Question

Problem 2 (60 POINTS)

(20 Points for each question)

Fig 1a shows a hollow tube with internal diameter of 230 mm and thickness of 30 mm, as shown in Fig 1b. The length of the tube is 1m.

Fig 1c shows a solid circular bar which also has the same length and the same weight as the hollow circular bar. Both the bars are made of same material (have same density or specific weight)

Fig 1a

Fig 1b Fig 1c

a) Calculate the radius of the solid circular bar.

b) Calculate the Polar moment of Inertia for both the bars.

c) Mention which one has a bigger value of Polar Moment of Inertia (J). Explain in details the significance

of having a bigger value of J considering both the bars are only under torsional load. Your explanation may be supported by formulas and illustrations.

Answer 1

D = 176.635 mm Solution page so Solid bar Hollow tube 2.1) 30mm Length Do Di +2+ = 230 + 2x30 = 240 mm bar. given weight of hollow tube = weight of dolid • (Areax Length x density) = (Area x length x densit is same for both But given length and density Hence (Avea) Howow = (Area) solid 9 I(A?- D;-) = 1202 1/4(2902_2304) > /D? Radius [R = 0/2 = 88.318om ans () (6) For Hollow bas polas moment of hertia 72 73 CD04m Dit) = 1/2 (2904- 23049 JH = 4.19638238X10 mt

1%, (176.839) m on bar we T 2) TE TJ which means hollow Shaft can dolution page no 2 For solid bar Is = 1/2D' = Jş = 9.5566 78x16s 4 Hollow tube has higher polar moment of berha than Solid shaft Consider torque T acting both bars let T allowable be the allowable shear stress for both material know 딨 T/R We can define polar modulus Ip = F/2. For Hollow Sectião (IDE 4.196382 38x10^4. 2.894x10 m (2999 =9.556678xiós CP for Hollow Section (TN) - Taux Zdrollon for Solid Section (Ts) = Tau r(2) dold Now (ZOH 2.894 x 103 TS (703 1.082x103 &, TH= 2.674 T& take more torque compared to So kid Shaje 3 For Solid Section (Zp) -3 1.08 2 Oxlo me .176635) T-TZp TH 2.674

Solution page no: 3 general cax Do Di tone7 0 @ consider hollow Shaft outer diameter la and diameter Di re Di De Similarly consider solid shape of D diameter By earlier case of Same weight & length Area of hollow - Area of Solid The (Do ² - D;?) 1/4 & Do²(1-02) 02 » bolo T%. Do3 (1-74) D2 bolo Now (ZD)n = nbs CD."-D:4) 32 16 Dolz %6 D3 (ZP s = 7/32" D/2 (TH) Tol. Zen (Te) (20) $ 1/16 Do" (1-04) / 16D3 De D 3 Co-ru) 3 TH TS G-vi) 1-24 is less than 1 (1-74) (1-74) 3/2 Now, as alway r= Di ::, The 7 Te which means Tosque Capacity of hollow shaft Solid Shaft of fame weight and longth please upvote ... Thank you ! Do is greater than