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(+) Modules 49,50/Topic 51
FUNCTIONS AND SCALES
This is the title of App. E of the author’s book (Kurian, 2005). He would like every
student of civil engineering in general, and geotechnical engineering in particular, to
study the contents carefully which he is not likely to find in any other textbook
sources. The author had diligently presented these topics in a few lectures whenever
he had met with a fresh batch of students, over several years, in IIT Madras.(My
student Sriram, who left for U.S. after his B.Tech. wrote to me in a letter that he has
not come across the like of it even in the U.S.) While all students may not share
Sriram’s or Purnanandam’s enthusiasm, the author strongly feels that all students
must put in an early effort to imbibe the contents thereof to their own academic and
professional advantage. These are simple facts involving basic mathematics which
the author feels should form part of the knowledge base of every student, particularly
of geotechnical engineering.
This Section excerpts only the most important parts, which every student must
first learn before going to the full Appendix mentioned above.
51.1 Direct proportionality
When y is directly proportional to x, we have
y = kx
(51.1)
where k is the constant of proportionality between y and x. x being in the first degree
(power), the above will plot as a straight line, as seen in Fig.51.1.
51.2 Slope
k, which is the constant of proportionality, is in other words the slope of this line
which is the same at any point on the line (see Fig.51.2 in which the slope is plotted
against x). The dimension of k is the dimension of y divided by the dimension of x.
Slope is not the tangent of the angle θ, i.e. bc/ab, which students normally tend to
assume. If it were so, it would change with the scales to which y and x are plotted,
and besides, it would have been dimensionless irrespective of the dimensions of y
and x. However, if by ‘tangent’ what is meant is the quantity on the y-axis
represented by the distance bc divided by the quantity on the x-axis represented by
the distanceab, then it correctly defines the slope k.
If it were a curve, for example,
y = kx2(51.2)
which is a parabola (Fig.51.3), the slope of the curve at any point is the slope of the
tangent to the curve at that point, taken in the same manner as above. In this
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respect, slope is the rate of change of y with respect to x which in this case is equal
to 2kx., which linearly varies (directly proportional) with x(see Fig.51.4 in which the
slope is plotted against x)
Note that in a physical problem, if the variation were of this kind, (dy/dx) would be
called the tangent modulus, and (y/x) the secant modulus (Fig.51.5) at a point (x,y)
specified on the curve. A secant is the line joining the origin to a given point on a
curve, and (y/x) is the slope of such a line and hence the name secant modulus.
The slope of the tangent at the origin of the curve is called the initial tangent
modulus. (Note that the terms tangent and secant have both geometrical and
trignometrical meanings.)
51.3 Inverse proportionality
When y is inversely proportional to x, it means that y is directly proportional to the
inverse of x. i.e.,
y𝛼
1
𝑥
y=
𝑘
𝑥
xy = k
(51.3)
Since x and y appear in the product, the relationship between them is not linear.
The variation of y with x is shown in Fig.51.6. The curve following Eq.(51.3) is a
rectangular hyperbola. In other words, while in direct proportion y increase linearly
with x, in inverse proportion y decreases hyperbolically with x. What is however
important to note in respect of inverse proportion is that the decrease in y with
increase in x is very fast in the initial ranges of values of x and very slow in the final
ranges of value of x.
If x and y appear in the sum, as
x+y=k
(51.4)
the variation is linear, with y decreasing as x increases, as shown in Fig.51.7.
It may be indicated at this stage that xy = k will plot as a straight line (like Fig.51.7)
in logarithmic scales (Sec.51.8.1).
Fig.51.8 is plotted with three different indices of x (1,0 and -1), in which point (1,1)
is seen to be common to the three plots.
51.4 Logarithm
Logarithm of a quantity to any base is the index (or power) by which the base is to
be raised to obtain the quantity.
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logab = 3
This means,
a3 = b
(51.5)
In relation to the base and the logarithm, the quantity itself is called the antilogarithm.
In the above b is the antilogarithm, 3, the logarithm and a, the base.
51.5 Unit of logarithm
The logarithm of a quantity id always dimensionless, whatever be the dimension
of the quantity. This can be simply explained by referring to Eq.(51.5).
If a has the dimension [L], b will have the dimension [L3]. Thus while the
antilogarithm, i.e., the quantity itself, and the base have dimensions, the logarithm
(3) has no dimension, it being merely the index of a. We further note that the
dimensions of the antilogarithm and the base are consistent with the value of the
logarithm.
We use log10 p in many situations in geotechnical engineering. It is interesting to
note that whereas p has the dimension [F/L2] and log p has no dimension, the base
10 has dimension, and what is more, it changes with the value of the logarithm.
𝐹 1
Thus, for example, if log p10 = 2.4, say, the base 10 has the dimension[ 2 ]2.4
𝐿
𝐹
such that when it is raised to the power2.4, we get the dimension of p[ 2 ].In this
𝐿
manner the dimension of the base 10 changes with the value of the logarithm, the
dimension of p being constant, even though we are not concerned with the
dimension of the base 10.
51.6 Logarithmic scales
The first scale at the top of Fig.51.9 shows the arithmetic scale in which each
succeeding quantity is obtained by adding a constant to the preceding quantity. In
the next scale it is seen that each succeeding quantity is obtained by multiplying the
preceding quantity by a constant (which is 10 in this case), with the zero appearing
at -∞ .It is noted in respect of this scale that the logarithm of the quantity to the base
10 is plotted to arithmetic scale. These are called logarithmic scales in which the
quantity, i.e., the antilogarithm, is plotted to the logscale, which means that its
logarithm to the base 10 (in this case) is plotted to arithmetic scale. As a matter of
fact, only arithmetic scale exists, the picture is that, either the quantity itself, or some
function of it, is plotted to the arithmetic scale, the function being logarithm in the
present case. One may also note that just as the base 10, we can have log scales to
any other base in the same manner. (Fig.51.9 also shows the variation of log x with x
over the cycle 1-10.)
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It is obvious on examining the logarithmic scale that the same distance or interval
represents increasingly smaller quantities as we go to the left and increasingly larger
quantities as we go to the right, from the middle. In other words, we are introducing
a wilfuldistortion by magnifyingsmaller differences and shrinking larger differences in
the representation of quantities on the logarithmic scale, which is necessary in
situations such as plotting the particle size in the ‘grain size distribution curve’, in
order to give significant representation to smaller particle sizes, even at the expense
of larger sizes, which in the absence of such distortion would have received very
insignificant representation, if plotted to arithmetic scale.
51.7 Fitting experimental results
y is a function of x and we have a large number of values of y at different values
of x obtained, say, from an experiment conducted in a laboratory, which are shown in
Fig.51.10. If the relationship is idealised as linear, we would plot the best fitting
straight line passing through the points (giving the least scatter of points on either
side of the line). The characteristics c (the y-intercept) and m (the slope) which fully
define the straight line are obtained as shown in the figure. (Note that, theoretically,
only two points are needed to plot a straight line.)
51.8 Logarithmic plots
Logarithmic scales enable us to prepare logarithmic plots of data. We have two
types of logarithmic plots: the log-log plots and the semi-log plots.
51.8.1 Log-log plots
If the relationship between y and x were of the power form, i.e.,
y = xa(51.6)
or y = x –a(51.7)
(Fig.51.11), instead of fitting a curve through the points, we can obtain the value of
the power a by plotting a straight line through the points, provided both y and x are
plotted to log scales (i.e., their logarithms plotted to arithmetic scales). This follows
from the fact that if we take logarithm,
log y = ± a log x
(51.8)
which means that log y will plot as a straight line against log x to the slopea
(Fig.51.12). In other words, the power a is obtained as the slope of the straight line
fitted on a log-log plot. (In the above, we are actually exploiting the convenience of
fitting a straight line by entering the points on the distorted log-log scales.)
In the case of the inverse proportion hyperbola,
xy = k
(51.3)
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log x + log y = log k (51.9)
we have a means of fitting a straight line provided the points are plotted on log-log
scales (Fig.51.13), in which the antilogarithm of either of the intercepts gives us the
unknown k.
51.8.2 Semi-log plots
Consider the exponential function (Fig.51.14)
y = ax(51.10)
or
y = a-x(51.11)
Taking logarithm,
log y = ±x log a
(51.12)
which means that log y will plot as astraight line against x to the slope loga
(Fig.51.15). Such a plot is called a semi-log plot, in which only one quantity is
plotted to logarithmic scale, the other quantity being plotted to arithmetic scale. The
basea, which is the unknown in the present case, is obtained as the antilogarithm of
the slope of the straight line fitted on the semi-log plot.
51.9 Other scales
Just as we plotted the logarithm of the quantity to arithmetic scale, to get the log
scale for the quantity, we can plot functions such as square, square root, inverse,
etc. to arithmetic scales, to get the corresponding quantities plotted to the square,
square root and inverse scales respectively. Figs. 51.16, 51.17 and 51.18 show the
above scales respectively. Among them, one observes that the inverse scale is a
very fast converging scale. (Note that the cycles 1-10,10-100 etc. are fast decreasing
here, compared to the log scales where they are constant.)
Apart from the uses of these scales, note that, in y = kx2 (Eq.51.2), if x is plotted
to the square scale, we get a straight line whose slope is k. The same applies to y =
kx ½ and y = kx-1 (xy = k – rectangular hyperbola) plotted respectively to square root
and inverse scales for x (see Figs. 51.19,20 and 21).
51.10 Hyperbolic variation
A hyperbolic variation of the type
𝑥
𝑦 = 𝑎+𝑏𝑥 (51.13)
where a and b are constants, is assumed in many problems in Soil Mechanics. In
the first place, the shape of this curve (Fig.51.23) follows by multiplying y =x and y
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=
1
𝑎+𝑏𝑥
(Fig.51.22). In order to determine the constantsa and b, the ultimate value of y
and the initial tangent modulus (Fig.51.23), one can manipulate the equation in the
following manner.
𝑥
𝑦
= a + bx
(51.14)
If we plot the data (x/y) against x we get the straight line shown in Fig.51.24. It is
seen that a is the y-interceptand b is the slope of this straight line.
(Note that from Fig.51.23, y = 0 at x = 0 following which (x/y) is indeterminate at x
= 0. However, from Eq.(51.14) we see that
lim 𝑥
( )
𝑥→0 𝑦
→𝑎
Fig.51.24 is called the ‘transformed’ plot.)
Now, dividing Eq.(51.14) by x,
1
𝑦
from which
Hence,
1
𝑦𝑢𝑙𝑡
𝑎
= + 𝑏(51.15)
𝑥
𝑎
lim
= 𝑥→∞
( + 𝑏) = 𝑏
𝑦
1
𝑦𝑢𝑙𝑡 = 𝑏
The above shows that y ult is obtained as the inverse of the slope of the straight line.
Further, reverting to Eq.(51.13), we write
𝑦
𝑥
=
1
𝑎+𝑏𝑥
𝑦
At x = 0
𝑥
=
1
𝑎
(51.16)
where (y/x) is the ‘initial tangent modulus’, which is obtained as the inverse of the yintercept a of the straight line in Fig.51.24.
(Note that (y/x) is the secant modulus. It becomes the initial tangent modulus,
since the secant coincides with the tangent at the origin.)
Eq.(51.15) reveals another possibility of plotting a straight line using inverse
scales, as in Fig.51.25 where a is the slope and b, the y-intercept. However,
between Figs.51.24 and 25, one would prefer the former because of the greater
ambiguity in plotting Fig.51.25 thanks to the fast convergence of the inverse scales.
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51.11 Stresses vs. Deformations
Structural design of flexural systems are based on criteria based on either
‘allowable stresses’ or ‘allowable deflections’. These may be called ‘structural design
parameters’ in line with ‘geotechnical design parameters’ – which apply both to
Working Stress Design and Limit State Design in structural engineering. The latter
refers to them as the limit states of ‘collapse’ (strength) and ‘serviceability’
(deflection). We have already noted that bearing capacity (strength) and settlement
are the corresponding criteria in geotechnical design.
Reverting to structural design and calling the parameters respectively as σ
(bending stress) and y, (bending deflection), it directly follows that σ is not directly
proportional to y, or vice versa,; since, if it were so, they would not have been
independent criteria and design for one criterion would have automatically taken care
of the other. It is therefore important for us to examine the relationship between the
two, and we shall do that by taking the case of a simply supported beam with a
central concentrated load, from which:
𝜎=
𝑀x𝑧
𝐼
𝛼𝑀
2
𝑑 𝑦
However,M = -𝐸𝐼 𝑑𝑥2
2
𝛼
𝑑 𝑦
𝑑𝑥2
This means that M is not proportional to y, but proportional to a function of y, which is
𝑑2𝑦
(
𝑑𝑥 2
), the second rate of change of y with respect to x. (The first rate of change is
the slope.) As a result M is maximum where (
𝑑2𝑦
𝑑𝑥 2
) is maximum, and not where y is
maximum, as beginners among students normally tend to assume.
The example illustrated in Fig.51.26 explains the difference in terms of design.
In the above, elastic supports can take the place of springs. If σ were the
criterion of design all the three systems would be equally acceptable; on the other
hand, if Y were the criterion, the last two, if not all the three, might not have been
acceptable.
Note: App. E cited (Kurian, 2005) presents many examples from geotechnical and
foundation engineering using the above concepts and scales.