fvcA-T/n-zn") NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM No. 1179 DETERMINATION OF THE STRESS CONCENTRATION FACTOR OF A STEPPED SHAFT STRESSED IN TORSION BY MEANS OF PRECISION STRAIN GAGES By A. Weigand TRANSLATION "Ermittlimg der Formziffer der auf Verdrehung beanspruchten abgesetzten Welle mit Hilfe von Feindehnungsmessungen." Luftfahrt-Forschung Band 20, Lieferung 7, pp. 21 7-219, Miinchen, July 20, 1943 Washington September 1947 -7 1^ _i !VU' NATIONAL ADVISOHi: COMMITTEE FOE AEEONAUTICS . TECroIICAL-^IEIMOEATOm M)'/ .1-179 DETEEMINAIIOW of the STBESS CONCENTRATION FACTOR OF A STEPPED SHAFT STRESSED. IN TORSION BY MEAl'IS OF PRECISION STRAIN GAGES*. By A» Weigsnd . The stress distribution. in stepped shafts" stressed in torsion is determined ty means of the electric precision strain gage by Lehr and Granacher [5I; the stress concentrati6n factor ajj as a function .of I and g for 0,5 ^ | S 0.9 and 0.1 ^ | < O.25 is ascertained . from the measui-'ements . 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 [l 1 by approximate integration of., -the differential equation of the stress function, E. Sonntag [2] gayg;; an. approximate formula for the stress function and the maximum BtreSs'- for 'the -case D - d = 2p, (See fig, 1.) L. S. Jacobsen, A. :Thvim>- and W, Bautz ([3] 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 /ensued [h], . ' ■ ^ ■■ ■ ■ I ■ — IIM.. —I I.I I I III. I . , .1— m il I . I I. .1 ■ ■ IM .— t I»» —» — ■■■■ ■ PM ' ■ I 1. -— ■ IM ■ ■■ I ■ ■ I ■■ IM ■■ i-» ^»^r^^^— •^"Eimittiung der Formziffer der auf Verdrehung beanspruchteri' ■ abgesetz ten Welle mit Hilfe von Feindehnuhgsmessiongsn." Zentrale fur wissenschaftliches Berichtsvesen der Luftfahrbforschung des General-r iui'tzeugmeisters, (ZWB) Berlin-Adlershof , Luf tfahrt-Forschijng .Band 20; Liefei-ung. 7, p, 217-219, Miinchen, July 20, 19'+3. NACA TM Wo . 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 E. Granacher [5], I. TEST SETUP AND PEOCEDUEE 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 ^5° 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 1, II, and III which were 90° to each other. The two principal strains ^2. ^^'^ ^2 , which must be equal and opposite for pure torsion, were measured. Actually, the values deviate from the average usually by 3 't'^ ^ percent; the reason is directional inaccuracy of the test section which was marked with a prick punch (I.3 mm) and a slight bending stress of the shaft caused by the fact that the twisting forces designated by a in figure 3 a^re not exactly equal. This error is essentially elimi- nated by measuring on three generatrices, figiare 5» From the three teat values which 3.ie on the same circle the mean value is taken. The strains were measured by means of the precision strain gage of 1.3-2Lillimeter gage length developed by E, Lehr and H. Granacher; its modus operandi is described in detail in [ 5l • 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 multiflexoalvanometer measuring the photocurrent which has a sensitivity of about h x 10~°mA per scale division. For the tests an initial load of Pq = 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 delection was referred to 400-kilogram load increment; in figure 6 on the average 53 scale sectors correspond to this load MCA TM No. 1179 ^ increment. Based on the (straight lined) calihration curve one now knows what strain corresponds to this load increment. II. EVAEQATIOW OF THE MEASUEEMEWTS AMD RESULTS From the 'kno\m relation 02 for the plane stress condition one obtains for the shear stress T in the cross section of the shaft T - ^r£2 _ £_ fl_lj2 /2) 2 " 1 + la 2 The moduli of elasticity and of shear of the test shafts which were manufactured of St. C U5. 6I., resulted as E = 2,l4 x lo6^ G = 0.82 X 10^ kg cm"2 and thus .i = -^ S .-. 1 = O.305 f» O.30. 2 G Thus T is for the present case T = 0.823 (?3_ - eg) X 10^ In figure 7 the distrihution of shear stress along the shaft is represented for the case p/d = 0.107^ d/D ■- O.7O. One can see that for the smooth part of the shaft the shear stress is constant except for slight scatter. The fillet hegins at test point 8 shortly hefore the shear stress starts to increase and reaches its maximum at ahout test point 9 which lies already within the fillet. The resulting stress concentration factor for the present case is: % - -7-- - 520 - ^-3^ In the same manner the stress concentration factor was detexmined for a series of values of p/d = a and d/D = p. Table 1 shows the result. Fairing these test values graphically, on& obtains the figures designated in table 1 as faired test values. These values are k MCA TM No. 11 '"9 plotted in figures 7 and 9 as functions of d/D and p/d. They are always somewhat lower than the va.li.ie3 calculated from the approximate formula hy Sonntag a-^ = p(l.5 + 3.0a) f-i4^ + (i _ p _ 2ap) {l + TT" ) (3) which is valid only for p < —- — , TaKin,^ the unavoidable errorra 1 + 2a of measurement into consideration (for instance, inaccuracy of the punch mark, not ideal tor^^icn) it will be pe:-Tiilseible to assume the accuracy of these faired valuej with about 5 pe^.-'cerit. III, GUlvMAF/ The stress distribution in stepped shafts stressed in torsion was determined by means of pei-cision strain gages, Comparison with an approximate foimula set up hi^ E, Sonntag showed that according to that formula the stress concentration factor av in the Interval 0.1 < ^ % 0.2'^ and 0.5 ^ ^ ^ 0.9 can be calculated with sufficient * d ^ - B ~ accuracy. Translated by Maiy L. i