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Say I have
the following beam:
I want to
calculate the reactions at the supports.
Step
0) Break all vectors into x and y-components
The first
thing I want to do is to orient all my forces in the x and y-directions.
The 600 N force is slanted at a 45 deg angle. So, I will
break this force up into it's x and y-components as follows:
Now I can
redraw my problem with the forces oriented only in the x and
y-directions:
Step
1) Draw the Free Body
Diagram (FBD)
A) Decide
upon a sign convention. It does not matter what sign convention
you choose, as long as you are consistent throughout your calculations.
B) Draw beam
with support reactions acting on it. Note that the arrows can
be oriented in any direction, because they are unknown at this
stage. When they are calculated, the correct orientation will
become obvious.
Step
2) Apply equilibrium
equations
A) The sum
the forces in the x-direction = 0
| Fx1
+ (424.3 N) = 0 |
Therefore, Fx1
= -424.3
N |
This means
that Fx1 is force of 424.3 N acting in the negative
x-direction.
B) The sum
the forces in the y-direction = 0
| Fy1
+ (-424.3 N) + (-100 N) + Fy2 = 0 |
Therefore, Fy1 = 524.3
N - Fy2 |
But,
we still don’t know Fy2.
C)
The sum of the moments about any point = 0. (Remember
moment equals force times distance!)
(a) Choose
a point O.
(b) Sum
moments. Note that Fx1 and both 424.3 N forces
do not contribute to the moment because the line they act
on goes directly through Point O. All the other forces
do contribute moments. The following table shows how:
| Force |
Direction
of rotation |
Moment |
| Fy1 |
Clockwise. |
-(2
m)Fy1 |
| 100
N |
Clockwise |
-(3
m)(100 N) = -300 N·m |
| Fy2 |
Counter-clockwise |
(5
m)Fy2 |
Now we
can sum the moments about Point O.
| -(2
m)Fy1 + (-300 N·m) + (5 m)Fy2
= 0 |
| Therefore,
(5 m)Fy2 = 300 N·m
+ (2 m)Fy1 |
| or,
Fy2 = 60 N + (0.4)Fy1 |
Now, we can
plug in the expression for Fy1 (Fy1
= 524.3 N - Fy2) from above.
| Fy2
= 60 N + (0.4)(524.3 N - Fy2) |
| Therefore,
1.4Fy2 = 269.7
N |
| Therefore,
Fy2 = 192.6 N |
This means
that Fy2 is a force of 192.6 N acting in the positive
y-direction. Now, we plug this value into our expression
for Fy1 to get its value:
| Fy1
= 524.3 N - Fy2 = 524.3 N - 192.6 N = 331.6
N |
This means
that Fy1 is a force of 331.6 N acting in the positive
y-direction.
Step
3) Draw final diagram
Go here
to see the shear force and bending moment diagram for this example.
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