Implementation of the matrix differential sweep method for non-self-adjoint spectral problems in tall buildings with elastic foundations and self-weight

The present paper examines the dynamic stability of high-rise buildings exposed to non-conservative loading and flexible boundary conditions, with an emphasis on the spectral problems associated with non-self-adjoint operators. Ill-conditioned matrix formation and extreme computational errors remain the main issues of high-order natural frequencies calculation using conventional direct algebraic formulations. To address this important drawback, the Matrix Differential Sweep Method (MDSM) was successfully implemented to study the non-conservative spectral problem. The numerical results show that including the self-weight P(x) reduces the fundamental natural frequency by 13.3%. At the same time, the inclusion of this parameter greatly influences the dynamic stability limit, causing the critical load (Pcr) to fall by 24.0%, while the total instability occurs at 0.95PEuler instead of 1,25PEuler. In addition, increasing kθ from 0.05 to 0.09 reduces λ1 from 1.707 to 1.600. These reductions are consistent with the geometric softening produced by the axial dead weight, which acts as a distributed compressive load and lowers the effective flexural stiffness, and with the relaxation of the base restraint caused by increased foundation compliance. Because the boundary value problem is integrated directly rather than reduced to a large algebraic eigenvalue problem, no such matrices are assembled. The results indicate that both self-weight and base compliance should be included when assessing the response of tall slender structures to low-frequency wind or seismic excitation, since neglecting them overestimates the fundamental frequency and the stability margin

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