Section
17 Buckling
17.1 General
17.1.1
Symbols. The symbols used in this Chapter are defined as follows:
|
= |
actual compressive stresses for plates, in N/mm2
|
|
= |
compressive axial stress in the stiffener, in N/mm2,
in way of the midspan of the stiffener |
τ |
= |
actual shear stress, in N/mm2
|
|
= |
reference stress, in N/mm2
|
= |
0,9E
|
E
|
= |
modulus of elasticity, 206 000 N/mm2
|
|
= |
net thickness of plate panel, in mm |
|
= |
specified minimum yield stress of the material, in
N/mm2
|
|
= |
reduction factors, as given in Table 1.18.1 |
s
|
= |
stiffener spacing, in mm |
|
= |
net flange thickness, in mm |
|
= |
net web thickness, in mm |
|
= |
flange breadth, in mm |
|
= |
Poissons ratio, 0,3. |
Figure 1.17.1 Stiffener
cross-sections
17.1.2
Scope
- This Section contains the methods for
determination of the buckling capacity, definitions of buckling utilisation
factors and other measures necessary to control buckling of plate panels,
stiffeners and primary support members.
- The buckling utilisation factor is to satisfy
the following criteria:
- For structural idealisation and definitions
see also Pt 10, Ch 1, 8 Structural idealisation. The thickness and section properties of plates and
stiffeners are to be taken as specified by the appropriate Rule
requirements.
Table 1.17.1 Buckling factor and
reduction factor for plane plate panels
Case
|
Stress ratio ψ
|
Aspect ratio α
|
Buckling factor K
|
Reduction factor C
|
|
1 ≥ ψ ≥ 0
|
α > 1
|
K =
|
= 1 for λ ≤
|
= c () for λ >
|
0 > ψ > 1
|
K =
7,63 ψ (6,26 10ψ)
|
where
|
ψ ≤ 1
|
K =
5,975 (1 ψ)2
|
c
=(1,25 0,12ψ) ≤ 1,25
|
=(1 + )
|
|
1 ≥ ψ ≥ 0
|
α > 1
|
K =
|
=
|
where
|
0 > ψ > 1
|
1 ≤ α ≤ 1,5
|
K =
|
c
=(1,25 0,12ψ) ≤ 1,25
|
R =λ
(1 λ/c) for λ <
|
R = 0,22 for λ ≥
|
α > 1,5
|
K =
|
=0,5c
|
|
F =
|
|
= 0,5 and 1 ≤ ≤ 3
|
|
=1 for due to direct loads (3)
|
ψ ≤ 1
|
1 ≤ α ≤
|
K = 5,975
|
=(1 1/α) ≥ 0 for due to bending (in general) (2)
|
=0 for σ due to bending in extreme load cases
(e.g. w/t.bhds.)
|
α
>
|
K =
|
H =
|
T=
|
|
1 ≥ ψ ≥ 0
|
α > 0
|
K =
|
= 1 for λ ≤ 0,7
|
0 > ψ ≥ 1
|
K =
|
= for λ > 0,7
|
|
1 ≥ ψ ≥ 1
|
α > 0
|
K =
|
|
|
|
|
K =
|
= 1 for λ ≤ 0,84
|
α ≥ 1
|
|
= for λ > 0,84
|
0 < α < 1
|
|
|
|
|
|
K =
K r
|
|
K =
K according to Case 5
|
|
r =
opening red. factor
|
|
r =
|
|
≤ 0,7 and ≤ 0,7
|
|
where
|
ψ = the ratio between smallest and
largest compressive stress, as shown for Cases 1 to 4
|
= length, in mm, of the shorter side of the
plate panel for Cases 1 and 2
|
= length, in mm, of the side of the plate
panel, as defined for Cases 3, 4, 5 and 6
|
α = aspect ratio of the plate
panel
|
Edge boundary conditions:
|
- - - - - - - - - plate edge
free
|
plate edge simply supported
|
NOTES
|
1. Cases listed are general cases. Each
stress component () is to be understood in local coordinates.
|
2. due to bending (in general) corresponds to
straight edges (uniform displacement) of a plate panel
integrated in a large structure. This value is to be applied for
hull girder buckling and buckling of web plate of primary
support members in way of openings.
|
3. for direct loads corresponds to a plate panel
with edges not restrained from pull-in which may result in
non-straight edges.
|
17.2 Buckling of plates
17.2.1
Uni-axial buckling of plates.
- The buckling utilisation factor for uni-axial stress is to be
taken as:
for compressive stresses in x-direction
for compressive stresses in y-direction
for shear stress.
- Reference degree of slenderness, to be taken as:
- The critical stresses, or , of plate panels subject to compression or shear,
respectively, is to be taken as:
17.3 Buckling of stiffeners
17.3.2 Column buckling mode.
- Stiffeners are to be verified against the
column buckling mode as given in Pt 10, Ch 1, 17.3 Buckling of stiffeners 17.3.2.(b) with the allowable buckling utilisation factor, , see
Pt 10, Ch 1, 17.1 General 17.1.2.(b). Stiffeners not subjected to lateral pressure and that
have a net moment of inertia, , complying with Pt 10, Ch 1, 17.3 Buckling of stiffeners 17.3.2.(d) have acceptable column buckling strength and need not be
verified against Pt 10, Ch 1, 17.3 Buckling of stiffeners 17.3.2.(b).
- The buckling utilisation factor for column
buckling of stiffeners is to be taken as:
where
- The bending stress in the stiffener is equal
to:
where
- if lateral pressure is applied to the stiffener:
|
= |
the section modulus calculated at flange if
the lateral pressure is applied on the same side as the
stiffener |
|
= |
the section modulus calculated at attached
plate if the lateral pressure is applied on the side
opposite to the stiffener |
- if no lateral pressure is applied on the stiffener:
|
= |
the minimum section modulus among those
calculated at flange and attached plate |
|
= |
bending moment, in Nmm, due to the lateral
load P
|
= |
|
P
|
= |
lateral load, in kN/m2
|
|
= |
span of stiffener, in metres, equal to
spacing between primary support members |
|
= |
bending moment, in Nmm, due to the lateral
deformation w of stiffener |
= |
|
|
= |
ideal elastic buckling force of the
stiffener, in N |
= |
|
|
= |
|
|
= |
net thickness of plate flange, to be taken
as the mean thickness of the two attached plate panels,
in mm |
|
= |
nominal lateral load, in N/mm2,
acting on the stiffener due to membrane stresses, and , in the attached plate in way of the
stiffener midspan: |
= |
|
|
= |
N/mm2
|
|
= |
|
with and taken equal to
= 1,47
|
= 0,49 for ≥ 2,0
|
= 1,96
|
= 0,37 for ≥ 2,0
|
|
= |
net sectional area of the stiffener without
attached plating, in mm2
|
|
= |
factor taking into account the membrane
stresses in the attached plating acting perpendicular to
the stiffeners axis |
= |
0,5 (1 + ψ) for 0 ≤ ψ ≤ 1 |
= |
for ψ < 0 |
|
= |
membrane compressive stress in the attached
plating acting perpendicular to the stiffeners axis, in
N/m2
|
τ |
= |
shear membrane stress in the attached
plating, in N/mm2
|
w
|
= |
deformation of stiffener, in mm |
= |
|
|
= |
assumed imperfection, in mm |
= |
min
|
For stiffeners sniped at both ends is not to be
taken less than the distance from the midpoint of attached
plating to the neutral axis of the stiffener calculated with the
effective width of the attached plating according to Pt 10, Ch 1, 17.3 Buckling of stiffeners 17.3.4.(a)
|
= |
deformation of stiffener at midpoint of
stiffener span due to lateral load P, in mm. In
case of uniformly distributed load is to be taken as: |
= |
|
|
= |
elastic support provided by the stiffener,
in N/mm2
|
= |
|
|
= |
|
- Stiffeners not subjected to lateral pressure
are considered as complying with the requirements of Pt 10, Ch 1, 17.3 Buckling of stiffeners 17.3.2.(b) if their net moments of inertia, in cm4,
satisfy the following requirement:
where
NOTE
Other parameters are as defined
in Pt 10, Ch 1, 17.3 Buckling of stiffeners 17.3.2.(c).
17.3.3 Torsional buckling mode.
- The torsional buckling mode is to be verified
against the allowable buckling utilisation factor, , see
Pt 10, Ch 1, 17.1 General 17.1.2.(b). The buckling utilisation factor for torsional buckling
of stiffeners is to be taken as:
where
|
= |
compressive axial stress in the stiffener, in
N/mm2, calculated at the attachment point of the
stiffener to the plate, in way of the midspan of the stiffener
measured along the global x-axis |
|
= |
torsional buckling coefficient |
= |
1,0 for ≤ 0,2 |
= |
for > 0,2 |
Φ |
= |
0,5 (1 + 0,21 ( 0,2) + ) |
|
= |
reference degree of slenderness for torsional
buckling |
= |
|
|
= |
reference stress for torsional buckling, in
N/mm2
|
= |
|
∊ |
= |
degree of fixation
|
|
= |
torsional buckling length to be taken equal the
distance between tripping supports, in metres, distance from
connection to plate (C in Figure 1.17.1 Stiffener
cross-sections) to centre
of flange, in mm |
|
= |
( 0,5) for bulb flats |
= |
( + 0,5) for angles and T Bars net web area, in
mm2
|
|
= |
( 0,5) net flange area, in mm2
|
|
= |
|
Table 1.17.2 Moments of
inertia
Section property
|
Flat bars
|
Bulb flats, angles and T bars
|
|
|
|
|
|
|
|
|
for
bulb flats and angles:
|
|
for
T bars:
|
|
17.4 Primary support members
17.5 Other structures
17.5.1
Struts, pillars and cross ties.
- The critical buckling stress for axially compressed struts,
pillars and cross ties is to be taken as the lesser of the column and
torsional critical buckling stresses. The buckling utilisation factor, η, is
to be taken as:
where
|
= |
average axial compressive stress in the member, in
N/mm2
|
- The critical buckling stress in compression
for each mode is to be taken as:
= for
= for
where
- The elastic compressive column buckling
stress of pillars subject to axial compression is to be taken as:
where
|
= |
net moment of inertia about the weakest axis of the
cross-section, in cm4
|
|
= |
net cross-sectional area of the pillar, in
cm2
|
|
= |
end constraint factor:
- 1,0 where both ends are pinned
- 2,0 where one end is pinned and the other
end is fixed
- 4,0 where both ends are fixed
- A pillar end may be considered fixed when
effective brackets are fitted. These brackets are to be
supported by structural members with greater bending
stiffness than the pillar
- Column buckling capacity for cross tie
shall be calculated using equal to 2,0
|
|
= |
unsupported length of the pillar, in metres. |
- The elastic torsional buckling stress, , with respect to axial compression of pillars is to be
taken as:
= N/mm2
where
G
|
= |
shear modulus |
= |
|
|
= |
net polar moment of inertia about the shear centre
of cross-section |
= |
+ + () cm4
|
|
= |
end constraint factor:
- 1,0 where both ends are pinned
- 2,0 where one end is pinned and the other
end is fixed
- 4,0 where both ends are fixed
- Elastic torsional buckling capacity for
cross tie shall be calculated using equal to 2,0
|
|
= |
unsupported length of the pillar, in metres |
|
= |
net cross-sectional area, in cm2
|
|
= |
net moment of inertia about y-axis, in
cm4
|
|
= |
net moment of inertia about z-axis, in
cm4
|
- For cross-sections where the centroid and the
shear centre do not coincide, the interaction between the torsional and
column buckling mode is to be examined. The elastic torsional/column
buckling stress with respect to axial compression is to be taken as:
=
where
|
= |
net cross-sectional area, in cm2
|
Table 1.17.4 Cross-sectional
properties
Double symmetrical
sections
|
|
|
|
Single symmetrical
sections
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
NOTE
|
All dimensions of thickness,
breadth and depth are in mm.
|
Cross-sectional properties not
covered by this Table are to be obtained by direct
calculation.
|
17.5.2
Corrugated bulkheads.
- Local buckling of a unit flange of corrugated bulkheads is to
be controlled according to Pt 10, Ch 1, 17.2 Buckling of plates 17.2.1, for Case 1, as shown in Table 1.17.1 Buckling factor and
reduction factor for plane plate panels, applying stress
ratio ψ = 1,0.
- The overall buckling failure mode of corrugated bulkheads
subjected to axial compression is to be checked for column buckling
according to Pt 10, Ch 1, 17.5 Other structures 17.5.1 (e.g. horizontally corrugated longitudinal bulkheads,
vertically corrugated bulkheads subject to localised vertical forces). End
constraint factor corresponding to pinned ends is to be applied, except for
fixed end support to be used in way of stool with width exceeding two times
the depth of the corrugation.
|