Novel Kuhn-Tucker conditions for vibration analysis in a functionally graded porous beam using the R-program

Functionally graded materials provide a flexible and individualized strategy for material design, allowing for optimization of properties and performance for particular purposes. The investigation considers the effects of simply supported (SS), clamped-clamped (CC) and clamped-free (CF) configurations. The study examines the vibration characteristics of bi-directional functionally graded porous beams (BDFGPB) using the third-order shear deformation theory, considering both even and uneven porosity conditions. The Hamilton method is used to derive equilibrium equations for beams, which are then solved using the Kuhn-Tucker technique and R-program. The BDFGPB's validity was verified by comparing it with open literature, revealing deviations of 3.19%, 1.25%, and 2.15% in non-dimensional natural frequency for SS, CC, and CF boundary conditions. Furthermore, as the porosity index increases, the dimensionless natural frequency decreases, reducing beam stiffness and rigidity. This study demonstrates that porosity plays a critical role in the design of modern structures, as its ratio greatly impacts their performance and responsiveness.