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Top School in Noida By: school.edhole.com
ME221 Lecture 14 2
ME 221 Statics
Lecture #14
Sections 7.1 – 7.4
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ME221 Lecture 14 3
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
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ME221 Lecture 14 5
Chapter 7:Internal Forces in Structures
• Review internal/external forces
• How to find internal forces
• Sample problems
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ME221 Lecture 14 6
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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ME221 Lecture 14 7
Exposing Internal Forces• To analyze the stress at a given location in a
part, we need to know the forces at that particular section.
100 lb
a a
b b
c cAt any given section a-a, b-b, orc-c, there is an internal force arisingfrom the 100 lb external force.
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ME221 Lecture 14 8
Method for Finding Internal Forces• Determine reaction forces
100 lb 100 lb
Ay=100 lb
Ax=0
Mz=0
– Use equilibrium equations• Section and solve second equilibrium
problem to find internal forces
a a
100 lb
Fy
Fx
Mz
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ME221 Lecture 14 9
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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ME221 Lecture 14 10
Example:
Determine the internal forces and moments in the bar built into the foundation as shown in the figure.
x
y
z
O
l
h
P
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ME221 Lecture 14 11
(l-x)P
x
y
z
O
l
h
P
x R
M
pr
kPixlMM
kPRF
ˆx ˆ)(0
ˆ0
jxlPM
kPR
ˆ)(
ˆ
Horizontal Portion
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ME221 Lecture 14 12
x
y
z
O
l
h
P
y
R
M
l
(h-y)
P
jyhilrp ˆ)(ˆ
iyhPjPlM
kPR
ˆ)(ˆ
ˆ
Vertical Portion
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ME221 Lecture 14 13
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
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ME221 Lecture 14 14
Types of Beams by Supports• Transversely loaded beams have several
standard configurations• Determinate beams have the same number
of reactions as nontrivial equilibrium eqns.IndeterminateDeterminate
Simple
Overhanging
Cantileverschool.edhole.com
ME221 Lecture 14 15
Internal Force Component Rolls• Force components
• Moment components
– Axial is along beamP
– Shearing forces are transverse componentsVy
Vz
– Torsion along beam T
– Bending for transverse components
My
Mz
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ME221 Lecture 14 16
Shear and Moment Diagrams
-Sectioning Method
-Integration
-Singularity Functions
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ME221 Lecture 14 17
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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ME221 Lecture 14 18
Generate a shear / bending diagram as follows:
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 / moments3. Repeat 2 along the length of the beam.
1. Find reaction forces
w(x) distributed load
V(x) shear force
M(x) moment
Shear and Moment Diagrams using Sectioning Method
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ME221 Lecture 14 19
V
M
V
M
Sign Convention
Positive Shear and Positive Moment
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ME221 Lecture 14 20
Positive Shear
M M
Positive Moment
Effect of External Forces
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ME221 Lecture 14 21
20 lb/in 125 lb 125 lb
9 in. 12 in. 12 in. 12in.
V
x
M
x-ve
tension up
+ve
Tension downschool.edhole.com
