Internal Forces in Structures

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ME 221 StaticsLecture 27Section 7.4
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Homework 9

• Chapter 5 problems
• 54, 56, 62, 64, 69, 71 73
• Due Today

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Homework 10

• Chapter 7 problems
• 5, 8, 19, 21, 24, 26 35
• Due Wednesday, November 13

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Last LectureInternal Forces in Structures

• Reviewed internal/external forces

• Found internal forces

• Started shear moment diagrams

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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
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Sign Convention
Positive Shear and Positive Moment
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Effect of External Forces
Positive Shear
Positive Moment
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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

w(x)
MdM
V
M
Thus,
VdV
dx
This means you can integrate the shear diagram
to obtain the moment diagram.
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Shear and Moment Diagrams Using Integration Method
1 - Distributed Load
w(x)
MdM
V
M
VdV
dx
Area under the load intensity diagram between xj
and x j1
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Similarly
Area under the shear diagram between xj and x j1
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2 - Concentrated Load
P
P
MdM
V
M
C
VdV
3 - Concentrated Moment
MdM
V
M
C
VdV
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Notes

• Concentrated force will cause a jump in the shear
  diagram by an amount equal to the applied load

• Concentrated moment will cause a jump in the
  moment diagram by an amount equal to the negative
  of the applied moment

Connecting points in the shear and moment diagrams
xj
Negative increasing slope
xj1
xj
Positive decreasing slope
xj1
Negative decreasing slope
xj1
Positive increasing slope
xj1
xj
xj
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Developing Shear and Bending Diagrams
1. Show FBD and statics for each section
2. Determine equation for V(x) and M(x)
3. Draw shear and bending diagrams indicating
linear or parabolic
4. Label end points of diagram as well as every
region endpoint
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