Lecture 5 Buckling and Ultimate Strength of Plates 05... · 2018. 4. 18. · Elastic and Inelastic Buckling Post-Buckling and Ultimate strength DNV Rule for Classification of Ships
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Topics in Ship Structural Design(Hull Buckling and Ultimate Strength)
Lecture 5 Buckling and Ultimate Strength of Plates
Reference : Ship Structural Design Ch.12
NAOE
Jang, Beom Seon
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Facts about MSC Napoli
One of the world’s largest container ships when built (1991)
Built to BV Class and changed to DNV 2002
Last renewal survey carried out in 2004 in Singapore
Built 1991
Length over all 275.66 m
Breadth 37.13 m
Draught 13.50 m
Gross tonnage 53,409 GRT
Capacity 4419 TEU
Slide 2
Failure of MSC Napoli Container ship – DNV Report by Olav Nortun
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Accident January 2007 – MSC Napoli
Ship left Antwerp 17 January 2007
heading for Sines in Portugal
18 January - water ingress in engine
room reported
All 26 crew members safely rescued
Ship beached in Lyme Bay near
Branscombe, UK on 19 January
2007
Slide 3
Failure of MSC Napoli Container ship – DNV Report by Olav Nortun
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Accident January 2007 – MSC Napoli
The vessels was split into two in
July 2007
Forward part was towed to
Belfast for recycling
Failure of MSC Napoli Container ship – DNV Report by Olav Nortun
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B200 Plate panel in uni-axial compression
Unstiffened Plate (Plating between stiffeners)
Elastic and Inelastic Buckling
Post-Buckling and Ultimate strength
DNV Rule for Classification of Ships Part 3 Chapter 1, Section 13
Classification Rule
Johnson-Ostenfeld plasticity correction formula
)N/mm(1000
9.0 2
2
s
tkEel
For plating with longitudinal stiffeners (in direction of compression stress): k=4
Ideal elastic buckling stress
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Buckling of a Wide Column
The plate is acting more as a wide column than as a plate. The product EI is
replaced by the plate flexural rigidity D.
The thickness / length ratio plays the same role as the slenderness ratio for
columns.
The width b plays no part, no support along the unloaded edge → It is
inefficient to use
6
12.1 Elastic Plates Subjected to Uniaxial Compression
2
2cr
DbP
a
22 2
2 212(1 )cr
D E t
a t v a
Buckling of wide column
Simple support along the loaded edges
With no support along the unloaded edges
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Large-Deflection Plate Theory by von Karman
Small-Deflection Plate Theory
Large-Deflection Plate Theory
9.2 Combined Bending and Membrane Stresses-Elastic Range
2
22
2
2
4
4
22
4
4
4
21
2y
wN
yx
wN
x
wNp
Dy
w
yx
w
x
wyxyx
2
22
2
24 2
1
y
wN
yx
wN
x
wNp
Dw yxyx
D
p
y
w
yx
w
x
w
4
4
22
4
4
4
2
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Plate is assumed to be free to move inward under the action of the in-plane
compression. → The strain energy of deformation is due to bending only
Buckling of a Simply Supported Plate
From large-deflection plate theory
Since the edges are simply supported, the
deflected shape can be expressed in the form:
which satisfies both the boundary conditions and
the general biharmonic equation.
8
12.1 Elastic Plates Subjected to Uniaxial Compression
24
2
at ww
D x
sin sinmn mn
m n m n
m x n yw w C
a b
24 2 2
2
2 28mn
m n
ab m nU D C
a b
22 2
2 20 0=
2
a bD w wU
x y
22 2 2
2 22(1 )
w w wv dxdy
x y x y
Buckled shape of long plate
0, xyyax NNptN
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Strain Energy Density for plane stress (σz=0)
Reference
Load applied
on dydz area
Elongation in
x-direction
dy
dxdz
dxεx
dyεy
dzεz
dxdydz
dydxdzdxdydzdu
yyxx
yyxx
)(2
1
))((2
1))((
2
11
dxdydz
dydxdzV
du
xyx
xyx
2
1
))((2
1
22
thickness t
dx
dy
dz
dxdydzdu xyxyyxx )(2
1
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Strain Energy for plane stress (σz=0)
In Chapter 9 (Lecture 03), Plate bending (Derivation of Plate
Bending Equation), the followings are derived
Reference
2
2
2
2
2)(
1 x
wv
y
wz
v
Ey
yx
wGz
2
2 = +u
x y
x
wzu
y
wzv
yx
wz
2
)2(
dzdxdyduUa b t
txyxyyxx
0 0
2/
2/)(
2
1
2
2
2
2
2)(
1 y
wv
x
wz
v
Ex
2
2
)(y
wzy
2
2
)(x
wzx
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Strain Energy Density for plane stress (σz=0)
Reference
dydxyx
wv
y
w
x
wv
y
w
x
wD
dydxyx
wt
vv
vE
y
w
x
wv
y
w
x
w
v
Et
dzdydxyx
wGzz
y
w
x
wv
y
w
v
E
zx
w
y
wv
x
w
v
EduU
a b
a b
a b t
t
22
0 0 2
2
2
22
2
22
2
2
23
0 0 2
2
2
22
2
22
2
2
2
3
222
2
2
2
2
2
2
2
0 0
2/
2/
2
2
2
2
2
2
2
2
)1(222
)1)(1(2
)1(
12
42
)1(12
4)()1(
)()1(2
1
)1(2 v
EG
dydxyx
w
y
w
x
wv
y
w
x
wDU
a b2
2
2
2
2
2
0 0
2
2
2
2
2
)1(22
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Work done for plane stress
Reference
dxdyx
wtdxdyNW
a b
a
a
x
b
x
0 0
2
0 0 2
1
dxx
wdx
x
w bb
x
0
2
0
2
2
111
For unit-width strip in Section 9.2 dx
w
dxx
ww
dxx
w
2
1
21)1(
aa for small a
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Buckling of a Simply Supported Plate
Likewise, the work done by the in-plane compressive stress is
Because of W=U, and hence,
The minimum value of σa is given by taking only one term, say Cmn,
where m and n indicate the number of half-waves in each direction in the
buckled shape.
When n=1, σa gives the smallest value. Hence the plate will buckle into only
one half-wave transversely.
13
12.1 Elastic Plates Subjected to Uniaxial Compression
42 2
8
amn
m n
b tW C m
a
2 22 2 2
2 2
2 2
mn
m n
a
mn
m n
m na D C
a b
t m C
2
0 02
a bat w
W dxdyx
22 2 2 2
2 2 2( )a cr
a D m n
tm a b
222
2
1( )a cr
D am
a t m b
)/,...,/,/max(...
...)/,...,/,/min( 2211
21
212211 nn
n
nnn dcdcdc
ddd
cccdcdcdc
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Buckling of a Simply Supported Plate
A buckling coefficient k is generally used. It depends on the type of
boundary support.
For design applications, in which the plate thickness is to be determined, it
is usually written like this:
12.1 Elastic Plates Subjected to Uniaxial Compression
2mb a
ka mb
2
( )a cr
tKE
b
2
212(1 )
kK
v
2
2( )a cr
Dk
b t
Q: Which critical stress will be higher?, which stiffener
arrangement is better against in-plane compression?
3
212(1- )
EtD
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Buckling of a Simply Supported Plate
For long simply supported plates it is
usually assumed that k=4.
Assuming v=0.3
15
12.1 Elastic Plates Subjected to Uniaxial Compression
2
2( ) 4a cr
D
b t
2
( ) 3.62a cr
tE
b
Classification Rule
)N/mm(1000
9.0 2
2
s
tkEel
k=4, s=b (m)
Homework #1 Plot this curve
2mb a
ka mb
2
2( )a cr
Dk
b t
Buckled shape of long plate
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Buckling of a Simply Supported Plate
16
12.1 Elastic Plates Subjected to Uniaxial Compression
a/b=1, m=1 a/b=2, m=2
a/b=3, m=3
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Buckling of a Simply Supported Plate
For a wide plate, in which the aspect ratio(a/b) is less than 1.0, m=1
For a general "wide plate“, in terms of a
because a<b
For design purposes it may be written as:
For v=0.30
17
12.1 Elastic Plates Subjected to Uniaxial Compression
2
2( )a cr
Dk
a t
2a
k kb
2
( )a cr
tKE
a
222
21
12(1 )
aK
v b
222
2( ) 1a cr
D a
a t b
22
0.905 1a
Kb
a
b
2
b
a
a
bk
)1(12 2
32
v
EtD
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Buckling of a Simply Supported Plate
Longitudinal stiffeners: a>>b(=s),
Transverse stiffeners: a<<b, a=s, b=B,
Longitudinally stiffened plating have the great advantage over
transversely stiffened plating in ship structures, and the former is used
wherever possible.
18
12.1 Elastic Plates Subjected to Uniaxial Compression
2
2
4( )a cr
D
s t
222
2( ) 1a cr
D s
s t B
m=1
k=4
<<1
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Reproducing the event in a computer model
Direct wave load calculations
Linear strength analysis
Non-linear strength analysis
Load and strength comparisons
Simulation of crack propagation
Slide 19
Failure of MSC Napoli Container ship – DNV Report by Olav Nortun
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Most severe wave for engine room area
Failure of MSC Napoli Container ship – DNV Report by Olav Nortun
Wave crest around midship
Vertical ”g” force
Aft ship out of water
Hull forces: Shear force and moment
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Structural arrangement in Engine room zone
21
Failure of MSC Napoli Container ship
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Not sufficient buckling capacity
The buckling capacity might not have been checked sufficiently
when the ship was built
Potentially insufficient buckling strength in the engine room
bulkhead
Failure of MSC Napoli Container ship – DNV Report by Olav Nortun
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Four stages of progressive collapse Outer shell
Failure of MSC Napoli Container ship – DNV Report by Olav Nortun
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Four stages of progressive collapse Inner structure
Failure of MSC Napoli Container ship – DNV Report by Olav Nortun
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Four stages of progressive collapse Inner structure
Failure of MSC Napoli Container ship – DNV Report by Olav Nortun
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Alternative correcting actions
The likelihood of reoccurrence is very low:
Damage statistics are very good
Little likelihood of such a harsh sea state
The ship’s strength was below the strength of similar ships
Maybe not all ships checked in this area
However – the consequences are major
Increase buckling strength
Minor modifications – small amount of steel to be added
Aft of the engine room bulkhead
Can be done while in service
Failure of MSC Napoli Container ship – DNV Report by Olav Nortun
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Solutions for Some Principal Cases
When unloaded edge (A) is replaced
by simply supported, the critical
buckling stress drops more than when
loaded edge (B) is by simply supported.
27
12.2 Other Boundary Conditions
Buckling coefficient k in the design formula
for flat plates in uniaxial compress
2
( )a cr
tKE
b
A, B fixed
B simply supported
A fixed
B fixed
A simply supported
A & B simply
supported
Q: Which edge is more effective to in-
plane buckling? Loaded edge or unloaded
edge?
A : Unloaded edge
B: Loaded edge
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Solutions for Some Principal Cases
28
12.2 Other Boundary Conditions
Buckling stress coefficient k for flat plates
in uniaxial compression
A1, A2 clamped
A1 pinned
A2 clamped
A1, A2 pinned
A1 free
A2 clamped
A1 free
A2 pinned
A1 A2 free
Loaded edges
clamped
Loaded edges
Simply supported
2
2( )a cr
Dk
b t
In general, b≈ 800mm, a≈3300mm, a/b≈3~4
Unloaded edge : clampedLoaded edge : clamped
Unloaded edge : clampedLoaded edge : simply supported
Unloaded edge : pinned and clampedLoaded edge : clamped
Unloaded edge : pinned and clampedLoaded edge : simply supported
Unloaded edge : simply supportedLoaded edge : clamped
Unloaded edge : simply supportedLoaded edge : simply supported
Unloaded edge : free & clampedLoaded edge : clamped
Unloaded edge : free & clampedLoaded edge : simply supported
Unloaded edge : free & pinnedLoaded edge : clamped
Unloaded edge : free & pinnedLoaded edge : simply supported
Unloaded edge : freeLoaded edge : simply supported
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Solutions for Some Principal Cases
29
12.2 Other Boundary Conditions
Buckling stress coefficient k for flat plates
in uniaxial compression
Buckling coefficient k in the design formula
for flat plates in uniaxial compress
A1, A2 clamped
A1 pinned
A2 clamped
A1, A2 pinned
A1 free
A2 clamped
A1 free
A2 pinned
A1 A2 free
Loaded edges
clamped
Loaded edges
Simply supported
2
2( )a cr
Dk
b t
2
( )a cr
tKE
b
A, B fixed
B simply supported
A fixed
B fixed
A simply supported
A & B simply
supported
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Clamped Edges
For in-plane loads, as in the case of lateral loads, it is not possible to obtain
finite expressions for the solution of clamped plates.
Numerical solutions by Faxen, Maulbetsch, and Levy.
30
12.2 Other Boundary Conditions
Buckling coefficient k for clamped plates under uniaxial compression
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Unloaded Edges Rotationally Restrained
Lundquist and Stowell have
investigated the case in which
the support along the unloaded
edges is intermediate between
simply supported and clamped.
The degree of rotational restraint
is specified in terms of a
coefficient of restraint, defined as
Cy : rotational stiffness of the
supporting structure along the
unloaded edge
31
12.2 Other Boundary Conditions
y
bC
D
Loaded edges
clamped
Loaded edges
Simply
supported
Buckling coefficient k for plates with loaded edges simply supported and longitudinal edges
rotationally restrained
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Loaded Edges Rotationally Restrained
The important boundary conditions are those along the longer edges of the
plate. Thus, for short wide plates the edge restraint along the loaded edges
becomes significant.
Similar to end conditions in a column, by using an effective length ae:
for clamped ends ae = 1/2a
for one end simply supported and
the other clamped ae = 0.707a
Using a coefficient of restraint ζ :
Cx : rotational stiffness of the supporting structure along the unloaded edge
The solution to this case is obtained from
in which K1 and K2 are related to the buckling coefficient k.
32
12.2 Other Boundary Conditions
222
2( ) 1 e
a cr
e
aD
a t b
x
aC
D
2 21 21 2 1 2tan tan ( ) 0
2 2
k kK K K K
1,2 ( 4)2
K k k
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Loaded Edges Rotationally Restrained
33
12.2 Other Boundary Conditions
a<b
Buckling coefficient 𝑘 for wide plates in compression elastically restrained on the loaded edges
kbak 2)/(
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Loaded Edges Rotationally Restrained
The corresponding coefficient in the
"design" version of the wide plate
formula
In ship structures the rotational restraint
is usually provided by flange-and-web
type of transverse stiffeners.
In this case ζ is given approximately by
d : depth of the web
I : second moment of area of the stiffener about
the midthickness of the web
J : Saint-Venant’s torsion constant for the
stiffener
34
12.2 Other Boundary Conditions
2
( )a cr
tKE
a
2 2
3 2 2
27
2.6
a Id J
t b b
Buckling coefficient k for plates with loaded edges simply supported and
longitudinal edges rotationally restrained
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All Edges Simply Supported
a is parallel to σax and b to σay. Aspect ratio α=a/b.
Applying the energy method yields the following expression for the critical
combination:
U=W
If we denote the square plate critical stress and nondimensional form
35
12.3 Biaxial Compression
22 22
2 2
2=ax ay
cr
m D mn n
b t
2
,1( ) 3.62ax cr
tE
b
m m
ayaxmn
m m
ayaxmn n
mC
b
atn
b
am
a
bC
tW
2
2
22
4222
4
88
2
2
2
22
4
42
2
2
2
22
4
88
m m
mn
m m
mn nm
CDb
ab
b
n
a
mCD
abU
22 2
2 2
,1 ,1
1=
( ) ( ) 4
ayax
ax cr ax cr cr
m mn n
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All Edges Simply Supported
36
12.3 Biaxial Compression
Buckling stresses of biaxially loaded simply supported plates
α=3,m=3a
b
σay
σay
ab α=1,m=n=1
ab
σay
α=0.5,m=1,n=2
α=3,m=1a
b
σay
σxα=3,m=3a
b
ab
α=1,m=1
σx
ab σx
α=0.5,m=1
2
,1( ) 3.62ax cr
tE
b
When σax ≈σay, α=1
2
2
62.3
62.3)5.05.0()(
b
tE
b
tEcrayax
m=1 m=2m=3
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All Edges Simply Supported
37
12.3 Biaxial Compression
a/b = 3 a/b = 5
Plate under biaxial load
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B400 Plate panel in bi-axial compression
For plate panels subject to bi-axial compression the interaction between the
longitudinal and transverse buckling strength ratios is given by
DNV Rule for Classification of Ships Part 3 Chapter 1, Section 13
1
n
cyy
ay
cycxyx
ayax
cxx
ax K
Homework #2 Plot DNV bi-axial interaction curve and compare with theprevious interaction curve (Fig. 12.8)
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All Edges Clamped
For plates subjected to approximately equal compressive stresses
(σax ≈σay)the interaction formula is
When α=1,
For square plates(α=1), critical combinations are given for particular
values of σax / σay, including cases in which σay is tensile.
When σax = σay
39
12.3 Biaxial Compression
2
2 2
2
3( ) 1.20 3 2ax ay cr
tE
b
2
6.9)(
b
tEcrayax
22
15.10905.0)61.561.5()(
b
tE
b
tEcrayax
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Pure Shear
In ship structures the plating is commonly subjected to large shear loads.
The shearing load can cause buckling since it gives rise to in-plane
compressive stress.
40
12.4 Other Types of In-plane Loads
For the case of pure shear, in-plane
compressive stress is equal to the shear stress
and acts at 45° to the shear axis.
In shear buckling, the coefficients are denoted
as ks and Ks.
For simply supported plates
For clamped plates
2
2cr s
Dk
b t
2
cr s
tK E
b
2=5.35+4( / )sk b a
2=8.98+5.6( / )sk b a
24 2 t ww
D x y
Buckling of an infinitely long, simply supported plate
tNNpN xyyx ,0
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For simply supported plates
Pure Shear
12.4 Other Types of In-plane Loads
B300 Plate panel in shear (DNV Rule for Classification of Ships Part 3 Chapter 1, Section 13)
2
2
2
434.5),N/mm(1000
9.0
l
sk
s
tEk ttel
2=5.35+4( / )sk b a2
2cr s
Dk
b t
22
2
2
90.0)1(12
b
tEk
b
tEk
vsscr
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Pure Shear
ks and Ks are given for various types of boundary conditions. Because of the
symmetry of the pure shear loading , the choice of a and b is independent of
the load.
12.4 Other Types of In-plane Loads
Buckling coefficient of flat plates in shear
Buckling coefficient of flat plates in shear (Design formula)
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Biaxial Compression and Shear
For long plates ks is given approximately by:
– All edges simply supported:
– All edges clamped:
where
43
12.4 Other Types of In-plane Loads
1/2 1/21/2 1/2
= 2 1 2 2 1 6ay ayax ax
s
e e e e
k
1/2 1/21/2 1/2
4 8 4= 4 4 8
33 3
ay ayax axs
e e e e
k
2
2e
D
b t
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In-plane Bending
σb denotes the largest or edge value of the applied stress.
44
12.4 Other Types of In-plane Loads
Some approximate formulas to calculate the values of kb
– simply supported edges:
for
for
– clamped edges:
for
– one unloaded edge clamped; the others simply supported
for
– unloaded edges clamped; loaded edges simply supported
for
2
2( )b cr b
Dk
b t
/ 2 / 3a b
/ 2 / 3a b
/ 1a b
/ 1/ 2a b
/ 0.4a b
2 215.87 1.87( / ) 8.6( / )bk b a a b
23.9bk
41.8bk
25bk
40bk
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In-plane Bending
The figure illustrates the case in which the bending is unsymmetric.
For simply supported edges the value of kb is given approximately by
( simply supported edges only)
45
12.4 Other Types of In-plane Loads
25 +4bk 3
2
b
a
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Combined In-plane Loads: Interaction Formulas
Uniaxial compression and in-plane bending
(σa)cr : critical values of axial loading
(σb)cr : critical values of and in-plane bend
Uniaxial load(compressive or tensile) and shear
For convenience we adopt the symbol R to denote a critical load ratio.
In the present case the strength ratios are
The interaction formula is
In-plane bending and shear
46
12.4 Other Types of In-plane Loads
1.75
1( ) ( )
a b
a cr b cr
( )
ac
a cr
R
s
cr
R
2 1 1c sR R
( )
bb
b cr
R
21 0.61 1
1.6c sR R
2 2 1 ( >1/2)b sR R
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Combined In-plane Loads: Interaction Formulas
Biaxial compression, in-plane bending, and shear
The two compression strength ratios are
By performing a series of four-variable curve-fitting solutions,
47
12.4 Other Types of In-plane Loads
( )
axx
ax cr
R
( )
ay
y
ay cr
R
2
4
2
0.625(1 0.6 / )1 1
1(1 0.625 ) 1
(1 )
y s
xbx
x
R R
RRR
R
B500 Plate panel in bi-axial compression and shear (DNV Rule for Classification of Ships Part 3 Chapter 1, Section 13)
1
n
cyy
ay
cycxyx
ayax
cxx
ax
qqK
q
2
1
a
aq
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Combined In-plane Loads: Interaction Formulas
48
12.4 Other Types of In-plane Loads
Interaction curves for biaxial compression, in-plane bending, and shear drawn for α=2
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Combined In-plane Loads: Interaction Formulas
49
12.4 Other Types of In-plane Loads
2
4
2
0.625(1 0.6 / )1 1
1(1 0.625 ) 1
(1 )
y s
xbx
x
R R
RRR
R
B500 Plate panel in bi-axial compression and shear (DNV Rule for Classification of Ships Part 3 Chapter 1, Section 13)
1
n
cyy
ay
cycxyx
ayax
cxx
ax
qqK
q
2
1
a
aq
Homework #3 Plot DNV bi-axial interaction curve likethe right figure and compare with the following curvefor Rb=0
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Plates Without Residual Stress
Uniaxially loaded, simply supported square plate, with sides free to
pull in. some typical initial distortion in the form of a half wave in
each direction.
Plate slenderness
The relationship between the applied load (σa) and the axial
shortening
12.6 Ultimate Strength of Plates
50
Plate strength without welding (σr=0)
Et
b Y
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Plates Without Residual Stress
Slender plate (β>2.4)
Buckling stress is well below yield stress and below the curve of collapse
stress.
After buckling (σa) a greater proportion of the load is taken by the region of
plating near the sides → Non-uniform compressive stress distribution
Deflected shape of the buckled portion → overall stiffness of the plate
(dσa/dεa) is reduced.
The center region becomes more pronounced and the maximum stress at
the sides increases. When the maximum stress = yield stress → collapse.
12.6 Ultimate Strength of Plates
51Plate strength without welding (σr=0) Post-buckling stress distribution
Ultimate strength
Large margin between buckling and collapse
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Plates Without Residual Stress
Plates of intermediate slenderness (1<β<2.4)
Buckling stress ≈ yield stress
For a rigorous analysis, elasto-plastic large deflection theory to be used.
As applied stress increases → magnification of the initial distortion → loss
of stiffness → some local yield → stress redistribution → yielding of the
sides → sudden collapse.
Pitched roof : allows large axial shortening with minimum strain
energy.
12.6 Ultimate Strength of Plates
Typical post buckling behavior
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Plates Without Residual Stress
Sturdy plates (1>β)
The initial distortion is smaller and the magnification is less because the
elastic buckling stress is very large.
Plates can carry a load equal to the full “squash load” σa,u= σY.
After the peak load, the load carrying capacity remains
approximately constant up to very large strains.
12.6 Ultimate Strength of Plates
Plate strength without welding (σr=0)
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Plates With Residual Stress
Departure from linearity occur at the stress which is less σa
less than for a stress-free plate.
Sturdy plate (1>β) : no load shedding, but large
reduction in stiffness → regarded collapse.
Intermediately slender and slender plate (1<β) : the
loss of ultimate strength ≈ σr
12.6 Ultimate Strength of Plates
2=
2
r
Yb
t
Plate strength with welding (σr=0.1σY)
Middle part
σa+ σr = σY
Edge part
σa- σY = σY
Idealized residual stress
distribution
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Effects of Other Parameters
Restraint at Sides
Clamping the sides of a plate increase the elastic buckling stress by
75%, however, the increase in buckling stress even in slender plate ≈
10% at most.
Stiffeners surrounding the panel is not clamped edge → this restraint
can be ignored.
Initial Deformation
The effect of initial deformation removes sharp knuckle in curve of σa
and εa. The increasing lateral deflection causes a progressive reduction
in the in-plane stiffness of the plate.
However, the ultimate strength is slightly decreased.
Shear stress
In –plane shear stress tends to lower the resistance to longitudinal
compression.
Reduced yield stress rτσY
12.6 Ultimate Strength of Plates
55
2
31
Y
r
Yxyyyxxeq 222 3
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Ultimate Strength of Uniaxial Loaded Plates
Plating of uniaxially loaded, longitudinally stiffened,
initial deformation (δp < 0.2βt), residual stress (σr ≈ 0.1σY)
side constrained to remain straight but free to pull in
For sturdy plate, first loss of stiffness is taken as collapse.
For plates of greater slenderness : loss of stiffness is gradual.
Secant modulus ratio
12.6 Ultimate Strength of Plates
56
Design curves of ultimate strength and secant modulus
22
2 75.21,
4.10225.0
E
ET s
OPen INteractive Structural Lab
Ultimate Strength of Uniaxial Loaded Plates
Faulkner’s formula for the ultimate strength of unwedded plates : good
agreement with extensive experimental data.
The effect of residual stress → strength reduction factor Rr
12.6 Ultimate Strength of Plates
57Curves for ultimate strength of plates
2
, 12
Y
ua
Restrained : the sides remain straight and do not pull in.
Unrestrained : both types of transverse deformations can occur.
Stress relived (σr=0), average welding (σr≤0.1σr), heavily welded (σr≤0.33σr)
E
ER
Y
tsrr
1
00.1for and5.2for
)5.21(12
tsts
ts
EEE
E
E
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