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A new analysis model for the effective stiffness of stiffened metallic panels under combined compression and shear stress☆
Aerospace Science and Technology 10 (2006) 316–326 www.elsevier.com/locate/aescte
A new analysis model for the effective stiffness of stiffened metallic panels under combined compression and shear stress ✩ Ein neues Analyseverfahren zur Ermittlung effektiver Steifigkeiten metallischer Strukturen unter kombinierter Druck- und Schubbeanspruchung Matthias Heitmann ∗ , Peter Horst 1 Institute of Aircraft Design and Lightweight Structures, TU Braunschweig Hermann-Blenk-Str. 35, 38108 Braunschweig, Germany Received 4 January 2005; received in revised form 16 December 2005; accepted 20 December 2005 Available online 17 February 2006
Abstract The paper deals with the analysis of the effective stiffness of stiffened metallic panels under combined compression and shear stress as used, e.g. in aircraft fuselages. An important criterion for sizing and certification of aircraft fuselages is the local and global buckling behaviour. For optimisation of stiffened metallic panels it is necessary to know the buckling and postbuckling behaviour as accurately as possible. Due to the fact that nonlinear FE analyses of a whole aircraft fuselage are too time consuming, a very fast quasi nonlinear FE analysis with a coarse mesh under consideration of semi-empirical methods for the effective skin-stiffness are used. At this point the effective stiffness method derived in this paper is used. Admittedly previous semi-empirical methods like the effective width method [J. Wiedemann, Leichtbau 1: Elemente, second ed., 1996; M.C.Y. Niu, Airframe Stress Analysis and Sizing, second ed., Commilit. Press Ltd., 1999] (only for pure compression load) or the method of Kuhn [P. Kuhn, J.P. Peterson, L.R. Levin, A summary of diagonal tension part I – methods of analysis, Technical Note 2661, NACA, 1952] (only for pure shear load) have disadvantages for the general combined compression and shear load case. This is improved in the current method. The first part of the paper deals with the realisation of the compression and shear test facility in a finite element model. The verification of the finite element model is important for subsequent parameter variations. The second part of the paper presents the approach of how to assess the effective skin-stiffness. In comparison to the paper in [M. Heitmann, P. Horst, D. Fitzsimmons, Effective stiffness of postbuckled stiffened metallic panels under combined compression and shear stress, J. Strain Anal. 38 (6) (2003) 534–555] many new parameters have been analysed. Therefore the new approach for the geometrically nonlinear analysis derived from the finite element results for combined compression and shear stress is considerably improved. At the end of the paper the great benefit of the new approach is shown. The results of very fast quasi nonlinear FE analyses under consideration of the new approach for the effective skin-stiffness on a coarse panel mesh agree well with the results of time expensive nonlinear FE analysis on a very fine panel mesh. Further studies are necessary to expand the new method to the influence of plasticity. © 2006 Elsevier SAS. All rights reserved. Zusammenfassung Das Paper beschäftigt sich mit der Berechnung effektiver Steifigkeiten versteifter metallischer Strukturen unter kombinierter Druck- und Schubbeanspruchung, wie sie z.B. in der Analyse von Flugzeugrumpfstrukturen zur Anwendung kommen. Für die Auslegung und Zulassung solcher Strukturen ist das lokale und globale Beulverhalten ein wesentliches Kriterium. Will man solche versteiften, dünnwandigen Strukturen optimieren, ist eine möglichst genaue Kenntnis vom lokalen und globalen Beulverhalten erforderlich. Auf Grund der Tatsache, dass nichtlineare Berechnungen einer gesamten Flugzeugrumpfstruktur zu zeitaufwändig sind, werden derzeit als Stand der Technik schnelle quasi nichtlineare FE-Berechnungen an groben FE-Netzen unter Berücksichtigung von semi-empirischen Verfahren zur Ermittlung der effektiven Hautsteifigkeiten verwendet. Hierbei ✩
This article was presented at the German Aerospace Congress 2004.
* Corresponding author. Tel.: +49 (0)531 2336005; fax: +49 (0)531 3919904.
E-mail addresses: [email protected] (M. Heitmann), [email protected] (P. Horst). 1 Tel.: +49 (0)531 3919901; fax: +49 (0)531 3919904.
1270-9638/$ – see front matter © 2006 Elsevier SAS. All rights reserved. doi:10.1016/j.ast.2005.12.008
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besitzen die bisherigen semi-empirischen Verfahren der Mittragenden Breite [J. Wiedemann, Leichtbau 1: Elemente, second ed., 1996; M.C.Y. Niu, Airframe Stress Analysis and Sizing, second ed., Commilit. Press Ltd., 1999] (nur für die reine Drucklast) oder das Verfahren nach Kuhn [P. Kuhn, J.P. Peterson, L.R. Levin, A summary of diagonal tension part I – methods of analysis, Technical Note 2661, NACA, 1952] (nur für die reine Schublast) Nachteile in der Beschreibung des allgemeinen Druck- und Schublastfalls. Der erste Teil der Arbeit beschäftigt sich mit der Übersetzung der Druck- und Schubtestanlage in ein finite Elemente Modell. Die Verifikation des FE-Modells ist wichtig für nachfolgende Parametervariationen. Der zweite Teil der Arbeit handelt von der Vorgehensweise, wie ein neues semi-empirisches Verfahren für die effektiven Hautsteifigkeiten abgeleitet werden kann. Im Vergleich zum Artikel in [M. Heitmann, P. Horst, D. Fitzsimmons, Effective stiffness of postbuckled stiffened metallic panels under combined compression and shear stress, J. Strain Anal. 38 (6) (2003) 534–555] sind viele zusätzliche Parameter untersucht worden. Deshalb konnte der neue Ansatz zur Beschreibung der effektiven Hautsteifigkeiten im geometrisch Nichtlinearen hergeleitet aus den FE-Ergebnissen der kombinierten Druck- und Schubspannungsberechnungen wesentlich verbessert werden. Am Ende des Artikels wird der große Vorteil des neuen Verfahrens gezeigt. Die Ergebnisse aus sehr schnellen quasi nichtlinearen FE-Analysen unter Berücksichtigung des neuen Verfahrens zur Bestimmung effektiver Hautsteifigkeiten an einem groben Teilschalennetz stimmen gut überein mit den Ergebnissen einer zeitaufwändigen nichtlinearen FE-Analyse an einem fein idealisierten Teilschalennetz. Weitergehende Studien sind notwendig, um den Plastizitätseinfluss im Ansatz berücksichtigen zu können. © 2006 Elsevier SAS. All rights reserved. Keywords: Buckling; Postbuckling; Compression and shear interaction; Effective width method; Method of Kuhn Schlüsselwörter: Beulen; Nachbeulverhalten; Druck- und Schub-Interaktion; Methode der Mittragenden Breite; Methode nach Kuhn
1. Comparison between test results and finite element calculations The Airbus compression and shear test facility [3,13] and the assumptions for the finite element model [5,6] are described in more detail in the mentioned literature. For a better understanding of the coordinates and degrees of freedom of the panel geometry the finite element model with boundary conditions is shown in Fig. 1. In principle the type of boundary condition is clamping. Multipoint constraints (MPCs) are used at boundary 2, 3 and 4 in order to achieve connected displacements along the edge in the t and/or z direction, as indicated in the figure in order to simulate combined shear and compression. The finite element results of a stiffened panel calculated with ABAQUS [1] are compared with test results. The comparison comprises several aspects: (a) the critical buckling loads (b) the postbuckling behaviour (c) the maximum load factor and failure behaviour. Fig. 2 represents the maximum load factors of a stiffened panel which result under combined compression and shear loading. The load introduction in the tests consisted in clamping by means of discrete clamps along the edges 3 and 4, which al-
Fig. 1. Finite element model with boundary conditions.
lowed compressive displacements in the z-direction. Edges 2, 3 and 4 have been loaded by hydraulic cylinders in tangential direction, while edge 2 is loaded by another cylinder in z-direction, too. In view of the finite element accuracy the maximum load factors of the test and finite element results show good agreement. The variations of the maximum load factors are lower than 7–8% and the maximum load factors of the finite element results are generally larger than the test results due to the fact that the finite element model does not include rivet connections. The failure criterion for the finite element analysis of the stiffened panels is on the one hand the maximum load reached in a load-displacement curve or on the other hand the exceedance of a plastic strain (in this case 2%) in the skin. The second criterion makes sense because the nonlinear finite element analysis is not able to represent a connection failure. By experience of Airbus from several test results of stiffened panels with rivet connection, the plastic strain of the skin is always lower than 2% at failure. Fig. 3 shows the stress-strain curve of a stiffened panel under compression and shear force (ratio 50/50). The analysis of four continuous measurements of the relative displacement of the central frame section results from two different stress-strain curves of the test-panel. The same test points are analysed in the finite element model. A comparison between the stress-strain
Fig. 2. Interaction diagram for a stiffened panel under combined compression and shear force. Comparison between the finite element results and test results.
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Nomenclature ad , as A b c D E F FEM Fshear g, h G kds ks l LF MPC q r, t, z Rr, Rt, S t u v ε
effective stiffness factors area width dimensionless parameter applied compressive force Young’s modulus function finite element method shear force functions shear module shear factor shear stress correction factor length load case, displayed in Fig. 2 multi-point constraint shear flow cylindrical coordinates Rz cylindrical rotations applied shear force thickness displacement in length-direction displacement in tangential-direction difference strain in lengthwise direction (definition: compression strain is positive)
ζ, η ν γ σ σR τ
functions Poisson ratio shear strain compressive stress (definition: compressive stress is positive) boundary stress shear stress
Subscripts cr denotes critical stress cr0 critical stress of plane panel eff denotes effective stiffness i, k, ii denote control variables l left max maximal value of a test Mod modified N number of discrete FE-values n set of skin sections r right skin variable referring to a skin stringer, st variable referring to a stringer test variable referring to a test Superscripts ¯
average of a variable
can also be explained by the locally different postbuckling behaviour. The arithmetic average of the results of both, the test result and the finite element analysis shows a good agreement of the postbuckling behaviour. Also the critical buckling load and the maximum load of the stiffened structure show a high accuracy of the finite element mode. It is obvious that the maximum of the average strain in the finite element calculation is significantly larger than the maximum of the average strain in the test. This fact supports the second failure criterion. In general, the displacement behaviour of the finite element simulation of the stiffened metallic panel is different from the test panel behaviour if compared in detail. But the global behaviour of the stiffened panel that means the critical buckling load, the load-displacement curve and the maximum load factor is described with accuracy. Therefore, it is acceptable to parameterise such a finite element model and analyse the effective skin stiffness of metallic panels under combined compression and shear force by this type of model. 2. Calculation of the effective stiffness Fig. 3. The σ –ε-diagram for a stiffened panel under combined compression and shear force (50/50). Comparison between finite element results and test results.
curves at each test point shows a difference in the postbuckling behaviour of the stiffened panel because of the different buckling modes of the test-panel in detail. The difference between the test result and the finite element model at the same position
This section deals with the approach used to derive the effective compression and the effective shear stiffness of the skin by an analysis of the finite element calculation. Both, the finite element calculations and the derived method for the assessment of the effective stiffness are presented in a purely elastic case. This may be the basis for a further development, which includes
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The average shear strain in the skin is calculated by γ¯i =
k
vrii − vlii k ii=1 urii − ulii ii=1 lskin −
(3)
with k as the number of node pairs in a skin section. The equation for the effective shear stiffness in the skin is τ¯i (4) Geffi = . γ¯i
Fig. 4. Schematical presentation of the centre frame section of a stiffened panel. Position of cross-sections and the most important skin sections.
The influence of the boundaries three and four (Fig. 1) on the effective shear stiffness is lower than the influence on the effective compression stiffness, but the calculation of the average effective shear stiffness is also performed for the skin sections described in Fig. 4 because in view of a combined loading case it makes sense to use the same skin sections. The average shear stiffness of the stiffened panel is described by n−1 Geffi . (5) Geff = i=2 n−2 3. Influence of parameters
Fig. 5. Finite element mesh of skin section. Position of cross sections for effective shear stiffness analysis and position of node pair.
plasticity, e.g. by means of knock-down factors. The calculation of the effective stiffness of the skin is described in detail in [6]. Fig. 4 shows the centre frame section of a stiffened panel. Only the skin sections in the centre frame sections are of higher interest because the critical buckling forces for the stringers near the lengthwise boundaries of the panel are higher. The equations for the calculation of the effective compression stiffness and the combined loading case are the same as in [6] but the equations for the calculation of the effective shear stiffness are modified compared to [6] at some points. Fig. 5 shows the positions of the cross sections in a skin section for calculation of the effective shear stiffness. In each cross section of the skin a separate shear flow is calculated. The average shear flow in a skin is described by 1 3 2 4 Fshear Fshear Fshear 1 Fshear (1) q¯i = + + + 4 bskin bskin lskin lskin Due to the fact that there are areas of different thickness (a) pure skin thickness (b) skin thickness and the base thickness of the stringer or frame the average shear stress in the skin has to be reduced by a factor. This factor cannot simply be calculated by the area ratio because of secondary effects. From linear finite element solutions it is possible to calculate this correction factor ks . τ¯i = ks ∗
q¯i . tskin
(2)
For development of a new approach a number of parameters such as skin thickness, skin width, stringer cross-section shape, stringer area, stringer material and panel radius are analysed by geometrical and physical nonlinear FEM. In Table 1 all panel configurations are given which will be discussed in this paper. These changes in parameters are used, in order to cover essential influences, which may occur in real aircraft structures. Special emphasis is on the skin thickness (4), stringer pitch (3), stringer area (3), panel curvature (4) and partly on stringer material. All nonlinear FEM curves displayed are modified because an imperfection has been introduced for all nonlinear FEM calculations. The influence of imperfections disturbs a mathematical approximation, but on the other hand the imperfections are necessary to overcome critical buckling or higher bifurcation points [2,6,12]. Fig. 6 and Eqs. (6), (7) and (8) describe the calculation of the modified FEM values for the effective shear stiffness. τ¯ Geff = 1, if 1, (6) G Mod τcr 5 τ¯ Geff Geff Geff τcr − 1 1− = + , (7) G Mod G FE G 2 τ¯ 3, if 1 < τcr Geff Geff τ¯ = , if > 3. (8) G Mod G FE τcr The calculation for the modified effective compression stiffness follows in a parallel way. 3.1. Influence of shear factor kds The shear factor kds is defined by: kds =
τ¯ . τ¯ + σ¯
(9)
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Table 1 Analysed finite element models under combined compression and shear force Panel number
Skin thickness t [mm]
Skin width b [mm]
Skin material
Panel radius R [mm]
Stringer area Ast [mm]2
Stringer material
TS01 TS02 TS03 TS04 TS05 TS06 TS07 TS08 TS09 TS10 TS11 TS12 TS13 TS14 TS15 TS16 TS17 TS18 TS19 TS20 TS21 TS22 TS23 TS24 TS25
2.0 2.0a 2.0 1.6 1.6 2.5 3.0 2.5 1.6 2.0 1.6 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
176 176 176 176 176 176 176 176 176 100 100 150 200 100 200 176 176 176 176 176 176 100 176 176 176
2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3
2820 2820 2820 2820 2820 2820 2820 ∞ ∞ 2820 2820 2820 2820 ∞ ∞ 2820 2820 2820 2820 ∞ ∞ 2820 ∞ 5640 1410
148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 111.7 223.3 297.8 297.8 297.8 111.7 297.8 148.9 148.9 148.9
2024T3 7349T76 7349T76 7349T76 2024T3 7349T76 7349T76 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 7349T76 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3
a With skin doubler under the stiffeners (b Dp = 24 mm; tDp = 1.5 mm).
Fig. 6. Example for the modification of the nonlinear FE values (TS01 LF5).
Fig. 7. Influence of the shear force to the effective compression stiffness (TS01 in Table 1).
Figs. 7 and 8 show the high influence of the shear factor kds exemplarily for the panel TS01 (Table 1). In consequence of an increasing shear force the effective compressive stiffness of the skin is decreasing (Fig. 7). One reason for the decrease of the effective compression stiffness is the influence of the diagonal tension field described
in [7]. As a result of the diagonal tension field the compression force in the stiffeners is increasing and therefore the ratio between the boundary stress and the average compression stress σ¯ R /σcr is also increasing. On the other hand, Fig. 8 shows that the effective shear stiffness of the panel is decreasing because of an increasing compression force. Different from that, in the
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Fig. 8. Influence of the compression force to the effective shear stiffness (TS01 in Table 1).
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Fig. 9. Influence of the skin thickness for the effective compression stiffness. Comparison of the panels TS03, TS04, TS07 (Table 1) in LF1.
state of the art [11] the influence of a shear factor kds is not considered. 3.2. Influence of skin thickness Fig. 9 shows, exemplarily for the panels TS03, TS04 and TS07 (Table 1) in the pure compression load case, the influence of the skin thickness for the effective compression stiffness. It is obvious that due to a decrease of the skin thickness the effective compression stiffness in the postbuckling range is also decreasing. When the stiffness ratio between the stringer stiffness and the skin stiffness decreases due to an increase of the skin thickness the stringers are less able to carry the compression flow in the postbuckling range. Due to this fact the global buckling of the panel with 3 mm occurs near the critical local buckling load. An influence of the skin thickness on the effective shear stiffness also exists. A comparison shows that the effective shear stiffness of thinner panels is greater than of thicker panels in the postbuckling range. This effect can not be shown in a diagram clearly but in Section 4 this behaviour is implemented in the new semi-empirical approach. 3.3. Influence of panel width Similar to the influence of the skin thickness an influence of the panel width is given. Fig. 10 shows, exemplary for the panels TS01, TS10, TS13 (Table 1) in LF 3, the effective compression stiffness in dependency of the ratio σ¯ R /σcr . By an increase of the panel width the effective compression stiffness is decreasing. The influence of the panel width for the postbuckling behaviour is smaller than the influence of the skin thickness. Otherwise the same analogical explanations can be found for the influence of the skin width as for the skin thickness. Accordingly the effective shear stiffness is increasing with decreasing panel width.
Fig. 10. Influence of the panel width for the effective compression stiffness. Comparison of the panels TS01, TS10, TS13 (Table 1) in LF3.
3.4. Influence of the stringer The influence of the stringer area can also be seen like the influence of the panel thickness. Fig. 11 shows, exemplary for the panels TS01, TS16, TS18 (Table 1) in the pure shear load case, the effective shear stiffness in dependency of the ratio τ¯ /τcr . Due to an increase of the stringer area the effective shear stiffness is increasing and the effective compression stiffness is decreasing. This fact is not shown in a figure but is also implemented in the semi-empirical approach.
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Fig. 11. Influence of the stringer area on the effective shear stiffness. Comparison of the panels TS01, TS16, TS18 (Table 1) in LF5.
Altogether the influence of the stringer area on the postbuckling behaviour is smaller than the influence of the panel width or skin thickness.
Fig. 12. Influence of the panel radius on the effective compression stiffness. Comparison of the panels TS01, TS23, TS25 (Table 1) in LF1.
3.5. Influence of the panel radius Fig. 12 shows, for the panels TS01, TS23, TS25 (Table 1) in the pure compression load case, the influence of the panel radius for the effective compression stiffness based on the ratio between the boundary compression stress and the critical compression stress of a plane panel σ¯ R /σcr0 . Obviously there is no significant influence of the panel radius in the higher postbuckling range. On the contrary the influence of the panel radius near the critical buckling loads is large [4,9]. The effect of the panel radius seems to be the same like a positive geometrical imperfection and therefore is only an influence near by the critical buckling load. The effective compression stiffness curve with a radius influence converges very fast in the postbuckling range to the curve of the plane panel. This behaviour can not be found for the effective shear stiffness (Fig. 13). In fact the critical buckling load increases due to a decrease of the panel radius, but in the higher postbuckling range the effective shear stiffness decreases by a decrease of the panel radius. This influence of the panel radius on the effective shear stiffness is also identified in shear panel tests by [7] and is implemented in this semi-empirical approach.
Fig. 13. Influence of the panel radius for the effective shear stiffness. Comparison of the panels TS01, TS23, TS25 (Table 1) in LF5.
4. Development of a new semi-empirical approach
4.1. Approach for the effective compression stiffness of the skin
In the following chapter the two methods for calculating the effective compression or shear stiffness of the skin is described. The influences of the parameters in Section 3 are implemented. In general the effective stiffness values of the skin are approximated by mathematical functions under consideration of the method of least squares.
An exponential function for the postbuckling behaviour of the effective stiffness depending on the ratio σ¯ R /σcr has been chosen, like by Marguerre and Karman in [10]. σ¯ R σcr ad σ¯ R = 1.2 − 0.2. (10) g σcr σcr σcr0
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The advantage of this function is the dependency on only one free parameter ad . Under consideration that the sum of the least squares must be minimized follows: 2 N Eeff σ¯ R −g , F= E v σcr v v=1 ∂F σ¯ R = ln ∂ad σcr v ad ad σ¯ R σ¯ R Eeff × −2.4 0.2 − 1.2 . + σcr v σcr v E v (11) Not all discrete values N of the FEM calculation are taken into account, because values σ¯ R /σcr lower than σ¯ cr /σ¯ cr0 are affected by the influence of the panel radius, and values at the higher postbuckling range influenced by the global buckling are also not considered. The free parameter ad is derived for all load cases (except the pure shear load case) and all panel configurations (Table 1). The next step for calculating the effective compression stiffness at a load level is to approximate this one free parameter ad in Eq. (10) depends on the geometrical values and the shear factor kds . Therefore, a dimensionless parameter c is introduced which includes the influence of the different parameters in Section 3. Ast b 1 Est Ast b Est · · · = 2 · . (12) c= bt t α E E lt The influence of the panel radius in the parameter c is not considered. Therefore it is suggested that the results between the critical buckling stress and the ratio σ¯ cr /σ¯ cr0 has to be modified with a variable ad . Due to a mathematical polynomial of the third order it is possible to describe the parameter ad by the dimensionless parameter c and the shear factor kds . 2 3 ad (c, kds ) = η0 (c) + η1 (c)kds + η2 (c)kds + η3 (c)kds
(13)
with η0 (c) = −0.0034c − 0.1634, η1 (c) = −0.0338c + 0.2323, η2 (c) = 0.1534c − 1.6131, η3 (c) = −0.1613c + 0.6676. 4.2. Approach for the effective shear stiffness of the skin The approach for the assessment of the effective shear stiffness follows in a parallel way to the approach for the effective compression stiffness. The only difference is that the influence of the panel radius could not be found in one single approach. Therefore, the approach for the calculation of the effective shear stiffness in the plane panel and with a panel radius of 2820 mm is separated. For the approximation of the effective shear stiffness values depending on the ratio τ¯ /τcr a logarithmic function has been chosen because the logarithmic function can also describe the semi-empirical approach of Kuhn very well (the error for the TS01 (Table 1) under pure shear load is lower than 1%). 1 τ¯ = . (14) h τcr 1 + as ln(τ¯ /τcr )
323
The advantage of this function is the dependency on only one free parameter as . In the same manner to Section 4.1 follows: 2 ln(xv ) yv − 1+as 1ln(xv ) ∂F = (15) ∂as (1 + as ln(xv ))2 with xv =
τ¯ τcr
and yv = v
Geff G
. v
Not all discrete values N of the FEM calculation are taken into account, because values τ¯ /τcr lower than 1 are not affected by buckling and values at a higher postbuckling range influenced by the global buckling are also not considered. Due to a mathematical polynomial of the second order it is possible to describe the parameter as with the dimensionless parameter c and the shear factor kds . 2 as _2820 (c, kds ) = η0_2820 (c) + η1_2820 (c)kds + η2_2820 (c)kds
(16) with η0_2820 (c) = 0.004c + 0.7823, η1_2820 (c) = −0.0286c − 0.3989, η2_2820 (c) = 0.0222c − 0.0489, 2 as _∞ (c, kds ) = η0_∞ (c) + η1_∞ (c)kds + η2_∞ (c)kds
(17)
with η0_∞ (c) = 0.0069c + 0.3634, η1_∞ (c) = −0.0209c + 0.0004, η2_∞ (c) = 0.0121c − 0.1222. For the calculation of the parameter as with an arbitrary panel radius the following procedure is suggested: (a) Calculation of as _∞ and as _2820 (b) Calculation of as∗_R according to a mathematical polynomial of the second order depending on the panel radius and the shear factor kds . 1000 1000 2 as∗_R = as _∞ + ξ1 (kds ) + ξ2 (kds ) (18) R R with ξ1 (kds ) = −0.2281kds + 0.2305, ξ2 (kds ) = −1.2553kds + 1.8161. (c) Calculation of a correction factor kR as _2820 kR = ∗ . as _2820
(19)
(d) Calculation of as _R as _R = as∗_R kR .
(20)
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4.3. Limits of the new semi-empirical approach In general, the new approach is only used for the approximation of the effective stiffness of the skin under combined compression and shear force of a rectangular stiffened metallic panel. Plasticity is not considered in the approach till now, but geometrically and physically FEM calculations have also been made for all panel configurations and load cases. Therefore, the calculation of the plasticity influence must be considered by state of the art methods described for example in [8]. One limit for the approach is the beginning of the global buckling. After global buckling occurs the effective stiffness can not be calculated correctly with the new approach. Therefore only panel configurations are analysed which are able to introduce local buckling in the skin section prior to global buckling. In general, it should be clear that on the one hand with an increase of the shear factor kds the relative error increases for the calculation of the effective compression stiffness and on the other hand with a decrease of the shear factor kds the relative error increases for the calculation of the effective shear stiffness. The influence of the panel radius on the postbuckling behaviour is significant. Further studies on the influence of high panel curvatures have to be performed.
Fig. 14. Comparison of the effective compression stiffness between new semiempirical method and modified FEM results for the TS01 (Table 1).
5. Comparison of the new approach with fem results and semi-empirical methods In this chapter the results for the effective stiffness calculated with the new approach are compared with the detailed FEM results. Furthermore, a comparison is made between the new approach and the semi-empirical methods described in [7,10]. Fig. 14 shows, exemplary for the panel TS01 (Table 1), the comparison between the new semi empirical approach and the modified FEM results for the effective compression stiffness. Additionally the semi-analytical results for the effective width method by Karman and Marguerre in [10] are displayed. It is obvious that the methods of the effective width can not represent the influence of different boundary conditions or the influence of the shear factor kds . The divergences between the values calculated by the effective width methods and the FEM values increase with an increase of the shear factor kds . A comparison between the new semi empirical approach and the modified FEM results for the effective shear stiffness is shown exemplarily for the panel TS01 (Table 1) in Fig. 15. Also the semi-empirical results by Kuhn [7] with and without a radius correction for the pure shear load case are displayed in Fig. 15. The calculated semi-empirical effective shear stiffness values with a panel radius correction show a great difference to the FEM results. Neglecting the panel radius, the difference between the FEM results for the pure shear load case and the method of Kuhn [7] is much smaller. This postbuckling behaviour is typical for all analysed panels. Furthermore, it is obvious that the method of Kuhn [7] can not represent the influence of the shear factor kds correctly. The divergences between the values calculated by the method of Kuhn without a radius correction and the FEM values increase with a decrease of the shear factor kds .
Fig. 15. Comparison of the effective shear stiffness between new semiempirical method and modified FEM results for the TS01 (Table 1).
Other interesting facts are shown in Figs. 17 and 18. Firstly, the values are calculated with the method described in Section 2 (FEM results) and secondly the values with the semi-empirical methods are calculated by linear FEM on a simplified panel model shown in Fig. 16. The steps for calculating the values with the new approach and the semi-empirical methods in [7,10] are described in detail:
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Fig. 16. Simplified FEM model of a panel.
Fig. 18. Comparison of σ¯ R and σ¯ skin in dependency of τ¯ /τcr exemplary for the TS01 (Table 1) LF3 between FEM results, new approach and semi-empirical methods described in [7,10].
to a linear FEM calculation with the exception seen above. For calculation, the stringer compression stress by the semi analytical methods described in [10] and [7] a subsequent calculation which considers the influence of the diagonal tension field is necessary. Furthermore, the difference between the new approach and the nonlinear FEM values is lower than the difference between the semi-empirical methods and the FEM results because the semi-empirical methods do not consider the influence of the shear factor kds . 6. Perspective and conclusion Fig. 17. Comparison of displacements in dependency of σ¯ /σcr exemplary for the TS01 (Table 1) LF3 between FEM results, new approach and semiempirical methods described in [8,11].
(a) Calculate the effective compression and shear stiffness of the skin for every nonlinear load step; (b) Reduction of the compression and shear stiffness of the membrane elements in the central frame section; (c) Execution of a linear FEM analysis. The comparison of the average displacements in the centre frame section displayed in Fig. 17 shows a large difference between the FEM values and the values calculated by the semiempirical methods in [10] and [7]. In contrary therefore the new approach can verify the nonlinear FEM results with the exception of high shear factors kds with a high accuracy. In Fig. 18 the stringer and skin compression stresses are shown in dependency of the ratio τ¯ /τcr . With the new approach it is possible to calculate the stringer compression stress due
The paper describes a new method for the assessment of the effective compression or shear stiffness of a rectangularity stiffened metallic panel under combined compression and shear force in the postbuckling range. Very fast quasi nonlinear FEM solutions with reduced skin stiffness represent the time expensive geometrical nonlinear FEM results very well with the exception of large shear factors kds . In the case of great shear factors kds a subsequent calculation for the stringer compression stress is necessary. Another limit of the new method is the start of global buckling. The start of global buckling is not discussed in detail in this paper. Also further studies are necessary to expand the new method to the influence of plasticity. References [1] ABAQUS/Standard, Version 6.3. Hibbitt, Karlsson & Sorensen, Inc., 2002. [2] M.A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures, vol. 2, John Wiley & Sons, 1998.
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[3] B. Evers, Versuchsergebnisse an längs- und querversteiften Leichtbauschalen unter kombinierter Druck-Schub-Belastung, in: German Conference of Shell Buckling in Darmstadt, 1979. [4] G. Gerard, H. Becker, Handbook of structural stability part III – Buckling of curved plates and shells, Technical Note 3783, NACA, 1957. [5] M. Heitmann, P. Horst, M. Haupt, D. Fitzsimmons, Numerische Simulation von Teilschalenversuchen an versteiften metallischen Strukturen unter kombinierten Druck- und Schubbeanspruchung, Deutscher Luft- und Raumfahrtkongress 2001, Hamburg, DGLR-Jahrbuch 2001 Band I, KennNr. DGLR-TJ2001-110. [6] M. Heitmann, P. Horst, D. Fitzsimmons, Effective stiffness of postbuckled stiffened metallic panels under combined compression and shear stress, J. Strain Anal. 38 (6) (2003) 534–555. [7] P. Kuhn, J.P. Peterson, L.R. Levin, A summary of diagonal tension, part I – methods of analysis, Technical Note 2661, NACA, 1952.
[8] M.C.Y. Niu, Airframe Stress Analysis and Sizing, second ed., Commilit. Press Ltd., 1999. [9] S. Timoshenko, J. Gere, Theory of Elastic Stability, McGraw-Hill, 1961. [10] J. Wiedemann, Leichtbau 1: Elemente, second ed., 1996. [11] J. Wiedemann, D. Fitzsimmons, Luftfahrtfor-schungsprogramm Globales Tragverhalten von Flugzeugrümpfen final report, Airbus Deutschland, August 2003. [12] K. Wolf, Untersuchungen zum Beul- und Nachbeulverhalten schubbeanspruchter Teilschalen aus kohlenstoffaserverstärkten Kunststoff, Dissertation TU Braunschweig, 1989. [13] K. Wolf, H. Kossira, An efficient test method for the experimental investigation of the postbuckling behaviour of curved composite shear panels, in: EACM-Conference in Amsterdam, 1992, pp. 133–142.
A new analysis model for the effective stiffness of stiffened metallic panels under combined compression and shear stress ✩ Ein neues Analyseverfahren zur Ermittlung effektiver Steifigkeiten metallischer Strukturen unter kombinierter Druck- und Schubbeanspruchung Matthias Heitmann ∗ , Peter Horst 1 Institute of Aircraft Design and Lightweight Structures, TU Braunschweig Hermann-Blenk-Str. 35, 38108 Braunschweig, Germany Received 4 January 2005; received in revised form 16 December 2005; accepted 20 December 2005 Available online 17 February 2006
Abstract The paper deals with the analysis of the effective stiffness of stiffened metallic panels under combined compression and shear stress as used, e.g. in aircraft fuselages. An important criterion for sizing and certification of aircraft fuselages is the local and global buckling behaviour. For optimisation of stiffened metallic panels it is necessary to know the buckling and postbuckling behaviour as accurately as possible. Due to the fact that nonlinear FE analyses of a whole aircraft fuselage are too time consuming, a very fast quasi nonlinear FE analysis with a coarse mesh under consideration of semi-empirical methods for the effective skin-stiffness are used. At this point the effective stiffness method derived in this paper is used. Admittedly previous semi-empirical methods like the effective width method [J. Wiedemann, Leichtbau 1: Elemente, second ed., 1996; M.C.Y. Niu, Airframe Stress Analysis and Sizing, second ed., Commilit. Press Ltd., 1999] (only for pure compression load) or the method of Kuhn [P. Kuhn, J.P. Peterson, L.R. Levin, A summary of diagonal tension part I – methods of analysis, Technical Note 2661, NACA, 1952] (only for pure shear load) have disadvantages for the general combined compression and shear load case. This is improved in the current method. The first part of the paper deals with the realisation of the compression and shear test facility in a finite element model. The verification of the finite element model is important for subsequent parameter variations. The second part of the paper presents the approach of how to assess the effective skin-stiffness. In comparison to the paper in [M. Heitmann, P. Horst, D. Fitzsimmons, Effective stiffness of postbuckled stiffened metallic panels under combined compression and shear stress, J. Strain Anal. 38 (6) (2003) 534–555] many new parameters have been analysed. Therefore the new approach for the geometrically nonlinear analysis derived from the finite element results for combined compression and shear stress is considerably improved. At the end of the paper the great benefit of the new approach is shown. The results of very fast quasi nonlinear FE analyses under consideration of the new approach for the effective skin-stiffness on a coarse panel mesh agree well with the results of time expensive nonlinear FE analysis on a very fine panel mesh. Further studies are necessary to expand the new method to the influence of plasticity. © 2006 Elsevier SAS. All rights reserved. Zusammenfassung Das Paper beschäftigt sich mit der Berechnung effektiver Steifigkeiten versteifter metallischer Strukturen unter kombinierter Druck- und Schubbeanspruchung, wie sie z.B. in der Analyse von Flugzeugrumpfstrukturen zur Anwendung kommen. Für die Auslegung und Zulassung solcher Strukturen ist das lokale und globale Beulverhalten ein wesentliches Kriterium. Will man solche versteiften, dünnwandigen Strukturen optimieren, ist eine möglichst genaue Kenntnis vom lokalen und globalen Beulverhalten erforderlich. Auf Grund der Tatsache, dass nichtlineare Berechnungen einer gesamten Flugzeugrumpfstruktur zu zeitaufwändig sind, werden derzeit als Stand der Technik schnelle quasi nichtlineare FE-Berechnungen an groben FE-Netzen unter Berücksichtigung von semi-empirischen Verfahren zur Ermittlung der effektiven Hautsteifigkeiten verwendet. Hierbei ✩
This article was presented at the German Aerospace Congress 2004.
* Corresponding author. Tel.: +49 (0)531 2336005; fax: +49 (0)531 3919904.
E-mail addresses: [email protected] (M. Heitmann), [email protected] (P. Horst). 1 Tel.: +49 (0)531 3919901; fax: +49 (0)531 3919904.
1270-9638/$ – see front matter © 2006 Elsevier SAS. All rights reserved. doi:10.1016/j.ast.2005.12.008
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besitzen die bisherigen semi-empirischen Verfahren der Mittragenden Breite [J. Wiedemann, Leichtbau 1: Elemente, second ed., 1996; M.C.Y. Niu, Airframe Stress Analysis and Sizing, second ed., Commilit. Press Ltd., 1999] (nur für die reine Drucklast) oder das Verfahren nach Kuhn [P. Kuhn, J.P. Peterson, L.R. Levin, A summary of diagonal tension part I – methods of analysis, Technical Note 2661, NACA, 1952] (nur für die reine Schublast) Nachteile in der Beschreibung des allgemeinen Druck- und Schublastfalls. Der erste Teil der Arbeit beschäftigt sich mit der Übersetzung der Druck- und Schubtestanlage in ein finite Elemente Modell. Die Verifikation des FE-Modells ist wichtig für nachfolgende Parametervariationen. Der zweite Teil der Arbeit handelt von der Vorgehensweise, wie ein neues semi-empirisches Verfahren für die effektiven Hautsteifigkeiten abgeleitet werden kann. Im Vergleich zum Artikel in [M. Heitmann, P. Horst, D. Fitzsimmons, Effective stiffness of postbuckled stiffened metallic panels under combined compression and shear stress, J. Strain Anal. 38 (6) (2003) 534–555] sind viele zusätzliche Parameter untersucht worden. Deshalb konnte der neue Ansatz zur Beschreibung der effektiven Hautsteifigkeiten im geometrisch Nichtlinearen hergeleitet aus den FE-Ergebnissen der kombinierten Druck- und Schubspannungsberechnungen wesentlich verbessert werden. Am Ende des Artikels wird der große Vorteil des neuen Verfahrens gezeigt. Die Ergebnisse aus sehr schnellen quasi nichtlinearen FE-Analysen unter Berücksichtigung des neuen Verfahrens zur Bestimmung effektiver Hautsteifigkeiten an einem groben Teilschalennetz stimmen gut überein mit den Ergebnissen einer zeitaufwändigen nichtlinearen FE-Analyse an einem fein idealisierten Teilschalennetz. Weitergehende Studien sind notwendig, um den Plastizitätseinfluss im Ansatz berücksichtigen zu können. © 2006 Elsevier SAS. All rights reserved. Keywords: Buckling; Postbuckling; Compression and shear interaction; Effective width method; Method of Kuhn Schlüsselwörter: Beulen; Nachbeulverhalten; Druck- und Schub-Interaktion; Methode der Mittragenden Breite; Methode nach Kuhn
1. Comparison between test results and finite element calculations The Airbus compression and shear test facility [3,13] and the assumptions for the finite element model [5,6] are described in more detail in the mentioned literature. For a better understanding of the coordinates and degrees of freedom of the panel geometry the finite element model with boundary conditions is shown in Fig. 1. In principle the type of boundary condition is clamping. Multipoint constraints (MPCs) are used at boundary 2, 3 and 4 in order to achieve connected displacements along the edge in the t and/or z direction, as indicated in the figure in order to simulate combined shear and compression. The finite element results of a stiffened panel calculated with ABAQUS [1] are compared with test results. The comparison comprises several aspects: (a) the critical buckling loads (b) the postbuckling behaviour (c) the maximum load factor and failure behaviour. Fig. 2 represents the maximum load factors of a stiffened panel which result under combined compression and shear loading. The load introduction in the tests consisted in clamping by means of discrete clamps along the edges 3 and 4, which al-
Fig. 1. Finite element model with boundary conditions.
lowed compressive displacements in the z-direction. Edges 2, 3 and 4 have been loaded by hydraulic cylinders in tangential direction, while edge 2 is loaded by another cylinder in z-direction, too. In view of the finite element accuracy the maximum load factors of the test and finite element results show good agreement. The variations of the maximum load factors are lower than 7–8% and the maximum load factors of the finite element results are generally larger than the test results due to the fact that the finite element model does not include rivet connections. The failure criterion for the finite element analysis of the stiffened panels is on the one hand the maximum load reached in a load-displacement curve or on the other hand the exceedance of a plastic strain (in this case 2%) in the skin. The second criterion makes sense because the nonlinear finite element analysis is not able to represent a connection failure. By experience of Airbus from several test results of stiffened panels with rivet connection, the plastic strain of the skin is always lower than 2% at failure. Fig. 3 shows the stress-strain curve of a stiffened panel under compression and shear force (ratio 50/50). The analysis of four continuous measurements of the relative displacement of the central frame section results from two different stress-strain curves of the test-panel. The same test points are analysed in the finite element model. A comparison between the stress-strain
Fig. 2. Interaction diagram for a stiffened panel under combined compression and shear force. Comparison between the finite element results and test results.
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Nomenclature ad , as A b c D E F FEM Fshear g, h G kds ks l LF MPC q r, t, z Rr, Rt, S t u v ε
effective stiffness factors area width dimensionless parameter applied compressive force Young’s modulus function finite element method shear force functions shear module shear factor shear stress correction factor length load case, displayed in Fig. 2 multi-point constraint shear flow cylindrical coordinates Rz cylindrical rotations applied shear force thickness displacement in length-direction displacement in tangential-direction difference strain in lengthwise direction (definition: compression strain is positive)
ζ, η ν γ σ σR τ
functions Poisson ratio shear strain compressive stress (definition: compressive stress is positive) boundary stress shear stress
Subscripts cr denotes critical stress cr0 critical stress of plane panel eff denotes effective stiffness i, k, ii denote control variables l left max maximal value of a test Mod modified N number of discrete FE-values n set of skin sections r right skin variable referring to a skin stringer, st variable referring to a stringer test variable referring to a test Superscripts ¯
average of a variable
can also be explained by the locally different postbuckling behaviour. The arithmetic average of the results of both, the test result and the finite element analysis shows a good agreement of the postbuckling behaviour. Also the critical buckling load and the maximum load of the stiffened structure show a high accuracy of the finite element mode. It is obvious that the maximum of the average strain in the finite element calculation is significantly larger than the maximum of the average strain in the test. This fact supports the second failure criterion. In general, the displacement behaviour of the finite element simulation of the stiffened metallic panel is different from the test panel behaviour if compared in detail. But the global behaviour of the stiffened panel that means the critical buckling load, the load-displacement curve and the maximum load factor is described with accuracy. Therefore, it is acceptable to parameterise such a finite element model and analyse the effective skin stiffness of metallic panels under combined compression and shear force by this type of model. 2. Calculation of the effective stiffness Fig. 3. The σ –ε-diagram for a stiffened panel under combined compression and shear force (50/50). Comparison between finite element results and test results.
curves at each test point shows a difference in the postbuckling behaviour of the stiffened panel because of the different buckling modes of the test-panel in detail. The difference between the test result and the finite element model at the same position
This section deals with the approach used to derive the effective compression and the effective shear stiffness of the skin by an analysis of the finite element calculation. Both, the finite element calculations and the derived method for the assessment of the effective stiffness are presented in a purely elastic case. This may be the basis for a further development, which includes
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The average shear strain in the skin is calculated by γ¯i =
k
vrii − vlii k ii=1 urii − ulii ii=1 lskin −
(3)
with k as the number of node pairs in a skin section. The equation for the effective shear stiffness in the skin is τ¯i (4) Geffi = . γ¯i
Fig. 4. Schematical presentation of the centre frame section of a stiffened panel. Position of cross-sections and the most important skin sections.
The influence of the boundaries three and four (Fig. 1) on the effective shear stiffness is lower than the influence on the effective compression stiffness, but the calculation of the average effective shear stiffness is also performed for the skin sections described in Fig. 4 because in view of a combined loading case it makes sense to use the same skin sections. The average shear stiffness of the stiffened panel is described by n−1 Geffi . (5) Geff = i=2 n−2 3. Influence of parameters
Fig. 5. Finite element mesh of skin section. Position of cross sections for effective shear stiffness analysis and position of node pair.
plasticity, e.g. by means of knock-down factors. The calculation of the effective stiffness of the skin is described in detail in [6]. Fig. 4 shows the centre frame section of a stiffened panel. Only the skin sections in the centre frame sections are of higher interest because the critical buckling forces for the stringers near the lengthwise boundaries of the panel are higher. The equations for the calculation of the effective compression stiffness and the combined loading case are the same as in [6] but the equations for the calculation of the effective shear stiffness are modified compared to [6] at some points. Fig. 5 shows the positions of the cross sections in a skin section for calculation of the effective shear stiffness. In each cross section of the skin a separate shear flow is calculated. The average shear flow in a skin is described by 1 3 2 4 Fshear Fshear Fshear 1 Fshear (1) q¯i = + + + 4 bskin bskin lskin lskin Due to the fact that there are areas of different thickness (a) pure skin thickness (b) skin thickness and the base thickness of the stringer or frame the average shear stress in the skin has to be reduced by a factor. This factor cannot simply be calculated by the area ratio because of secondary effects. From linear finite element solutions it is possible to calculate this correction factor ks . τ¯i = ks ∗
q¯i . tskin
(2)
For development of a new approach a number of parameters such as skin thickness, skin width, stringer cross-section shape, stringer area, stringer material and panel radius are analysed by geometrical and physical nonlinear FEM. In Table 1 all panel configurations are given which will be discussed in this paper. These changes in parameters are used, in order to cover essential influences, which may occur in real aircraft structures. Special emphasis is on the skin thickness (4), stringer pitch (3), stringer area (3), panel curvature (4) and partly on stringer material. All nonlinear FEM curves displayed are modified because an imperfection has been introduced for all nonlinear FEM calculations. The influence of imperfections disturbs a mathematical approximation, but on the other hand the imperfections are necessary to overcome critical buckling or higher bifurcation points [2,6,12]. Fig. 6 and Eqs. (6), (7) and (8) describe the calculation of the modified FEM values for the effective shear stiffness. τ¯ Geff = 1, if 1, (6) G Mod τcr 5 τ¯ Geff Geff Geff τcr − 1 1− = + , (7) G Mod G FE G 2 τ¯ 3, if 1 < τcr Geff Geff τ¯ = , if > 3. (8) G Mod G FE τcr The calculation for the modified effective compression stiffness follows in a parallel way. 3.1. Influence of shear factor kds The shear factor kds is defined by: kds =
τ¯ . τ¯ + σ¯
(9)
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Table 1 Analysed finite element models under combined compression and shear force Panel number
Skin thickness t [mm]
Skin width b [mm]
Skin material
Panel radius R [mm]
Stringer area Ast [mm]2
Stringer material
TS01 TS02 TS03 TS04 TS05 TS06 TS07 TS08 TS09 TS10 TS11 TS12 TS13 TS14 TS15 TS16 TS17 TS18 TS19 TS20 TS21 TS22 TS23 TS24 TS25
2.0 2.0a 2.0 1.6 1.6 2.5 3.0 2.5 1.6 2.0 1.6 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
176 176 176 176 176 176 176 176 176 100 100 150 200 100 200 176 176 176 176 176 176 100 176 176 176
2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3
2820 2820 2820 2820 2820 2820 2820 ∞ ∞ 2820 2820 2820 2820 ∞ ∞ 2820 2820 2820 2820 ∞ ∞ 2820 ∞ 5640 1410
148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 148.9 111.7 223.3 297.8 297.8 297.8 111.7 297.8 148.9 148.9 148.9
2024T3 7349T76 7349T76 7349T76 2024T3 7349T76 7349T76 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3 7349T76 2024T3 2024T3 2024T3 2024T3 2024T3 2024T3
a With skin doubler under the stiffeners (b Dp = 24 mm; tDp = 1.5 mm).
Fig. 6. Example for the modification of the nonlinear FE values (TS01 LF5).
Fig. 7. Influence of the shear force to the effective compression stiffness (TS01 in Table 1).
Figs. 7 and 8 show the high influence of the shear factor kds exemplarily for the panel TS01 (Table 1). In consequence of an increasing shear force the effective compressive stiffness of the skin is decreasing (Fig. 7). One reason for the decrease of the effective compression stiffness is the influence of the diagonal tension field described
in [7]. As a result of the diagonal tension field the compression force in the stiffeners is increasing and therefore the ratio between the boundary stress and the average compression stress σ¯ R /σcr is also increasing. On the other hand, Fig. 8 shows that the effective shear stiffness of the panel is decreasing because of an increasing compression force. Different from that, in the
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Fig. 8. Influence of the compression force to the effective shear stiffness (TS01 in Table 1).
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Fig. 9. Influence of the skin thickness for the effective compression stiffness. Comparison of the panels TS03, TS04, TS07 (Table 1) in LF1.
state of the art [11] the influence of a shear factor kds is not considered. 3.2. Influence of skin thickness Fig. 9 shows, exemplarily for the panels TS03, TS04 and TS07 (Table 1) in the pure compression load case, the influence of the skin thickness for the effective compression stiffness. It is obvious that due to a decrease of the skin thickness the effective compression stiffness in the postbuckling range is also decreasing. When the stiffness ratio between the stringer stiffness and the skin stiffness decreases due to an increase of the skin thickness the stringers are less able to carry the compression flow in the postbuckling range. Due to this fact the global buckling of the panel with 3 mm occurs near the critical local buckling load. An influence of the skin thickness on the effective shear stiffness also exists. A comparison shows that the effective shear stiffness of thinner panels is greater than of thicker panels in the postbuckling range. This effect can not be shown in a diagram clearly but in Section 4 this behaviour is implemented in the new semi-empirical approach. 3.3. Influence of panel width Similar to the influence of the skin thickness an influence of the panel width is given. Fig. 10 shows, exemplary for the panels TS01, TS10, TS13 (Table 1) in LF 3, the effective compression stiffness in dependency of the ratio σ¯ R /σcr . By an increase of the panel width the effective compression stiffness is decreasing. The influence of the panel width for the postbuckling behaviour is smaller than the influence of the skin thickness. Otherwise the same analogical explanations can be found for the influence of the skin width as for the skin thickness. Accordingly the effective shear stiffness is increasing with decreasing panel width.
Fig. 10. Influence of the panel width for the effective compression stiffness. Comparison of the panels TS01, TS10, TS13 (Table 1) in LF3.
3.4. Influence of the stringer The influence of the stringer area can also be seen like the influence of the panel thickness. Fig. 11 shows, exemplary for the panels TS01, TS16, TS18 (Table 1) in the pure shear load case, the effective shear stiffness in dependency of the ratio τ¯ /τcr . Due to an increase of the stringer area the effective shear stiffness is increasing and the effective compression stiffness is decreasing. This fact is not shown in a figure but is also implemented in the semi-empirical approach.
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Fig. 11. Influence of the stringer area on the effective shear stiffness. Comparison of the panels TS01, TS16, TS18 (Table 1) in LF5.
Altogether the influence of the stringer area on the postbuckling behaviour is smaller than the influence of the panel width or skin thickness.
Fig. 12. Influence of the panel radius on the effective compression stiffness. Comparison of the panels TS01, TS23, TS25 (Table 1) in LF1.
3.5. Influence of the panel radius Fig. 12 shows, for the panels TS01, TS23, TS25 (Table 1) in the pure compression load case, the influence of the panel radius for the effective compression stiffness based on the ratio between the boundary compression stress and the critical compression stress of a plane panel σ¯ R /σcr0 . Obviously there is no significant influence of the panel radius in the higher postbuckling range. On the contrary the influence of the panel radius near the critical buckling loads is large [4,9]. The effect of the panel radius seems to be the same like a positive geometrical imperfection and therefore is only an influence near by the critical buckling load. The effective compression stiffness curve with a radius influence converges very fast in the postbuckling range to the curve of the plane panel. This behaviour can not be found for the effective shear stiffness (Fig. 13). In fact the critical buckling load increases due to a decrease of the panel radius, but in the higher postbuckling range the effective shear stiffness decreases by a decrease of the panel radius. This influence of the panel radius on the effective shear stiffness is also identified in shear panel tests by [7] and is implemented in this semi-empirical approach.
Fig. 13. Influence of the panel radius for the effective shear stiffness. Comparison of the panels TS01, TS23, TS25 (Table 1) in LF5.
4. Development of a new semi-empirical approach
4.1. Approach for the effective compression stiffness of the skin
In the following chapter the two methods for calculating the effective compression or shear stiffness of the skin is described. The influences of the parameters in Section 3 are implemented. In general the effective stiffness values of the skin are approximated by mathematical functions under consideration of the method of least squares.
An exponential function for the postbuckling behaviour of the effective stiffness depending on the ratio σ¯ R /σcr has been chosen, like by Marguerre and Karman in [10]. σ¯ R σcr ad σ¯ R = 1.2 − 0.2. (10) g σcr σcr σcr0
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The advantage of this function is the dependency on only one free parameter ad . Under consideration that the sum of the least squares must be minimized follows: 2 N Eeff σ¯ R −g , F= E v σcr v v=1 ∂F σ¯ R = ln ∂ad σcr v ad ad σ¯ R σ¯ R Eeff × −2.4 0.2 − 1.2 . + σcr v σcr v E v (11) Not all discrete values N of the FEM calculation are taken into account, because values σ¯ R /σcr lower than σ¯ cr /σ¯ cr0 are affected by the influence of the panel radius, and values at the higher postbuckling range influenced by the global buckling are also not considered. The free parameter ad is derived for all load cases (except the pure shear load case) and all panel configurations (Table 1). The next step for calculating the effective compression stiffness at a load level is to approximate this one free parameter ad in Eq. (10) depends on the geometrical values and the shear factor kds . Therefore, a dimensionless parameter c is introduced which includes the influence of the different parameters in Section 3. Ast b 1 Est Ast b Est · · · = 2 · . (12) c= bt t α E E lt The influence of the panel radius in the parameter c is not considered. Therefore it is suggested that the results between the critical buckling stress and the ratio σ¯ cr /σ¯ cr0 has to be modified with a variable ad . Due to a mathematical polynomial of the third order it is possible to describe the parameter ad by the dimensionless parameter c and the shear factor kds . 2 3 ad (c, kds ) = η0 (c) + η1 (c)kds + η2 (c)kds + η3 (c)kds
(13)
with η0 (c) = −0.0034c − 0.1634, η1 (c) = −0.0338c + 0.2323, η2 (c) = 0.1534c − 1.6131, η3 (c) = −0.1613c + 0.6676. 4.2. Approach for the effective shear stiffness of the skin The approach for the assessment of the effective shear stiffness follows in a parallel way to the approach for the effective compression stiffness. The only difference is that the influence of the panel radius could not be found in one single approach. Therefore, the approach for the calculation of the effective shear stiffness in the plane panel and with a panel radius of 2820 mm is separated. For the approximation of the effective shear stiffness values depending on the ratio τ¯ /τcr a logarithmic function has been chosen because the logarithmic function can also describe the semi-empirical approach of Kuhn very well (the error for the TS01 (Table 1) under pure shear load is lower than 1%). 1 τ¯ = . (14) h τcr 1 + as ln(τ¯ /τcr )
323
The advantage of this function is the dependency on only one free parameter as . In the same manner to Section 4.1 follows: 2 ln(xv ) yv − 1+as 1ln(xv ) ∂F = (15) ∂as (1 + as ln(xv ))2 with xv =
τ¯ τcr
and yv = v
Geff G
. v
Not all discrete values N of the FEM calculation are taken into account, because values τ¯ /τcr lower than 1 are not affected by buckling and values at a higher postbuckling range influenced by the global buckling are also not considered. Due to a mathematical polynomial of the second order it is possible to describe the parameter as with the dimensionless parameter c and the shear factor kds . 2 as _2820 (c, kds ) = η0_2820 (c) + η1_2820 (c)kds + η2_2820 (c)kds
(16) with η0_2820 (c) = 0.004c + 0.7823, η1_2820 (c) = −0.0286c − 0.3989, η2_2820 (c) = 0.0222c − 0.0489, 2 as _∞ (c, kds ) = η0_∞ (c) + η1_∞ (c)kds + η2_∞ (c)kds
(17)
with η0_∞ (c) = 0.0069c + 0.3634, η1_∞ (c) = −0.0209c + 0.0004, η2_∞ (c) = 0.0121c − 0.1222. For the calculation of the parameter as with an arbitrary panel radius the following procedure is suggested: (a) Calculation of as _∞ and as _2820 (b) Calculation of as∗_R according to a mathematical polynomial of the second order depending on the panel radius and the shear factor kds . 1000 1000 2 as∗_R = as _∞ + ξ1 (kds ) + ξ2 (kds ) (18) R R with ξ1 (kds ) = −0.2281kds + 0.2305, ξ2 (kds ) = −1.2553kds + 1.8161. (c) Calculation of a correction factor kR as _2820 kR = ∗ . as _2820
(19)
(d) Calculation of as _R as _R = as∗_R kR .
(20)
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4.3. Limits of the new semi-empirical approach In general, the new approach is only used for the approximation of the effective stiffness of the skin under combined compression and shear force of a rectangular stiffened metallic panel. Plasticity is not considered in the approach till now, but geometrically and physically FEM calculations have also been made for all panel configurations and load cases. Therefore, the calculation of the plasticity influence must be considered by state of the art methods described for example in [8]. One limit for the approach is the beginning of the global buckling. After global buckling occurs the effective stiffness can not be calculated correctly with the new approach. Therefore only panel configurations are analysed which are able to introduce local buckling in the skin section prior to global buckling. In general, it should be clear that on the one hand with an increase of the shear factor kds the relative error increases for the calculation of the effective compression stiffness and on the other hand with a decrease of the shear factor kds the relative error increases for the calculation of the effective shear stiffness. The influence of the panel radius on the postbuckling behaviour is significant. Further studies on the influence of high panel curvatures have to be performed.
Fig. 14. Comparison of the effective compression stiffness between new semiempirical method and modified FEM results for the TS01 (Table 1).
5. Comparison of the new approach with fem results and semi-empirical methods In this chapter the results for the effective stiffness calculated with the new approach are compared with the detailed FEM results. Furthermore, a comparison is made between the new approach and the semi-empirical methods described in [7,10]. Fig. 14 shows, exemplary for the panel TS01 (Table 1), the comparison between the new semi empirical approach and the modified FEM results for the effective compression stiffness. Additionally the semi-analytical results for the effective width method by Karman and Marguerre in [10] are displayed. It is obvious that the methods of the effective width can not represent the influence of different boundary conditions or the influence of the shear factor kds . The divergences between the values calculated by the effective width methods and the FEM values increase with an increase of the shear factor kds . A comparison between the new semi empirical approach and the modified FEM results for the effective shear stiffness is shown exemplarily for the panel TS01 (Table 1) in Fig. 15. Also the semi-empirical results by Kuhn [7] with and without a radius correction for the pure shear load case are displayed in Fig. 15. The calculated semi-empirical effective shear stiffness values with a panel radius correction show a great difference to the FEM results. Neglecting the panel radius, the difference between the FEM results for the pure shear load case and the method of Kuhn [7] is much smaller. This postbuckling behaviour is typical for all analysed panels. Furthermore, it is obvious that the method of Kuhn [7] can not represent the influence of the shear factor kds correctly. The divergences between the values calculated by the method of Kuhn without a radius correction and the FEM values increase with a decrease of the shear factor kds .
Fig. 15. Comparison of the effective shear stiffness between new semiempirical method and modified FEM results for the TS01 (Table 1).
Other interesting facts are shown in Figs. 17 and 18. Firstly, the values are calculated with the method described in Section 2 (FEM results) and secondly the values with the semi-empirical methods are calculated by linear FEM on a simplified panel model shown in Fig. 16. The steps for calculating the values with the new approach and the semi-empirical methods in [7,10] are described in detail:
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Fig. 16. Simplified FEM model of a panel.
Fig. 18. Comparison of σ¯ R and σ¯ skin in dependency of τ¯ /τcr exemplary for the TS01 (Table 1) LF3 between FEM results, new approach and semi-empirical methods described in [7,10].
to a linear FEM calculation with the exception seen above. For calculation, the stringer compression stress by the semi analytical methods described in [10] and [7] a subsequent calculation which considers the influence of the diagonal tension field is necessary. Furthermore, the difference between the new approach and the nonlinear FEM values is lower than the difference between the semi-empirical methods and the FEM results because the semi-empirical methods do not consider the influence of the shear factor kds . 6. Perspective and conclusion Fig. 17. Comparison of displacements in dependency of σ¯ /σcr exemplary for the TS01 (Table 1) LF3 between FEM results, new approach and semiempirical methods described in [8,11].
(a) Calculate the effective compression and shear stiffness of the skin for every nonlinear load step; (b) Reduction of the compression and shear stiffness of the membrane elements in the central frame section; (c) Execution of a linear FEM analysis. The comparison of the average displacements in the centre frame section displayed in Fig. 17 shows a large difference between the FEM values and the values calculated by the semiempirical methods in [10] and [7]. In contrary therefore the new approach can verify the nonlinear FEM results with the exception of high shear factors kds with a high accuracy. In Fig. 18 the stringer and skin compression stresses are shown in dependency of the ratio τ¯ /τcr . With the new approach it is possible to calculate the stringer compression stress due
The paper describes a new method for the assessment of the effective compression or shear stiffness of a rectangularity stiffened metallic panel under combined compression and shear force in the postbuckling range. Very fast quasi nonlinear FEM solutions with reduced skin stiffness represent the time expensive geometrical nonlinear FEM results very well with the exception of large shear factors kds . In the case of great shear factors kds a subsequent calculation for the stringer compression stress is necessary. Another limit of the new method is the start of global buckling. The start of global buckling is not discussed in detail in this paper. Also further studies are necessary to expand the new method to the influence of plasticity. References [1] ABAQUS/Standard, Version 6.3. Hibbitt, Karlsson & Sorensen, Inc., 2002. [2] M.A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures, vol. 2, John Wiley & Sons, 1998.
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[3] B. Evers, Versuchsergebnisse an längs- und querversteiften Leichtbauschalen unter kombinierter Druck-Schub-Belastung, in: German Conference of Shell Buckling in Darmstadt, 1979. [4] G. Gerard, H. Becker, Handbook of structural stability part III – Buckling of curved plates and shells, Technical Note 3783, NACA, 1957. [5] M. Heitmann, P. Horst, M. Haupt, D. Fitzsimmons, Numerische Simulation von Teilschalenversuchen an versteiften metallischen Strukturen unter kombinierten Druck- und Schubbeanspruchung, Deutscher Luft- und Raumfahrtkongress 2001, Hamburg, DGLR-Jahrbuch 2001 Band I, KennNr. DGLR-TJ2001-110. [6] M. Heitmann, P. Horst, D. Fitzsimmons, Effective stiffness of postbuckled stiffened metallic panels under combined compression and shear stress, J. Strain Anal. 38 (6) (2003) 534–555. [7] P. Kuhn, J.P. Peterson, L.R. Levin, A summary of diagonal tension, part I – methods of analysis, Technical Note 2661, NACA, 1952.
[8] M.C.Y. Niu, Airframe Stress Analysis and Sizing, second ed., Commilit. Press Ltd., 1999. [9] S. Timoshenko, J. Gere, Theory of Elastic Stability, McGraw-Hill, 1961. [10] J. Wiedemann, Leichtbau 1: Elemente, second ed., 1996. [11] J. Wiedemann, D. Fitzsimmons, Luftfahrtfor-schungsprogramm Globales Tragverhalten von Flugzeugrümpfen final report, Airbus Deutschland, August 2003. [12] K. Wolf, Untersuchungen zum Beul- und Nachbeulverhalten schubbeanspruchter Teilschalen aus kohlenstoffaserverstärkten Kunststoff, Dissertation TU Braunschweig, 1989. [13] K. Wolf, H. Kossira, An efficient test method for the experimental investigation of the postbuckling behaviour of curved composite shear panels, in: EACM-Conference in Amsterdam, 1992, pp. 133–142.