Determination of the stress concentration factor of a stepped shaft stressed in torsion by means of precision strain gages

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Material Information

Title:
Determination of the stress concentration factor of a stepped shaft stressed in torsion by means of precision strain gages
Series Title:
TM
Physical Description:
12 p. : ill ; 27 cm.
Language:
English
Creator:
Weigand, A
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Torsion   ( lcsh )
Strains and stresses -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Bibliography:
Includes bibliographic references (p. 5).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by A. Weigand.
General Note:
"Report No. NACA TM 1179."
General Note:
"Report date September 1947."
General Note:
"Translation of "Ermittlung der Formziffer der auf Verdrehung beanspruchten abgesetzten Welle mit Hilfe von Feindehnungsmessungen." Zentrale für wissenschaftliches Berichtswesen der Luftfahrtforschung des General-luftzeugmeisters, (ZWB) Berlin-Aldershof, Luftfahrt-Forschung Band 20, Lieferung 7, p. 217-219, München, July 20, 1943."

Record Information

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003760230
oclc - 85852072
System ID:
AA00009341:00001


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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHITICAL' METMORATDUMt 1179


DETERMINATION OF THE STRESS CONCENTRATION FACTOR

OF A STEPPED SHAFT STRESSED. IN TORSION BY

MEANS OF PRECISION STPAIN GAGES*

By A. Weigand


The stress distribution.in stepped shafts stressed in torsion is
determined by means of the electric precision strain gage by Lehr
and Granacher [5]; the stress concentration factor ak as a function
.of and Q for 0.5 < < 0.9 and 0.1 < Qp < 0.25 is ascertained
D f D d
from the measurements. It is shown that the test values always are
slightly lower than the values resulting from an approximate formula
by Sonntag [2].

Outline: I. Test Setup and Measuring Procedure

II. Evaluation of the Measurements and Results

III. Summary

IV. References

The stress distribution in stepped shafts stressed in torsion
.was.first determined by F. A. Willers [1] by approximate integration
ofthe 'differential equation of the stress function. R. Sonntag [2]
gay.e.:an approximate formula for the stress function and the maximum
strees'.for-the-case D d 2 p. (See fig. 1.) L. S. Jacobsen,
A. Thum' and W. Bautz ([31 and the bibliography quoted there)
ascertained' the concentration factor by means of an electric model.
The-values found in this manner do not always agree with those calcu-
lated according to -Sonntag; a discussion between these scientists
Sensed [4].

S"Ermittlung der Formziffer der auf Verdrehung beanspruchten
Sabgesetzten Welle mit Hilfe von Fein'.ehnungsmessungen." Zentrale fur
wissenschaftliches BerichtrJesen der Luftfahrtforschung des General-
luftzeugmeisters, (ZWB) Berlin-Adlerahof, Luftfahrt-Forschung .Band
20" Lieferung 7, p. 217-219, Minchen, July 20, 1943.






NACA TM No. 1179


In order to definitely clear up this problem the stress distri-
bution of steel shafts was determined by means of the precision
strain gage developed by E. Lehr and H. Granacher [5].


I. TEST SETUP AND PROCEDURE


Figure 2 shows the apparatus which was used for twisting of the
test shaft. Its modus operandi is explained in figure 3. The test
shaft is obviously stressed by a pure torque. This fact was confirmed
by a brittle lacquer test; the principal stress curves had an
inclination of 450 toward the generatrices of the cylindric shaft.

The test points 1 to 11 for the strain measurements were
arranged according to figure 5 on three generatrices I, II, and III
which were 900 to each other. The two principal strains cl and E2,
which must be equal and opposite for pure torsion, were measured.
Actually,the values deviate from the average usually by 3 to 4 percent;
the reason is directional inaccuracy of the test section which was
marked with a prick punch (1.3 mm) and a slight bending stress of the
shaft caused by the fact that the twisting forces designated by a
in figure 3 are not exactly equal. This error is essentially elimi-
nated by measuring on three generatrices, figure 5. From the three
test values which lie on the same circle the mean value is taken.

The strains were measured by means of the precision strain
gage of 1.3-millimeter gage length developed by E. Lehr and
H. Granacher; its modus operandi is described in detail in [ 5.
Before starting and at the end of each test series,which lasted three
to four days per shaft,the instrument was calibrated on a tensile bar
of known modulus of elasticity; the values found never differed by
more than 1 to 2 percent and lay within the errors- of measurement.
It is true, the filament current for the small bulb of the strain
gage must be kept exactly constant; at an intensity of the filament
current of 120 mA fluctuations of 0.1 mA are already troublesome.
Loose contacts, in particular, must be carefully avoided since
irregular fluctuations would.result in the multiflexealvanometer
measuring the photocurrent which has a sensitivity of about 4 x 10-mA
per scale division.

For the tests an initial load of Po = 50 kilograms was selected
and the deflection of the multiflexgalvanometer measuring the photo
current in dependence on the loading was measured. This dependence is
linear within the errors of measurement as can be seen in figure 6.
The galvanometer election was referred to 400-kilogram load increment;
in figure 6 on the average 53 scale sectors correspond to this load








NACA TM No. 1179


increment. Based on the (straight lined) calibration curve one now
knows what strain corresponds to this load increment.


II. EVAlUATION OF THE MEASUREMENTS AND RESULTS


From the known relation


E= (EE + E2)
1 (1)
1- (1)
02 = (-2 + ILE)


between the principal strains le E62 and the principal stresses
a0, 02 for the plane stress condition one obtains for the shear stress
T in the cross section of the shaft
aT l -- C2 E El- 2 (2)
T = (2)
2 1 + 2

The moduli of elasticity and of shear of the test shafts which were
manufactured of St. C 45. 61., resulted as E = 2.14 x 106,
G = 0.82 x 106 kg cm-2 and thus i = 1 E 1 = 0.302 0.30.
2 G
Thus T is for the present case

T = 0.823 (l e2) 106

In figure 7 the distribution of shear stress along the shaft is
represented for the case p/d = 0.107, d/D = 0.70. One can see that
for the smooth part of the shaft the shear stress is constant except
for slight scatter. The fillet begins at test point 8 shortly
before the shear stress starts to increase and reaches its maximum at
about test point 9 which lies already within the fillet. The resulting
stress concentration factor for the present case i:
Tmax 715
k = T 520 = 1.38

In the same manner the stress concentration factor was determined for
a series of values of p/d = a and d/D = p. Table 1 shows the result.
Fairing these test values graphically, one obtains the figures
designated in table 1 as faired test values. These values are








NACA TM No. 1179


plotted in figures 7 and 9 as functions of d/D
always somewhat lower than the values calculated
formula by Sonntag -


and
from


p/d. They are
the approximate


ak = (1.5 + 3.0a)


which is valid only for p -1- Taking the unavoidable error:3
1 + 2a
of measurement into consideration (for instance,inaccuracy of the
punch mark, not ideal torsion) it will be permissible to assume the
accuracy of these faired values with about 5 percent.


III. SUMMAFPY .


The stress distribution in stepped shafts stressed in torsion
was determined by means of percision strain gages. Comparison with
an approximate formula set up by R. SoritaE showed that according to
that formula the stress concentration fact.2- Air in the interval
0.15 < A 0.25 and 0.5 5d A 0.9 can be calculated with sufficient
d D.~
accuracy.



Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics.


1 .+ (1; 1 )(i)







NACP TM No. 1179


IV. REFERENCES


1. Willers, F. A.: Die Torsion ein's rotationsk6rpers um sein3
Achse. Z. Math. u. Phys. Bd. 95 (2907) p. 225.

2. Sonntag, R.: Zur Torsion von runden Wellen mit veranderlichem
Durchmesser. Z. angre. Math. n. Mech. Bdi. 9 (1929) p. 1.

3. Thum, A. and Bautz: Zur Frage der Formztffer. Z. VDI Bd. 79
(1935) p. 1303.

4. Sonntag, R., Thum, A., and Bautz, JW.: Zur Frage der Formziffer.
Z. VTI Bd. 81 (1937) p. 56'..


5. Lehr, E., and
Messtrecke
Forsch. a.


Granarher, H.: Dehnungsmessgeriit mit sehr kleiner
unri Anze'i e mittels Szerrschicht-PhL ozolle.
d. Geb. 4. IngenieunTesens Bd. 7 (1936) p. 66.







6 NACA TM No. 1.179


2.' BIE I

TBE STRESS CONCENTI TION FACTOR ak "S A

FUNCTION OF p/d AND d/D


p/d d/D a% k ak
P Measured Faired Calculated

0.25 0.50 1.17 1.22 1.23
.667 1.18 1.19 1.20
.824 1.21 1.13 --
.90 1.13 1.09 ""

.214 .50 1.24 1.25 1.27
.70 1.22 1.20 1.22
.824 1.19 .1.19 -'-
.90 1.13 1.10 ----

.15 .50 1.33 1.36 1.37
.70 1.22 1.28 1.30
.80 1.20 1.23 ---
.90 1.13 1.15 ---

.107 .50 1.43 1.45 1.49
60 1. h3 1.41 1.44
.70 1.38 1.35 1.38
."0 1.26 1.30 1.32
.90 1.19 1.22 -
._ ^ ij njt ~ n -- 1 i ^ i .i m -. T I L-I J nl ...-- .. ...... -






NACA TM No. 1179


Figure 1.- Designations at the stepped shaft.


Figure 2.- Torsion machine.


Sketch explaining the operation method of the torsion machine.


Figure 3.-





































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in 2011 with funding from
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http://www.archive.org/details/determinationofs00unit







NACA TM No. 1179


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Figure 4.- Test shaft with strain gauge.


Figure 5.- Arrangement of the test points 1 to 11 on the generatrices
I to III of the test sh ..


*
1; .,.. ;*I
;I'
-I








NACA TM No. 1179


0 200 9oo kg
P = load reading on the testing machine.


Figure 6.- Dependence of the photo current (galvanometer deflection)
on the load reading of the testing machine for a certain test point.


Test points


Figure 7.- Distribution of the shear stress along a generatrix of the
stepped shaft for p/d = 0.107, d/D = 0.70.





NACA TM No. 1179


Figure 8.- The concentration factor ak as a function of d/D.


Figure 9.- The concentration factor ak as a function of p/d.









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