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Unformatted text preview: Mechanics of Aircraft structures C.T. Sun 1.1 The beam of a rectangular thinwalled section (i.e., t is very small) is designed to carry both bending moment M and torque T . If the total wall contour length (see Fig. 1.16) is fixed, find the optimum b/a ratio to achieve the most efficient section if ) ( 2 b a L + = T M = and allowable allowable τ σ 2 = . Note that for closed thinwalled sections such as the one in Fig.1.16, the shear stress due to torsion is abt T 2 = τ Figure 1.16 Closed thinwalled section Solution: (1) The bending stress of beams is I My = σ , where y is the distance from the neutral axis. The moment of inertia I of the crosssection can be calculated by considering the four segments of thin walls and using the formula for a rectangular section with height h and width w. ) Ad wh 12 1 ( I 2 3 ∑ + = in which A is the crosssectional area of the segment and d is the distance of the centroid of the segment to the neutral axis. Note that the Parallel Axis Theorem is applied. The result is ) 3 ( 6 ] ) 2 ( ) ( 12 1 [ 2 12 1 2 2 2 3 3 b a tb b at at tb I + ≈ ⋅ + ⋅ ⋅ + ⋅ = , assuming that t is very small. (2) The shear stress due to torsion for a closed thinwalled section shown above is abt T 2 = τ . 1.1.1 HW #1 Solutions AAE 352 S08 Mechanics of Aircraft structures C.T. Sun (3) Two approaches are employed to find the solution. (i) Assume that the bending stress reaches the allowable allowable σ first and find the corresponding bending maximum bending moment. Then apply the stated loading condition of M T = to check whether the corresponding max τ has exceeded the allowable shear stress allowable τ . If this condition is violated, then the optimized b/a ratio is not valid. (a) ) 3 ( 3 ) 3 ( 6 2  2 2 b a tb M b a tb b M I My b y + = + ⋅ = = = σ When given ) ( 2 b a L + = as a constant, a can be expressed in terms of b and L as b L a − = 2 . Then we can minimize 6 ) 4 3 ( 3 ) 3 ( b L tb b a tb S − = + = in order to maximize σ , i.e., 8 3 ) 8 3 ( 6 L b b L t b S = ⇒ = − ⇒ = ∂ ∂ , so 8 2 L b L a = − = where the optimum ratio is 3...
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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 UniversityWest Lafayette.
 Spring '08
 Chen

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