vulcanhammer.net

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:

• TAMWAVE: Pile Toe Resistance, and Some More on Pile Shaft Resistance
• TAMWAVE 1: Entering Basic Soil and Pile Properties
• TAMWAVE 2: Modifying the Soil Properties
• TAMWAVE 3: Basic Results of Pile Capacity Analysis
• TAMWAVE 4: Shaft Resistance Profile, ALP and CLM2
• TAMWAVE 5: Wave Equation Analysis, Overview and Initial Entry
• TAMWAVE 6: Results of Wave Equation Analysis
• TAMWAVE 7: Analysis for a Cohesive Soil

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

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.

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.

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.

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.

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.

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.

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

Shaft Resistance Profile

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 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.

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

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

To analyse lateral pile loading, the CLM2 Method is employed. Details on this method can be found with the CLM 2 spreadsheet here. Some notes about this are as follows:

• 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, Moment 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.

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.