Calculating Shear Strain: σx / E and -vσy / E

[chetzread]

Homework Statement

for the stress applied at x direction , the tensile strain εx should be = σx / E , right ?
and for the tensile strain εy should be = -vσx / E , am i right ?

Homework Equations

The Attempt at a Solution

I think so because in 1.7.1.3 , εy = σy / E - vσx / E ,
and εx = σx / E - vσy / E

 

[chetzread]

Refer to the bottom part of picture , σx ,
after rearraging , i didnt get what the author get .. my working is :
Eεx = σx - vσy
σx = Eεx + vσy
= E(εx + (V / E)(Eεy )
= E(εx + vεy )

 

[Chestermiller]

Homework Statement

for the stress applied at x direction , the tensile strain εx should be = σx / E , right ?
and for the tensile strain εy should be = -vσx / E , am i right ?

Homework Equations

The Attempt at a Solution

I think so because in 1.7.1.3 , εy = σy / E - vσx / E ,
and εx = σx / E - vσy / E

This is correct.

 

[Chestermiller]

Refer to the bottom part of picture , σx ,
after rearraging , i didnt get what the author get .. my working is :
Eεx = σx - vσy
σx = Eεx + vσy
= E(εx + (V / E)(Eεy )
= E(εx + vεy )

You did the algebra wrong.

 

[chetzread]

You did the algebra wrong.

after rearraging , i didnt get what the author get .. my working is :
Eεx = σx - vσy
σx = Eεx + vσy
= E(εx + (V / E)(Eεy )
= E(εx + vεy )

sorry , for the symbol , you said i am wrong , which part is wrong ?

 

[Chestermiller]

after rearraging , i didnt get what the author get .. my working is :
Eεx = σx - vσy
σx = Eεx + vσy
= E(εx + (V / E)(Eεy )
= E(εx + vεy )

sorry , for the symbol , you said i am wrong , which part is wrong ?

I'm going to let you figure that out. You have 2 linear algebraic equations in two unknowns, $\sigma_x$ and $\sigma_y$. You should have learned how to solve such equations in first year algebra.

If you continue to have trouble with this, please submit the two equations and your working to the Precalculus Mathematics section of the Homework Forums.

 

[chetzread]

I'm going to let you figure that out. You have 2 linear algebraic equations in two unknowns, $\sigma_x$ and $\sigma_y$. You should have learned how to solve such equations in first year algebra.

If you continue to have trouble with this, please submit the two equations and your working to the Precalculus Mathematics section of the Homework Forums.

Are you sure ? Can you please look at it again ?

 

[Chestermiller]

Are you sure ? Can you please look at it again ?

Yes, I'm sure that your answer is wrong, and I'm also sure that the book's answer is wrong? In the book answer, the terms in the numerators should have plus signs, not minus signs. Your answer should have a denominator which matches the book's denominator.

 

[chetzread]

not minus signs.

do you mean it should be σx = (Eεx + vσy ) / (1-(v^2)) ?

 

[Chestermiller]

do you mean it should be σx = (Eεx + vσy ) / (1-(v^2)) ?

It's time for you to learn to use LaTex.$$\sigma_x=\frac{E(\epsilon_x+\nu \epsilon_y)}{(1-\nu^2)}$$

 

[chetzread]

It's time for you to learn to use LaTex.$$\sigma_x=\frac{E(\epsilon_x+\nu \epsilon_y)}{(1-\nu^2)}$$

do you mean this ?
why there is $$\{(1-\nu^2)}$$ below it ?

 

[Chestermiller]

do you mean this ?
why there is $$\{(1-\nu^2)}$$ below it ?

C'mon man. You need to learn algebra.
$$E\epsilon_x=\sigma_x-\nu \sigma_y\tag{1}$$
$$E\epsilon_y=\sigma_y-\nu \sigma_x\tag{2}$$
Multiply Eqn. 2 by $\nu$: $$E\nu \epsilon_y=\nu \sigma_y-\nu^2 \sigma_x\tag{3}$$
Add Eqns. 1 and 3 to eliminate $\sigma_y$:$$E(\epsilon_x+\nu \epsilon_2)=(1-\nu^2)\sigma_x\tag{4}$$
Solve Eqn. 4 for $\sigma_x$

 

[chetzread]

C'mon man. You need to learn algebra.
$$E\epsilon_x=\sigma_x-\nu \sigma_y\tag{1}$$
$$E\epsilon_y=\sigma_y-\nu \sigma_x\tag{2}$$
Multiply Eqn. 2 by $\nu$: $$E\nu \epsilon_y=\nu \sigma_y-\nu^2 \sigma_x\tag{3}$$
Add Eqns. 1 and 3 to eliminate $\sigma_y$:$$E(\epsilon_x+\nu \epsilon_2)=(1-\nu^2)\sigma_x\tag{4}$$
Solve Eqn. 4 for $\sigma_x$

why my working is wrong ? I noticed that in your working , you used 2 equations which is $$E\epsilon_x$$ and $$E\epsilon_y$$

 

[Chestermiller]

why my working is wrong ?

Like I said: Put your working in the Precalculus Mathematics Homework forum and let them help you figure out what you did wrong.

I noticed that in your working , you used 2 equations which is $$E\epsilon_x$$ and $$E\epsilon_y$$

Of course you need to use two equations. There are two unknowns: $\sigma_x$ and $\sigma_y$

 

[chetzread]

This is correct.

sorry , just to verify my concept again... is εy = σy / E ?
From another book, the εy is σx / E ..Which is correct ?

 

[Chestermiller]

sorry , just to verify my concept again... is εy = σy / E ?
From another book, the εy is σx / E ..Which is correct ?

check a third book.

 

[chetzread]

check a third book.

It's really hard for me to get a third source . I already searched thru many websites , but i couldn't find this . I could only get simple explanation of poisson ratio, but not the the explanation about when the force is applied in a direction , find the strain in another direction .
Can you explain the concept ?

 

[chetzread]

check a third book.

here it is

 

[chetzread]

check a third book.

so, εx= σx / E is correct ?

http://www.tech.plym.ac.uk/sme/MECH115-web/Cylinders & Spheres(8).PDF

 

[Chestermiller]

so, εx= σx / E is correct ?

http://www.tech.plym.ac.uk/sme/MECH115-web/Cylinders & Spheres(8).PDF

Yes, as long as the stresses in the y and z directions are zero.