Bending Moment and Shear Force by Singularity Functions
Consider the following problem.
3 in
4 in
𝑀𝐴
3 in
𝑤 = 20 lbf/in
𝐴
𝐵
240 lbf in
𝑅𝐴
A beam is cantilevered (fixed) at one end called and free at the other end labeled . The
beam has a length of 10 in. Various distributed and concentrated loads are shown acting on
the beam. The reaction force and moments are also displayed acting in the positive directions.
Although we may solve this problem using conventional methods, solving it via the singularity
functions will help avoid many involved and tedious computations.
Apply Force Moment Equilibrium to obtain reactions at :
Also consider moment about point .
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Let
be the loading function then we define this in terms of singularity functions as:
When defining loading function from left to right signs of all external and applied forces remain
the same, on the contrary, signs of external and applied moments are reversed. Let us see that
with the help of the diagram below:
𝑥
𝑀𝐴
𝑀
𝐴
𝑉
𝑅𝐴
Cutting the shaft at a length we assign the positive internal shear force
moment . If we apply equilibrium to balance of forces then we get:
and bending
=
If we now apply balance of moments about point
we get:
=
Observe closely the positive sign with
in shear force and negative sign with
in bending
moment. In forming the loading function we take into account these signs and put them in the
exact same sense. This is the reason why in the loading function
shown above we have
as the contribution from the reaction moment at the cantilevered end and
as the
contribution from the reaction force at the fixed end.
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Singularity Function:
In general a singularity function is defined as:
Where is the point of application of load or point at which a distributed load starts and is an
exponent which depends on the type of loading. The following table gives values for the most
common types of loading:
Loading Type
Negative CW Point
Moment acting at a
point .
Diagram
Singularity Function
𝑀
⟨
⟩
⟨
⟩
⟨
⟩
⟨
⟩
𝑎
𝐹
Negative Point Load
acting at a point .
𝑎
𝑤
Negative Distributed Load
acting at a point .
𝑎
𝑚
Negative Load with slope
acting at a point .
𝑎
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Singularity functions are integrated using following rule:
It must be noted that singularity functions are dead before their point of application.
Mathematically speaking, the function does not take effect before = . The function only
becomes active after = .
If we now sum all of it up and rewrite the loading function it should make a lot of sense.
Now once we have established the loading function, it becomes very easy to determine the
shear and bending moments. Successively integrating the above we obtain:
Integrating it one more time will give us the bending moment:
We may continue integrating the above function to even get the angle of deflection and
deflection itself.
Next we will plot the shear force and bending moments in the beam. Note the change in graph
at loading points such as = 0, = in, = in. Animated versions of the same will be
shown in the class.
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