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Classes #9 & #10

Classes #9 & #10. Civil Engineering Materials – CIVE 2110 Buckling Fall 2010 Dr. Gupta Dr. Pickett. Buckling = the lateral deflection of long slender members caused by axial compressive forces. Buckling of Columns. Buckling of Diagonals.

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Classes #9 & #10

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  1. Classes #9 & #10 Civil Engineering Materials – CIVE 2110 Buckling Fall 2010 Dr. Gupta Dr. Pickett

  2. Buckling = the lateral deflection of long slender members caused by axial compressive forces Buckling of Columns Buckling of Diagonals Buckling of Beams

  3. Column Buckling Theory uses ASSUMPTIONS OF BEAM BENDING THEORY Column Length is Much Larger Than Column Width or Depth. so most of the deflection is caused by bending, very little deflection is caused by shear Column Deflections are small. Column has a Plane of Symmetry. Resultant of All Loads acts in the Plane of Symmetry. Column has a Linear Stress-Strain Relationship. Ecompression = Etension σyield compression = σyield tension σBuckle < (σyield ≈ σProportional Limit ). σBuckle

  4. Column Buckling Theory usesASSUMPTIONS OF BEAM BENDING THEORY Column Material is Homogeneous. Column Material is Isotropic. Column Material is Linear-Elastic. Column is Perfectly Straight, Column has a Constant Cross Section (column is prismatic). Column is Loaded ONLY by a Uniaxial Concentric Compressive Load. Column has Perfect End Conditions: Pin Ends – free rotation allowed, - no moment restraint Fixed Ends – no rotation allowed, - restraining moment applied P P

  5. Column Buckling Theory An IDEAL Column will NOT buckle. IDEAL Column will fail by: Punch thru Denting σ > σyield compressive . Fracture In order for an IDEAL Column to buckle a TRANSVERSE Load, F, must be applied in addition to the Concentric Uniaxial Compressive Load. The TRANSVERSE Load, F, applied to IDEAL Column Represents Imperfections in REAL Column P=Pcr P=Pcr F P=Pcr P=Pcr Pcr = Critical Load Pcr = smallest load at which column may buckle

  6. Column Buckling Theory Buckling is a mode of failure caused by Structural Instability due to a Compressive Load - at no cross section of the member is it necessary for σ > σyield . Three states of Equilibrium are possible for an Ideal Column Stable Equilibrium Neutral equilibrium Unstable Equilibrium P=Pcr P=Pcr

  7. Column Buckling Theory – Equilibrium States Stable Equilibrium Unstable Equilibrium Neutral Equilibrium F F F P>Pcr P<Pcr P<Pcr P<Pcr P>Pcr P>Pcr P>Pcr P=Pcr P=Pcr P=Pcr Δ=small F Δ=grows F F P<Pcr P<Pcr P<Pcr P=Pcr P=Pcr P=Pcr P>Pcr P>Pcr P>Pcr P>Pcr

  8. Column Buckling Theory – Equilibrium States Stable Equilibrium Unstable Equilibrium Neutral Equilibrium P P P Ideal Column Ideal Column Ideal Column Pcr Pcr Pcr Real Column Real Column Real Column Δ/L= Δ/L= Δ/L= 0 0 0 P>Pcr P<Pcr P<Pcr P<Pcr P>Pcr P>Pcr P>Pcr P=Pcr P=Pcr P=Pcr Δ=small F Δ=grows F F P<Pcr P<Pcr P<Pcr P=Pcr P=Pcr P=Pcr P>Pcr P>Pcr P>Pcr P>Pcr

  9. Deflection - BEAM BENDING THEORY When a POSITVE moment is applied, (POSITIVE Bending) TOP of beam is in COMPRESSION BOTTOM of beam is in TENSION. NEUTRAL SURFACE: - plane on which NO change in LENGTH occurs. Cross Sections perpendicular to Longitudinal axis Rotate about the NEUTRAL (Z) axis.

  10. Elastic Buckling Theory – Ideal Column From Moment curvature relationship; P P Tension Tension M M Compression P P y=(-) y=(+) Compression P P M M P x x Tension Tension P P Compression Tension Tension Tension Compression Tension

  11. Elastic Buckling Theory – Ideal Column

  12. Elastic Buckling Theory – Ideal Column x=L y=0 L x y x=0 y=0

  13. Elastic Buckling Theory – Ideal Column x=L y=0 L x y x=0 y=0

  14. Elastic Buckling Theory – Ideal Column x=L y=0 L x y x=0 y=0

  15. PC Fixed PB 0.5 of Half Sine Wave PA 0.5LA Fixed Pin Half Sine Wave Half Sine Wave LC LA LA=Leff 0.5LC=Leff LB 0.5LB=Leff 0.5 of Half Sine Wave Fixed Pin 0.5LA PC PB PA Fixed

  16. PE Free PD PA 0.7071LD=Leff Pin Pin 0.5 of Half Sine Wave LE =LA Half Sine Wave Half Sine Wave LA=Leff LD 2LE=Leff PE 0.414 of Half Sine Wave Fixed Pin PD Fixed PA Fixed

  17. Elastic Buckling Theory – Ideal Column x=L y=0 L x y x=0 y=0

  18. RADIUS OF GYRATIONB&J 8th, Section: 9.5 σcr σcr KL/r

  19. rX X Y = distance away from X-axis, that an equivalent area should be placed, to give the same second moment of area ( Ix ) about X-axis, as the real area. rY Y X = distance away from Y-axis, that an equivalent area should be placed, to give the same second moment of area ( Iy ) about Y-axis, as the real area

  20. Elastic Buckling – Ideal vs. Real Column

  21. Elastic Buckling Theory – Ideal Column Pcr=0.25PAcrfor L=LA L=LA PAcr Pcr = 2PAcrfor L=LA Pcr = 4PAcrfor L=LA Leff=2L Leff=0.7L L=LA Leff=0.5L L=LA LA L=LA

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