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Buckling of composite panels: A criterion for optimum stiffener design P. Weißgraeber a,∗ , C. Mittelstedt b , W. Becker a a b

Technische Universität Darmstadt, Fachgebiet Strukturmechanik, Hochschulstraße 1, D-64289 Darmstadt, Germany ELAN GmbH, Team Methods and Tools, Karnapp 25, D-21079 Hamburg, Germany

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 4 February 2010 Accepted 1 February 2011 Available online 16 February 2011 Keywords: Stability Composite plates Minimum stiffness Design criterion Buckling

In this paper a linear, closed-form analysis of the buckling behavior of an orthotropic plate with elastic clamping and edge reinforcement under uniform compressive load is presented. This is a typical structural situation found in aerospace engineering for instance as stiffeners in wings or the fuselage. All governing equations are transformed in a dimensionless system using common characteristic quantities to gain good analytical access. The buckling behavior is analyzed and generic buckling diagrams are presented. The solutions show excellent agreement with results from literature and numerical analyses. The minimum bending stiffness of the edge reinforcement needed to withstand buckling is examined and a minimum stiffness criterion is presented. Furthermore an absolute minimum bending stiffness is found which is suﬃcient to enable the reinforcement to act as a near-rigid support for arbitrarily long plates. These criteria are of interest for optimized lightweight design of stringers and stiffeners. © 2011 Elsevier Masson SAS. All rights reserved.

1. Introduction Composite materials are increasingly used in lightweight engineering. Due to the high strength to stiffness ratio of composite and common thin-walled design buckling is a typical design limit [13,14]. Therefore the stability behavior of such thin-walled composite structures has to be assessed carefully in the dimensioning procedure. Up to now many design rules and tools for predesign are only suitable for isotropic materials. A purely numerical buckling analysis of large structures as for example aircrafts in general, e.g. using the ﬁnite element method, is very time-consuming. Therefore, in particular in the predesign phase, closed-form analytical or semi-analytical design guidelines are preferable. In this paper a linear, closed-form buckling analysis of a rectangular plate with elastic clamping and edge reinforcement is presented and utilized to derive a design criterion for the bending stiffness of the edge reinforcement. Such a conﬁguration of an elastically clamped plate with edge reinforcement is found in various variants in aircraft construction, for instance in stringer reinforced sections of the wings or fuselage, see e.g. Fig. 1. As typical for the buckling behavior of reinforced plates, the bending stiffness of the edge reinforcement has an important inﬂuence on the buckling behavior of the plate [1,7,12,14,15]. In essence, two buckling modes are possible: A local buckling mode in which the plate buckles and the reinforcement has a zero deﬂection, or a global buckling mode in which the reinforcement

*

Corresponding author. E-mail address: [email protected] (P. Weißgraeber).

1270-9638/$ – see front matter doi:10.1016/j.ast.2011.02.002

© 2011 Elsevier Masson

SAS. All rights reserved.

buckles in the same buckling mode as the plate. To gain the highest possible buckling load a local buckling mode is aimed for [14]. To do so, the minimum bending stiffness of the reinforcement that allows the reinforcement to act as near-rigid support must be found. This is a common problem in the analysis of stiffened plates and the corresponding stiffness is usually referenced to as minimum stiffness. In this paper the minimum stiffness of the edge reinforcement is analyzed to derive appropriate criteria for lightweight design. Furthermore an absolute minimum stiffness which is suﬃcient to reinforce arbitrarily long plates is examined. In preceding investigations the special cases of a composite plate with edge reinforcement [7] and of a plate with elastic clamping [2,6,9] have already been examined. These cases are covered as special cases by the current, more general approach. 2. Analysis approach An elastic composite laminate plate with length a, width b and 0 is thickness d under unidirectional, uniform compressive load N 11 considered, as shown in Fig. 2. Both loaded edges are modeled as simply supported. One of the unloaded edges is simply supported but elastically clamped. The elastic clamping is modeled by a distributed torsional spring with the spring stiffness k. The other unloaded edge is unsupported and reinforced by a ﬂange of arbitrary geometry. This ﬂange in an idealized manner is modeled as an Euler–Bernoulli beam with extensional stiffness EA, bending stiffness EI and torsional stiffness GI T . A uniform compressive loading is a typical load case of stiffeners in an aeroplane fuselage whose diameter is much larger, than the height of a stiffener.

P. Weißgraeber et al. / Aerospace Science and Technology 16 (2012) 10–18

11

Π = ΠiP + ΠiB + ΠoP + ΠoB .

(5)

2.1. Boundary conditions On the loaded edges with simple support the deﬂection u 3 and the moment M 11 have to vanish. Hence the boundary conditions at the loaded edges are

Fig. 1. Examples of typical stiffener cross section designs.

u 3 (x1 = 0) = u 3 (x1 = a) = 0,

(6)

M 11 (x1 = 0) = M 11 (x1 = a) = 0.

(7)

The unloaded edge at x2 = 0 must have a zero deﬂection and the respective bending moment must be in equilibrium with the reactive moment of the distributed spring. The boundary conditions can thus be given as

u 3 (x2 = 0) = 0, Fig. 2. Compressively loaded plate with elastic clamping and reinforced edge. Dashed lines represent a simple support.

The plate behavior itself is described by Classical Laminate Plate Theory (CLPT) [4,11]. Thus in the case of a symmetrical layup with ﬂexural orthotropy the bending behavior can be written as

⎡

M 11

⎤

⎡

D 11

D 12

⎣ M 22 ⎦ = ⎣ D 12

D 22

0

M 12

0

κ11 ⎤ 0 ⎦ ⎣ κ22 ⎦ . D 66 κ12 0

⎤⎡

(1)

⎡

M 11

⎤

⎣ M 22 ⎦ = M 12

d 2

− d2

⎡

⎤

σ11 ⎣ σ22 ⎦ x3 dx3 σ12

(2)

and the curvatures result from the second-order derivatives of the deﬂection

⎡

⎡

κ11 ⎤ ⎢ ⎣ κ22 ⎦ = − ⎢ ⎢ ⎣ κ12

∂ 2 u3 ∂ x21 ∂ 2 u3 ∂ x22

⎤

⎥ ⎥ ⎥. ⎦

EI

dx21

d2 w

+F

dx21

d2 w dx21

−q=0

(10)

wherein F is the axial force and q the line load. The line load q must equal Kirchoff’s substitute transversal force of the plate

V = −2

∂ M 12 ∂ M 22 − . ∂ x1 ∂ x2

(11)

The axial force within the beam can be calculated from the condition of equal strain in plate and beam. It results in

F=

EA E p1 db

0 . N 11

(12)

With the known in-plane stiffnesses A i j from CLPT the respective modulus of elasticity E p1 is [4,11]:

E p1 =

A 11 −

A 212 A 22

d

(13)

.

With this the boundary condition of equal deﬂection can be written as:

(3)

∂ 4 u 3 EA ∂ 2 u 3 0 + N 11 E p1 db ∂ x21 x2 =b ∂ x41 x2 =b ∂ 3 u3 ∂ 3 u 3 − D 22 3 − ( D 12 + 4D 66 ) 2 ∂x ∂x ∂x

0 = EI

∂2u

2 ∂ x ∂ 3x 1 2

The buckling behavior of the plate is described by the following fourth-order partial differential equation [4,13]:

D 11

(9)

At the edge x2 = b compatibility of deﬂection and rotation of the beam and the plate is required. Hence the ﬁrst condition is that the shear force of the plate must be in equilibrium with the Kirchoff’s substitute transversal force in the beam. The second condition states that the torsional moment of the beam must be in equilibrium with the corresponding bending moment of the plate. The general differential equation of the beam deﬂection is

d2

With the simpliﬁcation of ﬂexural orthotropy the terms of bending torsion coupling D 16 , D 26 are neglected. This simpliﬁcation is made as these terms highly complicate or even impede a closed-form analytical access of the buckling behavior [11,13]. It is a typical simpliﬁcation in predesign analysis. These coupling terms can be negligible small if the layup has a suﬃcient mixing of thin plies. This is given in many practically relevant layups especially when fabrics are employed as single ply material. As known from the CLPT the moments M α β (α , β = 1, 2) result from the in-plane stresses integrated over the plate thickness

(8)

∂ u 3 . M 22 (x2 = 0) = −k ∂ x2 x2 =0

2 ∂ 4 u3 ∂ 4 u3 ∂ 4 u3 0 ∂ u3 + D + 2 ( D + 2D ) + N = 0. 22 12 66 11 ∂ x41 ∂ x42 ∂ x21 ∂ x22 ∂ x21

2

1

.

(14)

2 x2 =b

The boundary condition of equal rotation is based on the differential equation of uniform beam torsion with a distributed moment mx :

(4) This equation is used to derive a transcendental equation for the 0 . buckling load N 11 For the later use of the Ritz method the total potential of the reinforced plate with elastic clamping is needed. The total potential Π consists of the elastic potential of the plate ΠiP and the beam ΠiB plus the outer potential of the applied forces acting on the plate ΠoP and on the beam ΠoB :

x2 =b

GI T

d2 ϑ dx21

= mx .

(15) ∂u

With the slope of the plate ϑ = ∂ x 3 and the constitutive rela2 tion (1) the condition of equal rotation results in:

∂ 2 u 3 ∂ 2 u 3 ∂ 3 u 3 0 = D 12 + D 22 2 − GI T 2 . ∂ x21 x2 =b ∂ x2 x2 =b ∂ x1 ∂ x2 x2 =b

(16)

12

P. Weißgraeber et al. / Aerospace Science and Technology 16 (2012) 10–18

Table 1 Exemplary values of the orthotropy parameter ε of different laminates.

η and the generalized Poisson ratio

Material

Stacking

η

ε

HT CFRP

[0/90]s [±20]6s [±45]6s

0.23 1.37 2.41

0.21 0.30 0.32

[0/90]s [±20]6s [±45]6s

0.12 1.97 2.70

0.17 0.31 0.32

[0/90]s [±20]6s [±45]6s

0.19 1.38 2.50

0.32 0.33 0.33

UHM CFRP

E-glass

2.2. Transformation into a dimensionless system A complete transformation of the buckling problem into a dimensionless system has proven to be very advantageous as it leads to shorter expressions that can easily be used for generic buckling analysis. Let us begin with a transformation of the coordinate system onto the unit square

ξ1 =

x1 a

ξ2 =

,

x2 b

.

(17)

α¯ =

D 22

4

b

(18)

D 11

and the dimensionless buckling load 0 ¯ 11 N

=

0 2 N 11 b

π

√ 2

D 11 D 22

α¯

∂ξ14

∂ 4 u¯ 3 ∂ 4 u¯ 3 ∂ 2 u¯ 3 0 + 2η 2 2 + α¯ 2 4 + N¯ 11 π 2 2 = 0. ∂ξ1 ∂ξ2 ∂ξ2 ∂ξ1

(19)

(20)

(21)

A very decisive quantity for the buckling behavior of a plate is the so-called orthotropy parameter η [3,14]

D 12 + 2D 66

η= √

D 11 D 22

ε=

D 12 + 2D 66

(25)

EI

,

(26)

D 11 b GI T

Θ= √ b

D 11 D 22

(27)

.

Then the boundary conditions at the reinforced edge (14), (16) can be written as:

2¯ γ ∂ 4 u¯ 3 3 0 2 ∂ u ¯ 11 + N π δ α¯ 2 ∂ξ14 ξ2 =1 ∂ξ12 ξ2 =1

∂ 3 u¯ 3 ∂ 3 u¯ 3 + ( ε − 2 ) η = 0, ∂ξ12 ∂ξ2 ξ2 =1 ∂ξ23 ξ2 =1 ∂ 3 u¯ 3 ∂ 2 u¯ 3 ∂ 2 u¯ 3 ηε 2 + α¯ 2 2 −Θ 2 = 0. ∂ξ1 ξ2 =1 ∂ξ2 ξ2 =1 ∂ξ1 ∂ξ2 ξ2 =1 − α¯ 2

(22)

u¯ 3 (ξ1 , ξ2 = 0) = 0,

Π D 11 D 22

(30)

For all possible laminate setups the orthotropy parameter η takes values between η = 0 and η = 3.0 [14]. Typical values of the generalized Poisson ratio lie between 0.15 and 0.4. With this limited range for the dimensionless quantities it is possible to provide generic buckling diagrams that cover the buckling behavior of all possible laminates. Table 1 shows some laminate set ups and the values of the generic quantities of the plate. For the transformation of the boundary conditions, it is appropriate to introduce the following dimensionless quantities:

(31)

with

1 2

[0,1]2

= Π¯ iP + Π¯ iF + Π¯ oP + Π¯ oF

1

α¯ 2

1 + k¯ α¯ 2

∂ 2 u¯ 3 ∂ξ12

1

2

0

∂ u¯ 3 ∂ξ2

2

+ α¯ 2

2

∂ 2 u¯ 3 ∂ξ22

(32)

2 d(ξ1 , ξ2 )

d ξ1 + η

ξ 2 =0

[0,1]2

ε

∂ 2 u¯ 3 ∂ 2 u¯ 3 ∂ξ12 ∂ξ22

2

2 ∂ u¯ 3 + (1 − ε ) d(ξ1 , ξ2 ), ∂ξ1 ∂ξ2

2

2 1 1 γ ∂ 2 u¯ 3 2 ∂ u¯ 3 + Θ d ξ1 , Π¯ iF = √ 2 ∂ξ1 ∂ξ2 ξ2 =1 α¯ ∂ξ12 ξ2 =1 1

0 Π¯ oP = − π 2 N¯ 11

[0,1]2

1

0 Π¯ oF = − δ N¯ 11

1

2

(23)

(29)

The total potential (5) eventually can be given in the following transformed representation:

2

.

(28)

On the other hand, the boundary conditions of the elastically clamped edge (8), (9) result in

0

.

In connection with natural boundary conditions the so-called generalized Poisson ratio ε is needed as well to describe the buckling behavior

D 12

,

E p1 db

Π¯ iP =

Herein the transformation of the deﬂection u¯ 3 could be chosen arbitrarily, but for a dimensionless representation of the total potential (5) it is transformed to

u3 u¯ 3 = √ . ab

γ=

(24)

,

D 22 EA

Π¯ = √

the differential equation (4) can then be written as 4¯ 3 −2 ∂ u

δ=

kb

∂ 2 u¯ 3 ∂ u¯ 3 ¯ −k = 0. ∂ξ2 ξ2 =0 ∂ξ22 ξ2 =0

With the transformed aspect ratio

a

k¯ =

0

∂ u¯ 3 ∂ξ1

∂ u¯ 3 ∂ξ1

(33)

(34)

2

2 ξ 2 =1

d(ξ1 , ξ2 ),

(35)

d ξ1 .

(36)

With that, the buckling problem of the orthotropic plate is given in a fully dimensionless manner using the classic quantities of buckling analyses of orthotropic plates. 2.3. Exact transcendental solution of the differential equation The differential equation (4) can be solved by means of using a Levy-type solution that uses a single sine function with unknown number m of half waves to describe the deﬂection in ξ1 -direction ¯ (ξ2 ) in ξ2 -direction and an unknown function w

P. Weißgraeber et al. / Aerospace Science and Technology 16 (2012) 10–18

¯ (ξ2 ). u¯ 3 (ξ1 , ξ2 ) = sin(mπ ξ1 ) w

13

(37)

This function satisﬁes the boundary conditions (6), (7) at the edges ξ1 = 0 and ξ1 = 1 exactly. The buckling equation (20) results in an ordinary differential equation of fourth order with constant coeﬃcients

α¯ −2 −

¯0 N 11 m2

¯ − 2η(mπ )2 (mπ )4 w

4 ¯ ¯ ∂2 w 2∂ w ¯ + α = 0. ∂ξ22 ∂ξ24

(38)

This differential equation can be solved using the ansatz function ¯ = emπ λξ2 . The general solution is w

¯ = C 1 cosh(mπ k1 ξ2 ) + C 2 sinh(mπ k1 ξ2 ) w + C 3 cos(mπ k2 ξ2 ) + C 4 sin(mπ k2 ξ2 )

(39)

with

¯ 0 α¯ 2 N 1 k1,2 = ± ±η + η2 + 112 − 1 α¯ m

3. Buckling analysis

(40)

for 0 ¯ 11 N >

m2

α¯ 2

(41)

.

The unknown factors C 1 to C 4 can be determined from the boundary conditions (28)–(31) at the unloaded edges. This leads to a linear equation system of the kind A · C = 0. For a non-trivial solution the determinant of the coeﬃcient matrix A must be zero. ¯ 0 which This provides an equation for the critical buckling load N 11 is transcendental:

¯ 0 , m, η, ε , α¯ , δ, γ , Θ 0= f N 11

= F 1 + cosh(k1mπ ) F 2 cos(k2mπ ) + F 3 sin(k2mπ ) + sinh(k1mπ ) F 4 cos(k2mπ ) + F 5 sin(k2mπ )

(42)

with the abbreviations

Fig. 3. Buckling curve of a reinforced plate with elastic clamping. The lowest buckling load of all buckling modes is the relevant one.

¯ 1k2m k4 + k4 α¯ 6 + 2 −k2 + k2 α¯ 4 η F 1 = kk 1 2 1 2

0 − 2α¯ 2 (−2 + ε )εη2 + 2π 2 m2 γ − N¯ 11 α¯ 2 δ Θ , (43) 2 2 2 2 2 4 ¯ Θ F 2 = k1 k2 m k1 + k2 m π α 2 2 4 2 2 2 2 + 2k¯ −m π γ Θ + α¯ k1k2 α¯ + k1 − k22 α¯ 2 η 0 + (−2 + ε )εη2 + N¯ 11 π 2 δΘ , (44) 2 2 2 4 2 2 2 ¯ k2 α¯ + εη F 3 = −k1 k1 + k2 π k1 m α 2 2 2 2 + m α¯ (−2 + ε )η k2 α¯ + εη 0 + π 2 −m2 γ + N¯ 11 α¯ 2 δ Θ 0 α¯ 2 δ + m2 γ + k22 α¯ 2 Θ , (45) − k¯ α¯ 2 − N¯ 11 2 2 2 2 0 4 ¯ γ + N¯ 11 α¯ δ F 4 = k2 k1 + k2 π k¯ −m α 2 2 4 2 2 + k1m α¯ k2 α¯ − (−2 + ε )η + k¯ Θ + m2 α¯ 2 εη −k22 α¯ 2 + (−2 + ε )η 0 + π 2 −m2 γ + N¯ 11 α¯ 2 δ Θ , (46) 2 2 0 ¯ 2 m2 γ − N¯ 11 α¯ 2 δ F 5 = m k1 + k22 π 2 α 4 4 2 2 − k¯ k1 α¯ k2 α¯ + εη + k22 α¯ 2 εη k22 α¯ 2 + (2 − ε )η 0 + π 2 m2 γ − N¯ 11 α¯ 2 δ Θ + k21 −k42 α¯ 6 + 2k22 α¯ 4 (−2 + ε)η 0 + α¯ 2 (−2 + ε )εη2 + π 2 −m2 γ + N¯ 11 α¯ 2 δ Θ . (47) The solution of this transcendental equation provides the exact buckling load of the reinforced plate with elastic clamping.

The solution of the transcendental equation (42) can be found in a numerical manner, for example with common mathematical toolboxes, that have built-in routines for root searches. As typical in buckling analysis the buckling load depends on the number of half waves m, whereat the lowest buckling load is the relevant one for all analysis and design purposes. Based on the typical behavior of Eq. (42) the ﬁrst maximum above the lower limit (41) is searched and used as a starting point to ﬁnd the ﬁrst root. Typically reinforcement proﬁles of stringers used in aerospace engineering are open proﬁled. For this reason the torsional stiffness GI T of the reinforcing beam is neglected in the following. This is expected to be a conservative assumption. A typical buckling curve is shown in Fig. 3. It can be seen that ¯ ≈ 2.5) the curves show up to a certain aspect ratio (in Fig. 3: α the typical behavior e.g. as known from a simply supported plate. The number of half waves m increases with higher aspect ratios, which represents a change of the buckling mode. Above the forecited aspect ratio a new buckling behavior occurs and the number of half waves is again m = 1. This happens due to a change from local to global buckling. This change is illustrated in Fig. 4, which shows the global and local buckling mode of a plate with aspect ¯ = 2.63 respectively α¯ = 2.7. ratios of α Up to this change of local to global buckling the reinforcement can be considered to act as a near-rigid support. Above this change the critical buckling load decreases quickly with higher aspect ratios. The aspect ratio associated with this change depends on the elastic properties of the beam. Reinforcements with higher related bending stiffness γ can act as a near-rigid support for plates with ¯. higher aspect ratios α A higher extensional stiffness δ acts opposingly as it attracts axial force into the beam, which leads to column buckling of the reinforcement. Due to the elastic clamping the critical buckling load does not descend to arbitrarily small values (as in case of no elastic clamping [7]) but it converges to a non-zero value, see Fig. 3. Examples of the calculated generic buckling diagrams are shown in Figs. 5 and 6. These diagrams show the inﬂuence of the clamping stiffness k¯ and the orthotropy parameter η . The inﬂuence of the generalized Poisson ratio ε turns out to be very small. Fig. 5 shows generic buckling diagrams with different values ¯ With higher clamping stiffnesses the of the clamping stiffness k. buckling loads increase. The higher the clamping stiffness the lower the aspect ratios where the changes of buckling modes appear. Fig. 6 shows generic buckling diagrams with different values of the orthotropy parameter η . It can be seen that the buckling loads

14

P. Weißgraeber et al. / Aerospace Science and Technology 16 (2012) 10–18

Fig. 4. Local and global buckling modes of a reinforced plate with elastic clamping.

increase and that higher bending stiffnesses γ of the beam are needed to reinforce the plate with higher orthotropy parameters. The results of this closed-form buckling analysis have been approved by comparison with the results of other investigators [5, 7–9] and with numerical ﬁnite element analyses of the authors. Fig. 7 shows the comparison of the closed-form buckling analysis with a ﬁnite element analysis. The ﬁnite element analyses of the [0/90]s HT CFRP laminate were performed with Abaqus 6.7.4 using up to 110 000 degrees of freedom. 4. Minimum stiffness Choosing a suﬃcient stiffness of the reinforcement is an important task in the dimensioning process of stiffener design. As it can be seen in the buckling diagram, an insuﬃcient bending stiffness results in a major decrease of the bearable loads. Above a certain stiffness no further increase of the buckling load can be gained. This minimum stiffness that enables the reinforcement to act as a near-rigid support must be found to achieve an optimized lightweight design of the reinforcement. In many situations [12, 14,15] such a minimum stiffness is needed to get a local buckling behavior instead of a global buckling behavior in which the reinforcement follows the buckling eigenmode of the plate. Solving the transcendental equation (42) for the bending stiffness γ leads to an equation of the type

B 0 γ = f N¯ 11 , m, k¯ , η, ε , δ, α¯ =

A

(48)

with

0 π k¯ N¯ 11 α¯ 2 δ + k21m2 α¯ 2 k22 α¯ 2 − (−2 + ε)η

(49)

(50)

+ m2 εη −k22 α¯ 2 + (−2 + ε )η , 2 0 2 2 ¯ 11 π α¯ δ + mk¯ k21 (k1 − k2 )k22 (k1 + k2 )α¯ 4 B 5 = m k21 + k22 N 2 + α¯ 2 −4k21k22 + k21 + k22 ε η + (k1 − k2 )(k1 + k2 )(−2 + ε )εη2 .

With this equation it is possible to compute the bending stiffness required to achieve a deﬁned buckling load. Whereas it is necessary to ﬁnd the relevant number of half waves m that maximizes the bending stiffness. If the chosen buckling load is set to the buckling load of the plate with simple supports on all edges the minimum stiffness can be computed. For the use in practice it would be very advantageous to have an explicit equation for the buckling load resulting in an explicit criterion for the minimum stiffness. Therefore in the sequel an approximate solution is computed by the means of the Ritz method. To describe the plate deﬂection the same deﬂection function as employed by Qiao et al. [10] is used. Accordingly, a sine function in ξ1 -direction and a fourth-order polynomial function in ξ2 -direction are employed for the deﬂection of the plate. Using the transformed variables it is

u¯ 3 = Ω ξ2 + a1 ξ22 + a2 ξ23 + a3 ξ24 sin(mπ ξ1 ).

(51)

The free constants a1 to a3 can be determined from the boundary conditions at the unloaded edges (28)–(31), wherein the bearing on all edges changes the boundary conditions of the unloaded edge ξ2 = 1 to

u¯ 3 (ξ2 = 1) = 0.

A = A 1 cosh(k1 mπ ) sin(k2 mπ ) + A 2 cos(k2 mπ ) sinh(k1 mπ )

+ A 3 sin(k2mπ ) sinh(k1mπ ), B = B 1 + cosh(k1 mπ ) B 2 cos(k2 mπ ) + B 3 sin(k2 mπ ) + sinh(k1mπ ) B 4 cos(k2mπ ) + B 5 sin(k2mπ ) , ¯ 1, A 1 = k21 + k22 m2 π kk 2 ¯ 2, A 2 = − k1 + k22 m2 π kk 2 2 A 3 = k1 + k22 m3 π 2 , 4 ¯ 1k2m k + k4 α¯ 4 + 2 −k2 + k2 α¯ 2 η B 1 = −kk 1 2 1 2 − 2(−2 + ε )εη2 , ¯ 1 k2 k2k2 α¯ 4 + k2 − k2 α¯ 2 η + (−2 + ε )εη2 , B 2 = −2kk 1 2 1 2 0 2 ¯ 11 α¯ δ + k21m2 α¯ 2 k22 α¯ 2 + εη B 3 = k1 k21 + k22 π k¯ N + m2 (−2 + ε )η k22 α¯ 2 + εη ,

B 4 = −k2 k21 + k22

(52)

Taking into account the boundary conditions the deﬂection function eventually results in

1¯ 2 5¯ 3 u¯ 3 = Ω ξ2 + kξ2 − 2 + k ξ2 2 6

1 + 1 + k¯ ξ24 sin(mπ ξ1 )

(53)

3

and with this the total potential of the plate can be written as

0 Π¯ = f k¯ , η, α¯ , N¯ 11 =

Ω2 ¯2 90720α

1116 + 285k¯ + 19k¯ 2 m4 π 4

+ 4536 24 + 11k¯ + k¯ 2 α¯ 4 0 2 + m2 π 2 α¯ 2 − 1116 + 285k¯ + 19k¯ 2 N¯ 11 π 2 + 432 51 + 13k¯ + k¯ η .

(54)

P. Weißgraeber et al. / Aerospace Science and Technology 16 (2012) 10–18

¯ Fig. 5. Generic buckling diagrams with different values of the clamping stiffness k.

15

Fig. 6. Generic buckling diagrams with different values of the orthotropy parameter η .

16

P. Weißgraeber et al. / Aerospace Science and Technology 16 (2012) 10–18

Fig. 7. Comparison of the exact solution with a ﬁnite element analysis.

Fig. 8. The minimum stiffness of the edge reinforcement against the aspect ratio of orthotropic plate.

In linear elastic stability problems the buckling condition requires a vanishing ﬁrst variation of the total potential

δ Π¯ = 0.

(55)

As the only variable in the deﬂection function (53) is the amplitude Ω this reduces to

∂ Π¯ = 0. ∂Ω

(56)

To obtain the result for the buckling load, the ﬁrst variation of the total potential with respect to the quantity Ω is to be solved ˜0 for the approximated buckling load N 11 0 ˜ 11 N =

(1116 + 285k¯ + 19k¯ 2 )m4 π 4 (1116 + 285k¯ + 19k¯ 2 )m2 π 4 α¯ 2 + +

¯2 432(51 + 13k¯ + k¯ 2 )m2 π 2 ηα

(1116 + 285k¯ + 19k¯ 2 )m2 π 4 α¯ 2 ¯4 4536(24 + 11k¯ + k¯ 2 )α (1116 + 285k¯ + 19k¯ 2 )m2 π 4 α¯ 2

.

(57)

This explicit, approximate solution can be compared with the exact solution and shows a good agreement with the exact solution calculated with inﬁnite bending stiffness. Due to the character of the Ritz method the values of the approximation are always somewhat higher than the exact values. The largest error arises at high spring stiffnesses of the clamping and is still lower than 2.2%. A more sophisticated function for the plate deﬂection could reduce the error even more but would lead to longer, less handy equations or could even impede a closed-form analytical access. Due to this the small error resulting from the simple deﬂection function is tolerated.

Fig. 9. Buckling curves with different values of the bending stiffness showing the two possible modes of stability failure. γAbsMin denotes the absolute minimum stiffness. For reasons of clearness only the ﬁrst four buckling modes are shown.

The approximate buckling load is an upper bound, therefore it is important to multiply the approximate buckling load with a constant ρ (ρ < 1) before using it to obtain the minimum stiffness from Eq. (48). Otherwise a buckling load could result from the Ritz method which cannot be totally reached by the exact solution. This would result in erroneous, non-real solutions for the bending stiffness. Furthermore, a requirement of a very close approach to the maximum buckling load leads to extremely high values for the minimum bending stiffness. When the requirement is mildly relaxed the resulting bending stiffness is signiﬁcantly lower. Parameter studies have shown that a value ρ = 0.95 is reasonable for lightweight design and is chosen in this analysis. Eq. (48) can be used to generate very precise generic minimum stiffness curves, as for example the one shown in Fig. 8. Unlike in the case of plates without elastic clamping the minimum stiffness values do not increase in an unlimited manner with higher aspect ratios. The minimum stiffness curve reaches a maximum value to which it converges with higher aspect ratios. This upper limit of the minimum stiffness will be called absolute minimum stiffness. It is a bending stiffness that is suﬃcient to reinforce arbitrarily long plates and thus offering a near rigid support. A parameter study for the absolute minimum stiffness has been performed. A full set of absolute minimum stiffnesses is shown in Appendix A. The occurrence of the absolute minimum stiffness can be explained when the two possible stability failure modes of the reinforced plate are regarded separately: the buckling of the plate

P. Weißgraeber et al. / Aerospace Science and Technology 16 (2012) 10–18

(local buckling mode) and the column buckling of the reinforcement (global buckling mode). The reinforcement can be considered to act like a bedded column [1]. With an increasing bedding factor the critical load of the bedded column can exceed the critical load of the buckling plate. When the bending stiffness is set to the absolute minimum stiffness the critical loads of column buckling and plate buckling are equal. Those three possible cases are shown in Fig. 9. Unfortunately, it is not possible to obtain the absolute minimum stiffness in a closed-form analytical way as the derivative dγ ¯ . Thus ¯ leads to a transcendental equation for the aspect ratio α dα the maximum of the minimum stiffness curves has to be identiﬁed numerically. This identiﬁcation can be performed with any mathematical toolbox.

Table 2 Absolute minimum stiffnesses for elastic clamping stiffnesses k¯ = 0.6 and k¯ = 5.

η

Appendix A. Absolute minimum stiffnesses

δ

k¯ 0.6

5

ε = 0.2

ε = 0.4

0.05

0.0 0.1 0.2 0.3

21.18 36.68 56.47 80.57

21.31 36.85 56.70 80.84

6.05 11.17 17.92 26.30

6.08 11.22 17.99 26.39

0.5

0.0 0.1 0.2 0.3

36.28 65.00 102.23 147.98

38.01 67.36 105.23 151.60

9.67 18.36 30.02 44.66

10.15 19.08 30.98 45.85

1

0.0 0.1 0.2 0.3

57.75 105.85 168.81 246.65

62.14 111.91 176.54 256.05

14.66 28.41 47.08 70.68

15.85 30.20 49.48 73.68

1.5

0.0 0.1 0.2 0.3

84.17 156.57 251.97 370.35

92.14 167.66 266.17 387.67

20.68 40.62 67.94 102.62

22.79 43.84 72.27 108.06

2

0.0 0.1 0.2 0.3

115.52 217.17 351.69 519.08

128.00 234.62 374.11 546.47

27.71 55.00 92.59 140.49

30.97 60.00 99.34 148.98

2.5

0.0 0.1 0.2 0.3

151.82 287.65 467.99 692.84

169.72 312.79 500.36 732.44

35.77 71.53 121.04 184.28

40.39 78.67 130.70 196.45

3

0.0 0.1 0.2 0.3

193.05 368.00 600.87 891.63

217.31 402.17 644.92 945.58

44.85 90.24 153.29 233.99

51.05 99.87 166.34 250.46

5. Conclusions In this paper a linear, closed-form buckling analysis of an orthotropic plate with edge reinforcement and elastic clamping has been presented. A transformation into a dimensionless system with classic quantities is used and proved to be advantageous as it leads to compact equations and a good analytical access. The cases of a plate with edge reinforcement [7] and a plate with elastic clamping [2,6,9] are included as special cases of this more general approach. The buckling behavior and the interaction of the boundary conditions have been analyzed and generic buckling diagrams have ¯ 0 can been presented. Even though Eq. (42) for the buckling load N 11 only be solved numerically the solution is exact as it is based on the solution of the governing differential equation. The inﬂuence of the bending stiffness of the edge reinforcement has been studied and a minimum stiffness criterion has been provided. It enables an optimized lightweight composite stiffener design. Generic minimum stiffness diagrams have been generated and discussed. Furthermore an absolute minimum stiffness, that is suﬃcient to enable the reinforcement to act as a near-rigid support for arbitrarily long plates is found. A complete set of absolute minimum stiffnesses for all practically relevant parameter ranges has been given and can be readily used for predesign of reinforced panels.

17

ε = 0.2

ε = 0.4

Table 3 Absolute minimum stiffnesses for elastic clamping stiffnesses k¯ = 10 and k¯ = 20.

η

δ

k¯ 10

20

ε = 0.2

ε = 0.4

0.05

0.0 0.1 0.2 0.3

5.15 9.75 15.88 23.53

5.19 9.80 15.94 23.61

4.77 9.18 15.11 22.55

4.80 9.23 15.17 22.63

0.5

0.0 0.1 0.2 0.3

8.06 15.67 26.00 39.05

8.47 16.29 26.84 40.10

7.36 14.57 24.44 36.96

7.74 15.16 25.23 37.96

1.0

0.0 0.1 0.2 0.3

12.03 23.86 40.13 60.84

13.02 25.40 42.22 63.48

10.88 21.99 37.39 57.09

11.80 23.43 39.36 59.57

1.5

0.0 0.1 0.2 0.3

16.78 33.75 57.30 87.43

18.54 36.51 61.06 92.17

15.08 30.91 53.07 81.56

16.69 33.47 56.59 86.02

2.0

0.0 0.1 0.2 0.3

22.31 45.34 77.51 118.81

25.01 49.60 83.34 126.19

19.96 41.34 71.48 110.37

22.42 45.29 76.93 117.30

2.5

0.0 0.1 0.2 0.3

28.62 58.63 100.75 154.98

32.43 64.69 109.06 165.53

25.52 53.28 92.62 143.52

28.98 58.89 100.37 153.41

3.0

0.0 0.1 0.2 0.3

35.72 73.62 127.03 195.94

40.82 81.78 138.24 210.19

31.76 66.72 116.48 181.02

36.38 74.26 126.92 194.36

See Tables 2–4.

ε = 0.2

ε = 0.4

18

P. Weißgraeber et al. / Aerospace Science and Technology 16 (2012) 10–18

Table 4 Absolute minimum stiffnesses for elastic clamping stiffnesses k¯ = 50 and k¯ = 10 000.

η

δ

k¯ 50

10 000

ε = 0.2

ε = 0.4

ε = 0.2

ε = 0.4

0.05

0.0 0.1 0.2 0.3

4.57 8.93 14.82 22.23

4.60 8.98 14.88 22.30

4.47 8.83 14.73 22.19

4.50 8.87 14.79 22.26

0.5

0.0 0.1 0.2 0.3

7.01 14.09 23.83 36.22

7.38 14.66 24.60 37.19

6.84 13.91 23.66 36.11

7.20 14.46 24.42 37.07

1.0

0.0 0.1 0.2 0.3

10.33 21.17 36.31 55.72

11.20 22.56 38.21 58.14

10.06 20.88 36.03 55.51

10.92 22.24 37.91 57.90

1.5

0.0 0.1 0.2 0.3

14.27 29.68 51.39 79.4

15.80 32.15 54.80 83.73

13.9 29.24 50.97 79.05

15.39 31.67 54.33 83.34

2.0

0.0 0.1 0.2 0.3

18.84 39.61 69.08 107.24

21.18 43.42 74.35 113.97

18.34 39.01 68.48 106.73

20.62 42.75 73.68 113.38

2.5

0.0 0.1 0.2 0.3

24.05 50.97 89.37 139.25

27.34 56.37 96.87 148.84

23.39 50.17 88.56 138.54

26.60 55.48 95.96 148.03

3.0

0.0 0.1 0.2 0.3

29.88 63.75 112.27 175.42

34.27 70.99 122.36 188.36

29.05 62.74 111.22 174.50

33.33 69.86 121.18 187.28

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