Posted in STADYN

STADYN Wave Equation Program 12: A Case Study in the Modulus of Elasticity of Concrete Piles

It’s been a while since the last post on this subject; this has slowed things down.  But in the course of getting started again a little “side trip” shows a good illustration of how sometimes determining the engineering properties of a structural element–in this case a driven concrete pile–can be challenging.

The test case for this is the FHWA’s A Laboratory and Field Study of Composite Piles for Bridge Substructures.  All of the information in this piece comes from that report.  The report dates back to 2006 and the actual field work earlier in the decade.  One of the test cases involves a bridge replacement in Hampton, VA, as shown above.

The study involved the installation and testing of three different types of piles, as shown below.  We’ll concentrate on the prestressed concrete pile on the left.

Pile Profiles Figure 110.png
Pile cross-sections tested at the Route 351 Project

The prestressed concrete pile was a 610 mm square solid pile.  This means that the cross-sectional area is 0.3721\,m^2 .  The pile was 18 m long, as shown below.

Elevation view and instrumentation plan for concrete pile.

Stress-strain curves were developed for the three materials, and these are shown below.

Stress-strain curves for the pile materials.

From the stress-strain curve for the concrete alone (and we usually assume that the concrete governs the pile elastic properties for compression at least) the curve would indicate that the modulus of elasticity is somewhere around \frac{E}{\epsilon} = \frac {50\,MPa}{.002} = 25,000\,MPa .  The diagram below, however, indicates that those involved in the project determined the modulus of elasticity to be around \frac {EA}{A} = \frac {8.2 \times 10^3\,MN}{.3721\,m^2} = 22,037\,MPa .

Axial load-axial strain behavior of test piles.

The interest from the STADYN standpoint is to obtain a force-time and velocity-time curve from the Pile Driving Analyzer, and this is certainly forthcoming:

PDA Results Figure 130.png
PDA Results for Test Piles

The value of \frac{2L}{c} = 8.884\,ms was probably determined from the two force peaks.  The first force peak is the impact of the hammer on the pile and the second is the reflection of that impact from the toe.  Both are compressive and the second is strong, which indicates a high level of toe resistance.

As is typical with PDA output, the force and the velocity (multiplied by the impedance) are plotted together.  Unfortunately the document does not give the impedance for this case, so it’s necessary to back compute the impedance.  Since we have a reasonably good idea of \frac {2L}{c} from the PDA, and the impedance Z is

Z = \frac{EA}{c}

we can determine the impedance.  Solving for c from \frac {2L}{c} ,

c = \frac{2 \times 18}{8.884\,ms} = 4052 \frac{m}{sec}

We need to pause at this point and note that other values of acoustic speed are implied in the data.  For example, the following table states an acoustic or wave speed of 3800 m/sec.

Table 30.png
Acoustic speeds and other results of pile driving and dynamic testing.

Before and after the test, PIT (Pile Integrity Tests) were run on the pile.  The results are below.

Results of PIT tests.

Converted to SI units, the acoustic or wave speed becomes 4037 m/sec, which is fairly close to the PDA tests.  The PDA results will be used for the remainder of this piece.

In any case, using the EA values from the earliest part of the test, the impedance is

Z = \frac{EA}{c} = \frac {8.2 \times 10^6}{4052} = 2023 \frac{kN-sec}{m}

The data was extracted from the PDA results.  The force values could be used “as is.”  The velocities were in reality the product of the velocity and the impedance, so the dashed line values were divided by the impedance just obtained to yield a velocity.  Unfortunately, when this was put into STADYN, the velocities that resulted–even in the early stages of impact, where semi-infinite pile conditions predominate–the velocities of the program varied from the velocities extracted from the data by a factor of two.  Checks in the program did not show any change in the way the program executed the algorithm from earlier runs, but the impedance values the program was yielding were considerably different from the one above.

In an attempt to sort things out, it is good to start by noting that the acoustic speed is computed by the equation

c = \sqrt{\frac{E}{\rho}}

The report states that the pile was poured to normal Virginia DOT specifications.  A fair assumption is that the density or unit weight of the concrete is close to normal, or \rho = 2403\,\frac{kg}{m^3} .  That being the case, the computed acoustic speed from the values of Young’s modulus E (which is necessary to put into Pa for unit consistency) and the density assumed yields

c = \sqrt{\frac{E}{\rho}} = \sqrt{\frac{22.04 \times 10^9}{2403}} = 3,208\,\frac{m}{sec}

Something is clearly wrong here, and the most probable culprit is the modulus of elasticity of the concrete.  A common way to estimate the modulus of elasticity of concrete in MPa is to use the formula

E = 4700\sqrt{f'_c}

where {f'_c} is the 28-day compressive strength of the concrete.  The report gives this to us at the time of the load tests as 55 MPa, which yields a modulus of elasticity  of

E = 4700\sqrt{55\,MPa} = 34,856\,MPa

This is considerably higher than the earlier data would indicate.   It’s worth noting the the specifications for the pile set a minimum value for {f'_c} as 35 MPa; this indicates that the values of Young’s Modulus for concrete in piles can vary widely.

Another–and given the data probably a stronger–approach to compute the value of Young’s Modulus is to back compute it from the acoustic speed (which is known within reasonable values) and the density (see assumption above.)  Solving the basic equation for acoustic speed for Young’s Modulus yields

E = c^2 \rho

Substituting our values yields

E = 4052^2 \times 2403 = 39,454\,MPa

The impedance from this would be

Z = \frac{EA}{c} = \frac {39454\,MPa \times .3721\,m^2}{4052} = 3623 \frac{kN-sec}{m}

Applying values along this line and recomputing the velocities, the results of the STADYN program and the actual PDA results were much closer.


  1. The reason for the discrepancy in Young’s Modulus–and thus the pile impedance–is unclear.  It may be due to rate effects on the elastic response to concrete, or it may be due to other factors.
  2. Wave equation analysis are typically run according to “standard” material properties.  Those who run these should be aware that, with concrete and wood, those properties may not reflect the properties of what actually gets driven into the ground.
  3. Any force- and velocity-time data such as are produced by the PDA should have their axes labelled properly (with both force and velocity) or with the impedance reported.
  4. Even with controlled research projects, discrepancies can arise in the data which can impede (pun somewhat intentional) the use of the data, and careful analysis is necessary to avoid problems such as was seen in this situation.
Posted in STADYN

STADYN Wave Equation Program 11: Application of the STADYN Program to Analyze Piles Driven Into Sand

The newest update for the STADYN research project is available:

Application of the STADYN Program to Analyze Piles Driven Into Sand

The abstract is as follows:

Abstract: The STADYN program was developed for the analysis of driven piles both during installation and in axial loading. Up until now the test cases used were in predominantly cohesive soils. In this paper, the expansion of the program’s use into predominantly cohesionless stratigraphies has required consideration of two important factors. The first is the difference between strain-softening in clays as opposed to sands, and additionally static vs. dynamic strain effects. This requires a review of the whole concept of the “magical radius” for pile elasticity. The second is the effect of dilatancy on the response of the pile to axial loading. Both of these are discussed in this paper, and test cases are presented to illustrate the application of the program to actual driven piles.

It can also be found on Researchgate.

Posted in STADYN

STADYN Wave Equation Program 10: Effective Hyperbolic Strain-Softened Shear Modulus for Driven Piles in Clay

It’s been a while, but we hope it’s worth the wait: the monograph Effective Hyperbolic Strain-Softened Shear Modulus for Driven PIles in Clay is now available.  It was presented at the Research Dialogues for the University of Tennessee at Chattanooga 9-10 April 2019.  The abstract is as follows:

Abstract: Although it is widely understood that soils are non-linear materials, it is also common practice to treat them as elastic, elastic-plastic, or another combination of states that includes linear elasticity as part of their deformation. Assuming hyperbolic behavior, a common way of relating the two theories is the use of strain-softened hyperbolic shear moduli. Applying this concept, however, must be done with care, especially with geotechnical structures where large stress and strain gradients take place, as is the case with driven piles. In this paper a homogenized value for strain-softened shear moduli is investigated for both shaft and toe resistance in clays, and its performance in the STADYN static and dynamic analysis program documented. A preliminary value is proposed for this “average” value based upon the results of the program and other considerations.

The slide presentation for this follows:

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STADYN Wave Equation Program 9: Addition of Coefficient of Restitution to Cushion and Interface Properties

In our last STADYN post we discussed the addition of \alpha factors to take into account adhesion phenomena with cohesive soils.  In this post the addition of a more mundane but nevertheless important parameter for impact pile hammer systems is done: consideration of plastic losses in hammer and pile cushions, and interfaces as well.

Most impact pile hammers use some kind of hammer cushion; additionally, concrete piles are almost always driven with pile cushion at the pile head.  Cushions of both kinds are subject to significant plastic deformation and generation of heat.  There are several possibilities of modelling these elements in a simulation such as STADYN.

The first is to use velocity-dependent (viscous) damping to simulate the dissipation of energy.  STADYN in its current form has no velocity-dependent parameters; to add these would involve some major changes in the code, and in any case the testing of cushion material does not produce a result that would indicate such a property.

The second is to use an elastic-purely plastic approach similar to the one used in the soils.  The problem with this is that it would “flat-top” the impulse to the pile, and there is no evidence that the cushion material fails in this way.

The third is to use a “coefficient of restitution” approach, where the rebound of the cushion takes place at a different stiffness than the compression.  This is illustrated in two variants below.

Cushion Models for Plastic Cushions (from Warrington (1988))

The conventional model dates back to Smith, and is still used in GRLWEAP.  The ZWAVE model is described by Warrington (1988).  In both cases the energy lost in the cushion is represented by the shaded area.

For STADYN the conventional model was adopted.  Implementing this took a little more care in a finite element code than in finite-difference codes like WEAP and GRLWEAP but it was done.  To accomplish this, it was necessary to compute the force in the cushion incrementally, as with plasticity the response is now path-dependent.  When the cushion rebounded (i.e., the distance between the cushion faces increased from one step to the next) the rebounding stiffness is used.  In this way multiple rebounds can be modelled properly.

Since the inverse methods do not model the hammer, the Mondello and Killingsworth case is not considered here.  This leaves the other two cases, and these can be summarised very briefly.

The Finno (1989) case had a blow count increase from 15.8 to 17.0 blows/30 cm. For the SE Asia case, the blow count increased from 11.8 to 13.5 blows/30 cm.  Additionally for the latter case comparisons with the pile head force and ram velocity vs. time tracks were produced.

SE Asia Ram Force Comparison
Pile Head Force

The pile head force until peak was identical, and then decreased more rapidly afterwards. There was an additional “kick” at 2L/c not present in the previous run.

SE Asia Ram Velocity Comparison
Ram Point Velocity

The ram (point) velocity is the same until rebound, and then the ram is essentially stationary with the coefficient of restitution until 2L/c, after which the ram velocity in the two cases is very close.  The sawtooth effect is mostly due to the “ringing” of the ram, i.e., a stress wave going up and down the ram.

While it is evident that the method of energy transfer is different with the addition of the coefficient of restitution, the actual effect of plasticity on the blow count is not great.  This is probably due to two factors: most of the energy transfer takes place during compression of the hammer cushion, and both hammers are using micarta and aluminium, which has a relatively high coefficient of restitution (0.8).  Nevertheless cushion losses are greater in materials such as plywood, which is used with concrete piles.  It is to this type of pile that STADYN’s development now turns.


Posted in STADYN

STADYN Wave Equation Program 8: Modification of Adhesion Properties of Cohesive Soils

With the successful transition of the \xi-\eta soil property system, the time has come to consider how these soils interact with the pile shaft.  As was the case before, the work with the TAMWAVE project has proven helpful with this.

One of the things that makes STADYN more complex than either TAMWAVE or most other 1-D solutions is that soils are not considered as purely cohesive or cohesionless.  In most analysis of driven piles, soils are either on or the other, or at best alternately layered.  In reality the division between the two is not so clear-cut except for either clean sands on the one end or pure clays on the other.  STADYN’s soil system envisions soils as a continuum between one and the other; although this adds to the flexibility of the program (especially in the inverse mode) and its modelling of reality, it makes specifying soils a challenge.

As noted earlier, for soils between the purely cohesionless (\eta = -1 ) and cohesive (\eta = 1 ) interpolation is done so that soils have no cohesion in the former case, no friction in the latter, and are interpolatively mixed in between.  For example, for a middle case of \eta = 0 , the soil would have a reduced cohesion and friction for the same value of \xi and share these properties.  In this way any adjustments for adhesion of either type of soil would be made for each.

Cohesionless soils: there are two ways of looking at this problem.  We can assume a straight-up Coulomb friction failure between the pile and soil, or we can assume that the pile acts as a “direct shear” tester and thus forces the soil to fail at an apparent angle that is not the same as would be predicted by Mohr-Coulomb failure.  As with TAMWAVE, we have assumed the latter; this is explained in some detail here.  It is reasonable to assume that a continuum model such as is used by STADYN could predict such a failure; thus, no modification to the elements closest to the pile surface is done for cohesionless soils.

One thing that did change, however, was the way the lateral earth pressure on the pile was computed.  In an elastic-purely plastic system, lateral earth pressure varies in the elastic region, and with elastic theory that means with the variance of Poisson’s Ratio.  With a Mohr-Coulomb failure criterion, frictional cohesionless soils’ strength is mobilised by vertical effective stress acting laterally.  In recent code iterations Jaky’s Equation has been used to estimate Poisson’s Ratio; however, this has been changed to use the method given by Randolph, Dolwin and Beck (1994).  Once the lateral earth pressure coefficient is computed using this method, Poisson’s Ratio is determined.  At or below the pile toe Jaky’s Equation is used.

Cohesive soils: Mohr-Coulomb theory has no way of taking degradation of cohesion at an adhesion surface into account.  To do this the cohesion for the element(s) immediately adjacent to the pile is reduced by an \alpha factor as computed by the method of Kolk and van der Velde.  This is only done for the element immediately adjacent to the pile shaft surface.  This is the way STADYN does a pile-soil “interface.”  Doing it in this manner obviates the need for special interface elements between the pile and soil.

Implementing this is a little tricky, because the \alpha factor is dependent upon the effective stress.  It is necessary to thus generate the layers, compute the mid-point effective stresses in each, and then apply the factor to the cohesion of one set of elements only.

Results: Finno (1989) and Modello and Killingsworth (2014) Comparisons

The results of these two cases were most recently discussed here.  They can be discussed easily because the results varied little from the previous stage of the program.

For the first case, the Davisson capacity changed from 971 kN to 965 kN and the blow count from 17.6 to 15.8 blows/30 cm.

For the second (inverse) case, the Davisson capacity for the case of |\eta| < 3 the Davisson capacity changed from 269 kN to 274 kN and the blow count from 24.6 to 24.4 blows/30 cm.  The least squares difference actually increased from 0.00143 to 0.00149.

In both cases the soils were heavily cohesionless (at least that’s the way the pile looked at them) and the reduction in adhesion was minimal in impact.

Results: Notional Southeast Asia Case

Of all the test cases in the original study, the notional Southeast Asia case was the most problematic in the results, especially as they were compared to the GRLWEAP output.  The previous phase produced little difference in outcome; it was hoped that applying \alpha factors to the adhesion would at least solve the discrepancies of SRD estimates.  The results did not disappoint.

Since we have not presented too many results from this case, some graphical output is in order.  First, the force-time and velocity*impedance-time curves:

Force and Impedance*Velocity vs. Time Curves

The result above is a classic “offshore” pattern.  In the early part of impact (\frac{L}{c} < 1 ) both the actual pile head force and the product of the impedance are virtually identical.  This indicates an “infinite pile” condition; the theory behind this is discussed by Warrington (1997). Beyond this the two diverge; first the pile head moves upward in rebound from the pile shaft (indicated by the fact that the rebound takes place before \frac{2L}{c} ) and impacts the pile cap, producing a secondary force in the pile head.  Beyond \frac{2L}{c} the pile head force goes to zero and the velocity oscillates with the reflections from the pile; however, just after that time the compressive “kick” from the toe is evident.

Static Load Test Results

Now we have the result of the static load test.  As noted in the original study, static load tests are exceptional offshore, and for actual loading a tension test is probably of just as much interest (if not more) than a compressive one.  In the original study doubt was also expressed as to the relevance of Davisson’s criterion to offshore piles; the variation among different interpretation methods, however, were not that great.  In any case, the effect of reducing the adhesion of cohesive soils along the surface is evident: the Davisson ultimate load has dropped to 20,600 kN.  This is nearly identical to the Dennis and Olson (1983) method result, and below the API RP2A (2002) result.  This indicates that the application of the \alpha method to the soil elements along the pile shaft results in bringing the static results of STADYN more in line with those of static methods in use.

For the Dennis and Olson (1983) SRD, the GRLWEAP blow count varied from 18.4 blows/30 cm to 21.8 blows/30 cm, depending upon which value of damping was used (0.2 \frac{sec}{m} or 0.3 \frac{sec}{m} .  STADYN returned a blow count of 11.8 blows/30 cm.  This is a significant improvement.  There are two possibilities to explain the remaining difference:

  1. STADYN is modelling a lower effective damping value for the soils than is used in GRLWEAP.  As noted in the original study, STADYN has a different model for handing dissipative phenomena than GRLWEAP.
  2. The two programs have differing methods for arriving at the blow count.

Before we can make more definitive statements about this, we need to include cushion losses, which is our next step.  Nevertheless this result clears up a great deal of the difficulty with this case in the original study.