Question

(a) A criterion is required for selecting a material with the lowest cost in a strength-limited design for a cylindrical tie rod. Write the objective statement. Derive the criterion and identify the material index. The length of the tie rod, L, is fixed. (1 point) (b) A criterion is also required for selecting a material with the lowest cost in a strength-limited design for a square section beam (area b2). The length of the rod, L, is fixed. Write the objective statement and derive the criterion.

Answer 1

Answer:

For uniaxial tension the objective is to minimize cost:

C = mCm= ALrCm where m is mass, A is the cross-section area, r is density, and Cm is cost per unit mass. For strength limited design: F/A ≤sy, and A ≥ F/sy To minimize C = (F/sy) LrCm= (FL)(rCm/sy), minimize the quantity (rCm/sy). Maximize the material index,M =sy/(rCm)

b. The objective is to minimize cost C = mCm= b2LrCm, where A = b2 is the cross-section for strength limited design. It is necessary to eliminate the variable b from the equation.

Now if A= b2

Then b=A/2

Therefore cost C= mCm=A/2.2LrCm

= ALrCm

Answer 2

Answer:

look at the explanation section.

Explanation:

a) the function is cylindrical tie, the constraints is strength specified and the objective is minimum weight. To design structural elements the following elements are important:

-material properties

-the geometry

-the functional elements

solving we have the following:

objetive: minimize mass

function: tie rad. equation: mass = Area * Length * density

constraints: length must be specified; the material must have adequate fracture toughness.

b) objetive: minimize mass

function: tie rad. equation: mass = Area * Lenght * density = b² * length * density

constraint: the stiffness is

$s =\frac{CEI}{L^{3} }$

where I is the momentum of inertia, E is the Young`s modulus

the free variables are: the edge length b and the material choice

the equation is

$m=(\frac{125L^{5} }{C} )^{1/2} (\frac{P}{E^{1/2} } )$

you must select materials with low P/E^1/2, where P is the density