2. Deriving expression for energy stored in a beam in terms of internal contact force and moment which acts in the beam’s cross-section

Writing the stored energy in terms of the externally applied generalized forces is crucial if we want to use the Castigliano’s first theorem. Let us see how this can be achieved. Think of a general three-dimensional body which is clamped at some points as shown in Figure 1.

The beginning of our derivation emphasizes that to apply Castigliano’s first theorem effectively, we must express the energy stored in the beam as a function of the external forces acting on it. This approach is essential for analyzing how the beam will respond under load. Here, we visualize a generalized three-dimensional structure clamped at certain points, which plays a critical role in establishing boundary conditions for our calculations.

Imagine a tightrope walker balancing on a rope. If someone pulls on one end of the rope (the external force), we need to know how that force affects the tension (stored energy) in the rope. Just like the tightrope, a beam must respond to external forces, and expressing this relationship mathematically is vital.

We want to know the energy stored in this body. For linear stress-strain relation, the energy stored becomes.

When discussing energy storage in the beam, we assume a linear relationship between stress and strain, which is fundamental to materials under elastic deformation. It allows us to express this energy mathematically, paving the way for deriving expressions that will incorporate both the forces and the resulting displacements in a systematic manner.

Think of a rubber band. When you stretch the rubber band, energy is stored in it. The relationship between how much you stretch it (strain) and the tension you feel (stress) can be predictable in elastic materials like the rubber band.

Let us simplify this further in the context of beams which we will be mostly dealing with. Consider the beam shown in Figure 2 having cross-section Ω and length L. The integral in the parentheses above is called the cross-sectional strain energy as it is obtained by the integration of three-dimensional strain energy density over the beam’s cross-section. It is further integrated over the length of the beam to obtain the total energy. A beam can deform in several modes: it can undergo bending, stretching, shearing, twisting etc.

In this section, we refine our focus to beams specifically. By defining a beam's cross-section and length, we can express the total energy stored through integrals of strain energy density. It is essential to recognize that there are multiple deformation modes—each contributing uniquely to the overall energy in the system. Bending, stretching, twisting, and shearing can all affect how a beam stores energy based on the applied loads.

Consider a skateboard ramp. As the skater rolls down and up, the ramp bends (bending), stretches in certain directions (stretching), and can twist a bit. Each of these movements stores energy in the structure of the ramp, just like our beam will store energy under different loads.

Consider a beam which is being stretched by the application of an axial load P. 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.

When we consider axial extension, we analyze how a beam responds to a tensile load P. By assuming that this load is evenly applied across its cross-section, we can calculate the stress and strain distributions. This is critical for understanding how much energy is stored when the beam is extended. We use Hooke's Law to relate stress and strain, which allows us to quantify the energy experienced in this state.

Think of a classic slingshot. When you pull back the rubber band, you're applying force evenly along its length (similar to our axial load). The more you stretch it, the more energy is stored, making it ready to launch a projectile.

We can recall that the state of stress for such a case is given by. This is because the cross-section again relaxes freely in the transverse direction in case of pure bending.

In analyzing bending energy, we acknowledge that when a beam is bent, the distribution of stress varies along its height due to the curvature induced. The neutral axis experiences no stress, while other parts of the cross-section either compress or extend, creating a unique stress state. This understanding is essential when we derive expressions for energy associated with bending, which parallels the derivation for axial behavior but considers the different forces and moments.

Visualize bending a thin stick—when you apply a force at the middle, the top of the stick gets compressed while the bottom stretches. The amount of energy stored due to that bending is crucial for its return to the original shape once you release the force.

In case of torsion, the state of stress in cylindrical coordinates system is. We had also derived earlier that.

When considering torsion, we focus on how twisting a beam introduces shear stress. This unique stress distribution requires us to redefine our equations, similarly to how we addressed axial and bending cases. Here, we aim to express the energy stored due to torsion using the principles of mechanical behavior under shear stress, which is crucial for applications like twisting beams or shafts.

Picture twisting a towel to wring out water. As you apply torque to the towel, it resists and stores energy in that twist—this energy will release as the towel unwinds and returns to its original shape. Just like with our beam in torsion, we're observing energy storage through twist.

We learn earlier that the presence of transverse load leads to non-uniform bending of beams which also generates transverse shear stress in beams—this shear stress points in the direction of applied transverse load and is different from the one due to torsion.

When a transverse load is applied to a beam, it triggers shear forces which must be accounted for differently than bending or torsion. It is essential to distinguish these effects since each contributes independently to the energy stored. Understanding how shear stress behaves under these loads allows a comprehensive analysis of the beam's stored energy.

Think of a wide, flat broom when you sweep—applying force at one end causes a shear effect across the broom's body. The broom bends slightly, demonstrating how transverse loads introduce different stress states. This concept directly applies to beams under load, as we observe similar shear forces at work.

When we apply a general loading on a beam, force and moment both act on the cross-section. The internal force V in the cross-section can be decomposed into its three components as.

In summary, when a beam experiences multiple forces and moments simultaneously, we can combine the contributions from axial, bending, torsional, and shear stresses to find the total energy stored. This total energy becomes a simple expression due to the principle of superposition, which tells us that the total effects can be summed linearly in the context of linear elasticity.

Imagine a multi-layered cake where each flavor represents a different force acting on the beam. Just as combining the flavors creates a unique cake, combining the contributions of each force creates the total energy stored in the beam, giving a holistic picture of structural performance.

--