# HW#1 - Mechanics of Aircraft structures C.T Sun 1.1 The beam of a rectangular thin-walled section(i.e t is very small is designed to carry both

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Mechanics of Aircraft structures C.T. Sun 1.1 The beam of a rectangular thin-walled 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 thin-walled sections such as the one in Fig.1.16, the shear stress due to torsion is abt T 2 = τ Figure 1.16 Closed thin-walled 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 cross-section 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 cross-sectional 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 thin-walled 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...
View Full Document

## 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.

### Page1 / 9

HW#1 - Mechanics of Aircraft structures C.T Sun 1.1 The beam of a rectangular thin-walled section(i.e t is very small is designed to carry both

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online