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 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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UNITED STATES DEPARTMENT OF COMMERCE 
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 NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director 
 
 NBS 
 
 TECHNICAL NOTE 435 
 
 ISSUED NOVEMBER 1967 
 
 The Hyperbolic Character of Certain Experimental 
 Results Which Tend Toward Limiting Values 
 
 Arthur F. Kirstein 
 
 Metrology Division 
 
 Institute for Basic Standards 
 
 National Bureau of Standards 
 
 Washington, D.C. 20234 
 
 NBS Technical Notes are designed to supplement the 
 Bureau's regular publications program. They provide a 
 means for making available scientific data that are of 
 transient or limited interest. Technical Notes may be 
 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 
 
Digitized by the Internet Archive 
 
 in 2012 with funding from 
 
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 http://www.archive.org/details/hyperboliccharacOOkirs 
 
THE HYPERBOLIC CHARACTER OF CERTAIN EXPERIMENTAL 
 RESULTS WHICH TEND TOWARD LIMITING VALUES 
 
 by Arthur F. Kirs te in 
 
 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. Exper- 
 imental 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 Perry l.i f and later developed by 
 Southwelll^J to explain the behavior of eccentrically loaded thin elastic 
 struts. This method was also used by Rossi- -' 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 ■ j& w 
 
 which can be written in the form used by Southwell nearly one half 
 century later 
 
 c 
 
 6 - 0- + a = (2) 
 
 where 6 = deflection 
 
 a = stress 
 
 a = initial eccentricity 
 
 P = critical elastic stress. 
 
 When equation (2) is divided by 6 we recognize that a linear relation- 
 ship 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 hyper- 
 bola. Consider the rectangular hyperbola passing through the origin of 
 the cartesian coordinate system shown in figure 1 having as asymptotes 
 
 x + Oi = (3) 
 
 and y - (3 = 0; 
 
 the equation of this hyperbola is 
 
 xy - Px + o/y = 0. (4) 
 
 Equation (4) may be written as 
 
 x - Pv + a = (5) 
 
where 
 
 v-S. (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 ex, 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, P. 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 
 Reynolds L^J which produces curves that resemble Southwell's rectan- 
 gular 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 buck- 
 ling loads. Had he continued to examine this method when applied to 
 Karman s medium and thick struts 1 - -I 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 approxi- 
 mating 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 inher- 
 ent 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 use- 
 fulness 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 involv- 
 ing 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 South- 
 well's Method for Determining the Elastic General Instability 
 Pressure of Ring-Stiffened Cylinders Subject to External Hydro- 
 static Pressure", Proc. SESA, vol. XIII, no. 2, 1956. 
 
 5. von Karman T., "Untersuchungen liber 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: 
 
 ^dieted stress 
 
 No. 
 
 Observed 
 
 
 Predicted 
 
 served stress 
 
 
 kg/ cm 2 
 
 
 kg/ cm 2 
 
 
 
 
 
 Medium struts 
 
 
 
 7A 
 
 2760 
 
 
 2794 
 
 
 1.012 
 
 7B 
 
 2685 
 
 
 2796 
 
 
 1.041 
 
 8 
 
 2740 
 
 
 2724 
 
 
 0.994 
 
 9A 
 
 ; 3030 
 
 
 3058 
 
 
 1.009 
 
 9B* 
 
 
 
 -- 
 
 
 -- 
 
 10A 
 
 3185 
 
 
 3255 
 
 
 1.022 
 
 10B 
 
 1 3080 
 
 
 3120 
 
 
 1.013 
 
 11 
 
 3165 
 
 
 3185 
 
 
 1.006 
 
 12A* 
 
 i 
 
 
 -- 
 
 
 -- 
 
 12B 
 
 j 2960 
 
 
 3009 
 
 
 1.017 
 
 13 
 
 3060 
 
 
 3105 
 
 
 1.015 
 
 
 I 
 
 Thick struts 
 
 
 
 14B 
 
 | 3320 
 
 
 3439 
 
 
 1.036 
 
 15A 
 
 j 3395 
 
 
 3440 
 
 
 1.015 
 
 15B 
 
 3485 
 
 
 3360 
 
 
 0.964 
 
 16 
 
 | 3890 
 
 J 
 
 
 3989 
 
 
 1.025 
 
 17 
 
 4330 
 i 
 1 
 
 
 4510 
 
 
 1.042 
 
 *Insuf 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< 
 
 sdicted strength 
 
 Cylinder 
 
 Observed 
 
 Predicted 
 
 Ob 
 
 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 
 

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