Answered: FIND average normal stress specdify…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

FIND average normal stress

specdify the orientation of the element in each case , use mohrs circle to solve the question .

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Expert Solution

Step 1

Given that,

$S t r e s s o n e l e m e n t i n x d i r e c t i o n, σ_{x} = 0 S t r e s s o n e l e m e n t i n y d i r e c t i o n, σ_{y} = - σ S h e a r s t r e s s a c t i n g o n e l e m e n t, τ_{x y} = τ$

The Center of the circle is,

$σ_{n} = \frac{σ_{x} + σ_{y}}{2} σ_{n} = \frac{0 - σ}{2} σ_{n} = \frac{- σ}{2}$

And the radius of the circle is,

$R = \sqrt{{(\frac{σ_{x} - σ_{y}}{2})}^{2} + {(τ_{x y})}^{2}} R = \sqrt{{(\frac{0 - σ}{2})}^{2} + {(τ)}^{2}} R = \sqrt{\frac{σ^{2}}{4} + τ^{2}}$

Step 2

$Now by taking any value of σ and τ, we can get the center and radius of mohr circle .$

Drawing the mohr circle as shown below,

From mohr circle calculating principle stress,

$M a x i m u m p r i n c i p l e s t r e s s, σ_{1} = σ_{n} + R σ_{1} = \frac{- σ}{2} + \sqrt{\frac{σ^{2}}{4} + τ^{2}} M a n i m u m p r i n c i p l e s t r e s s, σ_{2} = σ_{n} - R σ_{2} = \frac{- σ}{2} - \sqrt{\frac{σ^{2}}{4} + τ^{2}}$

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