EXAMPLE: 3.1.3. MULTIPLE GEARS

Shaft BC is hollow with inner and outer diameters of 80mm and 120mm, respectively. Shafts AB and CD are solid and of diameter d. With the loading shown, determine:

a. The maximum and minimum shearing stress in shaft BC;

b. The required diameter d of shafts AB and CD if the allowable shearing stress in these shafts is 60 MPa.

Mechanics of Materials

Solution A:

The shearing stress in the shafts are measured with the transfer of stress from gears. Still using the same formula:

                                                

we should determine the values for the parameters T, c, and J.

For T, provide a cutting plane at shaft BC and determine the torque either starting from A or from D.

with both torque rotating the same.


If we consider the other end (starting at D)




with both torque rotating against each other.

For c, given the diameter of 120mm, radius will be computed as 60mm.

For J, take note that shaft BC is a hollow shaft.

                       

                       
                     

With all values already determined, we can solve the maximum shearing stress on shaft BC.

      

The minimum shearing stress will be solved using ratio and proportion.

                   
             
                   

                    

Solution B:

Using the same principle and process with the previous solution, use the general formula of the shearing stress to solve for the diameter. 

**Diameter = 2 x radius = 2r

For T:  Placing the cutting plane on either shaft AB or CD, 

With summation of torques:

Take note that for shafts AB and CD, the shafts are solid.

                          
                          

                         

                            

                             

Thus, the diameter can be determined: