HW#3 - Mechanics of Aircraft structures C.T. Sun 2.1...

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Unformatted text preview: Mechanics of Aircraft structures C.T. Sun 2.1 Consider a unit cube of a solid occupying the region 1 x , 1 y , 1 z After loads are applied, the displacements are given by x u = y v = = w (a) Sketch the deformed shape for 03 . = , 01 . = . (b) Calculate the six strain components. (c) Find the volume change V [ V = V (the volume after deformation) V 0 (the original volume) ] for this unit cube. Show that V zz yy xx + + . Solution: (a) Since , there is no deformation in the z-direction and the deformation can be represented in the x-y plane. = w The new position of point B after deformation is given by 03 . 1 1 03 . 1 | 1 ' , 1 = + = + = = = y x u x ) 01 . ( | ' , 1 = + = + = = = y x v y New coordinates of B = ( 1.03 , 0 ) Similarly, new positions of A, C, D can be obtained as follows: A = ( 0 , 0 ) C = ( 1.03, 0.99 ) D = ( 0 , 0.99 ) --- ANS A, A B 1 B 1.03 D, 1 D 0.99 C C x y A=(0,0) A=(0,0) B=(1,0) B=(1.03,0) C=(1,1) C=(1.03,0.99) D=(0,1) D=(0,0.99) 2.1.1 AAE 352 S08 HW #3 Solutions Mechanics of Aircraft structures C.T. Sun (b) Strain components are: nents, Normal strain compo 03 . = = = x u xx 01 . = = = y v yy = = z w zz Shear strain components, = + = = v u x y yx xy = + = = x w z u zx xz = + = = y w z v zy yz --- ANS (c) The volume change is defined by 03 . 1 1 ' z ' y ' x V V V 0197 . 1 1 1 1 99 . = = = = Also, 0197 . 02 . ) 01 . ( 03 . = = + + = + + V zz yy xx --- ANS It can also be verified by: zz yy xx zz yy zz xx yy xx zz yy xx zz yy xx zz yy zz xx yy xx zz yy xx zz yy + + + + + + = + + + + + + + = + + Since the deformation is very small, we have ) ( V ) 1 1 ( V 1 1 1 1 ) 1 ( 1 ) 1 ( 1 ) 1 ( V V V xx + = = 1 << => higher order terms can be dropped. Therefore we have V V zz yy xx zz yy xx + + = + + ) ( , since V 0 = 1 2.1.2 Mechanics of Aircraft structures C.T. Sun 2.2 Consider the following displacement field: y u = x v = = w Sketch the displaced configuration of a unit cube with the faces originally perpendicular to the axes, respectively. This displacement field does not yield perpendicular to the axes, respectively....
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This note was uploaded on 04/11/2008 for the course AAE 352 taught by Professor Chen during the Spring '08 term at Purdue University-West Lafayette.

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HW#3 - Mechanics of Aircraft structures C.T. Sun 2.1...

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