2. Polar moment of inertia As stated above, the polar second moment of area, J is defined as. Sample copy

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1 GATE PATHSHALA Polar moment of inertia As stated above, the polar second moment of area, is defined as z π r dr 0 R r π R π D For a solid shaft π (6) QP 0 π d Solid shaft π d Hollow shaft, " ( do d i ). The polar section modulus Z p / c, where c r D/ For a solid circular cross-section, Z p π D / 16 For a hollow circular cross-section, Z p π (D o - D i ) / (16D o ) Then, τ T / Z p If design shears stress, τ d is known, required polar section modulus can be calculated from: Z p T / τ d. Power Transmission (P) P (in Watt ) π NT 60 P (in hp) π NT 500 L NM O R For a hollow shaft of internal radius r: R R LrO π π z π r dr π ( R r ) D d 0 NM QP ch (7) r Where D is the external and d is the internal diameter. (1 hp 75 Kgm/sec). 5. Safe diameter of Shaft (d) Stiffness consideration T Gθ L Shear Stress consideration [Where N rpm; T Torque in N-m.] GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

2 GATE PATHSHALA - 9 T τ R We take higher value of diameter of both cases above for overall safety if other parameters are given. 6. In twisting Solid shaft, τ πd do Hollow shaft, τ π ( d d ) o i TL Diameter of a shaft to have a imum deflection "α " d.9x Gα [Where T in N-mm, L in mm, G in N/ mm ] 7. Comparison of solid and hollow shaft A Hollow shaft will transmit a greater torque than a solid shaft of the same weight & same material because the average shear stress in the hollow shaft is smaller than the average shear stress in the solid shaft If solid shaft dia D ( τ ) holloow shaft 16 D ( τ ) solid shaft 15 Hollow shaft, d o D, d i Strength comparison (same weight, material, length andτ ) Th n + 1 Externaldiameter of hollow shaft Where, n T s n n 1 Internaldiameter of hollow shaft Weight comparison (same Torque, material, length andτ ) W W h s ( n 1) ( n 1) n / / Externaldiameter of hollow shaft Where, n Internaldiameter of hollow shaft Strain energy comparison (same Torque, weight, material, length andτ ) Uh n Us n n 8. Shaft in series θ θ1 + θ Torque (T) is same in all section Electrical analogy gives torque(t) Current (I) [ONGC-005] [WBPSC-00] 9. Shaft in parallel θ1 θ and T T1 + T Electrical analogy gives torque(t) Current (I) 10. Combined Bending and GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

3 GATE PATHSHALA - 9 In most practical transmission situations shafts which carry torque are also subjected to bending, if only by virtue of the self-weight of the gears they carry. Many other practical applications occur where bending and torsion arise simultaneously so that this type of loading represents one of the major sources of complex stress situations. In the case of shafts, bending gives rise to tensile stress on one surface and compressive stress on the opposite surface while torsion gives rise to pure shear throughout the shaft. For shafts subjected to the simultaneous application of a bending moment M and torque T the principal stresses set up in the shaft can be shown to be equal to those produced by an equivalent bending moment, of a certain value M e acting alone. Figure Maximum direct stress ( σ x ) & Shear stress ( ( τ xy ) in element A M P σ x + π d A τ xy π d Principal normal stresses ( σ 1, ) & Maximum shearing stress ( τ ) σx σx σ 1, ± + τ Maximum Principal Stress ( σ 16 σ π d τ xy σ1 σ σ x ± + ( τxy ) ) & Maximum shear stress ( τ ) M + M + T 16 τ M + T π d Location of Principal plane (θ ) 1 1 θ tan T M Equivalent bending moment (M e ) & Equivalent torsion (T e ). M + M + T M e T M + T e GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

4 GATE PATHSHALA - 9 Important Note o Uses of the formulas are limited to cases in which both M & T are known. Under any other condition Mohr s circle is used. Safe diameter of shaft (d) on the basis of an allowable working stress. o o σ w in tension, d τ w in shear, d e πτ M e πσ 11. Shaft subjected to twisting moment only Figure Normal force ( F n ) & Tangential for ( F t ) on inclined plane AB F τ BC sin θ + AC cosθ n t w w [ ] [ ] F τ BC cos θ - AC sinθ Normal stress ( σ n ) & Tangential stress (shear stress) ( σ t ) on inclined plane AB. σ τ sin θ n σ τ cos θ t Maximum normal & shear stress on AB θ ( σ n ) τ 0 0 +τ 5 -τ τ 15 +τ 0 Important Note Principal stresses at a point on the surface of the shaft +τ, -τ, 0 i.e σ1, ± τsinθ Principal strains τ τ τ 1 (1 + μ); (1 + μ); (1 μ) E E E Volumetric strain, + v No change in volume for a shaft subjected to pure torque. 1. al Stresses in Non-Circular Cross-section Members GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

5 GATE PATHSHALA - 95 There are some applications in machinery for non-circular cross-section members and shafts where a regular polygonal cross-section is useful in transmitting torque to a gear or pulley that can have an axial change in position. Because no key or keyway is needed, the possibility of a lost key is avoided. Saint Venant (1855) showed that τ in a rectangular b c section bar occurs in the middle of the longest side b and is of magnitude formula Where b is the longer side and α factor that is function of the ratio b/c. The angle of twist is given by Where β is a function of the ratio b/c Shear stress distribution in different cross-section Rectangular c/s Elliptical c/s Triangular c/s 1. of thin walled tube For a thin walled tube Shear stress, τ T At 0 τ sl Angle of twist, φ AG O [Where S length of mean centre line, A O Area enclosed by mean centre line] Special Cases o For circular c/s π r t; A πr ; S πr o [r radius of mean Centre line and t wall thickness] T T. r T τ r π t At o TL τ L TL ϕ G Ao G π r tg o For square c/s of length of each side b and thickness t A b 0 S b o For elliptical c/s a and b are the half axis lengths. GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

6 A 0 π ab S π ( a+ b ) ab GATE PATHSHALA - 96 GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

7 GATE PATHSHALA - 97 Objective Questions (IES, IAS, GATE) Equation 1. Consider the following statements: [IES- 008] Maximum shear stress induced in a power transmitting shaft is 1. directly proportional to torque being transmitted.. inversely proportional to the cube of its diameter.. directly proportional to its polar moment of inertia. Which of the statements given above are correct? (a) 1, and (b) 1 and only (c) and only (d) 1 and only. A solid shaft transmits a torque T. The allowable shearing stress is τ. What is the diameter of the shaft? [IES-008] T T (a) (b) (c) (d) πτ πτ τ τ. Assertion (A): In theory of torsion, shearing strains increase radially away from the longitudinal axis of the bar. [IAS-001] Reason (R): Plane transverse sections before loading remain plane after the torque is applied.. The diameter of a shaft is increased from 0 mm to 60 mm, all other conditions remaining unchanged. How many times is its torque carrying capacity increased? (a) times (b) times (c) 8 times (d) 16 times [IES-1995; 00] 5. A circular shaft subjected to twisting moment results in imum shear stress of 60 MPa. Then the imum compressive stress in the material is [IES-00] (a) 0 MPa (b) 60 MPa (c) 90 MPa (d) 10 MPa 6. Angle of twist of a shaft of diameter d is inversely proportional to [IES-000] (a) d (b) d (c) d (d) d 7. A solid circular shaft is subjected to pure torsion. The ratio of imum shear to imum normal stress at any point would be [IES-1999] (a) 1 : 1 (b) 1: (c) : 1 (d) : 8. Assertion (A): In a composite shaft having two concentric shafts of different materials, the torque shared by each shaft is directly proportional to its polar moment of inertia. Reason (R): In a composite shaft having concentric shafts of different materials, the angle of twist for each shaft depends upon its polar moment of inertia. [IES-1999] 9. A solid circular shaft of 60 mm diameter transmits a torque of 1600 N.m. The value of imum shear stress developed is [GATE-00] (a) 7.7 MPa (b) 7.7 MPa (c) 57.7 MPa (d) 67.7 MPa GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

8 GATE PATHSHALA Maximum shear stress developed on the surface of a solid circular shaft under pure torsion is 0 MPa. If the shaft diameter is doubled then the imum shear stress developed corresponding to the same torque will be [GATE-00] (a) 10 MPa (b) 60 MPa (c) 0 MPa (d) 15 MPa 11. A steel shaft 'A' of diameter 'd' and length 'l' is subjected to a torque T Another shaft 'B' made of aluminium of the same diameter 'd' and length 0.5l is also subjected to the same torque 'T'. The shear modulus of steel is.5 times the shear modulus of aluminium. The shear stress in the steel shaft is 100 MPa. The shear stress in the aluminium shaft, in MPa, is [GATE-000] (a) 0 (b) 50 (c) 100 (d) A shaft is subjected to torsion as shown. [IES-00] Which of the following figures represents the shear stress on the element LMNOPQRS? 1. The shear stress at a point in a shaft subjected to a torque is (a) directly proportional to the polar moment of inertia and to the distance of the point form the axis [IAS-1995] (b) directly proportional to the applied torque and inversely proportional to the polar moment of inertia. (c) directly proportional to the applied torque and polar moment of inertia (d) inversely proportional to the applied torque and the polar moment of inertia 1. For a circular shaft of diameter d subjected to torque T, the imum value of the shear stress is [GATE-006] 6T T 8T ( a) ( b) ( c) ( d) πd πd πd πd GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

9 GATE PATHSHALA If two shafts of the same length, one of which is hollow, transmit equal torque and have equal imum stress, then they should have equal. [IAS 199] (a) polar moment of inertia (b) polar modulus of section (c) polar moment of inertia (d) angle of twist 16. A round shaft of diameter 'd' and length 'l' fixed at both ends 'A' and 'B' is subjected to a twisting moment 'T at 'C', at a distance of 1/ from A (see figure). The torsional stresses in the parts AC and CB will be (a) equal (b) in the ratio 1: (c) in the ratio :1 [IES-1997] (d) indeterminate Hollow Circular Shafts 17. A hollow circular shaft having outside diameter 'D' and inside diameter d subjected to a constant twisting moment 'T' along its length. If the imum shear stress produced in the shaft is as' then the twisting moment 'T' is given by [IAS-1999] π D d π D d π D d π D d (a) σ (b) σ (c) 8 s σ (d) D 16 s σ D s D 6 s D 18. One-half length of 50 mm diameter steel rod is solid while the remaining half is hollow having a bore of 5 mm. The rod is subjected to equal and opposite torque at its ends. If the imum shear stress in solid portion is or, the imum shear stress in the hollow portion is [IES-00] (a) τ (b) τ (c) 16 τ (d) 15 τ al Rigidity 19. Match List I with List II and select the correct answer using the codes given below the lists: List I (Mechanical Properties) List II ( Characteristics) [IAS-1996] A. al rigidity 1.Product of young's modulus and second moment of area about the plane of bending B. Modulus of resilience. Strain energy per unit volume C. Bauschinger effect. Torque unit angle of twist D. Flexural rigidity. Loss of mechanical energy due to local yielding Codes: A B C D A B C D (a) 1 (b) 1 (c) 1 (d) 1 0. Assertion (A): Angle of twist per unit length of a uniform diameter shaft depends upon its torsional rigidity. [IAS-00] Reason (R): The shafts are subjected to torque only. Power Transmitted by Shaft 1. In power transmission shafts, if the polar moment of inertia of a shaft is doubled, then what is the torque required to produce the same angle of twist? [IES-006] (a) 1/ of the original value (b) 1/ of the original value GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

10 GATE PATHSHALA - 00 (c) Same as the original value (d) Double the original value. While transmitting the same power by a shaft, if its speed is doubled, what should be its new diameter if the imum shear stress induced in the shaft remains same? (a) 1 of the original diameter (b) 1 of the original diameter [IES-006] (c) of the original diameter (d) ( ) 1 1 of the original diameter. For a power transmission shaft transmitting power P at N rpm, its diameter is proportional to: [IES-005] P (a) N 1/ P (b) N 1/ (c) P N / (d) P N. A shaft can safely transmit 90 kw while rotating at a given speed. If this shaft is replaced by a shaft of diameter double of the previous one and rotated at half the speed of the previous, the power that can be transmitted by the new shaft is [IES-00] (a) 90 kw (b) 180 kw (c) 60 kw (d) 70 kw 5. The diameter of shaft A is twice the diameter or shaft B and both are made of the same material. Assuming both the shafts to rotate at the same speed, the imum power transmitted by B is [IES-001; GATE-199] (a) the same as that of A (b) half of A (c) 1/8th of A (d) l/th of A 6. When a shaft transmits power through gears, the shaft experiences [IES-1997] (a) torsional stresses alone (b) bending stresses alone (c) constant bending and varying torsional stresses (d) varying bending and constant torsional stresses Combined Bending and 7. The equivalent bending moment under combined action of bending moment M and torque T is [IES-1996; 008; IAS-1996] 1 (a) M + T (b) M + M + T 1 (c) [ ] M + T (d) 1 M + T 8. A shaft is subjected to a bending moment M 00 N.m alld torque T 00 N.m The equivalent bending moment is [IAS-00] (a) 900 N.m (b) 700 N.m (c) 500 N.m (d) 50 N.m 9. A solid circular shaft is subjected to a bending moment M and twisting moment T. What is the equivalent twisting moment T e which will produce the same imum shear stress as the above combination? [IES-199; 007] (a) M + T (b) M + T (c) M + T (d) M T 0. A solid shaft can resist a bending moment of.0 knm and a twisting moment of.0 knm together, then the imum torque that can be applied is [GATE-1996] (a) 7.0 knm (b).5 knm (c).5 knm (d) 5.0 knm GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

11 GATE PATHSHALA A shaft is subjected to fluctuating loads for which the normal torque (T) and bending moment (M) are 1000 N-m and 500 N-m respectively. If the combined shock and fatigue factor for bending is 1.5 and combined shock and fatigue factor for torsion is, then the equivalent twisting moment for the shaft is [IES-199] (a) 000N-m (b) 050N-m (c) 100N-m (d) 16 N-m.. A member is subjected to the combined action of bending moment 00 Nm and torque 00 Nm. What respectively are the equivalent bending moment and equivalent torque? (a) 50 Nm and 500 Nm (b) 900 Nm and 50 Nm [IES-199; 00] (c) 900 Nm and 500 Nm (d) 00 Nm and 500 Nm. A shaft was initially subjected to bending moment and then was subjected to torsion. If the magnitude of bending moment is found to be the same as that of the torque, then the ratio of imum bending stress to shear stress would be [IES-199] (a) 0.5 (b) 0.50 (c).0 (d).0. A shaft is subjected to simultaneous action of a torque T, bending moment M and an axial thrust F. Which one of the following statements is correct for this situation? (a) One extreme end of the vertical diametral fibre is subjected to imum compressive stress only [IES-00] (b) The opposite extreme end of the vertical diametral fibre is subjected to tensile/compressive stress only (c) Every point on the surface of the shaft is subjected to imum shear stress only (d) Axial longitudinal fibre of the shaft is subjected to compressive stress only 5. For obtaining the imum shear stress induced in the shaft shown in the given figure, the torque should be equal to ( at ) ( bwl ) + T 1 1 wl wl () c ( Wl) + ( d) Wl+ + T [IES-1999] 6. Bending moment M and torque is applied on a solid circular shaft. If the imum bending stress equals to imum shear stress developed, them M is equal to [IES-199] T ( a) ( b) T ( c) T ( d)t GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

12 GATE PATHSHALA A circular shaft is subjected to the combined action of bending, twisting and direct axial loading. The imum bending stress σ, imum shearing force σ and a uniform axial stress σ(compressive) are produced. The imum compressive normal stress produced in the shaft will be [IES-1998] (a) σ (b) σ (c) σ (d) zero 8. Which one of the following statements is correct? Shafts used in heavy duty speed reducers are generally subjected to [IES-00] (a) bending stress only (b) shearing stress only (c) combined bending and shearing stresses (d) bending, shearing and axial thrust simultaneously Comparison of Solid and Hollow Shafts 9. The ratio of torque carrying capacity of a solid shaft to that of a hollow shaft is given by: ( ) ( ) 1 (a) 1 K (b) 1 K (c)k (d)1/ K [IES-008] D Where K i D ; D i Inside diameter of hollow shaft and D o Outside diameter of hollow shaft o Shaft material is the same. 0. A hollow shaft of outer dia 0 mm and inner dia of 0 mm is to be replaced by a solid shaft to transmit the same torque at the same imum stress. What should be the diameter of the solid shaft? [IES 007] (a) 0 mm (c) 10 (60) 1/ mm (b) 5 mm (d) 10 (0) 1/ mm 1. The diameter of a solid shaft is D. The inside and outside diameters of a hollow shaft D D of same material and length are and respectively. What is the ratio of the weight of the hollow shaft to that of the solid shaft? [IES 007] (a) 1:1 (b) 1: (c) 1: (d) 1:. What is the imum torque transmitted by a hollow shaft of external radius R and internal radius r? [IES-006] π π π π R r (a) ( R r ) f s (b) ( R r ) f s (c) ( R r ) f s (d) fs 16 R 8R R ( f s imum shear stress in the shaft material). A hollow shaft of the same cross-sectional area and material as that of a solid shaft transmits: [IES-005] (a) Same torque (b) Lesser torque (c) More torque (d) Cannot be predicted without more data GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

13 GATE PATHSHALA - 0. The outside diameter of a hollow shaft is twice its inside diameter. The ratio of its torque carrying capacity to that of a solid shaft of the same material and the same outside diameter is [GATE-199; IES-001] (a) (b) (c) 1 (d) Two hollow shafts of the same material have the same length and outside diameter. Shaft 1 has internal diameter equal to one-third of the outer diameter and shaft has internal diameter equal to half of the outer diameter. If both the shafts are subjected to the same torque, the ratio of their twists θ1/ θ will be equal to [IES-1998] (a) 16/81 (b) 8/7 (c) 19/7 (d) /56 6. Maximum shear stress in a solid shaft of diameter D and length L twisted through an angle θ is. A hollow shaft of same material and length having outside and inside diameters of D and D/ respectively is also twisted through the same angle of twist θ. The value of imum shear stress in the hollow shaft will be [IES-199; 1997] 16 8 τ τ τ τ 15 7 ( a ) ( b ) ( c ) ( d ) 7. A solid shaft of diameter 'D' carries a twisting moment that develops imum shear stress. If the shaft is replaced by a hollow one of outside diameter 'D' and inside diameter D/, then the imum shear stress will be. [IES-199] (a) (b) 1.1 (c) 1. (d) 8. A solid shaft of diameter 100 mm, length 1000 mm is subjected to a twisting moment 'T The imum shear stress developed in the shaft is 60 N/mm. A hole of 50 mm diameter is now drilled throughout the length of the shaft. To develop a imum shear stress of 60 N/mm in the hollow shaft, the torque 'T must be reduced by [IES-1998] (a) T/ (b) T/8 (c) T/1 (d)t/16 9. Assertion (A): A hollow shaft will transmit a greater torque than a solid shaft of the same weight and same material. [IES-199] Reason (R): The average shear stress in the hollow shaft is smaller than the average shear stress in the solid shaft. 50. A hollow shaft of length L is fixed at its both ends. It is subjected to torque T at a distance of L from one end. What is the reaction torque at the other end of the shaft? (a) T (b) T (c) T (d) T [IAS-007] 51. A solid shaft of diameter d is replaced by a hollow shaft of the same material and length. The outside diameter of hollow shaft d d while the inside diameter is. What is the ratio of the torsional stiffness of the hollow shaft to that of the solid shaft? GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

14 GATE PATHSHALA - 0 (a) (b) 5 (c) 5 (d) [IAS-007] 5. Two steel shafts, one solid of diameter D and the other hollow of outside diameter D and inside diameter D/, are twisted to the same angle of twist per unit length. The ratio of imum shear stress in solid shaft to that in the hollow shaft is [IAS-1998] (a) 9 τ (b) 8 16 τ (c) 7 15 τ (d) τ 5. A hollow shaft is subjected to torsion. The shear stress variation in the shaft along the radius is given by [IES-1996] Shafts in Series 5. What is the total angle of twist of the stepped shaft subject to torque T shown in figure given above? (a) l 8T (b) l πgd πgd 6T (c) l 66T (d) l πgd πgd [IES-005] 55. Two shafts having the same length and material are joined in series. If the ratio of the diameter of the first shaft to that of the second shaft is, then the ratio of the angle of twist of the first shaft to that of the second shaft is [IAS-1995; 00] (a) 16 (b) 8 (c) (d) 56. A circular shaft fixed at A has diameter D for half of its length and diameter D/ over the other half. What is the rotation of C relative of B if the rotation of B relative to A is 0.1 radian? [IAS 199] (a) 0. radian (b) 0.8 radian (c) 1.6 radian (d). radian GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

15 GATE PATHSHALA - 05 (T, L and C remaining same in both cases) 57. A torque of 10 Nm is transmitted through a stepped shaft as shown in figure. The torsional stiffness of individual sections of lengths MN, NO and OP are 0 Nm/rad, 0 Nm/rad and 60 Nm/rad respectively. The angular deflection between the ends M and P of the shaft is [GATE-00] (a) 0.5 rad (b) 1.0 rad (c) 5.0 rad (d) 10.0 rad Shafts in Parallel 58. For the two shafts connected in parallel, find which statement is true? (a) Torque in each shaft is the same (b) Shear stress in each shaft is the same (c) Angle of twist of each shaft is the same (d) al stiffness f each shift is the same [IES-199] 59. A circular section rod ABC is fixed at ends A and C. It is subjected to torque T at B. AB BC L and the polar moment of inertia of portions AB and BC are and respectively. If G is the modulus of rigidity, what is the angle of twist at point B? TL TL (a) (b) (c) TL (d) TL [IES-005] G G G G 60. A solid circular rod AB of diameter D and length L is fixed at both ends. A torque T is applied at a section X such that AX L/ and BX L/. What is the imum shear stress developed in the rod? [IES-00] 1T 8T T (a) (b) (c) (d) π D π D π D π D 61. A stepped solid circular shaft shown in the given figure is built-in at its ends and is subjected to a torque T o at the shoulder section. The ratio of reactive torque T 1 and T at the ends is ( 1 and are polar moments of inertia) [IAS-001] GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

16 GATE PATHSHALA - 06 (a) l l 1 1 (b) l l 1 1 (c) l l 1 1 (d) l l Steel shaft and brass shaft of same length and diameter are connected by a flange coupling. The assembly is rigidity held at its ends and is twisted by a torque through the coupling. Modulus of rigidity of steel is twice that of brass. If torque of the steel shaft is 500 Nm, then the value of the torque in brass shaft will be [IAS-001] (a) 50 Nm (b) 5 Nm (c) 500 Nm (d) 708 Nm 6. A steel shaft with bult-in ends is subjected to the action of a torque Mt applied at an intermediate cross-section 'mn' as shown in the given figure. [IAS-1997] Assertion (A): The magnitude of the twisting moment to which the portion BC is subjected is M t a+ b Reason(R): For geometric compatibility, angle of twist at 'mn' is the same for the portions AB and BC. 6. A steel shaft of outside diameter 100 mm is solid over one half of its length and hollow over the other half. Inside diameter of hollow portion is 50 mm. The shaft if held rigidly at two ends and a pulley is mounted at its midsection i.e., at the junction of solid and hollow portions. The shaft is twisted by applying torque on the pulley. If the torque carried by the solid portion of the shaft is 16000kg-m, then the torque carried by the hollow portion of the shaft will be (a) kg-m (b) kg-m (c) 1000 kg-m (d) 1000 kg-m [IAS-1997] 65. Two shafts are shown in the above figure. These two shafts will be torsionally equivalent to each other if their (a) polar moment of inertias are the same (b) total angle of twists are the same (c) lengths are the same (d) strain energies are the same [IES-1998] GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

17 GATE PATHSHALA - 07 Chapter-9 Answ e rs w it h Ex pla na t ion (Obje c t ive ). Ans. (a). Ans. (a) T r π d τ. Ans. (c) τπ d or T for same material τ const. 16 πd T α d 5. Ans. (b) 6. Ans. (d) T d 60 8 T1 d1 0 or T and normal stress πd πd.g at Sa ep m at pl hs e h co al py a.c 7. Ans. (b) Shear stress om 1. Ans. (d) τ :. Ratio of shear stress and normal stress 1: 8. Ans. (c) 9. Ans. (a) τ π d Ans. (c) τ, 0 if diameter doubled d d, then τ 0MPa 8 πd πd π ( d) 11. Ans. (c) τ 1. Ans. (b) 15. Ans. (b) T τ Here T & τ are same, so should be same i.e. polar modulus of section R τ T τ Gθ GRθ 1 or τ τ R L L L π σ s ( D d ) T Gθ τ τ will be same. w 16. Ans. (c) w w 17. ans. (b) is same T τ R 1. Ans. (d) 1. Ans. (b) As T & d both are same π d L R gives T R T τ τ or T r r τ s τ h h D or ; rs rh rs rh D π 16 σs (D d ) D 18. Ans. (d) GATE PATHSHALA Educational Services Llp. #15, II-floor, Karuneegar st., Adambakkam, Chennai-88 website :, contact: or

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