Numerically Stable Solutions to the State Equations for Structural Analyses

Abstract

The state space method has been widely used to analyze the static and dynamic characteristics of homogeneous, laminated, functionally graded, or even intelligent structures. However, the solution of the state equation using the traditional transfer matrix generally encounters the problem of numerical instability. This work, therefore, derives the general solution to the state equation by making use of similarity transformation to convert the system matrix into a matrix in Jordan canonical form (including the diagonal matrix as a special case), so as to avoid the previously stated problem. A special form of the exponential function is also introduced according to the characteristics of the eigenvalues of the system matrix. Furthermore, the undetermined coefficients in the general solution—rather than the original state variables—are considered as the primary unknowns. Consequently, a new solution with numerical robustness to the state equation is obtained. Finally, numerical examples for the free vibration analyses of beams and plates as well as interfacial shear stress analysis of fiber-reinforced polymer (FRP)-strengthened concrete beams are presented to verify that the proposed procedure can circumvent numerical instability completely.