2.5. Total energy stored in the beam

When we apply a general loading on a beam, force and moment both act on the cross-section (see Figure 5).

This chunk introduces the concept of total energy stored in a beam when it is subjected to loading. The load can be both a force and a moment acting together on the beam's cross-section. It sets the stage for understanding how these loads contribute to the energy stored in the material.

Think of a beam like a bridge. When cars drive over it, they exert forces downwards, and if they are turning, they can also create moments. Just like how the materials of the bridge absorb some energy when stressed, beams in various structures also absorb energy when loaded.

The internal force V in the cross-section can be decomposed into its three components as: (19). The component along the axis V is axial force P. The other components are shear forces.

In this portion, the internal forces acting on the beam's cross-section are discussed. The forces can be broken down into axial forces (acting along the length of the beam) and shear forces (acting parallel to the cross-section). Understanding these components is crucial for calculating the energy associated with the beam's deformation.

Imagine pushing a board from the side (shear force) and pulling it from the ends (axial force). Both actions exert different types of stress on the board, similar to how different loads affect a beam.

We can similarly resolve the internal moment into three components as: (20). The component along the axis M is the torque T. The other two components are bending moments.

Here, the internal moments that occur in the beam due to loading are detailed. Similar to how internal forces can be broken down, internal moments can also be decomposed. This understanding allows for better analysis of how moments from different directions affect the beam's stability and energy storage.

Consider twisting a stick (torque) while also bending it in various places. Each way you apply force and motion creates different types of stress patterns that the material must handle.

We derived energy due to each of the six components separately. As we are working in the regime of linear elasticity, the energy when all of these are present together can be obtained using the principle of superposition.

In this section, the computed energy contributions from each type of strain (three from normal strain and three from shear strain) are summed together to find the overall energy stored in the beam. The principle of superposition states that the combined effect of several loads can be determined by calculating the individual effects and adding them up. This approach simplifies the calculations involved when analyzing the energy of the beam under multiple loads.

Imagine stacking multiple blankets on top of each other. Each blanket contributes weight and warmth (energy) on its own, and together they create a greater overall effect. Similarly, the sum of energy contributions from each type of loading leads to the total energy stored in the beam.

This means that the total energy stored in the beam in the general case can be written as (21). The first three terms here are due to normal strain while the last three terms are due to shear strain.

This final chunk presents the formula that encapsulates the total energy stored in the beam due to various strains. By combining all the contributions, we get a comprehensive expression that takes into account both the normal strains (from axial and bending loads) as well as shear strains. This formula is essential for understanding how beams behave under different loading scenarios.

Think about a sponge that absorbs water. Each type of load on a beam can be thought of as different ways to fill the sponge (like pouring, squeezing, and twisting). The total amount of water the sponge can hold is like the total energy stored in the beam from all the different strains.

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