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

Posted on 24 July 201824 July 2018 by Don Warrington

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.

zwavefig1 — 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.

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

Posted on 23 July 201824 July 2018 by Don Warrington

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.

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:

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.
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.

Soil Mechanics

Relating Hyperbolic and Elastic-Plastic Soil Stress-Strain Models: A More Complete Treatment

Posted on 29 June 201829 June 2018 by Don Warrington

In an earlier post, we discussed this topic. This is meant as a follow-up to that post; in a sense we left the reader “hanging” because the solution, although informative, was incomplete. This should “tie some loose ends” and make the result, although it’s still theoretical, more useful. The concept for most of this is the same but the implementation more closely follows the physical reality of stress-strain.

Let us begin by considering a modified version of the original graphic which compares the hyperbolic and elastic-purely plastic stress-strain models.

We need to make a few definitions.

First, let’s begin by defining two strains. The first strain is the strain at failure (we’re assuming perfectly plastic failure here) if the small-strain elastic or shear modulus could be maintained to failure (i.e., if linear elasticity would hold until failure.) That strain is

$\epsilon_0=\frac{\sigma_u}{E_1}$

In this case we are making the dashed line a single failure stress $\sigma_u$ , the ordinate $\sigma$ and the strain $\epsilon$ . Although elastic modulus E is habitually used, this treatment could apply to shear modulus G as well.

The second is the failure strain at a reduced modulus assuming an elastic-purely plastic deformation characteristic, or

$\epsilon_1=\frac{\sigma_u}{E_2}$

If we use $\epsilon_0$ as a “reference” strain, we can make the problem dimensionless as follows:

$\hat \epsilon=\frac{\epsilon_1}{\epsilon_0}$

In any case the equation for the hyperbolic stress-strain curve for a given strain is

$\sigma=\frac{E_1 \epsilon_0^2}{\epsilon_0+\epsilon}$

Integrating the area above this curve to the failure stress and $\epsilon_1$ yields

$A_1 = \ln\left( \epsilon_0 + \epsilon_1 \right)E_1\epsilon_0^2-\ln(\epsilon0)E_1\epsilon_0^2$

Defining

$A = \frac{E_2}{E_1}$

the area above the elastic region of the elasto-plastic deformation line is

$A_2 = \frac {\epsilon_1^2AE_1}{2}$

We need to do the following:

Equate the areas.
Solve for the modulus ratio A.
Substitute the dimensionless strain ratio $\hat \epsilon$ .

Doing all of this yields

$A = 2\,{\frac {\ln (1+{\it \hat\epsilon})}{{{\it \hat\epsilon}}^{2}}}$

Plotting this yields the following:

Although the notation is different, this is basically the same result we got before. It also has the same problem: it “blows up” as the strain ratio approaches zero . For high-strain problems (which is our own chief field of interest) this is not a problem, but it still needs to be addressed. The basic problem is that the whole “area ratio” concept itself breaks down as the strains approach zero. At zero strain the moduli should be the same and the modulus ratio unity, but the area ratio does not represent this.

This can be seen if we look at a more experimentally-based treatment of the problem, which is summarised in this graph, taken from this publication:

Although it’s certainly possible to do the usual empirical correlation on a curve like this, the higher strain portion and our theoretical presentation resemble each other. The smaller strain region is the problem. In many ways this resembles the Euler column buckling problem familiar to structural engineers, where two regions are defined with two equations which meet at a point where both their slope and their value are the same.

But what equation to use for the small-strain region? Whatever equation we use needs to come to unity at zero strain and decrease from there. A simple function for this purpose is the cosine function, modified as follows:

$A = \cos(\beta \hat\epsilon)$

To find the meeting point, we need to find the point where both the values of A and the derivatives are the same. Without going into the algebra, for the second equation $\beta = .495$ and the meeting point is $\hat\epsilon = 1.947$ and $A = 0.571$ . This is plotted below.

Although a more rigourous analysis is necessary, the two plots look very similar. The biggest difference–and this is not insignificant–is that the empirical plot above is semi-logarithmic in nature, while the theoretical one is linear.

From all this, we can conclude the following:

The “area ratio” concept, while useful for larger strains, breaks down with smaller strains.
The quantities $\epsilon_1$ and $\hat\epsilon$ are very useful in generalising strains in soils, although the former is physically impossible.
“Stitching together” the two equations yields a theoretical construct that shows potential to representing reality in soil stress-strain relationships. The biggest difference, as noted, is the logarithmic vs. linear nature of the plots; this probably indicates an underlying principle that needs to be addressed.
The actual values of the ratio of small-strain shear or elastic modulus to elasto-plastic modulus is very application dependent. Since quantifying both elastic and shear modulus is more important in geotechnical engineering (primarily due to finite element analysis) than in the past, the need to establish values of this ratio for various applications is great.

STADYN

STADYN Wave Equation Program 7: Revision of Soil Properties (Results)

Posted on 25 June 201826 June 2018 by Don Warrington

In the last post we discussed the change in cohesion in the $\xi-\eta$ interpolation. Now we apply these to the program to see how they change the results.

In the original study there are three test cases, as noted here. For this stage the first and the third will be considered. The results for the second (after adjustments for changes in the $\xi-\eta$ interpolation) are little different from before; we will consider major changes to impact that case in our next round.

That leaves the first and third. For the first–comparison with a static load test–it was necessary to readjust the values for $\eta$ to reflect changes in the meaning of $\eta$ relative to soil properties, as discussed in the last post. The changes made resulted in layering with the following characteristics:

Layer	Bottom y-coordinate, m	$\xi$	$\eta$
1	5.18	-1	0
2	7.32	-1	0
3	15.2	0.5	0
4	30.5	0.5	0

The static load test results can be seen below.

Comparison with the original study show little improvement in the failure load correlation; however, the load-deflection relationship before failure is significantly improved. The system was much softer in the original study and the improvement reflects the better estimate of the elastic modulus. The blow count is very similar to the original study.

Turning to the inverse case that was also discussed in Warrington and Newman (2018), based on the previous results, it was decided to drop consideration of 1-norm results. The results are reproduced in outline below for the three cases run (refer to Warrington and Newman (2018) for details.)

$\eta$ Limiting	Difference Sum	Static Load, kN	Average Shaft $\xi$	Average Shaft $\eta$	Toe $\xi$	Toe $\eta$
+-1	0.0034	673	0.084395	-0.46825	-0.992	0.758
+-2	0.00327	449	-0.455	-1.3375	-0.57	-0.727
+-3	0.00143	269	-0.26775	-2.1775	0.113	1.12

Comments:

The difference sum for the highest $\eta$ variation was the best match we have obtained to date for this case; the velocity match is shown below. The situation around L/c = 1.5 is still difficult but the rest of the correlation is improved.
The static load decreases with broader $\eta$ variation and a lower difference sum. This is different than Warrington and Newman (2018), where the static load “settled down” to a close range of values.
The plot above does not reflect that, on the whole, the variation of $\eta$ along the shaft was less than experienced in the past, especially with the +-3 run. The stratigraphy of the site suggested relatively uniform soil properties along the shaft, and this is beginning to be seen in the inverse results.
The +- 3 run was very heavily “toe weighted” in resistance.

While both of these cases show progress, the time has come to consider the whole issue of pile-soil interface issues, which will be considered in our next updates.

STADYN

STADYN Wave Equation Program 6: Revision of Soil Properties (Cohesion)

Posted on 24 June 201824 June 2018 by Don Warrington

In our last post we discussed the overhaul of STADYN’s $\xi-\eta$ system relative to the modulus of elasticity, which additionally involved revising the way the program estimated dry unit weight and void ratio. The last is necessary because the modulus of elasticity is estimated using the Hardin and Black formulation. In this post we will discuss revision of another parameter, namely soil cohesion.

We based the relationship of $\rho$ to $\eta$ based on work for the TAMWAVE program. It would doubtless be useful to state the relationship between $\eta$ and the consistency/density of the soil, and this is as follows:

$\eta$	Cohesive Designation	Cohesionless Designation
-1	Very Soft	Very Loose
-0.6	Soft	Loose
-0.2	Medium	Medium
0.2	Stiff	Dense
0.6	Very Stiff	Dense
1	Hard	Very Dense

Doing it this way enabled us to have a linear relationship between $\rho$ and $\eta$ . It is too much to expect for the linear relationship to extend to other variables, and this is certainly the case with cohesion. Unfortunately, a conventional $\xi-\eta$ interpolation dictates such a relationship. The original $\xi-\eta$ function for cohesion can be seen below, for values of cohesion in kPa.

Note that the relationship between cohesion and $\eta$ is linear for the purely cohesionless state at $\xi = 1$ . If extended past the bounds of the graph for lower values of $\eta$ , the cohesion becomes negative. STADYN prevents this from happening but this essentially deprives soft soils of any cohesion.

Baseon on the TAMWAVE values, for purely cohesive soils the following approximate relationship can be established for cohesion:

$\frac{c}{p_{atm}} = 0.5e^{1.5\eta},\,\xi=1$

where $c$ is the soil cohesion and $p_{atm}$ is the atmospheric pressure. The left hand side of the equation is the “normalised” cohesion using the atmospheric pressure. Doing this for parameters such as effective stress makes for an interesting look at soil properties. The best known use of this is in the SPT correction for overburden.

For cases where $\xi < 1$ , the value can be reduced linearly so that $c = 0$ when $\xi = -1$ . The result of all this can be seen in the graph below.