HW3 - of x is 2 2 1 wx M z = . From basic mechanics of...

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MAE 3250 Analysis of Mechanical and Aerospace Structures Fall 2009 HW# 3, Due: 11:10 am, Monday, Sep. 21 st 1. Show that the dilation, z y x e ε + + = , satisfies the differential equation 0 2 2 2 2 2 2 = + + z e y e x e if the body forces are constants. 2. The differential equations: y x x y xy y x = + γ 2 2 2 2 2 z x x z xz z x = + 2 2 2 2 2 z y z y yz y z = + 2 2 2 2 2 are called conditions of compatibility . Use these equations to prove that: ) )( 1 ( ) 1 ( 2 z Z y Y x X + + + = Θ ν where z y x σ + + = Θ and X , Y , and Z are the body force per unit volume. 3. For the beam shown in the figure, if the weight is neglected, the bending moments as a function
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Unformatted text preview: of x is 2 2 1 wx M z = . From basic mechanics of materials, the bending stress is given by z z x I y M / = . Using the differential equations of equilibrium to determine how xy and y vary as functions of and . x y 4. Given zero body forces, determine whether the following stress distribution can exist for a body in equilibrium: xy c x 1 2 = , , 2 2 z c y = = z xz c y c c xy 3 2 2 1 ) ( + = , y c xz 3 = , = yz...
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