7.2.1 Basic relations for Torsion of Circular Members

Transcription

1 Section osion In this section, the geomety to be consideed is that of a long slende cicula ba and the load is one which twists the ba. Such poblems ae impotant in the analysis of twisting components, fo eample lug wenches and tansmission shafts. 7.. asic elations fo osion of icula Membes he theoy of tosion pesented hee concens toques which twist the membes but which do not induce any waping, that is, coss sections which ae pependicula to the ais of the membe emain so afte twisting. Futhe, adial lines emain staight and adial as the coss-section otates they meely otate with the section. Fo eample, conside the membe shown in Fig. 7.., built-in at one end and subject to a toque at the othe. he ais is dawn along its ais. he toque shown is positive, following the ight-hand ule (see 7..4). he membe twists unde the action of the toque and the adial plane D moves to D. D L Figue 7..: cylindical membe unde the action of a toque Wheeas in the last section the measue of defomation was elongation of the aial membes, hee an appopiate measue is the amount by which the membe twists, the otation angle. he otation angle will vay along the membe the sign convention is that is positive in the same diection as positive as indicated by the aow in Fig Futhe, wheeas the measue of stain used in the pevious section was the nomal stain, hee it will be the engineeing shea stain y (twice the tensoial shea stain y ). elationship between (dopping the subscipts) and will net be established. s the line defoms into, Fig. 7.., it undegoes an angle change. s defined in 4.., the shea stain is the change in the oiginal ight angle fomed by and a tangent at (indicated by the dotted line this is the y ais to be used in y ). If is small, then R ( L) tan (7..) L the tem toque is usually used instead of moment in the contet of twisting shafts such as those consideed in this section 79

2 Section 7. whee L is the length, R the adius of the membe and (L) means the magnitude of at L. Note that the stain is constant along the length of the membe although is not. onsideing a geneal coss-section within the membe, as in Fig. 7.., one has R ( ) (7..) E E F Figue 7..: section of a twisting cylindical membe he shea stain at an abitay adial location, 0 R, is ( ) ( ) (7..3) showing that the shea stain vaies fom zeo at the cente of the shaft to a R ( L) / L R( ) / on the oute suface of the shaft. maimum he only stain is this shea stain and so the only stess which will aise is a shea stess. Fom Hooke s Law G (7..4) whee G is the shea modulus (the of Eqn. 6..5). Following the shea stain, the shea stess is zeo at the cente of the shaft and a maimum on the oute suface. onsideing a fee-body diagam of any potion of the shaft of Fig. 7.., a toque acts on all coss-sections. his toque must equal the esultant of the shea stesses acting ove the section, as schematically illustated in Fig. 7..3a. he elemental foce acting ove an element of aea d is d and so the esultant moment about 0 is () d (7..5) d ut / is a constant and so theefoe also is / (povided G is) and Eqn can be e-witten as 80

3 Section 7. () () d (7..6) he quantity in squae backets is called the pola moment of inetia of the cosssection (also called the pola second moment of aea) and is denoted by : d Pola Moment of ea (7..7) whee d is an element of aea and the integation is ove the complete cosssection. Fo the cicula coss-section unde consideation, the aea element has sides d and d, Fig. 7..3c, so R R R D (7..8) dd d whee D is the diamete. () () d dd d d (a) (b) (c) Figue 7..3: Shea stesses acting ove a coss-section; (a) shea stess, (b,c) moment fo an elemental aea Fom Eqn. 7..6, the shea stess at any adial location is given by ( ) (7..9) Fom Eqn. 7.., 7..4, 7.6 and 7..9, the angle of twist at the end of the membe o the twist at one end elative to that at the othe end is L (7..0) G 8

4 Section 7. Eample onside the poblem shown in Fig.7..4, two tosion membes of lengths L, L, diametes d, d and shea moduli G,G, built-in at and subjected to toques and. Equilibium of moments can be used to detemine the unknown toques acting in each membe:, 0 (7..) 0 so that and. (a) D L L (b) (c) Figue 7..4: stuctue consisting of two tosion membes; (a) subjected to toques and, (b,c) fee-body diagams he shea stesses in each membe ae theefoe 4 4 whee / 3 and / 3. d d, (7..) Fom Eqn. 7..0, the angle of twist at is given by L / G. he angle of twist at is then L G (7..3) Statically indeteminate poblems can be solved using methods analogous to those used in the section 7. fo uniaial membes. 8

5 Section 7. Eample onside the stuctue in Fig. 7..5, simila to that in Fig only now both ends ae built-in and thee is only a single applied toque,. (a) D L L (b) (c) Figue 7..5: stuctue consisting of two tosion membes; (a) subjected to a oque, (b) fee-body diagam, (c) sepaate elements Refeing to the fee-body diagam of Fig. 7..5b, thee is only one equation of equilibium with which to detemine the two unknown membe toques: 0 (7..4) and so the defomation of the stuctue needs to be consideed. systematic way of dealing with this situation is to conside each element sepaately, as in Fig. 7..5c. he twist in each element is L L, (7..5) G G he total twist is zeo and so 0 which, with Eqn. 7..4, can be solved to obtain L G L G (7..6), LG LG LG LG he otation at can now be detemined,. 83

6 Section Stess Distibution in osion Membes he shea stess in Eqn is acting ove a coss-section of a tosion membe. Fom the symmety of the stess, it follows that shea stesses act also along the length of the membe, as illustated to the left of Fig Shea stesses do not act on the suface of the element shown, as it is a fee suface. ny element of mateial not aligned with the ais of the cylinde will undego a comple stess state, as shown to the ight of Fig he stesses acting on an element ae given by the stess tansfomation equations, Eqns : sin, sin, yy y cos (7..7) 0 y y y yy Figue 7..6: Stess distibution in a tosion membe o Fom Eqns , the maimum nomal (pincipal) stesses aise on planes at 45 and ae and. hus the maimum tensile stess in the membe occus at o 45 to the ais and aises at the suface. he maimum shea stess is simply, with Poblems. shaft of length L and built-in at both ends is subjected to two etenal toques, at and at, as shown below. he shaft is of diamete d and shea modulus G. Detemine the maimum (absolute value of) shea stess in the shaft and detemine the angle of twist at. L / 4 L / 4 L / 84