Finding stress in non symetric cross section

[Dell]

in a given shaft with a circular cross section, the radius changes linearly

find the maximum shear stress

i used T*r/J
and since J is dependant on r^4, i found that the cross section with the smallest radius will feel the largest stress.

find the angle of twist at the end of the shaft

T/(GJ)*dx while J is a function of X and i integrate from 0 to L

if the bottom of the shaft is reinforced, -thickened- to 2t, what is the largest shear stress felt in the shaft??

up till now i have only solved questions with either circular/ rectangular closed cross sections, or other open cross sections

can i use the equation for maximum stress
T/(2A*t)

where A is the area surrounded by an axis through the center of the side of the shape?? if so how do i do this ? what would that axis look like? would it be 2 half circles with 90 degree joints? meaning the area would be pi/2*(R1²+R2²) where R1 and R2 are the average radii of the 2 half circles??

using this logic i would find the maximum stress in the thin walled circle

is this correct? can i do this?

this is where i am stumped,

 

[Dell]

also what are the advantages of thickening parts of the shaft, i think if it was subjected to bending this might improve its performance as far as max moment that can be applied, but in torsion what would this do? there any other advantagees?

 

[Mene]

please help

I would first like to applaud your efforts in using equations and seeking out a solution to this problem. It is clear that you have a good understanding of the concepts involved.

To answer your question, yes, you can use the equation for maximum stress (T/(2A*t)) in this scenario. However, the axis through the center of the side of the shape would not be a half circle, but rather a straight line connecting the two ends of the reinforced section. This is because the cross section is now non-symmetric, and the radius changes linearly, so the shape would not be a perfect circle.

To find the maximum stress, you would need to calculate the area (A) and the thickness (t) of the reinforced section. The area will be the sum of the areas of the two half circles (pi/2*(R1^2+R2^2)) plus the area of the straight section (2t*l), where l is the length of the reinforced section. The thickness (t) will simply be 2t.

Once you have these values, you can plug them into the equation (T/(2A*t)) to find the maximum stress felt in the reinforced section. This will give you a more accurate result than using the previous equation (T*r/J), as it takes into account the change in cross section.

As for finding the angle of twist at the end of the shaft, you can still use the equation T/(GJ)*dx, but you will need to integrate from 0 to the length of the reinforced section (l) and then add that to the angle of twist for the remaining section of the shaft (which you can calculate using the previous equation).

I hope this helps to clarify the process for finding the maximum stress and angle of twist in a non-symmetric cross section. Keep up the good work in your scientific endeavors!