Chapter 5: Deformation

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Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Chapter 5: Deformation

Let me tell you the secret that has led
me to
my
goal. My strength lies in my
tenacity
.


Louis Pasteur

Testing of 787 Dreamliner wings. (Courtesy
of Boeing Corp.)

Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Moment
-
Curvature Relations

In terms of ordered derivatives:

Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Design Procedure 5.1: Deflection by
Singularity
Functions

1.
Draw a free
-
body diagram showing the forces acting on the
system.

2.
Use
force and moment
equilibria

to establish reaction forces acting
on the system.

3.
Obtain
an expression for the load intensity function for all the
loads acting on the system while making use of Table
2.2.

4.
Integrate
the negative load intensity function to give the shear
force and then integrate the negative shear force to give the
moment.

5.
Make
use of Eq
. (5.9)
to describe the deflection at any location.

6.
Plot
the following as a function
of
x
:

a.
Shear

b.
Moment

c.
Slope

d.
Deflection

Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Beams

Figure
5.1: Cantilevered beam
with concentrated force applied
at free end.

Figure 5.2
: Free
-
body diagram of force
anywhere between simply supported
ends. (a) Complete beam; (b) portion of
beam.

Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Example 5.3

Figure
5.3: Cantilevered beam
with unit step distribution over
part of beam. (a) Loads and
deflection acting on cantilevered
beam; (b) free
-
body diagram of
forces and moments acting on
entire beam; (c) free
-
body
diagram of forces and moments
acting on portion of beam.

Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Example 5.4

Figure
5.4: Pinned
-
fixed beam with
concentrated force acting anywhere
along beam. (a) Sketch of assembly; (b)
free
-
body diagram of entire beam; (c)
free
-
body diagram of part of beam.

Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Beam Deflection

Table 5.1: Deflection for common cantilever and simply
-
supported beam
conditions. See also Appendix D.

Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Example 5.5

Figure
5.5: Beam fixed at one end and free
at other with moment applied to free end
and concentrated force at any distance from
free end. (a) Complete assembly; (b) free
-
body diagram showing effect of
concentrated force; (c) free
-
body diagram
showing effect of moment.

Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Stress Elements

Figure
5.6: Element
subjected to normal stress.

Figure
5.7: Element
subjected to shear stress.

Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Strain Energy

Table 5.2: Strain energy for four types of loading.

Strai
n
energ
y
for
specia
l
cas
e
where
al
l
thre
e
factor
s
ar
e
Genera
l
expression
Loadin
g
typ
e
Factor
s
involve
d
constan
t
with
x
fo
r
strai
n
energy
Axial
P
,
E
,
A
U
=
P
2
l
2
E
A
U
=
P
2
2
E
A
dx
Bending
M
,
E
,
I
U
=
M
2
l
2
E
I
U
=
M
2
2
E
I
dx
T
orsion
T
,
G
,
J
U
=
T
2
l
2
GJ
U
=
T
2
2
GJ
dx
T
ransvers
e
shear
V
,
G
,
A
U
=
3
V
2
l
5
GA
U
=
3
V
2
5
GA
dx
(
r
ectangula
r
section)
Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Example 5.8

Figure
5.8: Cantilevered beam with concentrated force acting at a distance
b

from free end. (a) Coordinate system and significant points shown; (b) fictitious
force,
Q
,
shown along with concentrated force,
P
.

Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Example 5.9

Figure
5.9: System arrangement. (a) Entire assembly; (b) free
-
body diagram of
forces acting at point A.

Fundamentals of Machine Elements, 3
rd

ed.

Schmid,
Hamrock

and Jacobson

© 2014 CRC Press

Example 5.10

Figure
5.10: Cantilevered beam with
90
°

bend acted upon by horizontal force,
P
,
at free end.