Galerkin method References –Pilkey, W. & Wunderlich, W. (1994), Mechanics of Structures – Variational and computational methods, CRC Press, Boca Raton –Allen H. & Bulson, P. (1980), Background to buckling, McGraw-Hill, London Linear analysis of stability –Diferential equations in the form of: L(u, )=0 L is a diferencial operator u is the displacement field to be determined according to u and boundary conditions
Galerkin method Approximate solution Evaluation of residuals by replacing the vector with the approximate solution Orthogonallity obtained by choosing appropriated q parameters
Galerkin method - Example Clamped simply supported column Form function Residual Galerkin equation Critical Load
Rayleigh-Ritz method Potential energy of structural system V(u, ) –Only for conservative systems References –Pilkey, W. & Wunderlich, W. (1994), Mechanics of Structures – Variational and computational methods, CRC Press, Boca Raton –Dym, C. & Shames, I. (1973), Solid Mechanics – a variational approach, McGraw-Hill, New York.
Rayleigh-Ritz method 1.Approximate solution 2.Evaluation of variational 3.Stationarity of potencial energy 4.Linear problem of eigenvalues
Rayleigh-Ritz method Long columns –u=w(x) –Potential Energy – Rigidity Matriz –Geometric Matriz
Rayleigh-Ritz method - Example Clamped simply supported column Form function Rigidity Matriz Geometric Matriz Critical Load
Effective length of columns Effect of boundary conditions In general the critical load is: – due to boundary conditions –l is the length –EI is the structural rigidity