a) For the extruded aluminum cross-section below, what would the shape factor be
ID: 2086257 • Letter: A
Question
a) For the extruded aluminum cross-section below, what would the shape factor be? b) What would be the dimensions for a square cross-section of solid material of same weight as extruded cross-section? What would the cross-section need to be in order to have the same bending benefit if you were to have a solid square cross-section? How much (%) heavier would it be than the extruded cross- section? If you were to make a square hollow cross-section of same weight as the extruded cross-section shown below with web thickness equal to 1/8 the total outer square width, what would the shape factor be? c) HFS (Standard Type) NFS (Economy Type) 20 12 4-R1 04.2 6005A-T5 Aluminum Mass/length 0.5 kg/m Sectional area 183 mm2 y0.742E4 mm 2 4.8Explanation / Answer
(a) Shape factor of any cross-section for elastic bending,
eB = (Bending stiffness of a section) / (Bending stiffness of a circular section of same area and material)
= EI / EI0; where E = Young’s modulus and I = second moment of area of a section
= I / I0
Now, I0 = (?*D4)/64; where D = diameter of circular cross-section
=> I0 = (?*D4)/64 = (A2/4?); where A = cross-sectional area
So, eB = I / (A2/4?) = Ix / (A2/4?) = 7420 / (1832/4?) = 2.78
(b) Mass = 0.5 kg/m = (0.5/1000) kg/ mm = 0.0005 kg/ mm
Density = 0.0005/183 kg/ mm3
Let a square cross-section has each side = X mm
For 1 mm length, volume of the square = X2 * 1 = X2
So, mass of the square block = X2 * density = X2 * (0.0005/183)
Now, X2 * (0.0005/183) = 0.0005 => X = 13.53 mm
(c) Let the hollow square cross-section has outer width = b and web thickness = t
I for the hollow square cross-section = 2*b3*t/3 = 2*b3*(b/8)/3 = b4/12
Area of the hollow square cross-section = b2 – (b – b/4)2 = 7 b2/16
Since the weight of hollow square and given section are same,
(7 b2/16) * (0.0005/183) = 0.0005 => b = 20.45
I = b4/12 = (20.45)4/12 = 14574.44 mm4
Area = 7 b2/16 = 7*20.452/16 = 182.96
Hence, shape factor = I / (A2/4?) = 14574.44 / (182.962/4?) = 5.47
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