1. PAEN - 2nd Section Stability of structures Methods Initial Imperfections

2. Linear Analysis Aproximate methods –Finite differences method –Engesser-Newmark method –Galerkin method (differential equations) –Rayleigh-Ritz method (potential energy) –Finite element method

3. 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

4. Galerkin method Approximate solution Evaluation of residuals by replacing the vector with the approximate solution Orthogonallity obtained by choosing appropriated q parameters

5. Galerkin method - Example Clamped simply supported column Form function Residual Galerkin equation Critical Load

6. 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.

7. Rayleigh-Ritz method 1.Approximate solution 2.Evaluation of variational 3.Stationarity of potencial energy 4.Linear problem of eigenvalues

8. Rayleigh-Ritz method Long columns –u=w(x) –Potential Energy – Rigidity Matriz –Geometric Matriz

9. Rayleigh-Ritz method - Example Clamped simply supported column Form function Rigidity Matriz Geometric Matriz Critical Load

10. 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

11. Effective length of columns

12. Geometric Imperfections Initial deformation w o (x) Eccentricity of the axial load

13. Initial deformation Equilibrium Initial imperfections Deformed shape Magnification

14. Eccentricity Equilibrium Bending moment –Bending from the beginning of loading Critical load ?

15. Non linearity due to Material Stress strain curves –Proportional limit –Yield stress in steel –Hardening –Unloading –Yield plateau Residual stresses –Shape –Level Theories –Tangent modulus –Reduced modulus Shanley model –Perry-Robertson