Posted in Geotechnical Engineering, Soil Mechanics

Superposition, and Using Point Loads in Place of Distributed Ones

Elastic solutions to stresses induced by loads at the surface have their limitations, but they allow the use of the principle of superposition. The principle of superposition states that you can add the effects of different loads on a single point. Superposition requires that the stress state of the point be path independent, which is the case with elastic conditions. No matter how you load and unload a point in a system, if elasticity is maintained the result will be the same for a given set of loads.

This is illustrated for point loads in the graphic below, but it applies to distributed loads (such as this and this) as well.

Action of a Number of Concentrated Forces (from Tsytovich (1976))

The stresses that result from each load affect the total stress at the point of interest. They can be computed and added together. So, since “point” loads are physically impossible, is their computation be useful? The answer to this question is “yes” but it takes some judgement, like so many things in geotechnical engineering.

Let us consider the following case, from NAVFAC DM 7.01.

Separate Column Footings Problem, from NAVFAC DM 7.01

To solve this problem, DM 7 converted the square footings to circular ones and then used the chart shown in the post Going Around in Circles for Rigid and Flexible Foundations. This chart, like others, is hard to read. Is it possible to use point loads as a substitute?

There are three things we need to note here. The first is that the load on each column is 27 tons, and is the same for each column.

The second is that the “r” shown in the table above is not the same as it is for the point loads. The variable “r” for the point loads is the horizontal distance from the load to the point of interest.

The third is that there are three column positions shown in the diagram, with three corresponding values of r:

  • The column on top of the load, Column B2
  • The columns in the mid-point of the edges, Columns A2, B1, B3, and C2. These have an value of r of 15′ from the point of interest.
  • The columns in the corners of the square, Columns A1, A3, C1, and C3. These have a value of r of 21.2′ from the point of interest.

Now, instead of the chart, we apply the formula derived earlier for the influence coefficient, which is

K=3/2\,{\frac {1}{\pi \,\left (1+(\frac{r}{z})^{2}\right )^{5/2}}}

from which the stress is computed by the equation

\sigma_z = K \frac{P}{z^2}

Using this formula, we can construct the table below for this problem.

(1) Z, ft(2) r/Z for B2(3) r/Z for A2, B1, B3, C2(4) r/z for A1, A3, C1, C3(5) K for B2(6) K for A2, B1, B3, C2(7) K for A1, A3, C1, C3(8) Stress for B2, tsf(9) Stress for each of A2, B1, B3, C2, tsf(10) Stress for each of A1, A3, C1, C3, tst(11) Total Stress at Z, tsf
207.50010.6070.4770.0000.0003.2230.0000.0003.224
403.7505.3030.4770.0010.0000.8060.0010.0000.810
602.5003.5360.4770.0030.0010.3580.0030.0010.370
1001.5002.1210.4770.0250.0070.1290.0070.0020.163
1501.0001.4140.4770.0840.0310.0570.0100.0040.113
2000.7501.0610.4770.1560.0730.0320.0110.0050.094
2500.6000.8490.4770.2210.1230.0210.0100.0050.080
Results from the Separate Column Footings problem in NAVFC DM 7.01, using point loads. Note that the stresses in Columns 9 and 10 are due to a single column, and not all four of them. The total stress in column (11) is the sum of the stress in Column 8 plus 4 times the result in Column 9 plus four times the result in Column 10.

We could have opted to add the influence coefficients and then compute the stresses since, for each elevation Z, both the elevation and the column load were the same. We did not for clarity; it is certainly possible to have columns of different loads.

The results are conservative, they tend to be higher, especially at the lower value elevations. It’s worth noting that the total stress at Z = 2′ is higher than the distributed loads on the footings. One way to make this better is to use the center formulae for stress under circles for Column B2 and point loads for the rest. Given, however, the limitations of the method in general, and the considerably lower effort in obtaining the influence coefficients, the method is a reasonable one to use.

Superposition doesn’t only apply to point loads; the example given in the post Analytical Boussinesq Solutions for Strip, Square and Rectangular Loads uses it for rectangular and square loads. Nevertheless, with appropriate engineering judgement, using point loads in place of distributed loads can be a viable option.

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Posted in Geotechnical Engineering, Soil Mechanics

Soil Mechanics Now in Print

In the Soil Mechanics and Foundations courses on this site, we’ve relied on Verruijt’s text as the principal “theoretical” text. Last year we linked to download the Soil Mechanics Textbook by Tsytovich. Now we offer this text in print for sale.

Purchase Tsytovich’s Soil Mechanics

  • CHAPTER ONE. THE NATURE AND PHYSICAL PROPERTIES OF SOILS
    • Geological Conditions of Soil Formation
    • Components of Soils
    • Structural Bonds and Structure of Soils
    • Physical Properties and Classification Indices of Soils
  • CHAPTER TWO. BASIC LAWS OF SOIL MECHANICS
    • Compressibility of Soils. The Law of C o m p a c t i o n
    • Water Perviousness of Soils. The Law of Laminar Filtration
    • Ultimate Contact Shear Resistance of Soils. Strength Conditions
    • Structural-Phase Deformability of Soils
    • Features of the Physical Properties of Structurally Unstable
    • Subsidence Soils
  • CHAPTER THREE. DETERMINATION OF STRESSES IN SOIL
    • Stress Distribution in the Case of a Three-Dimensional Problem
    • Stress Distribution in the Case of a Planar Problem
    • Pressure Distribution over the Base of the Foundation of
    • Structures (Contact Problem)
  • CHAPTER FOUR. THE THEORY OF ULTIMATE STRESSED STA­TE OF SOILS AND ITS APPLICATION
    • Stressed State Phases of Soils with an Increase in Load
    • Equations of Ultimate Equilibrium for Loose and Cohesive
    • Soils
    • Critical Loads on Soil
    • Stability of Soils in Landslides
    • Some Problems of the Theory of Soil Pressure on Retaining Walls
    • Soil Pressure on Underground Pipelines
  • CHAPTER FIVE. SOIL DEFORMATIONS AND SETTLEMENT OF
  • FOUNDATIONS
    • Kinds and Causes of Deformations
    • Elastic Deformations of Soils and Methods for Their Determi­nation
    • One-Dimensional Problem of the Theory of Soil Consolidation
    • Planar and Three-Dimensional Problems in the Theory of Fil­tration Consolidation of Soils
    • Prediction of Foundation Settlements by the Layerwise Summa­tion Method
    • Prediction of Foundation Settlements by Equivalent Soil Layer Method
  • CHAPTER SIX. RHEOLOGICAL PROCESSES IN SOILS AND THEIR SIGNIFICANCE
    • Stress Relaxation and Long-Term Strength of Cohesive Soils
    • Creep Deformations in Soils and Methods for Their Description
    • Account of Soil Creep in Predictions of Foundation Settlements
  • CHAPTER SEVEN. DYNAMICS OF DISPERSE SOILS
    • Dynamic Effects on Soils
    • Wave Processes in Soils under Dynamic Loads
    • Changes in the Properties of Soils Subject to Dynamic Effects
    • The Principal Prerequisites for Taking the Dynamic Proper­ties of Soils into Account in Vibrational Calculations of Founda­tions

As with Verruijt, hopefully in the coming time we will feature interesting techniques in this book which may be of interest to our visitors.

Posted in Soil Mechanics

Fatal Landslide Triggered by Hurricane Ida

A portion of the embankment on MS26 in George County Mississippi failed after heavy rains and flooding caused by Hurricane Ida. It created a hole about 20 feet deep and 50 feet across late on Monday night, and I can imagine it was difficult for motorists to see until it was too late. There were […]

Fatal Landslide Triggered by Hurricane Ida
Posted in Geotechnical Engineering, Soil Mechanics

From Elasticity to Consolidation Settlement: Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve”

Note: I have extensively revised this to boost the rigor of the equation derivation and to clarify the relationship between void ratio and shear/elastic modulus.

Three years ago I posted Jean-Louis Briaud’s “Pet Peeve” on the Analysis of Consolidation Settlement Results. Since that time he has been elected President of the American Society of Civil Engineers and I am in the process of retiring from full-time teaching, so our trajectories are a little different. (He’ll catch up, don’t worry.)

Nevertheless his Presidency would go unfinished if some explanation of the pet peeve wasn’t given. To remind my readers it is as follows:

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.

Theory of Elasticity Considerations

Dr. Briaud notes that consolidation specimens are confined by a ring when they are tested in an odeometer. That’s to simulate the fact that one-dimensional consolidation theory (and the settlement theory that goes along with it) is based on the assumption that a) you have a uniform surcharge and b) the layer experiencing consolidation settlement is “confined” by the semi-infinite mass we assume the soil to be. That assumed, for theory of elasticity purposes we assume uniaxial strain conditions, which I discuss in my post Constitutive Elasticity Equations: Uniaxial Cases.

The uniaxial strain (assuming the x-direction is vertical) is

\epsilon_{x}={\frac{\sigma_{{x}}\nu}{\left(1-\nu\right)\lambda}} (1)

Since, from Constitutive Elasticity Equations: Three-Dimensional Formulation,

\lambda = \frac{\nu E}{(1+\nu)(1-2\nu)} (2)

Substituting for \lambda ,

\epsilon_x = \frac{\sigma_x(1+\nu)(1-2\nu)}{E(1-\nu)} (3)

which can be rewritten (Tsytovich (1976))

\epsilon_x = \frac{\sigma_x}{E} \beta (4)

where

\beta = 1-\frac{2\nu^2}{1-\nu} (5)

The variable \beta is a measure of the effect of lateral confinement of either the odeometer specimen or the compressed layer in the field. At \nu = 0 , \beta = 1 and there is no confinement effect. At \nu = 0.5 , \beta= 0 , and the confinement effect is total: the “fluid” (which what it is in reality) is incompressible.

If we define, assuming that \sigma_x is the uniaxial stress applied to the specimen/soil,

m_v = \frac{\epsilon_x}{\sigma_x} (6)

then combining Equations (4) and (6) yields

E = \frac{\beta}{m_v} (7)

Strain vs. Void Ratio

Using theory of elasticity involves stress vs. strain. Unfortunately, as Verruijt observes:

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.

So we must relate strain to void ratio. To do this, we need to consider the compression of the soil from one void ratio to another, as is shown in the diagram below (from Broms):

Manipulating the equation in the diagram, the relationship of strain to void ratio is as follows:

\epsilon=\frac{\delta_p}{H_o} = {\frac {{\it e_0}-{\it e_1}}{1+{\it e_0}}} (8)

From Equation (6),

m_v = \frac{e_0 - e_1}{\sigma_x(1+e_0)} (9)

Combining and rearranging Equations (8) and (9),

\delta_p = m_v H_o \sigma_x (10)

At this point we can make the following observations:

  • The modulus of elasticity E and the coefficient of volume expansion m_v are linearly, inversely proportional to each other.
  • The physical difference between the two is that coefficient of volume expansion includes the effect of confinement while the elastic modulus does not. (Consider that most tension and compression tests on materials are conducted in an unconfined way.) This is the reason why we can write Equation (10) the way we can.
  • The coefficient of volume expansion is also important when we consider time consolidation.
  • If we assume \nu to be constant and both of these parameters are constant for the range of void ratios under consideration, consolidation settlement can be considered in a linear elastic way.

Facing Reality About Consolidation Settlement

Unfortunately this is not the case. Let’s pick up where we left off in our last post on the subject, with the E vs. \epsilon plot below:

It was noted at the time that the apparent elastic modulus increased more or less linearly with strain. Since void ratio and strain are linearly dependent, we would expect a similar looking result if we did a void ratio plot.

To solve this problem, we first observe that the elastic and shear modulus G are linearly proportional via \nu (as are E and m_v .) From this, it can be noted that the shear modulus can be estimated for a soil (excluding strain-softening effects) as follows:

\frac{G_{0}}{p_{atm}}=SF\left(e\right)\left(\frac{\sigma_{0}}{p_{atm}}\right)^{\bar{n}} (11)

where the notation is shown in the source. Let’s make some assumptions:

  • Poisson’s ratio remains constant, thus the relationship between strain and elastic modulus is constant.
  • S is constant for a given soil type.
  • Effective stress \sigma_o for a given sample (that’s just about a given for triaxial tests in any event)
  • Other constants, such as p_{atm} and \bar{n} also remain constant.

That leaves the variable F(e) to change. The shear (and by extension the elastic) modulus of a material is a function of the void ratio. From the same source,

F\left(e\right)=\left(1+e\right)^{-3} (12)

We plot this as follows, including a logarithmic trend line:

Plot of Equation (12) with Logarithmic Trend Line

The logarithmic correlation isn’t perfect; however, as Verruijt notes in his commentary on his image below, in the caption:

When performing the test, it is observed, as expected, that the increase of vertical stress caused by a loading from say 10 kPa to 20 kPa leads to a larger deformation than a loading from 20 kPa to 30 kPa. The sample becomes gradually stiffer, when the load increases. Often it is observed that an increase from 20 kPa to 40 kPa leads to the same incremental deformation as an increase from 10 kPa to 20 kPa. And increasing the load from 40 kPa to 80 kPa gives the same additional deformation. Each doubling of the load has about the same effect. This suggests to plot the data on a semi-logarithmic scale. In this figure log(σ/σ0 ) has been plotted against ε, where σ0 denotes the initial stress. The test results appear to form a straight line, approximately, on this scale. The logarithmic relation between vertical stress and strain has been found first by Terzaghi, around 1930.

This, of course, leads us to the classic equation we use in one-dimensional consolidation theory for normally consolidated soils,

\delta_p = \frac{C_c H_o}{1+e_o} \log{\frac{\Delta p + \sigma_o}{\sigma_o}} (13)

The need to treat compression due to settlement completely differently than that of elastic (or elasto-plastic) settlement is one of the anomalies of geotechnical engineering. The observation that the elastic modulus decreases with increasing void ratio (or strain) is a start in putting the two together and presenting a more or less unified theory of soil deformation. Coupled with agreement on using strain in consolidation tests, this would bring us a long way to solving the dilemma of Jean-Louis Briaud’s–and some of the rest of our–pet peeves.

Posted in Soil Mechanics

Soil Mechanics Textbook by Tsytovich Available

Up until now the only comprehensive soil mechanics textbook we offered for download was Verruijt’s. We now add to that N. Tsytovich’s Soil Mechanics. Download is at the link or at the book cover below; information on the book is as follows:

This is a textbook in the course of Soil Mechanics for higher-school students of civil engineering and hydrotechnical engineering, and also for students of other specialties associated with construc­tion of engineering structures, such as road constructors, ameliorators, geologists, soil scientists.

The Author has made an attempt to write a concise course on the basis of a wide synthesis of natural sciences and to present the theore­tical data in the most simple and comprehensive form, without depreciating, however, the general scientific aspect of the problem; his other aim was to present a number of engineering solutions of problems in the theory of soil mechanics (calculations of strength, stability and deformability), which might be widely used in engi­neering.

Some problems in the book are discussed from new standpoints which take into account the principal properties of soils: contact shear resistance, structure-phase deformability (including creep of skeleton), compressibility of gas-containing porous water, and the effect of natural compaction of soils.

The book shows some new methods used for determination of characteristics of soils and gives some new solutions of the theory of consolidation and creep of soils, which can be used for predictions of settlement rates of foundations of structures and their time varia­tions; a separate chapter discusses rheological processes in soils and their significance.

Topics include the following:

  1. CHAPTER ONE. THE NATURE AND PHYSICAL PROPERTIES OF SOILS
    • Geological Conditions of Soil Formation
    • Components of Soils
    • Structural Bonds and Structure of Soils
    • Physical Properties and Classification Indices of Soils
  2. CHAPTER TWO. BASIC LAWS OF SOIL MECHANICS
    • Compressibility of Soils. The Law of C o m p a c t i o n
    • Water Perviousness of Soils. The Law of Laminar Filtration
    • Ultimate Contact Shear Resistance of Soils. Strength Conditions
    • Structural-Phase Deformability of Soils
    • Features of the Physical Properties of Structurally Unstable
      Subsidence Soils
  3. CHAPTER THREE. DETERMINATION OF STRESSES IN SOIL
    • Stress Distribution in the Case of a Three-Dimensional Problem
    • Stress Distribution in the Case of a Planar Problem
    • Pressure Distribution over the Base of the Foundation of
      Structures (Contact Problem)
  4. CHAPTER FOUR. THE THEORY OF ULTIMATE STRESSED STA­TE OF SOILS AND ITS APPLICATION
    • Stressed State Phases of Soils with an Increase in Load
    • Equations of Ultimate Equilibrium for Loose and Cohesive
      Soils
    • Critical Loads on Soil
    • Stability of Soils in Landslides
    • Some Problems of the Theory of Soil Pressure on Retaining Walls
    • Soil Pressure on Underground Pipelines
  5. CHAPTER FIVE. SOIL DEFORMATIONS AND SETTLEMENT OF
    FOUNDATIONS
    • Kinds and Causes of Deformations
    • Elastic Deformations of Soils and Methods for Their Determi­nation
    • One-Dimensional Problem of the Theory of Soil Consolidation
    • Planar and Three-Dimensional Problems in the Theory of Fil­tration Consolidation of Soils
    • Prediction of Foundation Settlements by the Layerwise Summa­tion Method
    • Prediction of Foundation Settlements by Equivalent Soil Layer Method
  6. CHAPTER SIX. RHEOLOGICAL PROCESSES IN SOILS AND THEIR SIGNIFICANCE
    • Stress Relaxation and Long-Term Strength of Cohesive Soils
    • Creep Deformations in Soils and Methods for Their Description
    • Account of Soil Creep in Predictions of Foundation Settlements
  7. CHAPTER SEVEN. DYNAMICS OF DISPERSE SOILS
    • Dynamic Effects on Soils
    • Wave Processes in Soils under Dynamic Loads
    • Changes in the Properties of Soils Subject to Dynamic Effects
    • The Principal Prerequisites for Taking the Dynamic Proper­ties of Soils into Account in Vibrational Calculations of Founda­tions