Size-Dependent Thermal Buckling Analysis of Functionally Graded Nanobeams Using a Fredholm Integral Solution Based on Eringen’s Nonlocal Elasticity Theory

This study analyses the free vibration and thermal buckling of size-dependent, axially functionally graded Euler–Bernoulli nanobeams using a nonlocal thermo-elastic Fredholm integral model based on Eringen’s nonlocal elasticity. The elastic modulus and mass density vary exponentially along the beam, so the thermal load, modeled as an equivalent axial force, follows the same exponential law. Four successive Fredholm integrations convert the governing equation into a weak-form integral equation that builds in the boundary conditions directly; this equation is then solved with a power series Galerkin method. The critical thermal buckling force is the axial force at which the fundamental frequency vanishes. Simple–simple (S–S), clamped–clamped (C–C), clamped–simple (C–S) and clamped–free beams are examined. Predicted frequencies agree with published solutions to within 0.11%. For a homogeneous beam, increasing the nonlocal parameter from zero to 0.2 lowers the fundamental frequency by 15.3% (S–S) and 18.3% (C–C). Shifting the thermal force from tension (−3) to compression (+3) lowers it by 26.9% (S–S) and 7.1% (C–C). Raising the nonlocal parameter from zero to 0.3 reduces the C–C buckling force by about 78% for a homogeneous beam and 94% for a gradient index of −2. Changing the gradient index from −2 to two lowers the first C–S frequency by 24.7% but the third by only 3.6%.