Choosing a Geometry for a Given Application Material Selection - PowerPoint PPT Presentation

Choosing a Geometry for a Given Application Material Selection Selection of Materials and Structures

Bristol F.2b Air Tractor AT-802A Photo by Alan Wilson CC BY-SA 2.0 Piper Super Cub Photo by flightlog CC BY 2.0 Photo by Ad Meskens CC BY-SA 3.0

Deflection of a cantilever beam L P 3 PL δ = 3 δ EI

Geometric variables What geometric properties appear in our characteristic equation? 3 Beam length PL δ = (fixed by overall geometry of bike frame) 3 EI Moment of inertia of beam section

What is the moment of inertia? A 1 A 2 P y 2 y 1 x Axis of bending (neutral axis) x y n = ∫ = ∑ 2 2 y dA I A y y A n i i Beam cross-section 1 = i A

Moment of inertia of common cross-sections 1 4 3 π π d td 3 = I bh = = I I 12 64 8

Solid square cross-section 24 mm 1 1 ( )( ) 3 3 24 24 = = I bh mm mm 12 12 □ 24 mm 2.76 10 mm 4 4 = ×

Solid circular cross-section We want to a circular cross section with an equivalent bending stiffness 4 π d d = 2.76 10 mm 4 4 = × I ○ 64 Equate to moment of inertia for square section 4 64 2.76 10 27.4 mm 4 4 = = × d mm π

Comparing area/weight 24 mm 27.4 mm 589 A 1.023 A = = ○ 576 □ 24 mm Area, and thus weight, of circular cross section is 2.3% larger than the square section π ( ) ( ) 2 2 27.4 589 2 24 576 2 = = = = A mm mm A mm mm 4 ○ □

Comparing area/weight Moment of inertia is larger for area located further from the neutral axis Neutral axis Circular section has more area closer to the neutral axis, thus needs a larger diameter to have the same stiffness as the square

Thin-walled circular cross-section (t = 10% d) We want to a thin-walled circular cross section with an equivalent bending stiffness 3 π td d = 2.76 10 mm 4 4 = × I ○ 8 Equate to moment of inertia for square section 8 29.0 mm 2.76 10 4 4 = = × d mm 4 t = 0.1d 0.1 π

Comparing area/weight 29 mm 24 mm 264 A 0.46 A = = ○ 576 □ 24 mm Area, and thus weight, of thin walled tube is 46% of the solid square section t = 2.9 mm ( )( ) ( ) 29 2.9 2 = π A mm mm 24 576 2 = = A mm mm ○ □ 264 2 = mm

Comparison for equal bending stiffness 34.4 mm 29 mm 24 mm 27.4 mm 24 mm t = 2.9 mm t = 1.7 mm 102% 46% Weight = 100% 32%

Examples of bike frames

Bending deflection Is there more than geometry? 200 ≈ E GPa steel 3 PL 72 ≈ E GPa δ = Al 3 EI Moment of inertia Material stiffness of beam section