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

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1 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 Buckling of Beams

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. E compression = E tension σ yield compression = σ yield tension σ Buckle < (σ yield ≈ σ Proportional Limit ). σ Buckle

Column Buckling Theory uses ASSUMPTIONS 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

Column Buckling Theory An IDEAL Column will NOT buckle. IDEAL Column will fail by: Punch thru Denting σ > σ yield 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 cr = Critical Load P cr = smallest load at which column may buckle F P=P cr

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 > σ yield. Three states of Equilibrium are possible for an Ideal Column Stable Equilibrium Neutral equilibrium Unstable Equilibrium P=P cr

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

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

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.

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

Elastic Buckling Theory – Ideal Column

x y L x=0 y=0 x=L y=0

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

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

Half Sine Wave 0.5 of Half Sine Wave Pin PBPB L A =L eff LBLB 0.5L B =L eff LCLC 0.5L C =L eff Fixed PAPA PBPB PCPC PAPA PCPC 0.5L A LALA

Half Sine Wave 0.414 of Half Sine Wave Pin PDPD L A =L eff Fixed Free PAPA PDPD PEPE PAPA Fixed L E =L A PEPE Pin LDLD 0.7071L D =L eff 2L E =L eff 0.5 of Half Sine Wave

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

RADIUS OF GYRATION B&J 8 th, Section: 9.5 σ cr KL/r

= distance away from Y-axis, that an equivalent area should be placed, to give the same second moment of area ( I y ) about Y-axis, as the real area = distance away from X-axis, that an equivalent area should be placed, to give the same second moment of area ( I x ) about X-axis, as the real area. rXrX rYrY X X Y Y

Elastic Buckling – Ideal vs. Real Column

Elastic Buckling Theory – Ideal Column P Acr P cr = 4P Acr for L=L A P cr = 2P Acr for L=L A P cr =0.25P Acr for L=L A L=L A LALA L eff =2L L eff =0.5L L eff =0.7L

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