MA4001_Practice_Questions_Answers(1).docx - School of...

School of Mechanical & Aerospace Engineering
MA4001 – Engineering Design
Solutions to Materials Selection Practice Questions
Problems on the derivation of Material Performance index
1.
A thermal connecting rod of given length, l, and unknown cross-section (square
cross-section of side length, t ), t x t , is set at T
1
and T
2
temperature at the two
extreme ends. We need to transfer less than a heat of Q
0
from one end to
another end having minimum weight. Derive an expression for performance
index.
2.
You are required to design a light weight Crutch (a column with padded top,
which goes under armpit, and loaded in compression) to support an applied
axial compressive load of 80N. For design perspective, based on length either
it shouldn’t crush or shouldn’t buckle (applied separately). The crutch is of
given length L, has a hollow circular cross-section of fixed wall thickness (t)
and varying radius r. For this design (a) identify the function, objective,
constraint, fixed and free variable. (b) Is this fully determined, over
constrained or underconstrained problem? And (c) Obtain material index to
maximize the performance of design.
!
1

(c) objective: minimum weight,
m = ALρ = ( 2 π ) rtLρ
--- (1)
We need to eliminate the free variable A or radius r from the objective equation (1)
Consider the constraint of uniaxial crushing :
σ =
P
A
≤σ
y

A =
P
σ
y
(2)
Substituting the expression for A from the constraint equation in (1) we get:
m = ALρ =
P
σ
y
Lρ =
(
P
) (
L
)
ρ
σ
y
Mass can be minimized by maximizing the index: M =
σ
y
ρ
, which is specifc strength.
Now, consider the constraint of buckling , P =
π
❑
2
EI
L
❑
2
≤P
c
 P
c
=
π
❑
2
E
(
π r
❑
3
t
)
L
❑
2
(2)

r =
(
P
c
L
❑
2
π
❑
3
Et
)
❑
1
3
(3)
Now, substituting the expression for radius (r) from (3) in the objective equation we get:
m = ( 2 π ) rtLρ = ( 2 π )
(
P
c
L
❑
2
π
❑
3
Et
)
❑
1 / 3
tLρ =
3
√
8 P
c
(
L
❑
5
t
❑
2
)
❑
1
3
ρ
3
√
E
Mass can be minimized by maximizing the index: M
3. Currently a light weight cantilever beam of given length (L) in solid rectangular cross-section
of prescribed width (w) and unspecified height (h) is made of balsa (wood, perpendicular to
=
3
√
E
ρ