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FE Review – Mechanics of
Materials
FE Review Mechanics of Materials 2
Resources
You can get the sample reference book:
www.ncees.org
– main site
http://www.ncees.org/exams/study_ma
terials/fe_handbook
Multimedia learning material web site:
http://web.umr.edu/~mecmovie/index.
html
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FE Review Mechanics of Materials 3
Normal Stress (normal to surface)
Shear Stress (along surface)
First Concept – Stress
FE Review Mechanics of Materials 4
Normal Strain – length change
Mechanical
Thermal
Shear Strain – angle change
Second Concept – Strain
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FE Review Mechanics of Materials 5
Material Properties
Hooke’s Law
Normal (1D)
Normal (3D)
Shear
FE Review Mechanics of Materials 6
Material Properties
Poisson’s ratio
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FE Review Mechanics of Materials 7
Axial Loading
Stress
Deformation
FF
P
L
A
E
δ
=
∑
x
P
A
σ
=
F
σ
x
FE Review Mechanics of Materials 8
Torsional Loading
Stress
Deformation
TL
J
G
θ
=
∑
θ
J
ρ
τ
=
TT
max
Tc
J
τ
=
ρ
τ
τ
max
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FE Review Mechanics of Materials 9
Bending Stress
Stress
Find centroid of crosssection
Calculate I about the Neutral Axis
r
x
M
y
I
σ
=−
max
r
M
c
I
σ
=
M
M
σ
x
FE Review Mechanics of Materials 10
Transverse Shear Equation
ave
V
A
τ
=
Average over entire crosssection
ave
VQ
Ib
τ
=
Average over line
V = internal shear force
b = thickness
I = 2
nd
moment of area
Q = 1
st
moment of area of partial
section
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FE Review Mechanics of Materials 11
Partial 1
st
Moment of Area (Q)
FE Review Mechanics of Materials 12
Max. Shear Stresses on Specific
CrossSectional Shapes
Rectangular CrossSection
max
3
2
V
A
τ
=
τ
Circular CrossSection
max
4
3
V
A
τ
=
τ
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FE Review Mechanics of Materials 13
Max. Shear Stresses on Specific
CrossSectional Shapes
WideFlange Beam
max
web
V
A
τ
≈
τ
FE Review Mechanics of Materials 14
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FE Review Mechanics of Materials 21
V & M Diagrams
dV
w
dx
=
V
M
dM
V
dx
=
FE Review Mechanics of Materials 22
Six Rules for Drawing V & M
Diagrams
1.w = dV/dx
The value
of the distributed load
at any point in the beam is equal to the
slope
of the shear force
curve.
2.V = dM/dx
The value
of the shear force
at any point in the beam is equal to the slope
of
the bending moment
curve.
3.The shear force curve
is continuous unless there is a point force
on the
beam. The curve then “jumps” by the magnitude of the point force (+ for
upward force).
4.The bending moment curve
is continuous unless there is a point moment
on
the beam. The curve then “jumps” by the magnitude of the point moment
(+ for CW moment).
5.The shear force
will be zero at each end of the beam unless a point force is
applied at the end.
6.The bending moment
will be zero at each end of the beam unless a point
moment is applied at the end.
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FE Review Mechanics of Materials 23
Deflection Equation
2
2
d y
M
EI
dx
=
y = deflection of midplane
M = internal bending moment
E = elastic modulus
I = 2
nd
moment of area with
respect to neutral axis
To solve bending deflection problems (find y):
1.Write the moment equation(s) M(x)
2.Integrate it twice
3.Apply boundary conditions
4.Apply matching conditions (if applicable)
FE Review Mechanics of Materials 24
Method of Superposition
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FE Review Mechanics of Materials 25
Stress Transformation
Plane Stress Transformation Equations:
cos2 sin2
2 2
x y x y
n xy
σ
σ σ σ
σ
θ τ θ
+
−
= + +
sin2 cos2
2
x y
xy
nt
σ
σ
τ
θ τ θ
⎛ ⎞
⎜ ⎟
⎝ ⎠
−
=− +
τ
xy
σ
x
σ
y
FE Review Mechanics of Materials 26
Stress Transformation
Principal Stresses:
2
2
1,2
2 2
x
y
x y x y
p p
σ σ σ σ
σ
τ
⎛ ⎞
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
+ −
=
+ +
( )
tan 2
2
xy
p
x
y
τ
θ
σ
σ
=
−
⎛ ⎞
⎜ ⎟
⎝ ⎠
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FE Review Mechanics of Materials 27
Stress Transformation
Max Shear Stress:
1 2
max
2
p p
σ
σ
τ
−
=
1
max
2
p
σ
τ
=
2
max
2
p
σ
τ
=
FE Review Mechanics of Materials 28
Stress Transformation
Mohr’s Circle
σ
τ
C
(
)
,
x
xy
σ
τ−
(
)
,
y xy
σ
τ
R
τ
xy
σ
x
σ
y
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FE Review Mechanics of Materials 29
Combined Loading
We have derived stress equations for four
different loading types:
x
P
A
σ
=
max
V
k
A
τ
=
FE Review Mechanics of Materials 30
x
M
c
I
σ = −
x
M
c
I
σ = +
Tc
J
τ
=
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FE Review Mechanics of Materials 31
Method for Solving Combined
Loading Problems
1.Find internal forces and moments at
crosssection of concern.
2.Find stress caused by each individual
force and moment at the point in
question.
3.Add them up.
FE Review Mechanics of Materials 32
ThinWalled Pressure Vessels
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FE Review Mechanics of Materials 33
Column Buckling
FE Review Mechanics of Materials 34
σ
Y
σ
Y
−σ
Y
−σ
Y
Failure occurs when:
1
p
Y
σ
σ>
where
σ
p1
is the largest principal stress.
if σ
p1
and σ
p2
have the same sign
1 2
p
p Y
σ
σ σ
−
>
if σ
p1
and σ
p2
have different signs
σ
p1
σ
p2
Maximum Shear Stress Theory
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FE Review Mechanics of Materials 35
σ
Y
σ
Y
−σ
Y
−σ
Y
Failure occurs when:
2 2 2
1 1 2 2
p
p p p Y
σ
σ σ σ σ
−
+ >
σ
p1
σ
p2
Maximum Distortion Energy
Theory
This theory assumes that failure
occurs when the distortion
energy
of the material is
greater than that which causes
yielding in a tension test.
FE Review Mechanics of Materials 36
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