Mathematical analysis on a model for the rectangular cantilever beam

The mechanical model for the rectangular cantilever beam proposed by Zhang et al is solved analytically by the series solution with mathematical properties investigated in detail. The derived series solution is proved convergent, and restrained only by the small deflection presumed by the Euler-Bernoulli beam theorem, and is applicable for calculating the deflection and curvature for any value of the exerted axial stress. The formulae estimating the accuracy of the coefficients and the series solution are developed from Stirling's approximation for the gamma function. The condition on the axial stress is developed, by which the genuinely nonlinear curvature can be approximated by a linear function and the deflection can be calculated from the boundary condition by a cubic polynomial. The additional redundant boundary condition used in Zhang's work is discussed, which should be removed since it fails to fit the model by inducing errors for calculating the deflection and the curvature. The present series solution approach provides formal deflection-stress and curvature-stress relations for the design of a MEMS micro-cantilever system as a bio-detection device. For self-assembly applications, the adsorbing material can be identified by solving the exerted axial stress from the series solution