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 p a r t i a l fulfilment of the requirements for the Degree o f Master of Applied Science . i n the Department of Mechanical and E l e c t r i c a l Engineering The University of B r i t i s h October, 1949 Columbia ABSTRACT This study was made to determine the e f f e c t of d i f f e r e n t width s l o t s on the bending and shear stresses, and on the r i g i d i t y of a s l o t t e d shaft subject to pure torque. As a f i r s t step i n the s o l u t i o n o f the problem, the stress distri- bution due to a pure torque, acting on the shaft segment,,and the t o r s i o n a l r i g i d i t y of the segment were obtained by use of a numerical method of solution and St. Venant^ P r i n c i p l e . After t h i s solution was obtained, the r e s u l t s were used i n the c a l c u l a t i o n of the actual stresses by combining the effects o f twisting and bending i n the segment. Formulas were developed f o r obtaining the t o r s i o n a l r i g i d i t y , maximum shear and bending stresses r e s u l t i n g from a known applied torque. A slotted shaft was then tested i n 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 s t r a i n measurement. ACKNOWLEDGMENT The author wishes to express his thanks to Professor W. 0. Richmond, who gave invaluable suggestions and c r i t i c i s m s throughout t h i s 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 BIBLIOGRAPHY GO g NOMENCLATURE Applied torque T Torque on segment M Bending moment Rectangular co-ordinates T Shearing stresses p a r a l l e l to x, y axes on the plane perpendicular to the z a x i s . T 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 . Moment of i n e r t i a of segment I A Area of segment f 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, ,k Shear stress coefficients 2 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. torque The T was considered to be acting as shown ( f i g . 1)„ 0 As a f i r s t step i n the solution of the problem, the stress d i s t r i b u t i o n due to pure torque T, acting on a shaft segment ( f i g . 2), was obtained by the use of a numeric a l method of solution and St. Venants' P r i n c i p l e . 1 After t h i s 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 t o r s i o n a l r i g i d i t y , maximum shear, and bending stresses resulting from a known applied torque. jected to A slotted shaft was then sub^ 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 . / ' / - T 1 \ > ' \ 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 a l s o found by experiment and the agreement between t h e o r e t i c a l and experimental r e s u l t s was found t o be w i t h i n the l i m i t s o f experimental e r r o r . THE TORSION PROBLEM St. Venant showed that the shear s t r e s s d i s - 2 t r i b u t i o n of a shaft of constant c r o s s - s e c t i o n subject t o torque could be obtained by the i n t r o d u c t i o n of a s t r e s s function (j) which i s a s o l u t i o n of the p a r t i a l d i f f e r e n t i a l equation ^ ^x + 2 ^ - e q e by = • x where x and y are rectangular l u s o f r i g i d i t y , and 0 co-ordinates, (1) Gj i s the modu- i s the angle of t w i s t per u n i t of length. In terms of t h i s s t r e s s f u n c t i o n , the shear stresses as i n d i c a t e d ( f i g . 3) are given by - - - - - - ca) =-c)<t> -r 7 5 dx (3) Equations 2 and 3 state that the change i n t o r s i o n function (J) i n the x and y d i r e c t i o n s i s equal to the shear 4 Y Fiq.4 5 stresses i n the x and y d i r e c t i o n s , respectively. On the boundary of the section ^ 4 = Q (4) is which follows from the condition that the r e s u l t i n g shear stress i s i n the d i r e c t i o n of the tangent to the boundary. Equation (4) must be s a t i s f i e d i f the l a t e r a l surface of the section i s to be free from external forces. If the stress function^ (|) Is put i n the form (|).Vp-(^Cx* y^ + . ... ( 5 ) then from Eq.(1) W T ~ O — (6) and from Eq.(4) IU - Q 6 ( x f y ) = a c o n s W \ i 2 • which hold l (7) 2 throughout the section and along the boundary, respectively. In the case of singly connected boundaries, e.g. f o r s o l i d bars, t h i s constant can be chosen a r b i t r a r 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 ) (8) z on the boundary. By considering the d i s t r i b u t i o n of shearing s t r e s s e s over the c r o s s - s e c t i o n s of the bar, i t i s found that the r e l a t i o n s h i p between the a p p l i e d torque and the t o r s i o n f u n c t i o n (j) f o r a uniform s e c t i o n I s given by the following T= where T zff(j)dxdy ( ) 9 equals the torque a c t i n g on the s e c t i o n . Summarizing, the f u n c t i o n (J) when p l o t t e d 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 s t r e s s at that p o i n t ; and twice the volume included between the- (j) surface and the xy plane gives the torque a p p l i e d t o the s e c t i o n . S t . Venant's equations have been solved e x p l i c i t l y f o r a number of d i f f e r e n t c r o s s - s e c t i o n s and. the chronologica l development i n the fundamental theory and i n the exact s o l u t i o n s e f f e c t e d up t o 1942 o f t h i s problem was given by 3 T . J . Higgins. However f o r an a r b i t r a r y shape of cross- s e c t i o n i t may not be p o s s i b l e t o o b t a i n a s o l u t i o n I n 7 a n a l y t i c a l 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) v a r i a t i o n a l , (2) graphic, (3) and arithmetic — according as they are developed from the theory of the calculus of v a r i a t i o n s , from the use of graphic analysis, or from the theory of l i n e a r difference equations. The arithmetic method Is the one used i n t h i s i n vestigation and the general scheme of solution involves replacement of the p a r t i a l d i f f e r e n t i a l equation and the boundary conditions defining the problem by an analogous p a r t i a l difference equation and equivalent boundary conditions, and development of a sequence of approximate numeric a l solutions of the difference problem. This general theory for solving the torsion problem by f i n i t e differences was f i r s t used by Runge, 5 w n o determined and plotted the stress l i n e s i n a cross-shaped area comprised of five squares. A.Thorn and J . Arf£_^ ex- tended this method for solving the t o r s i o n a l properties for c i r c u l a r shafts of varying r a d i i . J . OTT(J) also used the numerical method for the solution of the torsional propert i e s of an I-section and a key-wayed shaft and found close agreement between theoretical and experimental r e s u l t s . ^ ($ 8 THEORY OF NUMERICAL SOLUTION Before the torsion function (£> can be solved for the given region ( f i g . 3), the values of same region are required. throughout These values of i p the 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 ax by numerical methods. 2 ^ *. v Any numerical method of handling a d i f f e r e n t i a l equation Involves the replacement of points i n the i n t e r i o r and on the boundary of the region by a discrete set of points. Leibmann developed a convenient procedure for solving for unknown values of \\) fying Laplace's Equation. at i n t e r i o r points satis- 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 i n t e r i o r points. The net i s traversed repeatedly i n some definite order as A, B , C, D e t c . , and the value at each i n t e r i o r point i s replaced by the average of i t s values, neighbouring points ( f i g . 5). of f i n i t e at the four This follows from the theory 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 l d the true derivatives. S i m i l a r l y the Laplacian Equation = 4 ( ^ where m - i p ) do) 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 associated with each point inside the net and a single number with each boundary point. The top number at each i n t e r i o r . point i s a value a r b 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 order. Previously altered values are used whenever possible i n calculating the improved value at a given point. 10 Fiq. 5 jt>< \ 77.0 40.5. K \ !©0.0 64 0 \ \ 15.2 \ 7O.0 £6.7 40.8 k?-o Fiq.6 TS.O pro-4 _uoo \ \ 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 applicable. Figure 7 shows such a point. Here two neighbours l i e atdistances which are less than the net spacing h. value of The formula for the improvement at this point i s HUM S+t sc.\+s> lit i+s (ii) tu+l) The values of S and t here represent fractions of the normal net. spacing and as a check the sum of the coefficients C^, C » C^, and 2 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 f e r e n t i a 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 ^ l y In the improvement of the succeeding points. traversed u n t i l the values of L|J immediateThis net i s 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. of I f the values l p at the same point from the two nets d i f f e r 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 is • very slow and i n order to increase i t s r a p i d i t y and reduce the labour required, both the immediate and diagonal neighbours are used ( f i g , 9) \Z The formula t O l p .4(lJJ +^ + ^ + U 0 ^ * ^ + ^ ^ t l 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 \ \ j 0 M" i / /' / / 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 i n t e r i o r points. As t h i s represents the segment of a 2-inch diameter shaft, necessitated E q s . ( 5 ) and (8) it being changed to (J)»l|j-ei£LCx*+yO ~ (13) . (14) Zoo and *4J - QO (x^+y*) 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 i n t e r i o r 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 the finer net became stationary,, at points throughout 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, respectively, to bring to i t s stationary values at points throughout the segment. A £ - i n c h net was placed over the quadrant ( f i g . 12) and the values of were calculated for 200 all points of the net. These values were used i n Equation 13, l^J _ (^9 ( x + y ) r> each point ZOO ° of the segment where had been calculated for the same a < l t o s o l v e f o r a t point. The 17) f i n a l results are shown ( f i g s , 13, 14, 15, 16, for the different segments where each segment has an angle of twist of 200 ^ (q .radians per unit of length. -4\ i_ \ >—Tv ! so.o !^;o & 5 . 0 n^o'BTo 4<?).0 ^ 4 ? 3kO _ [ 4 2 \ J41-.0 Flq.12. VALUES OF zoo (r*) Foe SEM\-C»RCL&. -H ;G40 " [ 7 2 - 3 pl-d _ i ^ C L J 7 Z 3 _ J8L0. 9 0 3 19 o o ! o 0 4o 5.5 8-7 4.1 78 9.0 i'11.8 lit o 2-8 7Z 147 3~4" 177 15.4 147 2o.8 22 2 !Z4o !(4o 77 78 120.(0 323 3(-8 wr %0 354 .333 p 147 •2J2. 29=8 76 "TJ4.'8"!207 |25.6 "|3o.( "|'S.2~ " S ' " 3T4";38.l "j3gl ^~~^4>.T" 34*2 0 l7(o 237 %0 2^> (IT 204 : 2<b*> 343 25q I '348 34 3 7-4 13.5 Z27 277 3(^ ; 153 22^ 273 337 287 ;25.2 'il4£" 519 3o.<3 20.1 ! lb.8 27.^) 271 I 3* 23, | 2.8-8 13,1 97 j 172 233 22>3 14 2(4- 28.4- 253 I B.5 O |C<2. 25.o 25.5 2 5 3 181 20. 237 223 '2o.8 9* 'I'D- ft i 18.2 12.4- \3I 15. I04 |\7o 2.7.1 . J V l j|g3> /2-b n"3'"K^ !2o.^"l25^ ~SoT 33.4%\ "'^.7 "38? . 1|43 12(0 J2£,0 3o.S .'8U 318 Fiq. 13 of V^LUE.^ 357 3&D '.345 3 4 4 3 ( 4 4> FOR 5ELMI-CIRCI_E 28.0 237 ; I 8 T fox , 7 4 0 20 :7i 17V W 25.2 ~7_4.r 14^ 13*5 70 175 £|.4 Z5.8 W 15% 178""~W-4 £14 £4o IS.8 111 Wo %1 ii "Z42 ; \*1 jIB.b ~ 14£ " I ' "' j 1 r?o i t j it i !6.9 pro I'M 12^. ff5 Fiq.14 V A L U E S !Z73 O F 6 F O R W •-• . D & US 'i25.o .ZQ jZi-t ;i95 15.7 in 3 :.||.3 x:*> 21 0 2.1 0 / 24. | i ! 0 ! iI ti 1 \ \ - V i TG.4 1 O 79 i jk.o i i 3.8 i 9.3 .. 1 • jT.T 1 i 0 j i | I i 33 52 \4r3 "(T4~ I 144 i i i ITO ! t 3a 103 |I78 IG.O 3.6 M 74 574" )5.\ I | • i JI84 • m |»T7 $3 ( S T ji i i 57""" lo-4 140 16.4 \78 —-—>• i j 6.6 \J | - ' i t 131 Oi 58 i 11-8. s.z 0 H2 i \41 0 . | 12-8 ; 0 ! i 9.0 0 (23 \ 13 !• I 10.5 ; i ; ^ • ! 1 46 i i / i 1 3| / i (|.<3 ! 1 5-3 103 I5T-3 I37 K>& 143 11.4 , 72. \7( 15.1 ! I8.5 • » f 57 10-4- 4o |64"~ • i • ]4o~" 0 (8.3 i 58 (8.8 -I78 \5.3 |12- 6 • 1 lB.8 • I ]I4.1 171 J87 5"7 . lo.4 JI4-.I ICS; ^87 ... £ 8 o „ \0S FlQ.I5_. VALUED OPCJ) FOR 137""" I94 I7-8 ^ 8 . . >; .. W« .30 16-0.. ?-4 B-7 22 '•2.{ !0 3.6 .0 5.2 45 4 8 2-e 0 to" 7-T i 7<3 t.Y £.6' : 8* 56 44 ! 103 7-T 4 3 8.4 8 0 in 110-7 7Z (46 j K).*> 4 S &2 )0.5 12-Y \\Z .13.0 124 4.6 ss~ 107 |26 6.8 HI 125 as Hi "' ib.^ 14.1 14^ j o 4 7 147 ilo. I 3-3 9.0 1.7 122 ||5l 4^ i"5l "Tsi"~"ti4^~^B3" &io K 4-^ F\qJ6 =5 il4i iS^> VALUE.** (43 153 15 or r 10 I SIT 10.1 ii>7 0;4-n 4 ? 5 . 5 (J) F O R ^ 113 8- 1\4T TPS" 11-5 .40 5.1 23 1CJ 1-2 i.<5 2.1 2.0 • W- O 31 10 27" 5.4- 1 34 :3.G t.5' 27 ^ 6 " G.6" •8.8' 3s~ ; 8.^ 3.6" iG'.-T 87 3.5 37 70 ao 1103 3-1 i 103 ;3<b I0.3 10.1- jll.O 33 " 7.3 4-2 14, 75 t 17 10.1 71,5 4.4 '' 7^ !4A piq.17 ! 77 87 ; 13.9 V A L U E S O F ill* <$> 75 5^.2 70";"3.^T 3.6 9.<b 76 107 3.1 ; 103 K.O 9cT ,i7.k;ii.3" *3.6 |UG F O P , W ~ 71 18 74 . 5 0 14.6 0 24 From Eq, (9) but where Q T = C0 G i s "the t o r s i o n a l r i g i d i t y o f the s e c t i o n and i s the angle o f t w i s t per u n i t o f l e n g t h . As ( q 0 has been made e q u a l t o 200, then the t o r s i o n a l r i g i d i t y i s C \oo ffadx^y JJ — . (15) From Eq. (9) the torque producing shear i n t o r s i o n i s g i v e n by twice the volume i n c l u d e d between the (J) and the xy p l a n e . surface To compute the volume, i t was convenient t o i n t e g r a t e along a complete s e t o f p a r a l l e l l i n e s indivi- d u a l l y and then i n t e g r a t e the r e s u l t i n g sums i n the c r o s s direction, z^. (13 ) The area under the curve ( f i g . 18) i s g i v e n by\_y A (16) The volume under the (|) Apply Eq. (16) was computed as f o l l o w s : B-B , 1 C-C , 1 s u r f a c e and the xy plane e t c . ( f i g . 19) along l i n e s A-A , o b t a i n i n g a sum f o r each. Then i n t e g r a t e from top t o bottom u s i n g the sum along A-A 4>, , the sum along B-B 1 as (J) , e t c . ,in E q . ( l 6 ) . 1 1 as The 26 r e s u l t was the desired volume. The shear stresses at d i f f e r e n t points i n the seg,14 which defines the change In (b ment are equal to cm T when moving normal to the l i n e of shear stress at the point i n 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 i n t e r s e c t i o n with the boundary, where (£) equals zero, and the two nearest i n t e r i o r points along the l i n e A-B ary and ( f i g . 20). , be sh I f the distance between the bound- ( f i g . 21), where S i s a f r a c t i o n of the net i n t e r v a l h, the magnitude of the slope at the boundary i s then given by a>4>_ 15 i (s+t<t>, - _s_ 4> > ....— ( 1 7 ) 27 /sh h Fiq.21 28 RESULTS AND CALCULATIONS The previous calculations were repeated for shaft segments of slotted shafts having a r a t i o of slot width W to diameter of shaft D of . 2 , , 3 , . 4 , . 5 and for a semi- c i r c u l a r shaft, and the values of the stress function ^) for these are shown ( f i g s . 1 3 , 14, 1 5 , 16 and 1 7 ) . Lines passing through constant values of (J) give the d i r e c t i o n of the shear stress. The maximum shear stress which occurs at point A (fig. 20) can be expressed by the equation 16 Y where (q k-^ s — ». k,(q0.a (18) i s the factor depending on r a t i o ^ i s the modulus of r i g i d i t y , 0 ( f i g . 22), 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 r c u l a r boundary at point B ( f i g . 20) can be expressed by the equation ~Y = k (qO a C where k 2 (19) z i s a factor depending on the r a t i o 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 f o r the segment can be ex17 pressed by the equation (20) C =k^G^Ga where ko i s a f a c t o r depending on the r a t i o D ( f i g . 23) The values o f the f a c t o r s k^, k , and k^ are shown 2 i n the f o l l o w i n g t a b l e and can also be found from graphs ( f i g s . 2g and 2«). TABLE I Shear S t r e s s and R i g i d i t y C o e f f i c i e n t s f o r Various Segments. Segments o f Shafts having the f o l l o w i n g r a t i o o f s l o t width to diameter o f shaft Semi-Circular Shaft w D * k l k 2 0.865 0.725 0.298 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 s t r e s s d i s t r i b u t i o n i n torsion for a semi-circular shaft. H i s values are compared i n Table I I w i t h those obtained from t h i s investigation. 31 FOR SEQMEKIT 32 TABLE I I Comparison between C o e f f i c i e n t s o f S t . Venant and those from t h i s I n v e s t i g a t i o n f o r a SemiC i r c u l a r Shaft. ' k k l 2 k 3 St. Venant 0.849 0.719 0.296 This I n v e s t i g a t i o n 0.865 0,725 0.298 Percent V a r i a t i o n 1.89 0.84 0.67 A p p l i c a t i o n t o a S l o t t e d Shaft In applying the foregoing r e s u l t s t o the case o f 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 p o r t i o n o f the member. The equations o f S t . Venant were developed on the assumption that the s e c t i o n was f r e e t o warp. I f the sec•t t i o n i s prevented from warping, then bending s t r e s s e s are set up on each side of the s l o t as w e l l as shear stresses due t o the t w i s t i n g moment. The bending moments are c a l c u l a t e d by assuming that the metal on each side o f the s l o t behaves as a c a n t i l e v e r beam ^ w i t h a torque also a c t i n g and that the r o t a t i o n o f each segment i s the same ( f i g . 25). In the f o l l o w i n g c a l c u l a t i o n s the r o t a t i o n s of the segment were considered s m a l l . If T i s the torque c a r r i e d by each segment, then the r o t a t i o n o f each segment (D is 33 BEFORE ROTATION AFTER. ROTATION 34 given by <P • X ! C where 1 (2D i s the effective length of the s l o 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 rotations are small, where R i s the distance from the axis of the shaft to the centroid of the segment. a force P and a bending moment deflection ( f i g . 25). as the This introduces M tending to reduce the With these forces acting on a c a n t i - lever beam, the deflection i s given by S =vR<p»ELL - M J s BET — — - (22) 2 SET and the slope i s given by Slope = Ri ZE\ where (23) - Mi ET 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 i n e r t i a of segment about axis X-X ( f i g . 26). As the s o l i d shaft prevents a change i n slope of the cantilever at the ends of the s l o t , P i * -±41 2EI E I the slope here is ( ) 24 35 36 Thus (25) M = PI z cp--±ip^, p i n , P I * R<3E1 4 E 1 i 12'EIR Equating the two values of CD 5 12EIR C (26) P -~ 1 2 E . I R T and Let 6E.1R-T M= T 0 (27) he the externally applied torque acting on the shaft ( f i g . 22). Then, T = ZT t - e P R (28) o Therefore ~"T = From Eqs. (26) and ( 2 9 ) —• 3a (29) ? P= I Z E I R . T 0 C V- zw: — — oo 37 and f r o m E q s . (25) (30), and M=-£l z _ H I 4R (3D I2E1R' From E q . (25) therefore t h e moment i s z e r o at t h i s point at the middle the s t r e s s i s due o n l y of cantilever, t o shear. 38 EXPERIMENTAL STRESS ANALYSIS To verify the results of the numerical investigat i o n , 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 s t r a i n measuring apparatus employed consisted of SR-4 s t r a i n gauges and a Baldwin (SR-4) s t r a i n i n d i c a t o r . The SR-4 gauge 20 consists of a grid of small diameter wire (fig. 27) cemented to paper backing which i n turn i s cemented by "Duco" cement to the surface to which the s t r a i n i s to be measured. When the metal surface elongates due to s t r a i n , the wire becomes s l i g h t l y smaller and i t s resistance Increases. This Increase i n resistance due to s t r a i n i s measured by -the Baldwin (SR-4) s t r a i n indicator, which operates on the principle of the wheatstone bridge, and the s t r a i n experienced by the gauge i s read d i r e c t l y 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 r a t i o called s t r a i n - s e n s i t i v i t y or gauge factor. Thus, (32) where R i s the resistance of the gauge A R i s the change i n resistance, L i s the o r i g i n a l length of the gauge and 39 AL i s the change i n length due to s t r a i n . If € is the unit s t r a i n (inches per Inch) experienced by the gauge, then £ = A L , i AR The Baldwin (SR-4) Indicator i s primarily a fourarm wheatstone bridge and the s t r a i n gauge i s always used In such a way as to unbalance this bridge c i r c u i t when i t s resistance i s changed by the s t r a i n . In the usual s t r a i n gauge arrangement ( f i g . 2 8 ) balance or zero potential across the two resistances, A and C i n p a r a l l e l , i s regained by changing resistance zero current. D u n t i l the galvanometer Cq shows This establishes the fundamental relationship between the resistances A-D=B-C (33) To obtain the effect of a variable resistance and D, a s l i d e wire C 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 r a i n experienced by the different s t r a i n gauges was read from the s t r a i n indicator. In the case of simple tension or compression i t i s only necessary to make a single observation of s t r a i n i n the 40 ( ® ) BALDWIN INDICATOR, I.I. II" FIG,. 29 41 direction of tension or compression to determine the stress condition completely since °~ = E * £ <T^ i s the stress intensity i n l b / i n where E ( 3 4 ) 2 i s the modulus of e l a s t i c i t y of the material and £. i s the observed s t r a i n The following table gives the e l a s t i c 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 1 0 6 lb/in 2 G 2 2 3.85 x 1 0 6 lb/in 2 Diameter 2.312 inches Width of Slot 1.030 inches Length of Slot 12.0 inches Ratio W/D 0.445 11.0 Inches Effective length of slot J x 0.218 in 4 C 0.0475 G ( 1 . 1 5 6 ) R 0.791 inches k k k 0.512 l 2 3 1/R 0.490 0.0475 13.9 4 42 Method.of A p p l y i n g Torque Consider pure torque t h e method and a p p a r a t u s u s e d t o a p p l y to the s l o t t e d shaft measured t o r q u e was a p p l i e d pounds t o a 3 4 - i n c h ten clamped r i g i d l y of t h e s h a f t was a s l i p 3D, was p r e s s e d roller as p o s s i b l e with the r o l l e r moment a t X bearing hearing o f the shaft without having (fig. 3 D . W as n e a r t h e Thus t h e bending on t h e l e v e r arm was o u t by t h e b e n d i n g moment a t t h e same caused by the v e r t i c a l normal weight 3 0 and t h e arm i n c o n t a c t reaction a t the bearing. From f i g u r e 3 1 , t h e b e n d i n g moment a t the (figs. T h e l e v e r arm portion case cancelled was a n y r o t a t i o n and t h e o t h e r e n d due t o t h e n o r m a l w e i g h t approximately point One end o f t h e s h a f t torque n e g l i g i b l e . i n t o the s o l i d bearing arm. f i t i n a roller making t h e f r i c t i o n The by t h e a d d i t i o n o f i n c r e m e n t s o f lever t o prevent 3 0 and 3 1 ) . (figs. a X due t o is M* = W x - W / x As M then where and W W]_ a pure multiplied the X C i s the v e r t i c a l reaction r o t a t i o n was a t the b e a r i n g . considered t o r q u e w h i c h was e q u a l by the l e v e r small. H o n t h e l e v e r arm, s h a f t was t h e r e f o r e applied ^ O i s the normal f o r c e The to and V X »>AX arm l e n g t h t o be subjected t o t h e w e i g h t irV L ( f i g . 3 2 ) , provided 43 Fiq. 30 45 Determination of Stresses The SR-4 gauges were cemented to the duralumin shaft In the positions shown ( f i g s . 33 and 3 4 ) . Gauge 1 was used to indicate the" s t r a i n experienced by the segment of the shaft near the end of the s l o t . The gauge was placed two inches from the end of the slot i n order that the l o c a l stress d i s t r i b u t i o n 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 i n d i c a t i o n 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 r a i n there i s e n t i r e l y due to a bending stress,, Consider t h i s point A ( f i g . 3 6 ) . The bending moment at A i s M = A M - P(2) where the bending moment (Eq.(30)), and P the end of the segment (Eq. 2 9 ) . Therefore ^ _ 4 = , £ B 2 To . Z _ M is i s the force acting at .191 — (35) "the bending stress at t h i s point i s given by the equation (36) 46 FIQ. 33 47 48 where M A i s the bending moment (Eq.35)» I i s the moment of i n e r t i a of the segment about i t s l i n e of symmetry, and y i s the distance of the point from the l i n e of symmetry. This l i n e 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 r a i n gauges cannot indicate the shearing s t r a i n d i r e c t l y but can only indicate tensile or compressive s t r a i n . As pure shear 3 j . 2 sa combination of equal tensile and compressive stresses acting as shown ( f i g . 3 7 ) , then by placing the s t r a i n 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 where T «LQrr- ^ ) = CT{ (38) <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 (figs. 3 3 , 3 4 ) on the c i r c u l a r 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 l o 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 t h i s segment from Eq. (29) i s T» T o Z T T Q 7 —CT ^ - where t Z 4 £ l R " T -^ Q \ T (39) 0 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 .OSZGCq C - 4-rzjz> X GjO = 4 ^ T Therefore, ... ( 4 0 ) (41) a The maximum shear stress, point A ( f i g . 20) i s given by i s taken from ( f i g . 23) where for D ~T^s.512 (4.25 T ) 1.156 = 2.50 Q T ( 0 4 2 ) 51 The maximum shear stress on c i r c u l a r boundary, point B ( f i g . 20) i s given by where kg for i s taken from ( f i g . 23) W D - 04-45 l £ = : . 4 9 0 (4.25 T = 2.41 T Q ) 1.156 (43) 0 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 measured torques,acting as shown ( f i g s . 38 and 39). Flat-surface mirrors were fastened r i g i d l y on the s o l i d portion of the shaft at each end of the slot ( f i g s . 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 e v e l , ing rod, and 10-lb. weights. The equipment was positioned as shown ( f i g . 38) with the l e v e l l i n g rod placed 21 feet from the axis of the shaft being tested and the l e v e l between rod and shaft. Before the torque was applied, a sight was taken through the l e v e l onto the mirror and the scale on the l e v e l . i n g rod was brought into focus i n the mirror and the scale reading, i n 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 o r i g i n a l reading and one-half t h i s 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 t h i s rotation was small. Prom figure 39, the angular rotation of the shaft 2(Z\) is radians (43) where £> i s the difference between o r i g i n a l and f i n a l reading on the scale i n feet due to the applied torque. To give the r e l a t i v e rotation over the length of s l o t , the rotations of the mirrors at each end of the slot were subtracted after each torque was applied. As the distance between 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 T 6 0 (44) The r e s u l t s , from the experimental analysis, are compared with those from the numerical calculations i n tables I V and V which follow. 53 TABLE IV T Experimental Analytical A p p l i e d Torque inch-lbs 0 % Variation 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 21.40 x 10-3 2.74 TABLE V T Comparison between E x p e r i m e n t a l and A n a l y t i c a l Results for 9 x 10-3 Comparison between E x p e r i m e n t a l and A n a l y t i c a l R e s u l t s f o r T e n s i l e and Shear S t r e s s e s a t p o i n t s shown ( f i g . 30)4 f.£-34 Applied Torque inch-lbs Gauge 0 #1 lb/in* 470 % p Var. 2 169 An. Exp. 509 Exp. kn. 1440 1500 4.15 849 An. Exp. 2441 2400 1.70 kn. 3370 3300 4350 4300 1189 1529 Exp. kn. Bxp. 500 6.37 2.08 1.15 Gauge #2 . lb/in^ 394 400 1180 % Var. 1.52 1200 1980 2000 1.70 2770 2800 3560 3500 1.10 1.00 1.70 Gauge #3 % p Var. 408 400. 2.00 lb/in 1230 1300 2040 2100 2860 2900 3680 3600 2 5.65 2.95 1.40 .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 t r a i n 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 subjected to pure torque T . shaft The formulas are derived i n 0 Appendix B. Maximum shear stress point A ( f i g . 20) 7 T. T , - K.Qq ZC + 24-E. (46) l(& Shear stress at B ( f i g . 25) (47) Angle of twist"~per unit of length of slot e = x r•aadd i a n s - (48) Maximum bending stress^ (49) where W i s the width of s l o t . 55 ^PPENDIX 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 p a r h or the n e t ho double thn<=, value in i h e d r i e r part o-f the net. .f •ro • •c • 0 0 •a -ci • 'e •i *K ' • • O 0 •o • • 1— FIG. 4 0 •n Zh J • • 56 APPENDIX £> Consider ECJ. » k, <56Q .% b^low t h e " t o r s i o n a l r i g i d i t y E g . (21 ) i s of O se.^rrte s lout T = C G then -Tv-ono the -se^men-l: T = B £j.("29) the aclin^ on "tcrgue is X 2 4-24E.I R ere f o r e . 0 - Substitu-KSj I_ = _ C C ( Z f 2 4 E I "this value r- I o-C G in - r 1 o ^- ^' —I 2 C + 2 4 E . L ^ T- The tending t Y i a y i r 7 i u i ^ i YY\OYV\CO\ (21) I L L 4R . + C 1 I |2E.IR X c b/ "2. in £rovY\ 1c_^ <£G) 57 APPENDIX <3nd t h B fconfd) stress io e n d i n g is I From tine F»q.4-l v-r-a xIvYium where v& ^ tJ s i W — 7 radio width & of t h e s> Inqf-L £*nd W is tine 0-f s l o i . TV»ere-fohe , the n'la >arnuyY-) 4-R I4 CI 1 I2E.K 1 bending stress is 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 shear stress Is proportional to the distance :from the center of t w i s t . 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 r a t i o 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 NOTES (cont.) 20. 21. See reference No. 1, Bibliography, page 41. See reference No. 5, Bibliography, page 817. 23. See reference No. 5, Bibliography, page 817. See reference No. 10, Bibliography, page 9. 24. See reference No- 22. 8 j bibliograpnsy 5 pa<ge 34&. 60 BIBLIOGRAPHY, 1, Gibbons, C.H., "The Use of the Resistance Wire S t r a i n Gage i n S t r e s s Determination," Experimental S t r e s s A n a l y s i s v o l . 1 , n o . l . 2. Hathaway, CM., " E l e c t r i c a l Instruments f o r S t r a i n Anal y s i s , " Experimental Stress A n a l y s i s v o l . l , noo 1. 3. Higgins, T.J., "A Comprehensive Review of Saint Vena n t ^ Torsion Problem," Amer. J o u r n a l of P h y s i c s , Vol.16, No.5, (1942). 4. Higgins, T.J., "The Approximate Mathematical Methods of Applied Physics as Exemplified by A p p l i c a t i o n to Saint Venant's T o r s i o n Problem," Journal of Applied P h y s i c s , vol.12, (1943). 5. "Metals Handbook", 6. Orr, J . , 7. S o k o l n i k o f f , 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. S h o r t l e y , G.H. 9. Thorn, A and J . Orr, "The S o l u t i o n of the Torsion Problem f o r C i r c u l a r Shafts of Varying Radius," Proc.Roy.Soc. (A) v o l . 1 3 1 (193D. The American S o c i e t y f o r Metals (1948), "The T o r s i o n a l P r o p e r t i e s of S t r u c t u r a l 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). and R. Weller. "The Numerical S o l u t i o n of Laplace's Equation," J o u r n a l of Applied P h y s i c s , v o l . 9 , no.5 (1938). 61 BIBLIOGRAPHY (cont.) 10. Timoshenko, S., 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 . F r i e d , "The Solution of Torsion Problems by Numerical Integration of Poisson's Equation," Journal of Applied Physi c s v o l . 1 1 , (1940). "Theory of E l a s t i c i t y . " McGraw-Hill Book Company, Inc. 1934-•
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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 |
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UBCV |
Scholarly Level | Graduate |
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