Experimental and computational mathematics for vibration in concrete beams containing nanoparticles based mathematical modelling

In this work, computational mathematics framework demonstrates the analysis of vibration in nanoparticle-reinforced concrete beams via superior mathematical modeling. It derives a sinusoidal shear deformation beam theory (SSDBT) continuum mechanics model that is higher-order model with exact or true geometrical nonlinearity. A stochastic homogenization problem that models probabilistic agglomeration of the nanocomposite is used to derive its effective properties based on Mori-Tanaka micromechanics. The equations that govern this situation, as a partial differential equation, are obtained as a result of the variational calculus (also known as Hamilton principle) and expressed as an eigenvalue problem by means of the precise analytic techniques. To validate the accuracy of the proposed model, experimental studies are conducted to compare compressive strength. Since GO nanoparticles typically do not readily disperse in water, a thorough dispersion process is employed prior to concrete sample production. This involves utilizing a combination of mechanical shaking, magnetic stirring, ultrasonic treatment, and mechanical mixing. Computational mathematic algorithm is used to ensure the resulting transcendental frequency equation is sufficiently solved. In order to validate its model, the model is compared with the results obtained in the experiment as a benchmark case of the numerical solutions. The mathematical model is impressive in its predictive accuracy because the measured experimental data on compressive strength. The compressive strength exhibit a close alignment with the mathematical model and existing literature, with a maximum difference of 1.25%. The use of mathematical modeling, which forms the core of this study, has established a formal analytical mechanism to determine the vibrational characteristics and reduces the need for costly experimental trials in designing high-performance nanocomposite structures.