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CE 234
STRENGTH OF MATERIALS
CHAPTER 2
Stress and Strain – Axial Loading
Lecture By: Dr. Özgür KÖYLÜOĞLU, Yeditepe University
Book: Mechanics of Materials, 6th Edition, by Ferdinand P. Beer; E. Russel
Johnston, Jr.; John T. DeWolf, David F. Mazurek, Mc Graw Hill.
Presentation Reference: Lecture Notes by J. Walt Oler, Texas Tech University on
Mechanics of Materials, 3rd Edition, by Ferdinand P. Beer; E. Russel Johnston,
Jr.; John T. DeWolf, Mc Graw Hill.
Stress and Strain – Axial Loading
2
Normal Strain
Load-deformation diagram


P
 stress
A

L
 normal strain
3
Normal Strain


P
 stress
A

L
 normal strain


2P P

2A A

L
P
A
2 


2L L

4
Normal Strain - Definition
5
Stress-Strain Diagram
Tensile Test Machine
Typical Tensile-Test
Specimen, L0= gage length6
Stress-Strain Diagram
7
Stress-Strain Diagram of Ductile Materials
8
http://www.esm.psu.edu/courses/emch13d/design/animation/Fnecking.html
Stress-Strain Diagram of Brittle Materials
9
Determination of Yield Strength by
Offset Method
10
Ductility
11
Stress-Strain Diagram Under
Compression
12
Stress-Strain Diagram Under
Compression
Stress-Strain Diagram of Concrete
13
Hooke’s Law; Modulus of Elasticity
14
Isotropic vs. Anisotropic Materials
15
Elastic vs. Plastic Behavior
16
Elastic vs. Plastic Behavior –
Baushinger Effect
17
Repeated Loadings; Fatique
18
Deformations Under Axial Loading
19
Deformations Under Axial Loading
20
Example Problem 2.01
A=194 mm2
A=581 mm2
300 kN
180 kN
120 kN
Determine the deformation of the
steel rod shown under the given
loads (E=200 GPa).
300 mm
300 mm
300 mm
300 kN
180 kN
120 kN
120 kN
180 kN
300 kN
180 kN
120 kN
120 kN
21
Sample Problem 2.1.
The rigid bar BDE is supported by two links AB and CD.
Link AB is made of aluminum (E = 70 GPa) and has a cross-sectional area of
500mm2. Link CD is made of steel (E = 200 GPa) and has a cross-sectional area of
(600 mm2).
For the 30-kN force shown, determine the deflection a) of B, b) of D, and c) of E.
22
Sample Problem 2.1.
Deflection of B:
SOLUTION:
Free body: Bar BDE
Deflection of D:
23
Sample Problem 2.1.
24
Statically Indeterminate Problems
25
Example Problem 2.02
26
Example Problem 2.04
Determine the reactions at A and B.
27
Example Problem 2.04
28
Example Problem 2.04
29
Problems Involving Temperature
Changes (Thermal Stresses)
30
Example Problem 2.06
A=380 mm2
300 mm
A=750 mm2
300 mm
Determine the stress in
portions AC and CB of the
steel bar shown, when the
temperature of the bar is 45oC, knowing that a close fit
exists at both of the rigid
supports when the temperature
is +75oC. (E=200 GPa,
α=11.7x10-6 /oC)
31
Poisson’s Ratio
32
Example Problem 2.07
A 500-mm long, 16 mm
diameter rod made of
homogeneous, isotropic
material is observed to
increase in length by 300
μm, and to decrease in
diameter by 2.4 μm when
subjected to an axial 12-kN
load.
Determine the modulus of
elasticity and Poisson’s
ratio of the material.
33
Multiaxial Loading; Generalized
Hooke’s Law
34
Example Problem 2.08
The steel block shown is
subjected to a uniform
pressure on all its faces.
50 mm
100 mm
Knowing that the change in
length of edge AB is 30x10-3 mm, determine
75 mm
a)The change in length of
the other two edges
b)The pressure p applied to
the faces of the block.
Assume E=200 GPa and
ν=0.29)
35
Dilatation; Bulk Modulus
36
Example Problem 2.09
50 mm
100 mm
75 mm
Determine the change in
volume ΔV of the steel
block shown, when it is
subjected to the hydrostatic
pressure p=180 MPa.
Assume E=200 GPa and
ν=0.29)
37
Shearing Strain
38
Generalized Hooke’s Law for
General Stress Condition
 x  y  z
x  
E
y  
z  

 x
E

E

 y  z
E
 x  y
E

E
E


E
z
E
 xy  G  xy  yz  G  yz  zx  G  zx
39
Relation Among E, υ and G
40
Fiber-Reinforced Composite
Materials
41
Fiber-Reinforced Composite
Materials
• Equations similar to Hooke’s Law can be developed for composite
materials under multiaxial loading as follows:
ex =
s x u yxs y u zxs z
Ex
ey = -
ez = u xy
Ex
=
-
-
Ey
g xy =
Ez
u xys x s y u zys z
Ex
u xzs x
Ex
+
-
Ey
-
g yz =
Ez
u yzs y s z
u yx
u yz
Ey
Ey
Ey
=
+
g xz =
Ez
u zy
u zx
Ez
Ez
=
t xy
Gxy
t yz
Gyz
t xz
Gxz
u xz
Ex
42
Example Problem 2.11
A 60-mm cube is made from
layers of graphite epoxy with
fibers aligned in the x-direction.
The cube is subjected to a
compressive load of 140 kN in the
x-dir. Determine the changes in
the cube dimensions, knowing
(a) Cube is free to expand in yand z-directions
(b) The cube is free to expand in
z-direction but restrained in ydirection by two firctionless
plates.
43
Example Problem 2.5
.
44
Saint-Venant’s Principle
45
Stress Concentrations; Hole
Discontinuities of cross section may result in high localized or
concentrated stresses.
 max
K
 ave
K = Stress-concentration factor
46
Stress Concentrations; Fillet
Discontinuities of cross section may result in high localized or concentrated
stresses.
 max
K
 ave
47
Example Problem 2.12
Determine the largest axial load P
that can be safely supported by a
flat steel bar consisting of two
portions, both 10 mm thick, and
respectively 40 and 60 mm wide,
connected by fillets of radius r = 8
mm. Assume an allowable normal
stress of 165 MPa.
48
Example Problem 2.12
49
Elastoplastic Materials
50
Plastic Deformations

A
P   ave A  max
K
 A
PY  Y
K
PU = s Y A
PY =
PU
K
51
Residual Stresses
52
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