1. ME 221 Statics Lecture #14 Sections 7.1 – 7.4

2. Homework #6 Chapter 7 problems: Chapter 6 problems
6, 19 & 26
Chapter 6 problems
3 & 6
Due Monday, June 28
MatLab Group Problems
7.19, 7.26 & 6.15
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Lecture 14

3. Quiz #6
Friday, June 25
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4. Chapter 7: Internal Forces in Structures
Review internal/external forces
How to find internal forces
Sample problems
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5. 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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6. 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
At any given section a-a, b-b, or
c-c, there is an internal force arising
from the 100 lb external force. c b a
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7. Method for Finding Internal Forces
Determine reaction forces
Use equilibrium equations
Section and solve second equilibrium problem to find internal forces
100 lb
Ay=100 lb
Ax=0
Mz=0
100 lb a
100 lb Fy Fx Mz
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8. 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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9. Example: x y z O l h P
Determine the internal forces and moments in the bar built into the foundation as shown in the figure.
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10. x y z O l h P x
(l-x) P
Horizontal Portion
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11. x y z O l h P l
(h-y) P y
Vertical Portion
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12. 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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13. 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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14. Internal Force Component Rolls
Force components
Shearing forces are transverse components Vy Vz
Axial is along beam P
Moment components
Bending for transverse components My Mz
Torsion along beam T
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15. Shear and Moment Diagrams
-Sectioning Method
-Integration
-Singularity Functions
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16. 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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17. 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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18. Positive Shear and Positive Moment
Sign Convention V M V M
Positive Shear and Positive Moment
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19. Effect of External Forces
Positive Shear M
Positive Moment
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20. +ve -ve 125 lb 20 lb/in 9 in. 12 in. 12 in. 12in. V x M x tension up
Tension down
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