A nalyt ical Volume 1
Edition JULY 15, 1929
Number 3
On the Reduction of Rubber Stress-Strain Determinations‘ W. B. Wiegand and H. A. Braendle BINNEY& SMITHeo., 41 EAST42N0 ST., NEW YORK,N. Y.
In this paper the authors seek means of eliminating the personal equation in the selection of rubber stress-strain data. It became necessary to determine the most probable or true values for tensile strength, elongation, and intermediate values. The authors, therefore, tested a complete inner tube, securing ninety-five breaks, which were then reduced by statistical methods. Frequency distributions of rubber stress-strain data do not strictly obey the law of accidental error, being negatively skewed for breaking values, and positively skewed for intermediate tensiles.
The true values, therefore, are not the arithmetic means but the modes, which are found to differ from the‘arithmetic means in every case to an extent greatly exceeding the probable error of such means. Using these true values, and dividing the ninety-five breaks into successive groups of five, the latter were weighted according to four empirical methods and the results compared with the modal values. The method of selecting three highest elongations out of five together with corresponding tensiles was found most closely to approach modal values. This method is therefore tentatively advanced as a starting point for the standardized selection of rubber breaks.
.... ...... HE excellent report of the Physical Testing Committee (1)* illustrates the trend towards increased accuracy in rubber testing; as also the recent investigation of the effect of temperature on physical properties (9). The present study is confined to the experimental data themselves, and to the application of statistical methods in their analysis. Determinable (instrumental) errors are excluded from this survey. I n most laboratories it is customary to break three, five, or possibly seven test pieces of a rubber specimen, and to take the arithmetic mean of the resulting values, first rejecting such as appear to the operator to be clearly “wild.” In order to work out a scheme for the treatment, or weighting, of such testing data, that shall be free from the vagaries of personal judgment, it is first necessary to determine, with the utmost precision possible, what are the true values of the quantities in question. Since the reliability of the times as accurate as that of average value of N tests is one test, an increase in the number of test pieces broken suggests itself as one obvious step in arriving a t the desired ‘[true,” or more strictly, most probable, values.
T
d??
Experimental
A factory-run inner tube was cut longitudinally, and the total number of available test pieces cut across the grainninety-five-were died out. These pieces were then thoroughly mixed, selected a t random, and tested on a Scott Presented before the Boston Group Meeting of the Division of Rubber Chemistry of the American Chemical Society, January 12, 1929. * Italic nrzmbers in parenthesis refer t o literature cited a t end of article.
machine. The test pieces were of dumbbell shape, I/, inch wide, 1 inch between bench marks (machine of standard pattern), speed of separation 20 inches per minute. Intermediate as well as breaking values were secured. (See Table I for breaking data.) EFFECTOF NUMBEROF TESTBON ERROR OF ARITHMETIC MEAN-consider first breaking tensiles as shown in Figure 1. I n order to trace the effect of increasing the number of tests on the arithmetic mean, progressive values for such means are shown, each dot on the graph indicating the arithmetic mean of all tests up to that point. Note the irregular trend of the arithmetic mean values, particularly in the early stages. Only after sixty breaks does the mean settle down to its final value as measured by the ninety-five breaks. The probable error of the mean EM
=
* 0.67
d--
as indicated in the graph by the heavy bands, has decreased from a value of 36 pounds for five breaks, and 17 pounds for fifteen breaks, to 12 pounds for ninety-five breaks. Thus it is seen that when ninety-five test pieces are broken in place of the customary five, and the arithmetic means determined in each case, the value for tensile strength of the inner tube would seem to be 2637 pounds with a probable error of * 12 pounds. There is, however, a fallacy in this conelusion. This arises because the data have been treated on the assumption that they obey the law of accidental error. As shown in Figure 2, this law assumes symmetrical distribution about the true, or most probable, value. For such a distribution
Vol. 1, No. 3
ANALYTICAL EDITION
114
Table I-Breaking
GAGE OF
SAMPLE
Data
TENSILE TENSILE TENSILE ELoN G A AT
300%
AT
500%
AT
SAMPLE ELONG. ELONG. BREAK Inch Lbs./sq. in. Lbs./sq. in Lbs./sq. in.
TION AT
BREAK
GAGE OF
TENSILE TENSILE
AT300%
AT
500%
TENSILE AT
ELONG. BREAK SAMPLB SAMPLE ELONG. Lbs./sq. in. Lbs./sq. in. Lbs./sq. in. Inch
ELONGA TION AT BREAK
51 52 53 54 55
0.069 0.070 0.068 0.069 0.068
200 200 205 205 205
520 515 530 520 530
2585 2750 2800 2755 2520
Per cent 760 770 780 770 750
780 780 770 760 770
56 57 58 59 60
0.074 0.081 0 074 0.073 0.069
215 220 215 220 200
540 545 540 550 490
2675 2345 2455 2600 2375
770 750 750 760 760
2715 2685 2715 2745 2685
770 770 770 780 780
61 62 63 64 65
0.070 0.085 0.078 0.083 0.071
200 210 205 220 225
515 495 485 505 505
2660 2300 2425 2650 2480
770 750 770 770 760
590 535 500 520 550
2915 2670 2290 2430 2400
780 770 750 750 750
66 67 68 69 70
0,082 0.074 0.071 0.073 0.069
195 190 195 190 205
510 510 505 490 490
2610 2455 2515 2635 2755
770 760 770 780 770
220 215 215 230 220
520 530 510 540 520
2460 2685 2695 2575 2735
760 780 780 760 780
71 72 73 74 75
0.071 0.069 0.067 0.070 0.070
195 230 210 200 200
505 520 540 515 515
2755 2870 2755 2660 2685
780 780 780 760 780
0.072 0.071 0.076 0.068 0.070
220 225 210 205 200
525 505 525 530 515
2740 2815 2525 2910 2480
780 780 770 780 760
76 77 78 79 80
0.069 0.068 0.068 0.072 0,068
200 205 205 195 205
490 530 530 500 530
1880 2745 2515 2525 2720
720 780 750 760 760
31 32 33 34 35
0.069 0.070 0.071 0.072 0.069
230 200 195 195 200
520 515 535 500 520
2925 2745 2750 2635 2675
790 780 780 780 770
81 82 83 84 85
0.070 0.067 0.068 0.072 0.067
200 240 205 220 240
485 670 590 585 630
2520 2780 2570 2220 2840
760 760 7 50 720 770
36 37 38 39 40
0.069 0.069 0.072 0.070 0.070
230 230 220 200 230
520 550 555 515 540
2615 2895 2720 2630 2825
770
780 760 770 780
86 87 88 89 90
0.074 0.065 0.066 0.070 0.071
215 245 240 200 225
540 615 605 540 565
2800 2530 2910 2745 2420
770 740 770 760 740
41 42 43 44 45
0.071 0.073 0.070 0.068 0.071
195 190 200 205 195
505 465 515 550 505
2815 2600 2600 2600 2500
790 790 770 770 770
91 92 93 94 95
0.065 0.066 0.069 0,073 0.070
245 245 205 220 230
650 580 580 550 515
2865 2825 2550 2700 2630
750 770 750 770 760
46 47 48 49 50
0.072 0.071 0.068 0.069 0.070
195 225 205 200 200
500 505 530 490 515
2665 2560 2915 2695 2485
780 770 780 780 750
1 2 3 4 5
0.069 0.067 0.072 0.068 0.074
200 240 195 205 215
550 535 470 530 485
2670 2480 2665 2740 2480
Per cent 770 740 790 780 770
6 7 8 9 10
0.069 0.069 0,067 0.070 0,067
200 200 210 200 240
520 490 570 515 535
2785 2695 2755 2480 2685
11 12 13 14 15
0.068 0.067 0.068 0,070 0.072
205 210 205 230 220
530 535 530 515 555
16 17 18 19 20
0.068 0.071 0.072 0.073 0,073
235 225 220 220 220
21 22 23 24 25
0.073 0.075 0.074 0.070 0.073
26 27 28 29 30
I
The curve of Figure 4 is that generated by the histogram of Figure 3. The arithmetic mean is no longer the most frequently occurring and therefore the most probable or true value. This quantity is defined as the ‘(mode” and the arithmetic mean constitutes the true value, and the is seen to show a distinctly higher value (8). greater the number of tests the more reliable and accurate I n Figure 5 the “true” or modal value for tensile is comis this form of “average.” The data of the experiment, pared with the values for the arithmetic mean (Figure 1). however, do not follow this law. I n no case does the arithmetic mean reach the mode, all the DETERMINATION OF TRUE(MODAL) BREAKING TENSILE- values being lower, and in I n Figure 3 are shown the number of occurrences of each no case does the probable tensile value in the ninety-five tests. It will a t once be error of the arithmetic mean noted that this frequency curve does not follow the law of overtake this d e v i a t i o n . accidental error. Instead of being symmetrical, the curve Increasing the number of is lopsided or “skewed” (5, 4 ) towards the left or the low t e s t pieces, even up to values; there is a preponderance of low breaks, due pre- ninety-five, although reducsumably to flaws in the rubber. ing the probable error of the arithmetic mean, does not simultaneously reduce its Figure DistributionFrequency 2675-E FOR IS BREAKS W . inaccuracy. I n fact, after y ke-hzzz ninety-five tests, the arithmetic mean falls short of the A mode by more than five times its own probable error. DETERMINATION OF TRUE INTERMEDIATE TENSILESE FOR Examination of Figure 6 shows that, as in the case of breaking tensile, the frequency curve is asymmetrical, thus again precluding the use of the arithmetic mean. The “skewness” is here towards the right, or positive, this being clearly due to personal equation in the form of lag in the recording of loads. The “skewness” a t 500 per cent is less than a t 300 per cent because the bench marks are here separating more slowly. Because the “skewness” is positive, the Figure 1-Number of Tests US. Arithmetic Mean a n d Probable Error arithmetic means are now higher than the true or modal of Arithmetic Mean 3
7
I
5
INDUSTRIAL AND ENGINEERING CHEMISTRY
July 15, 1929
115
Table 11-Statistical values, and to a degree again exceeding their own probable ARITHCOEFF. errors. QUANTITY METIC STANDARD OB MEASURED MEAN DEVIA- VARIADETERMINATIONOF TRUE BREAKINGELONGATIONTensile at: (95) TION TION Figure 7 shows that the curve is here once more “skewed” 100 elong. 85 3.9 4.63 2 0 0 70 elong. 136 12.5 9.23 towards the left, presumably because of flaws in the test 300%elong. 212 14.6 6.92 pieces. This “skewness,” however, is much less evident 400 0 elong. 326 17.4 5.35 5 O O g elong. 528 3 2 . 3 6 .11 than in the case of tensile, partly because the bench marks 600% elong. 1014 700% elong. 1854 Break 2637 Elong.atbreak767
No.o f Tests
15-
a
lo
-
65.1 6.42 112.4 6.07 169.6 6.44 17.4 2.26 mean mode * Skewness = standard deviation
-
Summary
SKEWNESS~
MEAN MEDIANMODE
$0.58
212
210
204
$0.084
528
528
525
-0.39 -0.24
2637 767
2670 770
2703 771
Applkation t o Laboratory Test Procedure
5-
Figure 3-Frequency
Distribution, Tensile at Break
are separating very slowly, thus permitting more accurate observation, but chiefly because the stress is increasing much more rapidly than the elongation, and any shortening of the curve due to defects in the rubber has consequently less influence on the elongation than on the tensile. As with breaking tensiles, the arithmetic mean is lower than the mode, the difference, although small in absolute terms, being still many times the probable error (of the arithmetic mean).:
Figure 4-Frequency
The discuEsion thus far has been confined to the influence of increasing; the number of breaks upon the accuracy of the final result. This has brought out the necessity for recognizing and evaluating the “skewed” nature of rubber stress-strain tests. It is, of course, impracticable in the course of daily control, and even of research testing, to carry out suffrcienlt breaks or intermediate readings to generate and evaluate a frequency curve in each case. To do this would require upwards of sixty tests. The problem is one of weighting a small number, say five tests, so as to approximate the truth. It is common practice in laboratories to reject what are loosely defmed as “obviously wild” results, but there is a disagreement as to what constitutes a “wild” result. A rule favored by some is that only such values shall be rejected as show a deviation four times greater than the average deviation of the remainder. This rule fails to eliminate admittedly “wild” results. More specifically, of the ninetyfive tests described in the present study, this rule would have eliminated only one, thus leaving the “skewness” of the frequency curves substantislly uncorrected. A determination of the modal value would therefore still be necessary. Some other means of weighting is clearly required in oyder that “mean” values may yield more nearly true results. Several empirical methods of weighting have therefore been applied. The ninety-five tests on the inner tube mentioned above have been divided into nineteen successive groups of five, thus, in effect, constituting nineteen “laboratories” working
27v
Distribution, Tensile a t
MODE-TRUE
Break
Reliability of Various Determinations Figure 8 shows in comparative form the frequency distribution for four critical stress-strain values. Visual inspection will indicate that the degree of concentration around the most probable or modal value is, roughly speaking, similar for all the tensile values, but that the concentration for elongation at break is markedly greater. In other words, the number of “wild” results is much less for elongation than for tensile a t rupture or modulus values. (This statement of course applies entirely to bar test-piece technic and not to ring testing.) The exact measure of the reliability is known as the “coefficient of variation” (6). v = Q
M
x
100 per
where B = coefficient of variation, mean, and M = mean
u =
cent standard deviation from
It is seen from the coefficients of variation given in Table I1 that the reliability of the elongation a t break is approximately three times as great as that of the other values.
TENSILE. 2703
2676-
r L m *
-
Arithmetic
v
Mean 2637
A
I
Ln
r
y2625-
x
-
I
a IY
2600-
-e
2575
!W
5tirr4
-1 IO
20
Figure 5--Mode
40 60 N U M B E R or E R E A L S
30 US.
60
IO
80
90
Arithmetic Mean a n d Probable Error
under exactlg comparable conditions and on the same material. (See Tables I and 111.) It was thought that this procedure was the only way in which irrelevant variables could be eliminated so that h a 1 results should be amenable to statistical treatment. Treatment has been confined to tensile strength and
116
ANALYTICAL EDITIOM
Vol. 1, No. 3
can give effect to his rejections immediately the breaks are finished and without first working out all of the tensile strengths-in fact, without any preliminary arithmetic. It is suggested that other rubber laboratories take enough time to select, say two or three “type” compounds, and vulcanize samples of sufficient size to supply upwards of sixty test pieces; that these be broken under strictly comparable conditions of t e m p e r a t u r e , h u m i d i t y , e t c . ; that the frequency curves be plotted for intermediate tend e s , tensile a t break, and elongation a t break; that modal values be ascertained, and then that the tests be split up in some m 220 240 such manner as has been here outlined, Tensile at 500% E. Tensile at 300% E. and a scheme of weighting evolved which Figure 6-Frequency Distribution a t Intermediate Elongations shall, for the type compounds in question elongation at break, these being the most popular tests, and for the specific laboratory, yield the closest approxiand also because the intermediate values are free from mation to the truth. It is possible, and even probable, negative “skewness” due to flaws, and consequently greater that no two laboratories and that no two compounds will symmetry of distribution is to be obtained, not by weighting, of Methods of Weighting but by refinement of technic-e. g., reduction in the per- LABORA- Table 111-Comparison MErHoD METHon METHOD 4 sonal lag. As a means of evaluating the various weighting TORY Tensile Tensile Tensile Elong. Tensile Elong. schemes, the corresponding tensiles obtained from the Lbs./sq. in. Lbs./sq. in. Lbs./sq. in. Per cent Lbs./sq. in. Per cent 2607 2705 2705 775 2692 780 nineteen “laboratories” were averaged, and deviations 2770 2680 2740 780 2745 777 3 2709 2730 2715 780 2715 777 of the “laboratories” from the true or modal value noted. 2541 2792 2793 775 2672 760 Method I-Mean of All Five Tensiles, Equal Weights. The average for the nineteen “laboratories” is, of course, 2637 pounds-the arithmetic mean of the ninety-five tests-and as such lies well below the mode (2703). Of the nineteen “laboratories,” six were more than 100 pounds off the modal value, two being off by more than 200. The elongation by this method is 767 per cent against a mode of 771 per cent. Method 2-Mean of Best Two Tensiles. The average for the nineteen ”laboratories” here gives a value of 2769 which is well above the modal value. In this case eight “laboratories” were off more than 100 pounds, the elongation by this method being 775 per cent. Method 3-Mean of Tensiles Correspondingto Best Two Elongations. The average for the nineteen “laboratories” gave 2757 pounds; still high: seven “laboratories” off more than 100 pounds; elongation 776 per cent. Method &Mean of Tensiles Corresponding to Best Three Elongations. The average for the nineteen “laboratories” was 2717 pounds as compared to the mode of 2703, a difference of only 14 poun’ds. In this case only five “laboratories” were off more than 100. Elongation by this method stands at 772 per cent against a mode of 771 per cent.
Inspection of Table 1V clearly brings out the superiority of Method 4. It is necessary to emphasize that the above weightings are purely empirical. It is also necessary to emphasize that the scheme of weighting found best for the data of the present paper might not be applicable to another set of tests using different equipment and employing compounds of different type. With these limitations in mind, it is suggested that the selection of the three highest elongations out of five breaks, together with the corresponding tensiles, be given consideration in the development of some standard method of weighting rubber stress-strain data. Irrespective of the question of approximation to modal or true values, the scheme of the three best elongations has the important practical advantage, a t -least with the ordinary bar test-piece machine, that the operator
5
2630
2715
2710
780
2705
780
6
2694 2746 2737 2623 2664
2863 2838 2860 2708 2805
2863 2835 2860 2708 2805
780 785 780 790 780
2822 2807 2783 2672 2758
780 783 777 783 780
11 12 13
2682 2490 2503 2594 2745
2776 2638 2655 2695 2813
2778 2638 2655 2695 2818
775 765 770 775 780
2768 2550 2578 2666 2793
773 763 770 777 780
i;18
2477 2586 2681 2714
2733 2810 2855 2845
2635 2810 2855 2763
770 765 770 770
2663 2713 2818 8718
767 733 767 767
‘98
10
:$
19
TENSILE a t 3 0 0 % E .
720
760-
T E N S I L E at 5 0 0 % E
800
Figure 7-Frequency Distribution, Elongation a t Break
%E.at B R E A K . of Frequency Distribution8
TENSILE atBREAK.
Figure 8-Comparison
INDUSTRIAL A N D ENGINEERING CHEMISTRY
July 15, 1929
yield the desired results from the same scheme of weighting. on the other hand, there is a possibility that the scheme of the three best elongations out of five, here described, or some variation of this scheme, will be close enough to the truth to provide the basis for a standardized treatment, the urgent need for which is self-evident. T a b l e IV-Variation RELATION80 MOOS Above Below 1 5 0 pound@or more *lo0 pounds or more *I50 pounds or more *200 pounds or mOce
from Mode (Tensile a t Break) by Different Weighting Methods Method 1 5
14 9
6
4 2
NUMBER OF LABORATORIES Method B Method 3 Method 4 16
3
12 8 3 0
15 2 12
13
7
5
3
2 0
0
out upon ‘%we” rubber specimens using Schopper rings, in the course of which the author established the approximate validity of the law of accidental error as applied to 731 breaks. Those interested in the theory of probabilities will find in Fric’s paper an elegant alternative treatment of the breaking tensile data in the form of Ogive curves. By means of these, he clearly demonstrates the abnormal frequency of low values which he ascribes to flaws in the rubber. It is thus seen that the preliminary part of the present paper stands as an independent coniirmation of Fric’s observations. The present authors were, of course, primarily interested less in the ultimate reduction to most probable values than in the development of methods for weighting a small number of tests.
6 9
Addendum Since writing the above paper, the authors have noticed a study, “Essais mkcaniques du caoutchouc et probabilith,” in Chimie et kdustrie, April, 1928, by M. R. Fric. It is a pleasure to acknowledge this prior publication of experiments carried
117
Literature Cited (1) Physical Testing Committee, A. C. S. Division of Rubber Chemistry, IND.END. CHEM., 20, 1245 (1928). (2) Somerville and Cope, llzdia Rubber Worid, 79, No. 2, 64 (1928). (3) Sutcliff, “Elementary Statistical Methods,” pp. 157 and 158, McGrawHill Book Co. (4) Sutcliff, Ibid., Chap. XVI. (5) Yule, “Introduction t o Theory of Statistics,” Chap. V I I I , C. Griffin 8h co. (6) Yule, Ibid., p. 149.
Volumetric Estimation of Sulfur in Crude Petroleum‘,’ A n Adaptation of the Nikaido Method Gladys Woodwards CHEMICAL LABOR.4TORY OF THE COLLEGE OF LIBERALARTS,NORTHW~STERN UNIVSRSITY, EVANST6N, ~ L L .
LTHOUGH the lamp method ( I ) * for the determination of sulfur in petroleum light distillates is fairly rapid, there has been no rapid method, volumetric or gravimetric, for this determination in crude petroleum oils. The standard method (2) of determination as barium sulfate includes two time-consuming filtrations and weighings. A volumetric method which would avoid these would be highly desirable. Such was especially desired in the present work, where it was necessary to follow closely the change in sulfur content of the crude petroleum with each treatment. I n 1902 Nikaido (a) suggested that sulfate might be determined by titration with lead nitrate using potassium iodide as indicator. I n 50-70 per cent alcohol solution Iead iodide does not form permanently until lead sulfate is quantitatively precipitated. Thus, when the yellow color of lead iodide is permanent, the end point of the reaction is reached. Since Nikaido first suggested this method, it seems never to have been used; perhaps because the reaction must be carried out in 50-70 per cent alcohol solution, and perhaps on account of other limitations which he also mentions. However, the method may be easily applied to determine the amount of sulfuric acid present in the washings from the combustion of a sample of oil in an oxygen bomb. When the potassium iodide indicator is added to this solution, a small amount of iodine is liberated coloring the solution yellow.
A
Received March 29, 1929. This paper contains results of an investigation carried out as part of project No. 17 of the American Petroleurn Institute research program. Financial assistance in this work has been received from a research fund donated by the Universal Oil Products Company. This fund is being administered by the American Petroleum Institute with the cooperation of the Central Petroleum Committee of the National Research Council. Frank C. Whitmore is director of project No. 17. a Research Fellow, American Petroleum Institute. * Italic numbers in parenthesis refer to literature cited a t end of article. 1
9
This is caused by the ferric salts present in the washings. The solution cannot be titrated when it is colored, as the end point is obscured. The color is easily removed by boiling with a trace of aluminum powder. After concentration to a volume of about 50 cc., alcohol is added to produce the 50-70 per Gent alcohol solution necessary, The end point is very definite, but it may vary a few drops according to the individual. It becomes more accurate as the eye becomes accustomed to the intensity of the yellow, color desired. The end point is not easy to distinguish, however, in yellow light, and therefore the titration must be carried out in daylight or in a room lighted by blue bulbs. The method is accurate to two drops of lead nitrate solution. This means that, when 0.20 gram of sulfuric acid is present in the solution to be titrated, the error in the end point does not exceed 0.4 per cent; when 0.03 gram is present, the error does not exceed 3.0 per cent. There is, therefore, a smaller percentage of error as the amount of sulfuric acid increases. But there is a limit to this, for as the amount becomes greater another error enters in. When a larger amount of lead nitrate solution is required to react with this larger amount of sulfuric acid, the alcohol solution becomes more dilute and the value of the lead nitrate solution changes slightly. However, for as much as 0.30 gram of sulfuric acid the titration is sufficiently accurate. Table I contains results of analyses on oil samples of varying composition using both the present method and the standard method ( 2 ) . It will be seen that the greatest error occurs in cases where a very small sample of low sulfur content was used. With the crude oils, however, where the amount of sulfuric acid formed was from 0.1 to 0.25 gram, the deviation between the methods did not exceed 1.8 per cent, which is no greater than the error the barium sulfate determination allows between check determinations.
