On the Resistance to Torsion of Certain Forms of Shafting, with Special Reference to the Effect of Keyways . RTAIN FOBMS OF SHAFTING. 351 S, flTC tanh 2 a sinh a 8a (cosh 2a — COS 20) n=<o . 271+It?-^seen -^—~ 2a sinh a „ro (2« + 1)V- + 16«^ — cosh a (sec 2/3 — 1) I 8/3 (cosh 2«-cos 2;g) »= ^ ^ 2/3 smh a Zo 2n-\- W - 16/3 »/-ji) • * * t • • < ! O tJ li S c ^T* = ?— Binh a (1 + s^^h 2«) + 8a (cosh 2a -- COS 2/3) =^. cosh : / cosh -r- :a / ;a OOsh a n~Q til + 1 77^ + 16a^ :=:: •— sillh a (1 + ^ec 2^) . ^^ v„ . 2^/2/ 4- Itt^ , , 2?i + l7ra8/3 (cosh 2a - cos 2/3) ^^ ^~ ^ ^^^ l^S™ ~~W^ a 2 2»
On the Resistance to Torsion of Certain Forms of Shafting, with Special Reference to the Effect of Keyways . RTAIN FOBMS OF SHAFTING. 351 S, flTC tanh 2 a sinh a 8a (cosh 2a — COS 20) n=<o . 271+It?-^seen -^—~ 2a sinh a „ro (2« + 1)V- + 16«^ — cosh a (sec 2/3 — 1) I 8/3 (cosh 2«-cos 2;g) »= ^ ^ 2/3 smh a Zo 2n-\- W - 16/3 »/-ji) • * * t • • < ! O tJ li S c ^T* = ?— Binh a (1 + s^^h 2«) + 8a (cosh 2a -- COS 2/3) =^. cosh : / cosh -r- :a / ;a OOsh a n~Q til + 1 77^ + 16a^ :=:: •— sillh a (1 + ^ec 2^) . ^^ v„ . 2^/2/ 4- Itt^ , , 2?i + l7ra8/3 (cosh 2a - cos 2/3) ^^ ^~ ^ ^^^ l^S™ ~~W^ a 2 2» + 1, V - 16y3 • • (36), whence I find- Table of Stresses. a = 7r/6 a==7r/2 /3-7r/2. /3-37r/4. /3-7r. - -67631-2-0716 - -853973-6144 - GO CC - 7r/6 a = 7r/2 •402651-7276 •426252-0509 a = 7r/6 5C = 7r/2 00 •851062-2794 We see that the greatest of the three stresses occurs at A, ?.e., at the vertex of thehyperbola, and it is probable that A is the true fail-point. The following are the values of Sa/So, So having the same meaning as in § 12 :— Table of Sa/Sq,. f> 0 A ON THE EESISTANCE TO TOEBION OF CERTAm FOR]\iB OF SHAFTING. The values of the ^ efficiency are easily obtained. a ~ 7r/6 a^=^7r Table of E, /3 = 7r/2. /^-=37r/4. * •7361 7171 0 •6485 •4728 0 The above results do not need any detailed discussion. We see that in all casesthe maximum stress is greater than the maxim inn stress for the circular the efficiency is always less than unity. If wq compare these values of theefficiency with those in § 19, we see that, on the whole, the rule holds that the morecompact the section the higher its efficiency. On the other hand, by indenting asection we render it less efficient. § 24. Conclusion and Stmimarj/, Looking back upon the results of the pajDer, we see that the study of these specialforms of cross-section sheds new light upon several little-explored parts of the theoryof elasticity. It confirms to a great
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