Question

The function \( y = A \left(1 - \cos\left(\frac{2\pi x}{L}\right)\right) \) is an allowed approximate function for a

• A. fixed-fixed beam
• B. cantilever beam
• C. simply-supported beam
• D. propped cantilever beam

Hint:

For modal analysis or eigenvalue problems in beam theory, choosing the correct function that matches the boundary conditions of the beam type is crucial for accurate analysis. Fixed-fixed beams often show symmetrical mode shapes with zero deflections at both ends.

Answer 1

The Correct Option is A

The function \( y = A \left(1 - \cos\left(\frac{2\pi x}{L}\right)\right) \) is characteristic of a beam where the mid-point deflection is maximal while both ends remain at zero deflection, matching the behavior of a fixed-fixed beam. In a fixed-fixed beam, both ends of the beam are restrained against both rotation and translation, meaning the deflection (y) must be zero at \( x = 0 \) and \( x = L \). The cosine term in this function becomes zero at these points, ensuring no deflection at the supports, consistent with a fixed-fixed beam's boundary conditions. This function would not be suitable for beams like simply-supported, cantilever, or propped cantilever beams as their boundary conditions differ significantly.