Hello this is Professor Kitch.
Welcome to this webcast on sections 9.4 and 9.5 which cover computation of geostatic stresses.
When you finish this module you should be able compare the concepts of geostatic and
induced stresses. And compute geostatic stresses given a soil
profile with the appropriate data. Finally you should be able to describe the
concept of the coefficient of lateral earth pressure. We’ll cover this last subject in
more detail later. There are two major categories of stresses
in the ground: Geostatic and induced stresses. Geostatic stresses are due to the soil itself.
They’re caused by the weight of the soil above a point in the ground. .
They’re naturally occurring but can be affected by both geologic and human actions.
Induced stress, by contrast, are caused by external loads on the ground surface, load
such as those generated by a building foundation, a water tank, or a vehicle passing by.
So how do we compute geostatic stresses. Well we have to start out with a model of
the subsurface. It’s often possible to model the subsurface as a layer cake where each
layer represents a differ soil with different properties.
To compute the geostatic stresses, we’re going to consider just a column of soil from this
cake. A elemental column with and area dx by dy.
For each soil stratum in this column, let’s assume we know the basic properties, unit
weight, gamma, and layer thickness H as shown. Now imagine that you’re standing under this
column and holding the whole thing up. You have to hold up the entire weight of the soil
column. The weight of each layer will be equal to the volume of that layer times it’s unit
weight. And the weight of the entire soil column will be the summation of the volume
times unit weight of each soil layer or the summation of V-sub-I gamma-sub-i.
The volume of each layer will be the thickness of the layer H-sub-I times dx dy.
When we substitute this into the summation the dx dy term will come out of the summation
since it’s the same for every layer. Now the vertical stress at the bottom of the
column will be the total weight divided by the area which is equal to dx dy time the
summation of H-I gamm-I divided by dx dy . The dx dy terms cancel out [cross out dx dy]
and we’re left with sigma-z being equal to the summation of the thickness times the unit
weight of each layer. That’s how we calculate the vertical geostatic
stress. It really very simple. More exactly the vertical geostatic stress
is the integral of gamma dz from zero to z. However in practice, we general use the summation
formulation because we divide the soil into finite layers.
While calculating vertical geostatic stress is simple, determining horizontal geostatic
stress is not–it’s much more complicated. Fundamentally, horizontal geostatic stress
is caused by the lateral confinement of soil and the vertical load.
In uniaxial loading, such as when we test a concrete cylinder, the sample is free to
strain in the lateral direction and there are no lateral stresses.
However soil extends infinitely in all directions and is confined laterally. Any vertical loads
will Induce a lateral load due the confinement and Poisson’s effect.
In addition to the Poisson’s effect the horizontal geostatic stress is complicated by many other
factors including. Load history. When a soil is loaded and then
unloaded horizontal stresses can get locked in from the higher loads.
There can be significant horizontal stresses due to geologic conditions. For example near
a reverse fault there can be very high horizontal stresses due to tectonic processes.
Because of its complicated nature, the horizontal geostatic stress is estimated using the concept
of the coefficient of lateral earth pressure, represented by K.
We can’t compute the horizontal stress from fundamental principles as we did with vertical
stress where we summed the unit weight times layer thickness.
Instead we estimate K, the coefficient of lateral earth pressure, based on soil properties,
load history, geology, or using field measurements. We then compute the horizontal geostatic stress,
sigma-x, as K time sigma-z. Estimating K is rather complicated and an
inexact science and we will cover it more in Section 9.8 and Chapter 17. For now, what’s
important to understand is that we don’t compute the horizontal geostatic stress directly,
but estimate the coefficient of later earth pressure and then estimate sigma-x based on
the vertical stress and K. If you though determining the geostatic horizontal
normal stress was complicated, determining the in situ shear stresses is even more complex.
There is, however, one case where the boundary conditions simplify the process.
In the case where the ground surface is horizontal and there are no surface loads, then there
is no shear stress at the ground surface . And tau-zx and tau xz are zero.
In this case the vertical and horizontal stresses, sigma-z and sigma-x , are principle stress.
And we can represent the 2-D state of stress with a Mohr Circle.
In nearly every other case the problem is more complicated and it is difficult to determine
the principle axes. So, to summarize, there are two sources of
stress in soil. Geostatic stresses that are naturally occurring.
And stresses induced from surface loads. Vertical geostatic stress is simple to compute.
Techncially, is the integral of gamma dz from the ground surface to the point of interest.
From a practical standpoint this is always simplified and the summation of H-i time gamm-i
. Determining the horizontal geostatic stress
is much more complicated. We do this by estimating the coefficient of
horizontal earth pressure, K. And then compute sigma-x and K times sigma-z.
When the ground surface is horizontal and there aren’t any surface loads, then the horizontal
and vertical planes are principle planes and sigma z and sigma-x are principle stresses.
Finally, here’s a practice problem for you to try.
Using the soil profile shown, compute the vertical geostatic stress, sigma-z at point
A. You can check your answer against example
problem 9.2 in your text. Try to complete this example before class.
That’s all for this webcast.