51.1 Direct proportionality

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Lillian Carpenter
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1 1 (+) 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 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 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 = kx 2 (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

2 2 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., y α 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 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 It may be indicated at this stage that xy = k will plot as a straight line (like Fig.51.7) in logarithmic scales (Sec ). 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 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.

3 3 logab = 3 This means, a 3 = 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 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 [L 3 ]. 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/L 2 ] and log p has no dimension, the base 10 has dimension, and what is more, it changes with the value of the logarithm. Thus, for example, if log p10 = 2.4, say, the base 10 has the dimension[ F 1 L 2] 2.4 such that when it is raised to the power2.4, we get the dimension of p[ F L2].In this 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 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.)

4 4 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 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 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 Log-log plots If the relationship between y and x were of the power form, i.e., y = x a (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)

5 5 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 Semi-log plots Consider the exponential function (Fig.51.14) Taking logarithm, y = a x (51.10) or y = a -x (51.11) 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 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 , and 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, 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 = kx 2 (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 ,20 and 21) Hyperbolic variation A hyperbolic variation of the type y = x 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

6 6 = 1 (Fig.51.22). In order to determine the constantsa and b, the ultimate value of y a+bx and the initial tangent modulus (Fig.51.23), one can manipulate the equation in the following manner. x = a + bx (51.14) y If we plot the data (x/y) against x we get the straight line shown in Fig 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 0 (x y ) a Fig is called the transformed plot.) Now, dividing Eq.(51.14) by x, 1 y = a x + b(51.15) from which Hence, 1 = lim y ult x (a + b) = b y y ult = 1 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 = 1 x a+bx At x = 0 y x = 1 a (51.16) where (y/x) is the initial tangent modulus, which is obtained as the inverse of the y- intercept a of the straight line in Fig (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 where a is the slope and b, the y-intercept. However, between Figs and 25, one would prefer the former because of the greater ambiguity in plotting Fig thanks to the fast convergence of the inverse scales.

7 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 = α M However,M = -EI d2 y dx 2 α d2 y dx 2 M x z I This means that M is not proportional to y, but proportional to a function of y, which is ( d2 y dx2), 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 ( d2 y dx2) is maximum, and not where y is maximum, as beginners among students normally tend to assume. The example illustrated in Fig 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.

Appendix A. Common Mathematical Operations in Chemistry

Appendix A. Common Mathematical Operations in Chemistry Appendix A Common Mathematical Operations in Chemistry In addition to basic arithmetic and algebra, four mathematical operations are used frequently in general chemistry: manipulating logarithms, using