1. Timoshenko Beam Theory

In Euler-Bernoulli beam theory, it is assumed that the centerline tangent and the cross-section normal are aligned as shown in Figure 1a. So, we have only one kinematic variable which is the deflection of the beam y: we can obtain the cross-section orientation from its derivative, i.e., . However, in Timoshenko Beam Theory (TBT), we do not assume the centerline tangent and cross-section normal to be aligned as shown in Figure 1b. Thus, this is a more general theory than EBT: we now have deflection y and cross-section rotation θ as two independent unknowns.

The Timoshenko Beam Theory expands upon the classical Euler-Bernoulli beam theory by addressing some of its limitations. In Euler-Bernoulli theory, the assumption is made that the angle between the beam's central axis and the cross-section remains constant during bending, which simplifies the analysis. This means that only one variable, the deflection of the beam (y), is needed to determine the deformation. In contrast, Timoshenko theory recognizes that the cross-section can rotate independently of the central axis, resulting in two variables: deflection (y) and rotation (θ). This acknowledgment leads to a more accurate representation of short beams where shear deformations become significant.

Imagine a fishing rod when you cast a line. The rod bends as you apply force, but the top part of the rod (the tip) may twist slightly due to the force of the line pulling on it. The traditional theory would not account for that twist, while Timoshenko's theory would, allowing for a better understanding of how the rod behaves under load.

Let us consider a beam which is initially straight and gets deformed due to some arbitrary loading. The centerline tangent makes an angle α with the horizontal which implies (assuming the slope to be small enough). The deformed cross-section normal makes an angle θ with the horizontal. Thus, the angle between the cross-sectional line (shown as solid green line in Figure 2b: it should not be confused with cross-section normal) and the centerline tangent becomes . In the undeformed straight beam, the angle between the cross-sectional line and the centerline tangent was which, upon deformation, changes to . As shear strain measures the changes in angle (initial-final angle) between perpendicular line elements, we have.

When a beam is subjected to loading, it bends and rotates, changing the orientation of both its centerline and cross-sections. In Timoshenko Beam Theory, we analyze these changes by measuring the angles formed between the original and deformed shapes of the beam. The angle between the centerline's tangent and the cross-sectional line is critical because it helps in determining shear strains that develop as the beam bends. The theory allows for a relationship between shear force and cross-sectional shear strain, leading to equations that can be used to analyze the beam's loading effectively.

Think about a slinky toy. When you pull on one end, the coils not only get stretched (deflection) but also twist together (rotation). Just like in beams, the changes in position and angles of the coils can be studied to understand how they behave under force, similar to how Timoshenko Beam Theory analyzes bending and rotation in beams.

Consider the problem that we had solved using EBT in the previous lecture. A transverse force P is applied at the free end of a beam which is clamped at the other one end. This problem is also called a Cantilever Problem. We would like to solve for the deformation using TBT and compare the result with those of EBT.

In practical applications, like a cantilever beam, we can use Timoshenko Beam Theory to analyze how the beam responds to forces. By applying a load at the free end of the cantilever while it is fixed at the other end, we can calculate the resulting displacement and compare these values obtained from TBT with those from the simpler Euler-Bernoulli theory. This comparison helps in realizing the effects of shear deformation, which are significant in short and deep beams where EBT’s conservative assumptions would yield inaccurate results.

Picture a diving board held at one end. When a diver steps on the far end, the board bends, not just downward (deflection) but also slightly twists. Using Timoshenko theory helps in accurately predicting the board's behavior under this load as it combines both bending and twisting effects, unlike the simpler approach that might miss important details.

Let us now explore which of the two theories to use in a given scenario. If the result from TBT is very close to the one from EBT, then we can simply apply EBT and neglect the effect of shear.

Choosing the correct beam theory is essential for accurate analysis. When the analysis yields similar results between TBT and EBT, it's acceptable to use the simpler Euler-Bernoulli theory since it reduces the complexity of calculations. However, in applications where shear deformation is significant, as with short or thick beams, Timoshenko theory should be used for a more accurate representation of the actual behavior, ensuring safety and reliability in design.

Consider a narrow wooden plank versus a thick steel beam. For the thin plank, a simpler approach might work well. But if it were a thick steel beam subjected to heavy loads, neglecting shear effects could lead to unsafe designs. Using Timoshenko theory provides a more accurate understanding of how much the beam would bend and twist under stress, much like using a specialized tool for a specific job.

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