Strength and stability of geometrically nonlinear orthotropic shell structures

Thin-Walled Structures 106 (2016) 428–436

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Strength and stability of geometrically nonlinear orthotropic shell structures A.A. Semenov Saint Petersburg State University of Architecture and Civil Engineering, 2nd Krasnoarmeyskaya st., 4, Saint Petersburg, 190005, Russia

art ic l e i nf o

a b s t r a c t

Article history: Received 8 August 2015 Received in revised form 13 May 2016 Accepted 23 May 2016

The article presents a methodology for the study of shell structure strength and stability. The basis of the study is a geometrically nonlinear mathematical model, which takes into account the transverse shifts and orthotropy of material. The model is presented in dimensionless parameters in the form of the total energy potential functional and can be used for different types of shells of revolution. The model is studied by using an algorithm based on the Ritz method and the method of solution continuation according to the best parameter (MSCBP), which allows for obtaining the values of the upper and lower critical loads and examining the supercritical behavior of designs. In accordance with an algorithm, the computer program has been developed and a comprehensive study of the strength and stability of shallow shells (which are square in plane), cylindrical, and conical panels has been explored. The load loss of strength and buckling load values have been obtained, and their relationship to one another has been demonstrated. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Shells Strength Stability Orthotropy Cylindrical panels Conical panels

1. Introduction In modern building, as well as in shipbuilding, mechanical engineering, aviation, and other industries, structures in the form of shells have wide application [1–12]. Currently there are composite materials [13–19] (CFRP, GRP etc.), which have high strength, fire resistance, chemical and corrosion resistance, lightness, and their application in the design of shell structures deserves much attention. Given that the reinforcement elements in the material are often placed along curved axes of the shell’s coordinate system, such structures are to be considered orthotropic [20,21]. Renewed interest in the study of shell structures in recent years has arisen not only due to the emergence of promising new materials, but also, above all, due to the development of computer technologies, which allow one to take a fresh look at the nonlinear problems of shells [22,23]. Composite materials are widely used in machine building, shipbuilding and rocketbuilding, but in building, used as coatings of span structures, such materials are still not widely used because of their high cost and the lack of research on such structures. In considering the problems of shell stability it is important to analyze the strength of the material, given that after the flowing deformation or brittle fracture stress, irreversible changes occur, and study of the stability of the structure in the linear elastic E-mail address: [email protected] http://dx.doi.org/10.1016/j.tws.2016.05.018 0263-8231/& 2016 Elsevier Ltd. All rights reserved.

formulation is invalidated. Therefore the combined research of strength, stability and supercritical orthotropic materials shell behavior is topical, and it is based on the most accurate mathematical models of their deformation, effective computational algorithms, and specially developed software. Shell resistance has been studied by many authors, but almost all publications relate to the study of isotropic shells. The works of Rikards and Teters [20] are among the first studies on the stability of orthotropic shells. In these works a mathematical model based on the hypothesis of Kirchhoff – Love was used; however, as shown by experimental research, it is necessary to consider lateral shifts in the study of the stability of such shells. The works by Reddy, Guz and Babic, Maksimyuk and Chernyshenko and others have generated great theoretical interest. The present state of the different divisions of shell theory is well represented in review articles and monographs by Carrera [1], Qu [24], Reddy [25], Ventsel and Krauthammer [10], Golushko and Nemirovsky [4], Karpov [26,27], Grigolyuk and Kulikov [28], Maksimyuk and Chernyshenko [21] et al. [29–32]. A number of publications is devoted to the review on works on shell stability [33–35]. Composite material shell stability is examined in the works of Trushin [29]. Experimental data of the determination of the limit values of composite material shell stresses are presented in the works of Smerdov et al. [13]. Special attention should be paid to conical shells, which for the most part are widely used in aviation technology and engineering,

A.A. Semenov / Thin-Walled Structures 106 (2016) 428–436

but also used in construction [12]. In the field of the shell stability research, the works of Mushtari, Alumyae, Grigolyuk, Sachenkov et al., as well as works [36–39] should be noted. The problem of conical shell stability was reduced to finding the eigenvalues of the system of partial differential equations with variable coefficients, and it was shown that the solution must be sought approximately [40]. One of the previously used approaches to solve this problem was the reduction of the conical shell to the cylindrical shell. The radius of the cylindrical shell is adopted as a cross between a large and small radius of the conical shell. This technique proved successful in the calculation of shells with a small conical angle [transformed], but with its increasing conical shell structure specificity its stability begins to be affected, and this approach is no longer acceptable. Compared to calculating cylindrical shells, it is more difficult to explore such constructions. This is manifest primarily in the complication of the geometrical relationships which link displacements and deformations. If we substitute in them the formulas of curvature and the Lame parameters ( A = 1, B = x⋅ sin θ , ctgθ k x = 0, k y = x ), due to the formula’s dependence on the variable x , the complexity of further calculations increases significantly. Recent works in this area should include an article by Shadmehri et al. [38], where closed conical shells made of composite materials are considered; however the mathematical model is based on first order theory, and geometric nonlinearity is not taken into account. In most works concerning shell stability, cylindrical shells are the most studied, but supercritical behavior and the relationship between stability and strength are not investigated. The purpose of this work is to study the strength and stability of geometrically nonlinear orthotropic shell structures based on the proposed methodology.

2. Methodology used and related equations 2.1. Governing equations A mathematical model of deformation of shells consists of three groups of relations:

 geometric relations linking deformations and displacements;  physical relations between stresses and strains;  the total energy functional of deformation of the shell, from the minimum of which the equations of equilibrium are derived. Fig. 1 shows the general view of the thin-walled shell with axes of the local coordinate system x, y, z (orthogonal coordinate system in the middle surface of the shell structure; x, y – the curvilinear coordinates directed along the lines of the principal curvatures, z – coordinate directed towards the concavity of the shell surface perpendicular to the middle). For shells of revolution of the general form we can introduce the following dimensionless parameters [41]:

ξ=

x , a

a¯1 =

aUA U¯ = 2 , h σ¯x =

a1 , a

bVB V¯ = 2 , h

(

)

λ=

2

E1h

aA A¯ = , h

aA , bB

,

σ¯y =

kξ = hk x,

Ψ aA Ψ¯x = x , h

¯ = W, W h

2 2

σx 1 − μ12 μ21 a A

a4A4 q P¯ = 4 , h E1

y , b

η=

(

)

Ψ¯y =

bB B¯ = , h

E2h z¯ =

z , h

ΨybB h

2 2

σy 1 − μ12 μ21 a A 2

k η = hk y,

,

τ¯xy =

429

Fig. 1. General view of the thin-walled shell with axes of the local coordinate system.

x and y respectively [m], λ – dimensionless coefficient; ξ, η, z¯ – new (dimensionless) coordinate system of the shell; U = U ( x, y ) , V = V ( x, y ) , W = W ( x, y ) – displacements of points of the middle surface of the shell along axes x, y, z [m]; Ψx = Ψx( x, y ) , Ψy = Ψy( x, y ) – rotation angles of normal in the planes xOz and yOz respectively; k x =

1 , R1

ky =

1 R2

– the major curvatures of

shell along axes x and y [1/m]; R1, R2 – the major radii of curvature of shell along axes x and y [m]; A , B – Lame parameters, which characterize the geometry of the shell; h – shell thickness [m]; q = q( x, y ) – outward transverse load [MPa]. Geometric relations in the middle surface of the shell, taking into account the geometric nonlinearity [42], using the dimensionless parameters (1), take the following form

ε¯x =

γ¯xy =

¯ ¯ 2 ∂U¯ ¯ + 1 θ¯ 2; ε¯ = λ2 ∂V + 1 ∂B U¯ − k A¯ 2 W ¯ + 1 λ2θ¯ 2; − kξA¯ W 1 y η 2 2 B¯ ∂ξ ∂ξ 2 ∂η ∂V¯ ∂U¯ 1 ∂B¯ ¯ + − V + θ¯1θ¯2; γ¯xz = k f ( z¯ )⎡⎣ Ψ¯x − θ¯1⎤⎦; ∂ξ ∂η B¯ ∂ξ γ¯ = λk f ( z¯ )⎡⎣ Ψ¯ − θ¯ ⎤⎦. y

yz

2

Here f ( z¯ ) – function which characterizes the distribution of stresses τyz over the thickness of shell τxz and

f ( z¯ ) = 6

(

1 4

)

5 − z¯ 2 ; k = 6 ;

¯ ⎛ ∂W ⎞ ⎛ ∂W ⎞ ¯ θ¯1 = − ⎜ + kξU¯ ⎟; θ¯2 = − ⎜ + k ηV¯ ⎟. ⎝ ∂ξ ⎠ ⎝ ∂η ⎠ Functions of change of curvature and torsion

χ¯1 =

∂Ψ¯y ∂Ψ¯y ∂Ψ¯x ∂Ψ¯ 1 ∂B¯ ¯ 1 ∂B¯ ¯ Ψx; 2 χ¯12 = Ψy . + + x − ; χ¯2 = λ2 ¯ ∂ξ ∂η ∂ξ ∂η B ∂ξ B¯ ∂ξ

At the linearly elastic deformation, physical relations for orthotropic shells of revolution [43] with the dimensionless parameters (1) can be written as

,

τxya2A2 2

G12h

,

(

)

(

)

σ¯x = ε¯x + μ21ε¯y + z¯ χ¯1 + μ21χ¯2 ; σ¯y = ε¯y + μ12 ε¯x + z¯ χ¯2 + μ12 χ¯1 ;

τ¯xy = λ⎡⎣ γ¯xy − 2z¯χ¯12 ⎤⎦; τ¯xz = Gkf ( z¯ )A¯ ⎡⎣ Ψ¯x − θ¯1⎤⎦; τ¯yz = kf ( z¯ )A¯ λ⎡⎣ Ψ¯y − θ¯2⎤⎦. (1)

where a, b – linear dimensions of the shell in the directions of axes

Functionality of total potential energy of deformation [44] is written as follows:

430

E¯ p =

A.A. Semenov / Thin-Walled Structures 106 (2016) 428–436

1

η2( ξ )

1

1

This approach allows us to explore the strength and stability of shells, to bypass the singular points of curve “load - deflection” in order to obtain the values of upper and lower critical loads, find points of bifurcation, and investigate the supercritical behavior of design [45–47].

∫a¯ ∫η ( ξ) { ε¯x2 + G¯ 2ε¯y2+( μ21 + G¯ 2μ12)ε¯xε¯y + G¯ 12λ2γ¯xy2 + 1 2 χ¯ + G¯ 2χ¯22 + μ21 + G¯ 2μ12 12 1 2 2 2 + G¯ 13kA¯ ( Ψ¯x − θ¯1) + + G¯ 23kA¯ λ2

(

+

)χ¯1χ¯2 + 4G¯ 12λ2χ¯122 )

(

2

( Ψ¯y − θ¯2)

(

)

¯ ¯ ξ dη . ¯ ¯ }ABd − 2 1 − μ12 μ21 PW

(2) 3. Numerical results

In view of the fact that for orthotropic material E1μ21 = E2μ12, the notations below are introduced:

(

)

(

)

G12 1 − μ12 μ21 G13 1 − μ12 μ21 E G¯ 2 = 2 , G¯ 12 = , G¯ 13 = , G¯ 23 E1 E1 E1 =

(

).

G23 1 − μ12 μ21 E1

2.2. The Ritz method and the method of solution continuation according to the best parameter In order to minimize the total potential energy of deformation functional, we used the Ritz method. To solve problems in the dimensionless parameters, the unknown functions are: N

U¯ ( ξ , η) =

N

∑ U¯ (I )Z¯ 1(I );

V¯ ( ξ , η) =

∑ V¯ (I )Z¯ 2(I );

I=1

I=1 N

¯ ( ξ , η) = W

∑ W¯ (I )Z¯ 3(I ); I=1

N

Ψ¯x( ξ , η) =

N

∑ PS(I )Z¯ 4(I );

Ψ¯y( ξ , η) =

I=1

∑ PN (I )Z¯ 5(I ), I=1

¯ ( I ), PS( I ), PN ( I ) – unknown dimensionless here U¯ ( I ), V¯ ( I ), W numerical coefficients, Z¯ 1( I ) – Z¯ 5( I ) – known approximate functions of arguments ξ and η , which satisfy the given boundary conditions on the contour of the shell, N – the number of terms in the expansion. By substituting the expansion of unknown functions in the functional (2), and conducting the Ritz method procedure, we obtain a system of nonlinear algebraic equations. The resulting system will be solved by the method of solution continuation according to the best parameter [45–47].

3.1. About the materials of the constructions under consideration The mechanical characteristics of the construction materials are shown in Table 1. In addition to the orthotropic shell materials (CFRP [48] and fiberglass [49]), isotropic shells made from steel and plexiglass were considered for comparison. To substantiate the reliability of the results, calculations were made on the structures [50,51]. The material parameters of these designs are also given in Table 1 and designated as Material 1 and Material 2. For all designs hereinafter, the axes of orthotropy of material 1 and 2 coincide with the local coordinate system of shell x, y, z . The direction of orthotropy axis 1 coincides with the direction of axis x (dimensionless axis ξ ). In Table 1 E1, E2 – modules of elasticity in the directions of orthotropy 1 and 2 [MPa]; G12, G13, G23 – modules of shear in the planes xOy, xOz, yOz respectively [MPa]; μ12 , μ21 – Poisson’s ratios; F1+, F2+ – ultimate tensile strength in the directions of orthotropy 1, 2 [MPa]; F1−, F2− – ultimate compressive strength in the directions + − – ultimate shear strength of orthotropy 1, 2 [MPa]; F12, F12,45 , F12,45 in the plane of orthotropy and at an angle 45° to the axis of orthotropy [MPa]; σT – the yield stress for isotropic material [MPa]. The analysis of the material strength of orthotropic constructions is conducted according to the criterion of maximum stresses. We will use this criterion because the limiting loads of strength loss which were found by the criterion of maximum stresses are practically identical, with the same loads which were found by other criteria [52] and, moreover, in contrast to other criteria, it points to the component of stresses in which a loss of strength is found. Criterion of maximum stress is:

F1− ≤ σ11 ≤ F1+, Here

F2− ≤ σ22 ≤ F2+,

τ12 ≤ F12.

σ11, σ22 – the normal stresses in the directions of

Table 1 The mechanical characteristics of the materials of considered constructions. Characteristic

Isotropic Steel

E1, MPa μ12

2.1⋅105 0.3

Orthotropic Plexiglass

0.03⋅105 0.35

Material 1 [50,51]

CFRP [48]

Fiberglass [49]

LU-P / ENFB

T300/Epoxy

M60J/Epoxy

T300/976

T-10/UPE22-27

0.3⋅105 0.3

1.4⋅105 0.3

1.25⋅105 0.34

3.3⋅105 0.32

1.4⋅105 0.29

0.294⋅105 0.123

Material 2 [50,51]

0.2⋅105 0.1

E2, MPa

2.1⋅105

0.03⋅105

0.3⋅105

0.97⋅104

0.78⋅104

0.59⋅104

0.97⋅104

1.78⋅104

0.4⋅105

G12, MPa

0.807⋅105

0.012⋅105

1.154⋅104

0.46⋅104

0.44⋅104

0.39⋅104

0.55⋅104

0.301⋅104

1.0⋅104

G13, MPa

0.807⋅105

0.012⋅105

1.154⋅104

0.46⋅104

0.44⋅104

0.39⋅104

0.55⋅104

0.301⋅104

1.0⋅104

G23, MPa

5

5

4

4

4

4

4

4

1.0⋅104 –

F1+, MPa F1−, MPa F2+, MPa F2−, MPa F12, MPa + F12,45 , MPa − F12,45 , MPa

σT , MPa Density, kg/m3

0.807⋅10 –

0.012⋅10 –

– –

– –

– –

−600 27

−1570 80

−780 30

−1599 46

−209 246

– –

– – –

– – –

– – –

−184 55 –

−168 98 –

−168 39 –

−253 41.4 –

−117 43 130

– – –

–

–

–

–

–

–

–

−160

–

1720 7800

75 1190

– –

– 1500

– 1500

– 1500

– 1500

– 1800

– –

1.154⋅10 –

0.46⋅10 700

0.44⋅10 1760

0.39⋅10 1760

0.33⋅10 1517

0.301⋅10 508

A.A. Semenov / Thin-Walled Structures 106 (2016) 428–436

431

orthotropy axes 1, 2 [MPa]; τ12 – shear stresses in the plane xOy [MPa]. To analyze the structural strength of the isotropic materials we apply the criterion of von Mises:

σi ≤ σT , where σi – stress intensity, it can be represented as

σi =

(

)

2 2 2 2 2 σ11 − σ11σ22 + σ22 + 3 τ12 + τ13 + τ23

.

Here τ13, τ23 – shear stresses in the planes xOz , yOz [MPa]. Since the directions of orthotropy axes 1 and 2 coincide with the directions of axes x and y (dimensionless axes ξ and η ), then σ11 = σx, σ22 = σy, τ12 = τxy.. 3.2. Verification results To substantiate the reliability of the results, we calculated the structures that were considered in [50,51]. The parameters of structural materials are given in Table 1 and designated as Material 1 and Material 2. In the study referenced above [50], isotropic and orthotropic shallow shells of double curvature, square in plane and fixed by hinges on the contour, are studied. In the base of the algorithm, the DQM (differential quadrature method) is proposed. After reduction of the input data to the notation adopted in this work, we obtain the following characteristics of constructions: Version 1. Linear size a = b = 0.2 m, the radii of the principal curvatures R1 = R2 = 7.2727 m, shell thickness h = 0.00022 m, Material 1, isotropic. Version 2. Linear size a = b = 0.2 m, the radii of the principal curvatures R1 = 5 m, R2 = 3.33 m, shell thickness h = 0.00022 m, Material 2, orthotropic. It should be noted that the model presented in [50], in contrast to the proposed model, does not include the effect of transverse shear. Fig. 2a and b shows the combination of graphs “load P¯ – de¯ ” (after the transition to the dimensional parameters and flection W matching of the coordinate axes) for the test problems of version 1 and 2. Hereinafter, the points of maximum on the graphs correspond to the values of the upper critical buckling load P¯kr (in dimensional parameters – qkr ). In the article [50], the authors compare the obtained values of critical load to calculation results obtained by PC NASTRAN (FEM) and the work [51] (based on the algorithm, the partitioned solution method (PSM) and adjacent equilibrium method (AEM) are used). The author notes that the asymmetric orthotropic shell discrepancy between the results is several times more than for a symmetric isotropic. A comparison of the critical loads, obtained by the authors, is shown in Table 2. Fig. 3a and b show the deflection field obtained in [50] and in this work for the test problem of version 2. The scaling factors of deflection 5 and 10 are validated, respectively. As can be seen from the figures, the field distribution of the deflection shell has a similar character, which shows a good consistency in the results obtained by different authors. Fig. 3a and b shows the deflection field, which obtained in [50] and in this paper for the test problem of version 2. The scaling factors for deflection 5 and 10 are validated, respectively. As seen from the figures, the field of distribution of the deflection by the shell has a similar character, which shows a good consistency in the results, which have been obtained by different authors. Next, we will consider the thin-walled shells, hinged-fixedly mounted on the contour (at x = a1, x = a U = V = W = Mx = Ψy = 0; at y = y1( x ), y = y2 ( x ) U = V = W = Ψx = My = 0), being under the action of uniformly distributed load and made of different

¯ ”, which were obtained in [50] Fig. 2. Overlay of graphics of “load P¯ – deflection W and in this paper, for test problems 1 and 2, respectively.

432

A.A. Semenov / Thin-Walled Structures 106 (2016) 428–436

Table 2. Comparison of the critical load values with the works [50,51]. Version

1 2

Table 3. Parameters of the options considered for shallow shells. Version of shell

qkr , Pa van Campen et al. [51]

Wang [50]

A. A. Semenov

PSM

AEM

DQM

FEM

MCSBP

25.14 57.86

31.92 76.71

22.74 50.76

22.63 52.89

23.22 54.35

1

2

3

Linear size a = b, (m)

Radii of the principal curvatures R = R1 = R2, (m)

Thickness of the shell h, (m)

Dimensionless parameter a¯

18 27 36 54 18 27 36 54 18 27 36 54

45.3 67.95 90.6 135.9 22.65 34 45.3 67.95 11.325 17 22.65 34

0.03 0.045 0.06 0.09 0.03 0.045 0.06 0.09 0.03 0.045 0.06 0.09

600

1510

600

755

600

377.7

Dimensionless parameter R¯ = R¯ = R¯ 1

2

Table 4 The calculation results for all investigated variants of shallow shells. Version of shell

Version 1 R¯ = 1510

Version 2 R¯ = 755

Fig. 3. Fields of deflections of construction at the time of loss of stability, obtained in [50] and in this paper for test problem 2.

materials, mechanical characteristics of which are shown in Table 1. 3.3. Shallow shells, square in plane Lame parameters and curvatures for this type of shells have the following form:

A = 1, B = 1, k x = 1/R1, k y = 1/R2. Parameters of the considered versions of constructions are shown in Table 3. The resulting values of critical buckling load qkr (in dimensional parameters), found by the graphs “load – deflection”, and loads of strength loss qpr are shown in Table 4. Hereinafter, for strength analysis, stresses were evaluated on the outer surface of the shell at z = − h/2. From the data, presented in Table 4, it can be concluded that with decrease in the radius of curvature of a shallow shell at identical linear dimensions, its rigidity increases. Therefore the critical buckling load and the load loss of strength increase. For a shell with the least radius of curvature (version 3) for the two kinds of CFRP, strength loss occurs before buckling; for the rest, the loss of stability occurs before strength loss. In half of the cases,

Version 3 R¯ = 377.7

CFRP

Nonlinear analysis

qpr ( MPa)

Component of limit stress

qkr ( MPa)

LU-P/ ENFB T300/ Epoxy M60J/ Epoxy T300/976

0.023

F2+

0.013

0.030*(0.011)

F2−

0.011

0.031

F1−

0.013

0.032

F2+

0.013

LU-P/ ENFB T300/ Epoxy M60J/ Epoxy T300/976 LU-P/ ENFB T300/ Epoxy M60J/ Epoxy T300/976

0.043*(0.076)

F2+

0.076

0.145

F2+

0.064

0.057*(0.074)

F1−

0.074

0.063*(0.078) 0.406

F12 F1−

0.078 0.569

0.292*(0.487)

F2+

0.487

0.413

F1−

0.521

0.561*(0.586)

F2+

0.586

achievement of the limit stress value occurs on the descending branch of the graph “load q – deflection W ”, which is not physically implemented, and in Table 4 it is shown with “*”. Therefore, as the load of strength loss is taken as the load of buckling (value shown in parentheses), the loss of strength occurs simultaneously with the loss of stability. It should be noted that the points of initial breach of strength are defined here, while the construction may still not lose its efficiency. However, upon further loading, these areas are widened and this can lead to structural failure. 3.4. Panels of cylindrical shells Also, we will consider the panels of cylindrical shells, and the curvatures and Lame parameters for such constructions are the following:

A = 1, B = R2, k x = 0, k y = 1/R2. The parameters of the versions of cylindrical panels under

A.A. Semenov / Thin-Walled Structures 106 (2016) 428–436

433

shown on the graphs by the black curve; deflection curve Wc in

Table 5. Parameters of the considered versions of cylindrical shell panels.

the center of the field of construction

(x =

Parameter

Version 1

Version 2

Version 3

Version 4

by the red curve, curve W 4 in the quarter

Linear size, a (m) Angle of rotation, b (radian)

10

10 π

20

20 π

shown by the blue curve, and

Radius of curvature, R (m) Thickness of the shell, h (m) Ratio of linear dimensions λ1 Gravity load (MPa)

5.4 0.01 1.179 0.00015

5.4 0.01 0.589 0.00015

5.4 0.01 2.357 0.00015

π 2

π 2

5.4 0.01 1.179 0.00015

Table 6. Critical loads and loads of loss of strength for the considered panels of cylindrical shells. Ver.

CFRP

Nonlinear calculation

qpr ( MPa)

F

qkr ( MPa)

LU-P / ENFB

0.025 0.072 0.033

F2+ F2−

0.014

T300/Epoxy M60J/Epoxy T300/976 LU-P / ENFB T300/Epoxy M60J/Epoxy T300/976

0.089 0.167 0.095

Version 3

LU-P / ENFB

0.009

Version 4

T300/Epoxy M60J/Epoxy T300/976 LU-P / ENFB

0.189 0.060 0.016 0.034

T300/Epoxy M60J/Epoxy

0.072 0.043

T300/976

0.065

Version 1

Version 2

F2+

0.151 0.072

0.045

F2+

0.194

0.063

F2+ F2− F2−

0.443

F2+ F2+ F2− F12 F12 F2+ F2−

0.377 0.539 0.465 0.007 0.006 0.005 0.007 0.632

F2+

0.445 0.250

F2+

0.566

consideration are shown in Table 5. The obtained values of critical buckling load qkr (in dimensional parameters) and load loss of strength qpr are given in Table 6. It is convenient to use the ratio of cylindrical panel linear dimensions λ1 = a/( Rb), and depending on this parameter, they can be roughly divided into three categories: short, medium, and long. In this case, the panel of version 2 is classified as short; versions 1 and 4 – to the medium category; and version 3 relates to the long panels. Let's also consider separately the cylindrical mid-sized panel (version 4), made from fiberglass T-10/UPE22–27 [49]. Fig. 4 shows a fragment of the graph “load q – deflection W ” for the considered cylindrical panel. Hereinafter, maximum deflection W max which is calculated over the entire area of the shell is

Fig. 4. Graph “load q – deflection W ” for the considered cylindrical panel. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(x =

7a1 + a , 8

y=

b 8

a1 + a , 2

y=

(x =

b 2

) is shown y = ) is

3a1 + a , 4

b 4

W8 in the eighth part

)

, for cylindrical panels a1 = 0, is shown by the

orange curve. The deflection curve at the center and maximum deflection curve show the points of maxima and minima, which correspond to the upper and lower critical loads, respectively. The diamonds in these curves show the points corresponding to the load loss of strength. On the other curves, these points are not marked, so as not to clutter up the chart. These calculations assume that after reaching the limit stresses, construction has not changed its geometry and the nature of behavior. The further curve has more scientific than practical interest. Let's call attention to the stability of the study construction. Fig. 4 shows that the shell loses its stability under load qkr =0.066 MPa; this is point A on the chart. At this point a transition occurs to a new state of equilibrium in point B (shown in dotted line). Thus, the shell loses its stability through a “clap”. According to the criteria of maximum stresses, at the loss of strength in this case, the largest contribution comes from the component that is responsible for compression along axis y . Stress analysis of the considered construction showed that a significant failure of strength terms occurs in the central part of the construction, and it extends along axis x , which corresponds to the action of compressive stresses along axis y . In the majority of studied cases the stress component along axis y by tension was the first to reach the limit values, less frequently – with regard to y by compression and shear in the plane. In [27] it was shown that an increase in the angle of rotation increases the rigidity of the shell and, consequently, increases buckling load values and load of strength loss, which corresponds to the received data. It should be noted that the cylindrical panels of versions 1, 2 and 4 are characterized by the following behavior: first, at a very light load (about 0.005 MPa), force is gradually focused through the center of the shell (load-deflection curve in the middle of construction suddenly starts to develop in a direction, close to the horizontal), and the further curve begins to grow again; there is a loss of strength, and then the loss of stability occurs. For many options the formation of a large loop on the graph “load - deflection” is typical, which speaks to the loss of stability. 3.5. Conical shell panels A schematic representation of a conical shell panel is shown in Fig. 5, and the parameters of the considered constructions are shown in Table 7. Fig. 6 is a graph “load q – deflection W ” for conical orthotropic panel from CFRP T300/976 (version 3). As shown in Fig. 6, the structure remains stable, however, at the load of 0.023 MPa,

Fig. 5. Schematic representation of a conical shell panel and adopted local coordinate system.

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Table 7 . Parameters of the considered versions of conical panels. Parameter

Version 1

Version 2

Version 3

a1 – the beginning of the shell along axis x , m a – the end of the shell along axis x , m θ – cone angle, radian b – rotation angle, radian h – shell thickness, m Material

5

5

5

25

25

25

0.78 π 0.01 plexiglass, isotropic 0.00012

0.78 π 0.01 steel, isotropic 0.00078

0.78 π 0.01 CFRP T300/976, orthotropic 0.00015

Gravity load, MPa

Fig. 6. Graph “load q – deflection W ” for conical panel of carbon fiber T300/976 (version 3).

strength is lost. In Fig. 7a and b, the deflection field is shown being set up from the plane in the PC Maple at the moment of reaching the limiting load of strength loss (0.023 MPa). Fig. 7c shows the same field of deflection set up from the surface of the shell. Changes in the construction were clearly visible, as the scaling factor of deflection was k m = 2. The critical buckling loads and ultimate loads of strength loss, as well as their respective maximum values of deflections for all the considered versions of shells are shown in Table 8. For the considered conical orthotropic panel (version 3), due to the geometry of this structure, the distribution of values of deflections and stresses by the area of the shell is uneven, with offset of the dents to the wider part of the shell. For shells, which lose their stability, the number of dents can vary. Upon comparison with the results of calculations of similar geometry, for isotropic constructions, the advantage of modern composite materials is validated by a combination of strength properties and their weight. The considered panels of conical shells of isotropic materials (plexiglass, steel) lost their stability, but the panel of carbon fiber did not lose its stability. The loss of strength for such panels has come under load, twice more than load loss of strength for the shell made of plexiglass.

4. Conclusions

1. For the considered shallow shells, which are rectangular in plane (cylindrical and conical shells panels, in most cases), the stress component along axis y at tension was the first to reach limit values. For shallow shells with the smallest radius of curvature (version 3) for the two kinds of carbon fiber, strength loss occurs before buckling, for the rest – the loss of stability occurs before or simultaneously with the loss of strength. 2. Since the stress limits for the different constructions and

Fig. 7. Field of deflections for conical panel of CFRP T300/976 (version 3) at the time of loss of strength under load 0.023 MPa.

A.A. Semenov / Thin-Walled Structures 106 (2016) 428–436

Table 8 . Values obtained for the considered versions of conical shell panels. Parameter

Panels of conical shells Version 1, plexiglass

Version 2, steel

Version 3, CFRP Т300/976

Critical load qkr , MPa

0.0152

0.2847

–

Maximum deflection at qkr , m

0.9710

0.3515

–

Limit load of strength loss qnlin , MPa

0.0113

0.2847

0.0230

0.00078 σi

0.00015

Gravity load, MPa 0.000118 Component of limit stress σi

F2+

materials are obtained by different components of the stress vector for each specific structure, it is necessary to carry out a separate computer simulation and research of its stress-strain state. Exceptions are similar constructions: the data, obtained for the considered versions of shallow shells, can be scaled to a similar structure, but with a proportionally resize. Thus, our results are consistent with a series of similar shells. 3. As for the analysis of geometric nonlinearity, the study of the strength of shallow orthotropic shells must take into account this factor, or the maximum permissible loads will be significantly overstated. For the considered versions of constructions, this difference in values can reach the first order. This is due to the fact that the deformation of shells, made of composite materials (CFRP, GRP, etc.) is essentially nonlinear. When calculating the geometric nonlinearity version, the graph of “load q – deflection W” is a straight line; i.e., the deflection is directly proportional to the load. When we take into account the geometric nonlinearity, the deflection significantly deviates from the linear relationship with increasing load. Moreover, changes in the initial shape of the curved surface frequently occur. As a consequence of increasing the value of the deflection, the strains and stresses increase, leading to the obtainment of limit values of stresses at much lower loads. 4. It should be noted, that for the considered medium and short cylindrical panels (versions 1, 2 and 4) the following behavior is typical: first, a very light load (about 0.005 MPa) is gradually forcing through the center of the shell (curve of load-deflection in the middle of the shell suddenly begins to progress in a direction, which is close to the horizontal), then the curve begins to rise again, where there is a loss of strength, and then the loss of stability. 5. For the many versions of cylindrical shell panels, before total loss of stability occurs, multiple local buckling with the formation of small dents in various parts of the shell occur. The dents move around on the shells when the deformation occurs.

Acknowledgments The work was supported by the Ministry of Education and Science of the Russian Federation, project no. 3801.

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