In the stepped shaft shown, which has a full quarter-circular fillet, the allowable shearing str

In the stepped shaft shown, which has a full...

Key Concepts

Geometric Discontinuities in Shafts

Geometric discontinuities, such as steps in shaft diameter and the inclusion of fillets, play a significant role in the distribution of stresses. The presence of a step requires careful consideration of the fillet design to avoid high stress concentrations that can compromise the structural integrity of the shaft. Proper analysis of these features is essential in mechanical design to ensure that shafts can safely transmit torque.

Allowable Stress Design

Allowable stress design is an approach in engineering where components are designed to ensure that the operating stresses do not exceed specified safe limits, such as the material's allowable shear stress. This concept ensures safety by incorporating material properties and safety factors into the design process, allowing for reliable performance under expected loading conditions.

Polar Moment of Inertia

The polar moment of inertia is a geometrical property of the cross-section of a shaft that indicates its resistance to torsion. For circular cross-sections, it is proportional to the fourth power of the diameter. This parameter is critical because it directly affects how much twist and associated shear stress will occur for a given applied torque.

Stress Concentration

Stress concentration refers to the localized increase in stress that occurs at points where there is a geometric discontinuity, such as a step change in diameter or the presence of a fillet. The design of fillets, such as a quarter-circular fillet, helps to reduce abrupt changes in cross-sectional area and thereby mitigates the effect of stress concentrations, which can lead to premature failure.

Torsional Shear Stress

Torsional shear stress is the internal stress induced in a shaft when it is subjected to an applied torque. It is calculated using the relationship ? = T*c/J, where T is the applied torque, c is the outer radius of the shaft, and J is the polar moment of inertia. This concept is fundamental in ensuring that the stress remains within safe allowable limits for the material in use.

Transcript

00:03 In this question we have given diameter of shaft ab equals to 20 m m and diameter of soft c d equals to 25 m m and diameter of soft c d equals to 25 m and diameter of soft e f is equal to 40 m and allowable searing stress tau is equal to 60 mega pascal okay so now as the peripheral motions of the gears are equal so we can write between soft ab and cd as t cd divided by r c equals to t divided by r b okay so from here t cd will be equals to t divided by r b okay so from here t cd will be equals to r c by r b multiplied by t okay so by putting the value of r c and r b r c and r b are the radius of gear c and b so torque in cd will be equals to 75 divided by 30 these values are directly returned from the given diagram so this will be equals to 2 .5 times of okay similarly for ef and cd we can write t ef divided by r f equals to t cd d okay so from here torque in ef equals to rf divided by rf divided by rd multiplied by tcd so by putting value of tcd we get t ef equals to rf from the diagram is 90 and rd from the diagram is 30 and tcd is 2 .53 from 2 .5 t from this equation okay so from here torque in ef comes out to be 7 .5 t okay so now the magnitude of torque in soft a b will be equals to t ab equals to t equals to pi by 2 r a b squared by allowable shearing stress tau so by putting values t will be equals to pi multiplied by 10 cube this is q okay multiplied by allowable stress is 60 divided by 2 so from here we get capital t equals to 94 .248 kilo kilo kilo per kilo newton mm okay so now the magnitude of torque in the soft cd will be equals to t cd equals to pi by 2 r cd q multiplied by t so now putting values we get t cd equals to pi by 12 .5 multiplied by 60 divided by 2.

04:13 So from here, tcd is 2 .5 times of t equals to pi multiplied by 12 .5 q multiplied by 60 divided by 2.

04:34 So from here we get t equals to 73 .6...

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