Elastic buckling of selected flanges of cold-formed thin-walled beams
The paper is devoted to five different flanges of cold-formed thin-walled beams. Mathematical models of each of the flanges are formulated and solved. The theory of elastic stability of plates and cylindrical shallow shells is applied for this purpose. Critical stress for each flange of the beam is...
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| Veröffentlicht in: | Фізико-математичне моделювання та інформаційні технології |
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| Datum: | 2006 |
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Центр математичного моделювання Інституту прикладних проблем механіки і математики ім. Я.С. Підстригача НАН України
2006
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| Zitieren: | Elastic buckling of selected flanges of cold-formed thin-walled beams / K. Magnucki // Фіз.-мат. моделювання та інформ. технології. — 2006. — Вип. 3. — С. 116-128. — Бібліогр.: 24 назв. — англ. |
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| citation_txt | Elastic buckling of selected flanges of cold-formed thin-walled beams / K. Magnucki // Фіз.-мат. моделювання та інформ. технології. — 2006. — Вип. 3. — С. 116-128. — Бібліогр.: 24 назв. — англ. |
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| container_title | Фізико-математичне моделювання та інформаційні технології |
| description | The paper is devoted to five different flanges of cold-formed thin-walled beams. Mathematical models of each of the flanges are formulated and solved. The theory of elastic stability of plates and cylindrical shallow shells is applied for this purpose. Critical stress for each flange of the beam is determined. Results of analytical solutions are discussed and compared with numerical (FEM) and experimental investigations. The formulas of critical stresses may be used in practical applications.
У роботі розглянуто п’ять різних профілів холоднокатаних тонкостінних балок. Сформульовано та досліджено математичні моделі для кожного з п’яти профілів. У представлених моделях використано теорію пружної стійкості пластин та пологих циліндричних оболонок. Визначено критичні напруження для кожного типу профілю. Проаналізовано результати аналітичних досліджень та проведено їх порівняння із результатами числових досліджень методом скінченних елементів, а також експериментальними даними. Отримані у роботі формули для критичних напружень можуть бути використані на практиці.
В роботе рассмотрено пять разных профилей холоднокатаных тонкостенных балок. Сформулированы и исследованы математические модели для каждого из пяти профилей. В представленных моделях использовано теорию упругой устойчивости пластин и цилиндрических пологих оболочек. Определены критические напряжения для каждого типа профиля. Проведен анализ аналитических результатов и представлено сравнение их с результатами численного исследования методом конечных элементов, а также экспериментальными данными. Представленные формулы для критических напряжений могут быть использованы в практике.
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| fulltext |
Elastic Buckling of Selected Flanges
of Cold-Formed Thin-Walled Beams
Krzysztof Magnucki
Professor, Doctor Hab. Engineer, Institute of Applied Mechanics, Poznan University of Technology, ul. Piotrowo, 3,
PL 60-965, Poznan, Poland, e-mail: krzysztof.magnucki@put.poznan.pl
The paper is devoted to five different flanges of cold-formed thin-walled beams. Mathematical mo-
dels of each of the flanges are formulated and solved. The theory of elastic stability of plates and
cylindrical shallow shells is applied for this purpose. Critical stress for each flange of the beam is
determined. Results of analytical solutions are discussed and compared with numerical (FEM)
and experimental investigations. The formulas of critical stresses may be used in practical
applications.
Key words: thin-walled beams, cold-formed beams, elastic buckling, thin plates,
thin shells
Introduction. Cold-formed thin-walled beams of flat walls may be modeled by long
rectangular plates joined with adjacent edges. Investigation of local stability of such
beams converts itself to studying buckling of long rectangular plates, taking into
account appropriate conditions of supporting. Timoshenko and Gere (1961) and Vol-
mir (1967) presented a discussion of the elastic buckling problems. Detailed results of
contemporary analytical, numerical — FEM and experimental investigations of selec-
ted problems of strength, buckling and optimization of cold-formed thin-walled beams
are presented for example by: Bradford and Ge (1997), Bradford (1998), Put, Pi and
Trahair (1999), Davies (2000), Pi and Trahair (2000), Magnucki and Monczak (2000),
Rasmussen (2001), Magnucki (2002), Hancock (2003), Mohri F., Brouki A., Roth (2003),
Corte et al. (2004), Dinis P. B., Camotim D., Silvestre (2004), Stasiewicz et al. (2004),
Trahair and Hancock (2004), Magnucki et al. (2004), Magnucka-Blandzi and Mag-
nucki (2004), Magnucki (2005), Magnucki and Ostwald (2005), Magnucki and Mać-
kiewicz (2005). Elastic buckling problems of five selected flanges of cold-formed thin-
walled beams are presented based on the referred papers. Beams with these flanges are
under a pure bending state. The upper flange of each of the beams is compressed and the
lower flange is subject to tension.
1. A flat flange with a bend of the channel beam
Channel beams with flat flanges are reckoned among typical cold-formed thin-walled
beams. One edge of such a flange is free or stiffened by bends. Buckling of these flan-
ges was studied by Hancock (1997), Rogers and Schuster (1997), Bambach and Ras-
УДК 539.3
116
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117
mussen (2001, 2004), Corte G. D. et al. (2004). They proposed some formulas
designed for determining critical loads. Stasiewicz et. al. (2004), Magnucki and Ost-
wald (2005) analytically determined critical loads of flat flanges with bends. Their ana-
lytical solution was subject to numerical verification by means of the finite element method.
Approximate model of a flat flange with bends is considered as a long rectangu-
lar plate with three simply supported edges. The fourth edge with the bends is free
(fig. 1). Moreover, the flange-web joint of the beam is assumed as a hinged one.
The flange width b is small as compared to its length L. A cross section of the flange
rotates with respect to its supporting point (the flange-web joint) by the angle ψ(x).
Angle of rotation of a flange cross section is assumed in the form
( )
L
xx πsin1ψ=ψ , (1)
where 1ψ — parameter;
hence, the corresponding displacement
( ) ( )
L
xzxzzxv πsin, 1 ⋅ψ=ψ⋅= . (2)
Potential energy of elastic strain for the flange
dx
dx
vdEJ
dx
dx
dGJU
bz
L
zp
L
t
=
ε ∫∫
+
ψ
=
0
2
2
2
0
2
22
, or
2
1
2
2
2
π
4
π
ψ⋅
+=ε zpt EJ
L
bGJ
L
U , (3)
where
( )
−−+
+++
= 22
2
11 ecdcc
edcb
y p is location of central axis of the flange
cross section,
a) b)
b
c
d
e
t y
z
z
ψ
Fig. 1. Schemes: a) cross section of the flange, b) displacement of a buckled flange
t
e
d
c y
ψ
z
z
b
Krzysztof Magnucki
Elastic Buckling of Selected Flanges of Cold-Formed Thin-Walled Beams
118
( ) ( )
−−+
+++
−−−+=
2
22323
2
11
3
1
3
2 ecdcc
edcb
ecdcctJ zp is moment
of inertia,
( )edcbtJt +++= 3
3
1 is geometric torsion stiffness of the cross section.
The upper flange is compressed and the work of the load
( )
+++
σ= ∫∫ ∫
=
L
bz
b L
dx
dx
dvedcdxdz
dx
dvtW
0
22
0 0
02
1 ,
and, upon integrating
( )[ ] 2
10
22
3
12
π
ψ⋅+++σ= edcb
L
tbW , (4)
where
max0 21 σ
−=σ
H
yp is compressive stress in the flange, H depth of the beam.
Maximal stress in the bent beam H
J
M
z2max =σ , where maxM is maximal ben-
ding moment.
The principle of stationary total potential energy ( ) 0=−δ ε WU enabled determi-
ning critical stress of the beam flange subject to pure bending
( )[ ]
+
+++
=σ zptKR EJ
L
bGJ
edcbtb
2
2
2
)1(
,0 π
3
3 . (5)
In particular case of a flange without bends ( )0=== edc the critical stress
( )
22
12
ν+
=
=σ
b
tE
b
tGKR .
Example calculation of critical stress has been carried out (5) for the channel
beam flange, assuming the following numerical data: t = 1,5 mm, b = 100 mm,
0 ≤ c ≤ 20 mm, 0 ≤ d ≤ 10 mm, 0 ≤ e ≤ 4 mm, L = 800 mm, ν = 0,3. Results of the cal-
culation are shown in fig. 2. Enlarging the length of any bend results, of course, in gro-
wing critical stress. Very effective growth of the stress occurs while extending the first
bend up to a certain minimal value ( ) 05,0min =bc . Smaller growth of the stress, slight-
ly below linear, corresponds to extending the second bend ( )1,00 ≤≤ bd . On the other
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119
hand, extending the third bend ( )04,00 ≤≤ be appears to be of small meaning, as it
results only in insignificant growth of the flange critical stress. It should be noticed that
a free edge complemented with the bends of growing area must be finally considered
as a supported one. Hence, the value of critical stress corresponding to the flange with
bends exceeds the value for the flange/plate supported at four edges. For example, such
support of the considered flange was obtained when the value of the second bend
01,0=bd (fig. 2).
Moreover, critical stress of the flange was determined by means of FEM (The
COSMOS/M System). The computation was confined to the variant of the flange with
maximal bends. Hence, the following numerical data have been assumed: t = 1,5 mm,
b = 100 mm, c = 20 mm, d = 10 mm, e = 4 mm, L = 800 mm, ν = 0,3, MPaE 51005,2 ⋅=
with uniformly distributed compressive stress. Values of the stresses obtained this way
are marked with the points of the diagram (fig. 2). Differences between numerical re-
sults obtained with both methods do not exceed 1,5%.
2. A flat rectangularly corrugated channel beam
Flat flanges of thin-walled beams may be also stiffened by their corrugation. In such a
case the flange takes, in practice, a form of a long orthotropic rectangular plate, with
three edges simply supported and the fourth edge free (fig. 2). Magnucki and Ostwald
(2005) analytically determined critical loads of flat rectangularly corrugated flange.
Magnucki and Maćkiewicz (2005) the critical loads of flat cosinusoidally corrugated
flange determined.
0,00 0,05 0,10 0,15
b
d
b
e
b
c ,,
rectangular plate with four
simply supported edges
rectangular plate with three
edges simply supported and one edge free
—— 0 < bc < 0,2
- - - - 0 < bd < 0,1
- ·-· - 0 < be <0,04
• MES
c = 20,
d = 10
c = 20,
e = 0
УДК
Fig. 2. Critical stress of the flange as a function
of bending ratio of its free edge
E
KR
)1(
,0σ
0,0014
0,0012
0,0010
0,0008
0,0006
0,0004
0,0002
0,0000
d = e = 0
Krzysztof Magnucki
Elastic Buckling of Selected Flanges of Cold-Formed Thin-Walled Beams
120
Similarly, like for the single-bend flange, a simplified scheme of displacements
shown in fig. 3 is adopted here. The angle of rotation of the flange was assumed in the
form (1) and, consequently, the displacement in the form (2).
Potential energy of elastic strain (Volmir (1967))
dxdz
z
vD
zx
vD
x
vDU
b L
zxzx∫ ∫
∂
∂
+
∂∂
∂
+
∂
∂
=
0 0
2
2
2222
2
2
2
2
1
ε , (6)
where
( ) ( )
L
xzxzzxv πsin, 1 ⋅ψ=ψ⋅= is the displacement, ( )cb
b
tcE
c
EJD
x
z
x +==
4
2
,
( )cbtcJ z +=
12
2
,
3
bcx = ,
+==
b
cGt
c
sGtD
x
xz 31
33
2
33
,
cbs +=
3
, ( ) s
cEtD x
z 2
3
112 ν−
= .
Integrating of the expression (6) provides
2
1
222
2
3
π
4
π
ψ⋅
+
=ε xzx DD
L
b
L
bU .
Work of the load
2
1
32
0 0
2
12
π
2
1
ψ⋅=
∂
∂
= ∫ ∫ xx
b L
xx N
L
bdxdz
x
vNW , where ( ) ( )2
03 σ+=
b
tcbNxx . (7)
The principle of stationary total potential energy ( ) 0=−δ ε WU was a basis for
determining critical stress
( ) ( )
+
+
ν+π+
ν+
=σ
2
2
2
)2(
,0 3
12
14 L
c
cb
cb
b
tE
KR . (8)
In particular case of a flat flange without corrugated (c = 0) the critical stress
( )
22
12
ν+
=
=σ
b
tE
b
tGKR .
y
z
b
c
b/3 t
y
z
z
ψ
Fig. 3. Schemes: a) cross section of corrugated flange, b) displacement of a buckled flange
y
b
b/3
t
c
z
y
ψ
z
z
a) b)
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121
3. A flat three layer flange of channel beam
Stiffness of flat flange of a thin-walled beam may be increased by its double bending
and filling the space created this way with polyurethane foam. In such a case the flange
is considered, in practice, as a long three-layer rectangular plate with three simply
supported edges and one edge free (Fig. 4). Magnucki and Ostwald (2001, 2005) pre-
sented the problems of stability of three-layer structures.
A simplified scheme of displacements is applied here, similarly as for the case of
the bent or rectangularly corrugated flanges. The angle of rotation of a cross section
was assumed in the form (1) and, consequently, the displacement as (2).
Potential energy of elastic strain
dxdz
z
vD
zx
vD
x
vDU
L b
zxzx∫ ∫
∂
∂
+
∂∂
∂
+
∂
∂
=
0 0
2
2
2222
2
2
2
2
1
ε , (9)
where
( ) ( )
L
xzxzzxv πsin, 1 ⋅ψ=ψ⋅= is displacement,
02ttc += , tEcDx
2
2
1
= ,
bc
bctGc
L
c
b
ctGcDxz +
+
≅
++
+=
−−
1
211 2
11
2 , tEcDz
2
2
1
= ,
it was taken into account that 1<<Lc .
Integration of the expression (9) gives
2
1
222
2
3
π
4
π
ψ⋅
+
=ε xzx DD
L
b
L
bU . (10)
y
z
b
t
2t
0
t
y
z
ψ
z
Fig. 4. Schemes: a) flat three-layer flange, b) displacements of a buckled flange
b
y
y
z
t
t
z
z
2t
0
a)
b)
Krzysztof Magnucki
Elastic Buckling of Selected Flanges of Cold-Formed Thin-Walled Beams
122
Work of the load
2
1
32
0 0
2
12
π
2
1
ψ⋅=
∂
∂
= ∫ ∫ xx
b L
xx N
L
bdxdz
x
vNW . (11)
The principle of stationary total potential energy ( ) 0=−δ ε WU enabled deter-
mining intensity of critical load
22
2
2
, 1
26π
2
16π
+
+
+
=+
=
b
c
bc
bcGt
L
cEtD
b
D
L
N xzxKRxx ,
hence, the critical stress of the compressed flange
( )
+
+
+
ν++
=σ
22
, π
2
1
1
2
1
3
2 L
c
b
c
bc
bc
cb
Eb
KRx . (12)
4. A flat double flange of I-beam
Mathematical model for local buckling of the upper flange of the beam is assumed in
the form of a beam on an elastic foundation [Magnucki and Ostwald (2005)]. Scheme
of the deformation of the cross section of the beam is shown in fig. 5.
The differential equation for the beam on an elastic foundation is in the follo-
wing form
( ) 02
2
2
4
4
=⋅β++ xw
dx
wdk
dx
wd (13)
where
fzEJ
Fk
,
2 = ,
fzEJ
c
,
=β , 3
, 3
2 btJ fz = ,
3
8
=
b
tEc is module of the elastic
foundation, F is longitudinal compression force of
the upper flange.
The web is rigid as compared to the flange of
the beam. In consequence, the deflection function
determining the buckling shape is assumed in the
following form
( )
L
xwxw a
mπsin2⋅= , (14)
where aw is amplitude, m is natural number.
The differential equation (13) is solved with
the Galerkin method. The critical force is obtained
in the following form
Fig. 5. Deformed cross section
of the beam under pure bending
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123
b
tE
Y
Y
b
tEF
YCR
3
2
2
3
281
4
3
3
2min8 =
+= , where
L
bmY π= (15)
The critical stress for the compressed flange of the I-section of cold-formed
beam is in the following form
( )
2
24
2
==σ
b
tE
bt
FCRanal
CR . (16)
5. A circular cylindrical flange
Stability of a compressed cylindrical shell was extensively studied and described in li-
terature. The first solution to the problem was presented by R. Lorenz in 1908 and 1911,
S. P. Timoshenko in 1910, and R. V. Southwell in 1914. They determined critical
stress of an axially compressed cylindrical shell
( )
( ) R
tESTL
KR 213 ν−
=σ −− , (17)
where t is thickness of the shell, r is radius of the shell, E, ν are material constants.
Stability of cylindrical shells was studied very thoroughly, particularly in the
latter part of the 20th century. Nevertheless, stability of an axially compressed open
cylindrical shell with linear free edges was investigated only rarely. Chu et al. (1967),
Yang and Guralnick (1976) solved this problem for open cylindrical shells with sec-
torial angle 2π≤β . Magnucka-Blandzi and Magnucki (2004) determined analytically
and FEM-numerically the critical stress for open cylindrical shell of grater sectorial
angles π≤β≤π 2 . Magnucki and Maćkiewicz (2005) presented an extended study of
these shells. They assumed that two curvilinear edges of the shell are pivoted, while
two others free. The shell is loaded at both ends with a distributed force of the intensity
xxN along the curvilinear edges, giving rise to the stress tN xxx =σ (fig. 6).
Fig. 6. Schemes: a) cross section of the cylindrical flange, b) cross section of the flat flange
y
b
z
t
R→∞
R
β
z
t
a) b)
b = Rβ
Krzysztof Magnucki
Elastic Buckling of Selected Flanges of Cold-Formed Thin-Walled Beams
124
Equations of stability for the circular cylindrical shell are in the following form
02
2
4 =
∂
∂
−∇
x
w
R
EtF , 01
2
2
0
2
2
4 =
∂
∂
+
∂
∂
+∇
x
wN
x
F
R
wD x , (18)
where ( )xww ,ϕ= — displacement-deflection function, 224 ∇∇=∇ — linear operator,
( )xFF ,ϕ= — airy force function, ( )[ ]23 112 ν−= EtD — bending stiffness, t and R —
thickness and radius of the cylindrical shell.
The internal forces of the shell are as follows
ϕ∂
∂
+
∂
∂
∂
∂
= 22
2
2
2
R
w
x
w
x
DQx ,
∂
∂
+
ϕ∂
∂
ϕ∂
∂
=ϕ 2
2
22
2
x
w
R
w
R
DQ are shear forces,
22
2
ϕ∂
∂
=
R
FNxx , 2
2
x
FN
∂
∂
=ϕϕ ,
ϕ∂∂
∂
−=ϕ xR
FNx
2
are normal and tangent forces,
ϕ∂
∂
ν+
∂
∂
= 22
2
2
2
R
w
x
wDM xx ,
∂
∂
ν+
ϕ∂
∂
=ϕϕ 2
2
22
2
x
w
R
wDM ,
( )
ϕ∂∂
∂
ν−=ϕ xR
wDM x
2
1 are bending and twisting moments.
Boundary conditions of the cylindrical shell are as follows:
• two simply supported edges (x = 0 and x = L)
( ) 0, ,0 =ϕ
= Lxxw , 022
2
2
2
=
ϕ∂
∂
ν+
∂
∂
R
w
x
w , (19)
• one simply supported edge ( )0=ϕ
( ) 0, 0 =ϕ
=ϕ
xw , 02
2
22
2
=
∂
∂
ν+
ϕ∂
∂
x
w
R
w , (20)
• one free edge ( β=ϕ )
02
2
22
2
=
∂
∂
ν+
ϕ∂
∂
x
w
R
w , 02
2
22
2
=
∂
∂
+
ϕ∂
∂
ϕ∂
∂
x
w
R
w , 02
2
=
∂
∂
x
F , 0
2
=
ϕ∂∂
∂
x
F . (21)
The system of two differential equations (18) includes two unknown functions
( )xw ,ϕ and ( )xF ,ϕ . Confining the solution only to asymmetric buckling the unknown
functions are assumed in the following forms
( )
L
xmwxw πsin
2
π3sin
2
πsin, 31
ϕ
β
α+
ϕ
β
=ϕ ,
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2006, вип. 3, 116-128
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( )
L
xmEtRfxF πsinπ2sin
2
1πsin, 1
ϕ
β
+
ϕ
β
−=ϕ ,
where 2
2
3 49
41
k
k
ν+
ν+
=α ,
L
Rmk β= , m is natural number, 1w , 1f are function parameters.
The functions satisfy the boundary conditions (19), (20) and (21), while the
equations of stability (18) are not met. The Bubnov-Galerkin method enabled formula-
ting two orthogonal conditions
0πsinπ2sin
2
1πsin
0 0
2
2
4 =ϕ
ϕ
β
+
ϕ
β
∂
∂
−∇∫∫
β
dxd
L
xm
x
w
R
EtF
L
,
0πsin
2
π3sin
2
πsin1
3
0 0
2
2
2
2
4 =ϕ
ϕ
β
α+
ϕ
β
∂
∂
+
∂
∂
+∇∫∫
β
dxd
L
xm
x
wN
x
F
R
wD
L
o
x .
This, upon integrating, some simple transformations, and consideration of the
tN o
x
o
x =σ expression, enabled to express the critical stress in the form
( ) ( )
( )
+
=
σ
kC
kCkC
E k
o
KRx
3
21, min ,
where
( ) ( ) ( ) ( )
22
222
3
22
21
π4941
1192
1
β
+α++
ν−
=
R
tkkkC ,
( )
22
342
2
2 π7
91
π15
32
51620
4
β
α+
++
=
kk
kkC ,
( ) ( )2
3
2
3 1 α+= kkC are coefficients for the case const== o
xxx NN .
Numerical study of the expression shows two local minima. The first one is rela-
ted to classical shell buckling, being compatible with the value resulting from the
expression (17). The other is related to local buckling of the free shell edge that is
considerably smaller than the first classical one. Calculation shows that in this case the
value of the angle β only slightly affects the level of critical stress. Consideration of
the Lorenz, Timoshenko and Southwell formula (17) enabled to propose the following
form of the critical stress of an open cylindrical shell subject to axial compression
( )const== o
xxx NN
( ) R
tE
c
s
KR 21
)1(
13 ν−
α=σ , where
β
−=α
π
0146,01
11,8
1
1c , π
2
π
≤β≤ . (22)
Krzysztof Magnucki
Elastic Buckling of Selected Flanges of Cold-Formed Thin-Walled Beams
126
It should be noticed that buckling of an open cylindrical shell is of local cha-
racter and concentrates at its free edge. Hence, critical stress of an open cylindrical
shell subject to axial compression with the force varying at its length should be assu-
med to be equal to two above described cases with lengthwise constant force.
Moreover, buckling of the cylindrical shells was numerically studied with FEM.
The following data have been adopted: t = 1 mm, r = 302,6 mm, π6,5π3,2π,2π=β ,
ν = 0,3, MPa1005,2 5⋅=E . Comparison of the values of critical stress obtained from
analytical and FEM-numerical solutions indicates that the difference between them does
not exceed 6,5%.
Conclusions. The buckling problems of flat flanges of cold-formed thin-walled beams
are extensively investigated and described in many publications. Basic models of these
flanges are isotropic or orthotropic rectangular plates under longitudinal compression.
Nevertheless, the buckling problem of axially compressed cylindrical panel with three
simply supported edges and one edge free were only weakly recognized. The critical
stresses of thin-walled elements with free edges are significantly smaller than the criti-
cal stresses of simply supported elements.
References
[1] Timoshenko S. P., Gere J. M. Theory of elastic stability. — New York: McGraw-Hill, 1961.
[2] Volmir A. S. Stability of deformation systems. — M.: Fiz-Mat-Lit, 1967.
[3] Bradford M. A., Ge X. P. Elastic distortional buckling of continuous I-beams // Journal
of Constructional Steel Research. — 1997. — 41. — P. 249-266.
[4] Bradford M. A. Inelastic buckling of I-beams with continuous elastic tension flange rest-
raint // Journal of Constructional Steel Research. — 1998. — 48. — P. 63-77.
[5] Put B. M., Pi Y.-L., Trahair N. S. Lateral buckling tests on cold-formed channel beams //
Journal of Structural Engineering. —125, № 5. — P. 532-539.
[6] Davies J. M. Recent research advances in cold-formed steel structures // Journal of
Constructional Steel Research. — 2000. — 55. — P. 267-288.
[7] Pi Y-L., Trahair N. S. Distortion and warping at beam supports // Journal of Structural
Engineering. — 2000. — 126 (11). — P. 1279-1287.
[8] Magnucki K., Monczak T. Optimum shape of open cross section of thin-walled beam //
Engineering Optimization. — 2000. — 32. — P. 335-351.
[9] Rasmussen K. J. R. Bifurcation experiments on locally buckled Z-section columns //
Proc. of Third International Conference on Thin-Walled Structures. Zaras et al (Eds). —
Elsevier, 2001 — P. 217-224.
[10] Magnucki K. Optimization of open cross section of the thin-walled beam with flat web
and circular flange // Thin-Walled Structures. — 2002. — 40 (3). — P. 297-310.
[11] Hancock G. J. Cold-formed steel structures // Journal of Constructional Steel Research. —
2003. — 59(4). — P. 473-487.
[12] Mohri F., Brouki A., Roth J. C. Theoretical and numerical stability analysis of unrestrai-
ned, mono-symmetric thin-walled beams // Journal of Constructional Steel Research. —
2003. — 59. — P. 63-90.
ISSN 1816-1545 Фізико-математичне моделювання та інформаційні технології
2006, вип. 3, 116-128
127
[13] Corte G. D., Fiorino L., Landolfo R., De Martino A. Numerical modeling of thin-walled
cold-formed steel C-sections in bending, Proceedings of Fourth International Conference
on Coupled Instabilities in Metal Structures // CIMS’04. — Rome, 2004. — P. 153-162.
[14] Dinis P. B., Camotim D., Silvestre N. Generalised beam theory to analyse the buckling
behaviour of the thin-walled steel members with «branched» cross-sections // Proceedings
of Fourth International Conference on Thin-Walled Structures. — Loughborough UK,
2004. — P. 819-826.
[15] Stasiewicz P., Magnucki K., Lewiński J., Kasprzak J. Local buckling of a bent flange of a
thin-walled beam // Proc. in Applied Mathematics and Mechanics — PAMM, 2004. —
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[16] Trahair N. S. and Hancock G. J. Steel member strength by inelastic lateral buckling //
Journal of Structural Engineering. — 2004. — 130 (1). — P. 64–69.
[17] Magnucki K., Szyc W., Stasiewicz P. Stress state and elastic buckling of a thin-walled
beam with monosymmetrical open cross section // Thin-Walled Structures. — 2004. —
42 (1). — P. 25-38.
[18] Magnucka-Blandzi E., Magnucki K. Elastic buckling of an axially compressed open cir-
cular cylindrical shell // Proc. in Applied Mathematics and Mechanics. — PAMM, 2004. —
4. — P. 546-547.
[19] Magnucki K. Lower critical stress analysis of axially compressed cylindrical shells //
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Paper 52. — P. 1-10.
[20] Magnucki K., Ostwald M. Optimal design of selected open cross sections of cold-formed
thin-walled beams // Publishing House of Poznan University of Technology. — 2005.
[21] Magnucki K., Maćkiewicz M. Elastic buckling of an axially compressed cylindrical panel
with three edges simply supported and one edge free // Thin-Walled Structures (in re-
view, 2005).
[22] Magnucki K., Maćkiewicz M. Optimal design of a mono-symmetrical open cross section
of cold-formed beam with cosinusoidally corrugated flange // Thin-Walled Structures (in
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Пружне випучування вибраних профілів
холоднокатаних тонкостінних балок
Кшиштоф Магнуцкі
У роботі розглянуто п’ять різних профілів холоднокатаних тонкостінних балок. Сфор-
мульовано та досліджено математичні моделі для кожного з п’яти профілів. У представлених
моделях використано теорію пружної стійкості пластин та пологих циліндричних оболонок.
Визначено критичні напруження для кожного типу профілю. Проаналізовано результати
аналітичних досліджень та проведено їх порівняння із результатами числових досліджень
методом скінченних елементів, а також експериментальними даними. Отримані у роботі
формули для критичних напружень можуть бути використані на практиці.
Krzysztof Magnucki
Elastic Buckling of Selected Flanges of Cold-Formed Thin-Walled Beams
128
Упругое выпучивание некоторых профилей
холоднокатаных тонкостенных балок
Кшиштоф Магнуцки
В роботе рассмотрено пять разных профилей холоднокатаных тонкостенных балок.
Сформулированы и исследованы математические модели для каждого из пяти профилей.
В представленных моделях использовано теорию упругой устойчивости пластин и цилинд-
рических пологих оболочек. Определены критические напряжения для каждого типа профиля.
Проведен анализ аналитических результатов и представлено сравнение их с результатами
численного исследования методом конечных элементов, а также экспериментальными
данными. Представленные формулы для критических напряжений могут быть использо-
ваны в практике.
Отримано 18.10.05
|
| id | nasplib_isofts_kiev_ua-123456789-20980 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1816-1545 |
| language | English |
| last_indexed | 2025-12-07T15:18:49Z |
| publishDate | 2006 |
| publisher | Центр математичного моделювання Інституту прикладних проблем механіки і математики ім. Я.С. Підстригача НАН України |
| record_format | dspace |
| spelling | Magnucki, K. 2011-06-13T19:05:19Z 2011-06-13T19:05:19Z 2006 Elastic buckling of selected flanges of cold-formed thin-walled beams / K. Magnucki // Фіз.-мат. моделювання та інформ. технології. — 2006. — Вип. 3. — С. 116-128. — Бібліогр.: 24 назв. — англ. 1816-1545 https://nasplib.isofts.kiev.ua/handle/123456789/20980 539.3 The paper is devoted to five different flanges of cold-formed thin-walled beams. Mathematical models of each of the flanges are formulated and solved. The theory of elastic stability of plates and cylindrical shallow shells is applied for this purpose. Critical stress for each flange of the beam is determined. Results of analytical solutions are discussed and compared with numerical (FEM) and experimental investigations. The formulas of critical stresses may be used in practical applications. У роботі розглянуто п’ять різних профілів холоднокатаних тонкостінних балок. Сформульовано та досліджено математичні моделі для кожного з п’яти профілів. У представлених моделях використано теорію пружної стійкості пластин та пологих циліндричних оболонок. Визначено критичні напруження для кожного типу профілю. Проаналізовано результати аналітичних досліджень та проведено їх порівняння із результатами числових досліджень методом скінченних елементів, а також експериментальними даними. Отримані у роботі формули для критичних напружень можуть бути використані на практиці. В роботе рассмотрено пять разных профилей холоднокатаных тонкостенных балок. Сформулированы и исследованы математические модели для каждого из пяти профилей. В представленных моделях использовано теорию упругой устойчивости пластин и цилиндрических пологих оболочек. Определены критические напряжения для каждого типа профиля. Проведен анализ аналитических результатов и представлено сравнение их с результатами численного исследования методом конечных элементов, а также экспериментальными данными. Представленные формулы для критических напряжений могут быть использованы в практике. en Центр математичного моделювання Інституту прикладних проблем механіки і математики ім. Я.С. Підстригача НАН України Фізико-математичне моделювання та інформаційні технології Elastic buckling of selected flanges of cold-formed thin-walled beams Пружне випучування вибраних профілів холоднокатаних тонкостінних балок Упругое выпучивание некоторых профилей холоднокатаных тонкостенных балок Article published earlier |
| spellingShingle | Elastic buckling of selected flanges of cold-formed thin-walled beams Magnucki, K. |
| title | Elastic buckling of selected flanges of cold-formed thin-walled beams |
| title_alt | Пружне випучування вибраних профілів холоднокатаних тонкостінних балок Упругое выпучивание некоторых профилей холоднокатаных тонкостенных балок |
| title_full | Elastic buckling of selected flanges of cold-formed thin-walled beams |
| title_fullStr | Elastic buckling of selected flanges of cold-formed thin-walled beams |
| title_full_unstemmed | Elastic buckling of selected flanges of cold-formed thin-walled beams |
| title_short | Elastic buckling of selected flanges of cold-formed thin-walled beams |
| title_sort | elastic buckling of selected flanges of cold-formed thin-walled beams |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/20980 |
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