Posted in Academic Issues

The University of Tennessee at Chattanooga College of Engineering and Computer Science is Looking For a New Dean

Just a quick note to everyone that the institution I’ve taught at since 2001 is looking for a new Dean. If you’re interested, some links here:

There is one requirement that merits some comment:

A demonstrated record of intentional and successful actions to foster and enhance inclusion, equity, and diversity…

And this:

UTC takes its commitment to inclusion, equity, and diversity seriously.  Letters of interest and other application materials should specifically address the candidate’s intentional and systematic initiatives and accomplishments related to that commitment.

The College of Engineering and Computer Science (including the current Dean) has the most ethnically diverse faculty on campus. This is something the administration does little to publicize.

You can link to Parker’s website to submit an application. If you would like to ask me questions about this, you can go to the Contact page to reach me.

Posted in Academic Issues, Geotechnical Engineering

The Equivalent Thickness Method for Estimating Elastic Settlements

In our very popular post Analytical Boussinesq Solutions for Strip, Square and Rectangular Loads we discuss the use of the theory of elasticity (as originally formulated by Boussinesq) to estimate the stresses and settlements under foundations. We start by giving methods of estimating the stresses under various configurations of rectangular foundations (the circular ones are discussed in the post Going Around in Circles for Rigid and Flexible Foundations.) We then show the use of superposition to expand the use of these results for complex foundations (with further discussion in our post Superposition, and Using Point Loads in Place of Distributed Ones.) We then show the estimation of deflections for simple rigid and flexible foundations. But when it comes to deflections for more complex situations…crickets.

This post is an attempt to solve the “crickets” problem through the use of a method shown in Tsytovich (1976). It’s doubtless useful for preliminary calculations and to enhance our understanding of how settlements of foundations in one place can affect adjacent structures. It also uses some of the linkage between elastic and consolidation settlement theory which is discussed in From Elasticity to Consolidation Settlement: Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve”.

Let us start with Equation (4) of the last linked post, namely

\epsilon_{{x}}={\frac {p\beta}{E}} (1)

where we swap p for \sigma_x as the vertical pressure.

Now let us define an equivalent height heq. Keep in mind that we are assuming that the soil’s reaction to vertical pressure is that of a laterally confined specimen; the equivalent height is the height of that equivalent specimen. Multiplying both sides of Equation (1) by this equivalent height,

h_{{{\it eq}}}\epsilon_{{x}}={\frac {h_{{{\it eq}}}p\beta}{E}} (2)

Since by definition

\epsilon_x = {\frac {s}{h_{{{\it eq}}}}} (3)

where s is the settlement, Equation (1) becomes

s={\frac {h_{{{\it eq}}}p\beta}{E}} (4)

Equation (7) of the last linked post tells us that

E={\frac {\beta}{m_{{v}}}} (5)


\beta = 1-2\,{\frac {{\nu}^{2}}{1-\nu}} (7)

Combining Equations (4) and (5) yields

s=h_{{{\it eq}}}pm_{{v}} (8)

To compute the equivalent height, we turn to our post Analytical Boussinesq Solutions for Strip, Square and Rectangular Loads, where we modify the equation presented for the deflection of rectangles and squares with Equation (5) above to obtain

s={\frac {\omega\,pb\left (1-{\nu}^{2}\right )m_{{v}}}{\beta}} (9)

The value of \omega is discussed in that post for squares and rectangles and for circles in Going Around in Circles for Rigid and Flexible Foundations. These are very complex (and in the case of circles have no closed form solution.) For convenience the table for these is reproduced below.

If we equate the right hand sides of Equations (8) and (9) and solve for heq, we have at last

h_{eq} = {\frac {\omega\,b\left (1-{\nu}^{2}\right )}{\beta}} (10a)

We can also substitute Equation (7) into Equation (10a) and obtain

h_{eq} = {\frac {\left (-1+\nu\right )^{2}b\omega}{1-2\,\nu}} (10b)

If we define

A = {\frac {\left (1-\nu\right )^{2}}{1-2\,\nu}} (10*)

we can also write the equation thus

h_{eq} = A\omega b (10c)

It should be evident that there are several computational routes to obtain the equivalent height, which is then substituted into Equation (8) to obtain the settlement. Let us consider these options:

  • We could tabulate values of A \omega for various foundation configurations and then use these to compute the equivalent height using Equation (10c). This is given in Tsytovich (1976).
  • We could determine values for A (it is simply a function of Poisson’s Ratio \nu ), obtain \omega using the table above and then compute the equivalent height using the width of the foundation b . Values for both A and \beta are shown in graphical form as a check for computations.
  • We could perform direct substitution into Equations (10a) or (10b.) Equation (10a) is probably the best as it will be necessary to compute \beta using Equation (7).
Parameters for Tsytovich Equivalent Thickness Method as a function of Poisson’s Ratio. The red line is the parameter “A” and the green line is the parameter “β”.

Worked Example

As an example, let us consider the same example from the post Analytical Boussinesq Solutions for Strip, Square and Rectangular Loads. The foundation diagram is shown below; we are interested in the settlement at Point A.

The complete solution is given in the same spreadsheet as the solution for the stress problem, which you can access here. The geometry is the same and the loading is the same: 5 kPa on the yellow (2 m x 10 m foundation) and 15 kPa on the brown foundation (2 m x 6 m.) Keep in mind that, for this method, the value b is always the smaller of the two, and that goes for the “void” foundations as well.

We do not need the depth from the surface; we are only interested in surface deflections. We do need the elastic modulus and Poisson’s Ratio of the soil, which are E = 10,000 kPa and ν = 0.25.

The superposition is exactly the same as before, using the following diagram as before:

The superposition scheme is as follows:

  • Yellow Foundation Positive, corners ABFG, pressure +5 kPa
  • Yellow Foundation Negative, corners ABJH, pressure – 5kPa
  • Brown Foundation Positive, corners ACEG, pressure +15 kPa
  • Brown Foundation Negative, corners ABFG, pressure – 15 kPa

We will only go through the calculations for the first one; you can view the spreadsheet for the rest. We proceed as follows:

  • We compute A and β as follows:
    • From Equation (10*), A = (1-0.25)2/(1-(2)(0.25)) = 1.125
    • From Equation (7), β = 1-(2)(0.25)2/(1-0.25) = 0.833
    • You can verify these using the plot of these parameters.
  • We compute the coefficient of volume compressibility by using Equation (5), mv = 0.833/10000 = 0.0000833 1/kPa
  • We compute the value of α = l/b = 10/6 = 1.6667
  • We compute the corner value for ω (since we are dealing with corners as was the case with stresses.) We can use the table for ω or we can compute it using the formulae from Analytical Boussinesq Solutions for Strip, Square and Rectangular Loads, but it is ω = 0.71.
  • We then use Equation (10c) to compute the equivalent height, thus heq = (1.125)(0.71)(6) = 4.8 m. This value will be different for each corner point considered.
  • Using Equation (8), the settlement for this portion of the analysis is s = (4.8)(5)(0.000083333) = 0.002 m = 1.998 mm.

You repeat this process for all four “foundations.” Keep in mind that the negative foundations will result in negative settlements. A summary of the results is as follows:

Yellow Foundation
RectangleB, mL, malphaomega (corner)heq, mPressure p, kPaDeflection, mm
ABFG (+)6101.66670.714.805.001.998
ABJH (-)4102.50000.833.76-5.00-1.565
Brown Foundation
RectangleB, mL, malphaomega (corner)heq, mPressure p, kPaDeflection, mm
ACEG (+)6122.00000.775.1715.006.462
ABFG (-)6101.66670.714.80-15.00-5.994
Complete Total0.901
Posted in Deep Foundations, Pile Driving Equipment

About that “Warrington Method” For Vibratory Pile Drivability —

Every now and then something comes up that you really didn’t expect. That took place with a paper published this year cited “W.J. Lu, B. Li, J.F. Hou, X.W. Xu, H.F. Zou, L.M. Zhang, “Drivability of large diameter steel cylinders during hammer-group vibratory installation for the hong kong–zhuhai–macao bridge,” Engineering (2022), doi:” (You can […]

About that “Warrington Method” For Vibratory Pile Drivability —
Posted in Civil Engineering, Deep Foundations

The Paper “Vibratory and Impact-Vibration Pile Driving Equipment” Cited —

It’s happened again: the paper “Vibratory and Impact-Vibration Pile Driving Equipment” has been cited by Mohammed Al-Amrani and M Ikhsan Setiawan in their paper “Prefabricated and Prestressed Bio-Concrete Piles: Case Study in North Jakarta.” The abstract of their paper is here: In this research, we will talk about Prefabricated and Prestressed Concrete piles in general and […]

The Paper “Vibratory and Impact-Vibration Pile Driving Equipment” Cited —
Posted in Academic Issues, Geotechnical Engineering, Soil Mechanics

Determining the Degree of Consolidation

This is the last (hopefully) post in a series on consolidation settlement. We need to start by a brief summary of what has gone before. Note: the material for this derivation and those that preceded it have come from Tsytovich with some assistance from Verruijt.


In the post From Elasticity to Consolidation Settlement: Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve”, we discussed the issue of how much soils (especially cohesive ones) settle through the rearrangement of particles. We were able to start with the theory of elasticity and, considering the effects of lateral confinement, define the coefficient of volume compression m_v by

m_v = \frac{\beta}{E} (1)

where E is the modulus of elasticity and \beta is a factor based on Poisson’s Ratio and includes the effects of confinement, be that in an odeometer or in a semi-infinite soil mass. We also showed that, for a homogeneous layer,

\delta_p = m_v H_o \sigma_x (2)

where \delta_p is the settlement of the layer, H_o is the thickness of the layer and \sigma_x is the uniaxial stress on the layer. The problem is that m_v is not constant, and the settlement more accurately obeys the law

\delta_p = \frac{C_c H_o}{1+e_o} \log{\frac{\Delta p + \sigma_o}{\sigma_o}} (3)

where C_c is the compression index, e_o is the initial void ratio of the layer, \Delta p is the change in pressure induced from the surface, and \sigma_o is the average effective stress in the layer.

Turning to the post Deriving and Solving the Equations of Consolidation, we first determined that the change in porosity \Delta n could, for small deflections, be equated to the change in strain \epsilon . From this we could say that

\Delta n = m_v \Delta \sigma_x (4)

The change in porosity, for a saturated soil whose voids are filled with an incompressible fluid (hopefully water) induces water flow,

{\frac {\partial }{\partial x}}q(x,t)=-{\frac {\partial }{\partial t}} {\it n}(x,t) (5)

where q(x,t) is the flow of water out of the pores and n(x,t) is the porosity as a function of position and time. The flow of water is regulated by the overall permeability of the soil, and all of this can be combined to yield

{\frac {k{\frac {\partial ^{2}}{\partial {x}^{2}}}u(x,t)}{{\it \gamma_w }}}=m_{{v}}{\frac {\partial }{\partial t}}\sigma_{{x}}(x,t) (6)

where k is the permeability of the soil and \gamma_w is the unit weight of water. Defining

c_v = \frac{k}{m_v \gamma_w} (7)

and making some assumptions about the physics, we can determine the equation for consolidation as

c_{{v}}{\frac {\partial ^{2}}{\partial {x}^{2}}}u(x,t)={\frac {\partial }{\partial t}}u(x,t) (8)

where $latex u(x,t) is the pore water pressure. If we invoke the effective stress equation and solve this for the boundary and initial conditions described, we have a solution

\sigma_{x}(x,t)=p\left(1-\frac{4}{\pi}\left(\sin(1/2\,{\frac{\pi\,x}{h}}){e^{-1/4\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}+1/3\,\sin(3/2\,{\frac{\pi\,x}{h}}){e^{-9/4\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}+1/5\,\sin(5/2\,{\frac{\pi\,x}{h}}){e^{-{\frac{25}{4}}\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}\cdots\right)\right) (9)

The Degree of Consolidation

One thing that our theory presentation demonstrated was the interrelationship between pore pressure, stress and deflection. We know what the ultimate deflection will be based on Equation (3) above (or more complicated equations when preconsolidation is taken into consideration.) But how does the settlement progress in time?

We start by defining the degree of consolidation thus:

U = \frac{\delta(t)}{\delta_p} (10)

where \delta(t) is the settlement at any time before complete settlement. For the specific case (governing equations, initial equations and boundary conditions) at hand, the degree of consolidation–the ratio of settlement at a given point in time to total settlement–can be determined as follows:

U_{o}=\intop_{0}^{h}\frac{\sigma_{x}(x,t)}{ph}dx (11)

In this case the result is divided by the uniform pressure p and the height h. Let us further define the dimensionless time constant

T_{v}=\frac{c_{v}t}{h^{2}} (12)

That being the case, if we integration Equation (9) with Equation (11), we obtain

U_{o}=1-\sum_{n=1}^{\infty}4\,{\frac {{e^{-1/4\,{\it Tv}\,{n}^{2}{\pi }^{2}}}\left (\cos(n\pi )\cos(1/2\,n\pi )-\cos(n\pi )-\cos(1/2\,n\pi )+1\right )}{{n}^{2}{\pi }^{2}}} (13)

otherwise put

U_{o}=1-8\,{\frac {{e^{-1/4\,{\it Tv}\,{\pi }^{2}}}}{{\pi }^{2}}}-{\frac {8}{9}}\,{\frac {{e^{-9/4\,{\it Tv}\,{\pi }^{2}}}}{{\pi }^{2}}}-{\frac {8}{25}}\,{e^{-{\frac {25}{4}}\,{\it Tv}\,{\pi }^{2}}}{\pi }^{-2}\cdots (14)

As was the case with Equation (9), only the odd values of n are considered; the even ones result in zero terms.

It is regrettable that, in defining T_v , the value \frac{\pi^2}{4} was not included, as using Equation (14) would be much simpler. For certain cases, it is possible to use the first two or three terms. In any case the usual method for determining T_v –and by extension the degree of consolidation–is generally done either using a graph or a table, as is shown in the graph at the start of the post (repeated below:)

Degree of Consolidation for Instantaneous Uniform Loading and One-Dimensional Flow. From NAVFAC DM 7.1: Soil Mechanics

The notation is a little different. We use the variable U_o to emphasise that we are dealing with the “standard” case. The above graph also gives approximating equations; it is easy to see that, for T_v > 0.2 , the equation given is simply the first two terms of Equation (14). The distinction between the drainage length h (H_{dr} in the graph above) and the layer thickness H is clear.


We have covered the basic, classic case of consolidation settlement in this post and its predecessors From Elasticity to Consolidation Settlement: Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve” and Deriving and Solving the Equations of Consolidation. We trust that this presentation has been enlightening and informative.