6. Example 3

We will now discuss a problem which is difficult to solve using EBT/TBT. Consider a ring of radius R which is subjected to equal and opposite forces along its vertical diameter line as shown in Figure 12a. The magnitude of the force applied is P. The points at which the forces act are labelled A and B. Another point C is marked as shown in Figure 12a. We need to find answers to the following three questions:

• By how much do points A and B get closer? • By how much does point C move outward? • What is the internal bending moment at point C?

In this chunk, we're introduced to a problem involving a circular ring experiencing equal and opposite forces at its diameter's endpoints (A and B). The objective is to determine how these forces affect the positions of points A, B, and C on the ring. Since the ring's curvature complicates direct application of beam theories (EBT/TBT), we need to utilize energy methods which are more flexible for such problems. We focus on how the forces will lead to displacements: whether points A and B will move closer together and how point C will deflect outward due to the forces applied.

Imagine a rubber band stretched between two fingers where you pull on the ends. As you pull, the distance between your fingers changes, and the rubber band stretches in the middle. In a similar way, when equal forces pull on the ends of the circular ring, it alters the distances between points A, B, and C.

As the beam is curved and the beam theories developed in this course are applicable only for initially straight beams, we cannot use them directly here. Let us divide the ring into two equal parts about a diameter line passing through C and draw the free body diagram of the upper half as shown in Figure 12b.

Here, we acknowledge that conventional beam theories apply best to straight beams, not to curved structures like rings. Therefore, to analyze the forces and their effects on the ring, we divide it methodically into halves. This simplification allows us to consider only the upper half of the ring for our computations and analyze the forces acting there. Drawing a free body diagram helps visualize and mathematically model the situation we’re analyzing.

Think of slicing a piece of pizza in half. By analyzing just one half, it becomes easier to understand how the toppings (forces) above the slice (the ring) influence the entire slice's structure. Similarly, dividing the ring allows us to focus on the effects of the forces without getting overwhelmed by the whole shape.

The force P acts at the top. From symmetry, we can easily see that the vertical force that the bottom part applies on the top part is on both the ends. There will also be a force in the horizontal direction V and bending moment M at the two ends. If we apply force balance and moment balance, we will not be able to solve for V and M.

In this part, we recognize how the applied force P affects the upper half of the ring, creating reaction forces (V) and moments (M) at the ends. However, due to the symmetry of the problem, establishing balance equations for these forces complicates solving for V and M directly. This implies that merely balancing forces and moments isn't sufficient in this scenario; thus, an alternative approach (energy methods) will be more effective.

Consider a see-saw where one person pushes down on one side. To balance, if they push too hard, the other side needs not only to push back but also to tilt in place to maintain equilibrium. If you’re looking at how much force needs to be applied to balance both sides, analyzing each side separately becomes more efficient than trying to think about both sides at once in detail.

Here, energy method turns out to be a very useful and powerful tool. First of all, let us think of the shape that the deformed ring would take. We cannot note that due to symmetry, the cross-section at point A will not rotate. The point A also does not displace horizontally. Similarly, the cross-section at point C does not rotate. Neither does it displace vertically.

At this stage, we lean into the concept of energy methods for analyzing the deformation of the ring. Energy methods accommodate the complexity of shapes better than typical force analysis. The symmetry of the ring allows us to establish that certain points, like A and C, do not experience rotations or horizontal displacements. This insight simplifies our calculations as we set our boundary conditions by recognizing these fixed points in the model.

Imagine a suspension bridge with cables holding it in place. Just like how the anchors at either end don’t move sideways or tilt, points A and C, when considering the ring, remain stationary under the forces applied, allowing us to predict the shape changes much easier by observing the energy distribution instead of tracking every little force back and forth.

Let us now find the energy of the ring’s quarter for which we need the bending moment, shear force, and axial force profile. For this purpose, we cut a section in the upper quarter of the ring at an angular distance θ from point C and draw the free body diagram of its lower cut part as shown in Figure 15.

In this segment, we focus on determining the energy associated with a quarter of the ring's structure. To assess the energy effectively, we need to identify the shear and bending forces at various points. By examining an angular segment from point C, we write the force distribution and balances acting on that quarter, which ultimately allows us to derive the energy stored within that segment due to deformation.

Picture a section of an orange peeled away and held in your hand. To estimate how much juice (energy) that piece could hold when squeezed, you’d analyze that small segment to see where the pressure and forces are acting instead of trying to visualize the entire orange. Similarly, slicing the ring into quarters allows us to focus on just a part while understanding the whole structure's energy.

We can make an approximation here that the energy contribution due to shear force and axial force is insignificant when compared to contributions due to bending moment and torque. This is a good approximation whenever the beam is slender (i.e., the aspect ratio is large). Thus, we can drop the axial force and shear force energies from the expression of energy, which lead to the total energy equation.

This concluding chunk illustrates the final simplification we make when analyzing the energy of the system. When the ring is slender, the contributions from shear and axial energies become negligible compared to bending and torque moments. This allows us to base our energy calculations solely on bending moments and discard the less significant factors, ultimately leading to a more manageable total energy expression. This clarity simplifies our analysis significantly.

Think of a long piece of spaghetti that can bend easily. When you try to hold it, you’ll notice that its bending will hugely overshadow any other forces (like twisting). Thus, focusing on bending gives a clearer understanding of its behavior. In our ring, the same principle applies—concentrating on bending makes our calculations straightforward and effective.

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