Question

1. Consider a bar under tension. The bar has length L, rectangular cross-section of sides b and h, Young's modulus E, density p, and applied tensile force P. a) If all parameters are fixed except for h, find the value of h that minimizes the mass of the bar under stress, elongation, and stiffness constraints. Assume safety factors oo, Os, and Ok and failure values o*, 8*, and k* b) Also find the minimum mass of the bar m under these constraints 2. Using your results from Problem 1, plot the minimum value of m considering stress failure criterion for a range of tensile forces 100 NSP < 10000 N. The bar has a stress safety factor 0o = 1.5 and dimensions L = 1 m and b = 1 mm. Consider the following cases: a) The bar is comprised of titanium alloy (E = 120 GPa, p= 4500 kg/m3. o* = 1000 MPa) b) The bar is comprised of aluminum alloy (E = 70 GPa, p = 2700 kg/m3, o* = 300 MPa) c) The bar is comprised of steel (E = 200 GPa, p = 8000 kg/m3, o* = 500 MPa) Show all the 3 plots from a), b), and c) in the same figure.

Answer 1

PAGE 1

Sody Griven, Length of the bar = L . width of the cross section = b. Depth of the cross section =h Young's Modulus z E . Density = p. Tensile force. = P p >p - In this problem, all the parameters (bil, p) except 'is fixed. The stress induced in the bar due to the tessile load p is given by, : : Ooooh CY Under stress constraint, VIA II

PAGE 2

, minimum mass of the lay under stress constraint is given by . m s Density x Voluwie . : x 5 x 1. . . .. =($6L) h Substituting the minimum Velie of h im-bie above cuation, we get the minimum mass of -Ube lov under stress constraint as) .. mmin = $bLx pector Marconi = ($_do) The elongation of the bar due to the tensile load P is given by bo inder elongation constraint, iss S BEST. o Ihmin = PL&S / . best So, minimum mass of the bar under elongation constraint is given by, mula = (861) h = ($ 6L) PL AS mmin = shop ES*

PAGE 3

The' stiffness of the bar duie to tessile ecad. P is given by;' K: AE = (bdE = be Under Stifriess constraint) KZ 6. k* fuck* . (The bar sbould have a minimum stiffness of k* to prevent yielding. Otherwise, the bar would yield. under the tensile load P and eventually, it will be fail.) bar would yield. und =): h2 tv k*L, ЬЕ : 8 Thurn = Px K L BE iominimum mass of the bar under stiffness constraint is given by, Main = (SLL)h = 3bLHK = s[QK be