ME 221 Statics
Lecture #15a
Sections 7.3 - 7.4
ME221
Lecture 15
1
Homework #6
• Chapter 7 problems:
– 6, 19 & 26
• Chapter 6 problems
– 3&6
– Due Monday, June 28
• MatLab Group Problems
– 7.19, 7.26 & 6.15
– Due Monday, June 28
ME221
Lecture 15
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Last Lecture:
Internal Forces in Structures
• Reviewed internal/external forces
• Found internal forces
• Started shear & moment diagrams
ME221
Lecture 15
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Shear and Moment Diagrams using Sectioning Method
Generate a shear / bending diagram as follows:
1. Find reaction forces
2. Take a section on each side of an applied force or
moment and inside a distributed load
(take a new section whenever there is a change in the load
or shape of the beam)
- draw a FBD and sum forces / moments
3. Repeat 2 along the length of the beam
w(x) distributed load
V(x) shear force
M(x) moment
ME221
Lecture 15
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Sign Convention
M
M
V
V
Positive Shear and Positive Moment
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Lecture 15
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Effect of External Forces
Positive Shear
M
M
Positive Moment
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Lecture 15
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Relations Between w, V, and M
In balancing forces, we can come up with differential
equations relating w, V, and M. These are as follows:
F
 0  V  (V  dV )  w( x)dx  0
M
w(x)
M+dM
MV
 0 M  Vdx  ( M  dM )  0
Thus,
V  x 
dM  x 
dx
, w x 
dV  x 
dx
V+dV
dx
This means you can integrate the shear diagram to obtain
the moment diagram.
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Lecture 15
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Shear Forces
• Area under load curve
x
V ( x)   w(u )du
0
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Lecture 15
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Bending Moments
• Area under shear force curve
x
M ( x)   ( x  u ) w(u )du
0
ME221
Lecture 15
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