## Is the Principles and Practice of Engineering Exam a Barrier Against Women?

The intrepid Toni Airaksinen at Campus Reform has written an article highlighting the research of Drs. Julia Keen & Anna Salvatorelli on this subject.  The statistics are interesting and so are their recommendations for further research:

This study focused on pass rate, and the resultant disparity is only the first step. Additional research should be conducted to identify why women are not passing the PE exam at an equal percentage rate as men. This research should include:

• Identifying biases in the exam itself

• Examining the timing of administration of the exam in an engineer’s career progression

• Exploring the likelihood of women to retake the exam compared to men after failing since the number of attempts was not recorded within the data collected

• Identify factors that may contribute to higher pass rate for women in some states compared to others.

As someone who has taught civil engineering for more than a decade at the undergraduate level, this has more than a passing interest.  For me, it was also an interesting moment, because I saw this just after I had returned from the dedication of the new headquarters for Division 2 of the Tennessee Department of Transportation, where most of my students who work there are female.

Let me first set forth their “bottom line” cumulative statistics (I strongly urge those of you who can get access to their paper to do so):

1. About 20% of the people who take the “Principles and Practices” exam are women.  That tracks pretty well with the number of women in my classes.
2. 51.5% of the women pass the test on the first try, while 63.1% of the men do.

With that out of the way, I’d like to make some observations.

1. My female students tend to be a very diligent and competent group.  In many ways an engineering curriculum is more of an endurance match than anything else; the women “tough it out” at least as well as the men.
2. I’ve never noticed women having more difficulty with tests than men in my classes.  That’s saying a lot because my tests tend to be bizarre, as my students will attest.
3. Women in civil engineering have some built-in advantages because of the diffuse structure of the system by which structures get built and their socialization skills, as I explain this 2014 post.  Because of the nature of our society, engineers tend to get stuck in the caboose on the train of respectability; I think that women are a significant part of the key to change that situation.

Especially considering #2, I find it hard to believe that the test is intrinsically biased against women.  So why is this disparity so?  Our researchers give us four options, and my gut tells me that the second one is the most likely.

My reasoning is simple.  Generally speaking, most engineering students take their first exam (the FE exam) while they’re in undergraduate school.  After they they acquire four years of experience, they can apply for the privilege of taking the P&P exam.  If they pass it and meet other requirements, they obtain their Professional Engineers license.  For most people, that means that the critical moment takes place in their mid- to late twenties.  Millennials aren’t as “progressive” on sorting out tasks between spouses or partners as some might have you believe.  That time in life is also the same time when many marry, have children, etc., and the work associated with those events falls harder on women.  Thus the first opportunity to take the exam takes place at a point in life which is less opportune for women than it is for men.

So what is to be done?  Do we need a special accommodation?  The answer is “no.”  Since venting pet peeves seems to be the thing on this site these days, let me vent one of mine: there is no cogent reason why we should force people to wait several years out from their academic studies to take the P&P exam.  This exam is supposed to reflect experience, but a reality check is in order: it’s just another academic exercise like just most any other test.  Fortunately change is in the wind, as this statement from the National Society of Professional Engineers indicates:

Until relatively recently, candidates for licensure as a professional engineer have needed to gain four years of approved work experience before taking the Principles and Practice of Engineering (PE) Exam. In recent years, however, attitudes within the profession toward the early taking of the PE exam have begun to shift. In 2013, the National Council of Examiners for Engineering and Surveying (NCEES) removed from its Model Law the requirement that candidates earn four years of experience before taking the exam. Separating the experience requirement from eligibility for taking the PE exam is sometimes called decoupling. For the National Society of Professional Engineers, as stated in Position Statement No. 1778,

“Licensing boards and governing jurisdictions are encouraged to provide the option of taking the Principles and Practice of Engineering exam as soon as an applicant for licensure believes they are prepared to take the exam. The applicant would not be eligible for licensure until meeting all requirements for licensure— 4-year Accreditation Board for Engineering and Technology/Engineering Accreditation Commission accredited degree, passing the Fundamentals of Engineering exam and the Principles and Practice of Engineering exam, and 4 years of progressive engineering experience.”

The NSPE would have us think that this concept is a novelty, but that’s not really the case.  When I was an undergraduate at Texas A&M University in the 1970’s, Texas allowed people to take both exams before graduation; our own NSPE student chapter strongly encouraged that, and I did it myself.  Taking the P&P exam not only gets the exam away from major life events in early adulthood, it also eliminates a good deal of remedial work trying to remember things one learned in school but had forgotten in the years before the exam.

I think that, if we do not obscure our thinking with trendy concepts and look at things realistically, we can solve this disparity by making a change that will benefit both men and women and improve our profession.  If this disparity provides motivation to move the process of “decoupling” forward, then so be it.  It’s a change that’s overdue.

Posted in Soil Mechanics

## Jean-Louis Briaud’s “Pet Peeve” on the Analysis of Consolidation Settlement Results

In his recent, excellent article on the settlement (and subsidence) of the San Jacinto Monument east of Houston, Briaud (2018) takes an opportunity to vent a “pet peeve” of his relative to the way consolidation tests are reduced and consolidation properties reported:

### A Chance to Share a Pet Peeve

The consolidation e versus log p’ curve is a stress-strain curve.  Typically, stress-strain curves are plotted as stress on the vertical axis and strain on the horizontal axis.  Both axes are on normal scales, not log scales.  It’s my view that consolidation curves should be plotted in a similar fashion: effective vertical stresses on the vertical axis in arithmetic scale, and normal strain on the horizontal axis in arithmetic scale.  When doing so, the steel ring confining the test specimen influences the the measurements and skews the stiffness data.  Indeed the stress-strain curve, which usually has a downward curvature, has an upward curvature in such a plot. (p. 54)

Is this correct?  And is he the only one who thinks this way?  The two questions are neither the same nor linked.  Although this problem will certainly not be solved in one blog post, it deserves some investigation.

## Statement of the Problem

Let’s start with a text we use often here: Verruijt, A., and van Bars, S. (2007). Soil Mechanics. VSSD, Delft, the Netherlands. Early in the presentation on the subject, he presents the following plot:

As Jean-Louis would have us do, the strain (or negative strain, since we’re dealing with compression) is on the abscissa, and the dimensionless stress is on the ordinate.  The difference between the two is that the stress is plotted logarithmically.  But it’s a step.  We’ll come back to that later.

Verruijt defines the relationship between the strain and stress ratio as follows:

$\epsilon = -\frac{1}{C}\ln\frac{\sigma}{\sigma_0}$

This relationship goes back to Terzaghi’s original tests and formulation of settlement and consolidation theory almost a century ago.

From a “conventional” standpoint there are two things wrong with this formulation.  The first is that it is based on strain, not void ratio.  The second is that it uses the natural logarithm rather than the common one.  The last problem can be fixed by rewriting it as follows:

$\epsilon = -\frac{1}{C_{10}}\log\frac{\sigma}{\sigma_0}$

This formulation is essentially the same as is used in Hough’s Method for cohesionless soils, once the strains are converted to displacements by considering the thickness of the layer.  So it is not as strange as it looks.

The first problem can be “fixed” by noting the following:

$\epsilon = \frac{e-e_0}{1+e_0}$

We can substitute this into the equation before it and, with judicious changes of the constants and other subsitutions, come up with the familiar, non-preconsolidated formula for consolidation settlement, or

$\Delta H = \frac{C_c H_0}{1+e_0}\log\frac{\sigma}{\sigma_0}$

When we reverse the axes, we then get the “classic” plot as follows:

But is there a problem with using strain?  Verruijt explains the two conventions as follows:

In many countries, such as the Scandinavian countries and the USA, the results of a confined compression test are often described in a slightly different form, using the void ratio e to express the deformation, rather than the strain ε…It is of course unfortunate that different coefficients are being used to describe the same phenomenon. This can only be explained by the historical developments in different parts of the world. It is especially inconvenient that in both formulas the constant is denoted by the character C, but in one form it appears in the numerator, and in the other one in the denominator. A large value for $C_{10}$ corresponds to a small value for $C_c$. It can be expected that the compression index $C_c$ will prevail in the future, as this has been standardized by ISO, the International Organization.

As is often the case, the simplest way to help sort out this issue is with an example.  Briaud (2018) actually has one, but we will use another.

## Example of Settlement Plotting

An example we have used frequently in our teaching of Soil Mechanics is this one, from the Bearing Capacity and Settlement publication.  It is a little more complex than the theory shown above because it involves a preconsolidated soil.  The plot (with the simplifications for determination of $C_c$ and $C_r$ is shown below.

With this information in hand, we process the data as follows:

1. We convert the void ratio data to strains using the formula above.
2. We convert the stresses to dimensionless stresses by dividing them by the initial stress.
3. We “split” the data up into compression and decompression portions to allow us to develop separate trend lines for both.

First, the strain-dimensionless stress plot, using natural scales for both.

The result is similar to that in Briaud (2018).  The compression portion best fits a second-order polynomial fit.  (Not that we have thrown out the zero point to allow more fit options.)  The decompression portion fits an exponential trend line best.

Below is the same plot with the stress scale now being logarithmic.

This is basically the original graph with the axes reversed.  There is no effect using strain; we will discuss the advantages of doing so below.

Now let us look at the data from another angle: the tangent “modulus of elasticity,” defined of course by

$E = \frac{\Delta\sigma}{\Delta\epsilon}$

We consider natural scales for both modulus and strain.  To obtain the slope, we used a “central difference” technique except at the ends.

It’s interesting to note that, except for the “kink” caused by preconsolidation, in compression the tangent modulus of elasticity increases somewhat linearly with strain, as it does with the decompression.

## Discussion of the Results

There’s a great deal to consider here, and we’ll try to break it down as best as possible.

### Use of Strain vs. Void Ratio

The graphs above show that there is no penalty in using strain instead of void ratio to plot the results.  The advantage to doing so is both conceptual and pedagogical.

In the compression and settlement of soils, we traditionally conceive of it as a three-stage process: elastic settlement, primary consolidation settlement, and secondary consolidation settlement.  Consolidation settlement is nothing more than the rearrangement of particles under load; the time it takes to do so is based in part on the permeability of the soil and its ability to expel pore water trapped in shrinking voids.  Elastic settlement is due to the elastic modulus of the material, the strain induced in the material and the geometry of the system.  This distinction, however, obscures the fact that we are dealing with one soil system and one settlement.  Using strain for all types of settlement would both help unify the problem conceptually and ease the transition to numerical methods such as finite element analysis, where strain is used to estimate deflection.  In the past we were able to use a disparate approach without difficulty, but that option is not as viable now as before.

### The Natural Scale, Consolidation Settlement Stiffness, and the Ring

Both here and in Briaud (2018) the natural stress-strain curve experiences an upward curvature, which is obviously different from what we normally experience in theory of elasticity/plasticity.  This comes into better focus if we consider the variation of the tangent modulus of elasticity, which (except for the aforementioned preconsolidation effect) linearly increases with stress.  There are two possible explanations for this.

The first is to observe that, as soils compress in consolidation settlement, their particles come closer together, and thus more resistant to further packing.

The second, as suggested by Briaud (2018), is that the presence of the confining ring in the consolidation test augments the resistance of the particles to further compression.  The issue of confinement is an interesting one because in other tests (unconfined compression tests, triaxial tests) confinement is either very flexible or non-existent.  It should be observed that consolidation theory, as originally presented, is one-dimensional consolidation theory.  For true one-dimensional consolidation, we assume a semi-infinite case where the infinite boundary “confines” the physical phenomena.  The use of a confining ring assumes that the ring can replicate this type of confinement in the laboratory.  Conditions in the field, with finite loads and variations in the surrounding soils, may not reflect this.  While it would be difficult to replicate variations in confinement in the laboratory, these variations should be kept in mind by anyone using laboratory-generated consolidation data.

### The “Modulus of Elasticity” for Consolidation Settlement

This may strike many geotechnical engineers (especially those in areas where void ratio is used to estimate consolidation settlement) as an odd concept, but if we consider the material strain vs. its deflection, it is a natural one.  Varying moduli of elasticity are nothing new in geotechnical engineering; they have been discussed on this site in detail.  The situation here is somewhat different for a wide variety of reasons, not the least of which is that here we are dealing with a tangent modulus while previously we looked at a secant one.  Also, differing physical phenomena are at work; theory of elasticity implicitly assumes that particle rearrangement is at a minimum, while consolidation settlement (both primary and secondary) is all about particle rearrangement.

A more unified approach to settlement would probably reveal a process where the change in stress vs. the change in strain varies at differing points in the process along a stress path with multiple irreversibilities.  Such an approach would require some significant conceptual changes in the way we look at settlement, but would hopefully result in more accurate results.

## Conclusion

Consolidation settlement is a topic that has occupied geotechnical engineering for most of its modern history.  While the theory is considered well established, changes in computational methodology will eventually force changes in the way the theory is applied.  A good start of this process is to use strain (rather than void ratio) as the measure of the relative deflection of structures, and the example from Briaud (2018), along with the demonstration relative to natural scales, is an excellent start.

References

Briaud, J.-L. (2018) “The San Jacinto Monument.”  Geostrata, July/August.  Issue 4, Vol. 22, pp. 50-55.

## 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.

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.

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.

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

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:

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:

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.

Posted in Soil Mechanics

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

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:

1. Equate the areas.
2. Solve for the modulus ratio A.
3. 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:

1. The “area ratio” concept, while useful for larger strains, breaks down with smaller strains.
2. The quantities $\epsilon_1$ and $\hat\epsilon$ are very useful in generalising strains in soils, although the former is physically impossible.
3. “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.
4. 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.