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

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.

Posted in TAMWAVE

## TAMWAVE 4: Shaft Resistance Profile, ALP and CLM2

With the basic parameters established, we can turn to the static analysis of the pile, both axial and lateral.

### Shaft Resistance Profile

 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 48.4 0.163 0.0022 0.009 4.03 45.394 0.009 1.50 83.9 0.163 0.0038 0.027 6.99 19.911 0.027 2.50 108.3 0.163 0.0050 0.045 9.02 13.572 0.045 3.50 128.1 0.163 0.0059 0.063 10.68 10.543 0.063 4.50 145.3 0.163 0.0067 0.081 12.11 8.730 0.081 5.50 160.6 0.164 0.0074 0.098 13.38 7.509 0.098 6.50 174.6 0.164 0.0080 0.116 14.55 6.623 0.116 7.50 187.6 0.164 0.0086 0.134 15.63 5.948 0.134 8.50 199.7 0.164 0.0091 0.152 16.64 5.414 0.152 9.50 211.1 0.164 0.0097 0.170 17.59 4.980 0.170 10.50 222.0 0.164 0.0102 0.188 18.50 4.618 0.188 11.50 232.3 0.164 0.0106 0.206 19.36 4.313 0.206 12.50 242.2 0.164 0.0111 0.224 20.18 4.050 0.224 13.50 251.7 0.164 0.0115 0.242 20.98 3.822 0.242 14.50 260.9 0.164 0.0120 0.260 21.74 3.621 0.260 15.50 269.8 0.164 0.0124 0.278 22.48 3.444 0.278 16.50 278.4 0.164 0.0128 0.296 23.20 3.285 0.296 17.50 286.7 0.164 0.0132 0.314 23.89 3.142 0.314 18.50 294.8 0.164 0.0135 0.332 24.57 3.013 0.332 19.50 302.7 0.164 0.0139 0.351 25.22 2.895 0.351 20.50 310.4 0.164 0.0143 0.369 25.86 2.787 0.369 21.50 317.9 0.164 0.0146 0.387 26.49 2.688 0.387 22.50 325.2 0.164 0.0149 0.405 27.10 2.597 0.405 23.50 332.4 0.165 0.0153 0.423 27.70 2.512 0.423 24.50 339.4 0.165 0.0156 0.441 28.29 2.434 0.441 25.50 346.3 0.165 0.0159 0.460 28.86 2.361 0.460 26.50 353.1 0.165 0.0162 0.478 29.42 2.292 0.478 27.50 359.7 0.165 0.0166 0.496 29.98 2.228 0.496 28.50 366.3 0.165 0.0169 0.515 30.52 2.168 0.515 29.50 372.7 0.165 0.0172 0.533 31.06 2.112 0.533 30.50 379.0 0.165 0.0175 0.552 31.58 2.058 0.552 31.50 385.2 0.165 0.0178 0.570 32.10 2.007 0.570 32.50 391.3 0.166 0.0181 0.589 32.61 1.960 0.589 33.50 397.4 0.166 0.0183 0.607 33.11 1.914 0.607 34.50 403.3 0.166 0.0186 0.626 33.61 1.871 0.626 35.50 409.2 0.166 0.0189 0.645 34.10 1.830 0.645 36.50 415.0 0.166 0.0192 0.664 34.58 1.790 0.664 37.50 420.7 0.166 0.0195 0.683 35.06 1.753 0.683 38.50 426.4 0.166 0.0197 0.702 35.53 1.717 0.702 39.50 432.0 0.167 0.0200 0.721 36.00 1.682 0.721 40.50 437.5 0.167 0.0203 0.740 36.46 1.649 0.740 41.50 443.0 0.167 0.0206 0.759 36.92 1.618 0.759 42.50 448.4 0.167 0.0208 0.778 37.37 1.587 0.778 43.50 453.8 0.168 0.0211 0.798 37.82 1.558 0.798 44.50 459.1 0.168 0.0214 0.817 38.26 1.530 0.817 45.50 464.4 0.168 0.0216 0.837 38.70 1.502 0.837 46.50 469.6 0.168 0.0219 0.856 39.13 1.476 0.856 47.50 474.8 0.169 0.0221 0.876 39.56 1.450 0.876 48.50 479.9 0.169 0.0224 0.896 39.99 1.426 0.896 49.50 485.0 0.169 0.0227 0.916 40.42 1.402 0.916 50.50 489.1 0.169 0.0229 0.933 40.76 1.382 0.933 51.50 492.3 0.170 0.0231 0.947 41.03 1.367 0.947 52.50 495.5 0.170 0.0233 0.960 41.30 1.352 0.960 53.50 498.7 0.171 0.0234 0.974 41.56 1.337 0.974 54.50 501.9 0.171 0.0236 0.988 41.83 1.323 0.988 55.50 505.1 0.171 0.0238 1.002 42.09 1.308 1.002 56.50 508.3 0.172 0.0240 1.016 42.36 1.294 1.016 57.50 511.5 0.172 0.0242 1.031 42.63 1.280 1.031 58.50 514.7 0.173 0.0244 1.045 42.89 1.266 1.045 59.50 517.9 0.173 0.0246 1.060 43.16 1.252 1.060 60.50 521.1 0.174 0.0248 1.075 43.42 1.238 1.075 61.50 524.3 0.174 0.0250 1.091 43.69 1.224 1.091 62.50 527.5 0.175 0.0252 1.106 43.96 1.211 1.106 63.50 530.7 0.176 0.0254 1.122 44.22 1.197 1.122 64.50 533.9 0.176 0.0256 1.139 44.49 1.184 1.139 65.50 537.1 0.177 0.0258 1.155 44.76 1.170 1.155 66.50 540.4 0.178 0.0260 1.172 45.03 1.157 1.172 67.50 543.6 0.178 0.0262 1.189 45.30 1.144 1.189 68.50 546.9 0.179 0.0265 1.207 45.57 1.130 1.207 69.50 550.2 0.180 0.0267 1.224 45.85 1.117 1.224 70.50 553.5 0.181 0.0269 1.243 46.12 1.104 1.243 71.50 556.8 0.182 0.0272 1.262 46.40 1.091 1.262 72.50 560.1 0.183 0.0274 1.281 46.68 1.078 1.281 73.50 563.5 0.184 0.0277 1.300 46.96 1.065 1.300 74.50 566.9 0.185 0.0280 1.321 47.24 1.051 1.321 75.50 570.3 0.186 0.0282 1.341 47.52 1.038 1.341 76.50 573.7 0.187 0.0285 1.363 47.81 1.025 1.363 77.50 577.2 0.188 0.0288 1.385 48.10 1.012 1.385 78.50 580.7 0.190 0.0291 1.407 48.39 0.999 1.407 79.50 584.3 0.191 0.0294 1.431 48.69 0.985 1.431 80.50 587.9 0.193 0.0297 1.455 48.99 0.972 1.455 81.50 591.5 0.194 0.0300 1.479 49.29 0.959 1.479 82.50 595.2 0.196 0.0303 1.505 49.60 0.945 1.505 83.50 598.9 0.197 0.0307 1.532 49.91 0.932 1.532 84.50 602.7 0.199 0.0310 1.559 50.22 0.919 1.559 85.50 606.5 0.201 0.0314 1.587 50.54 0.905 1.587 86.50 610.4 0.203 0.0318 1.617 50.87 0.891 1.617 87.50 614.4 0.205 0.0322 1.647 51.20 0.878 1.647 88.50 618.4 0.207 0.0326 1.678 51.53 0.864 1.678 89.50 622.5 0.210 0.0330 1.711 51.87 0.850 1.711 90.50 626.7 0.212 0.0334 1.745 52.22 0.837 1.745 91.50 630.9 0.215 0.0339 1.781 52.58 0.823 1.781 92.50 635.2 0.217 0.0343 1.817 52.94 0.809 1.817 93.50 639.7 0.220 0.0348 1.856 53.30 0.795 1.856 94.50 644.2 0.223 0.0353 1.896 53.68 0.781 1.896 95.50 648.8 0.226 0.0358 1.937 54.07 0.767 1.937 96.50 653.5 0.229 0.0364 1.981 54.46 0.753 1.981 97.50 658.3 0.233 0.0369 2.026 54.86 0.739 2.026 98.50 663.3 0.236 0.0375 2.073 55.27 0.725 2.073 99.50 668.3 0.240 0.0381 2.122 55.69 0.710 2.122

The results should be self explanatory; however, some observations are in order.

• A 1′ increment was used for the analysis.  This will be carried over to both the static and dynamic axial analyses.  For this routine it’s probably overkill, but for a real system with multiple soil layers this eliminates a great deal of interpolation and adjustment.
• Both the shear modulus and the maximum shear stress on the shaft surface vary with effective stress.  This tends to homogenise the quake to some degree.  The increase of shear modulus with depth also increases the shaft element stiffness as well.
• Beta values are about 50% higher at the pile toe than at the pile head.  This is mostly due to the depth effect of the $K$ value computed by the method used.
• The resulting quakes are lower than the “traditional values.”  This varies from run to run.
• The Smith-type damping constants are considerably higher than is usually expected.  This will be discussed with the wave equation analysis itself.
• There is no difference between ultimate capacity and SRD with this run because of the cohesionless soils.  This will change with cohesive ones.

### ALP Program

The original routine used the PX4C3 routine to construct the axial load-deflection curve.  For this routine it was replaced by the ALP program, which is described in Verruijt.  The Turbo Pascal code in the text was converted to php and modified for the online application.  The ALP99 program, which allows for layered soils, has been used in a classroom setting, is a good program but has three serious weaknesses:

1. There is no guidance on what values of quake to use for either shaft or toe, and for beginners this is very confusing.
2. The guidance on entering shaft resistance properties is primitive, to say the least.
3. The program simply crashes if a resistance in excess of the ultimate resistance is entered, even though the latter is easily computed.

This online version of ALP addresses all of these by limiting the highest resistance during the “load test” and furnishing quake and resistance values all along the shaft and toe.

The basic parameters of ALP returned by TAMWAVE are shown below.

 Length of the pile, in. 1,200.0 Axial stiffness EA. lbs. 720,000,000 Circumference, in. 48.000 Point resistance, lbs. 202,673 Quake of the point, in. 0.879 Number of pile elements 100 Number of loading steps 20 Maximum pile load, lbs. 572,676.9 Load Increment, lbs. 57,267.7 Failure Load, lbs. 572,676.9

Some of these are repetitious from earlier data output.  The results of the actual “load test” are shown below.

 Load Step Force at Pile Head, kips Pile Head Deflection, in. Number of Plastic Shaft Springs 0 0.0 0.000 0 1 57.3 0.033 22 2 114.5 0.082 39 3 171.8 0.144 52 4 229.1 0.216 64 5 286.3 0.300 74 6 343.6 0.395 85 7 400.9 0.601 100 8 343.6 0.571 10 9 286.3 0.534 22 10 229.1 0.489 31 11 171.8 0.437 39 12 114.5 0.378 45 13 57.3 0.314 52 14 0.0 0.244 58

The program ceases to load the pile and begins to unload when all of the shaft friction is mobilised or the ultimate load is achieved, whichever comes first.  This is intended to prevent the routine from going unstable with the applied load too near the maximum capacity of the pile, thus violating static equilibrium.

ALP solves the system by constructing a tridiagonal matrix and then solving the non-linear problem.  In some cases it will achieve a result before coming to actual convergence according to the convergence criterion.  In such cases ALP will report that no convergence was achieved.

One new feature with the current version of TAMWAVE is the inclusion of two basic graphs of the results.  This is one of them.  Contrary to American practice, the deflection (y) axis is upward even though the actual deflection is downward.  For serious plotting purposes it is probably best for the student to copy and paste the results into a spreadsheet or other plotting program and then make the results look more presentable.

### CLM 2 Routine for Lateral Loads

• The analyser is for single piles only, no group or bent analysis.
• The following cases can be considered:
• Free (Pinned) Head, Lateral Force Only
• Free Head, Combined Force and Moment
• Fixed Head, Lateral Force Only
• Any lateral load or pile head moment is entered when the soil properties are confirmed. If zero load or moment is entered, the results are expanded or truncated accordingly.

For this example the results of the CLM 2 analysis are here.

 Nominal Soil Unit Weight, lb/in3 0.06944 Pile Moment of Inertia, in4 1,728.00 Pile Section Modulus, in3 288.00 Pile Solid Circle Moment of Inertia, in4 1,017.88 Moment of Inertia Ratio Ri 1.698 Pile Moment of Inertia Ratio Product, ksi 8,488.3 Pile-Soil Interaction Variable 97,803 Pile L/D Ratio 100.0 Characteristic Load, lbs. 2,745,232.8 Characteristic Moment, in-lbs. 196,821,533.6 Pile Head Fixity Free Pile Head Lateral Load, lbs. 5,000.0 Pt/Pc 0.00182 Yt/D 0.00800 Pile Head Deflection due to Load, inches 0.096 Maximum Moment Due to Pile Head Lateral Load, in-lbs 136,112.3 Maximum Bending Stress Due to Pile Head Lateral Load, in-lbs 472.6

The results are explained in the CLM 2 documentation.  The bending stresses are not really meaningful in concrete piles, as flexure is generally transmitted through the reinforcement.  Parametric studies must be run manually, i.e., one load at a time.

CLM 2 is a quick way to obtain estimates of lateral loads, shears and moments for groundline piles and simple soil profiles, and both of these are present in TAMWAVE.  Since all of the soil input is already done, this source of error is eliminated.

Once these results are complete, the user can proceed to run a wave equation analysis.

Posted in TAMWAVE

## TAMWAVE 3: Basic Results of Pile Capacity Analysis

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

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

### Pile Data

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

### Soil Data

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

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

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

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

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

$\beta = K tan \phi$

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

$\beta = K tan \delta$

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

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

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

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

### Pile Toe Results

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

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

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

So what value of $N_q$ should we adopt?  As is all too common in geotechnical engineering, there has been a proliferation of values for this parameter.  We experimented with several, including that of Vesic.  Taking into account both theoretical methods and empirical ones such as Dennis and Olson, for TAMWAVE the “basic” formula (from Verruijt) was chosen:

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

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

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

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

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

and so

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

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

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

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

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

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

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

$G=\frac{ 2}{ 3}\frac{\left( 3000 - 1000 e \right)^2}{ 1 + e} \sqrt{\frac{ I_1}{ 3 p_{atm}}}$

The same formula is used for the shaft friction, the main difference is that the $\frac {I_1}{3}$ is different because the lateral earth pressure coefficient/Poisson’s Ratio is different, thus the lateral/confining stresses are different.

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

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

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

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

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

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

### Final Results

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

### References

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

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

## TAMWAVE 2: Modifying the Soil Properties

With the first step out of the way, we can proceed to the second: allowing the user to modify the properties of the soil.  This option must be used with care since it is easily possible to put together a set of soil properties that is physically unrealistic if not impossible.

Also, if you have chosen a sand or clay, you have chosen the methodology you will use.  Adding cohesion to a sand or gravel, for example, will have no effect on the subsequent performance of the model.

Finally, depending upon the choice of a free or fixed head, you are given the option of entering lateral loads and/or moments for the pile head.  In this case we have opted to add a lateral load of 10 kips to the pile and no moment.  The default is zero for both load and moment; this will produce some coefficients but no result for lateral loading.

Once the properties are as required, you can accept the form and proceed to the next step.

Posted in TAMWAVE

## TAMWAVE 1: Entering Basic Soil and Pile Properties

With a few preliminaries out of the way, we can proceed to discuss the new TAMWAVE routine, which can be found here.

### What is TAMWAVE?

TAMWAVE stands for Texas A&M Wave Equation.  The TTI wave equation was developed at Texas A&M in the late 1960’s and early 1970’s, and was a successor to Smith’s original wave equation program.  In reality this is more than a wave equation program; it is a driven pile analyser which, in addition to the wave equation program, analyses the static performance of a driven pile for both axial and lateral loads.  It is not intended to be used on actual projects, but as an educational tool for students.  Most of the software in current use is expensive, and predecessors such as SPILE, WEAP87 or COM624 are hard to use (they’re DOS programs) or methodological obsolescence issues.  (With WEAP87, there are not as many of those as you might think, but that’s another post…)

### Limitations of TAMWAVE

Given that this is an educational tool, there are some significant limitations to TAMWAVE’s capabilities.  Some of these are as follows:

• Only one type and consistency/density of soil is permitted.  However, the phreatic surface can be anywhere between the head and toe of the pile.  (If at the toe, the soil is assumed to be dry for the entire length of the pile, an unlikely scenario.)
• Piles have uniform cross-section and material for the entire length.  Starting in 2010, piles must be picked from a database presented at the start of the routine.  This limits the types of piles available, but makes input a lot simpler.
• Hammers are likewise limited to air/steam hammers, currently Vulcan and Raymond models.  (We may add Conmaco ones, later.)  This excludes diesel and hydraulic impact hammers, which simplifies the code considerably.
• Both axial and lateral loads are analysed by assuming the soil is either completely cohesionless or cohesive.  Unfortunately this is also a limitation of most current driven pile analysis methods as well.  Generally speaking, soils are entirely neither, but they’re close enough for current methods.  We’ll explore how to deal with this in the STADYN project; for now, we’ll stick with the binary methodology.

### Test Case, and Entering the Data

For this series of explanations we’ll use a test case as follows:

• 12″ concrete pile
• 100′ long (all piles are groundline in TAMWAVE, so there is no “stick-up” permitted
• Water table 50′ below surface
• Soil cohesionless or cohesive; in both cases a “medium” soil

Starting with the initial page, the data form looks like this:

The pile data input is pretty straightforward; the option for closed or open-ended pile toe is only relevant for hollow piles (pipe or concrete cylinder piles.)  The soil type is entered using the two-letter Unified system code.  This is to accomplish two things:

1. To make it simple to match the soil description from boring logs, if the problem is stated using one; and
2. In the case of cohesionless soils, to vary the internal friction angle with the soil type.

Density or consistency choice depends upon whether the soil is cohesionless or cohesive; the following charts (from here) were used for the data:

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At this point we need to pause and consider something we discussed in our last post: the use of CPT data.

When we attempt to establish soil response to load our ultimate goal is to determine the engineering properties of the soil.  Assuming we’re still in a Mohr-Coulomb universe, and setting aside the issue of consolidation settlement, that means three properties: the internal friction angle $\phi$, the cohesion $c$, and the unit weight (dry, moist or saturated) $\gamma$.  Ideally we can establish these properties using undisturbed samples in the laboratory.  The tricky part comes in obtaining these samples: not only is it expensive, but getting a truly “undisturbed” sample out of the soil is next to impossible, although we can come close.  This is why, from the earliest years of geotechnical engineering as a science, we’ve resorted to either tests of disturbed samples (the Atterberg limits are the most prominent of these) or in situ tests such as SPT, CPT or vane shear.  In the United States the SPT test has pretty much reigned supreme and is still the most commonly used test, in spite of its manifest limitations and inconsistencies, and appears on many soil boring logs.

Unfortunately most of the “academic” work in pile capacity has centred on the use of CPT results.  It was the original idea to essentially convert the SPT results from the charts above to “typical” CPT results and then use these with more contemporary techniques.  Unfortunately, in the development of TAMWAVE, it became clear that the results from doing this–especially with the toe response–had problems.  Based on this, we decided to drop the use of the artificially generated CPT data and use methods which could be derived from other properties.  The reason for this is twofold: the buildup of pore water pressures around the cone tip during insertion made the relationship between $q_c$ and $q_t$ problematic, and this, combined with basic differences in the SPT and CPT methodologies, makes correlating the two not a straightforward proposition.  These are discussed in  and the geotechnical practitioner would do well to keep this in mind when dealing with the results of either test.

One good thing that resulted from this decision is that we did not have much recourse to the SPT “data” either.  We were able to use the “Mohr-Coulomb” triple directly for most of our calculations.

The last piece of data is for lateral piles only: it is whether the head of the pile is “fixed” or “free” for lateral load analysis.  The loads themselves will be entered in the next step, which is accomplished by completing the for and pressing the “Submit Pile Data” button.