Numerical Implementation

The solution of the buckling and vibration eigenvalue problems is obtained by means of either

(i) Rayleigh-Ritz’s or Galerkin’s method, in the case of simply supported members, i. e., members with locally and globally pinned and free-to-warp end sections, or (ii) a GBT-based beam finite element formulation that uses Hermitean cubic polynomials to approximate the buckling/vibration modes, for members with other end support conditions (Camotim et al., 2004; Silvestre, 2005; Gonqalves and Camotim, 2005; Dinis et al., 2006). As for the solution of system (11), it requires the development of another (non-linear) GBT-based beam finite element (also using Hermite and Lagrange cubic polynomials to approximate фк(х) and Vj(x)) and resorting to an incremental – iterative numerical technique to determine the member equilibrium path (Silvestre and Camotim, 2003; Silvestre, 2005). After the usual integrations, the FEM system of algebraic equations reads

(К0е) + К(!° + K2e))d(e) – (K0e) + K1e) + K2e))d(e) = f{ee), (12)

where (i) Kp^ are linear (p = 0) and non-linear (p = 1, 2) secant stiffness matrices, (ii) d(e), and d(e) are the vectors of generalised and initial imperfection nodal displacements and (iii) іЄе) is the external applied force vector. The well-known Newton-Raphson predictor-corrector iterative technique was employed to determine the member post-buckling equilibrium paths (e. g., Crisfield, 1991-1996).

Figure 11. Geometrical and material properties: (a) buckling and (b) vibration and post-buckling.