3. Beams – Internal Effects
The external load applied to a beam can cause changes in the shape of the beam, it can bend for example. We do not want to examine changes in the shape, but we do want to know something about the forces inside the beam that could result in these deformations. In order to do this we must examine how the load affects the internal forces in a beam. There are three sets of forces we can examine – Shear, Bending and Torsion.
Shear (V) is a translational force perpendicular to the long axis of the beam.
Bending is described using the Bending Moment (M), which is a rotational force where the ends of the beam are rotated in opposite directions towards the center of the beam.
Torsion is described using the Torsional Moment (T), which is a rotational force where the ends of the beam are rotated around the long axis of the beam.

4. To examine the internal shear, bending moments or torsional moments of a beam we analyze the situation as if the beam was cut and therefore make the internal forces external. The direction of the vectors representing these forces in one half of a cut end of the beam is shown in the image to the right.
We will primarily be working with shear forces and bending moments, therefore there will be no further discussion of torsion.
We do need to define a sign convention because the force and moment directions are reversed when discussing each half of the cut end of the beam. This is because the sum of the internal forces or moments must be zero, since they are not observed external effects. The standard sign convention is shown below.
Positive directions for shear force and bending moment at the cut section of the beam.

5. The I-beam is a common construction material that is used because of its nearly equal strength and reduced weight as compared to a solid bar. The flanges on the top and bottom of the I-beam support tension or compression (for the image below compression on top and tension on the bottom). This pair of forces is a couple, which results in a bending moment.
The thin center webbing provides a link between the two flanges but provides minimal additional support.
Any object subjected to the forces described above is under the influence of a bending moment. The distribution along the center would have a different profile.

6. Shear Force and Bending Moment Relationships
To examine the internal forces of a beam we will begin by analyzing a small segment of the beam. When we extract this segment the internal forces become external and we can analyze the situation using known techniques.
Analyzing the forces on the segment, assuming w is constant over the small segment:
𝐹 =𝑉−𝑤𝑑𝑥− 𝑉+𝑑𝑉 =0
→𝑤=− 𝑑𝑉 𝑑𝑥
→−𝑤𝑑𝑥−𝑑𝑉=0
Does not hold at the location of the concentrated load!
The shear force can then be determined by integrating w:
𝑉 0 𝑉 𝑑𝑉 =− 𝑥 0 𝑥 𝑤𝑑𝑥
𝑉− 𝑉 0 =− 𝑥 0 𝑥 𝑤𝑑𝑥
It is necessary to know the function w(x) to complete the integration.

7. Similarly, we can analyze the moments relative to position x:
𝑀 =𝑀+𝑤𝑑𝑥 𝑑𝑥 2 + 𝑉+𝑑𝑉 𝑑𝑥− 𝑀+𝑑𝑀 =0
→ 𝑤𝑑𝑥 𝑑𝑥 2 + 𝑉+𝑑𝑉 𝑑𝑥−𝑑𝑀=0
Second order differential terms, therefore they can be ignored.
→𝑉𝑑𝑥−𝑑𝑀=0
→𝑉= 𝑑𝑀 𝑑𝑥
Solving for M:
𝑀 0 𝑀 𝑑𝑀 = 𝑥 0 𝑥 𝑉𝑑𝑥
→𝑀− 𝑀 0 =− 𝑥 0 𝑥 𝑉𝑑𝑥
The bending moment can also be determined directly from w:
𝑤=− 𝑑𝑉 𝑑𝑥
→𝑤=− 𝑑 𝑑𝑥 𝑑𝑀 𝑑𝑥
→𝑤=− 𝑑 2 𝑀 𝑑𝑥 2
If w is continuous in x, M can be determined from two integrations.
If bending occurs in multiple planes it is possible to look at each plane independently and add the results vectorally.