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, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Moment-Curvature Relations
In terms of ordered derivatives:
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Design Procedure 5.1: Deflection by
Singularity Functions
1.
2.
3.
4.
5.
6.
Draw a free-body diagram showing the forces acting on the
system.
Use force and moment equilibria to establish reaction forces acting
on the system.
Obtain an expression for the load intensity function for all the
loads acting on the system while making use of Table 2.2.
Integrate the negative load intensity function to give the shear
force and then integrate the negative shear force to give the
moment.
Make use of Eq. (5.9) to describe the deflection at any location.
Plot the following as a function of x:
a.
b.
c.
d.
Shear
Moment
Slope
Deflection
Fundamentals of Machine Elements, 3rd 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, 3rd 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, 3rd 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, 3rd 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, 3rd 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) freebody diagram showing effect of
concentrated force; (c) free-body diagram
showing effect of moment.
Fundamentals of Machine Elements, 3rd 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, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Strain Energy
Loading type
Factors involved
Strain energy for
special case where
all three factors are
constant with x
P 2l
U = 2EA
General expression
for strain energy
P2
U=
dx
2EA
Axial
P, E, A
Bending
M, E, I
U =
M2l
2EI
U=
M2
dx
2EI
Torsion
T, G, J
U =
T2l
2GJ
U=
T2
dx
2GJ
Transverse shear
(rectangular section)
V , G, A
U =
3V 2 l
5GA
U=
3V 2
dx
5GA
Table 5.2: Strain energy for four types of loading.
Fundamentals of Machine Elements, 3rd 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, 3rd 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, 3rd 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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press