Vibration correlation technique for predicting the buckling load of imperfection-sensitive isotropic cylindrical shells: An analytical and numerical verification

Thin-Walled Structures 140 (2019) 236–247

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Vibration correlation technique for predicting the buckling load of imperfection-sensitive isotropic cylindrical shells: An analytical and numerical verification

T

Felipe Franzonia,b,∗, Richard Degenhardta,b,c, Jochen Albusd, Mariano Andrés Arbeloe a

DLR, Institute of Composite Structures and Adaptive Systems, Braunschweig, Germany University of Bremen, Faserinstitut Bremen e.V., Bremen, Germany c PFH, Private University of Applied Sciences Göttingen, Composite Engineering, Campus Stade, Stade, Germany d ArianeGroup GmbH, Bremen, Germany e ITA, Technological Institute of Aeronautics, Department of Aeronautics, São José dos Campos, Brazil b

ARTICLE INFO

ABSTRACT

Keywords: Nondestructive experiments Vibration correlation technique Free vibrations Cylindrical shells Buckling Imperfection-sensitive structures

This paper presents an analytical and numerical investigation of the relationship between the compressive load level and the natural frequency variation toward a vibration correlation technique for the buckling load calculation of imperfection-sensitive isotropic cylindrical shell structures. Firstly, a back-to-basic s study is proposed and the linear equation between the applied load and the square of the loaded natural frequency is revisited. Such review considers the Flügge-Lur'e-Byrne's linear shell theory for the free vibrations of an isotropic unstiffened cylindrical shell under uniform axial loading. The demonstrated linear equation is rearranged for expressing the square of the applied load as a quadratic function of the square of the loaded natural frequency. The suggested formulation provides the analytical support to a novel vibration correlation technique that has been empirically proposed and experimentally validated for unstiffened cylindrical shells. Aiming a numerical verification based on finite element models, two cylindrical shells are defined. At first, the critical buckling load and the fundamental natural frequency for different load levels are determined and compared to the analytical results for validation of the numerical models. The finite element models are extended considering geometric nonlinearities, more realistic boundary conditions and three magnitudes of a benchmark measured initial geometric imperfection. The numerical results are considered for analyzing the variation of the natural frequency in the surroundings of buckling and for verifying the vibration correlation technique.

1. Introduction Thin-walled cylindrical shell structures are broadly used in space applications due to their natural optimized strength-to-weight ratio. Considering the operational load envelope of launcher structures, the project is mostly driven by buckling, which represents a big challenge for the validation of the design, as the mentioned structures are extremely imperfection-sensitive. The challenges are even higher considering the current scenario of reusable launcher structures because the flight-worthiness needs to be evaluated before the reuse of the structure. In this context, there is interest in the development of nondestructive methods to estimate the buckling load from the prebuckling stage for imperfection-sensitive structures, like the vibration correlation technique (VCT). The VCT relates an initial model and measured data prior to

∗

buckling to estimate the buckling load of the structure, without causing further damage to it. Analytical or finite element (FE) analyses based on the geometrically perfect structure can be used as an initial model while the measured data consists of a sequence of vibration tests performed at different axial compression load levels. A VCT may be classified into indirect or direct methods, as proposed in Ref. [1]. Considering the first group, indirect methods, an assessment of the actual boundary conditions is made in order to update the initial model improving the estimation of the buckling load [2]. The second group, direct methods, extrapolates an experimentally determined functional relationship between the applied load and the loaded natural frequency to estimate the buckling load, see Refs. [3–6] among others. For simply supported columns, plates [3] and, cylindrical shells [7,8], the effects due to prestress on the natural frequencies lead to a linear relationship between the total applied load and the square of the

Corresponding author. DLR, Institute of Composite Structures and Adaptive Systems, Braunschweig, Germany. E-mail address: [email protected] (F. Franzoni).

https://doi.org/10.1016/j.tws.2019.03.041 Received 9 October 2018; Received in revised form 14 March 2019; Accepted 18 March 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.

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loaded natural frequency:

f2 + p = 1

(1)

where f = ¯ mn / mn , being ¯ mn the loaded natural frequency and mn the unloaded natural frequency, both associated with the same vibration mode defined by m axial half-waves and n circumferential waves (for cylindrical shells), and p = P / PCR , being P the applied load and PCR the linearized buckling load (from an eigenvalue buckling analysis or theoretical buckling equations). As the buckling load is associated with the applied load level where the natural frequency becomes zero, the classic VCT consists of plotting the characteristic chart f 2 versus p and extrapolating a linear best-fit relationship up to the load level correspondent to zero frequency [1]. The above-mentioned linear method is straightforward in the case of beam structures. To the best of the authors’ knowledge, the first experimental verification is accredited to Sommerfeld at the beginning of the 20th century [9]. Nevertheless, other experiments based on the VCT dated from the 1950s, see for instance Refs. [3,10]. Although the relationship between the applied load and the squared loaded natural frequency is not linear for other than fully simply supported boundary conditions, the VCT method based on the linear best-fit still presents appropriate estimations for columns with different boundary conditions, as demonstrated in the experimental campaigns described in Refs. [3,11,12]. The definition and validation of a VCT for predicting the buckling load of imperfection-sensitive structures like plates and shells consist of an open and important research area [13]. The applicability of the technique based on a linear extrapolation is limited to a few imperfection-insensitive structures. In the 1950s, Lurie [3] was not able to validate it considering simply supported flat plates. However, during the 1970s, Chailleux et al. [12] investigated simply supported flat plate specimens with small imperfections achieving good results. Recently, Chaves-Vargas et al. [14] succeed in applying the linear method for flat carbon fiber-reinforced polymer (CFRP) stiffened plates. Specific VCT approaches have been proposed addressing imperfection-sensitive structures like curved panels and cylindrical shells [1]. Radhakrishnan [4] proposed tracking the vibration mode similar to the buckling mode and extrapolating the final linear path of the classical characteristic chart to the applied load axis; the author obtained exact results for tubes made of Hostaphan®. Segal [15] conducted a study adjusting an optimal parameter q to raise the natural frequency so a linear best-fit with the applied load would lead the VCT estimated buckling load (assumed as the load level in which the natural frequency is zero) to exactly match the buckling load of the structure. The author investigated 35 specimens and a formulation relating the main geometric characteristics of the stiffened cylindrical shells and the optimal parameter q was proposed. This methodology achieved a considerable reduction in the scatter of the VCT estimated knock-down factors (KDF) when compared to the indirect VCT method based on Eq. (1). Plaut and Virgin [16] also investigated the mentioned method through an analytical study aiming to determine the upper and lower bounds of the optimal parameter q and, consequently, of the estimated buckling load. Souza et al. [5] suggested a novel approach based on a modified characteristic chart in the parametric form (1 p) 2 versus 1 f 4 . In such representation, a linear relationship should be obtained as illustrated in Fig. 1, which reproduces the results from Ref. [5] for a schematic view of the proposed VCT. The suggested linear equation can be identified through a best-fit procedure of the experimental results. The parametric form (1 p) 2 contains the ratio between the load level and the linear buckling load; therefore, it can be evaluated when the natural frequency is zero (1 f 4 = 1 in the proposed parametric form) for an estimation of the load level in which the structure is unstable leading to the following relationship:

Fig. 1. Schematic view of the VCT proposed in Ref. [5].

(1

p) 2 + (1

2)(1

f 4) = 1

(2)

where 2 represents the drop of the buckling load due to initial imperfections. Thus, the VCT estimation of the buckling load PVCT is expressed in terms of the positive value of as:

PVCT = PCR (1

)

(3)

Analyzing Eq. (3), 1 can be compared to the KDF of conventional sizing approaches [17]. In Souza and Assaid [18], the authors investigated a cubic parametric curve to represent the classic characteristic chart. Throughout this method, the Hermite form is considered to define the parametric equations. Both methods proposed by Souza and his colleagues [5,18] were validated considering the experimental results of stiffened cylindrical shells tested at Technion [19]. In Jansen et al. [20], the authors suggested an extension of the semiempirical VCT method proposed in Ref. [5] by combining it with semianalytical analysis tools. Through the mentioned analysis tools, the effects of initial imperfections were considered and, the authors illustrated their capabilities to improve the VCT estimations. A second-order best-fit relationship was proposed by Abramovich et al. [13] for representing the classic characteristic chart. The authors investigated this approximation to curved stiffened panels. The predictions accounting for load levels up to 50% of the linear buckling load of the structure are reasonable; however, the authors suggested load levels near the typical sharp bend of the classic characteristic chart for improving the accuracy. Arbelo et al. [6] proposed a novel empirical VCT modifying the work done by Souza et al. [5]. The authors suggested evaluating the second-order best-fit relationship of a modified characteristic chart in the parametric form (1 p) 2 versus 1 f 2 . Fig. 2 reproduces the results from Ref. [6] for a schematic view of the proposed VCT. The adjusted second-order equation is minimized and 2 estimated as the correspondent value of (1 p) 2 axis. The authors suggested

Fig. 2. Schematic view of the VCT proposed in Ref. [6]. 237

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estimating the buckling load based on the positive value of as proposed in Ref. [6] and here presented in Eq. (3). Furthermore, this method is based on the effects of the initial imperfections in the vibration response of the structure and, typically, the first two or three natural frequencies can be evaluated for estimating the buckling load. Until this point, the proposed methodology has been validated through 7 experimental campaigns presented in Refs. [21–27]. Arbelo et al. [21] presented an experimental campaign based on three identical CFRP unstiffened cylindrical shells. The authors tracked the variation of the first natural frequency for several load steps. The results are compared to the buckling load resulting in an agreement between 2.3% and 7.8%. In Kalnins et al. [22] the method is validated for two CFRP and two metallic unstiffened cylindrical shells. The authors considered the first and the second vibration modes for applying the VCT. The results are in good agreement with the buckling load, within 0% and 10% of deviation. A statistical evaluation of the estimated buckling load is presented in Skukis et al. [23]. The authors tested two identical CFRP unstiffened cylindrical shells for VCT and obtained estimations within 4.8% and 8.4% of deviation from the buckling load. Additionally, they concluded that using load steps up to 65% of the buckling load gives reliability close to 90% for the VCT estimations. An experimental campaign for thin-walled cylindrical shells with and without a cutout is presented in Skukis et al. [24]. The authors found a good correlation for specimens governed by a global failure mode (cylindrical shells without cutout and with cutout considering a reinforcement in the cut region). Conversely, they conclude that the VCT proposed in Ref. [6] is not applicable when a local failure mode governs. Recently, Shahgholian-Ghahfarokhi and Rahimi [25] further validated the method proposed in Ref. [6] considering a grid-stiffened composite cylindrical shell. The authors considered load levels up to 90% of the experimental buckling load and obtained 3.1% of deviation. Labans et al. [26] investigated a classical and a variable angle tow composite cylindrical shells considering load levels up to 65.48% and 69.23% of the corresponding linear buckling load, respectively. The authors obtained 4.0% and 1.4% as deviations from the respective experimental buckling loads for the classical and variable angle tow shells, respectively. Additionally, the authors explored the VCT predictions considering the buckling load obtained from a nonlinear analysis taking into account geometric initial imperfections and the stiffness variation (for the variable angle tow composite shell) as the reference for calculating the load ratio p ; nevertheless, the results considering the linear buckling load of the respective perfect shell as the reference presented better correlation. Franzoni et al. [27] validated the method for pressurized skindominated stiffened cylindrical shell considering a maximum applied load equals 72.34% of the linear buckling load. The authors obtained deviations within 4.03% and 9.98% of the experimental buckling load for different levels of internal pressure; moreover, the authors validated an algorithm based on the modal assurance criterion (MAC) [28] for tracking the variation of the natural frequencies of numerical models. In summary, the VCT has been applied to imperfection-sensitive structures as an empirical curve fitting procedure, where the methods based on the squared applied load in the form (1 p) 2 achieved notable results for estimating the square of the drop of the load-carrying capacity, being experimentally validated in Refs. [5,6,21–27]. In this context, there is interest in developing analytical-verified approaches for the estimation of the buckling load of imperfection-sensitive cylindrical shells through a nondestructive experimental campaign; therefore, the focus of this paper is to evaluate analytically the relationship between the squared applied load in the parametric form (1 p) 2 and the loaded natural frequency of a simple supported isotropic unstiffened cylindrical shell. The resulting equation can be regarded as the rearrangement of the well-known linear relationship presented in Eq. (1), paving the path to a novel understanding for the VCT definition applied to imperfection-sensitive structures.

Fig. 3. Geometry of a conventional cylindrical shell.

2. Review of the free vibration analysis of an axially loaded cylindrical shell This section reviews the free vibration of an axially loaded unstiffened cylinder based on the linearized Flügge-Lur'e-Byrne's theory of shells. The free vibration problem as presented in Ref. [29] is extended to account for an axially loaded prestress state, as proposed in Ref. [8]. This back-to-basics study allows the verification of Eq. (1) providing support for the contributions of section 3. 2.1. Free vibration of a simply supported axially loaded cylindrical shell Consider the conventional cylindrical shell structure depicted in Fig. 3, which has an axial length L , a constant thickness h , and a middle surface radius R . An orthogonal cylindrical coordinate system (x , , z ) is defined in the middle surface of the shell as a reference, where x , , and z are the axial, circumferential and radial directions, and u , v , and w are the corresponding displacements of the middle surface of the shell, respectively. The differential equations of motion can be expressed in matrix form as proposed in Refs. [8,29]: (4)

[L]{ui} = {0}

where ui is the displacement vector {u v w }T and [L] is a matrix differential operator. Considering Flügge-Lur'e-Byrne's theory of shells taking into account the prestress effects due to uniform axially membrane force N¯x , which the magnitude is positive for tension loads, the matrix differential operator [L] is expressed as [8]:

[L] = [LD] + k [LF ] +

1 [Li ] CN

(5)

2) , [L ] contains the where k is given by h2 /12R2 , CN is given by Eh/(1 D terms associated with Donnell's theory of shells, [LF ] contains the terms for extending Donnell's theory to Flügge-Lur'e-Byrne's theory and [Li ] contains the additional terms related to the initial stress state. The three operators are given by Ref. [8]:

238

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R2

2

(1

+

x2

2

)

2 (1 + ) R x 2

2

2

R

where ¯m = m R . To find a nontrivial solution, the determinant of the left-hand side matrix of Eq. (9) is equated to zero giving the following cubic characteristic equation for a pair of m and n :

x

F

[LD] =

2 (1 + ) R x 2

R (1

(1

) 2

2

3(1

0 x3

[Li ] =

) 2

(1

) 2

2

N¯x R2

F

2

3

R

(3 2

x

0

N¯x R2

0

0

)

R2

R2

2

0

x2

R3

0

2

[LF ] =

2

+

4)

(1 + k

)

3

x2

x

2

R3

2

R2

2

3

x3

+

(3

x2

(1 )

2

) 2

R2

R

¯ 3mn

F 2

T2 = H11 + H22 + H33

3

x2 2

x2

T0 = H11 H22 H33

2

0

x2

N¯x R2

is the Poisson's ratio and F and

F =

(1

4

R4

2)

2 ¯ mn =

E

=

4

are defined as:

4

x2

2

4

+

(7)

4

being the mass density, t the time variable and, E the Young's modulus. Considering a circular cylindrical shell with two simply supported SS3 edges, where w = v = Nx = Mx = 0 , the solution of the differential equations is straightforward. The displacement components can be assumed as:

u (x , , t ) = Amn cos( m x )cos(n )cos( mn t ) v (x , , t ) = Bmn sin( m x )sin(n )cos( mn t ) w (x , , t ) = Cmn sin( m x )cos(n )cos( mn t )

H11

H12 H13

H12 ¯ mn H22

H13 H23

¯ mn

H23

H33

Amn Bmn Cmn

) 2

n2 (1 + k )

3 H13 = ¯m + k ¯m

H22 = H23 =

(1 2 n

2

)¯ 2 mn

) ¯2 3(1 2 m + n + k k

(3 2 2

2

¯ mn = (1

2)

E

) ¯2 m

) ¯2 mn

(9)

2

2 R2 ¯ mn

2 N¯x ¯m

CN

2n2)

R2

(13)

¯m2 N¯ x

+

(14)

hR2

H11

H12 ¯ mn H22

Amn /Cmn = Bmn /Cmn

H13 H23

(15)

2.2. The applied load in terms of the square of the loaded natural frequency Through Eq. (14), the total applied load P , defined as 2 RN¯x , can be expressed in terms of the squared loaded natural frequency as: 2 P = Gmn ( ¯ mn

(16)

2 mn )

where Gmn is given by:

Gmn =

(10c)

2

hR3 ¯m2

(17)

Referencing [7,8], the natural frequency dropping to zero is sufficient to obtain the buckling load of the cylindrical shell.

(10d)

2 PCR = Gmn ( ¯ mn

(10e)

H33 = 1 + k ( ¯m + n2 ) + k (1

(12)

The ratios between amplitude coefficients determine the vibration mode and, the third coefficient Cmn , which is undetermined through Eq. (15), is a scaling factor of the vibration amplitude determined by the initial conditions or a normalization criterion [29].

(10b)

(1

2 mn

H12

(10a)

(1 + ) ¯ H12 = mn 2

2)

(1

¯ mn

where the terms Hij , which depend on m , n , and some mechanical and geometric properties of the cylindrical shell, and the frequency parameter ¯ mn are given by:

(1 2 H11 = ¯m +

2 H33 H12

2 where mn is the square of the unloaded natural frequency. Typically, the lowest natural frequency corresponds to a vibration mode associated mainly with radial displacement. The vibration modes can be calculated through two linear-dependent equations based on Eq. (9). For example, returning to Eq. (9) and dividing the first two equations by Cmn gives the following two lineardependent equations in terms of the ratios between the amplitude coefficients:

(8)

0 = 0 0

¯ ¯2 ¯ mn + Nx m CN

E

2 ¯ mn =

where Amn , Bmn , and Cmn are the amplitude coefficients and m given by m /L . Moreover, the range of axial half-waves m and the range of circumferential waves n are 1,2, … and 0,1,2, …, respectively. Substituting the proposed displacement functions into Eq. (4) one can obtain the following system of algebraic equations in terms of Amn , Bmn , and Cmn :

¯ mn

2 H22 H13

The negative values from Eq. (13) have no physical meaning and are neglected. The three positive values correspond to three natural frequencies where each one is mainly associated with one of the displacement components. To conclude, Eq. (13) can be expressed in a more appropriate form:

t2

+ 2R2

x4

4

2

R2

2H12 H13 H23

2 H23

2

x2

(6) where

2 H13

2 H11 H23

The three roots of Eq. (11) are always real for any pair of m and n [8,29]. The natural frequency of the loaded cylindrical shell ¯ mn can be calculated from Eq. (10g) as:

0 2

2 H12

T1 = H11 H22 + H11 H33 + H22 H33

(1 + 2 )

3

(11)

T0 = 0

where T2 , T1 and T0 are coefficients depending on Hij from Eq. (10a-f):

3

x

2 T2 ¯ mn + T1 ¯ mn

2 mn ) ¯ mn = 0

=

Gmn

2 mn

(18)

Finally, the load ratio p , as presented in Eq. (1), can be found dividing Eq. (16) by (18):

(10f)

p=1 (10g)

where f is the ratio between ¯ mn and 239

(19)

f2 mn .

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procedure of the experimental data. The minimum value of Eq. (21) leads to:

(1

f 2 ) Minimum =

B 2A

(1

p) 2Minimum =

2

=

B2 +C 4A

(22)

Given these points, a direct VCT for imperfection-sensitive cylindrical shells is defined and general steps can be established: 1 Calculate the linear buckling load and the first natural frequency of the perfect structure. These are reference values for the VCT evaluation. 2 Track the first natural frequency during axial loading. 3 Identify the second-order best-fit relationship between (1 p) 2 versus 1 f 2 , and minimize the quadratic equation for evaluating 2 , as proposed in Eq. (22). 4 Use the estimated and the linear buckling load PCR in Eq. (3) to assess the buckling load of the structure.

Fig. 4. The typical load-shortening curve of an unstiffened cylindrical shell.

3. VCT implementation for imperfection-sensitive cylindrical shells

4. Numerical verification of the proposed methodology

The linear relationship presented in Eq. (1) and reviewed in section 2 can be rearranged to express the squared applied load in the parametric form (1 p) 2 in terms of the square of the loaded natural frequency in the parametric form 1 f 2 :

(1

p) 2 = f 4 = [1

(1

4.1. Overview of the isotropic unstiffened cylindrical shells Two isotropic unstiffened cylindrical shells are proposed, named ZAL1 and ZAL2. The structures have the geometric characteristics presented in Table 1. Moreover, the study considers the cylinders made of aluminum alloy AL7075-TT7351 and Table 2 shows the correspondent mechanical material properties obtained from Ref. [30]. Aiming to numerically investigate the influence of real geometric imperfections, the measured imperfection signature of the stainless steel laser-welded cylindrical shell tested in Ref. [22], named cylinder SST-1, was considered. Fig. 5 presents the mentioned measured initial imperfection pattern considering its original magnitude as related to the best-fit radius of the SST-1 structure (in mm).

(20)

f 2 )]2

Eq. (20) shows analytically that the squared applied load in the parametric form (1 p) 2 is related to the square of the loaded natural frequency in the form 1 f 2 through a second-order equation, as empirically proposed by Arbelo et al. [6]. Considering unstiffened cylindrical shells, such structures are characterized by a highly imperfection-sensitive behavior, consequently, a substantial drop in the buckling load PIMP is expected when compared to the linear buckling load PCR . In this context, Fig. 4 depicts the typical load-shortening curve for an unstiffened cylindrical shell in terms of the load ratio p ; moreover, the chart presents the variation of the reaction load in the form (1 p) 2 and the KDF as defined in Ref. [17] and as related to . Considering its definition, the minimum value of the squared applied load (1 p) 2 occurs exactly at the instability point, at which its square root equates from Eq. (3), as shown in Fig. 4. This statement is based on the behavior of the parametric form (1 p) 2 and it allows the definition of a VCT approach that does not rely on the drop to zero of the natural frequency, but on the minimum value of the parametric form (1 p) 2 . Given that Eq. (20) provides a functional relationship between the squared applied load (1 p) 2 and the squared loaded natural frequency 1 f 2 , this equation can be minimized with respect to 1 f 2 to evaluate 2 leading to (1 p) 2 = 0 at 1 f 2 = 1 where the second derivate is 2, which is constant and greater than zero, indicating that this result is the minimum point of Eq. (20). The described result was expected because it is based on a perfect structure and there is no drop of the load-carrying capacity due to initial imperfections; moreover, for this case, in which the buckling and vibration modes exactly match [7,8], the natural frequency of the loaded structure should be zero at the buckling load. Toward a direct VCT, measured natural frequencies are available for different load steps in the prebuckling regime. If the found quadratic relationship between the squared applied load in the form (1 p) 2 and the squared loaded natural frequency in the form 1 f 2 holds in the presence of initial imperfections and other boundary conditions, the square of the drop of the load-carrying capacity can be estimated based on the measured data for practical scenarios:

(1

p) 2 = A (1

f 2 ) 2 + B (1

f 2) + C

4.2. Finite element analyses The numerical models were developed using the commercial FE solver Abaqus Standard 6.16®. The Newton-Raphson iterative procedure with artificial damping stabilization was used as the nonlinear solver for the axial loading step. For the eigenvalues problems, i.e. linear buckling and free vibration analyses, the default Lanczos solver was employed. The FE model is defined considering quadratic quadrilateral thinshell elements with 8 nodes, 6° of freedom per node and reduced integration; labeled S8R elements in Abaqus® library. Convergence analyses were performed resulting in an FE mesh with 140 and 224 elements over the circumference of ZAL1 and ZAL2, respectively. Elements in other directions are chosen automatically by Abaqus® considering a global size of approximately 11.22 mm. Thus, the FE models consider 6160 and 16,128 S8R shell elements associated with 18,760 and 48,832 nodes for ZAL1 and ZAL2, respectively. Fig. 6 presents an overview of the FE mesh for both models. Concerning the boundary conditions, two types of simply supported are considered on both edges: (1) known as shear diaphragm or SS3, where w = v = Nx = Mx = 0 , which is compatible with Eq. (8), and it is, therefore, compared to the analytical equations and, (2) known as SS4, where w = v = u = Mx = 0 , which is more appropriate for the VCT Table 1 Geometric properties of ZAL1 and ZAL2.

(21)

where A , B and C are the coefficients determined based on the best-fit 240

Property

Symbol

ZAL1

ZAL2

Free axial length [mm] Middle surface radius [mm] Shell thickness [mm] Radius over thickness ratio

L R h R /h

500 250 0.50 500

800 400 0.50 800

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points. Fig. 8 presents the disturbed FE mesh of cylinders ZAL1 and ZAL2 considering the original magnitude of Fig. 5 (in mm) and the deformation amplified by 20. This model is used for a nonlinear static analysis followed by free vibrations analyses.

Table 2 . Material properties of the AL7075-T7351 alloy [30]. Property

Symbol

Magnitude

Compressive elastic modulus [MPa] Poisson's ratio Mass density [kg/m3]

E

73,084 0.33 2,796

4.3. Comparison between analytical and numerical results The equations for the free vibration of an unstiffened cylindrical shell depend on the number of axial half-waves m and circumferential waves n . Thus, a sweep analysis in m and n is required for defining the first vibration mode, which is associated with the minimum magnitude within the natural frequencies. This procedure is performed using a Matlab® algorithm. Fig. 9 presents the results of the variation of m and n considering both cylinders and the formulation presented in Eq. (14). From Fig. 9, the first unloaded natural frequency F1 of the cylindrical shells ZAL1 and ZAL2 are associated with the following pairs (m , n ): (1,8) and (1,9), respectively; moreover, Fig. 10 shows the correspondent first vibration modes, which are calculated through Eq. (15). The reference FE model without initial geometric imperfection and considering SS3 boundary condition can be directly compared to the presented analytical results; therefore, Table 3 presents a comparison between the analytical and the numerical buckling loads and first natural frequencies. The loaded natural frequencies are calculated for load levels in terms of the buckling load from Eq. (21), which is also presented and compared to the numerical results in the same table. Moreover, the analytical results of the natural frequencies are based on Eq. (14). From Tab. 3, there is a good agreement between the analytical and numerical results, being the maximum deviation of 0.41% as related to the analytical result. Additionally, Fig. 11 presents the numerical results of the first unloaded vibration mode and the first buckling mode for the cylinder ZAL1 considering SS3 boundary condition. From Fig. 11, one may notice that the first unloaded vibration mode and the first buckling mode exactly match, which is expected for SS3 boundary conditions [7,8]; moreover, Fig. 11(a) exactly matches Fig. 10(a) validating the reference numerical models. It is worth mentioning that the same comparison considering cylinder ZAL2 provides analogous results.

Fig. 5. Measured initial imperfection for the cylinder SST-1, available in Ref. [22].

verification, once it is closer to the experimental boundary conditions. Important to mention that the FE models consider additional constraints in the edge laying in the middle of the length of the cylinder for implementing the SS3 boundary conditions, which keeps the symmetry and avoids rigid body motions. Fig. 7 shows the details of the herein defined boundary conditions considering the geometry of the cylinder ZAL1; the presented constraints are related to a cylindrical coordinate system defined at the center of the bottom edge. Additionally, for loading the structures, shell edge forces are applied on both edges in the FE models with SS3 boundary condition Fig. 7(a) and enforced displacement on the top edge of the cylinder in the FE models considering SS4 boundary condition Fig. 7(b). Based on the presented study cases, two main FE models were defined: 1. Reference model: Considers the nominal geometry without initial geometric imperfection and it is employed in two types of eigenvalue problems: linear buckling and free vibrations analyses.

4.4. Assessing the effects of geometric nonlinearities and initial geometric imperfections in the load-frequency relationship

2. Imperfect model: Considers the geometry disturbed by the initial imperfection pattern presented in Fig. 5. For each cylinder, the measured imperfection amplitudes are downscaled aiming the maximum deviation to be 50%, 75%, and 100% of the shell thickness. The initial positions of the FE mesh are disturbed using a python script, which uses the same inverse-weighted interpolation and downscale rules presented in Ref. [31] and, it was herein considered for the 5 closest measured

In Section 4.2, the reference model with SS3 boundary condition was validated considering the analytical equations; however, in practical scenarios, such boundary condition is not obtained. Moreover, real structures present initial geometric imperfections as shown in Fig. 5. Given these points, this section proposes to extend the numerical analysis to obtain the frequency variation considering geometric

Fig. 6. . Overview of the finite element mesh. 241

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F. Franzoni, et al.

Fig. 7. . Details of the boundary conditions considered in the FE models.

Fig. 8. Measured initial imperfection applied to the FE model. Fig. 9. . Natural frequencies for different values of m and n .

(a) Cylinder ZAL1.

(b) Cylinder ZAL2.

Fig. 10. . First vibration modes.

nonlinearities, initial geometric imperfections, and SS4 boundary condition, which is closer to an actual experimental campaign. Firstly, the reference model presented in Section 4.2 is solved considering SS4 boundary condition for the linear buckling and free

vibration analyses. The linear buckling load is 70.18 kN for both cylinders and the first natural frequencies are 250.65 Hz and 126.43 Hz for cylinders ZAL1 and ZAL2, respectively; the associated first buckling modes and first vibration modes are shown in Figs. 12 and 13, 242

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Analyzing Fig. 16(a and b), one may notice a slight variation on the first vibration mode of the cylinder ZAL1M050, which is associated with a reduction of 32.97% in frequency magnitude. In Fig. 16(c) note that the number of half-waves in the axial direction has changed in the initial postbuckling regime of the cylinder ZAL1M050, which does not allow following the vibration mode after buckling. Similar results were found for the other cylindrical shells, therefore, Table 6 presents the magnitudes of the first natural frequency for all cylindrical shells considering the unloaded, PNL , and PPB load steps, respectively. From Tab. 6, it is clear that the magnitude of the first natural frequency calculated at the buckling load PNL is not zero for all the cylindrical shells. Furthermore, comparing Tables 5 and 6 it can be noticed that the magnitude of the first natural frequency at buckling increases as the KDF decreases. Additionally, Fig. 17 presents the numerical results for the first natural frequency variation considering the characteristic chart proposed in Ref. [6]. Fig. 17 allows an assessment of the nonlinear effects in the first natural frequency triggered by initial imperfections and the SS4 boundary condition. Note that besides a significant difference between the behaviors, when compared to the analytical solution, the verified second-order relationship still holds, particularly considering the results associated with greater values of the KDF as defined in Table 5. Therefore, the second-order relationship from Eq. (20) can be identified considering measured frequency data for different load steps in the prebuckling regime and used to determine the buckling load of the structure.

Table 3 Numerical and analytical results of ZAL1 and ZAL2. ZAL1 Variable

PCR [kN] F1 [Hz] (m , PCR F1 [Hz] (m , PCR F1 [Hz] (m , PCR F1 [Hz] (m , PCR

n) at 0% of n) at 25% of n) at 50% of n) at 75% of

ZAL2

FEM

Analytical

FEM

Analytical

68.24 176.22 (1,8) 152.65 (1,8) 124.64 (1,8) 87.76 (1,8)

68.21 176.24 (1,8)

68.64 87.33 (1,9) 75.65 (1,9) 61.76 (1,9) 43.56 (1,9)

68.62 87.34 (1,9)

152.63 (1,8) 124.62 (1,8) 88.12 (1,8)

75.64 (1,9) 61.76 (1,9) 43.67 (1,9)

respectively. From Figs. 12 and 13, there are no similarities between the buckling mode and the first vibration mode for both cylinders, which is recurrent in practical applications; this fact endorses the need for developing a VCT approach independent of the drop to zero of the natural frequency. The nonlinear analysis considers the imperfect model, which allows an assessment of the previously mentioned effects on the second-order relationship presented in Eq. (20); furthermore, in Section 5 of this paper a numerical verification of the VCT is proposed based on these results. Table 4 shows the main parameters considered during the nonlinear static analysis for axial loading. Fig. 14 presents the load-shortening curves for the cylinders ZAL1 and ZAL2 considering all the initial imperfection magnitudes while Tab. 5 summarizes the respective nonlinear buckling load PNL and the KDF NL . Hereafter, 3 additional labels are defined for ZAL1 and ZAL2 by adding an “M” followed by three digits representing the maximum amplitude of the initial geometric imperfection: 050, 075 and 100 representing 50%, 75% and 100% of the thickness of the shell, respectively. From Fig. 14 and Tab. 5, it can be noticed that the considered initial geometric imperfections significantly affect the nonlinear buckling load of the cylindrical shells resulting in two similar ranges of KDF: from 0.71 to 0.83 and from 0.75 to 0.84 for ZAL1 and ZAL2, respectively. In the following, the nonlinear static results from Fig. 14 are divided into 41 preload steps. These load steps are considered for a sequence of free vibrations analyses equally distributed from 2.5% up to 100% of the buckling load PNL , besides an additional step in the first stable increment of the postbuckling regime PPB . Fig. 15 shows the load-shortening curve for the cylinder ZAL1M050 emphasizing the load steps followed by a linear frequency analysis. The vibration results of each load step were analyzed using a Matlab® algorithm based on the MAC [27]; this algorithm compares the vibration modes between subsequent load steps identifying the variation of each vibration mode through the axial loading. Fig. 16 shows a comparison between the first vibration modes of the cylinder ZAL1M050 at the following load steps: (a) unloaded condition, (b) at the buckling load PNL , and (c) at the first stable increment in the postbuckling regime PPB , respectively.

5. Numerical verification of the VCT. The numerical results of the cylinders ZAL1 and ZAL2 considering all the imperfection magnitudes simulated in Section 4 are considered up to 97.5% of the buckling load PNL for evaluating the VCT studied in this paper. The method is applied as presented in Section 3 considering the load level normalized by PCR calculated for SS4 boundary conditions (70.18 kN), and the maximum load level and the number of load steps are simultaneously increased. Fig. 18 shows the variation of the deviation calculated between the VCT estimated buckling load PVCT and the buckling load PNL (presented in percentage of the correspondent PNL ). From Fig. 18, one may notice that the VCT herein demonstrated provides a good agreement for the estimations of the nonlinear buckling load considering all the cylindrical shells; this is true even when applied using low load levels far from the actual buckling load, once the magnitudes of the deviations are below 10% for most of the estimations. In addition, VCT will provide a good estimation of the buckling load at lower load levels as the magnitude of the initial geometric imperfections increases. Furthermore, note that the estimations are conservative, i.e. characterized by a negative convergence of . The results presented in Fig. 18 are summarized in Table 7 considering two approaches: (1) all load steps up to the maximum load level associated with the minimum , and (2) all load steps up to 50% of PNL ; for both cases, the maximum load level considered PMAX (in terms of PNL ), the VCT estimations PVCT , the correspondent KDF VCT and

Fig. 11. . First unloaded vibration mode and first buckling mode for cylinder ZAL1. 243

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Fig. 12. . First buckling modes considering SS4 boundary condition.

Fig. 13. . First vibration modes considering SS4 boundary condition. Table 4 Parameters considered during the nonlinear static step. Damping factor

10–7

Initial increment Minimum increment Maximum increment

0.001 10–6 0.001

deviation are presented. Furthermore, Fig. 19 shows the second-order best-fit relationship associated with the results presented in Tab. 7 and the estimated 2 for all imperfection magnitudes of ZAL1 and ZAL2. Analyzing Table 7 for the first study, Fig. 19(a) and (c), the VCT estimations and their respective KDF are conservative and in good agreement with the correspondent values from Table 5, as the greater deviation is 2.9% in magnitude. Considering the second study, as can be noticed from Fig. 19(b) and (d), the estimated KDF are similar for different magnitudes of geometric imperfections; nevertheless, the results are conservative and in good agreement (within 3.0% and 9.9% in magnitude), which supports the applicability of the method as a truly

Fig. 15. Load steps followed by frequency analyses for ZAL1M050.

Fig. 14. . Comparison between load-shortening curves. 244

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Fig. 16. . Comparison between the first vibration modes for ZAL1M050. Table 5 Nonlinear buckling loads and respective KDF. Cylinder

PNL [kN]

ZAL1M050 ZAL1M075 ZAL1M100 ZAL2M050 ZAL2M075 ZAL2M100

58.51 54.88 49.74 58.97 57.38 52.85

Table 6 First natural frequency variation. NL

0.83 0.78 0.71 0.84 0.82 0.75

nondestructive experimental procedure for the suggested unstiffened cylindrical shells.

Cylinder

F1 [Hz]

FNL [Hz]

FPB [Hz]

ZAL1M050 ZAL1M075 ZAL1M100 ZAL2M050 ZAL2M075 ZAL2M100

250.65 250.65 250.65 126.43 126.43 126.43

167.84 176.26 189.02 84.55 87.05 92.88

53.84 57.74 73.50 33.65 27.93 23.94

considered, especially for greater values of the KDF to be estimated. The numerical results were used to verify the VCT method and the results were presented in two different approaches: (1) minimizing the deviation between the VCT estimation and the nonlinear buckling load, and (2) considering load steps up to 50% of the nonlinear buckling load. The proposed study corroborates the method herein analytically demonstrated as appropriate for nondestructive estimations of the buckling load of imperfection-sensitive isotropic unstiffened cylindrical shells considering load levels up to 50% of the actual buckling load. Since the magnitude of the initial geometric imperfection affects the VCT estimations, the next studies should focus on finding limitations and good practices for the proper application of the methodology. Furthermore, future researches should concentrate on enhanced analytical equations for addressing further effects here neglected such as damping and initial geometric imperfections.

6. Final remarks This paper analytically demonstrated the novel VCT that has been empirically developed by Arbelo et al. [6] and validated in Refs. [21–27] for an isotropic unstiffened cylindrical shell. The rearrangement of the classic linear relationship between applied load and squared natural frequency corroborates the anticipated second-order relationship between the parametric form (1 p) 2 versus 1 f 2 . The concept of relating the minimum value of (1 p) 2 with the KDF is true for perfect and imperfect structures and, it does not depend on the drop to zero of the natural frequency of the loaded structure. This step consists of a new insight for the definition of VCT approaches for imperfection-sensitive structures. Nonlinear FE models were defined based on a measured geometric imperfection signature. These models were used to simulate the vibration response of the cylindrical shells under axial loading considering several load steps up to the nonlinear buckling load. The analytical second-order relationship holds when SS4 boundary conditions, initial geometric imperfections, and geometric nonlinearities are

Acknowledgments The research leading to these results has received funding from the European Space Agency (ESA), Contract number 4000119184/17/NL/ MH/GM. All support is gratefully acknowledged. The information in this paper reflects only the authors' views and the European Space

(a) Cylinder ZAL1.

(b) Cylinder ZAL2.

Fig. 17. . Numerical results considering the characteristic chart proposed in Ref. [6]. 245

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F. Franzoni, et al.

(a) Cylinder ZAL1.

(b) Cylinder ZAL2.

Fig. 18. . Convergence of the deviation in the VCT estimations. Table 7 Summary of the VCT results. Cylinder ZAL1M050 ZAL1M075 ZAL1M100 ZAL2M050 ZAL2M075 ZAL2M100

Minimum deviation PMAX [%] 97.5 77.5 17.5 97.5 97.5 45.0

PVCT [kN] 57.14 54.88 49.46 57.28 56.59 52.76

VCT

0.81 0.78 0.70 0.82 0.81 0.75

[%] −2.4 −0.0 −0.6 −2.9 −1.4 −0.2

Up to 50% of PNL PMAX [%] 50.0 50.0 – 50.0 50.0 –

Fig. 19. . VCT estimations for cylinders ZAL1 and ZAL2.

246

PVCT [kN] 53.70 53.24 – 53.13 52.66 –

VCT

0.77 0.76 – 0.76 0.75 –

[%] −8.2 −3.0 – −9.9 −8.2 –

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Agency is not liable for any use that may be made of the information contained therein.

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