1. The diaphragm in a differential pressure gauge deflects proportionally with the pressure

difference across it. This deflection is used to measure the pressure difference, δp. To

operate well, the material must not fail under the loading, and it should have a large

deflection in order to increase the sensitivity of the measurement. Using the equations

below for maximum stress and deflection of a circular diaphragm of fixed radius, answer

the following questions. The diaphragm thickness is t, E is Young's Modulus, and µ is

Poisson's ratio.

(a) What is the measure of performance for this design?

(b) Derive the performance index, M.

First, solve the constraint equation of SIGMA for the free parameter, t:

Now substitute this into the measure of performance, and separate out the materials

performance index:

(c) For a performance index of M = σ3/2 / ρ1/3 , what slope will the decision line be for a

materials selection chart of log σ (Y axis) versus log ρ (X axis)? Show your work.

(d) Use the selection chart below to determine the subset of materials with a strength (σf)

greater than 1000 MPa AND a performance index M = σ2/3 / ρ > 100 (MPa)2/3 /

(Mg/m3). Show your work with a sketch on the selection chart.

The first step is to draw a horizontal line at the value of SIGMA=1000 MPa (red line

on chart). The second step is to determine the slope of the performance index line for

σ2/3/ρ

For each two decades in RHO, we rise three decades in σ. To find the position of the

line for which M=100 (MPa) 2/3/(Mg/m3), we need to find a combination of σ and ρ

that gives us a value of 100:

For σ = 1000MPa, σ2/3=100 (MPa)2/3. The correct ρ value will be ρ =1 Mg/m3, so the

search line has a slope of 3/2 and passes through the point σ=1000 MPa, ρ=1 Mg/m3

(green line on chart).

The search region is to the upper left of these two lines, and the only materials that

satisfy the requirements are the ENGINEERING CERAMICS.

2. The company you are working for has asked you to select the material for a new high speed

magnetic disk drive for use in computers to be sited in smart machinery for metals

processing, typically a difficult and noisy environment for electronic equipment. The main

part you are to design is the arm supporting the read/write head as it moves over the

magnetic disk. The arm is essentially a cantilever beam, and the most important feature is

that the deflection of the end of the beam should be as small as possible. There is one

constraint, that the natural vibration frequency of the arm, f, should be higher than fo. Using

the equations below, derive the materials performance index that you would use to select

the proper material for this job. ASSUME: L is fixed, the cross section of the beam is

SQUARE (b X b), and b is unknown.

(a) What is the measure of performance for this design?

Minimum deflection, or P = 1/δ.

(b) Is this design under constrained, fully determined, or over constrained?

(c) Derive the performance index, M, for the vibration constraint.

Substitute this constraint on b into the objective function:

(d) For a performance index of M = σ1/2/E1/3, what slope will the decision line be for a

materials selection chart of log E (Y axis) versus log s (X axis)? Show your work.

(e) Use the selection chart below to determine the subset of materials with a modulus (E)

less than 1.0 GPa AND a performance index M = σ3/2/E > 3162 (MPa) 3/2 / (GPa).

Show your work with a sketch on the selection chart.

3. You need to build a scaffolding to use for painting your house. Basically, the flooring of

the scaffold is a rectangular plank of length L, width W, and thickness t. To save time in

moving the scaffold around, you want to use the LONGEST possible plank. In order to feel

safe, you don't want the deflection at the center to be larger than a maximum value, δ max,

when you are standing in the center with a load of F. The values of W and t are fixed.

There is one constraint, that the beam must not deflect by more than δ max under the

application of the center load, F. Using the equation for center deflection of a rectangular

beam given below, derive the materials performance index that you would use to select the

proper material for this job. ASSUME: W, t are fixed, L as long as possible.

(a) What is the measure of performance, p, for this design?

The measure of performance is found from the problem statement as the ATTRIBUTE

THAT IS TO BE MAXIMIZED OR MINIMIZED. In this problem, the question asks you

to design the LONGEST beam, so the answer is p = L.

(b) Derive the materials selection criterion, M, using the deflection constraint.

(c) A particular design asks us to choose a material using . For a plot of log (σf)

versus log (ρ) (as in chart shown below), determine the slope of the selection line.

(d) Use the selection chart below to determine the subset of materials with a strength σf greater

than 100 MPa AND a performance index greater than . Show

your work with a sketch on the selection chart.

Part 1) Drawing a horizontal line at the positon of Strength = 100 MPa, as shown in the

plot below as the GREEN line.

Part 2) Take the logarithm of the selection index equation to find:

Part 3) To find the position of the line of constant M, we need only find an X-Y point that

lies on the line.

This point is shown on the plot as the RED DOT.

Part 4) Search in the area ABOVE the horizontal line, and to the LEFT of the diagonal line.

4. A bracket for holding a hummingbird feeder is needed that will support the feeder a

distance, L, from the house, and will be hidden around a corner. The feeder produces a

torque on the bracket, and the rotational deflection must be kept below a value. Assume

the length, L, is fixed, but the radius, r, is free to vary. For dynamic considerations, the

mass of the bracket must be kept as small as possible. Assume that ONLY a torque loading

is involved. Use the following information to answer the design questions below.

Rotational deflection of the cantilever beam bracket: where T is the torque,

and G is the shear modulus of the bracket material.

(a) What is the measure of performance, p, for this design?

The measure of performance, p, for this design is given in the problem statement as the

feature to be maximized. In this case, the weight or mass, m, is to be made as small as

possible, so that

(b) Derive the materials selection criterion, M, using the deflection constraint.

We know that the rotational deflection from the given equation. Solve this for the free

parameter, r, and plug into the perfomance equation.

(c) A particular design asks us to choose a material using For a plot of ,

, determine the slope of the selection line.

Take the log of both sides of the M equation and arrange as Y = A + BX:

(d) Use the selection chart below to determine the subset of materials with a strength

greater than 400 MPa AND a performance index greater than greater than M = 100

[dimensionless]. Show your materials with a sketch on the selection chart clearly

indicating the selection region.

Part a) Strength greater than 400 MPa. This is a vertical line at 400 MPa with the

selection region RIGHT of the line. This is shown as the BLUE line below.

Part b) The performance index gives us a line of slope = 1;

The position of the line is found from the value for M as follows.

This line is shown in GREEN below, and the selection region is for values greater than

that of the line, materials to the upper left. The materials of choice are highlighted.

5. In the filament winding process for advanced composites, the high strength fiber is

unwound from a spool and wound around a form to produce the final shape, which is then

impregnated with an epoxy matrix. The fiber passes through a guide that keeps the fiber

from tangling, and must elastically bend to even out the tension in the fiber as it is wound

onto the final form. The fiber guide can be modeled as a cantilever beam of circular cross

section with an end load. An important feature of the design is that the guide does not

permanently deform (yield) under the load applied by the fiber winding operation.

Assume the length, L, is fixed, but the radius, r, is free to vary. For dynamic considerations,

the mass must be kept as small as possible. Use the following information to answer the

design questions below.

Maximum root stress in cantilever beam:

(a) What is the measure of performance, P, for this design?

The measure of performance, p, for this design is given in the problem statement as the

feature to be maximized. In this case, the weight or mass, m, is to be made as small as

possible, so that

p = 1 / m

(b) Derive the materials selection criterion, M, using the load constraint.

We know that the maximum stress in a cantilever beam is given in the equation above.

To keep the guide material in elastic loading (to avoid plastic deformation), the

maximum stress must be less than the yield stress:

(c) A particular design asks us to choose a material using . For a plot of

For a plot of log (ρ) [X axis] versus log(E) [Y axis], determine the slope of the

selection line.

Take the log of both sides of the M equation and arrange as Y = A + BX:

(d) Use the selection chart below to determine the subset of materials with a strength less

than 30 MPa AND a performance index greater than . SHOW

your materials with a sketch on the selection chart clearly indicating the selection

region.

Part a) Strength less than 30 MPa. This is a vertical line at 30 MPa with the selection

region LEFT of the line. This is shown as the BLUE line below.

The performance index gives us a line of slope = 1/2;

The position of the line is found from the value for M as follows.

This line is shown in GREEN below, and the selection region is for values greater

than that of the line, materials to the upper left. The materials of choice are

highlighted.