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HW11 Solutions

# HW11 Solutions - Mechanics of Aircraft structures C.T Sun...

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Mechanics of Aircraft structures C.T. Sun 5.14 Show that the shear center for the section of Fig. 5.36 is at a distance ) 1 )( ( ) ( α + + + = b a b a a e to the left of stringer 1. Figure 5.36 Four-stringer thin-walled section Solution: (a) Assume that the thin sheets are ineffective in bending. The centroid of this four-stringer section is located at + = + = = 1 ) ( 2 ) )( ( 2 a A A a A A y A y i i i i i c 2 ) ( 2 ) )( ( b A A b A A A z A z i i i i i c = + + = = relative to stringer 3. We set up the (y,z) coordinate system with the origin placed at the centroid as shown in Fig. 5.36. The moments of inertia are A b b A A z A I i i i y 2 2 2 2 1 ) 2 )( ( 2 + = + = = + = + + + = = 1 2 ) 1 )( ( 2 ) 1 )( ( 2 2 2 2 2 A a a a A a A y A I i i i z (b) Shear flows Since this cross-section is symmetric with respect to y axis, the shear center is located on the y axis. Hence it is only necessary to determine the y position of the shear center. We can consider a fictitious cut section with shear flow plus the existing constant shear flow . ' q 0 q (1) First, calculate the shear flows by assuming a cut in the wall between stringers 1 and 4. Then 5.14.1

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Mechanics of Aircraft structures C.T. Sun 0 ' 41 = q b V A b b A V I Q V q z z y z ) 1 ( 2 1 ) 2 )( ( ' 2 1 12 α + = + = = b V A b b A V I Q V q z z y z = + + = = 2 2 23 2 1 ) 2 ( ) 1 ( ' b V A b b A b A V I Q V q z z y z ) 1 ( 2 1 )] 2 ( ) 2 ( ) 1 [( ' 2 3 34 + = + + + = = (2) The total shear flow and their resultant forces are, 0 12 12 ' q q q + = => a q b aV a q V z 0 12 1 ) 1 ( + + = = 0 23 23 ' q q q + = => b q V b q V z 0 23 2 + = = 0 34 34 ' q q q + = => a q b aV a q V z 0 34 3 ) 1 ( + + = = 0 0 41 41 ' q q q q = + = => b q b q V 0 41 4 = = Assume that the force is acting through the shear center and, thus, twist angle is produced. Consequently, we require z V 0 2 1 = = ds t q A G θ => 0 ) ( ) ) 1 ( ( ) ( ) ) 1 ( ( 0 0 0 0 = + + + + + + + + b q a q b aV b q V a q b aV z z z => z V b b a b a q ) )( 1 ( 2 ) 1 ( 2 0 + + + + = 5.14.2
Mechanics of Aircraft structures C.T. Sun z z V b a a a q b aV a q V ) )( 1 ( 2 ) 1 ( ) 1 ( 0 12 1 + + = + + = = α z z V b a b b a b q V b q V ) )( 1 ( 2 2 0 23 2 + + = + = = z z V b a a a q b aV a q V ) )( 1 ( 2 ) 1 ( ) 1 ( 0 34 3 + + = + + = = z V b a b a b q b q V ) )( 1 ( 2 ) 1 ( 2 0 41 4 + + + + == = = Take moment about stringer 1.The moment equivalence condition gives

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HW11 Solutions - Mechanics of Aircraft structures C.T Sun...

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