HW03_Solution

# HW03_Solution - EML 4507 Finite Element Analysis Design...

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EML 4507 Finite Element Analysis & Design Fall 2010 HW03 Solution 12. If the displacement field is given by 22 2 2 2 2 ( ) 2 x y z u x y u y x y z u z xy (a) Write down 3×3 strain matrix. (b) What is the normal strain component in the direction of (1,1,1) at point (1,–3,1)? Solution: (a) 3×3 symmetric strain matrix can be calculated from its definition as 2 2( ) 0 02 x y z y y z x y yz In addition, the unit normal vector in the direction of (1, 1, 1) is 1 {1 1 1} 3 T n b) Thus, the normal component of strain is 12 (2 2 2 2 ) 33 x y z y y z x y y z y nn Thus, the normal component of strain reduces as the y -coordinate of a point increases. At point (1, −3, 1), y = −3 3 2 y . 13. Consider the following displacement field in a plane solid: ( , ) 0.04 0.01 0.006 ( , ) 0.06 0.009 0.012 u x y x y v x y x y (a) Compute the strain components xx , yy , and xy . Is this a state of uniform strain? (b) Determine the principal strains and their corresponding directions. Express the principal strain directions in terms of angles the directions make with the x -axis.

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(c) What is the normal strain at Point O in a direction 45 o to the x -axis? Solution: (a) Strain components: 0.01 xx u x 0.012 yy v y 0.009 0.006 0.015 xy vu xy Yes, this is a state of uniform strain, because the strains are independent of position x,y,z .
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HW03_Solution - EML 4507 Finite Element Analysis Design...

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