(hull buckling and ultimate strength)
TRANSCRIPT
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Topics in Ship Structural Design(Hull Buckling and Ultimate Strength)
Lecture 4 Buckling and Ultimate Strength of Columns
Reference : Ship Structural Design Ch.09
NAOE
Jang, Beom Seon
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Goal
In Lecture 02 :
Euler buckling load
Load eccentricity → Secant formula
Elastic and Inelastic column behavior
Tangent-Modulus Theory
Reduced-Modulus Theory
Shanley Theory
In this Lecture:
Residual stress effect
Load eccentricity + column geometric eccentricity (initial deflection)
→ Perry-Robertson formula
Effect of lateral load
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Ideal Columns
The Euler buckling load PE is
(Le : Effective Length)
The Euler buckling load is the load for which an ideal column will first have
an equilibrium deflected shape.
Mathematically, it is the eigenvalue in the solution to Euler’s differential
equation
The “ultimate load” Pult is the maximum load that a column can carry.
It depends on
initial eccentricity of the column, eccentricity of the load
transverse load, end condition,
in-elastic action, residual stress.
Pult of a practical column is less than PE
Buckling - the sudden transition to a deflected shape - only occurs in the
case of "ideal" columns.
11. 1 Review of Basic Theory
2
2E
e
EIP
L
2
20
d w Pw
dx EI
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Ideal Columns
If the compressive stress exceeds proportional limit even in ideal column
→ Actual buckling load < Euler load
due to the diminished slope of the stress-strain curve.
For an ideal column (no eccentricity or residual stress) buckling will occur at
the tangent modulus load, given by
(Et : tangent modulus,
the slope of the stress-stain curve corresponding to Pt/A)
Since Et depends on Pt modulusthe calculation of Pt is generally an iterative
process.
It is more convenient to deal in terms of stress rather than load
→ The effects of yielding and residual stress to be included.
11. 1 Review of Basic Theory
2
2
tt
e
E IP
L
2
2=
( / )
EE
e
P E
A L
ρ : the radius of gyration , I= ρ2A
Le/ ρ : slenderness ratio
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Ideal Columns
Load-deflection behavior for practical columns
containing residual stress and eccentricity
11. 1 Review of Basic Theory
Slender Column (PE<PY) Stocky Column (PE>PY)
Q: Explain Pt?
2
2
tt
e
E IP
L
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Ideal Columns
The ultimate strength of a column is defined as the average applied
stress at collapse
11.1 Review of Basic Theory
Stress-strain curveTangent modulus
curveTypical experiment values for straight
rolled sections and welded sections
ultult =
P
A
2
tult ideal 2
( ) =
e
E
L
due to entirely the compressive residual stress σr
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Residual Stress-Rolled Sections
What is rolled section?
a metal part produced by rolling and having any one of a number of cross-
sectional shapes.
11.1 Review of Basic Theory
Rolled section
Welded section
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Residual Stress-Rolled Sections
The uneven cooling between Residual stresses will result after the cooling
because of the non-uniform temperature distribution through the cross
section during the cooling process at ambient temperature.
flange root : tesile residual stresses
flange tip : compressive residual stresses,
about 80 MPa for mild steel
11.1 Review of Basic Theory
Typical residual stresses in
rolled section
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Residual Stress-Rolled Sections
The parts of the cross section that have a compressive residual stress
may commence yielding when applied stress = σY - σr
→ particularly detrimental because it occurs in the flange tips.
The material is no longer homogeneous, and the simple tangent modulus
approach is no longer valid.
A comparatively simple solution can be achieved for the buckling strength in
the primary direction by assuming an elastic-perfectly plastic stress-strain
relationship.
The progressive loss of bending stiffens is linearly proportional to the extent
of the yielded zone → the use of an average value of tangent modulus.
11.1 Review of Basic Theory
Typical stub column
stress-strain curve
Structural tangent
modulus diagram
Q: What is σspl : Structural proportional limit?
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Residual Stress-Rolled Sections
Structural tangent modulus Ets by Ostenfeld-Bleich Parabola
σspl : Structural proportional limit
Substituting Ets in place of Et
For rolled wide flange section, typical value of σspl = 1/2 σY .
Johnson parabola for rolled section (essentially straight and pinned end)
11.1 Review of Basic Theory
)(for)(
)(Yavspl
splYspl
avYavts
E
E
E
L Ye
Y
spl
Y
spl
Y
ult
,11 2
2for,4
12
Y
ult2,
42
1,
41
2
1 222
when σult = 1/2 σspl
2
tult ideal 2
( ) =
e
E
L
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Residual Stress-Rolled Sections
11.1 Review of Basic Theory
Basic column curve (Johnson parabola and Perry-Robertson curve
σult = 1/2 σspl
YeYe
Ye
L
E
L
E
E
L
2
22
2
2
2
22
22
)/(
1
2
2=
( / )
EE
e
P E
A L
22
1)(
Y
ultE
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Residual Stress-Welded Sections
Residual thermal stress is induced by uneven internal temperature
distribution
11.1 Review of Basic Theory
Residual stress close to
yield stress
in welding direction
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Residual Stress-Welded Sections
The interaction between the different fibers results in a locked-in
tensile stress in and near the weld which is equal to approximately
equal to the yield stress.
11.1 Review of Basic Theory
The extent of the tension yield zone (3~6 times thickness) depends mainly
on the total heat input, the cross-sectional area of the weld deposit, the type
of welding and the welding sequence.
Typical residual stresses in welded section
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Residual Stress-Welded Sections
From the equilibrium
ηt : width of the tension yield zone, b : total flange width
11.1 Review of Basic Theory
2=
2
r
Yb
t
For narrow thick sections residual stress will be high, and this will seriously
diminish the strength of the section.
It should be noted that
the effect will be much worse for open sections than for box sections due
to the tensile stress at corners of box sections.
flame-cutting creates conditions similar to welding → high tensile residual
stress is beneficial.
the effect of residual stress is somewhat diminished for higher yield
steels due to narrower tension zone.
Yr ttb 2)2(
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Pinned column with an initial deflection δ(x) and addition deflection (x)
due to axial load. Bending moment is P(δ+w).
Deflection continues to grow and magnified as long as P increases.
→ no static equilibrium configuration and no sudden buckling.
Eccentricity: Magnification Factor
11.1 Review of Basic Theory
2
2+ ( +w)=0
d w P
dx EI
1
= sinn
n
n x
L
Let δ be represented by a Fourier series
Assume that the additional deflection
due to the bending is also a Fourier
seriesL
xnww n
sin
Eccentric columns
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Eccentricity: Magnification Factor
11.1 Review of Basic Theory
2 2 2
2 21
= sinn
n
d w n n xw
dx L L
The dominant term is n=1. When PE/P→1, wn /δn →∞
and the total deflection is
where ϕ is called the magnification factor.
,...3,2,1,
12
nwhere
P
Pnw
E
nn
0sin)(1
2
22
L
xnw
EI
Pw
L
n
n
nnn
0)(
2
22
nnn wEI
Pw
L
n
0)()( 2
2
22
nnEnn wPnPwL
EInP
,
1
P
Pw
E
PP
P
PP
Pww
E
E
E
T
PP
P
E
E
Load-deflection for eccentric columns
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Eccentricity: Magnification Factor
Eccentricity effects may also arise due to eccentricity of load. In this
case the magnification factor is given by
The two types of eccentricity (load application and column geometry)
can be combined linearly Δ=δ+e.
Eccentricity can be related to the slenderness ratio
11.1 Review of Basic Theory
=sec2 E
P
P
crP
PPeM
2secmax
Secant Formula in
Lecture 2
cA
Zr
rA
P
Z
A
A
P
Z
P
A
P
c
c
2
max
radius core
,11
2 ratioty eccentrici
c
rc
ratioty eccentrici
2
r
ec
Secant Formula in Lecture 2
resultsfor test boundlower for 0.003,L
cr
Column with eccentric load
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Perry-Robertson Formula
It adopts Robertson`s value for α and assumes that the column will
collapse when the maximum compressive stress reaches the yield
stress.
It is a quadratic, and may be rewritten in a nondimensional, factored form
where R is the strength ratio of the column
η is the eccentricity ratio
λ is the column slenderness parameter
For a perfectly straight column, having η=0, reduces to the Euler curve,
, providing that λ>1.
11.2 Column Design Formulas
18
ult
ult
11
Y
E
L
ult
ult
( )1 E
Y
E
P L P
A P P
2(1 )(1 )R R R
ult
Y
R
L
2
2 2 2
1 1 1 1 11 1
2 4R
2
1R
RRultYult
Y
22 11
1,
11
E
L Y
E
Y
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Accurate Design Curves
Perry-Robertson formula with different values of α being used for different
types of sections
11.2 Column Design Formulas
19Column design curve for α=0.002 α=0.0035 α=0.0055
fail by yield
σult
(MPa) σult
(MPa)
σult
(MPa)
Le /ρ Le /ρ Le /ρ
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Homework #1
Plot Secant curve shown below and Perry-Robertson
Formula in a graph and discuss the differences of two
approach associate with the plots.
Euler curve
Graph of the secant formula for
σmax = 250 MPa (σyield) and E = 200 GPa
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Accurate Design Curves
The curves have a horizontal plateau at values
of slenderness ratio less than some threshold
value (L/ρ)0 . To achieve this the value of the
eccentricity ratio is defined
for
for
where (L/ρ)0 is given by
For welded I-and box sections, the yield
stress should be reduced by 5% to allow
for welding residual stress.
11.2 Column Design Formulas
Curve selection(α) table for rolled
and welded sections
0
0c
L l
r
0 0
c
L l L L
r
0
0.2Y
L E
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Use of Magnification Factor
For a pinned column subjected to lateral pressure q and P, the wmax is
The maximum bending moment
→ different for deflection, end condition, load
The simpler magnification factor which was derived for sinusoidal
initial eccentricity is used.
Max. bending moment =due to lateral load(M0)+ due to eccentricity(δ0)
11.3 Effect of Lateral Load: Beam Columns
22
4 2
max 4
5 24sec 1
384 5 2
qLw
EI
2
L P
EI
2
max 2
2(1 sec )
8
qLM
E
E
P
P P
max 0 0( )M M P max
max
MP
A Z
Homework #2 Derive wmax
and Mmax.
Central deflection of a
laterally loaded memberThe effect of axial load
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Use of Magnification Factor
To demonstrate the accuracy of this approach, if P=0.5PE, Δ=0
We assume that the column will collapse when the maximum compressive stress reaches the yield stress. Therefore
It can be expressed in terms of nondimensional parameters
where
11.3 Effect of Lateral Load: Beam Columns
23
max 02.030M M
2 4 2
max 0 0 02
5 50.5 2 1 2.028
384 48
EI qLM M M M
L EI
ult 0 ult 0
ult
( )
1
Y
E
P M P
A Z PZ
P
2(1 )(1 )R R R
ult
Y
R
Y Y
E
L
E
0( )A
Z
0
Y
M
Z
2
max 2
2(1 sec )
8
qLM
2
L P
EI
Homework #3 Derive left formula
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Use of Magnification Factor
The solution becomes
11.3 Effect of Lateral Load: Beam Columns
24
A pinned beam column subjected to a specified lateral load.
0( )A
Z
0
Y
M
Z
Initial deflection and
load eccentricity
B.M due to lateral load
2
2 2 2
1 1 1 1 11 1
2 4R
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Use of Magnification Factor
11.3 Effect of Lateral Load: Beam Columns
25
0( )A
Z
0
Y
M
Z
Initial deflection and
load eccentricity
B.M due to lateral load