3. Verification of reciprocal relation

Let us verify the reciprocal relation through an example. Think of a beam which is clamped at one end and is being acted upon by a force P at a distance a from the clamped end.

In this scenario, we have a beam that is fixed or clamped at one end. A force is applied to the beam at a distance 'a' from the clamped end. This setup is common in structural analysis, as it allows us to investigate how the beam responds to applied loads. Specifically, we are interested in understanding how the application of force affects the rotation at the free end of the beam.

Imagine a seesaw (the beam) that is secured at one end. If you push down on one side (applying force P), you can observe how the opposite end (the free end) tilts or rotates up. This is similar to our beam setup where we want to measure that tilt or 'rotation'.

Our goal is to find the rotation of the cross-section at x = L, i.e., the tip rotation θ(L). We can solve this problem using beam theory.

Here, we need to determine the rotation at the tip of the beam, labeled as θ(L). To achieve this, we will use beam theory, which provides a framework for calculating how beams deform under various loads. The gesture of looking for θ(L) signifies our focus on the beam's response and the relationship between the applied force and the resulting rotational displacement at the free end.

Think of a tightly stretched rubber band that can twist and bend. If you pull one side of it (like our force P), you will notice how it starts to spin or rotate at the opposite end. Here, finding θ(L) is akin to figuring out how far and in what direction the rubber band twists when you pull on it.

The bending moment profile can be found by cutting a section in the beam and analyzing one of its portions. It turns out to be M = 0 (x > a) and M = F(a − x) (x < a).

To understand how the force influences the beam, we create a bending moment profile. This involves 'cutting' the beam at a point and analyzing the internal forces acting on that section. The results show that the moment (M) is zero for positions beyond the applied force (x > a) and varies linearly for positions before the force (x < a). This analysis is vital for calculating the rotation of the beam because it helps us understand the bending behavior of the beam under load.

Picture bending a uniform stick. Every time you apply force at one end, a specific amount of bending occurs nearer to where you apply the force, and it remains straight (with no bending) further down the length. This situation mimics our moment profile with an external force P applied.

Substituting the moment profile in equation (22) gives the expression for rotation θ( x ). Integrating the expression for x < a twice gives θ(x).

After determining the bending moment profile, we need to substitute this information into a relevant equation to find the rotation expression θ(x). By integrating this expression from point x < a, we can find the rotation of the beam at any given point along its length. This step is crucial in deriving how the entire beam behaves under load, especially at the free end where we are most interested.

Imagine continually pushing down different points along a long ruler. Each push causes a bend. By tracking the bend at various positions, especially the end of the ruler, we can better understand how each point influences the overall shape, resembling our integration of θ(x) to capture the entire bending profile.

The boundary condition at the clamped end is θ(0) = 0, which when applied gives C1 = C2 = 0.

Boundary conditions are constraints used while solving equations in physics and engineering. In our situation, since the beam is clamped at one end, it cannot rotate whatsoever, which is expressed as θ(0) = 0. When we apply this condition to our general solution, it results in constants C1 and C2 being equal to zero. This step finalizes our expression for rotation and allows us to derive specific values for θ at various x positions.

Think of a flagpole fixed to the ground. No matter how hard the wind blows, the base of the pole (the clamped end) does not sway. The rotation (or bending) cannot occur at this point, much like our boundary condition stating θ(0) = 0.

Thus, θ is constant in the segment from x = a to x = L, which implies that we find an influence coefficient relating θ at point 2 with the force at point 1.

From our earlier findings, we conclude that the rotation θ remains unchanged from point a to point L. This constancy in rotation signifies that we can derive a relationship (influence coefficient) connecting the rotation at the beam's tip (point 2) back to the applied force (point 1) that initiated the reaction.

Returning to our seesaw analogy, once the seesaw is tilted at one end (point 1), that angle remains as you move towards the far end (point 2). This relationship allows us to predict how much the end will rise or fall based on adjustments made at the beginning.

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