proposed Static Analysis Methods | ${Q}_{s}$ = 2.5 mm; ${Q}_{t}$ = D/120 | Recommended ${J}_{s}$ & ${J}_{t}$ from Back-calculation |

WEAP-SAR | SPT N-based (SA) Procedure | ${Q}_{s}$ = 2.5 mm; ${Q}_{t}$ = D/120 | Recommended ${J}_{s}$ & ${J}_{t}$ from Back-calculation |

_{s}= Shaft quake; Q

_{t}= Toe quake; J

_{s}= Shaft damping; J

_{t}= Toe damping; D = Pile Diameter (mm).

New Prediction Equations for S-IGMs | |
---|---|

S-IGM | Unit Shaft Resistance |

ML-IGM (H-pile & Pipe pile) | ${\widehat{q}}_{s}=\left[\frac{1.80}{1+44{e}^{-0.89\frac{{s}_{u}}{{P}_{a}}}}\right]{P}_{a}$ |

CL-IGM (H-pile & Pipe pile) | ${\widehat{q}}_{s}=\left[\frac{1.58}{1+47.6{e}^{-1.34\frac{{s}_{u}}{{P}_{a}}}}\right]{P}_{a}$ |

CH-IGM (H-pile & Pipe pile) | ${\widehat{q}}_{s}=\left[\frac{2}{1+50.4{e}^{-1.4\frac{{s}_{u}}{{P}_{a}}}}\right]{P}_{a}$ |

CG-IGM (H-Pile) | ${\widehat{q}}_{s}=\left[\frac{1.21}{1+12.62{e}^{-4.06\left(\frac{{{\sigma}^{\prime}}_{v}}{{P}_{a}}\frac{58}{{\left({N}_{1}\right)}_{60}}\right)}}\right]{P}_{a}$ |

CG-IGM (Pipe-Pile) | ${\widehat{q}}_{s}=\left[\frac{\frac{{{\sigma}^{\prime}}_{v}}{{P}_{a}}\frac{58}{{\left({N}_{1}\right)}_{60}}}{0.105+0.52\frac{{{\sigma}^{\prime}}_{v}}{{P}_{a}}\frac{58}{{\left({N}_{1}\right)}_{60}}}\right]{P}_{a}$ |

S-IGM | Unit End Bearing |

FG-IGM (H-pile & Pipe pile) | ${\widehat{q}}_{b}=\left[\frac{\frac{{s}_{u}}{{P}_{a}}\times \frac{D}{{D}_{B}}}{0.001+0.0027\frac{{s}_{u}}{{P}_{a}}\times \frac{D}{{D}_{B}}}\right]{P}_{a}$ |

CG-IGM (H-pile & Pipe pile) | ${\widehat{q}}_{b}=93.76{P}_{a}{\left[\frac{{p}_{a}}{{{\sigma}^{\prime}}_{v}}\frac{{(N}_{1)60}}{58}\right]}^{0.22}$ |

_{u}= undrained shear strength; ${{\sigma}^{\prime}}_{v}=$ Vertical effective overburden stress; P

_{a}= atmospheric pressure = 101.3 kPa; D = pile dimension or diameter; D

_{B}= total pile penetration; ${\widehat{q}}_{s}$ = predicted unit shaft resistance; and ${\widehat{q}}_{b}$ = predicted unit end bearing.

Subsurface Condition | Geomaterial | Q_{s} (mm) | Q_{t} (mm) | J_{s} (s/m) | J_{t} (s/m) |
---|---|---|---|---|---|

I | Soil | 2.5 | D/120 (very dense/hard soil); D/60 (soft soil) | 0.66 (Fine-grained); 0.16 (coarse-grained); 0.33 (silts) | 0.5 |

FG-IGM | D/120 | ${\widehat{J}}_{t}={\widehat{J}}_{s}=0.05\frac{{S}_{u}\times L}{{D}_{p}}-0.025$ | |||

II | Soil | D/120 (very dense/hard soil); D/60 (soft soil) | 0.66 (Fine-grained); 0.16 (coarse-grained); 0.33(silts) | 0.5 | |

CG-IGM | D/120 | ${\widehat{J}}_{t}={\widehat{J}}_{s}=0.07{e}^{9.3\frac{L}{{(N}_{1})60}}$ | |||

III | Soil | D/120 (very dense/hard soil); D/60 (soft soil) | 0.66 (Fine-grained); 0.16 (coarse-grained); 0.33(silts) | 0.5 | |

CG-IGM | D/120 | 0.59 | |||

FG-IGM | 0.33 |

_{s}= Shaft quake; Q

_{t}= Toe quake; J

_{s}= Shaft damping; J

_{t}= Toe damping; D

_{p}= Pile Dimension (mm); D = Pile Diameter (mm); ${S}_{u}$ = Undrained shear strength of the bearing layer at the bottom of pile tip (kPa); L = Embedded pile length (m); (N

_{1})

_{60}= Corrected N (b/0.3 m) of the bearing layer at bottom of pile tip; and FG-IGM = Fine grained soil-based IGM; CG-IGM = Coarse-grained soil-based IGM.

**Table 8.**Summary of independent test pile data and relevant End of Driving (EOD) information for validation.

Site | Location | Pile Type | L (m) | Bearing Geo-Material | EOD Information | |||
---|---|---|---|---|---|---|---|---|

S (m) | B | R_{m} | Hammer | |||||

Jules | SW wing, Pile 1 | HP 310×79 | 7.8 | FG-IGM | 2.5 | 48 | 589.6 | Delmag D 8-32 |

I-78 Over S. | HP | HP 360×132 | 19.5 | CG-IGM | 1.8 | 30 | 1337.7 | Delmag D 36-32 |

Greenville | Pile 3 | CEP 356 mm | 8.5 | CG-IGM | 1.9 | 42 | 843.7 | Delmag D 25-32 |

Pile 12 | CEP 356 mm | 6.7 | CG-IGM | 2.2 | 17 | 845.9 | Delmag D 36-32 | |

Pile 13 | CEP 356 mm | 15.2 | FG-IGM | 2.2 | 33 | 1438.7 | Delmag D 25-32 | |

Mahomet | North Abt. | HP 360×108 | 20.7 | CG-IGM | 2.1 | 126 | 2553.9 | Delmag D 30-32 |

South Abt. | HP 360×108 | 12.6 | CG-IGM | 2.2 | 42 | 1276.3 | Pileco D 19-42 | |

Pier 2 | HP 360×108 | 15.2 | CG-IGM | 2.5 | 81 | 2833.3 | Delmag D 30-32 | |

Godfrey | West Abt. | CEP 356 mm | 12.5 | FG-IGM | 2.6 | 52 | 1012.4 | Delmag D 12-32 |

Bloomington | K-pile | HP 310×93 | 30.8 | CG-IGM | 2.4 | 115 | 1367.9 | APE D 19-42 |

Panther creek | South Abt. | HP 250×85 | 12.0 | CG-IGM | 2.7 | 144 | 1599.8 | ICE 42-S |

Oquawka | East Abt. | CEP 356 mm | 17.7 | CG-IGM | 2.1 | 37 | 763.6 | MKT DE 42 |

Pier | CEP 356 mm | 16.0 | CG-IGM | 2.5 | 49 | 1416.9 | MKT DE 42 | |

West Abt. | CEP 356 mm | 20.1 | FG-IGM | 2.3 | 22 | 645.3 | MKT DE 42 | |

Stronghurst | North Abt. | CEP 356 mm | 16.6 | FG-IGM | 2.7 | 40 | 1367.9 | Delmag D 19-42 |

Pier 1 | CEP 356 mm | 21.0 | FG-IGM | 2.9 | 34 | 833.0 | Delmag D 19-42 | |

Pier 2 | CEP 356 mm | 17.2 | CG-IGM | 2.7 | 38 | 1288.7 | Delmag D 19-42 | |

Jacksonville | Pier 1 | HP 310×93 | 11.9 | FG-IGM | 2.4 | 120 | 1651.8 | Delmag D 19-32 |

Pier 2 | HP 310×93 | 19.0 | FG-IGM | 2.1 | 200 | 1376.8 | Delmag D 19-32 | |

RCS Godfrey | North Abt. | CEP 356 mm | 11.9 | CG-IGM | 2.2 | 90 | 1596.7 | Delmag D 19-32 |

North pier | CEP 356 mm | 9.0 | CG-IGM | 1.8 | 72 | 977.7 | Delmag D 19-32 | |

South pier | CEP 356 mm | 5.9 | CG-IGM | 2.1 | 81 | 1579.8 | Delmag D 19-32 |

Method | Sample Size | $\mathbf{Mean}\text{}\left(\overline{\mathit{x}}\right)$ | COV | Normal | Log-Normal | Log-Likelihood | |||
---|---|---|---|---|---|---|---|---|---|

SW | AD | SW | AD | ||||||

p-Value | Normal | Log-Normal | |||||||

WEAP SAD | 56 | 1.10 | 0.24 | 0.08 | 0.22 | 0.90 | 0.62 | −4.99 | −2.86 |

WEAP UWD | 56 | 1.11 | 0.23 | 0.10 | 0.13 | 0.46 | 0.24 | −2.21 | −1.05 |

WEAP UWR | 56 | 1.02 | 0.18 | 0.007 | 0.003 | 0.001 | 0.000 | NA | NA |

WEAP SAR | 56 | 1.01 | 0.19 | 0.005 | 0.03 | 0.03 | 0.02 | NA | NA |

WEAP SAD * | 54 | 1.08 | 0.21 | 0.76 | 0.70 | 0.32 | 0.33 | 3.28 | 4.23 |

WEAP UWD * | 54 | 1.08 | 0.21 | 0.08 | 0.25 | 0.19 | 0.11 | 2.58 | 2.90 |

WEAP UWR * | 50 | 1.02 | 0.12 | 0.08 | 0.13 | 0.26 | 0.15 | 33.49 | 34.04 |

WEAP SAR * | 51 | 1.00 | 0.14 | 0.23 | 0.28 | 0.65 | 0.52 | 29.07 | 29.80 |

Method | FOSM | FORM | MCS | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

β_{T} = 2.33 | β_{T} = 3.00 | β_{T} = 2.33 | β_{T} = 3.00 | β_{T} = 2.33 | β_{T} = 3.00 | |||||||

$\mathit{\phi}$ | $\frac{\mathit{\phi}}{\left(\overline{\mathit{x}}\right)}$ | $\mathit{\phi}$ | $\frac{\mathit{\phi}}{\left(\overline{\mathit{x}}\right)}$ | $\mathit{\phi}$ | $\frac{\mathit{\phi}}{\left(\overline{\mathit{x}}\right)}$ | $\mathit{\phi}$ | $\frac{\mathit{\phi}}{\left(\overline{\mathit{x}}\right)}$ | $\mathit{\phi}$ | $\frac{\mathit{\phi}}{\left(\overline{\mathit{x}}\right)}$ | $\mathit{\phi}$ | $\frac{\mathit{\phi}}{\left(\overline{\mathit{x}}\right)}$ | |

WEAP SAD | 0.70 | 0.64 | 0.57 | 0.52 | 0.82 | 0.75 | 0.70 | 0.64 | 0.82 | 0.75 | 0.69 | 0.63 |

WEAP UWD | 0.70 | 0.64 | 0.57 | 0.52 | 0.82 | 0.75 | 0.70 | 0.64 | 0.82 | 0.75 | 0.69 | 0.63 |

WEAP UWR * | 0.75 | 0.74 | 0.64 | 0.63 | 0.94 | 0.92 | 0.84 | 0.82 | 0.93 | 0.91 | 0.83 | 0.81 |

WEAP SAR | 0.72 | 0.72 | 0.61 | 0.61 | 0.88 | 0.88 | 0.79 | 0.79 | 0.88 | 0.88 | 0.77 | 0.77 |

_{T}= Reliability index; $\phi $ = Resistance factor; $\frac{\phi}{\left(\overline{x}\right)}$ = Efficiency factor; * = Selected proposed method.

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**MDPI and ACS Style**

Kalauni, H.K.; Masud, N.B.; Ng, K.; Wulff, S.S.
Improved Wave Equation Analysis for Piles in Soil-Based Intermediate Geomaterials with LRFD Recommendations and Economic Impact Assessment. *Geotechnics* **2024**, *4*, 362-381.
https://doi.org/10.3390/geotechnics4020020

**AMA Style**

Kalauni HK, Masud NB, Ng K, Wulff SS.
Improved Wave Equation Analysis for Piles in Soil-Based Intermediate Geomaterials with LRFD Recommendations and Economic Impact Assessment. *Geotechnics*. 2024; 4(2):362-381.
https://doi.org/10.3390/geotechnics4020020

**Chicago/Turabian Style**

Kalauni, Harish K., Nafis Bin Masud, Kam Ng, and Shaun S. Wulff.
2024. "Improved Wave Equation Analysis for Piles in Soil-Based Intermediate Geomaterials with LRFD Recommendations and Economic Impact Assessment" *Geotechnics* 4, no. 2: 362-381.
https://doi.org/10.3390/geotechnics4020020