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Quiz-Fall2010-1_1

Quiz-Fall2010-1_1 - Problem 1(i Write the three linear...

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EML 4507 Fall 2010 Quiz 1-1 If you want, write your name in the back. Solution

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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 10 I2 12 8 8 28 I3 24 Hence, the cubic equation for principal stresses is: - + - = σ3 I1σ2 I2σ I3 0 or - + - = σ3 10σ2 28σ 24 0 - - - = σ 2σ 2σ 6 0 Problem 3 : Consider a function f ( x,y ) , = [ ] - fx y 12xy A xy xy40 where = A 6440 . The eigen values of [ A ] are: -2 and 8. 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 one of the eigen values is positive and the other is negative, the extremum value is neither a maximum nor a minimum. Problem 2 : It is determined that σ n =2 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 =2, then show them in a parametric for Solution ( - )-- ( - ) ( - ) = 4 2 20 2 4 2 000 2 2 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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