Quiz-Fall2010-1_3 - Problem 1. (i) Write the three linear...

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EML 4507 Fall 2010 Quiz 1-3 If you want, write your name in the back. Solution
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Problem 3 : Consider a function f ( x,y ) , = [ ] - fx y 12xy A xy xy40 where = A 9443 . The eigen values of [ A ] are 1 and 11. It is found that the function is extremum at ( x,y )=(0,1). Determine if it is: (a) maximum; (b) minimum; (c) neither. Solution Since the eigen values are greater than zero ( > ) λ 0 , matrix A is positive definite and the function attains a minimum at (0,1). Problem 1 : Derive the (cubic) characteristic equation for determining the principal stresses of the stress matrix shown below. Do NOT solve the equation. Solution Calculate the invariants: = , = + + = , = I1 7 I2 5 3 3 11 I3 5 Hence, the cubic equation for principal stresses is: - + - = σ3 I1σ2 I2σ I3 0 or - + - = σ3 7σ2 11σ 5 0 - - - = σ 5σ 1σ 1 0 Problem 2 : It is determined that σ n =1 is one of the principal stresses for the state of stress in
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Unformatted text preview: Problem 1. (i) Write the three linear equations you will use to determine the corresponding principal direction. (ii) Solve the above equations to determine the principal direction. If there are multiple principal directions for σ n =1, then show them in a parametric form. Solution ( - )-- ( - ) ( - ) = 3 1 20 2 3 1 000 1 1 nxnynz 000 The first two equations are the same:-= → = 2nx 2ny 0 nx ny The last equation does not yield any useful information. Hence n z is arbitrary, say n z = α . Then the direction cosines of the principal stress direction take the form: +∝ , , 12 21 1 α where -∞≤ ≤+∞ α Some examples of the principal stress directions (eigen vectors) α n x n y n z-∞-1 1 1 ½ 2/3 2/3 1/3 +∞ +1...
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This note was uploaded on 06/07/2011 for the course EML 4507 taught by Professor Kim during the Fall '11 term at University of Florida.

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Quiz-Fall2010-1_3 - Problem 1. (i) Write the three linear...

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