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PAEN - 2nd Section

PAEN - 2nd Section. Stability of structures Methods Initial Imperfections. Linear Analysis. Aproximate methods Finite differences method Engesser-Newmark method Galerkin method (differential equations) Rayleigh-Ritz method (potential energy) Finite element method. Galerkin method.

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PAEN - 2nd Section

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  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,l)=0 • L is a diferencial operator • u is the displacement field • l 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,l) • 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 • Approximate solution • Evaluation of variational • Stationarity of potencial energy • 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: • g due to boundary conditions • l is the length • EI is the structural rigidity

  11. Effective length of columns

  12. Geometric Imperfections • Initial deformation wo(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

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