chapter 13 – buckling of columns
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Chapter 13 – Buckling of Columns. Failure mode of column = buckling (i.e. not stress) At P cr , stress less than yield strength!. At Pcr, column is on verge of being unstable so small lateral force, F will cause column to buckle. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 13 – Buckling of Columns
Failure mode of column = buckling (i.e. not stress)
At Pcr, stress less than yield strength!
Load P can be increased until failure occurs by yield or fracture
At Pcr, column is on verge of being unstable so small lateral force, F will cause column to buckle.
When F is removed, stays buckled. If P reduced to below Pcr will straighten out, if P increased beyond Pcr will increase lateral deflection
Derivation of Euler’s Equation:
PMdxdEI 2
2
02
2
EIP
dxd
Homogeneous, second order, linear DE with constant coefficients.
x
EIPCx
EIPC cossin 21
Solution to DE is:
C1 and C2 = constants of integration. =0 at x=0 so C2=0.
x
EIPC sin1
0sin
x
EIP
Which leaves:
Which is satisfied if: nLEIP
..........,3,2,12
22
nLEInP
Smallest value of P??
2
2
LEIPCR
Euler’s Equation
PCR = critical or maximum axial load on the column just before it begins to buckle.
E = modulus of elasticity for the material
I = least moment of inertia of the column’s cross-section
L = unsupported length of the column, whose ends are pinned
2
2
2
/
,
rLE
ArInote
CR
Examples: