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EMA 3702
Mechanics & Materials Science
(Mechanics of Materials)
Chapter 1 Introduction
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Course Information
Instructor: Dr. Zhe Cheng
Phone: 305-348-1973
Email: [email protected]
Office: EC3441
Prerequisites:
EGN3311 Statics
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
More About Dr. Zhe Cheng
Education & Experiences:
PhD in Materials Sci. & Eng., Georgia Tech 2008
Research scientist at DuPont 2008-2013
Research group: https://ac.fiu.edu
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Textbook and Other Materials
Mechanics of Materials, 7th ed, Beer, Johnston,
Dewolf, and Mazurek, McGraw Hill Education,
2015, ISBN 9780073398235
(Other earlier editions are OK!)
Via MHEducation.com:
http://www.mheducation.com/highered/product/
mechanics-materials-beer-johnston-jr/0073398233.html
Via Amazon:
https://www.amazon.com/Mechanics-Materials-7th-Ferdinand-eer/dp/0073398233
Course website:
https://ac.fiu.edu/teaching/ema3702/
WhatsApp chat group link (Supports Google Chrome, Firefox, Microsoft
Edge, Safari, etc.):
https://chat.whatsapp.com/7rUHtjl2bke9gZ5c2eufQc
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Grading Policy
Homework (10 points)
Attendance (10 points, in the form of random class exercises)
Three exams (25 points each for exam 1 & 2, 30 points for final exam)
Overall grade: A:>=85; A-:81-84.9; B+:77-80.9; B:71-76.9; B-:67-70.9;
C+:63-66.9; C:55-62.9; D: 50-54.9; F:<50
NOTE: At the beginning of the semester, if you perceive your other
obligations (e.g., outside employment/job, family responsibilities) will
prevent you from attending a significant number of classes (e.g.,
missing 30% or more of the classes), you may request to have the
overall grade calculated based only on homework and exams.
Past grade statistics
2016 sum: Average 60.4/C; Median 64.7/C+; Highest 94.1/A; <C including drop: 7 out of 34
2017 sum: Average 67.6/C+; Median 70.5/B-; Highest 92.6/A;<C including drop: 9 out of 30
Example:
mid-term ~25/100; final ~45/100; almost perfect (44/45) in hw/participation;
overall – almost pass, but not quite: should he/she get C or D? Dr. Cheng choice: D
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Grade vs. Work Hours
Grade = -0.249t + 76.444
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Time on Job-Employment (hour/week)
2017 summer data
Median # courses taken: 3
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Grade vs. Number of Courses Taken
y = -4.8202x + 85.608
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Number of Courses Taken
2017 summer data
Median # courses taken: 3
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Grade vs. Class Attendance
y = 2.2913x + 28.314
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Class Participation/Lectures Attended
2017 summer data
Median # courses taken: 3
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Grade vs. Homework Submission
y = 2.8983x + 20.1370
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Homework Submission
2017 summer data
Median # courses taken: 3
Introduction
Statics Mechanics of Materials
Rigid body Practical engineering materials
w/ Elasticity & Plasticity
No failure Failure (Fracture, Fatigue, Creep)
Important foundation for advanced courses such as
• Mechanical Design I & II (EML 3350 & 4501)
• Finite Element Analysis (EGM4350)
• Synthesis of Engineering Mechanics (EGM5615)
“Mechanics of Materials” is a branch of
Mechanics that develops relationships between :
The external
loads
Internal forces intensity (stress) &
resulting deformation/shape
change (strain) and even failure
https://en.wikipedia.org/wiki/Strength_of_materials
“Strength of materials, also called mechanics of
materials, is a subject which deals with the behavior of
solid objects subject to stresses and strains”
Statics: Equilibrium
Mechanics of Materials:
1. Stress & strain in practical engineering component
under load
2. Statically indeterminate situations
3. Deformation (deflection) of engineering components
under load
4. Yielding or failure of engineering components
Balance of
• Forces
• Moments of force or torque
0 xF 0 yF
0 oM
Stress & Normal Stress
Convention:
• Positive for tension
• Negative for compression
To describe intensity of load,
introduce the concept of Stress
For axial loading, define
Normal Stress
In statics: 10 N vs. 106 N
In reality:
• Can a design (materials and/or
geometry) sustain the load?
A
P
Beer et al. (2015)
Force and corresponding
stress perpendicular to
the area of interest
Units of (normal) stress
SI Units for
stress
English Units
for stress:
psi (pounds per square inch) = lb/in2 = 6895 Pa
ksi = 103 psi
Pa = l N/m2
kPa = 103 N/m2
MPa = 106 N/m2
GPa = 109 N/m2
Load P in unit of N (SI) or lb of force (English)
Cross-section area in unit of m2 (SI) or in2 (English) A
P
Average Stress & Local Stress
(Normal) stress may vary within
the same cross-section due to
variations in local conditions (e.g.
material properties, clamping)
Local stress at Q:
Average normal stress A
P
A
F
A
lim
0
Stress distribution generally
NOT uniform
For a small area:
A
dAdFP
External load P is related to
local stress σ by:
For simple axial loading, if loads
pass through centroid or centric
loading, normal stress can be
assumed to be uniform if not too
close to the ends
dAdF
For eccentric (i.e., off-center) loading, the distribution of
the internal stress will not be uniform (see Chapter 4)
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Class Exercise on Stress (1)
For a cylindrical sample
with tensile force of 10000 N
loaded along its long axis, if
the initial cross-section area is
10 cm2, please calculate the
average normal stress σ.
18
A
F
MPaPam
N
m
N
cm
N
A
F101010
1010
10
10
10000 7
2
7
24
4
2
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Class Exercise on Stress (2)
For a straight metal bar with
a square cross-section, if
knowing the tensile stress is
along its long axis and it is
10 MPa and the square cross-
section has initial edge length
of 1 cm, please calculate the
tensile force applied.
19
1 cm 1 cm
AF
NmPacmcmPabAF 10001010)11(1010 24762
Shearing Stress
A
F
Strictly, as defined here is also an average stress
Force and corresponding
stress parallel to the
area of interest
Situation of Double Shear
If cross-section
area of bolt is A
A
F
A
F
A
Pave
2
2/
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Shearing Stress Class Exercise (1)
For the pin as illustrated, if
cross-section area is 50 cm2,
and load of 50 kN is applied,
please calculate the (average)
shearing stress
MPaPam
N
m
N
cm
N
A
F101010
10
10
50
1050 7
2
7
24
3
2
3
A
F
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Shearing Stress Class Exercise (2)
For the pin as illustrated, if
cross-section area is 20 cm2,
and load of 40 kN is applied,
please calculate the (average)
shearing stress
MPaPam
N
m
N
cm
N
A
F101010
10
10
20
2/)1040( 7
2
7
24
3
2
3
Double shear situation
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Bearing Stress in Connections
Complex condition for
bearing surfaces
An approximation:
Bearing Stress
Bearing surface
td
P
A
Pb t Thickness
d Diameter
General Method of Problem Solution 1. Draw free body diagrams (FBD)
2. Apply equations based on equilibrium of force &
moment and additional geometry considerations (if
applicable)
3. Solve for reactions (force/load), and resulting stress,
strain, and deformation (e.g., deflections)
4. Check answers including units
By convention,
• Use 4 figures to record numbers beginnings with “1”
• Use 3 figures in all other cases.
Examples:
A force of 40 lb. should be read as 40.0 lb
A force of 15 lb. should be read as 15.00 lb
Accuracy criteria
• Accuracy of the given data
• Accuracy of the computations performed
For engineering, accuracy better than 0.2% is rare
Numerical Accuracy
(1.12)
Stress on an Oblique Plane under Axial
Loading P
cosPF
sinPV
For a plane at angle
from the normal plane
Normal force for that plane
Shearing force for that plane
(Average) normal stress &
shearing stress for that plane
A
F
A
V
Aθ Area of that plane
θ
θ P
F
V
Relationship between the area of the two planes:
Normal cross-section A0 and Oblique plane Aθ
cos0 AA
2
00
coscos/
cos
A
P
A
P
A
F
cossincos
sin
00 A
P
A
P
A
V
Average normal stress & shearing stress for that plane
(at angle from the normal cross-section) will be
θ
θ P
F
V
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Maximum Normal & Shearing Stress
Max shearing
stress at = 45o
Max normal
stress at = 0o 0
maxA
P
0
max2A
P
2
0
cosA
P
cossin0A
P
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Class Example
Two wooden pieces of uniform square cross-section with
edge length of 1 cm are joined by gluing along oblique
interface, as shown. Knowing tensile load of 2 kN. Please
calculate the normal and shearing stress along the glued
interface as illustrated. Knowing Sin30o=Cos60o=0.5,
Cos30o=Sin60o=0.866.
P P
NNPCosFN 10002
1200060
NNPSinV 17322
3200060
30o
In this case, θ = 90o - 30o = 60o !!!!
2
00 2260
cmACos
AA
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Class Example (continued)
Two wooden pieces of uniform square cross-section with
edge length of 1 cm are joined by gluing along oblique
interface, as shown. Knowing tensile load of 2 kN. Please
calculate the normal and shearing stress along the glued
interface as illustrated.
Pam
N
cm
N
A
FN 6
242105
10500
2
1000
Pam
N
cm
N
A
V 6
2421066.8
10866
2
1732
P P
30o
Stress under General Loading Conditions:
Components of Stress
The plane is ┴ to x-axis
The vector is // to y - direction
Notation:
A
F x
Ax
lim
0
A
V x
y
Axy
lim
0
A
V x
z
Axz
lim
0
x
yV
The surface is ┹ to
x - direction or axis
The direction of the
component, i.e., shear
stress is // to y - direction
Notation for
Shearing Stress
Three normal stress
components
Six shearing stress
components
xy
For a FBD diagram, all the forces in a system must
fulfill the equations of equilibrium:
0 xF
0 xM
0 yF 0 zF
0 yM 0 zM
Equation of equilibrium
around z axis
0)()( aAaA yxxy
Therefore,
Similarly,
0 zM
yxxy
zxxz zyyz
Implication #2: only six independent components are
required to uniquely define the stress state at a given
point:
Implication #1: if there is
shearing in one plane, there
must be shear on another
plane perpendicular to the
first one
yxxy
σx, σy, σz, τxy, τyz, and τxz
For small cube with = 0o
max normal stress and
zero shearing stress
For small cube w/ = 45o
Max shearing stress and
same magnitude of
normal stress
Same Loading – Different Stress Interpretations?
Addressed in Chapter 7
Concept of Factor of Safety:
Design Considerations
ultimate load
allowable load
Factor of Safety = F.S.
stress Allowable
stress Ultimate
load Allowable
load Ultimate
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Class Exercise
For a long cylinder component with
designed cross-section area of 1 cm2 to
bear maximum axial load of
2104 N, if factor of safety (FS) is at
least 3, which material listed in the
table could be used?
Material Tensile strength
(MPa)
Steel 1090 mild 841
Aluminum 2014-T6 483
Cu, 99.9% 220
Cast iron, 4.5% C, ASTM-A-48
200
MPaPam
N
cm
Nall 200102
10
102
1
102 8
24
4
2
4
3all
TSFS
MPaMPaallTS 60020033
Only 1090 mild steel could be used
Class Exercise For the loading condition on the right, if AB and BC cross-section area is both 4cm2, determine the normal stress in beam AB and BC Since AB is a two-force beam, reaction at A must be horizontal, i.e., FAy = 0 Consider moment around C
mmkNmmFAx 80030600 kNFAx 40 kNFF AxAB 40
MPaPam
N
cm
NAB 100101
10
1010
4
1040 8
24
3
2
3
Class Exercise Similarly, BC is also a two-force beam. For point B, the three force balance is below
kNkNkNFBC 50)40()30( 22
MPaPam
N
cm
NBC 1251025.1
10
105.12
4
1050 8
24
3
2
3
30kN
40kN
FBC
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Homework 1.0
Read textbook section 1.1 to 1.5 (you may skip the sections named
“sample problem”) and give an honor statement confirm reading.
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Homework 1.1
Two solid cylinder rods DE and EF are bounded together at E and bonded to base at F. Knowing diameter d1 = 30 mm and d2=50 mm. Given the load condition as illustrated, calculate the average normal stress at the midsection of
(a) Rod DE
(b) Rod EF
F1=30kN
F2=62.5kN F3=62.5kN
d1
D
E
F
d2
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Homework 1.2
Link FH consists of a single bar 1 inch wide and 1 inch thick. Each pin has a 0.375 inch diameter. Determine the value of the maximum average normal stress in bar FH if
(a) θ = 0 (Pushing to left)
(b) θ = 90o (Pushing down) Hint: Have to consider
that stress at the cross-section where the connection pin is located will be higher than that at center of the bar (see textbook examples)
12in
6in
F1= 4 kips
θ
30o E
F
G
H
Pins with
diameter of
0.375 inch
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Homework 1.3
Determine the diameter of the largest circular hole that can be
punched to a sheet of polystyrene 4 mm thick. Punch force applied
will be 22.5 kN and the average shearing stress of 55 MPa is needed
to cause the polystyrene material used to fail (i.e., for the punch to
cut through and make the hole)
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Homework 1.4
Two rectangular wood parts are bonded by glue as illustrated. The
wood parts are 2 inch long and 1 inch wide. Shearing force are applied
to the two bonded pieces. When shearing force P reaches 1600 lb, the
two pieces started to separate or fail along the bonding interface,
please calculate the average shearing stress at the time of failure
P
P
2 in 1 in
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Homework 1.5
Two wooden pieces of uniform square cross-section with edge length of
1 cm are joined by gluing along oblique interface, as shown. Knowing
tensile load of 10 KN. Please calculate the normal and shearing stress
along the glued interface as illustrated.
60o P P
EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 1 Introduction
Homework 1.6
Members of DE, EF, and DF of the truss shown are made of the same
material. It is known that the material has ultimate tensile strength
U = 200 MPa. If safety factor is 3.0 is to be achieved for both DE and
DF bars, please determine the required minimum cross-section area
of bar DE and bar DF
0.75 m
0.4 m
1.4 m
10 kN
D
E
F