BUCKLING EULER & COLUMN/LOAD IMPERFECTIONS AERSP 301 BUCKLING EULER & COLUMN/LOAD IMPERFECTIONS Jose...
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Transcript of BUCKLING EULER & COLUMN/LOAD IMPERFECTIONS AERSP 301 BUCKLING EULER & COLUMN/LOAD IMPERFECTIONS Jose...
AERSP 301BUCKLINGBUCKLING
EULER & COLUMN/LOAD IMPERFECTIONSEULER & COLUMN/LOAD IMPERFECTIONS
Jose Palacios
August 2008
TodayToday
• 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
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)
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
crcrcr PPPPPP
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
zw
xPcr
STRUCTURAL INSTABILITY (EULER)
• FROM BUCKLED SHAPE BENDING MOMENT AT ANY X LOCATION (show this)
wPx
wEIM cr
2
2
0
0
2
2
2
2
wEI
P
x
wor
wPx
wEI
cr
cr
STRUCTURAL INSTABILITY (EULER)
• SET
• SOLUTION TO THIS HOMOGENEOUS ODE IS OF THE FORM:
– w – LATERAL DISPLACEMENT– A, B – CONSTANTS
EI
Pcr2Eigenvalue Problem
02 ww
xBxAw cossin
STRUCTURAL INSTABILITY (EULER)
• USE BOUNDARY CONDITIONS TO DETERMINE CONSTANTS A & B:
xBxAw cossin
00 ,0@ Bwx
xAw sin
0sin0 ,@ LAwLx
STRUCTURAL INSTABILITY (EULER)
• POSSIBLE SOLUTIONS:– A = 0 TRIVIAL SOLUTION
– OR SIN(λL) = 0: Non-Trivial Solution
– λ: EIGENVALUES (ALL POSSIBLE SOLUTIONS TO ODE)
0sin LA
3... 2, 1,n
0sin
L
n
nLL
STRUCTURAL INSTABILITY (EULER)
• THEN:
3... 2, 1,n
0sin
L
n
nLL
EI
Pcr2
EIL
nP
EI
P
L
n
cr
cr
2
22
2
STRUCTURAL INSTABILITY (EULER)
• NOW:
x
L
nAw
sin sin
xL
nIs called the buckling mode shape
9
3
4
2
1
2
2
2
2
2
2
L
EIPn
L
EIPn
L
EIPn
cr
cr
cr
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
12
2
L
EIPn cr
4
22
2
L
EIPn cr
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
9
32
2
L
EIPn cr
STRUCTURAL INSTABILITY (EULER)
• DETERMINATION OF BUCKLING LOAD FOR A CLAMPED-FREE COLUMN
What will the moment be?
STRUCTURAL INSTABILITY (EULER)
• BENDING MOMENT AT X (show this):
• EQUILIBRIUM EQUATION:
ntdisplaceme lateral tip theis where
wPM cr
crcr
cr
PwPx
wEIor
wPMx
wEI
2
2
2
2
STRUCTURAL INSTABILITY (EULER)
• NON-HOMOGENOUS ODE SOLUTION (2 PARTS):– COMPLIMENTARY SOLUTION (SOLUTION TO HOMOGENOUS PART):
22
2
2
ww
EI
Pw
EI
P
dx
wd crcr
EI
Pcr2
02 ww
xBxAwh cossin
STRUCTURAL INSTABILITY (EULER)
– PARTICULAR SOLUTION:
– FULL SOLUTION:
• APPLY BOUNDARY CONDITIONS:
pw
xBxAwww ph cossin
xBxAw sincos
0 0 0@ wwx
STRUCTURAL INSTABILITY (EULER)
• ALSO, w(L) = δ
– THIS IMPLIES:
0 0 0@ wwx
00
0
AAw
BBw
L cos1
0cos L
STRUCTURAL INSTABILITY (EULER)
• SO, 0cos L
2
12
0cos
0
nL
L
22
22
2
12
2
12
nL
EI
PL
nL
crn
n
For n = 1, 2, 3, 4,…
STRUCTURAL INSTABILITY (EULER)
• BUCKLING LOAD – LOWEST VALUE FOR CLAMPED-FREE BEAM:
22
2
2
12
L
EInPcr
For n = 1, 2, 3, 4,…
22
4L
EIPcr
STRUCTURAL INSTABILITY (EULER)
• SIMILARLY, IT CAN BE SHOWN THAT FOR A
• FROM THE ABOVE RESULTS, WE CAN WRITE:
• FOR ANY COLUMN, WHERE THE EQUIVALENT LENGTH, Le, DEPENDS ON THE BOUNDARY CONDITIONS
Clamped-Clamped Beam:2
24L
EIPcr
Clamped-Pinned Beam: 22046.2
L
EIPcr
22
ecr L
EIP
STRUCTURAL INSTABILITY (EULER)
• Le DEPENDS ON BOUNDARY CONDITIONS:
22
ecr L
EIP
For a pinned-pinned: Le = LFor a clamped-clamped: Le = L/2For a clamped-free: Le = 2LFor a clamped-pinned: Le = 0.7L
STRUCTURAL INSTABILITY (EULER)
• WE COULD ALSO WRITE:
C: COEFFICIENT OF CONSTRAINT OR END FIXITY FACTOR2
2
L
EICPcr
For a pinned-pinned: C = 1For a clamped-clamped: C = 4For a clamped-free: C = 0.25For a clamped-pinned: C = 2.046
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).
STRUCTURAL INSTABILITY (IMPERFECTIONS)
• BENDING MOMENT ALONG COLUMN:
totPwdx
wdEIM
2
2
002
2
0 wwEI
P
dx
wdwwwtot
EI
Psetting 2
022
2
2
wwdx
wd
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)
L
xLaw
sin00
L
xLaww
sin022
L
xLa
L
LxBxAw
sincossin 0222
22
Homogenous SolutionParticular Solution
STRUCTURAL INSTABILITY (IMPERFECTIONS)
• APPLY BOUNDARY CONDITIONS TO DETERMINE A & B:
0 0 0@ Bwx
L
xLa
L
LxAw
sinsin 0222
22
00sin 0 @ ALAwLx
L
xLa
L
Lw
sin0222
22
STRUCTURAL INSTABILITY (IMPERFECTIONS)
• SINCE:
L
xLa
L
Lw
L
xLaw
www
tot
o
tot
sin1
sin
0222
22
0
0
STRUCTURAL INSTABILITY (IMPERFECTIONS)
• @ X = L/2, LATERAL DEFLECTION TAKES ITS MAX. VALUE (CALL IT )
• USING
LaL 0222
2
La
EIPL 02
2
2
EI
Pcr2
LaPLEI
EI022
2
STRUCTURAL INSTABILITY (IMPERFECTIONS)
• RECALL, FOR PERFECT COLUMN, EULER’S CRITICAL BUCKLING LOAD WAS:
LaPLLP
LP
cr
cr022
2
222
2
LPEIL
EIP crcr
0/1
1 ; :Defining a
PPa
L
δa
cr
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)
0/1
1 a
PPa
cr
00 1
1a
P/Pa
a
cr
0
1a
a
P
P
cr
or
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)
STRUCTURAL INSTABILITY (IMPERFECTIONS)
• BENDING MOMENT ON COLUMN:
• DETERMINE SOLUTION TO NON-HOMOGENEOUS ODE