ME 221 Statics
Lecture #26
Sections 7.1 – 7.4
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Homework #9
Chapter 5 problems:
– 54, 56, 62, 64, 69, 71 & 73
• Due Wednesday, November 6
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Homework #10
Chapter 7 problems:
– 5, 8, 19, 21, 24, 26 & 35
• Due Wednesday, November 13
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Chapter 7:
Internal Forces in Structures
• Review internal/external forces
• How to find internal forces
• Sample problems
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External/Internal Forces
• External forces arise from contact or
gravitational attraction
– Point and distributed loading
– Weight
• Internal forces are forces arising to hold
bodies together
– Internal stress is a form of an internal force
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Exposing Internal Forces
• To analyze the stress at a given location in a
part, we need to know the forces at that
particular section.
c
c
b
a
b
a
At any given section a-a, b-b, or
c-c, there is an internal force arising
from the 100 lb external force.
100 lb
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Method for Finding Internal Forces
• Determine reaction forces
– Use equilibrium equations
• Section and solve second equilibrium
problem to find internal forces
Mz=0
Ay=100 lb
Ax=0
Mz
a
100 lb
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a
100 lb
Lecture 26
Fy
Fx
100 lb
7
General Internal Forces
• In general, there is a force and moment
component for each coordinate direction at
a given section
– 6 possible unknowns
Sample problems:
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Example:
l
y
Determine the internal
forces and moments in
the bar built into the
foundation as shown in
the figure.
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P
x
h
z
O
9
(l-x)
l
y
P
x
h

M
x

R

rp
P
Horizontal Portion
z
O


 F  0  R  Pkˆ


 M  0  M  (l  x)iˆxPkˆ
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

R   Pkˆ

M  P(l  x) ˆj
10
l
l
y
P
P
x
h

M
y
z
O

R   Pkˆ

M  Plˆj  P(h  y )iˆ
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(h-y)

rp  liˆ  (h  y ) ˆj

R
Vertical Portion
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Shear and Bending Diagrams
(Secs. 7.3, 7.4)
• Topic is also called transversely loaded beams
• Beam classifications and boundary conditions
• Internal forces and the components’ specific rolls
• Relation between shear and bending
• Generation of shear and bending diagrams
• Sample problems
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Types of Beams by Supports
• Transversely loaded beams have several
standard configurations
• Determinate beams have the same number
of reactions as nontrivial equilibrium eqns.
Determinate
Indeterminate
Simple
Overhanging
Cantilever
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Internal Force Component Rolls
• Force components
P
– Axial is along beam
Vz
Vy
– Shearing forces are transverse components
• Moment components
– Torsion along beam
– Bending for transverse
components
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T
Mz
My
14
Shear and Moment Diagrams
-Sectioning Method
-Integration
-Singularity Functions
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What is expected for 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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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
M
M
V
V
Positive Shear and Positive Moment
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Effect of External Forces
Positive Shear
M
M
Positive Moment
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125 lb
20 lb/in
9 in.
12 in.
125 lb
12 in.
12in.
V
x
M
+ve
x
-ve
tension up
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Tension down
20
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