Finding stress in non symetric cross section

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In analyzing a shaft with a circular cross-section where the radius changes linearly, the maximum shear stress occurs at the smallest radius, calculated using the formula T*r/J. The angle of twist at the end of the shaft is determined by integrating T/(GJ) from 0 to L, considering J as a function of position. When the shaft is reinforced to a thickness of 2t, the maximum shear stress can be calculated using T/(2A*t), where A represents the area around the axis through the center. The discussion raises questions about the correct application of stress equations for non-symmetric sections and the implications of thickening the shaft, particularly in terms of performance under torsion and bending. Overall, understanding these factors is crucial for optimizing shaft design and performance.
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in a given shaft with a circular cross section, the radius changes linearly
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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??
Captu?re.JPG


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,
 
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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?
 

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