2. Euler-Bernoulli Beam Theory

In this theory, it is assumed that the centerline tangent and the cross-section normal are aligned (see Figure 4b).

The Euler-Bernoulli beam theory makes a crucial simplification by assuming that the direction in which the beam bends (the tangent of the centerline) is the same as the direction normal to the cross-section. This means when the beam bends, the cross-sections remain perpendicular to the centerline at any point, which simplifies the analysis significantly.

Imagine a straighter-than-usual drinking straw. When you bend it, the top surface (cross-section) remains at a right angle to the line formed by the bent straw (centerline). This is similar to how the beam behaves under the assumptions of the Euler-Bernoulli theory.

To get the required equations, we can invoke the following bending formula derived in an earlier lecture: M=EIκ.

The equation M = EIκ relates the bending moment (M) acting on the beam to its curvature (κ), where E is the modulus of elasticity and I is the moment of inertia of the beam's cross-section. This relationship is essential because it allows us to understand how the bending moment influences the beam's deformation.

Think of bending a rubber band. The more you stretch it (apply a moment), the tighter the curve (increased curvature) becomes. In the same way, as the moment increases for a beam, the curvature increases, leading to greater bending.

In the Euler-Bernouli beam theory, the axial displacement is also assumed to be negligible.

This assumption simplifies the analysis further by stating that the horizontal position (axial displacement) of any point along the beam does not change significantly when the beam bends. This means that the angle of the centerline bends without moving sideways.

Picture a tightrope walker. As they sway side to side, the rope itself doesn't elongate in the horizontal direction. It remains mostly straight; similarly, the beam's axial displacement is minimal during bending.

To linearize the above expression in deflection y, it is further assumed that the magnitude of the slope of the centerline is very small.

This means that when the beam bends, the angle of the slope (the tangent of the curve) is small enough that we can approximate the curvature using just the deflection of the beam without needing complex calculations.

Consider a gentle hill on a road. If the hill is subtle enough, you don't need to account for sharp changes; you can treat it as a straight line for simplicity when driving over.

Substituting the above relation in equation (2), we get d²y / dx² = M(X) / EI.

This derived equation, d²y / dx² = M(X) / EI, is a second-order ordinary differential equation that relates the deflection of the beam (y) at any point to the bending moment acting at that section (M(X)). It allows engineers to calculate how much the beam will deflect under a given load.

Think about a trampoline. The deeper you push down (the more force you apply), the more the surface dips. Similarly, this equation shows how the force applied to the beam relates to how much it bends.

We need two boundary conditions to solve it.

Boundary conditions specify the behavior of the beam at its supports or ends. For example, if one end of the beam is fixed, the deflection at that point will be zero. These conditions are crucial for getting specific solutions to the differential equation derived.

Imagine a swing. If one end is held tightly, the swing won’t droop there, just like a beam won’t deflect at a fixed support. The conditions help define how the beam behaves under loading.

Suppose we have a straight beam which is clamped at one end and is subjected to a transverse load P at its other end.

Here, the practical example illustrates how to use the Euler-Bernoulli beam theory to calculate the bending moment and deflection of the beam subjected to a transverse load. By cutting the beam at a certain point and analyzing equilibrium, we derive expressions that help in determining deflection due to the applied load.

Think of a diving board. If you stand on the end of it, the board bends down due to your weight, and by applying this theory, you can predict how much it will bend based on your weight and the board's stiffness.

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