- Email: [email protected]
Stiffness and critical buckling load of perforated sheeting
ARTICLE IN PRESS
Thin-Walled Structures 44 (2006) 1223–1230 www.elsevier.com/locate/tws
Stiffness and critical buckling load of perforated sheeting K. Kathagea, Th. Misiekb,, H. Saalb a
Deutsches Institut fu¨r Bautechnik (DIBt), KolonnenstraX e 30L, 10829 Berlin, Germany Versuchsanstalt fu¨r Stahl, Holz und Steine, Universita¨t Karlsruhe (TH), KaiserstraX e 12, 76128 Karlsruhe, Germany
b
Available online 16 February 2007
Abstract This paper deals with selected foundations of the design of perforated trapezoidal sheeting. Based on numerical analysis, graphs on effective stiffness values for perforated sheeting with different arrays of holes are provided. As an outlook on further research, the calculation of the buckling coefficient for a perforated plate under uniform in-plane compression loading and for an infinitely long perforated plate under shear loading is presented. r 2007 Elsevier Ltd. All rights reserved. Keywords: Perforated sheeting; Trapezoidal sheeting
1. Introduction For architectural and noise absorption reasons, trapezoidal sheeting profiles are sometimes provided with perforated webs or even completely perforated. New regulations like prEN 1993-1-3 [1] cite equations for consideration of the perforation of the sheeting on the bending and the membrane stiffness. Unfortunately, the cited equations derived by Schardt and Bollinger [2] are only valid for sheeting with a triangular array of the holes (Figs. 1 and 2(a)). For quadratic arrays of the holes, the orthotropy of the sheet stiffness has to be taken into account as well.
are subjected to torsion caused by the twisting moments. Torsion in turn does not affect the deformation behaviour under in-plane loading. As a consequence, to derive effective mechanical properties or an effective stiffness of perforated sheeting under in-plane loading, the parameters d for the diameter of the holes and c for the distance between the centres of the holes are sufficient, whereas for perforated sheeting under transverse loading (bending), the thickness t has to be considered, as well. For example, the effective membrane stiffness can be described as Dij;p ¼ f ðc; dÞ,
(1)
and the bending stiffness as 2. Stiffness of perforated sheeting 2.1. Deformation behaviour of perforated sheeting In addition to the array of holes, the type of loading has to be considered as well in order to obtain effective crosssection values, effective stiffness values or fictive material properties for perforated sheeting. For sheeting under inplane loading these values differ from the values obtained for transverse loading. This is a result of the behaviour of the ligaments, which under bending of the plate continuum Corresponding author. Fax: +49 721 608 4078.
E-mail address: [email protected] (Th. Misiek). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.01.009
K ij;p ¼ f ðc; d; tÞ.
(2)
The number of parameters can be reduced by one, i.e. the effective membrane stiffness of perforated sheeting is a function of the type d Dij;p ¼ f , (3) c if d is normalised with the distance c. When this stiffness is normalised with the stiffness of the unperforated sheeting a non-dimensional expression follows: Dij;p d d ij;p ¼ ¼g . (4) c Dij
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
1224
c c0
1.0
c c0
dij, kij
d
0.0 0.0
0.25
0.5
ligament Fig. 1. Nomenclature.
a
0.75
1.0
d/c Fig. 3. Influence of the parameter d/c on the effective stiffness dij and kij (three characteristic trends for different parameters dij and kij).
c
b 60°
c
c
c
d
c
d
kij
c
d
c Fig. 2. Arrays of holes: triangular (a), quadratic with 01 orientation (b) and quadratic with 451 orientation (c).
On the other hand, the effective bending stiffness is a function of the form d t ; , K ij;p ¼ f (5) c c or kij;p
K ij;p d t ¼ ¼g ; . c c K ij
0.1
1
10
t/c Fig. 4. Influence of the parameter t/c on the effective stiffness kij (schematic).
the ratio can be written in the form GI T 12 1 t a 4; 6 ¼ . 2ð1 þ nÞ 3 5c0 EI
(9)
For t/c0X1 follows (6)
A further reduction of the number of parameters is not possible, despite the fact that this was achieved in several publications. In these cases, particular attention has to be paid to the scope and field of application of the reported results. Figs. 3 and 4 show schematically the influence of the parameters d/c and t/c on the stiffness of the sheeting. Introducing c0 ¼ c–d, the influence of the thickness t becomes evident if the ratio of the torsional stiffness of a beam element with the height t and the width c0 to its bending stiffness is plotted versus the parameter t/c0. For t/c0p1 follows GI T E=ð2ð1 þ nÞÞa c0 t3 12 a. ¼ ¼ 3 2ð1 þ nÞ EI Eððc0 t Þ=12Þ
0.01
(7)
The ratio only depends on the parameter a, which is a function of the aspect ratio of the beam. Approximating a for t/c0 p 1 by 1 t a , (8) 3 5c0
c 2 c 2 GI T ðE=ð2 ð1 þ nÞÞÞ a c30 t 12 0 0 ¼ ¼ 4:6 a , a 3 EI t t Eððc0 t Þ=12Þ 2 ð1 þ nÞ (10)
and with the approximation 1 c0 a , 3 5t
(11)
follows
c 2 GI T 12 1 c0 c0 2 0 a ¼ 4:6 . 2ð 1 þ n Þ 3 5t EI t t
(12)
Plotted versus t/c0 (Fig. 5), the resulting graph is affine to the one obtained for the influence of the parameter t/c or t/ c0 on the effective bending stiffness of the perforated plate. Eqs. (7)–(12) and Fig. 5 can be interpreted as follows: For small ratios t/c0, which means thin sheeting with small bending stiffness, due to the large width of the ligament beam element with a high torsional stiffness, the effective bending stiffness is only a result of the reduction of the cross-sectional area and the associated reduction in bending stiffness. There is no or only little twisting of the
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
ligament beams. With increasing ratio t/c0, the torsional stiffness decreases compared with the bending stiffness. Therefore, the influence of the twisting of the ligaments on the effective bending stiffness increases. With any further increase of the ratio t/c0, the torsional stiffness increases again and compensates the further decrease of the bending stiffness. At this point, the effective bending stiffness remains on a stable but low level. For the membrane stiffness, the model of a framework can be used for the description of the behaviour perpendicular to the main loading direction which—for sheeting under in-plane loading—strongly depends on the array of the holes. By assuming an effective Poisson’s ratio, two basic behaviours can be identified. For a triangular array or a quadratic array which is orientated under 451 to the main loading direction, the effective Poisson’s ration increases with the ratio d/c, whereas for a quadratic array which is orientated under 01 to the main loading direction, the effective Poisson’s ration decreases (Fig. 6). The bars orientated with 301, 601 (triangular array) or 451 (quadratic array orientated with 451) to the main loading direction subjected to an adjustment by turning into the loading direction, which decreases with increasing ratio d/c because of the reduction of the stiffness of the nodes. For a quadratic array orientated with 01 to the main
1225
loading direction, the ligament bars perpendicular to the loading direction experience decreasing deformation with increasing d/c. For larger values of d/c, the influence of the deformation of these ligament bars tends to be negligible because they behave as a non deformable solid. The Poisson’s ratio np tends to zero. 2.2. Theoretical background and numerical investigations For the orthotropic plate under transverse loading, the following bending stiffness matrix [3] results: 2 3 2 3 2 3 K 11;p K 12;p 0 m11 k11 6m 7 6K 7 6 k 7 (13) 22 5 4 22 5 ¼ 4 12;p K 22;p 0 5 4 0 0 K 44;p m12 2k12 with K 11;p ¼
E1 t3 E t3 ¼ k11 K 11 ¼ k11 , 1 n2 12 1 n12 n21 12
(14)
K 22;p ¼
E2 t3 E t3 ¼ k22 K 22 ¼ k22 , 1 n2 12 1 n12 n21 12
(15)
K 12;p ¼
n12 E 2 t3 nE t3 ¼ k12 K 12 ¼ k12 1 n2 12 1 n12 n21 12
(16)
and 0.35
K 44;p ¼ G 12
2 (1+ν) G IT 12 E I
0.3 0.25
(17)
0.2
In the present case with identical values c for both main directions, one follows
0.15
K 11;p ¼ K 22;p .
0.1 0.05 0 0.001
0.01
0.1
1
10
100
For the membrane stiffness 3 2 D11;p D12;p 0 n11 6n 7 6D 4 22 5 ¼ 4 12;p D22;p 0
1.0
(a)
0.5 0.3 0.0 0.5
0.75
0
matrix follows 3 2 3 11 7 6 7 5 4 22 5 D44;p 212
1.0
d/c Fig. 6. Effective Poisson’s ratio for triangular array or quadratic array with 451 orientation (a) and a quadratic array under 01 orientation (b).
(19)
with D11;p ¼
E1 E t ¼ d 11 D11 ¼ d 11 t, 1 n2 1 n12 n21
(20)
D22;p ¼
E2 E t ¼ d 22 D22 ¼ d 22 t, 1 n2 1 n12 n21
(21)
D12;p ¼
n21 E 1 nE t ¼ d 12 D12 ¼ d 12 t, 1 n2 1 n12 n21
(22)
D44;p ¼ G 12 t ¼ d 44 D44 ¼ d 44 Gt ¼ d 44
(b) 0.25
0
n12
Fig. 5. Ratio of the bending stiffness to the torsional stiffness of ligament beam depending from the aspect ratio of the ligament beam.
0.0
(18)
2
t/c0
νp
t3 t3 E t3 ¼ k44 K 44 ¼ k44 G ¼ k44 . 2 ð1 þ nÞ 12 12 12
E t, 2 ð1 þ nÞ
(23)
and also D11;p ¼ D22;p .
(24)
The values kij and dij were determined in FEMcalculations according to [2] with the help of periodically
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
1226
recurring basic elements. The deflection or rotation of the basic elements under unit loading were used for calculating its effective stiffness. In addition to [2], quadratic arrays are considered as well. Also, the stiffness parameters K12,p und K44,p for the triangular array are determined. For selected values d/c the effective bending stiffness was also calculated for evaluation purposes by using a perforated plate under transverse load spanning over a distance which is a large multiple of that of the holes. Fig. 7 shows a good congruence for the different ways of determining the stiffness. The possibility of cross-checking the results for the different orientations of the loading direction by rotation of the coordinate system was used as well. For avoiding the need to determine parameters kij for different parameters d/c and t/c, the relation described in Eqs. (9) and (12) for the ratio of the torsional stiffness to the bending stiffness is used: For a given ratio d/c, only the parameters kij,min t and kij,max t for a minimum sheeting thickness and a maximum sheeting thickness were determined. For interpolation, the following equation can be used: kij
d t ; c c
3. Critical buckling loads 3.1. Buckling of a rectangular plate under uniform uniaxial compression loading According to [4], the critical buckling load of a homogenous orthotropic rectangular plate under uniform
k11 = K11,p/K11
1
m11 κ11
d
c
0.7 0.6
kmint
kmaxt
0.5 0.4 0.3
0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 8. Normalised effective stiffness k11,min t and k11,max t for a triangular array of holes.
1
κ22 60°
0.8
m11
m11 d
t=c0 p1;
(26)
t=c0 41;
where G is based on the ratio of the stiffnesses. The membrane stiffness only depends on d/c, therefore no interpolation between parameters dij for minimum and maximum sheet thickness t is necessary. The values kij,min t, kij,max t and dij obtained from the numerical investigation are plotted in Figs. 8–25.
k11 = K11,p/K11
κ11
0
k12 = K12,p/K12
for
0.8
0.1
with the geometry-function G (
m11
0.2
d ¼ kij c t min d d t þ kij , ð25Þ kij G c max t c min t c
8 t < 3 13 5ct 0 G ¼ : 3 1 c0 c0 2 c 3 5t t
60°
0.9
kmaxt
0.6
c
kmint
0.4 0.2 0 0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
d/c Fig. 9. Normalised effective stiffness k12,min t and k12,max t for a triangular array of holes.
1
0.54
0.9
0.53
0.8
0.52
0.7
0.51
0.6
0.50
0.5
0.49
0.4
0.48
0.3
0.47
0.2
0.46
0.1
0.45 0.001
0 0.1 t/c
0.1
-0.2
0.55
0.01
κ22
1
10
Fig. 7. Interpolation for different values t/c.
beam, rotation at the support q ϕ beam, deflection q w basic element, rotation M ϕ basic element, deflection M w approximation Γ( t ) c
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
m12
0.8
κ12
0.6
0.9
κ12
0.8
d
kmint
0.7
1
60° m12 c
d44 = D44,p/D44
k44 = K44,p/K44
1 0.9
kmaxt
0.5
1227
0.4 0.3
60° n12 ε12
n12 ε12
d
c
0.7 0.6 0.5 0.4 0.3
0.2
0.2
0.1
0.1 0
0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
0
1
Fig. 10. Normalised effective stiffness k44,min t and k44,max t for a triangular array of holes.
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 13. Normalised effective stiffness d44 for a triangular array of holes.
1 0.9
60°
0.9
n11
0.8
ε11
d
n11
0.8
ε11
0.7
k11 = K11,p/K11
d11 = D11,p/D11
1
c
0.7 0.6 0.5 0.4
0.4 0.3 0.2 0.1
0.1
0 0
0 0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
n11
n11
k12 = K12.p/K12
ε22
0.5
0.5 d/c
0.6
0.7
0.8
0.4 0.3 0.2
0.9
1
κ22 d
m11
0.8
c
0.6
0.4
0.9
d
0.7
0.3
1
ε22 60°
0.8
0.2
Fig. 14. Normalised effective stiffness k11,min t and k11,max t for a quadratic array of holes orientated with 01.
1 0.9
0.1
1
Fig. 11. Normalised effective stiffness d11 for a triangular array of holes.
d12 = D12,p/D12
kmaxt
0.5
0.2
0.1
m11 κ11
c
kmint
0.6
0.3
0
d
m11 κ11
0.7
kmaxt
0.6
m11
κ22
kmint
0.5 0.4 0.3 0.2
0.1
0.1
0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 12. Normalised effective stiffness d12 for a triangular array of holes.
uniaxial compression loading (Fig. 26) can be calculated by the following equation: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 K 11 K 22 , (27) ski ¼ ks 2 t b where the buckling coefficient is defined as 2 a¯ 2 m þ q 2 þ pB ks ¼ m a¯ 2
0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 15. Normalised effective stiffness k12,min t and k12,max t for a quadratic array of holes orientated with 01.
with the adjusted aspect ratio rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi a 4 K 22 4 K 22 ¼ , a¯ ¼ a K 11 b K 11
(29)
and the stiffness coefficient (28)
K 12 þ 2K 44 B ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . K 11 K 22
(30)
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
1228
m12
0.8
κ12
0.7
kmaxt
0.6
1
d
m12 κ12
kmint
0.8
c
0.5 0.4 0.3
c
0.6 0.5 0.4 0.3 0.2
0.1
0.1 0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 16. Normalised effective stiffness k44,min t and k44,max t for a quadratic array of holes orientated with 01.
0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 19. Normalised effective stiffness d44 for a quadratic array of holes orientated with 01.
1 n11
0.8
ε11
d
n11
0.9
ε11
0.8 k11 = K11,p/K11
c
0.7 0.6 0.5 0.4 0.3
0.7
0.7
0.8
m11 κ11
d
kmint
0.6
kmaxt
0.5 0.4 0.3
0.2
0.2
0.1
0.1
0
m11 κ11
c
0.9
c
1
0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 17. Normalised effective stiffness d11 for a quadratic array of holes orientated with 01.
0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
1 ε22 d
n11
0.8
kmaxt k12 = K12,p/K12
ε11
0.5 0.4 0.3
m11
d c
c
0.6
κ22
0.8
n11
0.7
1
Fig. 20. Normalised effective stiffness k11,min t and k11,max t for a quadratic array of holes orientated with 451.
1 0.9
0.9
c
d11 = D11,p/D11
n12 ε12
0.7
0.2 0
d12 = D12,p/D12
d
n12 ε12
0.9 d44 = D44,p/D44
k44 = K44,p/K44
1 0.9
0.6 kmint
κ22
0.4 0.2
0.2 0
0.1
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.2 d/c
d/c Fig. 18. Normalised effective stiffness d12 for a quadratic array of holes orientated with 01.
Fig. 21. Normalised effective stiffness k12,min t and k12,max t for a quadratic array of holes orientated with 451.
For simply supported edges, p ¼ 2,0 and q ¼ 1,0 applies. With the nomenclature from above and the assumptions K11 ¼ K22 and a ¼ b (aspect ratio a ¼ 1 for obtaining the minimal buckling coefficients), the critical buckling stress is
where the buckling coefficient is given by 2 a¯ 2 m þ q þ 2 ðk44 þ 0:3 ðk12 k44 ÞÞ ks;p ¼ k11 m2 a¯ 2
ski;p ¼ ks;p se ,
(31)
or ks;p ¼ 2 ðk11 þ k44 þ 0:3 ðk12 k44 ÞÞ
(32)
(33)
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
1 κ12
d
m12
0.9
κ12
0.8
0.6
d44 = D44,p/D44
kmint
0.7 kmaxt
0.5 0.4 0.3
n12 ε12
0.8
0.9
n12 ε12
0.7 0.6 0.5 0.4 0.3
0.2
0.2
0.1
0.1 0
0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
0
1
Fig. 22. Normalised effective stiffness k44,min t and k44,max t for a quadratic array of holes orientated with 01.
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
ε11
ε11
b
0.8
x
n11
d
c
n11
c
0.9
1
Fig. 25. Normalised effective stiffness d44 for a quadratic array of holes orientated with 451.
1
d11 = D11,p/D11
d c
0.8
c
m12
c
0.9
c
k44 = K44,p/K44
1
1229
0.7 0.6 0.5
σ
a
0.4 0.3
y
0.2 Fig. 26. Plate under in-plane loading.
0.1 0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1 4.00
Fig. 23. Normalised effective stiffness d11 for a quadratic array of holes orientated with 451.
60°
3.50
d
3.00
c
kmint
kσ,p
2.50 1 n11
0.7
1.00
n11
d c
0.50
c
0.8 d12 = D12,p/D12
1.50
ε22
0.9
0.6
0.00
ε22
0.5
kmaxt
2.00
0
0.1
0.2
0.4
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
0.3 Fig. 27. Effective buckling coefficient ks,p,min t and ks,p,max t for a triangular array of holes.
0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 24. Normalised effective stiffness d12 for a quadratic array of holes orientated with 451.
with the Euler-stress se for the unperforated plate. The chosen method of normalisation with respect to the unperforated plate is just for convenience since all the parameters affected by the perforation are merged to one parameter. The effective buckling coefficient can be plotted
as a function of d/c (Figs. 27–29). For interpolation for different values t/c, the Eqs. (25) and (26) have to be used. For the isotropic case of a triangular array of holes, the effective buckling coefficient can be calculated in a simplified way as follows: ks;p ¼ ks k11 ¼ 4k11 .
(34)
The Eqs. (27)–(34) are confirmed by the numerical results from [5].
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
1230
the buckling coefficients for an infinitely long plate with a stiffness coefficient zp1 can be calculated by
4.00 d
3.50 3.00 2.50 kσ,p
kt;p ¼ k11 kt ¼ k11 ð3:293 þ 2:286 B 0:24 B2 Þ,
c
kmaxt
2.00
and for an infinitely long plate with a stiffness coefficient z41 by pffiffiffi 0:874 0:283 (39) kt;p ¼ k11 kt ¼ k11 B 4:757 þ 2 4 . B B
1.50 1.00 0.50 0.00 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 28. Effective buckling coefficient ks,p,min t and ks,p,max t for a quadratic array of holes orientated with 01.
Unfortunately, interpolation between different values of kt,p using Eq. (25) is not possible, so interpolation has to be done for the different values of kij. But for simplification, instead of using Eq. (30), the following equation can be used for calculating the stiffness coefficient: B¼
4.00 3.50
k44 þ 0:3 ðk12 k44 Þ . k11
(40)
d
4. Conclusions
c
c
3.00
kmint
2.50 kσ,p
(38)
kmint
kmaxt
2.00 1.50 1.00 0.50 0.00 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 29. Effective buckling coefficient ks,p,min t and ks,p,max t for a quadratic array of holes orientated with 451.
This paper presents some foundations of a procedure to calculate the load-bearing capacity of perforated trapezoidal sheeting. Based on numerical investigations and some additional basic considerations regarding the deformation behaviour of perforated sheeting, effective stiffness values are derived. Also, the buckling coefficients ks,p for perforated plates under uniform uniaxial compression-loading and kt,p for infinitely long perforated plates under shear loading are derived as a base for the calculation of the effective width of a cross-section according to [1]. Further research is currently being conducted on buckling under different loading conditions and postbuckling behaviour and will be published in the near future.
3.2. Buckling of a rectangular plate under shear loading According to [4,6], the critical shear buckling stress of an infinitely long orthotropic plate with a stiffness coefficient zp1,0 can be calculated by tki;p ¼
p2 Ek11 t 2 ð3293 þ 2286 B 0:24 B2 Þ, 12 ð1 n2 Þ b
and with a stiffness coefficient z41 by p2 Ek11 t 2 pffiffiffi 0:874 0:283 B 4:757 þ 2 4 . tki;p ¼ B B 12ð1 n2 Þ b
(35)
(36)
For using the common expression tki;p ¼ kt;p se
(37)
References [1] DIN V ENV 1993-1-3:2002-09: Eurocode 3: Design of steel structures—Teil 1-3: general rules—supplementary rules for coldformed thin gauge members and sheeting. [2] Schardt R, Bollinger K. Zur Berechnung regelma¨Xig gelochter Scheiben und Platten, Der Bauingenieur 1981;56:S227–39. [3] Altenbach H, Altenbach J, Naumenko K. Ebene Fla¨chentragwerke, Berlin, Heidelberg, 1998. [4] Blomm F, Coffin D. Handbook of thin plate buckling and postbuckling, Boca Raton. 2001. [5] Saal H, Misiek Th. Tragverhalten du¨nnwandiger Bauteile aus perforierten Blechen – Bericht Nr. 061501, Versuchsanstalt fu¨r Stahl, Holz und Steine, 2007, to be published. [6] Seydel E. U¨ber das Ausbeulen von rechteckigen, isotropen oder orthogonal-anisotropen. Platten bei Schubu¨bertragung: IngenieurArchiv 1933;4:S169–91.
Thin-Walled Structures 44 (2006) 1223–1230 www.elsevier.com/locate/tws
Stiffness and critical buckling load of perforated sheeting K. Kathagea, Th. Misiekb,, H. Saalb a
Deutsches Institut fu¨r Bautechnik (DIBt), KolonnenstraX e 30L, 10829 Berlin, Germany Versuchsanstalt fu¨r Stahl, Holz und Steine, Universita¨t Karlsruhe (TH), KaiserstraX e 12, 76128 Karlsruhe, Germany
b
Available online 16 February 2007
Abstract This paper deals with selected foundations of the design of perforated trapezoidal sheeting. Based on numerical analysis, graphs on effective stiffness values for perforated sheeting with different arrays of holes are provided. As an outlook on further research, the calculation of the buckling coefficient for a perforated plate under uniform in-plane compression loading and for an infinitely long perforated plate under shear loading is presented. r 2007 Elsevier Ltd. All rights reserved. Keywords: Perforated sheeting; Trapezoidal sheeting
1. Introduction For architectural and noise absorption reasons, trapezoidal sheeting profiles are sometimes provided with perforated webs or even completely perforated. New regulations like prEN 1993-1-3 [1] cite equations for consideration of the perforation of the sheeting on the bending and the membrane stiffness. Unfortunately, the cited equations derived by Schardt and Bollinger [2] are only valid for sheeting with a triangular array of the holes (Figs. 1 and 2(a)). For quadratic arrays of the holes, the orthotropy of the sheet stiffness has to be taken into account as well.
are subjected to torsion caused by the twisting moments. Torsion in turn does not affect the deformation behaviour under in-plane loading. As a consequence, to derive effective mechanical properties or an effective stiffness of perforated sheeting under in-plane loading, the parameters d for the diameter of the holes and c for the distance between the centres of the holes are sufficient, whereas for perforated sheeting under transverse loading (bending), the thickness t has to be considered, as well. For example, the effective membrane stiffness can be described as Dij;p ¼ f ðc; dÞ,
(1)
and the bending stiffness as 2. Stiffness of perforated sheeting 2.1. Deformation behaviour of perforated sheeting In addition to the array of holes, the type of loading has to be considered as well in order to obtain effective crosssection values, effective stiffness values or fictive material properties for perforated sheeting. For sheeting under inplane loading these values differ from the values obtained for transverse loading. This is a result of the behaviour of the ligaments, which under bending of the plate continuum Corresponding author. Fax: +49 721 608 4078.
E-mail address: [email protected] (Th. Misiek). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.01.009
K ij;p ¼ f ðc; d; tÞ.
(2)
The number of parameters can be reduced by one, i.e. the effective membrane stiffness of perforated sheeting is a function of the type d Dij;p ¼ f , (3) c if d is normalised with the distance c. When this stiffness is normalised with the stiffness of the unperforated sheeting a non-dimensional expression follows: Dij;p d d ij;p ¼ ¼g . (4) c Dij
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
1224
c c0
1.0
c c0
dij, kij
d
0.0 0.0
0.25
0.5
ligament Fig. 1. Nomenclature.
a
0.75
1.0
d/c Fig. 3. Influence of the parameter d/c on the effective stiffness dij and kij (three characteristic trends for different parameters dij and kij).
c
b 60°
c
c
c
d
c
d
kij
c
d
c Fig. 2. Arrays of holes: triangular (a), quadratic with 01 orientation (b) and quadratic with 451 orientation (c).
On the other hand, the effective bending stiffness is a function of the form d t ; , K ij;p ¼ f (5) c c or kij;p
K ij;p d t ¼ ¼g ; . c c K ij
0.1
1
10
t/c Fig. 4. Influence of the parameter t/c on the effective stiffness kij (schematic).
the ratio can be written in the form GI T 12 1 t a 4; 6 ¼ . 2ð1 þ nÞ 3 5c0 EI
(9)
For t/c0X1 follows (6)
A further reduction of the number of parameters is not possible, despite the fact that this was achieved in several publications. In these cases, particular attention has to be paid to the scope and field of application of the reported results. Figs. 3 and 4 show schematically the influence of the parameters d/c and t/c on the stiffness of the sheeting. Introducing c0 ¼ c–d, the influence of the thickness t becomes evident if the ratio of the torsional stiffness of a beam element with the height t and the width c0 to its bending stiffness is plotted versus the parameter t/c0. For t/c0p1 follows GI T E=ð2ð1 þ nÞÞa c0 t3 12 a. ¼ ¼ 3 2ð1 þ nÞ EI Eððc0 t Þ=12Þ
0.01
(7)
The ratio only depends on the parameter a, which is a function of the aspect ratio of the beam. Approximating a for t/c0 p 1 by 1 t a , (8) 3 5c0
c 2 c 2 GI T ðE=ð2 ð1 þ nÞÞÞ a c30 t 12 0 0 ¼ ¼ 4:6 a , a 3 EI t t Eððc0 t Þ=12Þ 2 ð1 þ nÞ (10)
and with the approximation 1 c0 a , 3 5t
(11)
follows
c 2 GI T 12 1 c0 c0 2 0 a ¼ 4:6 . 2ð 1 þ n Þ 3 5t EI t t
(12)
Plotted versus t/c0 (Fig. 5), the resulting graph is affine to the one obtained for the influence of the parameter t/c or t/ c0 on the effective bending stiffness of the perforated plate. Eqs. (7)–(12) and Fig. 5 can be interpreted as follows: For small ratios t/c0, which means thin sheeting with small bending stiffness, due to the large width of the ligament beam element with a high torsional stiffness, the effective bending stiffness is only a result of the reduction of the cross-sectional area and the associated reduction in bending stiffness. There is no or only little twisting of the
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
ligament beams. With increasing ratio t/c0, the torsional stiffness decreases compared with the bending stiffness. Therefore, the influence of the twisting of the ligaments on the effective bending stiffness increases. With any further increase of the ratio t/c0, the torsional stiffness increases again and compensates the further decrease of the bending stiffness. At this point, the effective bending stiffness remains on a stable but low level. For the membrane stiffness, the model of a framework can be used for the description of the behaviour perpendicular to the main loading direction which—for sheeting under in-plane loading—strongly depends on the array of the holes. By assuming an effective Poisson’s ratio, two basic behaviours can be identified. For a triangular array or a quadratic array which is orientated under 451 to the main loading direction, the effective Poisson’s ration increases with the ratio d/c, whereas for a quadratic array which is orientated under 01 to the main loading direction, the effective Poisson’s ration decreases (Fig. 6). The bars orientated with 301, 601 (triangular array) or 451 (quadratic array orientated with 451) to the main loading direction subjected to an adjustment by turning into the loading direction, which decreases with increasing ratio d/c because of the reduction of the stiffness of the nodes. For a quadratic array orientated with 01 to the main
1225
loading direction, the ligament bars perpendicular to the loading direction experience decreasing deformation with increasing d/c. For larger values of d/c, the influence of the deformation of these ligament bars tends to be negligible because they behave as a non deformable solid. The Poisson’s ratio np tends to zero. 2.2. Theoretical background and numerical investigations For the orthotropic plate under transverse loading, the following bending stiffness matrix [3] results: 2 3 2 3 2 3 K 11;p K 12;p 0 m11 k11 6m 7 6K 7 6 k 7 (13) 22 5 4 22 5 ¼ 4 12;p K 22;p 0 5 4 0 0 K 44;p m12 2k12 with K 11;p ¼
E1 t3 E t3 ¼ k11 K 11 ¼ k11 , 1 n2 12 1 n12 n21 12
(14)
K 22;p ¼
E2 t3 E t3 ¼ k22 K 22 ¼ k22 , 1 n2 12 1 n12 n21 12
(15)
K 12;p ¼
n12 E 2 t3 nE t3 ¼ k12 K 12 ¼ k12 1 n2 12 1 n12 n21 12
(16)
and 0.35
K 44;p ¼ G 12
2 (1+ν) G IT 12 E I
0.3 0.25
(17)
0.2
In the present case with identical values c for both main directions, one follows
0.15
K 11;p ¼ K 22;p .
0.1 0.05 0 0.001
0.01
0.1
1
10
100
For the membrane stiffness 3 2 D11;p D12;p 0 n11 6n 7 6D 4 22 5 ¼ 4 12;p D22;p 0
1.0
(a)
0.5 0.3 0.0 0.5
0.75
0
matrix follows 3 2 3 11 7 6 7 5 4 22 5 D44;p 212
1.0
d/c Fig. 6. Effective Poisson’s ratio for triangular array or quadratic array with 451 orientation (a) and a quadratic array under 01 orientation (b).
(19)
with D11;p ¼
E1 E t ¼ d 11 D11 ¼ d 11 t, 1 n2 1 n12 n21
(20)
D22;p ¼
E2 E t ¼ d 22 D22 ¼ d 22 t, 1 n2 1 n12 n21
(21)
D12;p ¼
n21 E 1 nE t ¼ d 12 D12 ¼ d 12 t, 1 n2 1 n12 n21
(22)
D44;p ¼ G 12 t ¼ d 44 D44 ¼ d 44 Gt ¼ d 44
(b) 0.25
0
n12
Fig. 5. Ratio of the bending stiffness to the torsional stiffness of ligament beam depending from the aspect ratio of the ligament beam.
0.0
(18)
2
t/c0
νp
t3 t3 E t3 ¼ k44 K 44 ¼ k44 G ¼ k44 . 2 ð1 þ nÞ 12 12 12
E t, 2 ð1 þ nÞ
(23)
and also D11;p ¼ D22;p .
(24)
The values kij and dij were determined in FEMcalculations according to [2] with the help of periodically
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
1226
recurring basic elements. The deflection or rotation of the basic elements under unit loading were used for calculating its effective stiffness. In addition to [2], quadratic arrays are considered as well. Also, the stiffness parameters K12,p und K44,p for the triangular array are determined. For selected values d/c the effective bending stiffness was also calculated for evaluation purposes by using a perforated plate under transverse load spanning over a distance which is a large multiple of that of the holes. Fig. 7 shows a good congruence for the different ways of determining the stiffness. The possibility of cross-checking the results for the different orientations of the loading direction by rotation of the coordinate system was used as well. For avoiding the need to determine parameters kij for different parameters d/c and t/c, the relation described in Eqs. (9) and (12) for the ratio of the torsional stiffness to the bending stiffness is used: For a given ratio d/c, only the parameters kij,min t and kij,max t for a minimum sheeting thickness and a maximum sheeting thickness were determined. For interpolation, the following equation can be used: kij
d t ; c c
3. Critical buckling loads 3.1. Buckling of a rectangular plate under uniform uniaxial compression loading According to [4], the critical buckling load of a homogenous orthotropic rectangular plate under uniform
k11 = K11,p/K11
1
m11 κ11
d
c
0.7 0.6
kmint
kmaxt
0.5 0.4 0.3
0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 8. Normalised effective stiffness k11,min t and k11,max t for a triangular array of holes.
1
κ22 60°
0.8
m11
m11 d
t=c0 p1;
(26)
t=c0 41;
where G is based on the ratio of the stiffnesses. The membrane stiffness only depends on d/c, therefore no interpolation between parameters dij for minimum and maximum sheet thickness t is necessary. The values kij,min t, kij,max t and dij obtained from the numerical investigation are plotted in Figs. 8–25.
k11 = K11,p/K11
κ11
0
k12 = K12,p/K12
for
0.8
0.1
with the geometry-function G (
m11
0.2
d ¼ kij c t min d d t þ kij , ð25Þ kij G c max t c min t c
8 t < 3 13 5ct 0 G ¼ : 3 1 c0 c0 2 c 3 5t t
60°
0.9
kmaxt
0.6
c
kmint
0.4 0.2 0 0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
d/c Fig. 9. Normalised effective stiffness k12,min t and k12,max t for a triangular array of holes.
1
0.54
0.9
0.53
0.8
0.52
0.7
0.51
0.6
0.50
0.5
0.49
0.4
0.48
0.3
0.47
0.2
0.46
0.1
0.45 0.001
0 0.1 t/c
0.1
-0.2
0.55
0.01
κ22
1
10
Fig. 7. Interpolation for different values t/c.
beam, rotation at the support q ϕ beam, deflection q w basic element, rotation M ϕ basic element, deflection M w approximation Γ( t ) c
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
m12
0.8
κ12
0.6
0.9
κ12
0.8
d
kmint
0.7
1
60° m12 c
d44 = D44,p/D44
k44 = K44,p/K44
1 0.9
kmaxt
0.5
1227
0.4 0.3
60° n12 ε12
n12 ε12
d
c
0.7 0.6 0.5 0.4 0.3
0.2
0.2
0.1
0.1 0
0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
0
1
Fig. 10. Normalised effective stiffness k44,min t and k44,max t for a triangular array of holes.
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 13. Normalised effective stiffness d44 for a triangular array of holes.
1 0.9
60°
0.9
n11
0.8
ε11
d
n11
0.8
ε11
0.7
k11 = K11,p/K11
d11 = D11,p/D11
1
c
0.7 0.6 0.5 0.4
0.4 0.3 0.2 0.1
0.1
0 0
0 0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
n11
n11
k12 = K12.p/K12
ε22
0.5
0.5 d/c
0.6
0.7
0.8
0.4 0.3 0.2
0.9
1
κ22 d
m11
0.8
c
0.6
0.4
0.9
d
0.7
0.3
1
ε22 60°
0.8
0.2
Fig. 14. Normalised effective stiffness k11,min t and k11,max t for a quadratic array of holes orientated with 01.
1 0.9
0.1
1
Fig. 11. Normalised effective stiffness d11 for a triangular array of holes.
d12 = D12,p/D12
kmaxt
0.5
0.2
0.1
m11 κ11
c
kmint
0.6
0.3
0
d
m11 κ11
0.7
kmaxt
0.6
m11
κ22
kmint
0.5 0.4 0.3 0.2
0.1
0.1
0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 12. Normalised effective stiffness d12 for a triangular array of holes.
uniaxial compression loading (Fig. 26) can be calculated by the following equation: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 K 11 K 22 , (27) ski ¼ ks 2 t b where the buckling coefficient is defined as 2 a¯ 2 m þ q 2 þ pB ks ¼ m a¯ 2
0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 15. Normalised effective stiffness k12,min t and k12,max t for a quadratic array of holes orientated with 01.
with the adjusted aspect ratio rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi a 4 K 22 4 K 22 ¼ , a¯ ¼ a K 11 b K 11
(29)
and the stiffness coefficient (28)
K 12 þ 2K 44 B ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . K 11 K 22
(30)
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
1228
m12
0.8
κ12
0.7
kmaxt
0.6
1
d
m12 κ12
kmint
0.8
c
0.5 0.4 0.3
c
0.6 0.5 0.4 0.3 0.2
0.1
0.1 0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 16. Normalised effective stiffness k44,min t and k44,max t for a quadratic array of holes orientated with 01.
0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 19. Normalised effective stiffness d44 for a quadratic array of holes orientated with 01.
1 n11
0.8
ε11
d
n11
0.9
ε11
0.8 k11 = K11,p/K11
c
0.7 0.6 0.5 0.4 0.3
0.7
0.7
0.8
m11 κ11
d
kmint
0.6
kmaxt
0.5 0.4 0.3
0.2
0.2
0.1
0.1
0
m11 κ11
c
0.9
c
1
0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 17. Normalised effective stiffness d11 for a quadratic array of holes orientated with 01.
0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
1 ε22 d
n11
0.8
kmaxt k12 = K12,p/K12
ε11
0.5 0.4 0.3
m11
d c
c
0.6
κ22
0.8
n11
0.7
1
Fig. 20. Normalised effective stiffness k11,min t and k11,max t for a quadratic array of holes orientated with 451.
1 0.9
0.9
c
d11 = D11,p/D11
n12 ε12
0.7
0.2 0
d12 = D12,p/D12
d
n12 ε12
0.9 d44 = D44,p/D44
k44 = K44,p/K44
1 0.9
0.6 kmint
κ22
0.4 0.2
0.2 0
0.1
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.2 d/c
d/c Fig. 18. Normalised effective stiffness d12 for a quadratic array of holes orientated with 01.
Fig. 21. Normalised effective stiffness k12,min t and k12,max t for a quadratic array of holes orientated with 451.
For simply supported edges, p ¼ 2,0 and q ¼ 1,0 applies. With the nomenclature from above and the assumptions K11 ¼ K22 and a ¼ b (aspect ratio a ¼ 1 for obtaining the minimal buckling coefficients), the critical buckling stress is
where the buckling coefficient is given by 2 a¯ 2 m þ q þ 2 ðk44 þ 0:3 ðk12 k44 ÞÞ ks;p ¼ k11 m2 a¯ 2
ski;p ¼ ks;p se ,
(31)
or ks;p ¼ 2 ðk11 þ k44 þ 0:3 ðk12 k44 ÞÞ
(32)
(33)
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
1 κ12
d
m12
0.9
κ12
0.8
0.6
d44 = D44,p/D44
kmint
0.7 kmaxt
0.5 0.4 0.3
n12 ε12
0.8
0.9
n12 ε12
0.7 0.6 0.5 0.4 0.3
0.2
0.2
0.1
0.1 0
0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
0
1
Fig. 22. Normalised effective stiffness k44,min t and k44,max t for a quadratic array of holes orientated with 01.
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
ε11
ε11
b
0.8
x
n11
d
c
n11
c
0.9
1
Fig. 25. Normalised effective stiffness d44 for a quadratic array of holes orientated with 451.
1
d11 = D11,p/D11
d c
0.8
c
m12
c
0.9
c
k44 = K44,p/K44
1
1229
0.7 0.6 0.5
σ
a
0.4 0.3
y
0.2 Fig. 26. Plate under in-plane loading.
0.1 0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1 4.00
Fig. 23. Normalised effective stiffness d11 for a quadratic array of holes orientated with 451.
60°
3.50
d
3.00
c
kmint
kσ,p
2.50 1 n11
0.7
1.00
n11
d c
0.50
c
0.8 d12 = D12,p/D12
1.50
ε22
0.9
0.6
0.00
ε22
0.5
kmaxt
2.00
0
0.1
0.2
0.4
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
0.3 Fig. 27. Effective buckling coefficient ks,p,min t and ks,p,max t for a triangular array of holes.
0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 24. Normalised effective stiffness d12 for a quadratic array of holes orientated with 451.
with the Euler-stress se for the unperforated plate. The chosen method of normalisation with respect to the unperforated plate is just for convenience since all the parameters affected by the perforation are merged to one parameter. The effective buckling coefficient can be plotted
as a function of d/c (Figs. 27–29). For interpolation for different values t/c, the Eqs. (25) and (26) have to be used. For the isotropic case of a triangular array of holes, the effective buckling coefficient can be calculated in a simplified way as follows: ks;p ¼ ks k11 ¼ 4k11 .
(34)
The Eqs. (27)–(34) are confirmed by the numerical results from [5].
ARTICLE IN PRESS K. Kathage et al. / Thin-Walled Structures 44 (2006) 1223–1230
1230
the buckling coefficients for an infinitely long plate with a stiffness coefficient zp1 can be calculated by
4.00 d
3.50 3.00 2.50 kσ,p
kt;p ¼ k11 kt ¼ k11 ð3:293 þ 2:286 B 0:24 B2 Þ,
c
kmaxt
2.00
and for an infinitely long plate with a stiffness coefficient z41 by pffiffiffi 0:874 0:283 (39) kt;p ¼ k11 kt ¼ k11 B 4:757 þ 2 4 . B B
1.50 1.00 0.50 0.00 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 28. Effective buckling coefficient ks,p,min t and ks,p,max t for a quadratic array of holes orientated with 01.
Unfortunately, interpolation between different values of kt,p using Eq. (25) is not possible, so interpolation has to be done for the different values of kij. But for simplification, instead of using Eq. (30), the following equation can be used for calculating the stiffness coefficient: B¼
4.00 3.50
k44 þ 0:3 ðk12 k44 Þ . k11
(40)
d
4. Conclusions
c
c
3.00
kmint
2.50 kσ,p
(38)
kmint
kmaxt
2.00 1.50 1.00 0.50 0.00 0
0.1
0.2
0.3
0.4
0.5 d/c
0.6
0.7
0.8
0.9
1
Fig. 29. Effective buckling coefficient ks,p,min t and ks,p,max t for a quadratic array of holes orientated with 451.
This paper presents some foundations of a procedure to calculate the load-bearing capacity of perforated trapezoidal sheeting. Based on numerical investigations and some additional basic considerations regarding the deformation behaviour of perforated sheeting, effective stiffness values are derived. Also, the buckling coefficients ks,p for perforated plates under uniform uniaxial compression-loading and kt,p for infinitely long perforated plates under shear loading are derived as a base for the calculation of the effective width of a cross-section according to [1]. Further research is currently being conducted on buckling under different loading conditions and postbuckling behaviour and will be published in the near future.
3.2. Buckling of a rectangular plate under shear loading According to [4,6], the critical shear buckling stress of an infinitely long orthotropic plate with a stiffness coefficient zp1,0 can be calculated by tki;p ¼
p2 Ek11 t 2 ð3293 þ 2286 B 0:24 B2 Þ, 12 ð1 n2 Þ b
and with a stiffness coefficient z41 by p2 Ek11 t 2 pffiffiffi 0:874 0:283 B 4:757 þ 2 4 . tki;p ¼ B B 12ð1 n2 Þ b
(35)
(36)
For using the common expression tki;p ¼ kt;p se
(37)
References [1] DIN V ENV 1993-1-3:2002-09: Eurocode 3: Design of steel structures—Teil 1-3: general rules—supplementary rules for coldformed thin gauge members and sheeting. [2] Schardt R, Bollinger K. Zur Berechnung regelma¨Xig gelochter Scheiben und Platten, Der Bauingenieur 1981;56:S227–39. [3] Altenbach H, Altenbach J, Naumenko K. Ebene Fla¨chentragwerke, Berlin, Heidelberg, 1998. [4] Blomm F, Coffin D. Handbook of thin plate buckling and postbuckling, Boca Raton. 2001. [5] Saal H, Misiek Th. Tragverhalten du¨nnwandiger Bauteile aus perforierten Blechen – Bericht Nr. 061501, Versuchsanstalt fu¨r Stahl, Holz und Steine, 2007, to be published. [6] Seydel E. U¨ber das Ausbeulen von rechteckigen, isotropen oder orthogonal-anisotropen. Platten bei Schubu¨bertragung: IngenieurArchiv 1933;4:S169–91.