Simple Flexibility Test DIETRICH G . STECHERT, The Gates Rubber Co., Denuer, Colo. A simple test for flexural rigidity is proposed. Flexural rigidity is not an arbitrary quantity but a standard engineering measure of flexibility. Because of the extreme simplicity of the equipment, technique, and calculation of flexural rigidity, the test may be of value wherever a quantitative measurement of the flexibility of rubber products is needed. An important advantage of this test is that these products may be bent to a degree en-
F"L
mme time there has been a need for a simple but meaningful flexibility test for rubber and plastic products. Tests proposed include simple beam and cantilever-type tests and various modifications. Generdlv these are considered to be small deflection tests. The flexibility of rubber and plastic products depends on the curvature to which they are bent: If information on the flexibility of belting or hose, for example, is needed for design or develapment purposes, it should be measured a t the desired degree of curvature. Force F
I t
.0
countered in service. The test is sensitive to small variations in flexibility. It may be of value in engineering design and development work, and its simplicity would suit it particularly well for control work. Continuous lengths of hose or belting 'can be tested without being cut. Tests may be E o n ducted at extreme temperatures using an insulated box equipped with hand holes, and used in the laboratory for measuring stock modulus.
Figures 2 and 3 show the proposed test used on a belt and on hose. Continuous lengths of hose or belting can be tested. As shown in Figure 3, the upper free end of the hose must be held parallel to the upper plme without coming in contact with it. The degree of bend of the test piece must not be M severe as to alter the shape of the cross section appreoiably. Hose, for example, must not be bent to the point where it collaptpses. In general, rubber products should be bent to the extent encountered in service. Other types of linkages c ~ be n devised for the test equipment. A linkage designed to allow the two parallel surfaces always to move in a direction perpendicular to those surfaces may be considorel desirable.
I
THEORY OF TEST
The test piece is bent into the shape shown in Figure 4 under the colinear forces F, F , until the free ends are parallel. To determino the relationship of flexural rigidity to F and h we start with the well-known equation for the bending moment of a heam:
._
0
v ! / / / / / / / / / / / / / / / / / / / / / / Figure 1. Pantograph Flexibility Tester
dJ = E I / r
(1)
.. whore 81 = flexural rigidity of the beam and r = radius oI curva"
t,ure of the centroids1 suilace of the beam. The proposed test measures a distance and a force using a aimple, inexpensive piece of equipment. A quick calculation yields the flexural rigidity, a standard measure of flexibility in the study of the strength of m&n& Although the equipment is simple, the test is sensitive to small variations in flexibility.
.. t
'..
DESCRIFTION OF TEST
The test piece, originally straight, is bent into a U-shape in B piece of equipment consisting of two surfaces that are always maintained parallel to each other through a pantograph linkage, as shown in Figure 1. The force, F , and the distance, (I, are measured. The distance between centroidal surfaces, b, is calculated, knowing the geometrical shape of the cross section of the test piece. The flexural rigidity, EI, which is the product of modulus of elasticity and moment of inertia of the cross section, can be computed from the formula:
.e 2.
Test of Flexihility of V-Belt
E l = 0.34830 Fh* If F is measured in pounds and b in inches, the units of EI are (pounds) (inches)*. In this test the beam is bent to a radius of ourvature of approximately one half b. Actually, the minimum radius is 0.41731h a t the mid-point of the test piece. If the test is conducted in a. vertical plme and weights w e used for force F, the weight of the test piece may introduce a slight error in EI. This error would be least for the most Severe bend, or the smallest value of 6.
Figure 3.
1730
Test of Flexibility of Hose
.
:r.
V O L U M E 25, NO. 1 1 , N O V E M B E R 1 9 5 3
1731
Xow a t any point P (z, y ) on the centroic!al surface T = -ds/dO where tan e is the slope a t P and s is the distance from 0 along the surface to P . The minus sign is necessary here, as r is always positive, and &/de is always negative (as s increases, 0 decreases). The moment M a t P of forces to the left of P is F y . Then Equation l becomes:
S o w since F = 2r,, EZ can be solved for in terms of b and F :
EI
=
1 2(1.19814)2Fb2
= 0.34830 Fb2
This is the relationship we have been seeking. Certain other quantities may be of interest, such as ro. I),, can be determined as follows:
Differentiating with respect to s and then integrating:
(5) (la, sa,
md
dy = s i n e d s
The negative root is the csorrect one here, as dB - is always negads 2 F 112 tive, and cos1/*e is always positive.
(n)
Then b=2x,
d
Analysis of Test
Figure 4.
Now to find zawe must use the relationship: ds =
COS
The following relationship is known:
e ds
The integral on the right can be obtained by making use of the following relationship (see any good advanced calculus text or table or integrals):
where m and n = - _1. 2'
>
-1 and B is the \)eta function. If m = 1 and
77,
where m has any value m = 1- * 2'
> - 1 and r is the gamma function.
JVhen = 2
Then
But according to Equation 5 : 2
=
s
(*)1'2-
= 1.19814
0.9 1906 0.90640
Then yo =
Then Equation 4 becomes:
b 1.19814
~
Or =
0.83463 b
ANALYTICAL CHEMISTRY
1732 The distance along the beam from 0 to A , sa, can be found as follolvs:
0.90640 0.91906 2.62206
=
Then
b = 2'62206 2(1.19814) =
( 7)
1.09422 b
ro can be found from Equation 3:
At 0, 9
=
i
r 1 5, and - = 0.
-4s we move along the centroidal surface
from 0 to A , r decreases until i t is a minimum a t A :
Thus the flexural rigidity of steel, which, unlike rubber, has a linear stress-strain curve, can be obtained within fairly narrow limits over a range of test conditions. An upper bound for the error in EI due to the weight of the test piece may be found as follows. The greatest possible error that could be introduced into the value for F is the weight of the spring steel test piece itself, which was 0.041 pound. Then the error in F , as well as in EI, in this experiment was less than 0.041/1.164 = 3.5% for the greatest value of b, and less than 0.041/3.369 = 1.2% for the smallest value of b. The flexural rigidity of the spring steel test piece was also measured in a cantilever-type test. A series of concentrated loads, P, up to a maximum of 0.077 pound, was applied a t B distance L = 6 inches from the supported end. The maximum deflection, d, at the 0.077-pound load, was 0.70 inch. Values of EI were computed according to the formula for a cantilever beam: EI = (PLs)/(3d). The average for four loadings was found to be 7.936 pounds sq. inch. The difference in average values of flexural rigidity obtained by the pantograph and cantilever 7.936 - 7.885 methods is = 0.6%. 7.885 The proposed test may also be used for the determjnation of modulus of elasticity. Thus for the spring steel the moment of inertia, I , of the cross section is:
Since the flexural rigidity, EI, was found to be 7.885 pounds sq. inch, it follows that the modulus of elasticity, E, is:
E = -EI I
-
b 2( 1.19814)
=-
= 29.89
= 0.41731b
Thus the shape of the centroidal surface is such that in Figure 4 r.
OAB
=
7.885 0.26382 (
0.41731b
(106) pounds sq. inch
An investigation of the flexibility of tubing, of l/c-inch inside and '/pinch outside diameter, made from three types of stock, yielded the following results a t 76" F.
yo = 0.83463b (along beam) = 2.18844b
These dimensions were verified experimentally using the flat spring steel test piece mentioned below.
Type of Stock
F , Lb.
b , Inches
0.570
2.40 2.40 2.38 2.35 2.39
EI, Average +, Limits of Uncertainty (n = 5 , P = 0.95), Sq. Inch Lb. Sq. Inch
Et,Lb.
APPLICATION OF TEST
A test piece of flat spring steel, 0 5 inch wide, 0.0185 inch thick, and about 16 inches long, was tested. The tare weight of the upper plane and connecting arms of the equipment was 1.164 pounds. An analysis of the forces involved shows that the tare weight is equal to the weight of the upper plane plus one half of the sum of the weights of the connecting arms. The diktances, a, between the planes were measured for various weights. ?;ow b is the distance between centroidal surfaces. For the rectangular cross section of the spring steel test piece the centroid lies a t the middle of the cross section. Then in this case it follows that b = a - t, where t is the thickness of the test piece. Weights were applied in 0,441-pound (200-gram) increments. LIeasurements and computed quantities were as follows: Weight, Lb 0 0.441 0.882 1.323 1.764 2.205
F , Lb.
1.164 1.605 2.046 2.487 2.928 3.369
a,
Inches 4.41 3.78 3.36 3.04 2.81 2.60
b , Inches
4.39 3.76 3.34 3.02 2.79 2.58
EI
Lb. Sq.'Inch 7.81 7.90 7.95 7.90 7.94 7.81
The average EI value is 7.885 pounds sq. inch. The dif7.95 - 7.81 ference in largest and smallest values is = 1.8%. 7.81
GR-S
1.14 1.14 1.12 1.10 1.13
1.126 i 0.021
Different values for F were used for the three stocks in order to obtain approximately the same degree of bend-that is, values of h. Values in the last column were computed according to the method recommended by the American Society for Testing Materials ( 2 ) . The limits of uncertainty of average values of EI reflect test error and variations among test pieces. The magnitude of the test error depends largely on the extent to which friction can be minimized in the bearings of the pantograph linkage. For the piece of test equipment used in this investigation, the error due to friction was on the order of zl11~~ when the connecting arms and the base were a t an angle of about 30" or less. Rubber and rubberlike materials have time-dependent elastic, or viscoelastic, properties. When rubber or plastic products are
V O L U M E 25, NO. 11, N O V E M B E R 1 9 5 3 tested, it is well always to obtain readings a t a given time interval, sag 30 seconds, after load application. This was done in the above evaluation of tubing. The viscoelastic nature of rubbers and plastics is responsible for the nonlinearity of the stress-strain curve. Consequently in a test such as was conducted on the steel spring, where weights were applied in increments, a t a certain rate, the flexural rigidity of rubberlihe products will not remain constant with increasing degree of bend, but will vary. The amount of variation depends on the material. This nould be true in any kind of bend test, however. iz completely satisfactory solution could be obtained by taking the viscoelastic spectrum of the material into consideration. The
1733 pertinent theoretical development, taking time or rate into consideration, for cantilever bending has been worked out by hlfrey ( 1 ) . The theoretical development for the proposed test method is beyond the scope of this paper. LITERATURE CITED
(1) ;ilfrey, T., “11echanical Behavior of High Polymers,” pp. 215-18, Xew York, Interscience Publishers, 1948. (2) Am. SOC. Testing Materials, “A.S.T.M. Manual on Quality Control of Materials,” Part 2, 1951. R E C E I V Efor D review March 19, 1953. Accepted August 25, 1953. Presented before the Division of Rubber Chemistry a t the 123rd Meeting of the Los Bngeles, Calif. AXERICAV CHEJIICAL SOCIETY,
Universal Anticipation of End-Point System for Automatic Titrations W. N. CARSON, J R . General Electric Co., Hanford Atomic Products Operation, Richland, Wash. This work was done to find a means for preventing overtitration in automatic titrations. Overtitration arises principally because of a time lag in the response of the indicator system. The proposed system circumvents this time lag and can be used with almost any indicator system. It consists of withdrawing part of the sample, titrating the remainder at a rapid rate until an end point is reached, then returning the withdrawn portion and titrating at a slow rate to the final end point. Variation in the amount of sample withdrawn, and in the rate of the slow titration, permits adaptation to a specific titration. Such an anticipation system has been designed for an automatic titrator used to titrate microsamples of oxidizing agents. The results of titration of dichromate samples show a standard deviation of less than 1%. The system extends the applicability of automatic titrators by making i t possible to use sluggishly responding indicator systems. It also permits faster titrations without danger of overtitration.
0
S E of the major problems of automatic titrations is stopping
the addition of titrant in time to prevent overshooting the end point. This generally calls for some means of anticipating the end point, either by the behavior of the indicator system itself or by some external system. This problem is discussed in a previous paper (1). The need for this anticipation of end point arises because the indicator response is never instantaneous. The response is affected by the rate of stirring of the solution, the rate of addition of the reagent, the type of indicator, and the physical form of the titration vessel with respect to the location of indicator electrodes, reagent addition point, nonuniform mixing of the solution and sluggish, but eventually stoichiometric, reactions. The more sluggish the response of the indicator system, the more necessary is the need of anticipation to obtain accurate results. In view of the numerous indicator systems that are useful in automatic titrations, a desirable anticipation system should be independent of the indicator system, and should also be independent of the speed of indicator response in the sense that either fast or slow indicator systems can be used. Muller (3) has discussed such an anticipation system in his column on instrumenta-
tion, but up to this time the writer is unaware of any reported titrator that has actually used the system. The device described in this paper operates essentially in the manner proposed by 3Iuller. The svstem consists of withdrawing part of the sample before titration, titrating the remainder at a rapid rate until an end point is obtained, returning the withdrawn portion, and titrating the remaining sample by the addition of small increments of titrant. The first end point is overshot by a small excess of ti-‘ trant; the amount of solution withdrawn must contain sufficient sample to react with this excess. The remaining sample is titrated m-ith small increments of titrant; sufficient time is allowed between increments to allow the indicator system to come t o equilibrium. The titration can then proceed to an end point that is not overshot by more than a nominal excess of titrant. Variation in the amount of withdrawn sample, increment size, and time b e h e e n addition of increments permits adaptation t o any indicator system, no matter how sluggish its behavior. An automatic titrator using this anticipation system has been tested for the titration of oxidizing agents with electrolytically generated ferrous ion according to the method of Cooke and Furman ( 2 ) . The basic design of the titrator has been given ( 1 ) . The controller circuit is given in Figure 1. At the start of a titration, all of the relays are in the open position shown in the figure, the trigger relay (not shown) operates when the indicator potential is above the end-point potential (excess oxidant present). Closure of the “operate” switch t o the titration position starts the titration The circuit (via K-5) to the anticipation system solenoid is energized, as is the circuit (via K-5 and the trigger relay) to the output rela! s, K-2, K-3, which control the addition of titration current and the stopwatch, Energizing the solenoid withdran s a portion of the sample solution from the titration zone (see Figure 2). The titration continues until the remaining portion of the sample has been titrated t o the stage where the indicator potential decreases below the end-point potential, which opens the trigger relay contacts. The output relays, K-2, K-3, now open and the addition of titration current is stopped. Opening K-3 closes a duration of end-point timing circuit (the 2050 thyratron and the R-C network C-1, R-l), which is set for 1 to 2 seconds. When an end point persists for this time, the thvratron fires, and relay K-6 closes, 15 hich in turn closes K-5, which locks in electrically. Closure of K-5 activates the increment addition circuit (K-7 and the 12AU7 network), and opens the solenoid circuit, permitting the M ithheld portion of the sample to return to the titration zone Part of this portion of the sample
