HW8 Solutions

# HW8 Solutions - Mechanics of Aircraft structures C.T Sun...

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Unformatted text preview: Mechanics of Aircraft structures C.T. Sun 4.1 A uniform beam of a thin-walled angle section as shown in Fig. 4.19 is subjected to the bending ( y M = z M ). Find the neutral axis and bending stress distribution over the cross-section. Figure 4.19 Thin-walled angle section Solution: (a) For finding the location of the centroid, we select the corner of the thin-walled section as the origin of a Cartesian coordinate system with the horizontal and vertical distances between the centroid and the origin denoted by and , respectively. c y c z 4 2 ) 2 / ( h ht h t h y c = ⋅ ⋅ = 4 2 ) 2 / ( h ht h t h z c = ⋅ ⋅ =--- ANS (b) Set up a Cartesian coordinate system (y, z) in the pane of the section with the origin at the centroid. The moments of inertia with respect to this coordinate system are (assume t << h ) 3 2 c 3 2 c 3 y th 24 5 thz 12 ht ) z h ( th 12 th I = + + − + = in which parallel axis theorem for moments of inertia has been employed and the term 12 ht 3 has been neglected. 3 2 c 3 2 c 3 z th 24 5 thy 12 ht ) y h ( th 12 th I = + + − + = 3 2 1 8 1 th yzdA yzdA yzdA I A A yz = + = = ∫ ∫ ∫ where, 3 2 1 16 1 | ) 2 ( th z t y ztdz y yzdA c c c c z h z c z h z c A = = = − − − − ∫ ∫ 4.1.1 Mechanics of Aircraft structures C.T. Sun 3 2 2 16 1 | ) 2 ( th y t z ytdy z yzdA c c c c y h y c y h y c A = = = − − − − ∫ ∫ (c) Using equation (4.25) in the textbook, z I I I M I M I y I I I M I M I yz z y z yz y z yz z y y yz z y xx 2 2 − − + − − = σ By substituting the known values we obtain ) 9 15 ( 2 ] ) 8 / 1 ( ) 24 / 5 )( 24 / 5 [( ) 24 / 5 ( ] ) 8 / 1 ( ) 24 / 5 )( 24 / 5 [( ) 8 / 1 ( 3 3 2 3 2 y z th M z th M y th M y y y xx − = − + − − = σ--- ANS Maximum positive stress: At h z h z c 4 3 = − = and 4 h y y c − = − = 2 3 4 27 ) 9 15 ( 2 th M y z th M y y xx = − = σ Maximum negative stress: At 4 h z z c −...
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HW8 Solutions - Mechanics of Aircraft structures C.T Sun...

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