2. Axial Force, Shear Force, Torque and Bending Moment
Diagrams
In this section, we learn how to summarize the internal actions (shear force and bending
moment) that occur throughout an axial member, shaft, or beam. Later, we learn how to
convert these internal forces to stresses. (Note: Axial force and torsion bending diagrams are
easy to derive. Will show an example to illustrate.)
.: mention that axial force and torsion bending diagrams are easy to derive. Will showcouple
of examples to ill
Outline
- Purpose of Axial Force, Shear Force and Bending Moment Diagrams
- Sign Convention
- Basic Method Examples
- Mathematical Relationship Between External Loading, Internal Shear, and Internal Bending
- Integration/Graphical Method Examples
Purpose of Axial Force, Shear Force and Bending Moment Diagrams
We learned that we can get the internal forces at any point in a structure by: (1) taking an
imaginary cut at the point; (2) drawing the free-body diagram; and (3) using the equations of
equilibrium to calculate the internal forces. The challenge with a beam is that the internal
forces can vary a great deal along the length of the beam. Therefore, we draw what are called
shear force and bending moment diagrams, which show the internal forces along the
length of the beam. This helps us visualize where the maximum stresses will occur.
We will learn two methods for drawing shear force and bending moment diagrams: 1) the
Basic Method; and 2) the Integration (also called Graphical) Method. In each case,
mathematical functions describing the shear force and bending moment throughout the beam
are derived and plotted along the length of the beam. In the Basic Method, we derive these
functions from first principles. In the Integration/Graphical Method, the relationships
between the applied load, shear force and bending moment are used to rapidly sketch out the
diagrams.
2-1
The approach used depends on the problem and what the designer requires from the
diagrams. For example, many problems require only the maximum values of the shear and
moment, and the locations at which these values occur. The Graphical Method is most useful
for these situations. In other cases, a beam may be subjected to a loading that is a fairly
complicated function. For these situations, the Basic Method may be needed.
You should know how to use both methods and recognize when to apply them!
Sign Convention (Section 6.1 in Mechanics of Materials)
We define the sign convention for internal shear force and bending moment:
Shear Force
Bending Moment
Positive internal shear force tends to rotate the free-body diagram clockwise.
Positive internal bending moment causes the beam to sag. Also known (informally) as the
smile rule. Bending moment is drawn on the compression side of the member.
Summary of Steps for Basic Method:
1) Determine the support reactions for the beam.
2) Specify an origin for a co-ordinate x along the length of the beam.
3) Section the beam with an imaginary cut at a distance x, and draw the free-body diagram.
4) Determine shear and bending moment as a function of x using equilibrium equations.
5) Repeat steps 3 and 4 for all regions between any two discontinuities of loading.
6) Draw, to scale, the functions on a sketch of the beam.
2-2
Basic Method Example
Consider beam ABC from the example in Section 1.
D
Pin
1.0 m
E
A
θ
60 N/m
B
C
F
0.5 m
0.5 m
0.5 m
0.5 m
Determine the axial force, shear force, and bending moment diagrams for the beam ABC.
2-3
2-4
Mathematical Relationship Between External Loading, Internal Shear, and Internal Bending
The previous example shows that:
1) axial force (and shear force) diagrams change abruptly at the location of a concentrated
axial force (or applied force);
2) for a region of a beam without external applied forces, the shear force has a constant
value, and the bending moment is a function of x;
[see section AB of the beam in the previous example]
3) for a region of the beam subjected to a uniformly distributed load (UDL), the shear force is
a function of x, and the bending moment is a function of x2.
[see section BC of the beam in the previous example].
In fact, we can prove that (see Section 6.2 of the textbook):
dV ( x)
= − w( x)
dx
(2-1)
i.e. the slope of the shear force diagram at x is equal to the negative of the value of the
loading function at x; and,
dM ( x)
= V ( x)
dx
(2-2)
i.e. the slope of the moment diagram at x is equal to the value of the shear function at x.
From Eq. 2-1 we have:
dV ( x)
= − w( x) through integration:
dx
(2-3).
(note what each side of the equation represents)
From Eq. 2-2 we have through integration:
(2-4).
(note what each side of the equation represents)
2-5
Graphically:
2-6
The following table, taken from your textbook, illustrates a number of common loading
cases. It shows how shear and moment diagrams can be constructed on the basis of knowing
the variation of the slope from the load and shear diagrams. Make sure that the relationships
make sense, but you should not memorize the table!! You should always work from the
basic relationships, Eqs. 2.1 to 2.4.
(show overhead of table)
2-7
Graphical Method Example 1
Consider the cantilever beam subjected to the loading w(x). Draw the internal forces on a
typical element of the beam, and hence derive the differential equations relating bending
moments, shear forces, and applied loading, i.e. Eqs. 2-1, 2-2. Use these differential
equations to draw shear force and bending moment diagrams for the following cases: (a) w(x)
= 20 kN/m; (b) w(x) = 20x kN/m
w(x)
(add soln from Campbell notes)
L
2-8
2-9
Example
Draw the shear force and bending moment diagrams.
35 kN/m
1m
40 kN/m
3m
2m
2-10
2-11
Finally, we end this section with a discussion of torque diagrams. These are usually simpler
than shear and bending moment diagrams, and can be illustrated with the following example.
Example
Draw the torque diagram for the cantilever shaft shown. Determine the maximum torque in
the shaft.
1m
1m
1m
fixed
end
5 kNm
7 kNm
12 kNm
2-12
2-13