You are to design for an automobile a hollow shaft baving an...

You are to design for an automobile a hollow shaft baving an outside diameter of 75 mum to transmit a maximum torque of $1000 \mathrm{~N}-\mathrm{m}$. What is the thickness $t$ for a steel shaft with a yield stress of $3.5 \times 10^8 \mathrm{~Pa}$ ? Employ a safety factor of 5 .

Key Concepts

Safety Factor

A safety factor is an engineering principle that provides a margin of safety in design by requiring that loads and stresses be kept significantly below the material’s failure limits. This ensures that even in cases of unexpected loads or material imperfections, the component remains safe and functional.

Geometry of Hollow Circular Shafts

Hollow circular shafts are used in applications requiring strength with reduced weight. Understanding the geometric properties, such as the outer and inner diameters and wall thickness, is critical to calculating the polar moment of inertia, which influences the torsional rigidity and capacity to resist torque.

Torsion in Shafts

This concept deals with how shafts transmit torque and the resulting shear stresses induced by the twisting action. It involves understanding that when a torque is applied to a shaft, the material experiences shear stress, which must be kept within safe limits to prevent failure.

Shear Stress and Yield Strength

Shear stress is a measure of the force per unit area acting parallel to the cross-section of a material. Yield strength is the maximum stress that a material can withstand without permanently deforming. In design, the applied stress must be kept below the yield strength, often by using a safety factor to account for uncertainties and to ensure durability.

Transcript

00:01 The outer radius of the soft cd, so it will be rcd, is equal to diameter of cd divided by 2.

00:09 Diameter of cd is given as 0 .08 divided by 2, so it will be equal to 0 .04 heater.

00:19 Similarly, now we can find out the value of polar moment of inertia of soft cd.

00:26 So we will write jcd is equal to 5 divided by 2 r2 to depotority.

00:31 Of 4 it will be equal to pi divided by 2 deployed with 0 .04 to the power of 4 so polar moment of inertia will come out to be 4 .0 to 1 multiplied by 10 to the power of minus 6 meter to the power of 4 now let us calculate the maximum shear stress the tau max will be equals to the torque multiplied by radius divided by polar moment of inertia.

01:04 The value of maximum shear stress given in our portion is 75 megapascal that is 75 multiplied by 10 to the power of 6 pascal multiplied by the maximum torque torque in cd section cd multiplied by radius that is 0 .04 divided by the polar moment of inertia that is 4 .021 multiplied with 10 to the power of minus 6.

01:30 So from this we get the torque in section cd is equal to 7 .54 multiplied with 10 to the power of 3 newton meter we can further write it as eccd is equal to 7 .54 kilo newton meter so this is the maximum torque that can be applied on the shaft this is maximum torque that can be applied on sap maximum torque to be applied on soft.

02:22 For section bc we can write the outer radius of section bc is equal to the outer diameter of section bc divided by 2.

02:33 The outer diameter of section bc is given as 0 .1 divided by 2 so it will be 0 .05 meter.

02:42 Similarly the inner radius of section bc are bc i this will be equals to diameter b c i divided by 2 diameter b c i is given as 0 .08 divided by 2 so it will be equals to 0 .04 meter now let us calculate the polar moment of inertia of section bc so it will be j bc this will be equals to pi divided by 2 r o bc 2d bc 2 minus r i bc 2d bc 2 so it will be equal to pi divided by 2, ro is 0 .05 to the power of 4 minus rri is 0 .04 to the power of 4.

03:31 So from this we will get the polar moment of inertia for the section bc is equal to 5 .76, 5 .76 multiplied with 10 to the power of minus 6 meter to the power of 4.

03:49 Now let's calculate the cre stress in 6.

03:53 Section bc, maximum serious stress in section bc, it will be torque in section bc multiply by outer radius of section bc...

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