ABSTRACT
In the above equations, the cross-sectional properties are defined as
in which ω is the warping coordinate. The boundary conditions arising from the variational principle are
4 GENERAL EXACT SOLUTION
The homogeneous solution for Equations (1-4) are assumed to take the form
in which 〈C〉Ti,1 × 4 =〈c1 c2 c3 c4〉i is a vector of unknown integration constants. From Equation (10), by substituting into Equations (1-4), a nontrivial solution is obtained by setting the determinant of the matrix to zero. The resulting quadratic eigenvalue problem is solved for the eight eigen-pairs (mi, Ci) which are then used to express the solution of the coupled system as
where [C¯]4×8 = [C1|C2|C3|C4| · · · |C8]4×8 is the matrix of eigen vectors, [X (Z)]8×8 = Diag[emz1 |emz2 | em
z 3 |emz4 | · · · |emz8 ]8 × 8 is a diagonal matrix of expo-
nential functions, and {A}8×1 is a vector of integration constants to be determined from the boundary conditions.
