AERSP 301 BUCKLING EULER & COLUMN/LOAD IMPERFECTIONS

1. AERSP 301BUCKLINGEULER & COLUMN/LOAD IMPERFECTIONS Jose Palacios August 2008

2. Today • BUCKLING • EULER BUCKLING • COLUMN IMPERFECTIONS AND LOAD MISALIGNMENT • ENERGY METHODS AND APPROXIMATE SOLUTIONS FINAL: Thursday, August 14 from 10:00 am – 12 noon @ RCOE Tentative Schedule: M – Beam Buckling T – Plate Theory W – Hw # 7 Review R – Intro to Vibration F – Final Exam Review

3. STRUCTURAL INSTABILITY • STRUCTURAL MEMBERS IN COMPRESSION ARE SUSCEPTIBLE TO FAILURE BY BUCKLING WHEN THE COMPRESSIVE LOAD EXCEEDS A CRITICAL LOAD (BUCKLING LOAD) • THERE ARE MULTIPLE TYPES OF BUCKING EULER BUCKLING OF COLUMNS • FOR SMALL, ELASTIC DEFLECTIONS OF PERFECT, SLENDER COLUMNS • VARIETY OF BOUNDARY CONDITIONS • PHYSICALLY – IF YOU APPLY A COMPRESSIVE LOAD TO A COLUMN, AT SOME VALUE OF LOAD IT WILL SUDDENLY BOW (OR BUCKLE)

4. STRUCTURAL INSTABILITY (EULER) • IN THEORY – FOR A PERFECT COLUMN LOADED PERFECTLY ALONG THE CENTROIDAL AXIS: • THERE WILL ONLY BE A SHORTENING, NO BOWING (BUCKLING). • BUT WHAT HAPPENS IF A SMALL LATERAL LOAD IS APPLIED? • DEPENDS ON THE LEVEL OF THE COMPRESSIVE LOAD… • FOR: • ADDITION OF LATERAL LOAD RESULTS IN DIFFERENT BEHAVOIR • EULER BUCKLING – BEFORE AND AT CRITICAL LOAD, COLUMN IS RELATIVELY UNDEFORMED • WHEN BUCKLING LOAD IS SURPASSED, SUDDEN, LARGE, DEFORMATION OCCURS

5. STRUCTURAL INSTABILTY (EULER) • DETERMINATION OF BUCKLING LOAD FOR A PINNED-PINNED COLUMN: • AT THE CRITICAL LOAD, Pcr, ANY ADDITIONAL LOAD WILL BUCKLE THE COLUMN AS SHOWN z w Pcr x

6. STRUCTURAL INSTABILITY (EULER) • FROM BUCKLED SHAPE BENDING MOMENT AT ANY X LOCATION (show this)

7. STRUCTURAL INSTABILITY (EULER) • SET • SOLUTION TO THIS HOMOGENEOUSODE IS OF THE FORM: • w – LATERAL DISPLACEMENT • A, B – CONSTANTS Eigenvalue Problem

8. STRUCTURAL INSTABILITY (EULER) • USE BOUNDARY CONDITIONS TO DETERMINE CONSTANTS A & B:

9. STRUCTURAL INSTABILITY (EULER) • POSSIBLE SOLUTIONS: • A = 0 TRIVIAL SOLUTION • OR SIN(λL) = 0: Non-Trivial Solution • λ: EIGENVALUES (ALL POSSIBLE SOLUTIONS TO ODE)

10. STRUCTURAL INSTABILITY (EULER) • THEN:

11. STRUCTURAL INSTABILITY (EULER) Is called the buckling mode shape • NOW:

12. STRUCTURAL INSTABILITY (EULER) • IN REALITY, BUCKLING OCCURS AT THE LOWEST VALUE • HIGHER MODES WILL BE OBSERVED ONLY IF THERE ARE RESTRAINTS AT NODES OF THOSE MODES

13. STRUCTURAL INSTABILITY (EULER) • LATERAL RESTRAINT AT MID-POINT SUPPRESSES THE 1ST MODE AND CRITICAL BUCKLING LOAD • LATERAL RESTRAINTS AT L/3 AND 2L/3 SUPPRESSES THE 1ST AND 2ND MODES AND CRITICAL BUCKLING LOAD IS INCREASED TO

14. STRUCTURAL INSTABILITY (EULER) • DETERMINATION OF BUCKLING LOAD FOR A CLAMPED-FREE COLUMN What will the moment be?

15. STRUCTURAL INSTABILITY (EULER) • BENDING MOMENT AT X (show this): • EQUILIBRIUM EQUATION:

16. STRUCTURAL INSTABILITY (EULER) • NON-HOMOGENOUS ODE SOLUTION (2 PARTS): • COMPLIMENTARY SOLUTION (SOLUTION TO HOMOGENOUS PART):

17. STRUCTURAL INSTABILITY (EULER) • PARTICULAR SOLUTION: • FULL SOLUTION: • APPLY BOUNDARY CONDITIONS:

18. STRUCTURAL INSTABILITY (EULER) • ALSO, w(L) = δ • THIS IMPLIES:

19. STRUCTURAL INSTABILITY (EULER) • SO, For n = 1, 2, 3, 4,…

20. STRUCTURAL INSTABILITY (EULER) • BUCKLING LOAD – LOWEST VALUE FOR CLAMPED-FREE BEAM: For n = 1, 2, 3, 4,…

21. STRUCTURAL INSTABILITY (EULER) • SIMILARLY, IT CAN BE SHOWN THAT FOR A • FROM THE ABOVE RESULTS, WE CAN WRITE: Clamped-Clamped Beam: Clamped-Pinned Beam: • FOR ANY COLUMN, WHERE THE EQUIVALENT LENGTH,Le, DEPENDS ON THE BOUNDARY CONDITIONS

22. STRUCTURAL INSTABILITY (EULER) • Le DEPENDS ON BOUNDARY CONDITIONS: For a pinned-pinned: Le = L For a clamped-clamped: Le = L/2 For a clamped-free: Le = 2L For a clamped-pinned: Le = 0.7L

23. STRUCTURAL INSTABILITY (EULER) • WE COULD ALSO WRITE: C: COEFFICIENT OF CONSTRAINT OR END FIXITY FACTOR For a pinned-pinned: C = 1 For a clamped-clamped: C = 4 For a clamped-free: C = 0.25 For a clamped-pinned: C = 2.046

24. STRUCTURAL INSTABILITY (IMPERFECTIONS) COLUMN IMPERFECTIONS & LOAD MISALIGNMENT • FORCE IS P, NOT Pcr • UNLIKE PERFECTLY STRAIGHT COLUMN (WHERE BENDING OCCURS ONLY AFTER Pcr), WITH IMPERFECTIONS BENDING OCCURS IMMEDIATLEY ON APPLICATION OF COMPRESSIVE FORCE (DUE TO ITS OFFSET FROM THE SLIGHTLY CURVED CENTER LINE).

25. STRUCTURAL INSTABILITY (IMPERFECTIONS) • BENDING MOMENT ALONG COLUMN:

26. STRUCTURAL INSTABILITY (IMPERFECTIONS) • INITIAL SHAPE OF THE COLUMN IS A SINE FUNCTION: • SOLUTION TO THIS NON-HOMOGENEOUS ODE: (aoL IS THE AMPLITUDE. a0 IS THE DIMENSIONLESS IMPERFECTION AMPLITUDE –VERY SMALL NUMBER) Homogenous Solution Particular Solution

27. STRUCTURAL INSTABILITY (IMPERFECTIONS) • APPLY BOUNDARY CONDITIONS TO DETERMINE A & B:

28. STRUCTURAL INSTABILITY (IMPERFECTIONS) • SINCE:

29. STRUCTURAL INSTABILITY (IMPERFECTIONS) • @ X = L/2, LATERAL DEFLECTION TAKES ITS MAX. VALUE (CALL IT ) • USING

30. STRUCTURAL INSTABILITY (IMPERFECTIONS) • RECALL, FOR PERFECT COLUMN, EULER’S CRITICAL BUCKLING LOAD WAS:

31. STRUCTURAL INSTABILITY (IMPERFECTIONS) • a  NON-DIMENSIONAL MID-PT. DISP MAGNITUDE OF INITIAL IMPERFECTION, ao, AFFECTS THE AMPLITUDE OF DEFLECTION, BUT NOT THE LIMITING (BUCKLING) LOAD IF ao = 0, BUCKLES LIKE EULER COLUMN (NO BENDING UNTIL LOAD PASSES Pcr) or

32. STRUCTURAL INSTABILITY (IMPERFECTIONS) • PREVIOUS COLUMN, LOADED PERFECTLY BUT GEOMETRICALLY IMPERFECT • NOW COLUMN IS GEOMETRICALLY PERFECT, BUT COMPRESSIVE LOAD P IS NOT ALIGNED WITH CENTROIDAL AXIS (LOAD IMPERFECTION, OFFSET BY ECCENTRICITY, e)

33. STRUCTURAL INSTABILITY (IMPERFECTIONS) • BENDING MOMENT ON COLUMN: • DETERMINE SOLUTION TO NON-HOMOGENEOUS ODE

34. STRUCTURAL INSTABILITY (IMPERFECTIONS) • ODE SOLUTION: • @ X = 0,

35. STRUCTURAL INSTABILITY (IMPERFECTIONS) • @ X = L,

36. STRUCTURAL INSTABILITY (IMPERFECTIONS) • MAX LATERAL DEFLECTION, , AT THE MID-PT. (X=L/2)

37. STRUCTURAL INSTABILITY (IMPERFECTIONS) • a, ae,  DIMENSIONLESS MID-SPAN DEFLECTION AND ECCENTRICITY

38. STRUCTURAL INSTABILITY (IMPERFECTIONS) • SOLVING FOR

39. STRUCTURAL INSTABILITY (IMPERFECTIONS) • FIGURE SHOWS THAT EVEN IF THERE IS A SMALL LOAD ECCENTRICITY, THE LOAD CAPACITY OF THE COLUMN IS DECREASED.