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## PowerPoint Slideshow about ' Classes #9 & #10 ' - gaille

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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.

Ecompression = Etension

σyield compression = σyield tension

σBuckle < (σyield ≈ σProportional Limit ).

σBuckle

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

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

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

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

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

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

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

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

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

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

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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