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1. Video Lectures for B.tech
By:
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2. ME 221 Statics
Lecture #14
Sections 7.1 – 7.4
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ME221 Lecture 14 2

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 3

4. Quiz #6
Friday, June 25
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ME221 Lecture 14 4

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 5

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 6

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.
c c At any given section a-a, b-b, or
b b
a a
100 lb
c-c, there is an internal force arising
from the 100 lb external force.
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ME221 Lecture 14 7

8. Method for Finding Internal Forces
• Determine reaction forces
– Use equilibrium equations
• Section and solve second equilibrium
problem to find internal forces
Ay=100 lb
Mz=0
Ax=0
a a
100 lb 100 lb
Fy
100 lb
Mz
Fx
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ME221 Lecture 14 8

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 9

10. Example:
Determine the internal
forces and moments in
the bar built into the
foundation as shown in
the figure.
x
y
ME221 Lecture 14 10
z
O
l
h
P
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11. (l-x)
P
M 
x R x

y
z
O
l
h
P
rp 
 
0 ˆ
= Þ + -
å å  
F = Þ R +
Pk
Horizontal Portion
Þ
0 ( )ˆx ˆ
M M l x i Pk
ˆ

R Pk
( ) ˆ
= -

M = P l -
x j
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ME221 Lecture 14 11

12. x
y
z
O
l
h
P
y
(h-y)
R 
M 
l
P
rp = liˆ + (h - y) ˆj
 Vertical Portion
ˆ
R Pk
ˆ ( )ˆ
= -

M = Plj - P h -
y i
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ME221 Lecture 14 12

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 13

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.
Determinate Indeterminate
Simple
Overhanging
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ME221 Lecture 14 14

15. Internal Force Component Rolls
• Force components
– Axial is along beam
Vy
Vz
– Shearing forces are transverse components
• Moment components
P
– Torsion along beam T
– Bending for transverse
Mz
components My
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ME221 Lecture 14 15

16. Shear and Moment Diagrams
-Sectioning Method
-Integration
-Singularity Functions
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ME221 Lecture 14 16

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 17

18. Shear and Moment Diagrams using Sectioning Method
Generate a shear / bending diagram as follows:
1. Find reaction forces
2. Ta k e 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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ME221 Lecture 14 18

19. V
Sign Convention
M
V
M
Positive Shear and Positive Moment
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ME221 Lecture 14 19

20. Effect of External Forces
Positive Shear
M M
Positive Moment
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ME221 Lecture 14 20

21. 20 lb/in 125 lb 125 lb
9 in. 12 in. 12 in. 12in.
ME221 Lecture 14 21
V
x
M
x -ve
tension up
+ve
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