International Journal of Computational Engineering Research||Vol, 03||Issue, 4|| www.ijceronline.com ||April||2013|| Page 17 Application of Matrix Iterative-Inversion in Solving Eigenvalue Problems in Structural Engineering 1, O. M. Ibearugbulem, 2, L. O. Ettu, J. C. Ezeh, 3, U. C. Anya 1,2,3,4, Civil Engineering Department, Federal University of Technology, Owerri, Nigeria I. INTRODUCTION The dynamic equation in structural dynamicsis given by Fullard(1980) as in equation (1). {F(t)} = [M]{X 11 } + [C]{X 1 } + [K]{X} -------------------- (1) where {F(t)} is the time-dependent loading vector; [M] is the mass matrix; {X 1 } and {X 11 } are the first and second time derivatives of the response vector,{X}; [C] is the damping matrix; and [K] is the stiffness matrix. For the case of free vibration in nature where {F(t)} = [c] = 0, the dynamic equation reduces to equation (2). [M]{X11} + [K]{X} = 0 --------------------- (2) When a static stability case is considered,the equation further reduces to equation (3). {F} = [K - Kg]{X} --------------------- (3) Where K is the material stiffness and Kg is the geometric stiffness. {F} is the vector of bending forces (shear force and bending moment). The continuum (beam or plate) can bucklein cases of axial forces only with no bending forces; the equation becomes as written in equation (4). 0 = [K - Kg]{X} --------------------- (4) Equation (2) is used in finding the natural frequencies in structural dynamics, while equation (4) is used to determine the buckling loads in structural stability. The solutions in both cases are called eigenvalue or characteristic value solutions. In structural mechanics, shape functions are usually assumed to approximate the deformed shape of the continuum. If the assumed shape function is the exact one, then the solution will converge to exact solution. The stiffness matrix [K] is formulated using the assumed shape function. In the same way, the mass matrix [M] and the geometric stiffness matrix [Kg] will be formulated. The mass matrix and the geometric stiffness matrix formulated in this way are called consistent mass matrix and consistent geometric stiffness matrix respectively (Paz, 1980 and Geradin, 1980). Abstrct: There are many methods of solving eigenvalue problems, including Jacobi method, polynomial method, iterative methods, and Householder’s method. Unfortunately, except the polynomial method, all of these methodsare limited to solving problems that have lump mass matrices. It is difficult to use them when solving problems that have consistent mass or stiffness matrix. The polynomial method also becomes very difficult to use when the size of the matrix exceeds 3 x 3. There is, therefore,a need for a method that can be used in solving all types of eigenvalue problems for allmatrix sizes. This work provides such a method by the application of matrix iterative-inversion, Iteration-Matrix Inversion (I-MI) method,consisting in substituting a trial eigenvalue, λ into (A – λB) = 0, and checking if the determinant of the resultant matrix is zero. If the determinant is zero then the chosen eigenvalue is correct; but if not, another eigenvalue will be chosen and checked, and the procedure continued until a correct eigenvalue is obtained. A QBASIC program was written to simplify the use of the method. Five eigenvalue problems were used to test the efficiency of the method. The results show that the newly developed I-MI method is efficient in convergence to exact solutions of eigenvalues. The new I-MI method is not only efficient in convergence, but also capable of handling eigenvalue problems that use consistent mass or stiffness matrices. It can be used without any limit for problems whose matrices are of n X n order, where 2 ≤ n ≤ ∞. It is therefore recommended for use in solving all the various eigenvalue problems in structural engineering. Keywords: Eigenvalue, Matrix, Consistent mass, Consistent Stiffness, Determinant