2.1. Axial extensional energy

Consider a beam which is being stretched by the application of an axial load P as shown in Figure 3.

In this chunk, we introduce the concept of axial extensional energy. When an axial load (a load applied along the length of the beam) is applied to a beam, it results in stretching. This is the starting point for understanding how energy gets stored in the beam due to axial tension.

Think of a rubber band. When you pull both ends, it stretches. The energy you use to pull it comes from your muscles and is stored as potential energy in the stretched rubber band. Similarly, when a beam is stretched, energy is stored within its material.

The total load P can be assumed to be uniformly distributed over the cross-section Ω. To find the energy stored in the cross-section, we need to first find the stress and strain components.

Here, we discuss the distribution of load across the beam's cross-section. Stress is the force applied per unit area, and strain is the deformation response of the material. To accurately calculate the energy stored in the beam due to the axial load, we need to derive these values using the basic principles of mechanics.

Imagine you are slowly pushing on a sponge. The pressure you exert spreads evenly across the sponge, causing it to compress. The force you use relates directly to how squished the sponge gets, similar to how stress and strain work in the beam.

We can recall from our previous lecture that, in case of axial extension of a beam, the cross-section is allowed to relax completely. So, the stress matrix has only one non-zero stress component.

This section explains that in axial loading scenarios, the beam's cross-section experiences uniform stress, meaning stress is primarily directed along the length of the beam. This uniformity simplifies our calculations because we only need to focus on the axial stress component when deriving energy stored in the beam.

Consider a very long elastic band under tension. If you pull only at the ends, the middle section uniformly experiences the pull. This can be likened to how the stress acts uniformly along the beam when axial loads are applied.

Using three-dimensional Hooke’s Law, we can then write the equations for the stress and strain associated with the axial load. The cross-sectional energy will thus become a function of the internally developed stress.

According to the principles of mechanics, when we apply an axial load, the stress can be expressed using Hooke’s Law, which relates stress and strain through the material's properties. The energy associated with this cross-section can be expressed mathematically using these stress values, allowing us to compute the energy stored as the beam stretches.

Think of a sponge again. When you squeeze it, it can only take so much pressure before it starts to deform, similar to how a material has limits on the stress it can handle before yielding.

Upon final integration over the cross-section, we get the total energy stored due to stretching of a beam.

After determining the cross-sectional energy, we integrate this energy across the entire cross-section of the beam. By doing so, we account for all the energy stored in the beam due to the applied axial load. This integration allows us to sum up all individual contributions across the entire area to find the total energy stored.

Imagine filling a swimming pool with water. The total amount of water represents the total energy stored in the beam after all contributions over its entire length are accounted for. Each scoop of water corresponds to energy being added as you stretch the beam.

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