5. Example 2

Consider a beam clamped at one end. The other end of the beam has a roller support. A transverse force P is applied at a distance a from the clamped end as shown in Figure 9.

In this scenario, we analyze a beam that is fixed at one end and supported by rollers at the other end. A transverse force is applied at a point along the beam. This situation is used to find the reaction force at the roller support. The roller support can only exert a vertical force because it cannot resist any moment (it can move horizontally).

Think of this beam setup as a seesaw. If you sit on one end and someone pushes your end down, the other end (where the support is) pushes back up. That upward force is similar to the reaction force we're calculating in this beam example.

We want to find the reaction from the roller support. The contact between the roller support and the beam is a line contact, so this support cannot exert any moment on the beam. Hence, the roller support only exerts a vertical reaction force (say R) on the beam which we need to obtain.

When we consider the roller support, it provides a single vertical force, R. This is an essential part of our energy method analysis because we need to express the beam's energy in terms of external forces, which include R.

Picture a rolling chair at the edge of a table. If the chair rolls freely in one direction (horizontal), it can't push against the table sideways without tipping over; it can only push up vertically when weight is applied.

To obtain the energy of the beam in terms of external forces (P and R), we need to first find the shear force and bending moment profiles. We cut two sections, one at x > a and another at x < a.

To calculate the beam's energy, we analyze its internal forces: the shear force, V, and the bending moment, M. By creating a cut in the beam at two sections, we can derive equations that describe how these forces change along the length of the beam. We can then express the total energy stored in the beam using these shear and moment profiles.

Imagine slicing a loaf of bread. Each slice represents a section of the beam, and by checking the strength and density of each slice, we can determine how the whole loaf (the beam) behaves when weights are placed on it.

Notice that the corresponding displacement of the unknown R is tip displacement in the y direction. This displacement must be zero because the roller support restricts any vertical movement.

In using Castigliano’s theorem, we assert that if the beam's tip cannot move vertically (as it is supported), the change in energy due to the reaction force, R, must also result in a zero displacement. This means we can derive an equation for R by setting the derivative of the total energy with respect to R equal to zero.

Think of trying to push a door that is locked. If you apply force but the door does not move, the potential energy you would have gained by opening the door (if it weren’t locked) remains unchanged. Similarly, the reaction force at the roller does not cause vertical movement of the beam.

We can use Castigliano’s first theorem to write the condition that the displacement due to R must be zero, leading to this equation: Taking the derivative of the energy expression with respect to R yields.

By differentiating the energy expression previously formulated in terms of R and P, we derive a mathematical relation that will allow us to solve for the unknown reaction force R at the roller support. This gives us an effective method to quantify the forces acting on the beam.

Picture a balance scale. If one side holds a known weight and the other side adjusts according to a spring, knowing that the scale remains in balance helps us find out what the unknown weight is—this is analogous to how we find R through our calculations.

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