Internal Forces

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Internal Forces

• ENGR 221
• March 19, 2003

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Lecture Goals

• Internal Force in Structures
• Shear Forces
• Bending Moment
• Shear and Bending moment Diagrams

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Internal Forces and Bending Moment
The bending moment, M.
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Internal Forces and Bending Moment
The shear force, V.
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Example Internal Forces in a Frame Problem
Determine the internal forces in member ACF at
point J and in member BCD at point K.
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Example Internal Forces in a Frame Problem
Determine the forces at
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Example Internal Forces in a Frame Problem
Look at the member BCD
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Example Internal Forces in a Frame Problem
Look at the member ACF
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Example Internal Forces in a Frame Problem
Look at the member ABE
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Example Internal Forces in a Frame Problem
Take a look at section ACF at point J
Which section would you like to have to compute
the moments?
q
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Example Internal Forces in a Frame Problem
Take a look at section ACF at point J
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Example Internal Forces in a Frame Problem
Determine the internal forces in ember BCD at
point K.
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Beams Definition
A beam is defined as a structural member designed
primarily to support forces acting perpendicular
to the axis of the member. The principal
difference between beams and the axially loaded
bars and torsionally loaded shafts is in the
direction of the applied load.
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Beams Supports
A beam have a variety of supports. - roller (
1-DOF) - pinned ( 2-DOF) - fixed ( 3-DOF)
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Beams Loadings
A beam have a variety of loads. - point loads
- distributed loads - applied moments
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Beams Types
A beam can be classified as statically
determinate beam, which means that it can be
solved using equilibrium equations, or it is ...
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Beams Types
A beam can be classified as statically
indeterminate beam, which can not be solved with
equilibrium equations. It requires a
compatibility condition.
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Beams Types
A combination beam can be either statically
determinate or indeterminate. These two beams
are statically determinate, because the hinge
provides another location, where the moment is
equal to zero.
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Shear and Bending moment Diagrams
In order to generate a shear and bending moment
diagram one needs to

• Draw the free-body diagram
• Solve for reactions
• Solve for the internal forces (shear, V, and
  bending moment, M)

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Beam Sections
A beam with a simple load in the center of the
beam. Draw the free-body diagram.
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Beam Sections
Starting from the left side a take a series of
section of the beam a compute the shear and
bending moment of the beam.
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Beam Sections
The plot of the resulting series of shear and
bending moment are the shear and bending moment
diagrams. The technique is a cutting method.
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Example Shear and Bending Moment Diagram
Obtain the shear and bending moment diagram for
the beam.
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Example Shear and Bending Moment Diagram
Draw the free body diagram and solve for
equilibrium.
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Example Shear and Bending Moment Diagram
Look at the sections 1-1.
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Example Shear and Bending Moment Diagram
Look at the section 2-2.
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Example Shear and Bending Moment Diagram
Look at the section at 3-3.
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Example Shear and Bending Moment Diagram
Look at the section 4-4.
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Example Shear and Bending Moment Diagram
Look at section 5-5.
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Example Shear and Bending Moment Diagram
Look at section 6-6
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Example Shear and Bending Moment Diagram
Look at section end
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Example Shear and Bending Moment Diagram
Draw the shear and bending moment diagrams.
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Class Shear and Bending Moment Diagram
Draw the shear and bending moment diagram.
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Relations between Load, Shear, and Bending Moment
Look at a section
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Relations between Load, Shear, and Bending Moment
Multiply by dx Integrate over Vc to Vd
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Relations between Load, Shear, and Bending Moment
Integrate over Vc to Vd
The difference of the shear is the area under the
load curve between c and d.
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Relations between Load, Shear, and Bending Moment
Look at a section
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Relations between Load, Shear, and Bending Moment
Multiply by dx Integrate over Mc to Md
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Relations between Load, Shear, and Bending Moment
Integrate over Mc to Md
The difference of the moment is the area under
the load curve between c and d.
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Example - Shear and Bending Moment Diagram
Draw the shear and bending moment diagram.
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Example - Shear and Bending Moment Diagram
Draw free-body diagram and use equilibrium
equations.
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Example - Shear and Bending Moment Diagram
Shear diagram.
Note that the area under the load diagram.
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Example - Shear and Bending Moment Diagram
Remember that Where will the maximum moment
occur?
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Example - Shear and Bending Moment Diagram
The maximum will occur where The maximum moment
is the positive area under the curve
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Example - Shear and Bending Moment Diagram
The moment equation
Note that the slope of the moment diagram is
equal to the shear.
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Example - Shear and Bending Moment Diagram
Draw the shear and bending moment diagram
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Example - Shear and Bending Moment Diagram
Free-body diagram
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Example - Shear and Bending Moment Diagram
Look at the shear
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Example - Shear and Bending Moment Diagram
Look at the shear diagram
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Example - Shear and Bending Moment Diagram
Find the moments
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Example - Shear and Bending Moment Diagram
Draw the moment diagram
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Class Shear and Bending Moment Diagram
Draw the shear and bending moment diagram for the
know reactions
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Class Shear and Bending Moment Diagram
Draw the shear and bending moment diagram
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Class Shear and Bending Moment Diagram
Draw the shear and bending moment diagram
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Homework (Due 3/26/03)
Problems
8-15, 8-20, 8-37, 8-41, 8-51
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Bonus Problem Shear and Bending Moment Diagram
Draw the shear and bending moment diagram
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Bonus Problem Shear and Bending Moment Diagram
Draw the shear and bending moment diagram