1/8/2016
Chapter seven
Laith Batarseh
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internal forces
Definitions
When a member is subjected to external load, an internal forces
and/or moment are generated inside this member.
The value of the generated internal load is important to see if this
member can resist it.
Method of sections is used to determine these loads
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Method of sections
To illustrate how we can use the method of sections to find the internal
loads, firs, we will start with the following cantilever beam
F2
A
F1
B
Now section this beam from point B as shown in the next figure
As you can see, the internal loads in the beam is now become external
loads at the section area.
internal forces
Method of sections
It is noted from applying this method on the beam that:
There is a perpendicular force (NB) to the section plane. This force
is called normal force. this force is the main cause of tensile stress in
the beam
There is a tangential force (VB) to the section plane. This force is
called shear force. this force is the main cause of shear stress in the
beam
Finally, the couple moment (MB) is called bending moment. this
moment is the main cause of bending stress in the beam
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internal forces
Method of sections
These forces and moment prevents the beam from the
translational and rotational motions( i.e. elongation and bending
deflection)
The direction of these forces and moment on both segments
satisfy Newton's 3rd law and can be determined using equilibrium
equations at one segment.
For our case, the right segment is the preferred segment because
it reduce the unknown reactions at the supporting point A.
 ∑Fx = 0 to find NB
 ∑Fy = 0 to find VB
∑MB = 0 to find MB
internal forces
Three dimensional cases
In three dimensional cases, a new shear and moment components as
shown in the figure below
z
Mz
Mx
Vz
Vy
My
y
Vx
Mx
x
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internal forces
Sign convection
To assign the sign of normal force and shear force and the moment, we
can assume:
 if the normal force tray to tensile the segment (tension),
then its is positive.
If the shear force tends to rotate the segment clockwise then
its is positive.
If the moment tends to concave the segment upward manner
then its is positive.
internal forces
Analysis procedures
To use the method of sections in finding the internal loads, you can
follow the following procedures:
Find the supports reactions using the equilibrium equations for
F.B.D of the whole system
Leave all the acting loads and moments on the graph and section it
at the point that you desire to find the internal loads at it.
Draw F.B.D for the segment that has the least number loads on it.
This ensure that you will not face statically in-determent situation.
Apply the equilibrium of moment at the section directly. This
eliminate the normal and shear forces at the sections. then apply
the equilibrium of forces equation.
Finally, verify the direction of normal and shear forces beside the
internal loads.
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Example
Determine the internal normal force, shear force, and moment in the beam
at points C and D. Point D is just to the right of the 5-kip load
internal forces
Example
Solution
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Example
Solution
internal forces
Example
Solution
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Example
Solution
shear and moment diagrams
Shear and moment
As noted from the previous analysis, shear and
moment vary along the loaded body which was in most
of our cases the beam. beams are important element in
structure because it can support both shear and
moment stresses.
The variation in moment and shear can be
determined using the method of sections as we learned
before.
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shear and moment diagrams
Shear and moment
However, the values of shear and moment are not
changed suddenly when we moves across the point of
force action. Therefore, we must develop a method to
study this variance with respect to the distances from
the supporting points and the points of forces action.
Simply, we can assume that the change in shear and
moment through the body is a function in terms of
body dimensions (mainly in the x-direction)
shear and moment diagrams
Shear and moment
In general, each segment of the loaded body is
studied separately and the functions of the shear and
moments are determined for the under study segment.
The definition of segment is given as: the part of the
body that have two consecutive loadings.
The type of loading could be concentrated force
(single force) or distributed load( f(x)).
To illustrate the process of drawing shear and
moment diagrams, see the following examples.
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shear and moment diagrams
Example
Draw both shear and moment diagrams for the following beam
shear and moment diagrams
Example
Solution
F.B.Ds
Equilibrium equations
 M  0  Pa  Pa  b  D 2a  b  0  D
   Fy  0  A  D  2P  A  P
A
y
y
y
y
P
y
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Example [2]
Solution
internal forces
Example
Solution
Shear and moment diagrams
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Example
internal forces
Example
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Example
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Example
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Example
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Example
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