THE EFFECT OF A SLOT ON THE STRESSES AND RIGIDITY" OF A SHAFT SUBJECT TO PURE TORQUE by FREDERICK JOSEPH SBUMAS ' A Thesis submitted i n partial fulfilment of the requirements for the Degree of Master of Applied Science . i n the Department of Mechanical and Electrical Engineering The University of British Columbia October, 1949 ABSTRACT This study was made to determine the effect of different width slots on the bending and shear stresses, and on the r i g i d i t y of a slotted shaft subject to pure torque. As a f i r s t step i n the solution of the problem, the stress d i s t r i -bution due to a pure torque, acting on the shaft segment,,and the torsional r i g i d i t y of the segment were obtained by use of a numerical method of solution and St. Venant^ Principle. After this solution was obtained, the results were used in the calculation of the actual stresses by combin-ing the effects of twisting and bending i n the segment. Formulas were developed for obtaining the torsional r i g i d i t y , maximum shear and bending stresses resulting from a known applied torque. A slotted shaft was then tested in torsion and the bending, shear stresses, and r i g i d i t y were determined experimentally. The agreement between calculated and experimental results was good; the error being of the same order as the inaccuracy i n the strain measurement. ACKNOWLEDGMENT The author wishes to express his thanks to Professor W. 0. Richmond, who gave invaluable suggestions and cr i t ic isms throughout this investigation. CONTENTS NOMENCLATURE STATEMENT OF PROBLEM 1 THE TORSION PROBLEM 3 THEORY OF NUMERICAL SOLUTION 8 NUMERICAL SOLUTION APPLIED TO SHAFT SEGMENTS 15 RESULTS AND CALCULATIONS 28 EXPERIMENTAL STRESS ANALYSIS 38 Strain Gauge.Theory 38 Method of Applying Torque 42 Determination of Stresses 45 Determination of the Angle of Twist. 51 CONCLUSIONS 54 APPENDIX A 55 APPENDIX B ^ . . , 56 NOTES 5 g BIBLIOGRAPHY GO NOMENCLATURE Applied torque T Torque on segment M Bending moment Rectangular co-ordinates T T Shearing stresses para l le l to x, y axes on the plane perpendicular to the z axis . Stress function c Torsional r i g i d i t y e Angle of twist per unit length E Modulus of e l a s t i c i t y i n tension and compression Q Modulus of r i g i d i t y a Radius of shaft . I Moment of ine r t i a of segment A f Area of segment Tensile or compressive stress due to bending r Shearing stress € Unit elongation l Effective length of slot R Distance of centroid of segment from axis of shaft S Deflection L Distance of scale from axis of shaft cp Angle of twist Normal stresses y Distance of fibre from neutral axis of segment K, ,k2Shear stress coefficients Ko Torsional r i g i d i t y coefficient 1 THE EFFECT OF A SLOT ON THE STRESSES AND RIGIDITY OF A SHAFT SUBJECT TO PURE TORQUE STATEMENT OF PROBLEM This study was made to determine the effect of d i f -ferent width slots on the bending and shear stresses and on the r i g i d i t y of a slotted shaft subject to torque. The torque T 0 was considered to be acting as shown ( f i g . 1)„ As a f i r s t step i n the solution of the problem, the stress dis t r ibut ion due to pure torque T, acting on a shaft segment ( f i g . 2), was obtained by the use of a numeri-ca l method of solution and St. Venants' P r i n c i p l e . 1 After th is solution was obtained, the results were used i n the calculation of the actual stresses by combining the effects of twisting and bending i n the segment. Formulas were developed for obtaining the torsion-a l r i g i d i t y , maximum shear, and bending stresses result ing from a known applied torque. A slotted shaft was then sub^ jected to measured torques and the maximum shear and bend-ing stresses on the shaft segment were determined 1 Superscripts refer to notes at the end of the thesis. -1 / 1 . / ' / 1 - T \ > ' \ I - -1 Fiq.2 3 experimentally. The t o r s i o n a l r i g i d i t y was also found by experiment and the agreement between t h e o r e t i c a l and ex-perimental r e s u l t s was found to be within the l i m i t s of experimental error. THE TORSION PROBLEM St. Venant 2 showed that the shear stress d i s -t r i b u t i o n of a shaft of constant cross-section subject to torque could be obtained by the introduction of a stress function (j) which i s a solution of the p a r t i a l d i f f e r e n t i a l equation ^ + ^ = - e q e • ^ x 2 byx (1) where x and y are rectangular co-ordinates, Gj i s the modu-lus of r i g i d i t y , and 0 i s the angle of twist per unit of length. In terms of t h i s stress function, the shear stresses as indicated ( f i g . 3) are given by ------ ca) - r =-c)<t> 7 5 dx (3) Equations 2 and 3 state that the change i n torsion function (J) i n the x and y directions i s equal to the shear 4 Y Fiq.4 5 stresses in the x and y directions, respectively. On the boundary of the section ^ 4 = Q (4) i s which follows from the condition that the resulting shear stress i s i n the direction of the tangent to the boundary. Equation (4) must be satisfied i f the lateral surface of the section i s to be free from external forces. If the stress function^ (|) Is put in the form (|).Vp-(^Cx* +y^ . ... ( 5 ) then from Eq.(1) T W ~ O — (6) and from Eq.(4) IU - Q 6 ( x 2 f y l ) = a c o n s W \ i (7) • 2 which hold throughout the section and along the boundary, respectively. In the case of singly connected boundaries, e.g. for solid bars, this constant can be chosen arbitrar-i l y , and i n the following discussion i t i s taken equal to zero. 6 then from Eq.(7) ^ * ^ p ( x * + y z) (8) on the boundary. By considering the d i s t r i b u t i o n of shearing stresses over the cross-sections of the bar, i t i s found that the relationship between the applied torque and the to r s i o n function (j) for a uniform section Is given by the following T = z f f ( j ) d x d y ( 9 ) where T equals the torque acting on the section. Summarizing, the function (J) when plotted over the xy plane, gives a pillow-shaped surface, the boundary values of which are zero; the magnitude of the gradient of dj) at any point give's the maximum shear stress at that point; and twice the volume included between the- (j) surface and the xy plane gives the torque applied to the section. St. Venant's equations have been solved e x p l i c i t l y for a number of di f f e r e n t cross-sections and. the chronologic-a l development i n the fundamental theory and i n the exact solutions effected up to 1942 of t h i s problem was given by 3 T . J . Higgins. However for an a r b i t r a r y shape of cross-section i t may not be possible to obtain a solution In 7 analyt ical form. In such instances recourse may be had to experimental methods of solution such as the membrane analogy 4 or to the numerical method of solution which has received considerable attention the past few years. The more important of the numerical methods that have been used comprise: (1) var ia t iona l , (2) graphic, (3) and arithmetic — according as they are developed from the theory of the calculus of variat ions, from the use of graphic analysis, or from the theory of l inear difference equations. The arithmetic method Is the one used i n this i n -vestigation and the general scheme of solution involves replacement of the pa r t i a l d i f fe rent ia l equation and the boundary conditions defining the problem by an analogous pa r t i a l difference equation and equivalent boundary condi-t ions, and development of a sequence of approximate numeri-ca l solutions of the difference problem. This general theory for solving the torsion prob-lem by f in i t e differences was f i r s t used by Runge, 5 w n o determined and plotted the stress l ines i n a cross-shaped area comprised of five squares. A.Thorn and J . Arf£_^ ex-tended this method for solving the torsional properties for c i rcular shafts of varying r a d i i . J . OTT(J) also used the ^ ($ numerical method for the solution of the torsional proper-t ies of an I-section and a key-wayed shaft and found close agreement between theoretical and experimental resul ts . 8 THEORY OF NUMERICAL SOLUTION Before the torsion function (£> can be solved for the given region ( f i g . 3), the values of throughout the same region are required. These values of i p must satisfy Eq.(6) and Eq.(7) throughout the region and on the boundary, respectively. An approximation to the exact solution i s ob-* tained by the solution of Laplace's Equation a x 2 ^ v * . by numerical methods. Any numerical method of handling a d i f fe ren t ia l equation Involves the replacement of points i n the in ter ior and on the boundary of the region by a discrete set of points. Leibmann developed a convenient procedure for solving for unknown values of \\) at in ter ior points sat is-fying Laplace's Equation. The procedure involves laying down a square netv/ork over the region ( f i g . 4) and assigning known values at the boundary and assumed values at the in ter ior points. The net i s traversed repeatedly i n some definite order as A, B, C, D e tc . , and the value at each in ter ior point i s replaced by the average of i t s values, at the four neighbouring points ( f i g . 5). This follows from the theory of f i n i t e differences. Thus, f i r s t and second derivatives, such as ^\\) are approximated, respectively, by 9 the following quotients 9 whose l i m i t s , as h approaches zero, y i e ld the true derivatives. Similar ly the Laplacian Equation = 4 ( ^ m - i p ) d o ) where i s the mean of the values of at the four points obtained by adding or subtracting i t from x or y of the point i n question ( f i g . 5). For example ( f i g . 6), a column of numbers i s as-sociated with each point inside the net and a single number with each boundary point. The top number at each in ter ior . point i s a value a rb i t r a r i l y assigned and the second number i n the column represents the value obtained after the f i r s t traverse over the net i n some definite o r d e r . Previously altered values are used whenever possible in calculating the improved value at a given point. 10 Fiq. 5 jt>< 40.5. k?-o 77.0 !©0.0 64 0 7O.0 £6.7 40.8 \ K \ \ \ 15.2 \ TS.O pro-4 _uoo \ \ F iq .6 11 In the case of a point which i s not equidistant from i t s four nearest neighbours, e .g . , near the boundary of the section, the simple improvement formula i s not ap-pl icable . Figure 7 shows such a point. Here two neighbours l i e atdistances which are less than the net spacing h. The formula for the improvement value of at this point i s HUM S+t sc.\+s> l i t i+s tu+l) ( i i ) The values of S and t here represent fractions of the normal net. spacing and as a check the sum of the coefficients C^, C2» C^, and equals one. The fineness of the net placed over the region' determines the accuracy with which the solution of the d i f -ference equation agrees with the d i f fe ren t ia l equation. A coarse net ( f i g . 8) i s f i r s t placed over the segment and the net i s traversed, using the new values of ^ immediate-l y In the improvement of the succeeding points. This net i s traversed u n t i l the values of L|J become stationary; then a net with one-half the spacing ( f i g . 8) i s placed over the segment and a similar procedure i s repeated. I f the values of l p at the same point from the two nets differ by less than three times the permissible error, then the value of l^i from the finer net i s satisfactory. 13 This convergence to a constant value of i s • very slow and i n order to increase i t s rapidi ty and reduce the labour required, both the immediate and diagonal neigh-bours are used ( f i g , 9) \Z The formula t O l p . 4 ( l J J l + ^ + ^ + U 0 ^ * ^ + ^ ^ t C12) i s applied here. A l i s t of improvement formulas, which can also be used for solving for values of lJJ when values at other than neighbouring points are known, i s given i n Appendix A. 14 1 (KM A \ 0 \ \ \ j i M" B / /' / / FiG*IO 15 NUMERICAL SOLUTION APPLIED TO SHAFT SEGMENT In using the numerical method of solution for the shaft segment* the size of the cross-section was magnified ten times ( f i g . 10) and a square, net with a spacing of 2 inches was f i r s t used* From the symmetry of the segment, i t was only necessary to solve for the values of tjJ for half the segment and these values were symmetricial about the line" of symmetry of the segment. The boundary values were f i r s t calculated and then arbitrary values were assigned to the in ter ior points. As this represents the segment of a 2-inch diameter shaft, i t necessitated Eqs.(5) and ( 8 ) being changed to ( J ) » l | j - e i £ L C x * + y O ~ (13) Zoo and *4J - QO ( x ^ + y * ) . (14) Zoo throughout the segment and on the boundary, respectively, i n order that the calculated results apply to the segment of a 2-inch diameter shaft. The angle of twist per unit length of the segment was taken as 200 f (q where Cq i s the modulus of r i g i d i t y and the following procedure was used for solving for l | i throughout the segment* As an example consider the semi-c i r c l e ( f i g . 11) where a net with a 2-inch spacing was 16 17 placed over the segment. The boundary values were f i r s t calculated (Eq.14) and then arbitrary values were assigned to the in ter ior points. The Leibmann procedure was then applied by t r a -versing the net i n a definite order u n t i l the net values became stationary. These stationary values were then transferred to a net of 1-inch spacing ( f i g . 11) and the Leibmann procedure was repeated u n t i l the values of at points throughout the finer net became stationary,, These values i n turn were transferred to a net of Jr-Inch spacing ( f i g . 11) and the previous procedure was repeated u n t i l the values of ljja.ga.iri became stationary. Equations 11 and 12 were used where the net intersects the boundary and throughout the segment, re-spectively, to bring to i t s stationary values at points throughout the segment. A £- inch net was placed over the quadrant ( f i g . 12) and the values of were calculated for 2 0 0 a l l points of the net. These values were used i n Equation 13, l^ J _ (^9 ( x a + y < l ) r> t o s o l v e f o r a t each point ZOO ° of the segment where had been calculated for the same point. The f i n a l results are shown (f igs , 13, 14, 15, 16, 17) for the different segments where each segment has an angle of twist of 200 ^ (q .radians per unit of length. -4\ i _ \ >—Tv ! -H so.o !^;o & 5 . 0 n^o'BTo 4<?).0 ^4? ;G40 " [ 7 2 - 3 pl-d 3kO _ [ 4 2 \ J41-.0 _ i^CL J 7 Z 3 _ J8L0. 9 0 3 F l q . 1 2 . V A L U E S OF ( r * ) Foe S E M \ - C » R C L & . z o o 19 o o o 4.1 o 7 8 3~4" 147 12.4-B.5 ! 0 5.5 8-7 9.0 i'11.8 lit 15.4 18.2 '2o.8 181 4o 147 ft i 177 2o.8 22 2 223 2 5 3 253 I 2.8-8 2 - 8 15. 20. 237 25 .5 28.4-22>3 7 Z \3 I 'I'D-|C<2. j 2(4-25.o 233 9* 23, | 273 27.^ ) 14 97 172 3 * lb.8 22^ Z 2 7 ;7-4 153 13.5 (IT 2^> O 13,1 I 271 519 337 3(^ 277 : 204 !(4o !Z4o 3o.<3 34 3 '348 3(-8 2<b*> l7(o 77 78 20.1 ! 287 %0 wr %0 237 147 'il4£" ;25.2 323 343 29=8 •2J2. I 0 4 120.(0 I 25q 354 .333 |\7o 76 "TJ4.'8"!207 |25.6p"|3o.( "|'S.2~ " S ' " 3T4";38.l "j3gl ^~~^4>.T" 34*2 2.7.1 .JVl j|g3> 0 n"3 '"K^ !2o.^ "l25^ ~SoT 33.4%\ "'^ .7 "38? . 1|43 12(0 J2£,0 3o.S .'8U 318 /2-b 357 3&D '.345 3 4 4 3 ( 4 2 8 . 0 237 ;I8T fox , 7 4 0 Fiq. 13 V^LUE.^ of 4> FOR 5ELMI-CIRCI_E 20 :7i i i 17V W 70 j 175 £|.4 r?o i 1 j 178" "~W-4 i t i pro I'M t !6.9 12^ . ff5 25.2 ~7_4.r 14^ 13*5 Z5.8 £1 £4o IS.8 111 \*1 W 15% "Z4;2 jIB.b ~ 14£ " I ' "' Wo %1 !Z73 US 'i25.o jZi-t ;i95 15.7 in 3 x:*> :.||.3 Fiq .14 V A L U E S O F 6 F O R W & . Z Q •-• . D 21 0 0 2.1 / i / i 24. | i ! i / ! i -t i 0 1 V I \ ; \ i \ ; O 3 | 1 i 1 i TG.4 i 79 jk.o i ^ ; Oi • \J —-—>• 3 . 8 i .. 1 • 9 . 3 ! ! • I 13 0 jT.T 1 i 1 j i 10.5 i . 0 4 6 | I 3 3 (23 | ! i i H2 58 5 7 4 " 0 9.0 12-8 i "(T4~ 52 11-8. \4r3 I i t 1 4 4 | - ' i \41 i i )5.\ i I 0 s.z 131 6.6 j ! t I T O | i j ( S T $ 3 i 3a IG.O |I78 i • i ! (|.<3 1 5-3 0 57""" 103 3.6 M 74 JI84 • i m |»T7 ! I5T-3 I 3 7 103 lo-4 140 16.4 • \78 » I8.5 K>& 143 11.4 , 72. f 0 o 57 £ 8 5"7 . 10-4-| 64"~ 4o ]4o~" • i i (8.3 • \7( 15.1 lB.8 • 58 (8.8 -I78 \5.3 | • 1 I 12- 6 137""" 16-0.. ?-4 B-7 \0S ]I4.1 lo.4 JI4-.I 171 J87 ICS; ^ 87 ... „ I94 .. I7-8 > ; ^8 . . FlQ.I5_. VALUED OPCJ) FOR W« .30 22 '•2.{ 3.6 5 .2 4 8 2-e i :t.Y 56 7-T (46 j j 4 S & 2 4 . 6 ss~ o 4 7 as !0 4 5 to" 7 -T 7<3 ! 8* 110-7 K).*> )0.5 12-Y .0 £.6' 103 i n \\Z .13.0 44 8.4 0 4 3 8 0 124 ilo. I 7Z 3-3 4 ? 107 |26 6.8 H I 125 1.7 9.0 H i "' ib.^ 0 ; 4 - n &io 4^ K 4-^ iS^ > ii>7 =5 i l 4 i 14.1 14^ 122 10 I SIT 147 | | 5 l i"5l "Tsi"~"ti4^~^B3" 153 1 5 r 5 . 5 ( 4 3 10.1 113 1\4T 11-5 8-TPS" 5.1 F \ q J 6 V A L U E . * * or (J) FOR ^ .40 2 3 2.0 27" 1-2 27 1CJ 2.1 3s~ ; 3.6" iG'.-T i.<5 3 1 • W-O 10 5.4- 1 34 t . 5 ' G . 6 " ^ 6 " •8.8' 8 . ^ ; 87 :3.G 87 75 5^.2 3 . 5 70";"3.^T 37 70 ao 1103 3.6 3 3 " 7.3 3-1 i 103 103 4-2 75 ;3<b 14, t 17 4.4 '' 7^ I0.3 K .O 10.1- jll.O 9.<b 107 7 6 ; 3.1 9cT 10.1 71,5 ,i7.k;ii.3" *3.6 !4A ! 7 7 13 .9 i l l * | U G 71 18 74 14.6 0 p i q . 1 7 V A L U E S O F <$> F O P , W ~ . 5 0 24 From Eq, (9) but T = C 0 where Q i s "the t o r s i o n a l r i g i d i t y of the section and G i s the angle of twist per unit of length. As (q0 has been made equal to 200, then the t o r s i o n a l r i g i d i t y i s C f f a d x ^ y — (15) \ooJJ . From Eq. (9) the torque producing shear i n t o r s i o n i s given by twice the volume included between the (J) surface and the xy plane. To compute the volume, i t was convenient to integrate along a complete set of p a r a l l e l l i n e s i n d i v i -dually and then integrate the r e s u l t i n g sums i n the cross d i r e c t i o n , z^. ( 1 3 ) The area under the curve ( f i g . 18) i s given by\_y A (16) The volume under the (|) surface and the xy plane was computed as follows: Apply Eq. (16) along l i n e s A-A1, B-B 1, C-C 1, etc. ( f i g . 19) obtaining a sum for each. Then integrate from top to bottom using the sum along A-A 1 as 4>, , the sum along B-B 1 as (J) , etc. ,in E q . ( l 6 ) . The 26 result was the desired volume. The shear stresses at different points in the seg-, 1 4 ment are equal to which defines the change In (b c m T when moving normal to the line of shear stress at the point in question. Maximum shear stress occurs at the middle of the straight side of the segment ( f i g . 20). A parabola was passed through the intersection with the boundary, where (£) equals zero, and the two nearest interior points along the line A-B ( f i g . 20). If the distance between the bound-ary and , be sh ( f i g . 21), where S i s a fraction of the net interval h, the magnitude of the slope at the bound-15 ary i s then given by a>4>_ i (s+t<t>, - _s_ 4> > . . . . — ( 1 7 ) 27 / s h h Fiq.21 28 RESULTS AND CALCULATIONS The previous calculations were repeated for shaft segments of slotted shafts having a ra t io of slot width W to diameter of shaft D of .2 , , 3 , .4 , . 5 and for a semi-circular shaft, and the values of the stress function ^) for these are shown ( f igs . 1 3 , 14, 1 5 , 16 and 1 7 ) . Lines passing through constant values of (J) give the direct ion of the shear stress. The maximum shear stress which occurs at point A ( f i g . 20) can be expressed by the equation 16 Y s ». k,(q0.a — (18) where k-^ i s the factor depending on ra t io ^ ( f i g . 22), (q i s the modulus of r i g i d i t y , 0 the angle of twist 'V>-v per unit of length, and a i s the radius of the shaft. The maximum shear stress on the c i rcular boundary at point B ( f i g . 20) can be expressed by the equation ~YC = kz (qO a ( 19 ) where k2 i s a factor depending on the ra t io Vs/ ( f i g . 22). D Co £ F F 1C i ENiT S 30 The t o r s i o n a l r i g i d i t y for the segment can be ex-17 pressed by the equation C = k ^ G ^ G a (20) where ko i s a factor depending on the r a t i o ( f i g . 23) D The values of the factors k^, k 2, and k^ are shown i n the following table and can also be found from graphs ( f i g s . 2g and 2«). TABLE I Shear Stress and R i g i d i t y C o e f f i c i e n t s for Various Segments. Segments of Shafts having the following r a t i o of s l o t width to diameter of shaft * k l k 2 Semi-Circular Shaft 0.865 0.725 0.298 w D 0.765 0.640 0.158 1 .3 0.640 o. 545 0.099 ; . 4 0.540 0.505 0.059 \ .5 0.480 0.460 0.035 St. Venant 18 obtained a n a l y t i c a l l y the stress d i s t r i b u t i o n i n torsion for a semi-circular shaft. His values are compared i n Table I I with those obtained from t h i s i n v e s t i g a t i o n . 31 FOR SEQMEKIT 32 TABLE I I Comparison between Coefficients of St. Venant and those from t h i s Investigation for a Semi-C i r c u l a r Shaft. ' k l k 2 k 3 St. Venant 0.849 0.719 0.296 This Investigation 0.865 0,725 0.298 Percent V a r i a t i o n 1.89 0.84 0.67 Application to a Slotted Shaft In applying the foregoing r e s u l t s to the case of a s l o t t e d shaft i n t o r s i o n , i t i s assumed that the planes A and B ( f i g . 24) are prevented from warping by the s o l i d portion of the member. The equations of St. Venant were developed on the assumption that the section was free to warp. I f the sec-•t t i o n i s prevented from warping, then bending stresses are set up on each side of the s l o t as well as shear stresses due to the twisting moment. The bending moments are calculated by assuming that the metal on each side of the s l o t behaves as a cantilever beam ^ with a torque also acting and that the r o t a t i o n of each segment i s the same ( f i g . 25). In the following calculations the rotations of the segment were considered small. I f T i s the torque carried by each segment, then the r o t a t i o n of each segment (D i s 3 3 B E F O R E R O T A T I O N AFTER. R O T A T I O N 3 4 given by <P • X ! ( 2 D C where 1 i s the effective length of the s lo t , and C i s the torsional r i g i d i t y of the segment, Eq.(20). Each segment i s deflected an amount R(p as the rotations are small, where R i s the distance from the axis of the shaft to the centroid of the segment. This introduces a force P and a bending moment M tending to reduce the deflection ( f i g . 25). With these forces acting on a canti-lever beam, the deflection i s given by S =vR<p»ELLs - M J 2 — — - (22) BET SET and the slope i s given by Slope = Ri - M i ( 2 3 ) ZE\ E T where E i s the modulus of e l a s t i c i t y of the material of the shaft, 1 i s the effective length, and I i s the mom-ent of ine r t i a of segment about axis X-X ( f i g . 26). As the so l id shaft prevents a change i n slope of the cantilever at the ends of the s lo t , the slope here i s P i * -±41 (24) 2EI E I 3 5 36 Thus M = PI (25) z cp--±ip^, p i n , PI* R<3E1 4E1 i 12'EIR Equating the two values of CD 5 12EIR C P -~ 12E.IRT (26) and M= 6E.1R - T (27) Let T 0 he the externally applied torque acting on the shaft ( f i g . 22). Then, T = ZT t - e P R (28) o Therefore ~"T = 3a — • (29) From Eqs. (26) and ( 2 9 ) ? P= I Z E I R . T 0 C V-- zw: — — oo 37 and from Eqs. (25) and (30), M=-£l z H I _ 4 R ( 3 D I2E1R' From Eq. (25) the moment i s zero at the middle of c a n t i l e v e r , t h e r e f o r e at t h i s p o i n t the s t r e s s i s due only t o shear. 38 EXPERIMENTAL STRESS ANALYSIS To verify the results of the numerical investiga-t ion , a slotted duralumin shaft was tested i n torsion and the stresses and r i g i d i t y were determined experimentally. Strain Gauge Theory The strain measuring apparatus employed consisted of SR-4 s t ra in gauges and a Baldwin (SR-4) s train indicator. The SR-4 gauge 20 consists of a grid of small diameter wire ( f i g . 27) cemented to paper backing which i n turn i s cemented by "Duco" cement to the surface to which the strain i s to be measured. When the metal surface elongates due to s t ra in , the wire becomes s l igh t ly smaller and i t s resistance Increases. This Increase i n resistance due to s train i s measured by -the Baldwin (SR-4) s train indicator, which operates on the prin-ciple of the wheatstone bridge, and the strain experienced by the gauge i s read d i rec t ly i n micro-inches from the i n d i -cator. The relationship between "change i n resistance" and "change i n strain" i s a rat io called s t ra in-sens i t iv i ty or gauge factor. where R i s the resistance of the gauge A R i s the change i n resistance, L i s the or ig ina l length of the gauge and Thus, (32) 39 A L i s the change i n length due to s t ra in . I f € i s the unit s train (inches per Inch) experienced by the gauge, then £ = AL , i AR The Baldwin (SR-4) Indicator i s primarily a four-arm wheatstone bridge and the strain gauge i s always used In such a way as to unbalance this bridge c i r cu i t when i t s re-sistance i s changed by the s t ra in . In the usual s t ra in gauge arrangement ( f i g . 28) balance or zero potential across the two resistances, A and C i n pa ra l l e l , i s regained by changing resistance D u n t i l the galvanometer Cq shows zero current. This establishes the fundamental relationship between the resistances A-D=B-C (33) To obtain the effect of a variable resistance C and D, a sl ide wire P i s introduced ( f i g . 2 9 ) , which shows the arrangement i n the Baldwin Indicator. After each measured torque was applied the bridge was balanced again and the corresponding s t ra in experienced by the different s train gauges was read from the strain indicator. In the case of simple tension or compression i t i s only necessary to make a single observation of s t ra in i n the 40 ( ® ) I.I. BALDWIN INDICATOR, II" FIG,. 29 41 direction of tension or compression to determine the stress condition completely since ° ~ = E * £ ( 3 4 ) where <T^ i s the stress intensity i n l b / i n 2 E i s the modulus of e l a s t i c i ty of the material and £. i s the observed strain The following table gives the e las t ic properties and the dimensions of the slotted duralumin shaft tested. TABLE I I I Properties of Test Specimen Form Slotted Shaft Material Duralumin E 2 1 10.3 x 10 6 l b / i n 2 G 2 2 3.85 x 10 6 l b / i n 2 Diameter 2.312 inches Width of Slot 1.030 inches Length of Slot 12.0 inches Ratio W/D 0.445 Effective length of 11.0 Inches slot J x 0.218 i n4 C 0.0475 G (1 .156) 4 R 0.791 inches k l 0.512 k 2 0.490 k 3 0.0475 1/R 13.9 4 2 Method.of A p p l y i n g Torque Consider the method and apparatus used t o apply a pure torque to the s l o t t e d s h a f t ( f i g s . 3 0 and 3 1 ) . The measured torque was a p p l i e d by the a d d i t i o n o f increments of ten pounds t o a 3 4 - i n c h l e v e r arm. One end of the s h a f t was clamped r i g i d l y t o prevent any r o t a t i o n and the other end of the s h a f t was a s l i p f i t i n a r o l l e r h e a r i n g ( f i g s . 3 0 and 3 D , making the f r i c t i o n torque n e g l i g i b l e . The l e v e r arm was pressed i n t o the s o l i d p o r t i o n o f the s h a f t as near the r o l l e r b e a r i n g as p o s s i b l e without having the arm i n contact w i t h the r o l l e r b e a r i n g case ( f i g . 3 D . Thus the bending moment at X due t o the normal weight on the l e v e r arm was approximately c a n c e l l e d out by the bending moment at the same po i n t caused by the v e r t i c a l r e a c t i o n at the b e a r i n g . From f i g u r e 3 1 , the bending moment at X due to the normal weight W i s M* = W x - W / x As X » > A X and V ^ H then M X C O where W i s the normal f o r c e on the l e v e r arm, and W]_ i s the v e r t i c a l r e a c t i o n at the b e a r i n g . The s h a f t was t h e r e f o r e c o n s i d e r e d t o be sub j e c t e d to a pure a p p l i e d torque which was equal to the weight irV m u l t i p l i e d by the l e v e r arm l e n g t h L ( f i g . 3 2 ) , provided the r o t a t i o n was s m a l l . 4 3 Fiq. 30 45 Determination of Stresses The SR-4 gauges were cemented to the duralumin shaft In the positions shown ( f igs . 33 and 34) . Gauge 1 was used to indicate the" s t ra in experienced by the segment of the shaft near the end of the s lo t . The gauge was placed two inches from the end of the slot i n order that the loca l stress dis t r ibut ion around the radius at the end of the slot would not cause a faulty reading to be taken. This reading would not give a true indication of the stresses at that point. At the edge of the segment, point A ( f i g . 35)» the shear stress i s zero and any s t ra in there i s entirely due to a bending stress,, Consider this point A ( f i g . 36) . The bending moment at A i s M A = M - P(2) where M i s the bending moment (Eq .(30)), and P i s the force acting at the end of the segment (Eq. 29). Therefore ^ _ 4 Z _ .191 = , £ B 2 To . — (35) "the bending stress at this point i s given by the equation (36) 46 FIQ. 3 3 47 48 where M A i s the bending moment (Eq.35)» I i s the moment of ine r t i a of the segment about i t s l ine of symmetry, and y i s the distance of the point from the l ine of symmetry. This l ine of symmetry i s the neutral axis of the cantilever beam. T h u S • I2lg-- Z.&S T A — (37) These SR-4 s t ra in gauges cannot indicate the shear-ing s t ra in d i rec t ly but can only indicate tensile or com-pressive s t ra in . As pure shear 2 3 j . s a combination of equal tensile and compressive stresses acting as shown ( f i g . 3 7 ) , then by placing the s t ra in gauge at 4 5 ° to the axis of the shaft, the measured stress i s equal to the pure shear stress. From f i g . 3 7 , ^ure shear T « L Q r r - ^ ) = CT{ ( 38 ) where <r( i s the tensile stress, (J^. i s the compressive stress "V i s the shear stress and 0"s = - o ^ Gauges 2 , 3 were placed at 4 5 ° to the axis of the shaft (f igs. 3 3 , 34) on the c i rcular boundary and on the straight side of the segment respectively, at the positions where the shaft segments were i n pure shear, e .g . , half way between the ends of the s lo t . 50 The tensile or compressive stress thus determined i s equal to the pure shear stress acting at right angles to the axis of the shaft. The maximum torque on this segment from Eq. (29) i s T » T o Z t Z 4 £ l R 7 " — C T T ^ T Q - - ^ \ T 0 (39) where TQ i s the applied torque on the shaft. The angle of twist per unit of length of the seg-ment from Eq.(21) i s C . O S Z G C q - 4-rzjz> X . . . ( 4 0 ) Therefore, GjO = 4 ^ T a (41) The maximum shear stress, point A ( f i g . 20) i s given by where i s taken from ( f i g . 23) for D ~T^s .512 (4.25 T Q) 1.156 = 2.50 T 0 ( 4 2 ) 51 The maximum shear stress on ci rcular boundary, point B ( f i g . 20) i s given by where kg i s taken from ( f i g . 23) for W - 0 4 - 4 5 D l£= : .490 (4.25 T Q ) 1.156 = 2.41 T 0 (43) Determination of the Angle of Twist ' The following outline gives the apparatus and the procedure which were used to determine the angle of twist of the shaft over the length of the slot for various meas-ured torques,acting as shown (f igs . 38 and 39). Flat-surface mirrors were fastened r i g i d l y on the sol id portion of the shaft at each end of the slot ( f igs . 30 and 33) so as to rotate at the same angle as the shaft when the torque was applied. The remainder of the apparatus used consisted of a l e v e l , l eve l , ing rod, and 10-lb. weights. The equipment was positioned as shown ( f i g . 38) with the leve l l ing rod placed 21 feet from the axis of the shaft being tested and the l eve l between rod and shaft. Before the torque was applied, a sight was taken through the l eve l onto the mirror and the scale on the leve l . ing rod was brought into focus i n the mirror and the scale reading, in hundredths of a foot was noted. After the torque was applied, the f i n a l 52 reading i n feet was subtracted from the or ig inal reading and one-half this difference i n feet divided by the distance of the scale from the axis of the shaft gave the angular rotation of the shaft i n radians, provided this rotation was small. Prom figure 39, the angular rotation of the shaft is 2(Z\) radians (43) where £> i s the difference between or ig ina l and f ina l reading on the scale i n feet due to the applied torque. To give the relat ive rotation over the length of s lo t , the -rotations of the mirrors at each end of the slot were sub-tracted after each torque was applied. As the distance be-tween the mirrors was thirteen inches, the theoretical angle of twist between them i s from Eq. (41) (0 . 4 . 25 T «Q V ) 5 =- 1 4 - 4 x l c T 6 T 0 (44) The results , from the experimental analysis, are compared with those from the numerical calculations i n tables I V and V which follow. 53 TABLE IV Comparison between Experimental and A n a l y t i c a l Results for 9 T 0 Applied Torque inch-lbs A n a l y t i c a l Experimental % V a r i a t i o n 169 2.44 x 10-3 2.61 x 10-3 7.00 509 7.30 x 10-3 7.10 x 10-3 2.71 849 12.20 x 10-3 12.51 x 10-3 2.54 1189 17.00 x 10-3 16.60 x 10-3 2.36 1529 22.0 x 10-3 21.40 x 10-3 2.74 TABLE V Comparison between Experimental and A n a l y t i c a l Results for Tensile and Shear Stresses at points shown ( f i g . 30)4 f.£-34 T 0 Applied Torque inch-lbs Gauge #1 p l b / i n * 2 % Var. Gauge #2 . l b / i n ^ % Var. Gauge #3 p l b / i n 2 % Var. 169 An. Exp. 470 500 6.37 394 400 1.52 408 400. 2.00 509 kn. Exp. 1440 1500 4.15 1180 1200 1.70 1230 1300 5.65 849 An. Exp. 2441 2400 1.70 1980 2000 1.00 2040 2100 2.95 1189 kn. Exp. 3370 3300 2.08 2770 2800 1.10 2860 2900 1.40 1529 kn. Bxp. 4350 4300 1.15 3560 3500 1.70 3680 3600 .2.18 54 CONCLUSIONS The agreement between calculated and experimental results i s good; the error being of the same order as the inaccuracy i n the s train measurement. Therefore the follow-ing equations can be used for determining the maximum stresses and the angle of twist per unit of length of a slotted shaft subjected to pure torque T 0 . The formulas are derived i n Appendix B. Maximum shear stress 7point A ( f i g . 20) T , - K.Qq T. ZC + 24-E. l(& Shear stress at B ( f i g . 25) Angle of twist"~per unit of length of slot (46) (47) e = x • a d rad ians Maximum bending stress^ - (48) where W i s the width of s lo t . (49) 55 ^ P P E N D I X A The {ollowing is a list of suitable improvement formulas which can be used for a region shown Cfig.40) where the interval i*o beina cnarxged fromn the value h in one parh or the net ho double thn<=, value in ihe d r i e r part o-f the net. . f •ro • •c • i • a -ci •n • • 'e • 0 0 • *K ' • O 0 • •o • • 1 — Zh J F I G . 4 0 5 6 A P P E N D I X £> C o n s i d e r E C J . . % » k, <56Q b ^ l o w t h e " t o r s i o n a l r i g i d i t y of O se.^rrte^ -from E g . (21 s ) i s lout T = C G t h e n -Tv-ono B £j.("29) the "tcrgue a c l i n ^ o n t h e -se^men-l: is T = X 2 4 - 2 4 E . I R e r e f o r e . 0 - I_ = _ C C ( Z f 2 4 E I S u b s t i t u-KSj " t h i s v a l u e o-C G in ^- c^' I r- - r - —I 1 o 2 C + 2 4 E . L ^ T-b/ <£G) "2. in T h e t Y i a y i r 7 i u i ^ i t e n d i n g YY\OYV\CO\ £rovY\ 1c_^ (21) I L L 4 R . 1 + C I | 2 E . I R X <3nd t h 57 APPENDIX B fconfd) io e n d i n g s t r e s s i s I F r o m F»q.4-l tine v-r-a xIvYium v& w h e r e ^ t J s i W 7 — r a d i o & of t h e s> Inqf-L £*nd W i s t i n e w i d t h 0-f s l o i . TV»ere-fohe , the n'la >arnuyY-) bending s t r e s s is 4-R I 4 C I 1 I2E.K1 58 NOTES 1. See reference No. 10, Bibliography, page 31. 2. See reference No. 10, Bibliography, Chap. 9, page 228. 3. See reference No. 3, Bibliography, pp. 248-259. 4. See reference No. 10, Bibliography, Chap. 9, page 260. 5. See reference No. 4, Bibliography, pp. 469-480. 6. See reference No. 9, Bibliography, page 30. 7. See reference No. 6, Bibliography, paper No. 128. 8. See reference No. 11, Bibliography, page A-71. 9. See reference No. 11, Bibliography, page A-71. 10. See reference No. 11, Bibliography, page A-71. 11. See reference No. 11, Bibliography, page A-71. 12. See reference No. 7, Bibliography, Chap. V. 13. See reference No. 12, Bibliography, page 283. 14. See reference No. 10, Bibliography, page 230. 15. See reference No. 12, Bibliography, page 283. 16. The the shear stress Is proportional to center of twis t . the distance : from 17. As the volume between the torsion function and the section i s proportional to the torsional r i g i d i t y of the section, then a change i n radius w i l l change the r i g i d i t y by a constant equal to the ra t io of the new radius to the previous radius raised to the fourth power. > 18. See reference No. 10, Bibliography, Chap. 9* page 250. 19. This method of solution was suggested by Prof. W.O. Richmond. 59 20. See reference No. 21. See reference No. 22. See reference No. 23. See reference No. 24. See reference No-NOTES (cont.) 1, Bibliography, page 41. 5, Bibliography, page 817. 5, Bibliography, page 817. 10, Bibliography, page 9. 8 j b i b l i o g r a p n s y 5 pa<ge 3 4 & . 6 0 BIBLIOGRAPHY, 1, Gibbons, C.H., "The Use of the Resistance Wire S t r a i n Gage i n Stress Determination," Exper-imental Stress Analysis vol.1, n o . l . 2. Hathaway, CM., " E l e c t r i c a l Instruments for S t r a i n Ana-l y s i s , " Experimental Stress Analysis v o l . l , noo 1. 3. Higgins, T.J., "A Comprehensive Review of Saint Ven-a n t ^ Torsion Problem," Amer. Journal of Physics, Vol.16, No.5, (1942). 4. Higgins, T.J., "The Approximate Mathematical Methods of Applied Physics as Exemplified by Application to Saint Venant's Torsion Problem," Journal of Applied Physics, vol.12, (1943). 5. "Metals Handbook", The American Society for Metals (1948), 6. Orr, J . , "The Torsional Properties of Structural and other Sections," The I n s t i t u t i o n of C i v i l Engineers, Selected Engineering Papers No.128 (1932). 7. Sokolnikoff, I.S. and R.D. Specht, "Mathematical Theory of E l a s t i c i t y . " McGraw-Hill Book Company Inc. (194b) 8. Shortley, G.H. and R. Weller. "The Numerical Solution of Laplace's Equation," Journal of Applied Physics, vol.9, no.5 (1938). 9. Thorn, A and J . Orr, "The Solution of the Torsion Prob-lem for C i r c u l a r Shafts of Varying Radius," Proc.Roy.Soc. (A) vol . 1 3 1 ( 1 9 3 D . 6 1 BIBLIOGRAPHY (cont.) 10. Timoshenko, S., "Theory of E l a s t i c i t y . " McGraw-Hill Book Company, Inc. 1934-• 11. Weller, R. and G.H. Shortley, "Calculations within the boundary of Photoelastic Models," Journal of Applied Mechanics, Trans. A .S .M.E . , v o l . 6 , no.2 (1939). 12. Weller, R . , G.H. Shortley and B. Fr ied, "The Solution of Torsion Problems by Numerical Integration of Poisson's Equation," Journal of Applied Phys-ics vol .11, (1940).
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The effect of a slot on the stresses and rigidity of a shaft subject to pure torque Shumas, Frederick Joseph 1949
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Title | The effect of a slot on the stresses and rigidity of a shaft subject to pure torque |
Creator |
Shumas, Frederick Joseph |
Publisher | University of British Columbia |
Date Issued | 1949 |
Description | This study was made to determine the effect of different width slots on the bending and shear stresses, and on the rigidity of a slotted shaft subject to pure torque. As a first step in the solution of the problem, the stress distribution due to a pure torque, acting on the shaft segment, and the torsional rigidity of the segment were obtained by use of a numerical method of solution and St. Venant’s Principle. After this solution was obtained, the results were used in the calculation of the actual stresses by combining the effects of twisting and bending in the segment. Formulas were developed for obtaining the torsional rigidity, maximum shear and bending stresses resulting from a known applied torque. A slotted shaft was then tested in torsion and the bending, shear stresses, and rigidity were determined experimentally. The agreement between calculated and experimental results was good; the error being of the same order as the inaccuracy in the strain measurement. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-03-08 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0106680 |
URI | http://hdl.handle.net/2429/41237 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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