Free vibration of non-uniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods

Thin-Walled Structures 44 (2006) 82–90 www.elsevier.com/locate/tws

Free vibration of non-uniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods A.A. Jafari a, M. Bagheri b,c,* a

Department of Mechanical Engineering, K. N. Toosi University of Technology, P.O. Box 16765-3381, Tehran, Iran b Mechanical Engineering, K. N. Toosi University of Technology P.O. Box 16765-3381, Tehran, Iran c Faculty of Aerospace Engineering, Shahid Sattari Air University, Tehran, Iran Received 21 February 2005; accepted 17 August 2005 Available online 10 November 2005

Abstract In this research, the free vibration analysis of cylindrical shells with circumferential stiffeners, i.e. rings with non-uniform stiffeners eccentricity and unequal stiffeners spacing is investigated using analytical, experimental and finite elements (FE) methods. Ritz method is applied in analytical solution while stiffeners treated as discrete elements. The polynomial functions are used for Ritz functions and natural frequency results for simply supported stiffened cylindrical shell with equal rings spacing and constant eccentricity is compared with other’s analytical and experimental results, which showed good agreement. Also, a stiffened shell with unequal rings spacing and non-uniform eccentricity with free–free boundary condition is considered using analytical, experimental and FE methods. In experimental method, modal testing is performed to obtain modal parameters, including natural frequencies, mode shapes and damping in each mode. In FE method, two types of modeling, including shell and beam elements and solid element are used, applying ANSYS software. The analytical and the FE results are compared with the experimental one, showing good agreements. Because of insufficient experimental modal data for non-uniformly stiffeners distribution, the results of modal testing obtained in this study could be as useful reference for validating the accuracy of other analytical and numerical methods for free vibration analysis. q 2005 Published by Elsevier Ltd. Keywords: Cylindrical shell; Ring stiffener; Free vibration; Natural frequency; Modal testing

1. Introduction Ring stiffened cylindrical shells are used in many structural applications such as pressure vessels, submarine hulls, aircraft, launch vehicles and offshore drilling rigs. Most of these structures are required to operate in a dynamic environment. Therefore, investigating the dynamic characteristics of these shells is very critical in developing a strategy for modal vibration control for specific operating conditions and determination of structural integrity and their fatigue life. The natural frequencies of vibration are of special interest for these structures. In the considerable literature on this subject, there are two main types of analysis, depending upon whether the stiffening rings are treated by averaging their properties over the surface * Corresponding author. Address: Mechanical Engineering, K. N. Toosi University of Technology, P.O. Box 16765-3381, Tehran, Iran. Tel.: C98 21 77343300; fax: C98 21 77334338. E-mail address: [email protected] (M. Bagheri).

0263-8231/$ - see front matter q 2005 Published by Elsevier Ltd. doi:10.1016/j.tws.2005.08.008

of the shell or by considering them as discrete elements. When ring stiffeners of equal strength are closely and evenly spaced, the stiffened shell can be modeled as an equivalent orthotropic shell. This is also called smearing method. However, as the stiffener spacing increases or the wavelength of vibration becomes smaller than the stiffener spacing, the determination of dynamic characteristics of stiffened shell is not accurate. Thus, for a more general model, the ring stiffeners have to be treated as discrete elements. When modeled in this respect, it is advantageous to use non-uniform eccentricity, unequally spaced and different materials for ring stiffeners. The free vibration of stiffened cylindrical shells has been investigated since 1950s by a number of researchers. Hopmann [1] investigated the free vibration of orthogonally stiffened cylindrical shells with simply supported ends, analytically and experimentally. In this study, smearing method for stiffeners was used in the analytical investigation. Mikulas and McElman [2] investigated the free vibration of eccentrically stiffened simply supported cylindrical shells by averaging the stiffeners properties over the surface of the shells and found that the eccentricity could have significant effects on natural frequencies. Egle and Sewall [3] extended this study with stiffeners

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treated as discrete elements. The effects of in-plane and rotary inertia on the natural frequencies of eccentrically stiffened shells were examined by Parthan and Johns [4]. A theoretical and experimental investigation of the vibration of axially loaded stiffened cylindrical shells was provided by Rosen and Singer [5] using Donell and Flugge theories. Mustafa and Ali [6] presented an energy method for free vibration analysis of stiffened cylindrical shells. The analysis took into account the flexure and extension of the shell and the flexure, extension and torsion of the stiffeners. Swaddiwudhipong et al. [7] presented the free vibrations of cylindrical shells with rigid intermediate supports. An automated Rayleigh–Ritz method was adopted to evaluate the natural frequencies and the mode shapes. A special polynomial unified set of Ritz function was used to span the displacement fields of various types and combinations of end boundary conditions. Wang et al. [8] extended the Ritz method for solving the free vibration problem of cylindrical shells with varying ring stiffener distributions. The Ritz formulation also allows stiffeners material to be different from one another and from the material of the parent shell. Ruotolo [9] presented a comparison of some thin shell theories used for the dynamic analysis of stiffened cylinders while smearing method used for stiffeners. As can be seen in literature, a few numbers of experimental data were found in published papers. Moreover, these data relate to evenly spaced and uniform stiffeners eccentricity. In this study, free vibration analysis of ring stiffened cylindrical shells with non-uniform stiffeners distribution is performed using analytical, experimental and finite elements (FE) methods. In analytical method, Ritz method with polynomial functions is applied for free vibration analysis. Modal testing is performed experimentally to obtain the modal characteristics of the ring stiffened cylindrical shell. Modal

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analysis is done computationally, using FE method in ANSYS software. Because of insufficient experimental modal data for non-uniformly stiffeners distribution, the results of modal testing obtained by authors could be a useful reference for free vibration considerations of stiffened cylinders. 2. Analytical formulation The cylindrical shell as shown in Fig. 1 is considered to be thin with uniform thickness h, radius R, length L, mass density r, modulus of elasticity E, Poisson’s ratio v, shear modulus GZE/2(1Cn). The shell is circumferentially stiffened by N number of rings, which may be placed internally or externally. The ith ring stiffener has rectangular cross-section with constant width bri and depth of dri and is located at aiL measured from one end of the shell. The rings spacing and rings depth may be varied along the length of the shell. The ring stiffeners may be constructed from different materials from one another and also from the parent shell material. The ith stiffener properties are defined as mass density pri, modulus of elasticity Eri, Poisson’s ratio vri and shear modulus Gri. 2.1. Shell energy Adopting Sander’s [10] thin shell theory, the strain energy of stretching and bending of the aforementioned cylindrical shell without stiffeners is expressed as:  2  2 Eh vu 1 vv UZ Kw C 2 vx 2ð1Ky2 Þ R vq 0 0       2y vu vv 1Ky vv 1 vu 2 Kw C C C R vx vq 2 vx R vq "  2 2 Eh3 v2 w 1 v2 w vv C C C vq vx2 R4 vq2 24ð1Ky2 Þ   2  2 2y v w v w vv C C 2 vq R vx2 vq2  2 2 #) 2ð1KyÞ v w 3 vv 1 vu C K C Rdqdx; vxvq 4 vx 4R vq R2 ðL 2p ð

(1)

where u,v,w are displacements in the longitudinal, tangential and radial directions, respectively, x, q are longitudinal and circumferential coordinates, respectively. Neglecting the effect of rotary inertia since the shell under consideration is thin; the kinetic energy of a cylindrical shell without stiffeners is expressed as: ðL 2p ð  2  2  2

1 vu vv vw T Z rh C C Rdqdx: (2) 2 vt vt vt 0 0

2.2. Ring stiffener energy

Fig. 1. The ring stiffened cylindrical shell with non-uniform stiffeners distribution.

In this analysis, geometric characteristic and material of rings may be different from one another. Also rings spacing and rings eccentricity can have non-uniform distributions.

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The strain energy of the ith ring stiffener with the effects of stretching, biaxial bending and wrapping is given by: 2p  2 ð( Eri Izri 1 vwr 1 v2 uri C Uri Z R C eri vq2 2 R C eri vx 0



2

2

Eri Ixri 1 1 v wri wr iC R C eri vq2 2 ðR C eri Þ3   Er Ar 1 vvr Kwr C i i 2 R C eri vq  2 2 ) Gri Jri 1 v wr 1 vuri C C dq: K vxvq R C eri vq 2 R C eri C

The following functions are adopted to separate the spatial variable x, q and the time variable t uðx; q; tÞ Z uðxÞsinðnq C utÞ vðx; q; tÞ Z vðxÞcosðnq C utÞ wðx; q; tÞ Z wðxÞsinðnq C utÞ;

ð3Þ

The kinetic energy of ith ring stiffener with the effects of triaxial translational inertia and rotary inertia about x and z axes is given by: 2p      ð   1 vuri 2 vvri 2 vwri 2 Tri Z rri Ari C C 2 vt vt vt 0  2 2 ) v wri CðIxri C Izri Þ ð4Þ ðR C eri Þdq; vtvx

where n is the number of circumferential waves and u is circular frequency of vibration. For generality and convenience, the following nondimensional terms are defined: u u Z ; h

h C dri ; 2

(5)

(6)

wri Z w:

w Z

h ; R

w ; R

 iZ er

Eri Z

Eri ; E

 iZ rr

rri ; r

Ari Z

Ari ; h2

Jri Z

Jri Rh3

2

U2 Z

x x Z ; L

(10)

eri h  i Z Izri ; Izr Rh3

 i Z Ixri ; Ixr Rh3

ð1Ky2 ÞrR2 2 u; E

using Eqs. (9) and (10), the nondimensional total energy functional can be expressed as:  du  K2yax  a x Cðxnv C wÞ ðxnv C wÞ dx 0 "   2 2 1Ky 2 dv x2 4 d2 w C x a Cnu C a 2 dx 12 dx2  2  d w   vÞ2 K2ya2 vÞ Cðn2 wCnx ðn2 wCnx dx2    dw 3 dv nx 2 2 C2ð1KyÞa n C x K u dx 4 dx 4a N  X 1Ky2 KU2 ½x2 u 2 Cx2 v 2 C w 2 dx C Eri ax2 3 ð1Cx e r Þ i iZ1

 FZ

where the sign (C) represents external stiffening and sign (K) is used for internal stiffening. From geometrical considerations, the relationships between the displacements (uri, vri, wri) of the ith stiffener and the displacements (u, v, w) of the shell at the position of the stiffener are given by: vw (7) uri Z u C eri vx

er  er vw vri Z v 1 C i C i R R vq

ð1 

2 2



du dx

2

2



  1i C Tr  2i  ; ½yr1i Cyr2i Cyr3i Cyr4i KU2 r ri ð1Cxeri Þax½Tr (11)

where

Substituting Eqs. (5)–(7) into Eqs. (3) and (4), the ring stiffener energy can be written in the form of shell middle surface displacement. Therefore, the energy functional of ring stiffened cylindrical shell can be written as: F Z U KT C

xZ

2ð1Ky Þ F; F Z phRLE

and the eccentricity of the ring stiffener is expressed as: eri ZG

v v Z ; h

R ; L

aZ

where the second moments of areas Izri, Ixri, cross-sectional area Ari and torsional rigidity Jri are determined by: br3 dr br dr 3 Ixri Z i i ; Ari Z bri dri ; Izri Z i i ; 12 12 " # N 1 192bri X 1 npdri 1K 5 Jri Z tanh br3i dri ; 3 2bri p dri nZ1;3;5;. n5

(9)

N X iZ1

ðUri KTri Þ:

(8)



2   dw 2 2  Kn xu y r1i Z Izri a 1Cxeri Kn xeri dx

(12.1)

 i fð1Kn2 Þwg  2 y r2i Z Ixr

(12.2)

y r3i ZAri

ð1Cxeri Þ2  2 nxð1Cx f eri Þv Cð1Cn2 xeri Þwg x

(12.3)

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y r4i ZJri



1 dw 2 Knxu Can ; 2ð1Cyi Þ dx

2.5. Equations of motion (12.4)

and  1i ZAri Tr



dw xu Cx e ri a dx 2

2

2

Cxð1Cxeri Þv Cxeri nw C w

2



(13.1)  2  2i Zxa2 ðIxr  i CIzr  i Þ dw : Tr dx

(13.2)

2.3. Geometric boundary conditions For simply supported cylindrical shells, four kinds of boundary conditions can be designated as follows: S1 : w Z v Z 0 S2 : w Z 0 S3 : w Z u Z0 S4 : w Z v Z u Z0: (14) For free boundary condition all of the displacements functions are non-zero. 2.4. Ritz functions In view of satisfying the foregoing geometric boundary conditions, the proposed Ritz functions for approximating the displacements are: ! NS NS X X 0 1 jK1  Pu Z  Pu ð1KxÞ u Z pj x pj u j (15) ðxÞ jZ1

y Z

NS X

Applying the Rayleigh–Ritz method (minimization of nondimensional energy functional with respect to Ritz functions coefficients), the equations of motion are derived as follows: 9 vF > Z0> > > vpj > > > > > = vF Z0 j Z 1; 2; ..; NS: (16) vqj > > > > > > vF > > Z0 > ; vrj Substituting Eq. (15) into Eq. (11) and then into Eq. (16) yields the following eigenvalue equation: " !# N N X X 2 ½K C ½Kri KU ½M C ½Mri  fCg Z f0g; (17) iZ1

iZ1

where [K] and [M], are stiffness and mass matrices of cylindrical shell, respectively, and [Kri] and [Mri] are corresponding matrices of the ith ring stiffener. Also fCgZ fp1 ; ..; pNS ; q1 ; ..; qNS ; r1 ; ..; rNS gT is the column vector of Ritz coefficients andU2 Z ð1Kn2 ÞrR2 u2 =Eis the non-dimensional frequency parameter. 3. Results and discussions 3.1. Cylindrical shells with uniform rings spacing and eccentricity

jZ1

! 0

1

 Pv Z  Pv ð1KxÞ qj xjK1 ðxÞ

NS X

jZ1

w Z

85

NS X

qj vj

jZ1

! rj x

jK1

0

1

 Pw Z  Pw ð1KxÞ ðxÞ

jZ1

NS X

rj w j ;

jZ1

where the powers of P are as shown in Table 1. The superscripts of P, i.e. 0 and 1, denote the cylindrical shell  0 and xZ  1, respectively. ends at xZ These forms of Ritz functions allows easy exact differentiation and integration and by increasing the number of polynomials sentences NS, better convergence to exact solution can be achieved. Table 1 Powers of P for Ritz functions Boundary condition

S1

S2

S3

S4

F

Pu Pv Pw

0 1 1

0 0 1

1 0 1

1 1 1

0 0 0

Rayliegh–Ritz method (Eq. (17)) with proposed displacement functions (Eq. (15)) is used to determine the dynamic characteristic of a simply supported (S1–S1) ring stiffened cylindrical shell. The geometrical dimensions and material properties of the shell model are given in Table 2. This model is an externally ring stiffened cylindrical shell with evenly spaced and uniform stiffeners eccentricity. Table 3 shows the comparison of predicted analytical results of natural frequencies with the experimental results of Hoppmann and also the analytical results of Mustafa and Ali Table 2 Geometrical and material properties of a stiffened shell Characteristics

Physical dimensions and values M1 model

Number of rings N Shell radius R (m) Shell thickness h (m) Shell length L (m) Rings height dr (m) Rings width br (m) Modulus of elasticity E (Gpa) Mass density r (Kg/m3) Poisson’s ratio n Stiffening type

19 0.049759 0.001651 0.3945 0.005334 0.003175 68.95 2762 0.3 External

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Table 3 Convergence and comparison of natural frequencies with other references for model M1 Mode number

Present analysis results natural frequencies (HZ)

m

n

Number of polynomials: NS 4 6

8

10

1

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1199.85 1564.52 4400.60 8405.96 13556.7 3790.78 2365.13 4444.75 8449.92 13568.8 6543.24 4111.98 4935.99 8496.70 13624.4

1199.58 1564.48 4387.68 8378.10 13500.4 3493.61 2113.87 4400.59 8392.89 13509.8 5840.25 3378.59 4596.06 8449.93 13567.2

1199.58 1564.47 4387.59 8377.75 13490.7 3493.59 2113.84 4400.58 8392.63 13508.9 5839.89 3378.17 4595.79 8449.89 13555.4

2

3

a

1199.58 1564.48 4388.49 8381.31 13510.3 3499.62 2118.10 4400.60 8393.05 13566.7 5868.62 3409.15 4607.88 8449.94 13624.3

Experimental [1]

Discrepancya (%)

Analytical [6]

– 1530 4080 – – – 2040 4090 – – – 3200 4520 7520 –

– 2.2 7.0 – – – 3.5 7.0 – – – 5.2 1.6 11.0 –

1204 1587 4462 8559 13780 3498 2129 4437 8482 13695 5844 3386 4627 8438 13595

Discrepancy between the presented analytical and the experimental [1] results.

Fig. 2. The M2 model physical dimensions.

for various modes of vibrations for M1 model. It could be observed that using polynomial terms of NSZ8 for each displacement function is adequate for converged results. It should be noted that the results are in good agreement. In addition, the agreements of the presented analytical results with the experimental values of Hoppmann are better than the analytical values of Mustafa and Ali. Therefore, applying this method with the proposed Ritz functions has adequate accuracy for determining the natural frequencies of ring stiffened cylindrical shell. Generally, in cylindrical shells, the lowest natural frequency does not necessarily correspond to the lowest wave index. In fact, the natural frequencies do not fall in ascending order of the wave index either. Mode shapes associated with each natural frequency are combination of Radial (flexural); Longitudinal (axial); Circumferential (torsional) modes. On the other words, fundamental frequency index is related to L/R, h/R, physical dimensions and number of stiffeners, stiffeners distribution along the shell length, and boundary conditions.

Fig. 3. Excitation points in modal testing.

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87

Fig. 4. Driving point FRF.

For M1 model, the fundamental frequency is obtained corresponding to mZ1, nZ1. Integer m refers to the number of longitudinal half wave and integer n refers to the circumferential wave number. 3.2. Non-uniformly ring stiffened cylindrical shells with FFBC1 Here, a model of ring stiffened cylindrical shell with unequal rings spacing and non-uniform eccentricity is considered. The geometric dimensions of the shell model are shown in Fig. 2. It should be noted that, both the ring stiffeners and the shell are made of same material. Different approaches for this shell model with FFBC are applied for the analysis, namely, analytical, experimental, and FE methods. In experimental method, modal testing is performed. To this end, an experimental stiffened shell model with dimensions as shown in Fig. 2 is machined from a thick steel pipe. The model is hanged with rubber rope to simulate FFBC. The modal testing is performed to obtain the modal characteristics of the stiffened cylindrical shell between 0 and 3200 Hz. The model is excited at predetermined points with an impact hammer in radial direction of the shell. Then the response is measured using an accelerometer at specified fixed point in radial direction (Roving hammer and fixed response method). To avoid hiding of some modes and also obtaining accurate longitudinal and circumferential mode shapes, the model is divided into 24 points in circumferential direction and 12 points along the shell length which their arrangements are shown in Fig. 3. The shell is excited by a hammer on these points. A piezoelectric accelerometer is attached to the model at point 1 by wax, to measure the output acceleration. The analysis of frequency response functions (FRF) is performed, applying the STAR MODAL software. Outputs of this analysis are natural frequencies, mode shapes and damping in each mode. Fig. 4 shows the FRF of point 1 which is called driving 1

Free–Free Boundary Condition.

point. Table 4 shows the obtained experimental results including natural frequencies and damping ratios for each mode. Totally, eleven natural frequencies are found within that range with corresponding damping ratios and mode shapes. In numerical method, FE analysis is done using ANSYS software. Two types of modeling including shell and solid modeling are performed. In shell modeling, SHELL 93 and BEAM 189 elements type are used for shell and ring stiffeners, respectively. In solid modeling, element type SOLID 45 is used for both shell and ring stiffeners (Fig. 5). Table 5 shows the comparison of predicted analytical and FE results of natural frequencies with obtained experimental results of modal testing which are in good agreement. Comparing results of these methods showed good agreement. The results show that analytical and FE results have enough accuracy and the maximum discrepancy between the analytical and experimental results is less than 7%. Therefore, these methods are useful for free vibration analysis. It should be noted that the fundamental frequency is obtained corresponding to mZ1, nZ2. Figs. 6–9 show some mode shapes of vibration obtained from analytical solution, modal testing and FE analysis. In experimental mode shapes, dashed lines represent the Table 4 Modal testing results include natural frequencies and damping for each mode Mode

Frequency (Hz)

Damping (Hz)

Damping (%)

1 2 3 4 5 6 7 8 9 10 11

326.283 347.7278 877.7521 920.8615 1597.2084 1697.7704 2295.2971 2384.1682 2516.2180 2642.9446 3066.7891

0.6868 4.1598 1.2028 1.0412 7.0756 2.1497 12.9315 6.7627 4.4320 2.0020 4.4320

0.2105 1.1962 0.1370 0.1131 0.4430 0.1266 0.5634 0.2836 0.1761 0.0757 0.1445

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Fig. 5. Ring stiffened cylindrical shell FE models: (a) SHELL and BEAM elements, (b) SOLID element.

Table 5 Comparison of experimental, analytical and FE results Mode number m

N

1

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

2

2

a

Experimental results (Hz)

326.3 920.9 1703.4 2643

347.7 877.7 1601.9 2516

2384 2295 3066

Analytical results (Hz)

0 323.8 911.3 1700 2584 3639 2 324.3 874 1622 2543 3639 6472 3715 2357 2252.6 3036 4144

Discrepancya (%)

K0.76 K1 K0.2 K2.23

K6.73 K0.42 1.25 1

K1.13 K1.85 K0.98

FE results (Hz) Shell and beam elements

Solid element

0 336 925 1687 2556 3604 4 329.5 880 1620 2523 3596 6603 3719 2370 2265 2974 3975

0 326.2 907.2 1677.5 2572.7 3644 0 328.3 878.4 1622.9 2541 3630 6614 3724.7 2371.4 2258.3 2960 3978

Discrepancy between the analytical and the experimental results.

Fig. 6. Extracted mode shape; (a) Analytical, (b) Modal testing, (c) FE method mZ2, nZ2.

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Fig. 7. Extracted mode shape; (a) Analytical, (b) Modal testing, (c) FE method mZ2, nZ3.

Fig. 8. Extracted mode shape; (a) Analytical, (b) Modal testing, (c) FE method mZ3, nZ3.

Fig. 9. Extracted mode shape; (a) Analytical, (b) Modal testing, (c) FE method mZ3, nZ5.

89

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un-deformed shape. The radial component of amplitude for different points is: constant along the shell length for mZ1; a linear function of x for mZ2; a characteristic curve, similar to that obtained for a free–free beam, with (mK1) axial nodes for mR3. 4. Conclusions The free vibration analysis of the ring stiffened simply supported cylindrical shells is investigated analytically using Ritz method and stiffeners treated as discrete elements. The eigenvalue equation results are validated by comparing with studies on well-known natural frequency results for uniformly ring stiffened cylindrical shell. Also, free vibration of ring stiffened cylindrical shell with non-uniform eccentricity and unequal rings spacing for free–free boundary condition is considered with analytical, experimental and FE approaches. Modal testing is performed experimentally, using fixed response approach. FE analysis results are obtained using ANSYS software. Comparing results of these methods showed good agreement. The results show that analytical and FE results have enough accuracy and these methods are useful for free vibration analysis. As modal testing data for nonuniformly ring stiffened cylindrical shells are not available,

these obtained data could be very useful for other researcher in this subject. References [1] Hoppmann WH. Some characteristics of the flexural vibrations of orthogonally stiffened cylindrical shells. J Acoust Soc Am 1958;30: 77–82. [2] Mikulas Jr MM, McElman JA. On the free vibration of eccentrically stiffened cylindrical shells and plates NASA TN-D 3010; 1965. [3] Egle DM, Sewall JL. Analysis of free vibration of orthogonally stiffened cylindrical shells with stiffeners treated as discrete elements. AIAA J 1968;6(3):518–26. [4] AL-Najafi AMJ, Warburton GB. Free vibration of ring stiffened cylindrical shells. J Sound Vib 1970;13(1):9–25. [5] Rosen A, Singer J. Vibrations of axially loaded stiffened cylindrical shells. J Sound Vib 1974;34(3):357–78. [6] Mustafa BAJ, Ali R. An energy method for free vibration analysis of stiffened circular cylindrical shells. Comput Struct 1989;32(2):335–63. [7] Swaddiwudhipong S, Tian J, Wang CM. Vibrations of cylindrical shells with intermediate supports. J Sound Vib 1995;187(1):69–93. [8] Wang CM, Swaddiwudhipong S, Tian J. Ritz method for vibration analysis of cylindrical shells with ring stiffeners. J Eng Mech 1997;123: 134–42. [9] Ruotolo R. A comparison of some thin shell theories used for the dynamic analysis of stiffened cylinders. J Sound Vib 2001;243(5):847–60. [10] Sanders JL. An improved first-approximation theory for thin shells NASA TR R-24; 1959.