Question

BRASS STEEL ads Problem 1. Torque To is applied at the midpoint of a solid steel bar with diameter ds and length 1.5 L which is bonded to two brass collars, each with wall thickness tp and length 0.25 L. The twist angle at point C is specified to be an allowable angle oca. Т. To * B 썰 С D E र VN ^^ 4 The steel bar and brass collars are attached to fixed supports at A and B. Equal reaction torques T are applied by the supports to keep the system in equilibrium. The steel and brass have shear moduli Gs and Gp respectively. (a) Find an expression for the twist angle ob at point B in terms of To, L, G5, Gp and the polar moments of inertia Ips and Iph of the steel bar and brass collars, respectively. (b) Find an expression for Iph in terms of Gs, Gb, Ips, To, L, and the allowable twist angle oca: (c) Find an expression for the thickness tp of the brass collars in terms of ds and Ipb

Answer 1

Braus + steel ds T th 4 T → 44 To IB ct from dymmenicity we con conclude that Toque applied by supports (T) = To shown in figure, 2 as in the same direction B • Bray Steel A R Consider this part of steel bor. let To= Torque carried by Brans Steel Ts= Tot Ts=T TB + Ts = To -0 2 = twist at B in Steel rod. (PBA) steel (QBA) brain twist at B in bras collar

(PBA) steel (Por brais Ts* / id= TL TB x 44 Gg Ipb GJ Gs Ips To To Gs Ips TiB Iph from eq. (I) Tst TB - To 2 To Gs Ips GB Ipb + To=To TB = To GB Ipo 2 tbs Ips + GB Iph Now we son find twist at point B. braus TB* Yu GB * Ipb pusting to (dian braus don) brows To the The author Tot L 2 4 L Gg Ips + GB Ipb ToL 8 Gg Ins+ GB Ipb Answer

Solving eqh 4 *[ds + 20] = " Ipb ㅈ 2x² + rds ² – 4 Ipb=0 W = -ds? I dgr _ 4* 2* (-4 Iph) 2*2 W = - dg & ds4 & 32 Ipb 4 no u -ds²+ d +32 Tpb C 4 Now to² & tods = = 6 to²+ ds to -c=0 tB = -ds & J d² +40 2

for 6 > to = -ds + Sds ² + 4C 2 to = -dst. ds² +4 (-ds² + √ds" + 32 Tpb pulting value of c 4 2 to = -dst de _d8+ Jelsh + 32 Ipo 2 to ds" + 32 Tpb 32 Jpk ds Answer 2 ya te - (ds4 + 32 Tp) ds Anywan 2