This document discusses single-mode loading conditions that will be covered in the class, including axial loading, direct shear, torsional loading, and beam bending. It provides examples and equations for calculating stresses and deformations for each loading condition. For axial loading of a rod, it shows how to calculate the load at yield and ultimate failure. For direct shear, it discusses shear area and provides examples. For torsional loading, it introduces the torsion formula and shows how to calculate shear stresses. It also discusses stress concentrations and provides an example problem calculating maximum shear stress in a lawnmower drive shaft.

1. Lecture 1: Single-Mode Loading Part A
For the first few weeks of this class, we will focus on identifying and analyzing machine elements that
are subjected to “single-mode loading.” WHAT IS SINGLE MODE LOADING???
Here are the loading modes that we will consider:
Tearout
Direct shear (review) Section 4.8
Section 4.7
Beam bending (review)
– Sections 4.9, 4.10
Bearing Stress
Column
Torsion (Review)
Buckling
Axial loading (review) Section 4.8 Section 4.16
Section 4.7 Section 4.12
1

2. A Good Place to Start: Review of a Few
Foundational Ideas from Materials Science
The uniaxial tension test yields data that
defines a few different material properties
A stress strain diagram for low carbon
steel (which is a ductile material)
Applied Force –
Elongation -
Stress –
Strain –
Engineering Stress vs. True Stress -
https://www.youtube.com/watch?v=D8U4G5k
cpcM 2

3. A Good Place to Start: Review of a Few
Foundational Ideas from Materials Science
A stress strain diagram for low carbon
steel (which is a ductile material)
Elastic Deformation –
Plastic Deformation –
Elastic Modulus E –
Tensile Yield Strength Syt –
Ultimate Tensile Strength – Sut
Ductility -
https://www.youtube.com/watch?v=D8U4G5k
cpcM
3

4. Single Mode Loading Condition #1: Axial Loading
Consider the state of stress when a mass is hanging at the bottom of a massless rod:
What type of loading does the rod see?
What is the state of stress within the rod?
Governing Equation for stress within the rod:
Elongation of the rod:
Strain in the rod:
Massless L
Question: Is this “true strain”
Weight = F
or “engineering strain”?

5. Examples of Axial Loading

6. Single Mode Loading Condition #1 - Axial Load
Suppose the rod is fabricated from 2024 heat treated aluminum and that the diameter of the rod is 0.5 in.
Assuming that the mass of the rod is negligible compared to weight F:
1) Find the value of F at which the rod will yield:
2) Find the value of F at which the rod will break:
Negligible L
Mass
NEW CONCEPT: FACTOR OF SAFETY (aka safety factor)
F.S. =
Weight = F
For the problem above, what is the largest value of F that is allowed in
order to maintain a safety factor of 1.5 against yielding?
For the value of F just found, what is the factor of safety against total breakage of the rod?

7. Single Mode Loading Condition #2 – Direct Shear
Direct shear (also known as “pure shear”) occurs when opposing forces act to slide atoms over each
other, creating new material faces.
The key to analyzing direct shear problems is figuring out what new faces will be created if
the part breaks. The dimensions of these shear faces are used to find the shear area.
Loading: Direct Shear
Stress state: Shear stress parallel to
the applied force
Equation for shear stress:
A blanking operation
creates shaped holes
in sheet metal by shearing
through the material
What is the shear area
for this blanking operation?

8. Single Mode Loading Condition #2 – Direct Shear
EXAMPLES OF DIRECT SHEAR.
For each example, what is the shear area?
Snowblower shear pins protect the blower drivetrain
The pin connecting the
hook to the clevis fork
is in direct shear.
Shaft keys are used to protect a
drivetrain against overload. The key
should fail before other drive
elements in the event of an overload.

9. Single Mode Loading Condition #2 – Direct Shear
A COMMON MISTAKE: DON’T CONFUSE DIRECT SHEAR WITH
BENDING Direct shear due to a directly applied force as shown
below in Figure (a) is requires that there is almost no clearance between
the two shearing faces.
If there is much distance between the faces, the
problem turns into a beam bending problem.
c) Bending
a) Pure shear b) Mixed shear + bending
d) More Bending
NOT DIRECT SHEAR!

10. Shear Stress vs. Shear Strain
Shear stress-strain relationship:
Yield strength in shear:
Ultimate shear strength:
NOTE: (THIS IS HUGE!!!!): For ductile materials (most steels and aluminums, brass, titanium,
many plastics, etc.):
You should have this equation tattooed on your forearm right after class.

11. An Example of Pure Shear
“Blanking” Operation to punch sheet metal
A Blanking Punch
EXAMPLE PROBLEM: Let’s calculate the force required
to punch out this blank given the following:
- The diameter of the punch is 2 in.
- The thickness of the sheet is 0.12 in.
- The material is AISI 1020 hot rolled steel
http://www.custompartnet.com/wu/sheet-metal-shearing

12. Example: Pure Shear in a Clevis
The drawings on this slide show a type of connection
called a “clevis.”
A clevis is used to connect two axially-loaded rods.
Suppose that the rods and the clevis joint are carrying 500 kips 500 kips 500 kips
an axial tensile load of 500 kips.
Also suppose that the clevis pin has a diameter
of 0.500 inches.
1) Look at the drawings and convince yourself
that the pin is in pure shear. Where are the planes
on which the pin would shear?
2) Calculate the shear stress on the shear
planes of the pin.
500 kips 500 kips 500 kips

13. Single Mode Loading Condition #3 - Torsional Loading
• Torque – moment that tends to twist a member about its longitudinal axis.
• This causes a twisting deformation, called torsion.
Assumptions:
- Radius does not change
- Ends remain flat
- Part is twisted
13

14. Examples of Torsional Loading
• Torque –
moment that
tends to twist a
member about its
longitudinal axis.
• This causes a
twisting
deformation,
called torsion.

15. Examples of Torsional Loading

16. Examples of Torsional Loading

17. = � 2
=
=
due to torsion
Max shear stress
the section) =
Shear stress due to
torsion (anywhere on
Torsion
formula
Shear Due to Torsion
This is a property of the
geometry of the shaft and is
called as the polar (area)
moment of inertia, J.
T – internal torque at the cross-section
J – polar moment of inertia of the
cross-sectional area
c – outer radius of the shaft
ρ – radius between 0 and c
τmax – max. shear stress (at the
outer surface)
NOTE: τ = 0 at the center
17

18. Example – Shear Stress Due to Torsion
• A pipe with an inner diameter of 80 mm and an outer diameter of 100 mm is tightened against the
support at A by a pair of disembodied hands. Determine the shear stress developed at the inner and
outer walls along the central portion of the pipe.
18

19. Stress concentration
• If there is a change in the cross section, or any other discontinuity on a shaft under
torsion, the shear stress can be higher than the one calculated by the torsion formula.
• This is the concept of stress concentration that we discussed for axial loading as well.
• In such cases, we will need to find the maximum shear stress as a result of the
discontinuity by finding the stress concentration factor (K).
19

20. Power transmission
• Shafts and tubes are commonly used to transmit power.
• This results in torque (T) due to the power that needs to be transmitted and the angular speed (ω) of the shaft.
• Power is defined as work done per unit time. Work transmitted by a shaft is equal to the product of torque and angle of rotation.
= = Nm/s (with ω in rad/s)
In US units, hp is the most commonly used unit, I hp = 550
= = 2
In the SI units, power is expressed in Watts (W), 1 W = 1
Frequency can be expressed in cps (or Hz), 1 cycle = 2π
ft-lb/s
Note: f is frequency in cycles per
radians
second, ω is in rad/s
20

21. Torsional Loading With a Stress Raiser: Example
The picture below shows the drive shaft of a lawnmower (IGNORE THE DISEMBODIED HAND HOLDING A WRENCH. NEITHER THE HAND NOR THE
WRENCH HAVE ANTHING TO DO WITH THIS PROBLEM).
To a good approximation, the shaft is loaded in pure torsion. (WHY? WHAT ASSUMPTIONS ARE BEING MADE IF WE SAY THAT THE SHAFT IS IN PURE
TORSION?)
Suppose that the lawnmower engine can generate 5 Hp of power while running at a speed of 3000 rpm. What is the maximum torsional shear stress
that is generated in the shaft while the lawnmower is running at full power?
Front and side views of the shaft. All dimensions in inches.
THIS IS THE PIECE THAT WE ARE ANALYZING!
21

22. Torsional Loading With a Stress Raiser: Example (cont)
22

23. Angle of twist
• Angle of twist (ϕ) is the angle by which a shaft is twisted when subject to a torsion load.
• If the material is homogeneous and if the behavior is linear and elastic, then the shaft will go back
to its original shape when the load is removed.
T – internal torque carried by
the shaft
J – polar moment of inertia G –
modulus of rigidity L –
length of shaft
=
=
�
0 =
3

24. Example: Windup of the lawnmower shaft
If the lawnmower shaft from the earlier example is fabricated from a medium- carbon steel, what is the total angular deformation (aka the windup of
the shaft) of the shaft when the mower is running at full power at a speed of 3000 rpm?
Front and side views of the shaft. All dimensions in inches.
24