71
2-19 Calculate the specific strength and specific stiffness of the following materials and pick one for use in an aircraft wing spar.
(a)
Steel
Sut = 80 kpsi (552 MPa)
2
(b)
Aluminum
Sut = 60 kpsi (414 MPa)
(c)
Titanium
Sut = 90 kpsi (621 MPa)
2-20 If maximum impact resistance were desired in a part, which material properties would you look for?
2-21 Refer to the tables of material data in Appendix A and determine the strength-to-weight ratios of the following material alloys based on their tensile yield strengths: heat-treated 2024 aluminum, SAE 1040 cold-rolled steel, Ti-75A titanium, type 302 cold-rolled stainless steel.
2-22 Refer to the tables of material data in Appendix A and determine the strength-to-weight ratios of the following material alloys based on their ultimate tensile strengths: heat-treated 2024 aluminum, SAE 1040 cold-rolled steel, unfilled acetal plastic, Ti-75A titanium, type 302 cold-rolled stainless steel.
2-23 Refer to the tables of material data in Appendix A and calculate the specific stiffnesses of aluminum, titanium, gray cast iron, ductile iron, bronze, carbon steel, and stainless steel.
Rank them in increasing order of this property and discuss the engineering significance of these data.
2-24 Call your local steel and aluminum distributors (consult the Yellow Pages) and obtain current costs per pound for round stock of consistent size in low-carbon (SAE 1020) steel, SAE 4340 steel, 2024-T4 aluminum, and 6061-T6 aluminum. Calculate a strength/
dollar ratio and a stiffness/dollar ratio for each alloy. Which would be your first choice on a cost-efficiency basis for an axial-tension-loaded round rod (a)
If maximum strength were needed?
(b)
If maximum stiffness were needed?
2-25 Call your local plastic stock-shapes distributors (consult the Yellow Pages) and obtain current costs per pound for round rod or tubing of consistent size in plexiglass, acetal, nylon 6/6, and PVC. Calculate a strength/dollar ratio and a stiffness/dollar ratio for each alloy. Which would be your first choice on a cost-efficiency basis for an axial-tension-loaded round rod or tube of particular diameters. (Note: material parameters can be found in Appendix A.)
(a)
If maximum strength were needed?
(b)
If maximum stiffness were needed?
2-26 A part has been designed and its dimensions cannot be changed. To minimize its deflections under the same loading in all directions irrespective of stress levels, which of these materials would you choose: aluminum, titanium, steel, or stainless steel? Why?
*
2-27 Assuming that the mechanical properties data given in Appendix Table A-9 for some carbon steels represents mean values, what is the value of the tensile yield strength for 1050 steel quenched and tempered at 400F if a reliability of 99.9% is required?
2-28 Assuming that the mechanical properties data given in Appendix Table A-9 for some carbon steels represents mean values, what is the value of the ultimate tensile strength for 4340 steel quenched and tempered at 800F if a reliability of 99.99% is required?
2-29 Assuming that the mechanical properties data given in Appendix Table A-9 for some carbon steels represents mean values, what is the value of the ultimate tensile strength for 4130 steel quenched and tempered at 400F if a reliability of 90% is required?
2-30 Assuming that the mechanical properties data given in Appendix Table A-9 for some carbon steels represents mean values, what is the value of the tensile yield strength for
* Answers to these problems are
4140 steel quenched and tempered at 800F if a reliability of 99.999% is required?
provided in Appendix D.
72
MACHINE DESIGN -
An Integrated Approach
2-31 A steel part is to be plated to give it better corrosion resistance. Two materials are being considered: cadmium and nickel. Considering only the problem of galvanic action, which would you choose? Why?
2
2-32 A steel part with many holes and sharp corners is to be plated with nickel. Two processes are being considered: electroplating and electroless plating. Which process would you choose? Why?
2-33 What is the common treatment used on aluminum to prevent oxidation? What other metals can also be treated with this method? What options are available with this method?
*2-34 Steel is often plated with a less noble metal that acts as a sacrificial anode that will corrode instead of the steel. What metal is commonly used for this purpose (when the finished product will not be exposed to saltwater), what is the coating process called, and what are the common processes used to obtain the finished product?
2-35 A low-carbon steel part is to be heat-treated to increase its strength. If an ultimate tensile strength of approximately 550 MPa is required, what mean Brinell hardness should the part have after treatment? What is the equivalent hardness on the Rockwell scale?
2-36 A low-carbon steel part has been tested for hardness using the Brinell method and is found to have a hardness of 220 HB. What are the approximate lower and upper limits of the ultimate tensile strength of this part in MPa?
2-37 Figure 2-24 shows “guide lines” for minimum weight design when failure is the criterion. The guide line, or index, for minimizing the weight of a beam in bending is
2/3 / , where is the yield strength of a material and is its mass density. For a given f
f
cross-section shape the weight of a beam with given loading will be minimized when this index is maximized. The following materials are being considered for a beam application: 5052 aluminum, cold rolled; CA-170 beryllium copper, hard plus aged; and 4130
steel, Q&T @ 1200F. The use of which of these three materials will result in the least-weight beam?
2-38 Figure 2-24 shows “guide lines” for minimum weight design when failure is the criterion. The guide line, or index, for minimizing the weight of a member in tension is
f / , where f is the yield strength of a material and is its mass density. The weight of a member with given loading will be minimized when this index is maximized. For the three materials given in Problem 2-37, which will result in the lowest weight tension member?
2-39 Figure 2-23 shows “guide lines” for minimum weight design when stiffness is the criterion. The guide line, or index, for minimizing the weight of a beam in bending is E 1/2 / , where E is the modulus of elasticity of a material and is its mass density. For a given cross-section shape the weight of a beam with given stiffness will be minimized when this index is maximized. The following materials are being considered for a beam application: 5052 aluminum, cold rolled; CA-170 beryllium copper, hard plus aged; and 4130 steel, Q&T @ 1200F. The use of which of these three materials will result in the lowest-weight beam?
2-40 Figure 2-24 shows “guide lines” for minimum weight design when stiffness is the criterion. The guide line, or index, for minimizing the weight of a member in tension is E / , where E is the modulus of elasticity of a material and is its mass density. The weight of a member with given stiffness will be minimized when this index is maxi-
* Answers to these problems are
mized. For the three materials given in Problem 2-39, which will result in the lowest-provided in Appendix D.
weight tension member?
3
LOAD
DETERMINATION
If a builder has built a house for a man and his
work is not strong and the house falls in and
kills the householder, that builder shall be slain.
FROM THE CODE OF HAMMURABI, 2150 BC
3.0
INTRODUCTION
This chapter provides a review of the fundamentals of static and dynamic force analysis, impact forces, and beam loading. The reader is assumed to have had first courses in statics and dynamics. Thus, this chapter presents only a brief, general overview of those topics but also provides more powerful solution techniques, such as the use of sin-gularity functions for beam calculations. The Newtonian solution method of force analysis is reviewed and a number of case-study examples are presented to reinforce understanding of this subject. The case studies also set the stage for analysis of these same systems for stress, deflection, and failure modes in later chapters.
Table 3-0 shows the variables used in this chapter and references the equations, sections, or case studies in which they are used. At the end of the chapter, a summary section is provided which groups all the significant equations from this chapter for easy reference and identifies the chapter section in which their discussion can be found.
3.1
LOADING CLASSES
The type of loading on a system can be divided into several classes based on the character of the applied loads and the presence or absence of system motion. Once the general configuration of a mechanical system is defined and its kinematic motions calculated, the next task is to determine the magnitudes and directions of all the forces and couples present on the various elements. These loads may be constant or may be varying over time. The elements in the system may be stationary or moving. The most general class is that of a moving system with time-varying loads. The other combinations are subsets of the general class.
73
74
MACHINE DESIGN -
An Integrated Approach
Table 3-0
Variables Used in This Chapter
Symbol
Variable
ips units
SI units
See
a
distance to load
in
m
Sect. 3.9
b
distance to load
in
m
Sect. 3.9
3
d
damping
lb-sec/in
N-sec/m
Eq. 3.6
E
energy
in-lb
joules
Eq. 3.9, 3.10
F
force or load
lb
N
Sect. 3.3
fd
damped natural frequency
Hz
Hz
Eq. 3.7
fn
natural frequency
Hz
Hz
Eq. 3.4
g
gravitational acceleration
in/sec2
m/sec2
Eq. 3.12
Ix
mass moment of inertia about x axis
lb-in-sec2
kg-m2
Sect. 3.3
Iy
mass moment of inertia about y axis
lb-in-sec2
kg-m2
Sect. 3.3
Iz
mass moment of inertia about z axis
lb-in-sec2
kg-m2
Sect. 3.3
k
spring rate or spring constant
lb/in
N/m
Eq. 3.5
l
length
in
m
Sect. 3.9
m
mass
lb-sec2/in
kg
Sect. 3.3
N
normal force
in
m
Case 4A
M
moment, moment function
lb-in
N-m
Sect. 3.3, 3.9
q
beam loading function
lb
N
Sect. 3.9
R
position vector
in
m
Sect. 3.4
R
reaction force
lb
N
Sect. 3.9
v
linear velocity
in/sec
m/sec
Eq. 3.10
V
beam shear function
lb
N
Sect. 3.9
W
weight
lb
N
Eq. 3.14
x
generalized length variable
in
m
Sect. 3.9
y
displacement
in
m
Eq. 3.5, 3.8
deflection
in
m
Eq. 3.5
correction factor
none
none
Eq. 3.10
coefficient of friction
none
none
Case 4A
rotational or angular velocity
rad/sec
rad/sec
Case 5A
d
damped natural frequency
rad/sec
rad/sec
Eq. 3.7
n
natural frequency
rad/sec
rad/sec
Eq. 3.4
Table 3-1 shows the four possible classes. Class 1 is a stationary system with con-
stant loads. One example of a Class 1 system is the base frame for an arbor press used
in a machine shop. The base is required to support the dead weight of the arbor press
which is essentially constant over time, and the base frame does not move. The parts
brought to the arbor press (to have something pressed into them) temporarily add their Title-page photograph courtesy of
weight to the load on the base, but this is usually a small percentage of the dead weight.
Chevrolet Division of General
Motors Co., Detroit, Mich.
A static load analysis is all that is necessary for a Class 1 system.