Posted in TAMWAVE

## TAMWAVE: Cavity Expansion Theory and Soil Set-Up

One of the things that was attempted in the TAMWAVE project is the use of cavity expansion theory to estimate soil set-up in cohesive soils.  Doing this, however, brought some complications that need some explanation.

Cavity expansion theory is basically the study of what happens when one body expands inside of another.  When this takes places, additional radial stresses (most analyses center around a cylinder or sphere) are generated.  In the case of driven piles, these additional stresses add to the pile’s resistance to load.  It can be argued that cavity expansion is one of the key advantages of driven piles.  In the case of drilled piles such as drilled shafts or auger-cast piles, this does not take place, as the soil is removed either before or during the actual pile installation.  The result of this is that driven piles, for the same length and diameter as a corresponding drilled pile, a driven pile will have a greater resistance to load (ultimate capacity.)

Applying cavity expansion theory to piles has a long history and is detailed in documents such as Randolph, Carter and Wroth (1979) and Yu and Houlsby (1991).  Our particular interest is with clays because, in addition to the changes in the soil from cavity expansion, the pore water pressure increases.  This is the primary (but not the only) reason why the SRD (soil resistance to driving) in cohesive soils is significantly less than the ultimate capacity; this fact inevitably complicates drivability studies.

The increase in pore water pressure is a dynamic phenomenon; it experiences a sudden increase during driving and then gradually dissipates after installation.  How gradually the latter takes place depends on many factors such as the permeability of the soil.  Study of this phenomenon is well represented in the literature; however, for the TAMWAVE project it is not really of interest.  The primary interest here is the value of SRD after the immediate increase of pore water pressures during driving.

Both cavity expansion theory and practice show that excess pore water pressures can easily exceed the effective stresses.  In principle, considering that we have established the beta (effective-stress dependent) method for both cohesionless and cohesive soils, this can mean a complete loss of SRD.  Although dramatic drops in SRD are not unknown, for most piles this is unrealistic.  The reason for this is that, like the dissipation, the build-up of pore water pressures is a dynamic phenomenon, albeit in a much smaller time frame.  Dissipation, hindered as it is by the low permeability of cohesive soils, begins immediately.  Pore water pressures (along with any other stresses induced by cavity expansion) also vary with the distance from the pile.

The result of all this is that prediction of both the increase of pore water pressure and its effect on the SRD of the pile during driving is a complicated phenomenon that is not completely represented by closed-form cavity expansion based solutions.  For a project such as this, what we need is something that will give a reasonable representation of soil set-up for cohesive soils.  To accomplish this, we stick with computing the excess pore water pressure, but with a different methodology.  We assume the following:

1. The basic validity of our effective-stress based beta methods of shaft friction calculation.
2. All of the decrease from static ultimate capacity to SRD takes place due to pore water pressure increase.
3. The excess pore water pressure increases affect the effective stress used to compute the shaft friction.
4. Only those pile segments under the water table are subject to this analysis.

Assuming all that, for a soil set-up factor (from this source, loosely adapted) $S_r$, the excess pore water pressures that affect the effective stress (which in turn determine the shaft friction) are computed by the equation

$\Delta u = \sigma'_{vo}\left( 1 - \frac{1}{S_r} \right)$

This gives identical results to those when the $S_r$ are applied directly.

In practice this phenomenon is still subject to investigation.  Some of the research involves use of numerical methods (such as finite element methods) to simulate cavity expansion effects.  This is doubtless an advance over the closed-form solutions of the past, and a necessity given the complexity of the physics of the problem.  Empirical methods are also still being developed, such as are documented by Wang, Verma and Steward (2009).

Advertisements
Posted in TAMWAVE

## New Version of TAMWAVE Online Wave Equation Program Now Available

The completely revised TAMWAVE program is now available.  The goal of this project is to produce a free, online set of routines which analyse driven piles for axial and lateral load-deflection characteristics and drivability by the wave equation. The program is not intended for commercial use but for educational purposes, to introduce students to both the wave equation and methods for estimating load-deflection characteristics of piles in both axial and lateral loading.

We have a series of posts which detail the theory behind and workings of the program:

This program replaces the original routine which was originally written in 2005 and updated in 2010. The documentation for that effort is here.

Posted in TAMWAVE

## TAMWAVE 7: Analysis for a Cohesive Soil

With the analysis of the concrete pile in cohesionless soils complete, we turn to an example in cohesive soils.

The analysis procedure is exactly the same.  We will first discuss the differences between the two, then consider an example.

### Differences with Piles in Cohesive Soils

• The unit weight is in put as a saturated unit weight, and the specific gravity of the soil particles is different (but not by much.)
• Once the simulated CPT data was abandoned, the “traditional” Tomlinson formula for the unit toe resistance, namely $q_t = N_c c$, where $N_c = 9$, was chosen.
• The ultimate resistance along the shaft is done using the formula of Kolk and van der Velde (1996).  This was used as a beta method, for compatibility with the method used for cohesionless soils.  Unless the ratio of the cohesion to the effective stress is constant, the whole concept of a constant lateral pressure due to cohesion needs to be discarded.
• For saturated cohesive soils, an estimate of pile set-up is done using cavity expansion methods.  Originally excess pore pressure due to cavity expansion during driving was estimated using the method described by Randolph (2003); however, this ran into difficulties and a different method was substituted, which is described here.  This excess pore pressure is then added to the existing pore pressure and a new effective stress is computed at each point for the Kolk and van der Velde method.  The results are within reasonable ranges.

### Test Case

This slideshow requires JavaScript.

The only change in basic parameters from the other case was the change to a CH soil.  We opted not to perform a lateral load test this time, although the program is certainly capable of using the CLM 2 method with cohesive soils.

 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 CH Specific Gravity of Solids 2.7 Void Ratio 0.84 Dry Unit Weight, pcf 91.5 Saturated Unit Weight, pcf 120.0 Soil Internal Friction Angle phi, degrees Cohesion c, psf 750 SPT N60, blows/foot 6 CPT qc, psf 12,696 Distance of Water Table from Soil Surface, ft. 50 Penetration of Pile into Water Table, ft. 50 Pile Toe Results Effective Stress at Pile Toe, ksf 7.454 SPT (N1)60 at pile toe, blows/foot 3 Unit Toe Resistance qp, ksf 6.8 Shear Modulus at Pile Toe, ksf 474.8 Toe Spring Constant Depth Factor 1.366 Toe Spring Constant, kips/ft 2,358.0 Pile Toe Quake, in. 0.034 Poisson’s Ratio at Pile Toe 0.500 Toe Damping, kips-sec/ft 14.0 Toe Smith-Type Damping Constant, sec/ft 2.069 Total Static Toe Resistance Qp, kips 6.75 Pile Toe Plugged? Yes Final Results Total Shaft Friction Qs, kips 219.92 Ultimate Axial Capacity of Pile, kips 226.67 Pile Setup Factor 2.0 Total Pile Soil Resistance to Driving (SRD), kips 115.44

 Depth at Centre of Layer, feet Soil Shear Modulus, ksf Beta Quake,inches Maximum Load Transfer, ksf Spring Constant for Wall Shear, ksf/in Smith-Type Damping Constant, sec/ft Maximum Load Transfer During Driving (SRD), ksf 0.50 34.9 2.541 0.0400 0.116 2.91 2.709 0.116 1.50 60.4 1.180 0.0322 0.162 5.03 2.559 0.162 2.50 78.0 0.827 0.0291 0.189 6.50 2.489 0.189 3.50 92.2 0.655 0.0273 0.210 7.69 2.443 0.210 4.50 104.6 0.550 0.0260 0.227 8.72 2.407 0.227 5.50 115.6 0.479 0.0250 0.241 9.64 2.378 0.241 6.50 125.7 0.427 0.0243 0.254 10.48 2.353 0.254 7.50 135.0 0.387 0.0236 0.266 11.25 2.332 0.266 8.50 143.8 0.356 0.0231 0.277 11.98 2.312 0.277 9.50 152.0 0.330 0.0226 0.287 12.66 2.294 0.287 10.50 159.8 0.308 0.0222 0.296 13.31 2.278 0.296 11.50 167.2 0.290 0.0219 0.305 13.93 2.262 0.305 12.50 174.3 0.274 0.0216 0.313 14.53 2.248 0.313 13.50 181.2 0.260 0.0213 0.321 15.10 2.234 0.321 14.50 187.8 0.248 0.0210 0.329 15.65 2.221 0.329 15.50 194.1 0.237 0.0208 0.336 16.18 2.208 0.336 16.50 200.3 0.228 0.0206 0.344 16.69 2.196 0.344 17.50 206.3 0.219 0.0204 0.351 17.19 2.184 0.351 18.50 212.1 0.211 0.0202 0.357 17.67 2.173 0.357 19.50 217.7 0.204 0.0201 0.364 18.14 2.162 0.364 20.50 223.2 0.197 0.0199 0.370 18.60 2.151 0.370 21.50 228.6 0.191 0.0198 0.377 19.05 2.141 0.377 22.50 233.9 0.186 0.0196 0.383 19.49 2.130 0.383 23.50 239.0 0.181 0.0195 0.389 19.92 2.120 0.389 24.50 244.1 0.176 0.0194 0.395 20.34 2.110 0.395 25.50 249.0 0.172 0.0193 0.401 20.75 2.100 0.401 26.50 253.8 0.168 0.0192 0.406 21.15 2.091 0.406 27.50 258.6 0.164 0.0191 0.412 21.55 2.081 0.412 28.50 263.2 0.160 0.0190 0.418 21.94 2.072 0.418 29.50 267.8 0.157 0.0190 0.423 22.32 2.062 0.423 30.50 272.3 0.154 0.0189 0.429 22.69 2.053 0.429 31.50 276.7 0.151 0.0188 0.434 23.06 2.044 0.434 32.50 281.1 0.148 0.0188 0.439 23.42 2.034 0.439 33.50 285.4 0.145 0.0187 0.445 23.78 2.025 0.445 34.50 289.6 0.143 0.0186 0.450 24.13 2.016 0.450 35.50 293.8 0.140 0.0186 0.455 24.48 2.007 0.455 36.50 297.9 0.138 0.0186 0.461 24.82 1.998 0.461 37.50 301.9 0.136 0.0185 0.466 25.16 1.989 0.466 38.50 305.9 0.134 0.0185 0.471 25.49 1.980 0.471 39.50 309.9 0.132 0.0184 0.476 25.82 1.971 0.476 40.50 313.8 0.130 0.0184 0.481 26.15 1.962 0.481 41.50 317.6 0.128 0.0184 0.487 26.47 1.953 0.487 42.50 321.4 0.126 0.0184 0.492 26.79 1.944 0.492 43.50 325.2 0.125 0.0183 0.497 27.10 1.935 0.497 44.50 328.9 0.123 0.0183 0.502 27.41 1.926 0.502 45.50 332.6 0.122 0.0183 0.507 27.72 1.917 0.507 46.50 336.2 0.120 0.0183 0.513 28.02 1.908 0.513 47.50 339.8 0.119 0.0183 0.518 28.32 1.898 0.518 48.50 343.4 0.118 0.0183 0.523 28.61 1.889 0.523 49.50 346.9 0.117 0.0183 0.528 28.91 1.880 0.528 50.50 349.7 0.116 0.0183 0.533 29.15 1.871 0.000 51.50 351.9 0.115 0.0183 0.537 29.33 1.862 0.005 52.50 354.1 0.115 0.0184 0.541 29.51 1.853 0.011 53.50 356.2 0.114 0.0184 0.546 29.69 1.844 0.018 54.50 358.4 0.114 0.0184 0.550 29.87 1.835 0.023 55.50 360.5 0.113 0.0185 0.555 30.04 1.826 0.029 56.50 362.6 0.113 0.0185 0.559 30.22 1.816 0.035 57.50 364.7 0.113 0.0185 0.564 30.39 1.807 0.041 58.50 366.8 0.112 0.0186 0.568 30.57 1.797 0.047 59.50 368.9 0.112 0.0186 0.573 30.74 1.788 0.053 60.50 371.0 0.112 0.0187 0.578 30.92 1.778 0.059 61.50 373.0 0.111 0.0187 0.583 31.09 1.768 0.064 62.50 375.1 0.111 0.0188 0.588 31.26 1.757 0.070 63.50 377.1 0.111 0.0189 0.593 31.43 1.747 0.076 64.50 379.1 0.111 0.0189 0.598 31.60 1.736 0.082 65.50 381.2 0.110 0.0190 0.603 31.76 1.726 0.088 66.50 383.2 0.110 0.0191 0.609 31.93 1.715 0.093 67.50 385.2 0.110 0.0191 0.614 32.10 1.703 0.099 68.50 387.1 0.110 0.0192 0.620 32.26 1.692 0.105 69.50 389.1 0.110 0.0193 0.626 32.43 1.680 0.111 70.50 391.1 0.110 0.0194 0.632 32.59 1.668 0.117 71.50 393.0 0.110 0.0195 0.638 32.75 1.656 0.123 72.50 395.0 0.110 0.0196 0.645 32.91 1.643 0.129 73.50 396.9 0.110 0.0197 0.652 33.07 1.630 0.135 74.50 398.8 0.110 0.0198 0.659 33.23 1.617 0.141 75.50 400.7 0.110 0.0199 0.666 33.39 1.603 0.147 76.50 402.6 0.110 0.0201 0.673 33.55 1.589 0.153 77.50 404.5 0.111 0.0202 0.681 33.71 1.575 0.159 78.50 406.4 0.111 0.0203 0.689 33.87 1.560 0.166 79.50 408.3 0.111 0.0205 0.698 34.03 1.544 0.172 80.50 410.2 0.112 0.0207 0.707 34.18 1.528 0.179 81.50 412.0 0.112 0.0209 0.716 34.34 1.512 0.186 82.50 413.9 0.113 0.0211 0.726 34.49 1.494 0.193 83.50 415.7 0.113 0.0213 0.737 34.64 1.476 0.200 84.50 417.6 0.114 0.0215 0.748 34.80 1.457 0.207 85.50 419.4 0.115 0.0217 0.760 34.95 1.437 0.215 86.50 421.2 0.116 0.0220 0.773 35.10 1.416 0.223 87.50 423.0 0.117 0.0223 0.787 35.25 1.394 0.232 88.50 424.8 0.118 0.0227 0.802 35.40 1.370 0.241 89.50 426.6 0.120 0.0230 0.819 35.55 1.345 0.250 90.50 428.4 0.121 0.0235 0.838 35.70 1.318 0.260 91.50 430.2 0.123 0.0239 0.859 35.85 1.288 0.271 92.50 432.0 0.126 0.0245 0.882 36.00 1.256 0.283 93.50 433.8 0.129 0.0252 0.910 36.15 1.220 0.297 94.50 435.5 0.132 0.0260 0.944 36.29 1.179 0.313 95.50 437.3 0.137 0.0270 0.985 36.44 1.133 0.331 96.50 439.0 0.143 0.0284 1.038 36.58 1.077 0.354 97.50 440.8 0.152 0.0303 1.113 36.73 1.006 0.385 98.50 442.5 0.168 0.0335 1.235 36.87 0.908 0.433 99.50 444.2 0.181 0.0363 1.343 37.02 0.837 0.477

 Length of the pile, in. 1,200.0 Axial stiffness EA. lbs. 720,000,000 Circumference, in. 48.000 Point resistance, lbs. 6,750 Quake of the point, in. 0.034 Number of pile elements 100 Number of loading steps 20 Maximum pile load, lbs. 226,672.5 Load Increment, lbs. 22,667.3 Failure Load, lbs. 226,672.5
 Load Step Force at Pile Head, kips Pile Head Deflection, in. Number of Plastic Shaft Springs 0 0.0 0.000 0 1 22.7 0.012 0 2 45.3 0.025 0 3 68.0 0.039 18 4 90.7 0.058 33 5 113.3 0.082 44 6 136.0 0.109 55 7 158.7 0.140 64 8 181.3 0.175 74 9 204.0 0.214 84 10 226.7 0.271 100 11 204.0 0.259 0 12 181.3 0.246 0 13 158.7 0.234 0 14 136.0 0.221 0 15 113.3 0.209 7 16 90.7 0.193 18 17 68.0 0.175 27 18 45.3 0.154 33 19 22.7 0.132 39 20 -0.0 0.108 44

Although the cohesive soils yield very different results from the cohesionless ones, the presentation is the same.  Note the significant difference between the element/segment SRD for the static resistance and with the pore pressure increase included.  The pile set-up factor is about 2, which is within an acceptable range.  This does not apply to the toe.

The input for the wave equation is identical, except for the hammer selected, which is much smaller than for the cohesionless soils.  This is not due to set-up but to the lower capacity of the pile; the hammer selection does not account for set-up.  The user will have to select a smaller hammer size to take full advantage of this, depending upon the results.

 Time Step, msec 0.04024 Pile Weight, lbs. 15,000 Pile Stiffness, lb/ft 600,000 Pile Impedance, lb-sec/ft 57,937.5 L/c, msec 8.04688 Pile Toe Element Number 102 Length of Pile Segments, ft. 1 Hammer Manufacturer and Size VULCAN 65C Hammer Rated Striking Energy, ft-lbs 19175 Hammer Efficiency, percent 50 Length of Hammer Cushion Stack, in. 18.5 Soil Resistance to Driving (SRD) for detailed results only, kips 115.4 Percent at Toe 5.85 Toe Quake, in. 0.009 Toe Damping, sec/ft 2.07

 Element Element Weight, lbs. Element Stiffness, kips/in Element Cross-Sectional Area, in2 Element Soil Resistance, kips Element Coefficient of Restitution Element Initial Velocity, ft/sec Element Soil Shaft Stiffness, kips/in Element Quake, in. Element Damping, sec/ft Ram 6,500.0 1,880.5 99.40 0.0 0.80 9.74 0.0 1,000.000 0.00 Driving Accessory 1,100.0 711.5 144.00 0.0 0.51 0.00 0.0 1,000.000 0.00 Pile Head 150.0 60,000.0 144.00 0.5 1.00 0.00 11.6 0.040 2.71 4 150.0 60,000.0 144.00 0.6 1.00 0.00 20.1 0.032 2.56 5 150.0 60,000.0 144.00 0.8 1.00 0.00 26.0 0.029 2.49 6 150.0 60,000.0 144.00 0.8 1.00 0.00 30.7 0.027 2.44 7 150.0 60,000.0 144.00 0.9 1.00 0.00 34.9 0.026 2.41 8 150.0 60,000.0 144.00 1.0 1.00 0.00 38.5 0.025 2.38 9 150.0 60,000.0 144.00 1.0 1.00 0.00 41.9 0.024 2.35 10 150.0 60,000.0 144.00 1.1 1.00 0.00 45.0 0.024 2.33 11 150.0 60,000.0 144.00 1.1 1.00 0.00 47.9 0.023 2.31 12 150.0 60,000.0 144.00 1.1 1.00 0.00 50.7 0.023 2.29 13 150.0 60,000.0 144.00 1.2 1.00 0.00 53.3 0.022 2.28 14 150.0 60,000.0 144.00 1.2 1.00 0.00 55.7 0.022 2.26 15 150.0 60,000.0 144.00 1.3 1.00 0.00 58.1 0.022 2.25 16 150.0 60,000.0 144.00 1.3 1.00 0.00 60.4 0.021 2.23 17 150.0 60,000.0 144.00 1.3 1.00 0.00 62.6 0.021 2.22 18 150.0 60,000.0 144.00 1.3 1.00 0.00 64.7 0.021 2.21 19 150.0 60,000.0 144.00 1.4 1.00 0.00 66.8 0.021 2.20 20 150.0 60,000.0 144.00 1.4 1.00 0.00 68.8 0.020 2.18 21 150.0 60,000.0 144.00 1.4 1.00 0.00 70.7 0.020 2.17 22 150.0 60,000.0 144.00 1.5 1.00 0.00 72.6 0.020 2.16 23 150.0 60,000.0 144.00 1.5 1.00 0.00 74.4 0.020 2.15 24 150.0 60,000.0 144.00 1.5 1.00 0.00 76.2 0.020 2.14 25 150.0 60,000.0 144.00 1.5 1.00 0.00 78.0 0.020 2.13 26 150.0 60,000.0 144.00 1.6 1.00 0.00 79.7 0.020 2.12 27 150.0 60,000.0 144.00 1.6 1.00 0.00 81.4 0.019 2.11 28 150.0 60,000.0 144.00 1.6 1.00 0.00 83.0 0.019 2.10 29 150.0 60,000.0 144.00 1.6 1.00 0.00 84.6 0.019 2.09 30 150.0 60,000.0 144.00 1.6 1.00 0.00 86.2 0.019 2.08 31 150.0 60,000.0 144.00 1.7 1.00 0.00 87.7 0.019 2.07 32 150.0 60,000.0 144.00 1.7 1.00 0.00 89.3 0.019 2.06 33 150.0 60,000.0 144.00 1.7 1.00 0.00 90.8 0.019 2.05 34 150.0 60,000.0 144.00 1.7 1.00 0.00 92.2 0.019 2.04 35 150.0 60,000.0 144.00 1.8 1.00 0.00 93.7 0.019 2.03 36 150.0 60,000.0 144.00 1.8 1.00 0.00 95.1 0.019 2.03 37 150.0 60,000.0 144.00 1.8 1.00 0.00 96.5 0.019 2.02 38 150.0 60,000.0 144.00 1.8 1.00 0.00 97.9 0.019 2.01 39 150.0 60,000.0 144.00 1.8 1.00 0.00 99.3 0.019 2.00 40 150.0 60,000.0 144.00 1.9 1.00 0.00 100.6 0.019 1.99 41 150.0 60,000.0 144.00 1.9 1.00 0.00 102.0 0.018 1.98 42 150.0 60,000.0 144.00 1.9 1.00 0.00 103.3 0.018 1.97 43 150.0 60,000.0 144.00 1.9 1.00 0.00 104.6 0.018 1.96 44 150.0 60,000.0 144.00 1.9 1.00 0.00 105.9 0.018 1.95 45 150.0 60,000.0 144.00 2.0 1.00 0.00 107.1 0.018 1.94 46 150.0 60,000.0 144.00 2.0 1.00 0.00 108.4 0.018 1.93 47 150.0 60,000.0 144.00 2.0 1.00 0.00 109.6 0.018 1.93 48 150.0 60,000.0 144.00 2.0 1.00 0.00 110.9 0.018 1.92 49 150.0 60,000.0 144.00 2.1 1.00 0.00 112.1 0.018 1.91 50 150.0 60,000.0 144.00 2.1 1.00 0.00 113.3 0.018 1.90 51 150.0 60,000.0 144.00 2.1 1.00 0.00 114.5 0.018 1.89 52 150.0 60,000.0 144.00 2.1 1.00 0.00 115.6 0.018 1.88 53 150.0 60,000.0 144.00 0.0 1.00 0.00 0.0 0.018 1.87 54 150.0 60,000.0 144.00 0.0 1.00 0.00 1.2 0.018 1.86 55 150.0 60,000.0 144.00 0.0 1.00 0.00 2.5 0.018 1.85 56 150.0 60,000.0 144.00 0.1 1.00 0.00 3.8 0.018 1.84 57 150.0 60,000.0 144.00 0.1 1.00 0.00 5.1 0.018 1.84 58 150.0 60,000.0 144.00 0.1 1.00 0.00 6.4 0.018 1.83 59 150.0 60,000.0 144.00 0.1 1.00 0.00 7.6 0.018 1.82 60 150.0 60,000.0 144.00 0.2 1.00 0.00 8.9 0.019 1.81 61 150.0 60,000.0 144.00 0.2 1.00 0.00 10.1 0.019 1.80 62 150.0 60,000.0 144.00 0.2 1.00 0.00 11.3 0.019 1.79 63 150.0 60,000.0 144.00 0.2 1.00 0.00 12.6 0.019 1.78 64 150.0 60,000.0 144.00 0.3 1.00 0.00 13.8 0.019 1.77 65 150.0 60,000.0 144.00 0.3 1.00 0.00 14.9 0.019 1.76 66 150.0 60,000.0 144.00 0.3 1.00 0.00 16.1 0.019 1.75 67 150.0 60,000.0 144.00 0.3 1.00 0.00 17.3 0.019 1.74 68 150.0 60,000.0 144.00 0.4 1.00 0.00 18.4 0.019 1.73 69 150.0 60,000.0 144.00 0.4 1.00 0.00 19.6 0.019 1.71 70 150.0 60,000.0 144.00 0.4 1.00 0.00 20.7 0.019 1.70 71 150.0 60,000.0 144.00 0.4 1.00 0.00 21.8 0.019 1.69 72 150.0 60,000.0 144.00 0.4 1.00 0.00 23.0 0.019 1.68 73 150.0 60,000.0 144.00 0.5 1.00 0.00 24.1 0.019 1.67 74 150.0 60,000.0 144.00 0.5 1.00 0.00 25.2 0.019 1.66 75 150.0 60,000.0 144.00 0.5 1.00 0.00 26.2 0.020 1.64 76 150.0 60,000.0 144.00 0.5 1.00 0.00 27.3 0.020 1.63 77 150.0 60,000.0 144.00 0.6 1.00 0.00 28.4 0.020 1.62 78 150.0 60,000.0 144.00 0.6 1.00 0.00 29.4 0.020 1.60 79 150.0 60,000.0 144.00 0.6 1.00 0.00 30.5 0.020 1.59 80 150.0 60,000.0 144.00 0.6 1.00 0.00 31.5 0.020 1.57 81 150.0 60,000.0 144.00 0.7 1.00 0.00 32.6 0.020 1.56 82 150.0 60,000.0 144.00 0.7 1.00 0.00 33.6 0.021 1.54 83 150.0 60,000.0 144.00 0.7 1.00 0.00 34.6 0.021 1.53 84 150.0 60,000.0 144.00 0.7 1.00 0.00 35.6 0.021 1.51 85 150.0 60,000.0 144.00 0.8 1.00 0.00 36.6 0.021 1.49 86 150.0 60,000.0 144.00 0.8 1.00 0.00 37.6 0.021 1.48 87 150.0 60,000.0 144.00 0.8 1.00 0.00 38.6 0.021 1.46 88 150.0 60,000.0 144.00 0.9 1.00 0.00 39.6 0.022 1.44 89 150.0 60,000.0 144.00 0.9 1.00 0.00 40.6 0.022 1.42 90 150.0 60,000.0 144.00 0.9 1.00 0.00 41.5 0.022 1.39 91 150.0 60,000.0 144.00 1.0 1.00 0.00 42.5 0.023 1.37 92 150.0 60,000.0 144.00 1.0 1.00 0.00 43.4 0.023 1.34 93 150.0 60,000.0 144.00 1.0 1.00 0.00 44.4 0.023 1.32 94 150.0 60,000.0 144.00 1.1 1.00 0.00 45.3 0.024 1.29 95 150.0 60,000.0 144.00 1.1 1.00 0.00 46.2 0.025 1.26 96 150.0 60,000.0 144.00 1.2 1.00 0.00 47.2 0.025 1.22 97 150.0 60,000.0 144.00 1.3 1.00 0.00 48.1 0.026 1.18 98 150.0 60,000.0 144.00 1.3 1.00 0.00 49.0 0.027 1.13 99 150.0 60,000.0 144.00 1.4 1.00 0.00 49.9 0.028 1.08 100 150.0 60,000.0 144.00 1.5 1.00 0.00 50.8 0.030 1.01 101 150.0 60,000.0 144.00 1.7 1.00 0.00 51.7 0.034 0.91 102 150.0 786.0 144.00 1.9 1.00 0.00 52.6 0.036 0.84 Pile Toe 0.0 786.0 144.00 6.8 0.00 0.00 0.0 0.009 2.07

 Element Time Step for Maximum Compressive Stress Maximum Compressive Stress, ksi Time Step for Maximum Tensile Stress Maximum Tensile Stress, ksi Maximum Deflection, in. Final Deflection, in. Final Velocity, ft/sec 1 183 2.90 592 0.00 0.818 0.277 -9.74 2 119 1.55 538 0.00 0.696 0.681 0.12 3 121 1.56 2 0.00 0.270 0.265 -0.02 4 123 1.56 3 0.00 0.270 0.265 -0.03 5 125 1.55 465 0.01 0.270 0.265 -0.02 6 127 1.55 467 0.05 0.270 0.265 -0.02 7 128 1.55 469 0.10 0.269 0.265 -0.02 8 130 1.55 471 0.14 0.269 0.265 -0.02 9 132 1.55 471 0.18 0.268 0.265 -0.01 10 134 1.55 473 0.22 0.268 0.265 -0.00 11 136 1.54 475 0.26 0.268 0.265 0.00 12 138 1.54 477 0.30 0.267 0.265 0.01 13 140 1.54 476 0.34 0.267 0.265 0.02 14 142 1.54 477 0.37 0.267 0.266 0.03 15 144 1.53 478 0.40 0.267 0.266 0.05 16 146 1.53 477 0.43 0.267 0.266 0.08 17 148 1.53 477 0.46 0.267 0.267 0.11 18 150 1.52 476 0.48 0.267 0.267 0.14 19 152 1.52 477 0.50 0.268 0.268 0.17 20 154 1.52 478 0.51 0.269 0.269 0.20 21 156 1.51 476 0.53 0.269 0.269 0.23 22 158 1.51 476 0.54 0.270 0.270 0.26 23 160 1.50 475 0.55 0.271 0.271 0.30 24 162 1.50 476 0.55 0.271 0.271 0.34 25 164 1.49 476 0.55 0.272 0.272 0.37 26 166 1.49 476 0.54 0.273 0.273 0.41 27 168 1.48 475 0.53 0.274 0.274 0.45 28 170 1.48 475 0.51 0.274 0.274 0.48 29 172 1.47 476 0.48 0.275 0.275 0.53 30 174 1.47 475 0.45 0.276 0.276 0.58 31 176 1.46 474 0.41 0.276 0.276 0.63 32 178 1.46 472 0.37 0.277 0.277 0.68 33 180 1.45 471 0.32 0.278 0.278 0.71 34 182 1.44 472 0.28 0.278 0.278 0.72 35 184 1.43 466 0.23 0.278 0.278 0.72 36 185 1.42 516 0.24 0.279 0.279 0.70 37 186 1.41 524 0.26 0.279 0.279 0.65 38 188 1.40 529 0.28 0.279 0.279 0.58 39 190 1.38 532 0.31 0.279 0.279 0.51 40 192 1.37 533 0.34 0.279 0.279 0.44 41 194 1.36 542 0.38 0.279 0.279 0.38 42 196 1.35 541 0.42 0.279 0.279 0.33 43 198 1.33 544 0.45 0.279 0.279 0.28 44 200 1.32 543 0.49 0.279 0.279 0.23 45 203 1.31 542 0.52 0.279 0.279 0.18 46 205 1.30 545 0.55 0.278 0.278 0.12 47 207 1.28 544 0.58 0.278 0.278 0.08 48 209 1.27 542 0.60 0.277 0.277 0.03 49 211 1.26 544 0.63 0.277 0.277 -0.01 50 213 1.24 543 0.65 0.277 0.277 -0.05 51 216 1.23 542 0.67 0.276 0.276 -0.10 52 217 1.22 540 0.69 0.276 0.276 -0.14 53 218 1.22 539 0.69 0.277 0.275 -0.18 54 220 1.22 540 0.69 0.278 0.275 -0.22 55 222 1.22 539 0.69 0.279 0.274 -0.25 56 224 1.22 538 0.69 0.281 0.274 -0.28 57 226 1.22 538 0.68 0.282 0.274 -0.32 58 228 1.22 538 0.66 0.283 0.273 -0.36 59 230 1.22 537 0.65 0.285 0.273 -0.41 60 232 1.23 536 0.63 0.286 0.273 -0.46 61 235 1.23 534 0.60 0.287 0.273 -0.52 62 237 1.23 535 0.57 0.288 0.273 -0.56 63 239 1.23 533 0.54 0.290 0.273 -0.61 64 241 1.23 532 0.50 0.291 0.273 -0.63 65 244 1.23 530 0.46 0.292 0.273 -0.66 66 246 1.23 531 0.41 0.293 0.273 -0.69 67 248 1.23 531 0.35 0.294 0.273 -0.72 68 250 1.23 530 0.29 0.294 0.274 -0.74 69 253 1.23 532 0.23 0.295 0.274 -0.75 70 255 1.23 470 0.18 0.296 0.274 -0.75 71 253 1.23 474 0.21 0.296 0.275 -0.75 72 255 1.23 473 0.24 0.296 0.275 -0.75 73 257 1.23 476 0.27 0.296 0.276 -0.74 74 260 1.23 476 0.30 0.296 0.276 -0.74 75 262 1.23 478 0.33 0.296 0.277 -0.72 76 264 1.23 478 0.35 0.295 0.277 -0.71 77 266 1.23 480 0.38 0.295 0.278 -0.70 78 268 1.22 479 0.39 0.294 0.278 -0.68 79 271 1.22 478 0.41 0.294 0.279 -0.66 80 273 1.22 480 0.43 0.293 0.279 -0.65 81 275 1.21 478 0.44 0.292 0.280 -0.64 82 277 1.21 477 0.46 0.292 0.280 -0.62 83 279 1.20 479 0.47 0.291 0.281 -0.60 84 280 1.19 477 0.48 0.290 0.282 -0.58 85 279 1.18 474 0.49 0.290 0.282 -0.55 86 280 1.17 474 0.50 0.289 0.283 -0.53 87 281 1.15 476 0.50 0.289 0.284 -0.51 88 281 1.12 469 0.51 0.288 0.284 -0.48 89 280 1.10 471 0.51 0.288 0.285 -0.45 90 281 1.06 471 0.51 0.288 0.285 -0.42 91 281 1.02 472 0.50 0.288 0.286 -0.40 92 280 0.97 473 0.49 0.288 0.287 -0.37 93 281 0.92 474 0.46 0.288 0.287 -0.34 94 282 0.87 474 0.42 0.288 0.288 -0.30 95 283 0.81 475 0.37 0.289 0.288 -0.27 96 282 0.75 476 0.31 0.289 0.289 -0.25 97 283 0.68 478 0.25 0.289 0.289 -0.23 98 289 0.62 480 0.19 0.289 0.289 -0.21 99 294 0.56 482 0.12 0.290 0.290 -0.19 100 302 0.51 485 0.07 0.290 0.290 -0.17 101 307 0.47 489 0.02 0.290 0.290 -0.15 102 316 0.46 532 0.00 0.290 0.290 -0.12
 Soil Resistance, kips Permanent Set of Pile Toe, inches Blows per Foot of Penetration Maximum Compressive Stress, ksi Element of Maximum Compressive Stress Maximum Tensile Stress, ksi Element of Maximum Tensile Stress Number of Iterations 23.1 (45.3) 1.541 7.8 1.53 4 1.21 24 2000 46.2 (90.7) 0.744 16.1 1.54 4 1.05 54 1149 69.3 (136.0) 0.494 24.3 1.54 4 0.97 54 872 92.3 (181.3) 0.349 34.4 1.55 4 0.86 54 740 115.4 (226.7) 0.281 42.7 1.56 4 0.69 54 592 138.5 (272.0) 0.228 52.6 1.58 3 0.52 56 588 161.6 (317.3) 0.184 65.2 1.61 3 0.30 92 480 184.7 (362.7) 0.144 83.3 1.64 3 0.20 94 477 207.8 (408.0) 0.108 111.1 1.67 4 0.11 95 474 230.9 (453.3) 0.077 155.4 1.70 4 0.07 92 471

The bearing graph data is complete.  The only difference with the cohesionless soils is the way the soil resistance is reported; the values in parentheses are ultimate resistance without set-up and those outside are the SRD with set-up.  The blow count indicates that a smaller hammer may be in order.

Posted in TAMWAVE

## TAMWAVE 6: Results of Wave Equation Analysis

With the data entered for the wave equation analysis, we can now see the results.  There’s a lot of tabular data here but you need to read the notes between it to understand what the program is putting out.  If you are not familiar at all with the wave equation for piles, you need to review this as well.

 Time Step, msec 0.04024 Pile Weight, lbs. 15,000 Pile Stiffness, lb/ft 600,000 Pile Impedance, lb-sec/ft 57,937.5 L/c, msec 8.04688 Pile Toe Element Number 102 Length of Pile Segments, ft. 1 Hammer Manufacturer and Size VULCAN O16 Hammer Rated Striking Energy, ft-lbs 48750 Hammer Efficiency, percent 67 Length of Hammer Cushion Stack, in. 16.5 Soil Resistance to Driving (SRD) for detailed results only, kips 572.7 Percent at Toe 35.39 Toe Quake, in. 0.220 Toe Damping, sec/ft 0.07

For those familiar with the wave equation, there are few surprises.  Some explanation of the parameters can be found with the documentation for the TTI program.

 Element Element Weight, lbs. Element Stiffness, kips/in Element Cross-Sectional Area, in2 Element Soil Resistance, kips Element Coefficient of Restitution Element Initial Velocity, ft/sec Element Soil Shaft Stiffness, kips/in Element Quake, in. Element Damping, sec/ft Ram 16,250.0 4,957.5 233.71 0.0 0.80 11.37 0.0 1,000.000 0.00 Driving Accessory 3,800.0 711.5 144.00 0.0 0.51 0.00 0.0 1,000.000 0.00 Pile Head 150.0 60,000.0 144.00 0.0 1.00 0.00 16.1 0.002 45.39 4 150.0 60,000.0 144.00 0.1 1.00 0.00 28.0 0.004 19.91 5 150.0 60,000.0 144.00 0.2 1.00 0.00 36.1 0.005 13.57 6 150.0 60,000.0 144.00 0.3 1.00 0.00 42.7 0.006 10.54 7 150.0 60,000.0 144.00 0.3 1.00 0.00 48.4 0.007 8.73 8 150.0 60,000.0 144.00 0.4 1.00 0.00 53.5 0.007 7.51 9 150.0 60,000.0 144.00 0.5 1.00 0.00 58.2 0.008 6.62 10 150.0 60,000.0 144.00 0.5 1.00 0.00 62.5 0.009 5.95 11 150.0 60,000.0 144.00 0.6 1.00 0.00 66.6 0.009 5.41 12 150.0 60,000.0 144.00 0.7 1.00 0.00 70.4 0.010 4.98 13 150.0 60,000.0 144.00 0.8 1.00 0.00 74.0 0.010 4.62 14 150.0 60,000.0 144.00 0.8 1.00 0.00 77.4 0.011 4.31 15 150.0 60,000.0 144.00 0.9 1.00 0.00 80.7 0.011 4.05 16 150.0 60,000.0 144.00 1.0 1.00 0.00 83.9 0.012 3.82 17 150.0 60,000.0 144.00 1.0 1.00 0.00 87.0 0.012 3.62 18 150.0 60,000.0 144.00 1.1 1.00 0.00 89.9 0.012 3.44 19 150.0 60,000.0 144.00 1.2 1.00 0.00 92.8 0.013 3.28 20 150.0 60,000.0 144.00 1.3 1.00 0.00 95.6 0.013 3.14 21 150.0 60,000.0 144.00 1.3 1.00 0.00 98.3 0.014 3.01 22 150.0 60,000.0 144.00 1.4 1.00 0.00 100.9 0.014 2.89 23 150.0 60,000.0 144.00 1.5 1.00 0.00 103.5 0.014 2.79 24 150.0 60,000.0 144.00 1.5 1.00 0.00 106.0 0.015 2.69 25 150.0 60,000.0 144.00 1.6 1.00 0.00 108.4 0.015 2.60 26 150.0 60,000.0 144.00 1.7 1.00 0.00 110.8 0.015 2.51 27 150.0 60,000.0 144.00 1.8 1.00 0.00 113.1 0.016 2.43 28 150.0 60,000.0 144.00 1.8 1.00 0.00 115.4 0.016 2.36 29 150.0 60,000.0 144.00 1.9 1.00 0.00 117.7 0.016 2.29 30 150.0 60,000.0 144.00 2.0 1.00 0.00 119.9 0.017 2.23 31 150.0 60,000.0 144.00 2.1 1.00 0.00 122.1 0.017 2.17 32 150.0 60,000.0 144.00 2.1 1.00 0.00 124.2 0.017 2.11 33 150.0 60,000.0 144.00 2.2 1.00 0.00 126.3 0.017 2.06 34 150.0 60,000.0 144.00 2.3 1.00 0.00 128.4 0.018 2.01 35 150.0 60,000.0 144.00 2.4 1.00 0.00 130.4 0.018 1.96 36 150.0 60,000.0 144.00 2.4 1.00 0.00 132.5 0.018 1.91 37 150.0 60,000.0 144.00 2.5 1.00 0.00 134.4 0.019 1.87 38 150.0 60,000.0 144.00 2.6 1.00 0.00 136.4 0.019 1.83 39 150.0 60,000.0 144.00 2.7 1.00 0.00 138.3 0.019 1.79 40 150.0 60,000.0 144.00 2.7 1.00 0.00 140.2 0.019 1.75 41 150.0 60,000.0 144.00 2.8 1.00 0.00 142.1 0.020 1.72 42 150.0 60,000.0 144.00 2.9 1.00 0.00 144.0 0.020 1.68 43 150.0 60,000.0 144.00 3.0 1.00 0.00 145.8 0.020 1.65 44 150.0 60,000.0 144.00 3.0 1.00 0.00 147.7 0.021 1.62 45 150.0 60,000.0 144.00 3.1 1.00 0.00 149.5 0.021 1.59 46 150.0 60,000.0 144.00 3.2 1.00 0.00 151.3 0.021 1.56 47 150.0 60,000.0 144.00 3.3 1.00 0.00 153.0 0.021 1.53 48 150.0 60,000.0 144.00 3.3 1.00 0.00 154.8 0.022 1.50 49 150.0 60,000.0 144.00 3.4 1.00 0.00 156.5 0.022 1.48 50 150.0 60,000.0 144.00 3.5 1.00 0.00 158.3 0.022 1.45 51 150.0 60,000.0 144.00 3.6 1.00 0.00 160.0 0.022 1.43 52 150.0 60,000.0 144.00 3.7 1.00 0.00 161.7 0.023 1.40 53 150.0 60,000.0 144.00 3.7 1.00 0.00 163.0 0.023 1.38 54 150.0 60,000.0 144.00 3.8 1.00 0.00 164.1 0.023 1.37 55 150.0 60,000.0 144.00 3.8 1.00 0.00 165.2 0.023 1.35 56 150.0 60,000.0 144.00 3.9 1.00 0.00 166.2 0.023 1.34 57 150.0 60,000.0 144.00 4.0 1.00 0.00 167.3 0.024 1.32 58 150.0 60,000.0 144.00 4.0 1.00 0.00 168.4 0.024 1.31 59 150.0 60,000.0 144.00 4.1 1.00 0.00 169.4 0.024 1.29 60 150.0 60,000.0 144.00 4.1 1.00 0.00 170.5 0.024 1.28 61 150.0 60,000.0 144.00 4.2 1.00 0.00 171.6 0.024 1.27 62 150.0 60,000.0 144.00 4.2 1.00 0.00 172.6 0.025 1.25 63 150.0 60,000.0 144.00 4.3 1.00 0.00 173.7 0.025 1.24 64 150.0 60,000.0 144.00 4.4 1.00 0.00 174.8 0.025 1.22 65 150.0 60,000.0 144.00 4.4 1.00 0.00 175.8 0.025 1.21 66 150.0 60,000.0 144.00 4.5 1.00 0.00 176.9 0.025 1.20 67 150.0 60,000.0 144.00 4.6 1.00 0.00 178.0 0.026 1.18 68 150.0 60,000.0 144.00 4.6 1.00 0.00 179.0 0.026 1.17 69 150.0 60,000.0 144.00 4.7 1.00 0.00 180.1 0.026 1.16 70 150.0 60,000.0 144.00 4.8 1.00 0.00 181.2 0.026 1.14 71 150.0 60,000.0 144.00 4.8 1.00 0.00 182.3 0.026 1.13 72 150.0 60,000.0 144.00 4.9 1.00 0.00 183.4 0.027 1.12 73 150.0 60,000.0 144.00 5.0 1.00 0.00 184.5 0.027 1.10 74 150.0 60,000.0 144.00 5.0 1.00 0.00 185.6 0.027 1.09 75 150.0 60,000.0 144.00 5.1 1.00 0.00 186.7 0.027 1.08 76 150.0 60,000.0 144.00 5.2 1.00 0.00 187.8 0.028 1.06 77 150.0 60,000.0 144.00 5.3 1.00 0.00 189.0 0.028 1.05 78 150.0 60,000.0 144.00 5.4 1.00 0.00 190.1 0.028 1.04 79 150.0 60,000.0 144.00 5.5 1.00 0.00 191.2 0.029 1.03 80 150.0 60,000.0 144.00 5.5 1.00 0.00 192.4 0.029 1.01 81 150.0 60,000.0 144.00 5.6 1.00 0.00 193.6 0.029 1.00 82 150.0 60,000.0 144.00 5.7 1.00 0.00 194.8 0.029 0.99 83 150.0 60,000.0 144.00 5.8 1.00 0.00 196.0 0.030 0.97 84 150.0 60,000.0 144.00 5.9 1.00 0.00 197.2 0.030 0.96 85 150.0 60,000.0 144.00 6.0 1.00 0.00 198.4 0.030 0.95 86 150.0 60,000.0 144.00 6.1 1.00 0.00 199.6 0.031 0.93 87 150.0 60,000.0 144.00 6.2 1.00 0.00 200.9 0.031 0.92 88 150.0 60,000.0 144.00 6.3 1.00 0.00 202.2 0.031 0.90 89 150.0 60,000.0 144.00 6.5 1.00 0.00 203.5 0.032 0.89 90 150.0 60,000.0 144.00 6.6 1.00 0.00 204.8 0.032 0.88 91 150.0 60,000.0 144.00 6.7 1.00 0.00 206.1 0.033 0.86 92 150.0 60,000.0 144.00 6.8 1.00 0.00 207.5 0.033 0.85 93 150.0 60,000.0 144.00 7.0 1.00 0.00 208.9 0.033 0.84 94 150.0 60,000.0 144.00 7.1 1.00 0.00 210.3 0.034 0.82 95 150.0 60,000.0 144.00 7.3 1.00 0.00 211.7 0.034 0.81 96 150.0 60,000.0 144.00 7.4 1.00 0.00 213.2 0.035 0.79 97 150.0 60,000.0 144.00 7.6 1.00 0.00 214.7 0.035 0.78 98 150.0 60,000.0 144.00 7.7 1.00 0.00 216.3 0.036 0.77 99 150.0 60,000.0 144.00 7.9 1.00 0.00 217.8 0.036 0.75 100 150.0 60,000.0 144.00 8.1 1.00 0.00 219.4 0.037 0.74 101 150.0 60,000.0 144.00 8.3 1.00 0.00 221.1 0.038 0.72 102 150.0 922.6 144.00 8.5 1.00 0.00 222.8 0.038 0.71 Pile Toe 0.0 922.6 144.00 202.7 0.00 0.00 0.0 0.220 0.07

A detailed output of the parameters for each segment/element.  TAMWAVE no longer uses the simplifications used in the past for resistance distribution along the shaft, i.e., uniform, triangular, etc., but constructs one based on the soil properties.  Much of this data is repeated from the static analysis.

 Element Time Step for Maximum Compressive Stress Maximum Compressive Stress, ksi Time Step for Maximum Tensile Stress Maximum Tensile Stress, ksi Maximum Deflection, in. Final Deflection, in. Final Velocity, ft/sec 1 50 3.70 164 0.00 1.299 1.299 -0.11 2 176 2.64 1 0.00 1.300 1.261 -2.56 3 178 2.64 2 0.00 0.650 0.646 -1.01 4 180 2.65 3 0.00 0.646 0.643 -0.93 5 182 2.66 4 0.00 0.641 0.639 -0.85 6 184 2.66 5 0.00 0.637 0.635 -0.78 7 186 2.67 6 0.00 0.632 0.631 -0.70 8 187 2.67 7 0.00 0.628 0.627 -0.62 9 190 2.68 8 0.00 0.623 0.622 -0.53 10 192 2.69 9 0.00 0.619 0.618 -0.45 11 194 2.69 10 0.00 0.614 0.613 -0.37 12 196 2.69 11 0.00 0.609 0.609 -0.30 13 198 2.70 12 0.00 0.604 0.604 -0.22 14 359 2.71 13 0.00 0.599 0.599 -0.14 15 361 2.72 14 0.00 0.594 0.594 -0.06 16 363 2.73 15 0.00 0.588 0.588 0.01 17 365 2.74 16 0.00 0.583 0.583 0.07 18 367 2.75 17 0.00 0.578 0.578 0.13 19 369 2.75 18 0.00 0.572 0.572 0.19 20 372 2.76 19 0.00 0.567 0.567 0.24 21 374 2.77 20 0.00 0.561 0.561 0.27 22 376 2.78 21 0.00 0.556 0.556 0.29 23 378 2.79 22 0.00 0.550 0.550 0.30 24 379 2.80 23 0.00 0.544 0.544 0.29 25 381 2.80 24 0.00 0.539 0.539 0.28 26 384 2.81 25 0.00 0.533 0.533 0.26 27 386 2.82 26 0.00 0.527 0.527 0.23 28 388 2.82 27 0.00 0.522 0.522 0.19 29 390 2.83 28 0.00 0.516 0.516 0.15 30 392 2.83 29 0.00 0.511 0.511 0.11 31 393 2.84 30 0.00 0.505 0.505 0.07 32 395 2.84 31 0.00 0.500 0.500 0.03 33 397 2.84 32 0.00 0.496 0.494 -0.01 34 399 2.84 33 0.00 0.491 0.489 -0.05 35 399 2.84 34 0.00 0.487 0.483 -0.08 36 400 2.84 35 0.00 0.483 0.478 -0.11 37 401 2.83 36 0.00 0.479 0.473 -0.14 38 400 2.82 37 0.00 0.474 0.468 -0.17 39 401 2.81 38 0.00 0.470 0.463 -0.19 40 400 2.80 39 0.00 0.466 0.457 -0.21 41 401 2.78 40 0.00 0.462 0.452 -0.24 42 399 2.76 41 0.00 0.458 0.447 -0.26 43 400 2.74 42 0.00 0.454 0.442 -0.27 44 399 2.71 43 0.00 0.449 0.437 -0.29 45 398 2.68 44 0.00 0.445 0.432 -0.30 46 397 2.65 45 0.00 0.441 0.427 -0.31 47 267 2.64 46 0.00 0.437 0.422 -0.32 48 270 2.64 47 0.00 0.433 0.417 -0.33 49 272 2.63 48 0.00 0.429 0.412 -0.33 50 275 2.62 49 0.00 0.425 0.407 -0.34 51 277 2.61 50 0.00 0.420 0.402 -0.34 52 279 2.60 51 0.00 0.416 0.397 -0.35 53 282 2.59 52 0.00 0.412 0.393 -0.35 54 284 2.58 53 0.00 0.407 0.388 -0.36 55 283 2.57 54 0.00 0.403 0.383 -0.36 56 286 2.56 55 0.00 0.398 0.378 -0.36 57 288 2.55 56 0.00 0.393 0.373 -0.36 58 290 2.54 57 0.00 0.389 0.368 -0.36 59 293 2.53 58 0.00 0.384 0.363 -0.36 60 295 2.52 59 0.00 0.379 0.358 -0.35 61 298 2.51 60 0.00 0.374 0.353 -0.35 62 300 2.50 61 0.00 0.368 0.349 -0.35 63 303 2.49 62 0.00 0.363 0.344 -0.35 64 301 2.47 63 0.00 0.358 0.339 -0.34 65 304 2.46 64 0.00 0.352 0.334 -0.34 66 306 2.45 65 0.00 0.347 0.329 -0.33 67 309 2.44 66 0.00 0.341 0.324 -0.32 68 311 2.43 67 0.00 0.336 0.319 -0.32 69 478 2.42 68 0.00 0.330 0.315 -0.31 70 480 2.43 69 0.00 0.324 0.310 -0.31 71 479 2.44 70 0.00 0.319 0.305 -0.30 72 481 2.44 71 0.00 0.313 0.300 -0.29 73 482 2.44 72 0.00 0.307 0.296 -0.29 74 481 2.43 73 0.00 0.302 0.291 -0.28 75 482 2.42 74 0.00 0.296 0.286 -0.28 76 480 2.40 75 0.00 0.290 0.282 -0.27 77 482 2.38 76 0.00 0.285 0.277 -0.26 78 479 2.35 77 0.00 0.280 0.273 -0.26 79 482 2.32 78 0.00 0.274 0.269 -0.25 80 483 2.28 79 0.00 0.269 0.264 -0.25 81 481 2.25 80 0.00 0.264 0.260 -0.24 82 483 2.21 81 0.00 0.259 0.256 -0.24 83 485 2.17 82 0.00 0.255 0.252 -0.23 84 483 2.13 83 0.00 0.250 0.248 -0.22 85 485 2.09 84 0.00 0.246 0.244 -0.21 86 487 2.05 85 0.00 0.241 0.240 -0.20 87 490 2.00 86 0.00 0.237 0.236 -0.19 88 487 1.95 87 0.00 0.233 0.232 -0.18 89 489 1.91 88 0.00 0.229 0.229 -0.18 90 492 1.86 89 0.00 0.226 0.225 -0.17 91 489 1.80 90 0.00 0.222 0.221 -0.16 92 492 1.75 91 0.00 0.218 0.218 -0.15 93 495 1.69 92 0.00 0.215 0.215 -0.15 94 497 1.63 93 0.00 0.212 0.211 -0.14 95 494 1.57 94 0.00 0.208 0.208 -0.15 96 497 1.51 95 0.00 0.205 0.205 -0.14 97 506 1.45 96 0.00 0.202 0.202 -0.15 98 508 1.39 97 0.00 0.199 0.199 -0.13 99 517 1.33 98 0.00 0.196 0.196 -0.16 100 521 1.28 99 0.00 0.193 0.193 -0.14 101 529 1.23 100 0.00 0.190 0.190 -0.15 102 532 1.24 101 0.00 0.188 0.187 -0.12

This table shows the end results of the run for the “target” SRD of the pile.  “SRD” is “soil resistance to driving,” and in TAMWAVE for cohesionless soils, SRD and the ultimate capacity are the same.  That’s not the case with cohesive soils, as we will see.  In any case TAMWAVE always does a “bearing graph” analysis, which proportionally varies the SRD and obtains different results for the blow count, maximum tensile and compressive stresses.  The bearing graph method isn’t perfect but it’s probably the best way we have to account for varying site conditions and to make judgments about the effect of those on our hammer selection.

The adoption of “Smith-type” damping was originally done for comparison purposes but for bearing graph analysis has one important advantages: it varies the soil radiation damping with the SRD, which is more realistic than just assuming fixed damping.

The table above only appears if the target SRD is actually achieved during bearing graph analysis.  If it doesn’t come up, the bearing graph analysis could not achieve net pile penetration at the target SRD, which means you need to revisit your hammer selection.

Here we see the second graphical output: the force-time history at the target SRD.  There are actually two histories: the actual pile head force (blue) and the pile head velocity multiplied by the impedance (red.)  For semi-infinite piles, the two should be the same; they will differ for actual finite piles, as is easily seen.  Although a “semi-infinite pile” may seem a very theoretical concept, the relationship of the two plots is very important in the field application of pile dynamics.

 Soil Resistance, kips Permanent Set of Pile Toe, inches Blows per Foot of Penetration Maximum Compressive Stress, ksi Element of Maximum Compressive Stress Maximum Tensile Stress, ksi Element of Maximum Tensile Stress Number of Iterations 114.5 1.707 7.0 2.61 30 0.67 43 1590 229.1 0.754 15.9 2.64 29 0.20 25 1124 343.6 0.355 33.8 2.67 28 0.00 102 719 458.1 0.111 108.2 2.71 32 0.00 102 567 572.7 0.000 0.0 2.84 34 0.00 102 549

The final results are shown here.  In this case, at the target SRD, no permanent set of the pile is recorded.  It will be necessary to vary the size of the hammer, being mindful of the stresses (whose allowable values are described here.)

At this point the analysis of this pile is complete.  The program gives you the choice of simply trying another hammer or starting over.  The latter is what we will do next with a sample case for cohesive soils.

Posted in TAMWAVE

## TAMWAVE 5: Wave Equation Analysis, Overview and Initial Entry

With the static analysis complete, we turn to the wave equation analysis.  TAMWAVE (as with the previous version) was based indirectly on the TTI wave equation program.  Although the numerical method was not changed, many other aspects of the program were, and so we need to consider these.

### Shaft and Toe Resistance

Most wave equation programs in commercial use still use the Smith model for shaft and toe resistance during impact.  Referencing specifically their use in inverse methods, Randolph (2003) makes the following comment:

Dynamic pile tests are arguably the most cost-effective of all pile-testing methods, although they rely on relatively sophisticated numerical modelling for back-analysis. Theoretical advances in modelling the dynamic pile-soil interaction have been available since the mid-1980s, but have been slow to be implemented by commercial codes, most of which still use the empirical parameters of the Smith (1960) model. In order to allow an appropriate level of confidence in the interpretation of dynamic pile tests, and estimation of the static response, it is high time that appropriate scientific models were used for pile-soil interaction, including explicit modelling of the soil plug for open-ended piles.

And that was in 2003…and the use of the Smith model in inverse methods was proceeded by its use in forward methods such as this one.  The model he is referring to from the mid-1980’s is, of course, the Randolph and Simons (1986) model, which was used in the ZWAVE program in the late 1980’s.  The details of this model were discussed in Warrington (1997).

The Randolph and Simons model is the one which is being used for the wave equation portion of this routine, as the static component was used for the ALP static axial pile analysis.  In converting the code from the Smith model to this one, there are some things that need to be understood.  We have discussed some of these earlier but others are as follows:

• Randolph and Simons (1985) used a visco-elastic-plastic model for both shaft and toe, the major difference being the location of the plastic slider for the shaft resistance (as is evident in the ZWAVE poster.)  Some contemporary “experimental” codes (such as Salgado, Loukidis, Abou-Jaoude and Zhang (2015)) add a series of springs and masses to replicate the soil mass that surrounds the piles.  While these doubtless enhance the performance of the models, we stuck with the simple visco-elastic-plastic model in TAMWAVE because these are better replicated in true 3D continuum models like STADYN.  1D code is good because of its simplicity, especially with an online routine like TAMWAVE.
• The 1′ segment/element lengths are carried over to the wave equation.  This is shorter than is customarily used even in commercial work but it saves interpolation of the properties along the shaft.
• The “Smith-type” damping constants are simply the damping of the element computed divided by its ultimate/plastic resistance.  Unlike the Smith model, however, the damping force does not vary with the instantaneous static resistance, but is simply the velocity multiplied by the damping constant and the ultimate resistance of that element, be it shaft or toe.  Thus different Smith type constants should be expected from the model being used.  Additionally, with the shaft resistance, the resistance of a shaft segment is limited to its ultimate static resistance.  This means that all additional damping forces must take place during elastic shearing of the soil surface.  Implicit in the Randolph and Simons model is that, once plasticity is achieved, the soil closest to the pile is effectively decoupled from the soil mass, and thus the pile movement can no longer radiate additional energy into the soil.  The result of this is that, as seen here, the Smith-type damping constants are much higher than one would normally assign.  Corte and Lepert (1985), in a direct comparison of the two models, note that the two give nearly the same result if the original Smith damping constants are multiplied by 7.5 for the new model.  Dividing the new result by this brings the damping constants much closer, especially in the lower reaches of the pile where most of the shaft resistance is found, although the ratio of 7.5 should be regarded as study-specific.  Bringing some rationality to the issue of damping constants would go a long way to improve the results of pile dynamics, forward and inverse, since variations of these have a significant impact on the results.
• We mentioned earlier that the toe quakes that resulted seemed high for this size of pile.  This may be due to the fact that “significant residual pressures are locked in at the pile base during installation (equilibriated by negative shear stresses along the pile shaft, as if the pile were loaded in tension.)  This will lead to a stiffer overall pile response in compression, and significantly higher end-bearing stresses mobilised at small displacements.”  (Randolph, 2003)  He goes on to state that “(f)or driven closed ended-piles the residual stress will be lower, but may still be as high as 75% of the base capacity…”  There are two ways to deal with this.  The first is to run the ALP program first and preload the base and shaft before using the resulting prestressed deflections to run the wave equation analysis.  This would be in effect a residual stress analysis (RSA,) which has been used in this field for many years.  The second is to use a “quick and dirty” method, i.e., to reduce the toe quake and thus simulate the higher toe stiffness and lower quake.  The latter was adopted in TAMWAVE, although one motivation from switching from P4XC3 to ALP was to make an RSA easier.  This is a possible point of future modification of the code.
• A change not related to the pile-soil interaction is the elimination of slack computation, as the pile is uniform and continuous (the hammer-cap and cap-pile interface is obviously inextensible.

### Initial Wave Equation Input

For our example the initial input of the wave equation is shown below.

Most of the data required has been carried over from the static analysis.  The hammer database was added in 2010; however, it was reordered in ascending rated striking energy order and a hammer was suggested using the “initial guess” criterion in the Soils and Foundations Handbook, which essentially suggests to set the initial hammer energy in ft-lbs at 8% of the ultimate capacity in pounds.  This is a “rule of thumb” designed to help students who, faced with a wave equation program for the first time, will have some idea of where to start, although there is no guarantee that the hammer will be either too large or small.  Since the energies are sorted, the user can move up or down the list to try another hammer.

The cushion material properties of the hammer, and the coefficient of restitution used to model cushion plasticity, are discussed (with sample properties) in the WEAP87 documentation.  No attempt was done to either convert coefficients of restitution to viscous damping or alter the rebound curve as was done in ZWAVE.  Pile cushion thickness is only input for concrete piles; the input is not shown for others.

### References

In addition to those already cited, the following is included:

Corte, J.-F., and Lepert, P. (1986) “Lateral resistance during driving and dynamic pile testing.”  Proceedings of the Third International Conference on Numerical Methods in Offshore Piling, Nantes, France, 21-22 May.  Paris: Éditions Technip, pp. 19-34.