ureau "of Standards

.01 Admin. BWg.

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435

The Hyperbolic Character of Certain

Experimental Results Which Tend

Toward Limiting Values

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U.S. DEPARTMENT OF COMMERCE

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NBS TECHNICAL NOTE 435

ISSUED NOVEMBER 1967

The Hyperbolic Character of Certain Experimental

Results Which Tend Toward Limiting Values

Arthur F. Kirstein

Mechanics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

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listed or referred to in the open literature.

For sale by the Superintendent of Documents, Government Printing Office

Washington, D.C, 20402 - Price 15 cents

THE HYPERBOLIC CHARACTER OF CERTAIN EXPERIMENTAL

RESULTS WHICH TEND TOWARD LIMITING VALUES

by Arthur F. Kirstein

ABSTRACT

The hyperbolic character of certain

experimental data obtained from structural and

mechanical testing provides an extremely simple

graphical method for use in approximating the

limiting values associated with the data. The

method, developed by Southwell to explain the

behavior of eccentrically loaded thin elastic

struts, is shown to have much wider application,

A limited number of illustrations are given to

describe the method and to show how it might be

used as a research tool.

Key words: Experimental, graphical solution,

hyperbolic character, limiting

values

1. Introduction

It is a common occurrence in experimental investigations to find

that plotted test results may form a curve which passes through the

origin and tends toward limiting values in much the same manner as a rectangular hyperbola having horizontal and vertical asymptotes. Experimental data possessing this hyperbolic character will approach limiting

values asymptotically. Thus an extremely simple graphical method is

provided for use in approximating the limiting values of asymptotes.

The equation for the rectangular hyperbola used in this method of

approximation was given by Ayrton and PerryL-'-J' and later developed by

Southwelll-2 J to explain the behavior of eccentrically loaded thin elastic

struts. This method was also used by RossL-^l in a slightly modified

form to represent the creep of concrete under load.

This investigation explored a number of different instances where

the hyperbolic character of experimental data was used to illustrate

^Figures in brackets indicate the literature references at the end of

this paper.
the method of approximation as a useful research tool. The basic

method was explained and some of the variations or extensions were also

shown.

2. Background

To the writer's knowledge the first published account recognizing

this hyperbolic character in the load vs deflection data obtained from

eccentrically loaded struts was given by Ayr ton and Perry. They found

that when the reciprocals of deflection and stress from tests of oak

struts were plotted on rectangular coordinates the points fell very

nearly on a straight line. This was expressed as

6 - fh a>

which can be written in the form used by Southwell nearly one half

century later

6 - p- + a = (2)

where 6 = deflection

<J = stress

0! = initial eccentricity

3 = critical elastic stress.

When equation (2) is divided by 6 we recognize that a linear relationship exists between the reciprocals of o~ and 6.

The use of this graphical method of approximating limiting values

can be explained more fully by examining the properties of the hyperbola. Consider the rectangular hyperbola passing through the origin of

the cartesian coordinate system shown in figure 1 having as asymptotes

x + a = (3)

and y - (3 = 0;

the equation of this hyperbola is

xy - gx + ay = 0. (4)

Equation (4) may be written as

x - 3v + a = (5)
where

v = 5 . (6)

Equation (5) is recognized as the equation for the straight line

shown in figure 2. The inverse slope and the intercept on the x axis

of this line are measures of (3 and a, respectively, which are the

limiting values or asymptotes of the original hyperbola shown in

figure 1.

The application of this method to experimental data is accom- plished graphically by plotting the data in the form shown in figure 2.

The linearity of the resulting curve is a measure of how well the data

approximates the rectangular hyperbola and the limiting values or asymptotes may be determined as described previously.

Southwell found that the load-deflection relationship for an eccentrically loaded thin elastic column approximated this hyperbola

and compared the critical load of the column to the predicted limiting

value, (3 . The results of this comparison were very good. No doubt,

this along with other factors accounts for the assumption that the

method would apply only so long as the system remained elastic.

Conversely, Ross used this method in a slightly modified form to

represent the creep of concrete under load. Thus Ross indicated that the

method could be used to predict limiting values other than critical

elastic loads as supposed by Southwell.

Certain other experimental data which may be observed from

mechanical or structural testing may be typified by the curves shown

in figure 3. The curves shown in figure 3a were modified as indicated

to produce the curves representing stress vs nonlinear strain in

figure 3b. This procedure is similar to that used by Galletly and

ReynoldsL^J which produces curves that resemble Southwell's rectangular hyperbola of figure 1. The modified results shown in figure 3b can be used in the graphical method of approximating the limiting

values of the curves given in figure 3a. The success or failure of

this method in predicting the limiting value is not dependent upon

elastic action, but is dependent on how well the experimental results

approximate the hyperbola.

3. Limiting Values and the Hyperbolic Character

of Experimental Results

It was indeed unfortunate that Southwell arbitrarily restricted

the use of his method to the examination of the critical elastic buckling loads. Had he continued to examine this method when applied to

Karman s medium and thick struts 1- •• he would have found that it would

have served to predict with a fair degree of accuracy the ultimate load
carrying capacities of the struts. Table 1 contains the data from

these tests of medium and thick struts. Column three of this table

contains the values predicted from Southwell's rectangular hyperbola

and column four gives the ratio of these predicted limiting values to

those observed by Karman. The good agreement shown in table 1 has

little or nothing to do with the elasticity of the medium and thick

struts. It merely shows that the load vs deflection curves possess

this hyperbolic character which is useful for the purpose of approximating the values of the horizontal asymtotes or limiting values which

in this case correspond to the ultimate loads.

Using the previously discussed modification of typical stress vs strain data as it pertains to figures 2 and 3, consider the stress vs strain curves for compressive tests on standard 6 x 12 inch concrete

cylinders shown in figure 4. Applying the linearity test which is inherent in the graphical method shown in figure 2 to these data produced the

results given in figure 5 . Note that the curves in figure 5 are linear

for the larger values of nonlinear strain, which indicates that the data

does at least approximate the rectangular hyperbola. It should be

noted that in the graphical method illustrated the straight line is

selected so that it coincides more closely with data associated with the

larger values of nonlinear strain, and no attempt was made to apply the

method of least squares or other mathematic method of curve fitting. It

has been found from experience L-> J that the graphical method produces

results which more closely represent the limiting values. A comparison

of the limiting values obtained from these curves and the compressive

strengths determined from the test results is made in table 2. These

results show that the method can be used to make reasonable predictions

of the compressive strength.

Ross found that the creep in concrete when plotted as the ordinate

against the corresponding time as the abscissa produced a curve similar

In appearance to the rectangular hyperbola shown in figure 1. Further,

he plotted a curve of time/creep vs time (t/c vs t) which he referred

to as the "reduction of creep curve". This curve was similar to that

shown in figure 2, however, Ross elected to write the linear equation

for the curve as

- = a + bt (7)

c

where c = creep per unit length in time, t

a = a constant (t/c intercept)

b = a coefficient (slope)

Ross found It useful to correlate a and b of equation (7) with

factors or conditions which affected the results of his studies of

creep
4. Summary

It has been observed that the graphical method of predicting the

limiting values or asymptotes for experimental data that exhibit this

hyperbolic character has a wider application than was recognized in the

past. It is obvious from the results presented in table 1 that the usefulness of the method extends beyond that ascribed to it by Southwell

when dealing with the stability of struts. Further, it was shown in

table 2 that the method has some degree of usefulness in predicting

mechanical properties associated with materials testing. Ross also

found this method to be of use in predicting the limiting values of

creep in concrete.

As noted it is fairly common in experimental investigations involving structural and mechanical testing to obtain test results which tend

toward limiting values asymptotically. Furthermore, from the previously

cited examples it would appear that the method described could be

applied as a very useful tool in the structural or mechanical testing

laboratory to predict results from limited investigation and to plan

testing programs which would yield a greater amount of useful results

with a lesser amount of testing. It should also be recognized that the

graphical method can be used to write empirical equations which show the

relationship of the variables involved.

5 . Acknowledgments

Acknowledgment is made to Mr. R. G. Mathey of the NBS Structures

Section for furnishing the data presented in figure 4.

6. References

1. Ayrton, W. E., and Perry, J., "On Struts", The Engineer, Dec. 10,

1886, pp. 464-465 and Dec. 24, 1886, pp. 513-515.

2. Southwell, R. V., "On the Analysis of Experimental Observations in

Problems of Elastic Stability", Proc . Royal Society, vol. 135A,

pp. 601-616 (London), 1932.

3. Ross, A. D., "Concrete Creep Data", The Structural Engineer, vol.

15, no. 8, pp. 314-326, 1937.

4. Galletly, G. D., and Reynolds, T. E., "A Simple Extension of Southwell's Method for Determining the Elastic General Instability

Pressure of Ring-Stiffened Cylinders Subject to External Hydrostatic Pressure", Proc. SESA, vol. XIII, no. 2, 1956.

5. von Karman T., "Untersuchungen uber Knickfestigkeit" (also printed

in "Mitteilungen uber Forschungsarbeiten" , published by the Ver.

D.euts.Ing., Berlin, 1909).
Table 1. - Comparison of Karman s Observed Maximum

Stress and the Predicted Limiting Value

Strut Maximum stress Pr< Ob;

sdicted stress

No. Observed Predicted served stress

kg/ cm kg/ cm

2

Medium struts

7A 2760 2794 1.012

7B 2685 2796 1.041

8

:

i 2740 2724 0.994

9A 3030 3058 1.009

9B* 1

! -- — --

10A

1

1 3185 3255 1.022

10B i

i 3080 3120 1.013

11 3165 3185 1.006

12A* ; -- -- --

12B

!1 2960 3009 1.017

13

J 3060 3105 1.015

j Thick struts

14B

1 3320 3439 1.036

15A !

! 3395 3440 1.015

15B

if

i 3485 3360 0.964

16

I

1 3890 3989 1.025

17

1i4330 4510 1.042

xInsuf f icient data recorded for determination of

predicted value
Table 2.- Observed and Predicted Compressive Strength

of 6 by 12 in. Concrete Cylinders

Compressive strength Pr<^dieted strength

Cylinder Observed Predicted 0b served strength

(psi) (psi)

1 3360 3566 1.061

2 3077 3271 1.063

3 2911 3090 1.061

4 2844 2922 1.027

5 2741 2960 1.080
Asymptote

Figure 1. Rectangular hyperbola

y / /

a

Figure 2. Relationship between v and x of equation (5),
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