1、a MIL-HDBK-776 (AR) 15 September 198 1 MILITARY HANDBOOK SHAFTS, ELASTIC TORSIONAL STRESS ANALYSIS OF NO DELIVERABLE DATA REQUIRED BY THIS DOCUMENT - / THIS DOCUMENT CONTAINS gg/ PAGES . AREA CDNC Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-H
2、DBK-776 CD 7797770 OOq22O E MIL-HDBK-77 6 (AR) 15 September 1981 DEPARTMENT OF DEFENSE WASHINGTON, DC 20360 Shafts, Elastic Torsional Stress Analysis Of 1. This standardization handbook is approved for use by the Armament Research and Development Command, Department of the Army, and is available for
3、 use by all departments and agencies of the Department of Defense. 2. Beneficial comments (recommendations, additions, deletions) and any per- tinent data which may be of use in improving this document should be addressed to: NJ 07801. US Army Armament Research and Development Command, ATTN: DRDAR-T
4、ST-S, Dover, F Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-HDBK-776 CD W 7777970 0042283 b R MIL-HDBK-776(AR) 15 September 1981 FOREWORD Design charts and tables have been developed for the elastic torsional stress analyses of free prisnatic
5、shafts, splines and spring bars with virtually all common industrially encountered thick cross sections. Circular shafts with rect- angular and circular keyways, external- splines! and mille flats along with rectangular and X-shaped torsion bars are presented. A computer prolrrarn (ISIIAFT“) was dev
6、eloped which provides a finite difference solution to the governing (POISSONS) partial differential equation which defines the stress functions for solid and hollow shafts with generalized contours. Using the stress function solution for the various shapes, and Prancltls membrane analoqy, dimensionl
7、ess design charts (and tables) have been generated for transmitted torque and maximum shearing stress. The design data have been normalized for a unit dimension of the cross section (radius or length) and are provided for solid shapes. The eleven solid shapes presented, along with the classical circ
8、ular cross section solution, provides the means for analyz- ing 144 combinations of hollow shafts with various outer and inner contours. Hollow shafts may be analyzed by using the computer program directly or by using the solid shape charts in this paper and the principles.of superposition based on
9、the concept of parallel shafts. Stress/toryue ratio curves are presented as beinq more intuitively recognizable and useful than those of stress alone. Sample problems illustrating.thQ use of the charts and tables as desisn tools and the validity of the superposition concept are included. iii _ _ - P
10、rovided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.-_I_ MIL-HDBK-776 CD W 7777970 0042282 B R. MIL-HDBK-776(AR) 15 September 1981 TABLE OF CONTENTS The Torsion Problem Design Charts and Tables Accuracy of the Computerized Solution Parallel Shaft Concept
11、 Bibliography Appendixes A Mathematical Model Used in the SHAFT Computer Program B Extension of Model to Hollow Shafts Page No. 1 4 54 55 6G 67 75 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-HDBK-776 CD 7779770 0042283 T MIL-HDBK-776(AR) 15 S
12、eptember 1981 Tables , 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Element nomenclature Split shaft, volume factor (v) Split shaft, stress factor (f) Single keyway shaft, volume factor (VI Single keyway shaft, stress factor (f) Two keyway shaft, volume factor (V) Two keyway shaft, stress factor (f)
13、 Four keyway shaft, volume factor (V) Four keyway shaft, stress factor (f) Single square keyway with inner fillets Single spline shaft, volume factor (V) Single spline shaft, stress factor (f) TWO spline shaft, volume factor (V) Two spline shaft, stress factor (f) Four spline shaft, volume factor (V
14、I Four spline shaft, stress factor (f) Square keyways and external splines volume factor (V) square keyways and external splines, stress factor (f) . c, *. 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS
15、MIL-HDBK-776 CD 9999770 0042284 L 111 MIL-HDBK-776( only relative values or differences are meaningful. The solutions to this equation required complicated math- ematics. Even simple, but commonplace, practical cross-sec- tions could not be easily reduced to manageable mathematical formulae, an
16、d numerical approximations or intuitive methods had to be used. One of the most effective numerical methods to soive . for Saint-Venants torsion stress function is that of finite differences and the best intuitive method, the membrane ana-. locry, came from Prandtl. IIe showed that the compatibility
17、 equation for a twisted bar was the “same“ as the equation or a membrane stretched over a hole in a flat plate, then in- flated. This concept provides a simple way to visualize the torsional stress characteristics of shafts of any cross- section relative to those of circular shafts for which an exac
18、t analytical solution is readily obtainable. A computer pro- qram called SHAFT was written and applied to produce the dimenslon- less desiqn charts on the following pages. The three-dimensional plot of over the cross-section is a surface and, with set to zero (a valid constant) along the periphery,
19、the surface is a domb or membrane. The trans- mitted torque (T,) is proportional to twice the volume under the membrane and the stress (Ss) is proportional to the slope of the membrane in the direction perpendicular to the measured slope. Neqlectinq the stress concentration of sharp re-entrant cor-
20、ners, which are relieved with generous fillets, the maximum stress for bars with solid cross sections is at the point on the periphery nearest the center. The design data have been normalized for a unit dimension (radius or lencrth) of the shaft cross-section and are in di- mensionless format. The d
21、ata and charts may, therefore, be used for shafts of any dimensions, materials and twist (loading). 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-HDBK-776 CD 7977770 0042288 7 PI MIL-HDBK-77 6 (AR) 15 September 1981 NODES FINITE DIFFERENCE ME
22、MBRANE SOLUTION CONTOURS CROSS-SECTIONS Figure 1. Membrane analogy. 3 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-HDBK-776(AR) 15 September 1981 DESIGN CHARTS AND TABLES Design charts and related data which support the elastic torsional stres
23、s analyses conducted by MISD are shown in figures 2 through 25 and tables 2 through 25, respectively. The item nomenclature used in the analyses is given in table 1. These data are based on the stress function solution for various shapes provided by the SHAFT computer program and on Prandtis membran
24、e ana logy. * Since the design charts are dimensionless, they can be used for shafts of any material and any dimensions. 4 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-HDBK-776 CD W 7777370 OOLI2270 7 W MIL-HDBK-776 (AR) 15 September 1981 Tabl
25、e 1. Element nomenclature TORSIONAL PROPERTIES OF SOLID, NON-CIRCULAR SHARS T = TRANSMITTED TORQUE, N - m (Ib - in.) 8 = ANGLE OF TWIST PER UNIT LENGTH, rad/mm (radlin.) G = MODULUS OF RIGIDITY OR MODULUS OF ELASTICITY IN SHEAR, kPa (Ib/in.2) R = OUTER RADIUS OF CROSS-SECTION, mm (in.) “,*,f = VARIA
26、BLES FROM CHARTS (OR TABLES) ds RELATED TO VOLUME UNDER “SOAP FILM MEMBRANE“ AND SLOPE OF “MEMBRANE“ SS .- SHEAR STRESS, kPa (lh/in.2) 5 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-MDBK-776(AR) 15 September -1981 .40 .35 .30 .25 V .2(3 .OC .o
27、 I I I I I .I .2 .3 .4 .5 .6 Fig ire 2. Split shaft, torque. 6 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-HDBK-776 CD W 7779770 OOi-12272 I MIL-HDBK-776(AR) 15 September 1981 Table 2. Split shaft, volume factor (VI Y/Ri Ri/Ro O. 6 - O. 5 - O
28、 4 - O. 3 - o. 2 - o. 1 - o. 1 .3589 ,2802 ,2068 .1422 .O891 .0491 o. 2 ,3557 .2762 .2030 ,1391 .O870 .0478 0.3 3525 .2722 .1991 .1360 .O848 .0464 O. 4 .3492 .2680 ,1952 .1328 .O825 .0450 O. 5 .3457 ,2637 .1911 .1294 .O801 .0436 O. 6 ,3423 .2593 ,1869 .1260 .O777 .0421 0.7 .3387 ,2548 .1824 1223 .O
29、750 ,0405 O. 8 ,3350 ,2499 .1776 ,1183 . O722 .0387 o. 9 .3312 .2447 ,1725 ,1139 .O689 .0367 1 .o .3269 .2389 ,1665 ,1087 ,0649 .0340 , 7 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-_-I_- MIL-HDBK-77b CD - 9777770 00112273 2 - MIL-HDBK-7 7 6 (AR)
30、 15 September 1981 6.5 6.0 5.5 5. O f 4.5 4.0 3.5 3.0 2.5 2.0 SHEAR STRESS I MAXIMUM AT INNER EDGE, 0.9 AT X S, = T (/ROI - - - - - - - - - 1 1 I 1 1 I o. I 0.2 0.3 0.4 0.5 0.6 Figure 3. Split shaft, stress. 8 0.7 0.5 0.3 o. 1 Provided by IHSNot for ResaleNo reproduction or networking permitted with
31、out license from IHS-,-,-_ MIL-HDBK-776 CD m 7977770 OOq2294 i.I m MIL-HDBK-776(AR) 15 September 1981 Table 3. Split shaft, -stress factor (f) 0.6 - 0.5 - 0.4 - 0.3 - 0.2 - 0.1 2.2140 2.2742 2.5771 3.2178 4.4650 0.2 2.2447 2 . 3162 2.6336 3.2977 4.5865 3 0.3 2.2767 2.3608 2.6942 3.3838 4.7182 0.4 2.
32、3103 2.4082 2.7597 3.4771 4.8620 - 0.5 2.3461 2.4594 2.8304 3 . 5795 5.0233 0.6 2.3883 2.5142 2.9084 3.6930 5 . 2016 I 0.7 2.4233 2.5750 2.9955 3.8232 5 . 4082 I 0.8 2.4670 2.6423 3 . 0952 3.9744 5.6550 I 0.9 2.5142 2.7197 3.2141 4 . 1618 5 . 9691 1.0 2.5672 2.8140 3.3690 4.4218 6.4392 9 Provided by
33、 IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MIL-HDBK-7 7 6 (AR) 15 September 1981 .70 .65 .60 .56 ./B = 0.3 0.4 - + I 2.0 f +B I 1 1 I I I o. 1 o. 2 0.3 0.4 0.5 BIR Figure 9. Four keyway shaft, stress 20 - _ - _- Provided by IHSNot for ResaleNo reproduction
34、 or networking permitted without license from IHS-,-,-MIL-HDBK-776 CD 7779770 0042306 7 MIL-HDBK-776(AR) 15 September 1981 Table 9. Four keyway shaft, stress factor (f) 0.2 0.3 0.4 0.5 a 0.6 0.7 0.8 0.9 1.0 1.2 1.5 2.0 0.1 - 1.0371 1.0252 1.0102 09910 .9661 .9331 ,9232 .8940 . 8796 0.3 1.7365 1.3566
35、 1.6214 1.3011 1.5468 1.2130 1.5046 1.1979 1.5115 1.1737 1.5175 1.1541 1.5382 1.1493 1.5609 1.1398 1.5899 1.1422 1.6424 1.1511 1.7092 1.1717 1.7512 - 0.2 - 2.1206 1.9971 2.0591 2.1568 2.2660 2.3834 2.4993 2.7301 0.5 2.4931 2.5784 2.8271 3.1882 3.6139 4.0368 - 21 Provided by IHSNot for ResaleNo repro
36、duction or networking permitted without license from IHS-,-,-_- MIL-HDBK-776 -O 777TW OOi.12307 7 W MIL-HDBK-776(AE) 15 September 1981 I75 .70 .65 .60 V .55 .50 .45 .40 TRANSMITTED TORQUE T = 2 * G *6(V) R4 SHEAR STRESS f2 7- 4- 1 1 O. 1 o. 2 0.3 0.4 0.5 BIR t- Figure 10. Single square keyway with i
37、nner fillets. 22 I .30 1.20 1.10 f 1 .o0 -90 .80 .70 P Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I MIL-HDBK-776 CD W 7777770 0092308 O W MIL-HDBK-776(AR) 15 September 1981 Table 10. Single square keyway with tight inner fillets Stress factor (f) Volume At keyway At inner factor (VI center (i) fillet (2) B/R 0.1 -7703 .7804 0.2 .7206 .9715 . 9777 0.3 .6504 .9941 1.0817 0.4 .5690 1.0735 1.1641 0.5 .4840 1.1977 1.2245 23 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
