Vibration and stability of ring-stiffened Euler–Bernoulli tie-bars

Applied Mathematical Modelling 30 (2006) 261–277 www.elsevier.com/locate/apm

Vibration and stability of ring-stiffened Euler–Bernoulli tie-bars S. Naguleswaran

*

Department of Mechanical Engineering, University of Canterbury, Private Bag 4800, Christchurch 8020, New Zealand Received 1 September 2003; received in revised form 1 March 2005; accepted 23 March 2005 Available online 4 June 2005

Abstract Tie-bars are frequently used in structural and mechanical engineering applications. To satisfy requirements like weight reduction, stability improvement, etc., the tie-bars are stiffened with rings. In this paper a method is developed to calculate the natural frequencies, buckling axial force, etc., of the ring-stiffened tie-bars. The dynamics of the ringed and the unringed portions of the beam are treated separately. The mode shape of the first portion was expressed as the superposition of two functions multiplied by constants. Consideration of continuity of deflection and of slope and compatibility of bending moment and shearing force at the first step enabled the mode shape of the second portion to be expressed as the superposition of two functions but multiplied by the same constants as in the first portion. This procedure was recursively carried out up to the last portion. The frequency equation was then derived from the boundary conditions at the end. Buckling of the tie-bar was considered as the case when one of the natural frequencies is zero. The first three frequency parameters and the first two buckling dimensionless axial forces are tabulated for tie-bars stiffened with various number of rings and for various combinations of boundary conditions. The calculation procedure can handle any number and any type of ring-stiffeners. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Vibration; Stepped beams; Tie-bars; Buckling

*

Tel.: +64 3 364 2987x7215; fax: +64 3 364 2078. E-mail address: [email protected]

0307-904X/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2005.03.025

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Nomenclature cl, pn, sl, fr clamped, pinned, sliding, free (a), (b), (c) stiffener arrangement Fig. 2a–c aL, bL axial length of beam portion and stiffener ÔactiveÕ dimension dj D, Dnj operators d/dx, dn =dxnj EIj, EIR, EIB flexural rigidity of jth portion, stiffener, beam G2,1..H2,4 coefficients Eq. (17) L, Lj length of beam, of jth portion mj, mR, mB mass per unit length of jth portion, stiffener, beam Mj(xj), Qj(xj) bending moment, shearing force Eq. (1) Mj(Xj), Qj(Xj) dimensionless defined in Eq. (2) dimensionless length of jth portion Rj axial force T0 Uj(Xj), Vj(Xj) functions Eq. (16) xj, Xj abscissa, dimensionless Eq. (2) yj(xj), Yj(Xj) deflection, dimensionless Eq. (2) reference frequency parameter Eq. (2) a0 nth frequency parameter a1,n b axial length constant Eq. (22) /j, lj Eq. (2) /R, lR equal to EIR/EIB, mR/mB x a natural frequency nth dimensionless frequency Eq. (2) Xn dimensionless axial tension s0 sc,1, sc,2 first and second buckling forces

1. Introduction Several publications are available on the vibration of beams with one-step change in cross-section (not carrying an axial force)—the most important being Jang and Bert [1] who expressed the frequency equation for classical end supports as 4th order determinant equated to zero. Transverse vibration of uniform beams carrying a constant axial force is covered in text books e.g. McCallion [2]. Bokaian [3] presented the frequencies (in graphical form) of a uniform beam under axial compressive force and discussed buckling conditions for classical boundary conditions. Bokaian [4] extended the work in [3] to tensile axial loads. Timoshenko and Gere [5] derived the transcendental equation from which the critical end force of one-step cantilevers may be obtained. The reference also considered the buckling of a simply supported one-step beam under axial forces at the ends and another force at the step. Arabi and Li [6] used the integral equation approach to formulate the buckling of a one-step beam and presented some results. Girijavallabhan [7] and Schreyer and Shih [8] presented methods to

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obtain lower bounds of the critical end load of one-step cantilever. OÕRouke and Zebrowski [9] used a finite difference based scheme to obtain the lower bound of the critical end force of onestep cantilevers and simply supported beams. The results presented in Refs. [7–9] were substantially different from the ÔexactÕ results presented in [5]. Vibration and stability of beams with one-step change in cross-section and under different axial forces in the two portions was considered by Naguleswaran [10]. It was shown that a zero natural frequency (which initiates onset of instability or Euler buckling), is possible for certain critical combinations of the axial forces—at least one of which must be compressive. Naguleswaran [11] extended the study in [10] to beams with three step changes in cross-section. The number of publications on multi-step beams is not extensive when compared to those on one-step beams. Sato [12] used the transfer matrix method together with finite element method to study the vibration of a uniform beam with one thin ring-grove. Bapat and Bapat [13] used the transfer matrix to formulate a solution to the vibration of a multi-step beam but did not present any results. The vibration of clamped–free, clamped–clamped, clamped–pinned uniform beams stiffened by one or more rings and under constant conservative or follower axial force was addressed by Dube et al. [14]. The frequency equation was expressed as a (2n)th order determinant equated to zero where n is the total number of ring-stiffeners and beam segments. RidderÕs method [15] was used to obtain the roots. The first three roots (accuracy to 3 places after the decimal place) of ring-stiffened beams with up to 4 (in some cases 6) rings were presented. Buckling of the ring-stiffened beams was treated separately. Au et al. [16] used C1 and C2 modified beam vibration functions to study the vibration and stability of beams with abrupt changes in cross-section and for example calculations/comparison chose the same ring-stiffened beams in Ref. [14]. The method in [16] requires considerable mathematical concepts. Some results obtained off C2 functions are of limited accuracy. Fan et al. [17] presented a kind of Gibbs-phenomenon-free Fourier series and demonstrated its applications to study the vibration and stability of uniform ring-stiffened beams and beams with open cracks. In the present paper, a tie-bar stiffened with several rings is considered. Using the boundary conditions at the left end, the mode shape of the first portion was expressed as the superposition of two functions each multiplied by an arbitrary constant. Consideration of the continuity of deflection and of slope at the first step and compatibility of moment and shearing force of the element at the step enabled the mode shape of the second portion to be expressed as the superposition of two other functions but multiplied by the same constants. This procedure was continued till one reached the last portion. The frequency equation was then expressed as a 2nd order determinant equated to zero. Schemes are presented to progressively calculate the elements of the frequency equation and a scheme to calculate the roots. Results are presented for beams stiffened with up to 25 rings. The procedure can handle any number of ring-stiffeners but excessive number of rings implies rings of small axial dimension which is not realistic in engineering applications. Euler buckling occurs when the axial force (compressive) reaches a critical value where a natural frequency is zero. Dube et al. [14] and Au et al. [16] presented the first critical value. The accuracy of the critical axial force obtained in [16] with C2 modified functions were mediocre. In the present paper, the first two critical axial forces are presented of beams under 16 combinations of classical boundary conditions and stiffened with up to 25 rings. The theory developed is applicable to any type of step change in cross-section but in the present paper particular attention was paid to three types which occur in common engineering

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applications. Type 1 beam is of constant depth and with step changes in breadth, Type 2 is of constant breadth and with step changes in depth and Type 3 is with step changes in depth and breadth i.e. with similar cross-sections—for example, a beam of circular cross-section with step changes in diameter. The ÔactiveÕ dimension of the three types of beams is the breadth, depth and diameter of the beam portion, respectively. The examples considered in [14,16] are Type 1. Ring-stiffened beams are Type 3. The results may be used as bench marks to judge the accuracy of results obtained by any numerical methods.

2. Theory Fig. 1 shows the Euler–Bernoulli beam A1An+1 with step change in cross-section at A2, A3, . . . , An. The end A1 is axially restrained and An+1 is axially free. The ends A1 and An+1 are on classical clamped (cl), pinned (pn), sliding (sl) or free (fr) supports. The flexural rigidity, mass per unit length, the length of the jth portion AjAj+1 (j = 1, 2, . . . , n) are EIj, mj and Lj and the centroidal axial force in the beam is T0 (positive if tensile). Coordinate systems are chosen with origin at Oj which coincides with Aj when the beam is in the undeflected position. The dynamics of each beam portion is treated separately. 2.1. The general expression of the mode shape of AjAj+1 Using the sign convention in Ref. [2], for free vibration at frequency x, if the ordinate yj(xj) is the amplitude of vibration at abscissa xj (0 6 xj 6 Lj), then the amplitude of bending moment Mj(xj) and shearing force Qj(xj) are d2 y j ðxj Þ M j ðxj Þ ¼ EIj ; dx2j

d3 y j ðxj Þ dy j ðxj Þ Qj ðxj Þ ¼ EIj þ T0 ; 3 dxj dxj

d4 y j ðxj Þ d2 y j ðxj Þ  T  mj x2 y j ðxj Þ ¼ 0. EIj ¼ 0 dx4j dx2j

ð1Þ

To express the set of Eq. (1) in dimensionless form, a beam of flexural rigidity EI0, mass per unit length m0 and length L is used as ÔreferenceÕ and one defines the dimensionless abscissa Xj, amplitude Yj(Xj), step position parameter Rj, the operators dj, Dnj , the dimensionless bending moment

Fig. 1. The multi-step beam and the coordinate systems.

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Mj(Xj), shearing force Qj(Xj), axial force s0, flexural rigidity ratio /j, mass per unit length ratio lj, dimensionless frequency parameters a0 and aj as follows: Xj ¼

xj ; L

M j ðX j Þ ¼ a20 ¼

Y j ðX j Þ ¼

Rj ¼

Lj ; L

Qj ðxj ÞL2 ; EIR ! lj 4 mj x2 L 4 4 a. aj ¼ ¼ EIj /j 0

M j ðxj ÞL ; EIR

m0 x4 L4 ; EI0

y j ðxj Þ ; L

Qj ðX j Þ ¼

Dj ¼

d ; dX j

s0 ¼

Dnj ¼

T 0 L2 ; EIR

dn ; dX nj

/1 ¼

EIj ; EIR

lj ¼

mj ; mR

ð2Þ

In Eq. (2), a0 is the natural frequency parameter. The nth natural frequency parameter is denoted by a0,n. The ÔactiveÕ dimension dj of the Type 1, 2 and 3 beams are the breadth, depth and diameter, respectively, and one has for Type 1 beam lj ¼ d j =d 0

and /j ¼ d j =d 0 ;

for Type 2 beam lj ¼ d j =d 0 for Type 3 beam l1 ¼ ðd j =d 0 Þ

ð3aÞ 3

and /j ¼ ðd j =d 0 Þ ; 2

ð3bÞ 4

and /j ¼ ðd j =d 0 Þ ;

ð3cÞ

where d0 is the ÔactiveÕ dimension of the ÔreferenceÕ beam. In Refs. [14,16], for example, calculations Type 1 cross-section change was considered. Eq. (1) in dimensionless form are M j ðX j Þ ¼ /j D2j ½Y j ðX j Þ; Qj ðX j Þ ¼ /j D3j ½Y j ðX j Þ þ s0 Dj ½Y j ðX j Þ;

/j D4j ½Y j ðX j Þ  s0 D2j ½Y j ðX j Þ  lj a40 Y j ðX 1 Þ ¼ 0. ð4Þ

The solution of the dimensionless mode shape differential Eq. (4) is Y j ðxj Þ ¼ C 1;j sin aj X j þ C 2;j cos aj X j þ C 3;j sinh bj X j þ C 4;j cosh bj X j ; where C1,j through to C4,j are the four constants of integration and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs20 þ 4lj /j a40 Þ  s0 ðs20 þ 4lj /j a40 Þ þ s0 a2j ¼ ; b2j ¼ . 2/j 2/j

ð5Þ

ð6Þ

2.2. The mode shape of the first portion A1A2 The general expression for the mode shape of A1A2 is Y 1 ðX 1 Þ ¼ C 1;1 sin a1 X 1 þ C 2;1 cos a1 X 1 þ C 3;1 sinh b1 X 1 þ C 4;1 cosh b1 X 1 .

ð7Þ

The need for Eq. (7) to satisfy the boundary conditions at A1 was used to eliminate two of the constants and the mode shape of the portion A1A2 was expressed as Y 1 ðX 1 Þ ¼ AU 1 ðX 1 Þ þ BV 1 ðX 1 Þ;

ð8Þ

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where A and B are constants and the functions U1(X1) and V1(X1) for classical boundary conditions at A1 are a1 V 1 ðX 1 Þ ¼ cos a1 X 1  cosh b1 X 1 ; if cl: U 1 ðX 1 Þ ¼ sin a1 X 1  sinh b1 X 1 ; b1 ð9aÞ if pn: U 1 ðX 1 Þ ¼ sin a1 X 1 ; V 1 ðX 1 Þ ¼ sinh b1 X 1 ; ð9bÞ V 1 ðX 1 Þ ¼ cosh b1 X 1 ; ð9cÞ if sl: U 1 ðX 1 Þ ¼ cos a1 X 1 ; 2 2 ð/ a þ s0 Þa1 a if fr: U 1 ðX 1 Þ ¼ sin a1 X 1 þ 1 12 sinh b1 X 1 ; V 1 ðX 1 Þ ¼ cos a1 X 1 þ 12 cosh b1 X 1 . ð/1 b1  s0 Þb1 b1 ð9dÞ The derivatives of U1(X1) and V1(X1) are obtained easily by straight forward differentiation. 2.3. The mode shape of the second portion A2A3 The general expression of the mode shape of the second portion A2A3 is Y 2 ðx2 Þ ¼ C 1;2 sin a2 X 2 þ C 2;2 cos a2 X 2 þ C 3;2 sinh b2 X 2 þ C 4;2 cosh b2 X 2 .

ð10Þ

Continuity of deflection and slope at A2 and compatibility of bending moment and shearing forces on the beam element at A2 leads to Y 1 ðR1 Þ ¼ Y 2 ð0Þ;

D1 ½Y 1 ðR1 Þ ¼ D2 ½Y 2 ð0Þ;

/1 D21 ½Y 1 ðR1 Þ ¼ /2 D22 ½Y 2 ð0Þ;

/1 D31 ½Y 1 ðR1 Þ  s0 D1 ½Y 1 ðR1 Þ ¼ /2 D32 ½Y 2 ð0Þ  s0 D2 ½Y 2 ð0Þ.

ð11Þ

From the equations which result when Eqs. (7) and (10) are substituted into the four Eq. (11), the four constants C1,2, C2,2, C3,2 and C4,2 may be eliminated (as described in the next section) and the mode shape of A2A3 expressed as Y 2 ðX 2 Þ ¼ AU 2 ðX 2 Þ þ BV 2 ðX 2 Þ.

ð12Þ

The expressions for the functions U2(X2) and V2(X2) are obtained from the general case treated in the next section. This procedure may be recursively extended out portion by portion. 2.4. The mode shape of the jth portion AjAj+1 The general expression of the mode shape of AjAj+1 (j = 2, . . . , n) is Eq. (5). Continuity of deflection and slope at Aj and compatibility of bending moment and shearing forces on the beam element at Aj leads to Y j1 ðRj1 Þ ¼ Y j ð0Þ;

Dj1 ½Y j1 ðRj1 Þ ¼ Dj ½Y j ð0Þ;

/j1 D3j1 ½Y j1 ðRj1 Þ ¼ /j D3j ½Y j ð0Þ.

/j1 D2j1 ½Y j1 ðRj1 Þ ¼ /j D2j ½Y j ð0Þ; ð13Þ

Suppose the mode shape of Aj1Aj is Y j1 ðX j1 Þ ¼ AU j1 ðX j1 Þ þ BV j1 ðX j1 Þ.

ð14Þ

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267

Eqs. (5) and (14) must satisfy Eq. (13) and hence C1,j, C2,j, C3,j, C4,j may be eliminated and the mode shape of portion AjAj expressed as Y j ðX j Þ ¼ AU j ðX j Þ þ BV j ðX j Þ; ð15Þ in which the functions Uj(Xj) and Vj(Xj) (j = 2, 3, . . . , n) are U j ðX j Þ ¼ G1;j sin aj X j þ G2;j cos aj X j þ G3;j sinh aj X j þ G4;j cosh aj X j ; V j ðX j Þ ¼ H 1;j sin aj X j þ H 2;j cos aj X j þ H 3;j sinh aj X j þ H 4;j cosh aj X j ;

ð16Þ

in which the coefficients G1,j through to G4,j are G1;j ¼ G2;j ¼ G3;j ¼ G4;j ¼

/j b2j Dj1 ½U j1 ðRj1 Þ  /j1 D3j1 ½U j1 ðRj1 Þ /j aj ða2j þ b2j Þ /j b2j ½U j1 ðRj1 Þ  /j1 D2j1 ½U j1 ðRj1 Þ /j ða2j þ b2j Þ

;

/j a2j Dj1 ½U j1 ðRj1 Þ þ /j1 D3j1 ½U j1 ðRj1 Þ /j bj ða2j þ b2j Þ /j a2j ½U j1 ðRj1 Þ þ /j1 D2j1 ½U j1 ðRj1 Þ /j ða2j þ b2j Þ

;

ð17Þ ;

.

The coefficients H1,j through to H4,j are obtained by replacing U with V in above expressions. 3. The frequency equation The mode shape of AnAn+1 may be expressed as Y n ðX n Þ ¼ AU n ðX n Þ þ BV n ðX n Þ;

ð18Þ

in which the functions Un(Xn) and Vn(Xn) were recursively established as outlined earlier. The frequency equation eventuates from the need for this equation to satisfy the boundary conditions at An+1. The frequency equations are if An+1 is cl: pn: sl: fr:

U n ðRn ÞDn ½V n ðRn Þ  Dn ½U n ðRn ÞV n ðRn Þ ¼ 0; U n ðRn ÞD2n ½V n ðRn Þ  D2n ½U n ðRn ÞV n ðRn Þ ¼ 0; Dn ½U n ðRn ÞD3n ½V n ðRn Þ  D3n ½U n ðRn ÞDn ½V n ðRn Þ ¼ 0; D2n ½U n ðRn ÞfD3n ½V n ðRn Þ  s0 Dn ½V n ðRn Þg  fD3n ½U n ðRn Þ  s0 Dn ½U n ðRn ÞgD2n ½V n ðRn Þ ¼ 0.

ð19aÞ ð19bÞ ð19cÞ ð19dÞ

3.1. Natural frequency calculations In this paper the ÔreferenceÕ beam in the set of Eq. (2) was chosen with EI0 = EI1 i.e. /1 = 1 and m0 = m1 i.e. l1 = 1 and natural frequency parameters were expressed (without loss of generality) via the frequency parameter a0 = a1. Without loss of generality one may choose R1 þ R2 þ þ Rn ¼ 1.

ð20Þ

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S. Naguleswaran / Applied Mathematical Modelling 30 (2006) 261–277

The system parameters are lj, /j, Rj (j = 1, 2 . . . , n) and s0. The roots of the frequency equations (19a, b, c or d) were determined by a ÔsearchÕ to bracket an approximate range within which a root is present followed by an iterative procedure based on linear interpolation. The procedure is as follows: U1(X1) and V1(X1) were chosen from Eq. (9a, b, c or d) taking account of the boundary conditions at A1. A trial frequency parameter (a0 = 0.1 say) was assumed and U1(R1), V1(R1), D1[U1(R1)], D1[V1(R1)], etc., were calculated. One proceeded to calculate the coefficients G1,2 through to H4,2 from Eq. (17) and hence U2(R2), V2(R2), D2[U2(R2)], D2[V2(R2)], etc., from Eq. (16). This was continued till Un(Rn), Vn(Rn), Dn[Un(Rn)], Dn[Vn(Rn)], etc., were established. These were inserted into the appropriate frequency equations (19a, b, c or d which was chosen taking the boundary condition at An+1 into account) to yield a ÔremainderÕ. The value of a0 was increased in steps of 0.1 and the calculations described were repeated till a sign change in the ÔremainderÕ occurred. The sign change indicated the presence of a root within this range. A ÔsearchÕ was made within this range but with change of 0.01 in a0 to narrow the range within which the root lies. At this stage an iterative procedure based on linear interpolation was invoked to calculate the root to the pre-set accuracy. The ÔsearchÕ procedure was continued (from the value of the first root) to locate the second root and so on. 3.2. Numerical examples For example, calculations a beam with several regular rectangular ripple profile was chosen to represent the ring-stiffeners. The ring-stiffeners are equi-spaced of flexural rigidity EIR, mass per unit length mR and constant axial length LR = bL. The flexural rigidity, mass per unit length and the length of the beam segments are EIB, mB and LB = aL. If there are n ring-stiffeners clearly nb < 1. The three types of ring-stiffener arrangement considered in this paper are shown in Fig. 2a, b and c and are referred to as (a), (b) and (c), respectively. Mathematically, arrangements (a) and (c) are identical. For the arrangement (a), the total number of portions is (2n  l) and

Fig. 2. The ring-stiffener arrangements (a), (b) and (c).

S. Naguleswaran / Applied Mathematical Modelling 30 (2006) 261–277

a ¼ ð1  nbÞ=ðn  1Þ

ðn P 2Þ;

269

ð21aÞ

for (b), the total number of portions is 2n and a ¼ ð1  nbÞ=n

ðn P 1Þ;

ð21bÞ

and for (c), the total number of beam portions is (2n + 1) and a ¼ ð1  nbÞ=ðn þ 1Þ

ðn P 1Þ.

ð21cÞ

3.3. Example 1: Vibration of ring-stiffened beams ÔActiveÕ dimension of dR = 2.0 and dB = 1.0 were chosen for the ring-stiffener and beam, respectively. From Eqs. (3a)–(3c) lR, lB, /R and /B were calculated for the Type 1, 2 and 3 change in cross-section. Ring-stiffener length b = 0.05 (which limits the number of ring-stiffeners to 19), the beam segment length a was calculated from Eq. (21a, b or c) taking account of the ring arrangement. The first three frequency parameters of the beam stiffened with up to 18 rings were calculated. In Table 1, to economise on space, only the frequency parameters of beams with ring-stiffener arrangement (a) under clncl, clnpn and pnnpn boundary conditions are tabulated for various n and those of the 16 classical boundary condition with n = 10 arrangement (b). Note that for arrangement (b), clncl and frnfr Type 1 cross-section change beams have the same frequency parameters and this property hold for clnpn and slnfr and for clnsl and slnfr beams. This relationship does not hold for Type 2 or 3 beams. For arrangement (a), the 2 ring system provided the maximum increase in the frequency parameters. Increase in the frequency parameters for increase in n is small. To illustrate the influence of ring-stiffener arrangement, the first three frequency parameters of clnfr and frncl beams with ring-stiffeners arrangement (a), (b) or (c) are tabulated in Table 2. Note that because of symmetry the frequency parameters of clnfr and frncl identical for arrangement (a) and for (c). 3.4. Example 2: Vibration of ring-stiffened tie-bars The frequency parameters of tie-bars (same beam parameters of Table 1) under axial force s0 = 10.0 (tensile) were calculated and are tabulated in Table 3. The frequency parameters in Table 3 are greater than the corresponding values in Table 1. 3.5. Euler buckling of ring-stiffened beams Physical considerations suggest that decrease in the axial force s0 will result in a decrease in the frequency parameters. For some combinations of s0 and beam parameters, a frequency parameter may be zero. This is Euler buckling or the condition for the onset of instability. A necessary but not sufficient condition for buckling is s0 must be negative (i.e. compressive). Further decrease in s0 will render the mode unstable. To calculate the buckling axial force sc, a0 is set to zero, a trial (negative) sc was assumed and the ÔsearchÕ and iteration procedure was invoked to calculate the first two critical values sc,1 and sc,2. The first two critical axial forces sc,1, and sc,2 of beams with stiffener arrangement (a) under various boundary conditions are tabulated in Table 4. Because of

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S. Naguleswaran / Applied Mathematical Modelling 30 (2006) 261–277

Table 1 The first three frequency parameters of beam with n stiffeners, ÔactiveÕ stiffener dimension dR = 2.0, ÔactiveÕ beam dimension dB = 1.0, stiffener length b = 0.05, beam segment length a from Eqs. (21a) and (21b), stiffener arrangement in first column

(a)

(b)

BC

n

Type 1

Type 2

Type 3

a1,1

a1,2

a1,3

a1,1

a1,2

a1,3

a1,1

a1,2

a1,3

clncl

2 3 4 5 8 12 18

4.94733 4.84893 4.80593 4.76963 4.70202 4.66606 4.69701

8.1782 8.1766 7.9951 7.9489 7.8194 7.7489 7.7969

11.4022 11.3379 11.3795 11.1273 10.9672 10.8531 10.9148

5.16688 5.10752 5.05146 5.03094 5.06079 5.24453 6.03149

8.5575 8.5568 8.5456 8.4598 8.4530 8.7282 10.0107

11.9508 12.0550 11.9521 12.0770 11.9184 12.2540 14.0119

5.21008 4.89666 4.80524 4.72229 4.61454 4.69375 5.55884

8.6383 8.6303 8.0483 7.9741 7.7317 7.8248 9.2302

12.0772 11.9232 11.9764 11.2393 10.9533 11.0113 12.9253

clnpn

2 3 4 5 8 12 18

4.01615 3.95401 3.92493 3.90287 3.86120 3.84435 3.89052

7.2128 7.1945 7.0887 7.0449 6.9587 6.9222 7.0030

10.3927 10.3429 10.3463 10.2024 10.0662 10.0026 10.1147

4.10088 4.06577 4.05075 4.05022 4.09731 4.26431 4.95471

7.3745 7.3731 7.3554 7.3388 7.4001 7.6871 8.9183

10.6405 10.6934 10.6637 10.6826 10.7334 11.1231 12.8807

4.11362 3.91899 3.84243 3.78960 3.72374 3.80189 4.54285

7.3912 7.3176 7.0080 6.8977 6.7409 6.8612 8.1790

10.6509 10.5448 10.4664 10.1022 9.8148 9.9446 11.8164

pnnpn

2 3 4 5 8 12 18

3.14096 3.10809 3.09305 3.08071 3.05694 3.05279 3.10567

6.2784 6.2778 6.1979 6.1694 6.1177 6.1071 6.2114

9.4102 9.3452 9.4052 9.2788 9.1866 9.1644 9.3174

3.14145 3.14113 3.14508 3.15305 3.20250 3.34447 3.92287

6.2825 6.2823 6.3016 6.3161 6.4114 6.6924 7.8462

9.4242 9.4770 9.4240 9.4958 9.6336 10.0472 11.7704

3.13901 3.02036 2.97433 2.94156 2.90199 2.97145 3.57894

6.2641 6.2616 5.9952 5.9153 5.8194 5.9498 7.1589

9.3680 9.1941 9.3491 8.9683 8.7705 8.9425 10.7409

clncl clnpn clnsl clnfr

10

4.59643 3.84610 2.32157 1.85188

7.6379 6.9271 5.3879 4.6384

10.7060 10.0127 8.4744 7.7661

4.95445 4.18599 2.51510 2.01360

8.2542 7.5517 5.8380 5.0503

11.6038 10.9386 9.2009 8.4692

4.42936 3.75019 2.26050 1.81572

7.3925 6.7748 5.2487 4.5607

10.4190 9.8343 8.2921 7.6639

pnncl pnnpn pnnsl pnnfr

3.78286 3.05079 1.52541 3.84610

6.8145 6.1037 4.5783 6.9271

9.8532 9.1611 7.6359 10.0127

4.02589 3.27878 1.63043 4.13349

7.2619 6.5629 4.8964 7.4478

10.5170 9.8582 8.1725 10.7724

3.58760 2.92805 1.45898 3.71890

6.4792 5.8660 4.3863 6.7108

9.4013 8.8248 7.3338 9.7281

slncl slnpn slnsl slnfr

2.27333 1.52541 3.05224 2.32157

5.3007 4.5783 6.1090 5.3879

8.3377 7.6359 9.1732 8.4744

2.42852 1.64871 3.28313 2.51335

5.6810 4.9526 6.5789 5.8290

8.9547 8.2709 9.8946 9.1804

2.15708 1.46936 2.93498 2.25912

5.0591 4.4184 5.8916 5.2414

7.9898 7.3908 8.8838 8.2747

frncl frnpn frnsl frnfr

1.79118 3.78286 2.27333 4.59643

4.4873 6.8145 5.3007 7.6379

7.5166 9.8532 8.3377 10.7060

1.90468 4.06582 2.42672 4.94014

4.7774 7.3285 5.6703 8.2135

8.0136 10.6069 8.9263 11.5227

1.68575 3.60931 2.15576 4.41846

4.2333 6.5146 5.0514 7.3612

7.1127 9.4481 7.9694 10.3557

symmetry, clnpn and pnncl will have the same buckling axial force and the same is true for the group clnsl and slncl, clnfr and frncl and for the group pnnsl and slnpn, pnnfr and frnpn, slnfr

S. Naguleswaran / Applied Mathematical Modelling 30 (2006) 261–277

271

Table 2 The first three frequency parameters of ring-stiffened cantilevers for stiffener arrangement (a), (b) or (c) BC

n

Type 1

Type 2

Type 3

a1,1

a1,2

a1,3

a1,1

a1,2

a1,3

a1,1

a1,2

a1,3

(a)

clnfr

2 3 4 5 8 12 18

1.83913 1.83714 1.83283 1.82881 1.82132 1.82354 1.85548

4.6502 4.5986 4.5944 4.5848 4.5642 4.5670 4.6449

7.8260 7.8203 7.6983 7.6899 7.6483 7.6463 7.7723

1.87466 1.87894 1.88484 1.89267 1.92962 2.02102 2.36307

4.7471 4.7423 4.7467 4.7612 4.8450 5.0668 5.9153

7.9985 7.9951 8.0260 8.0241 8.1380 8.4932 9.8978

1.75153 1.74858 1.73970 1.73290 1.73318 1.79219 2.16465

4.5969 4.4121 4.3953 4.3731 4.3607 4.4980 5.4200

7.8602 7.8207 7.4460 7.4100 7.3474 7.5509 9.0718

(b)

clnfr

1 2 3 4 5 8 12 18

1.92145 1.91407 1.90280 1.89201 1.88232 1.86053 1.84766 1.86124

4.7994 4.7561 4.7546 4.7350 4.7138 4.6608 4.6269 4.6593

8.0149 8.0147 7.9017 7.9132 7.8879 7.8046 7.7454 7.7964

1.96007 1.95777 1.95600 1.95631 1.95899 1.98271 2.05994 2.37986

4.9014 4.9119 4.9118 4.9140 4.9204 4.9759 5.1636 5.9572

8.1929 8.1935 8.2633 8.2524 8.2631 8.3501 8.6534 9.9677

1.96688 1.94150 1.90381 1.87216 1.84762 1.81172 1.84285 2.18775

4.9209 4.7594 4.7664 4.7076 4.6507 4.5557 4.6244 5.4777

8.2294 8.2238 7.8427 7.9044 7.8329 7.6665 7.7611 9.1679

(c)

clnfr

1 2 3 4 5 8 12 18

1.86958 1.86031 1.85147 1.84364 1.83696 1.82381 1.82191 1.85370

4.6491 4.6513 4.6355 4.6184 4.6026 4.5693 4.5626 4.6404

7.8535 7.7336 7.7545 7.7340 7.7101 7.6538 7.6385 7.7650

1.87350 1.87240 1.87315 1.87601 1.88111 1.91126 1.99649 2.34194

4.7018 4.7023 4.7053 4.7121 4.7239 4.7956 5.0042 5.8625

7.8545 7.9138 7.9040 7.9148 7.9333 8.0466 8.3862 9.8101

1.85255 1.81838 1.78933 1.76652 1.74958 1.72969 1.77305 2.13673

4.5425 4.5505 4.4972 4.4449 4.4025 4.3469 4.4480 5.3501

7.8495 7.4854 7.5479 7.4834 7.4154 7.3099 7.4629 8.9552

(b)

frncl

1 2 3 4 5 8 12 18

1.79634 1.79560 1.79268 1.79006 1.78812 1.78729 1.79884 1.84793

4.5495 4.4982 4.4963 4.4895 4.4839 4.4792 4.5052 4.6260

7.6699 7.6599 7.5419 7.5374 7.5241 7.5069 7.5435 7.7409

1.79634 1.80048 1.80612 1.81349 1.82275 1.86372 1.96070 2.32525

4.5495 4.5440 4.5483 4.5619 4.5818 4.6779 4.9150 5.8208

7.6702 7.6669 7.6911 7.6892 7.7110 7.8542 8.2383 9.7406

1.67658 1.67371 1.66512 1.65833 1.65440 1.66221 1.72794 2.11450

4.3896 4.2190 4.2042 4.1831 4.1691 4.1789 4.3354 5.2946

7.5036 7.4659 7.1160 7.0847 7.0461 7.0331 7.2753 8.8626

System parameters as in Table 1.

and frnsl beams. It was found that pnnfr and frnfr had the same buckling force as pnnpn. The pnnsl and slnfr had the same buckling force as clnfr. Hence it was sufficient to tabulate the buckling forces of clncl, clnpn, clnslnclnfr and pnnpn beams. For arrangement (b), the buckling forces of clncl, clnpn, clnslnclnfr, pnncl and pnnsl beams are tabulated in Table 5. It was found that the group clnsl, pnnpn, pnnfr, slncl, slnsl, frnpn and frnfr, group clnfr, slnpn, slnfr and the group pnnsl, frncl and frnsl beams had the common buckling forces. Arrangement (c) is symmetrical and the results are tabulated in Table 6. It was found that there was a substantial increase in the numerical value of the critical axial force with increase in n.

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Table 3 As in Table 1 for tie-bar under (tensile) axial force s0 = 10.0 BC (a)

(b)

n

Type 1

Type 2

Type 3

a1,1

a1,2

a1,3

a1,1

a1,2

a1,3

a1,1

a1,2

a1,3

clncl

2 3 4 5 8 12 18

5.19886 5.08544 5.04303 5.00087 4.91296 4.85154 4.84707

8.3927 8.3909 8.1859 8.1418 7.9977 7.9054 7.9228

11.5732 11.4957 11.5500 11.2724 11.1084 10.9773 11.0145

5.41095 5.32966 5.27864 5.24858 5.24548 5.38650 6.10707

8.7645 8.7635 8.7165 8.6403 8.6089 8.8473 10.0732

12.1151 12.2012 12.1134 12.2010 12.0428 12.3488 14.0611

5.45347 5.10667 5.01744 4.92263 4.77668 4.81087 5.60994

8.8448 8.8363 8.2051 8.1390 7.8691 7.9233 9.2724

12.2412 12.0619 12.1381 11.3503 11.0635 11.0901 12.9585

clnpn

2 3 4 5 8 12 18

4.41176 4.33127 4.29345 4.26099 4.18884 4.13397 4.12612

7.4902 7.4708 7.3440 7.2947 7.1864 7.1216 7.1629

10.5995 10.5382 10.5501 10.3868 10.2360 10.1509 10.2330

4.49125 4.43307 4.40463 4.38806 4.38721 4.49066 5.07741

7.6470 7.6406 7.6004 7.5744 7.6002 7.8404 8.9988

10.8431 10.8833 10.8553 10.8585 10.8835 11.2372 12.9397

4.50319 4.26750 4.17358 4.10036 3.97834 3.98913 4.62648

7.6624 7.5864 7.2340 7.1145 6.9169 6.9879 8.2336

10.8521 10.7270 10.6591 10.2600 9.9476 10.0392 11.8565

pnnpn

2 3 4 5 8 12 18

3.74102 3.67847 3.64781 3.62051 3.55406 3.49612 3.47083

6.6419 6.6411 6.5335 6.4941 6.4125 6.3653 6.4183

9.6628 9.5841 9.6571 9.5066 9.3917 9.3428 9.4591

3.74132 3.69966 3.68092 3.66673 3.65029 3.70165 4.12286

6.6454 6.6450 6.6255 6.6233 6.6724 6.8934 7.9521

9.6760 9.7090 9.6748 9.7140 9.8156 10.1856 11.8420

3.73836 3.55402 3.47648 3.41464 3.29637 3.26860 3.71698

6.6258 6.6229 6.2990 6.1987 6.0489 6.1162 7.2311

9.6179 9.4179 9.5974 9.1693 8.9312 9.0573 10.7896

clncl clnpn clnsl clnfr

10

4.79446 4.14963 2.66006 2.51225

7.8057 7.1367 5.6435 5.1193

10.8396 10.1686 8.6571 8.0436

5.11829 4.43622 2.80401 2.60646

8.3926 7.7222 6.0507 5.4515

11.7146 11.0657 9.3523 8.6955

4.56843 3.96210 2.50766 2.32707

7.5102 6.9192 5.4306 4.9045

10.5135 9.9423 8.4217 7.8583

pnncl pnnpn pnnsl pnnfr

4.09071 3.51394 2.16938 2.08290

7.0280 6.3752 4.9292 4.4785

10.0127 9.3490 7.8621 7.2717

4.28454 3.67056 2.20904 2.09013

7.4393 6.7862 5.1917 4.6731

10.6498 10.0125 8.3612 7.7321

3.80757 3.26135 1.95887 1.84908

6.6300 6.0552 4.6386 4.1817

9.5145 8.9558 7.4951 6.9547

slncl slnpn slnsl slnfr

2.60644 2.15579 3.51546 3.22869

5.5502 4.9187 6.3807 5.8370

8.5171 7.8546 9.3613 8.7460

2.70962 2.20379 3.67509 3.31656

5.8852 5.2305 6.8028 6.1996

9.1015 8.4476 10.0494 9.4011

2.39527 1.94330 3.26856 2.94829

5.2312 4.6526 6.0817 5.5580

8.1139 7.5398 9.0157 8.4637

frncl frnpn frnsl frnfr

2.44991 2.06098 3.19212 2.96861

4.9634 4.4058 5.7485 5.2711

7.7898 7.1501 8.6078 8.0167

2.50002 2.06751 3.25545 2.97362

5.1809 4.5967 6.0472 5.5121

8.2407 7.6050 9.1511 8.5260

2.19743 1.81303 2.86964 2.61870

4.5737 4.0587 5.3698 4.9055

7.3034 6.7465 8.1591 7.6267

3.6. Comparison with published results The ring-stiffeners used by Dube et al. [14] and Au et al. [16] were not of constant axial length but were related by b = ab where b is constant. The axial length of the stiffeners decrease with

S. Naguleswaran / Applied Mathematical Modelling 30 (2006) 261–277

273

Table 4 The first two critical axial forces of beam under classical boundary conditions, stiffener arrangement (a) and for various n b

BC

n

Type 1

Type 2

Type 3

sc,1

sc,2

sc,1

sc,2

sc,1

sc,2

0.05

clncl

2 3 4 5 10 15

43.70128 46.08575 44.93650 46.14224 53.53649 63.81698

88.814 88.902 99.193 95.268 109.549 130.490

47.40698 52.30013 49.58218 52.41467 72.68302 118.29852

96.705 96.896 119.210 107.939 148.398 241.324

48.06729 53.45345 50.39357 53.58179 77.22212 137.75816

98.184 98.396 123.222 110.128 157.499 280.635

0.05

clnpn

2 3 4 5 10 15

21.22638 21.87289 22.44519 23.07513 26.90895 32.29653

62.815 64.599 65.936 67.987 79.447 95.427

22.10444 23.22122 24.43207 25.78929 35.80329 58.62307

65.526 69.452 70.459 75.256 105.170 172.641

22.25937 23.45138 24.79096 26.29857 37.88075 67.81467

66.003 70.372 71.174 76.550 111.074 199.311

0.05

pnnpn

2 3 4 5 10 15

9.87765 10.39107 10.69884 11.01822 12.92733 15.62499

39.604 39.620 42.487 43.886 51.644 62.477

9.88365 10.81089 11.40979 12.06304 16.82824 27.75439

39.696 39.725 44.885 47.737 66.969 110.693

9.88465 10.88336 11.53673 12.25582 17.71739 31.87310

39.711 39.742 45.295 48.429 70.422 126.940

0.05

clnfr

2 3 4 5 10 15

2.59577 2.66649 2.74074 2.81915 3.28941 3.94765

23.377 23.918 24.625 25.341 29.589 35.523

2.69889 2.83660 2.98753 3.15512 4.38314 7.17435

24.328 25.238 26.725 28.261 39.332 64.453

2.71667 2.86692 3.03289 3.21888 4.63996 8.30493

24.492 25.459 27.100 28.805 41.600 74.536

0.1

clncl clnpn clnsl clnfr pnncl pnnpn

6

58.19374 28.79136 14.54738 3.52327 28.79135 13.64324

119.238 84.736 58.194 31.653 84.736 54.269

88.14042 42.02699 22.37900 5.17909 42.02699 19.07432

177.679 120.714 88.140 46.046 120.714 74.583

95.88498 45.44723 24.53888 5.61716 45.44723 20.41912

191.453 129.279 95.885 49.725 129.279 79.382

Stiffener length b in first column.

increase in the number of rings and theoretically there is no restriction on the number of ring-stiffeners. For n rings one has if arranged as in Fig. 2a:

a ¼ 1=½nð1 þ bÞn  1;

ð22aÞ

Fig. 2b:

a ¼ 1=½nð1 þ bÞn;

ð22bÞ

Fig. 2c:

a ¼ 1=½nð1 þ bÞn þ 1.

ð22cÞ

Most of the results tabulated in the two references were for EIR/EIB = 4 and mR/mB = 4 which is the Type 1 change in cross-section used in the present paper. In Ref. [14], the frequency equation

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S. Naguleswaran / Applied Mathematical Modelling 30 (2006) 261–277

Table 5 Same as Table 4 for stiffener arrangement (b) b

BC

n

Type 1

Type 2

Type 3

sc,1

sc,2

sc,1

sc,2

sc,1

sc,2

0.05

clncl

1 2 3 4 5 10 15

41.51005 43.69962 42.56002 43.77444 45.03909 52.58790 63.14434

84.656 84.800 94.236 90.084 92.395 107.532 129.098

43.17115 47.40658 45.01389 47.49988 50.19729 69.82404 114.45907

88.203 88.363 107.684 97.680 102.913 142.479 233.455

43.45640 48.06716 45.43286 48.16671 51.15644 73.82343 132.29188

88.840 89.007 110.220 98.972 104.825 150.527 269.513

0.05

clnpn

1 2 3 4 5 10 15

21.20661 21.98564 22.42196 22.98796 23.60600 27.37363 32.62516

62.654 63.788 66.932 68.081 69.815 80.876 96.412

22.06651 23.44488 24.38279 25.59830 26.98431 37.28042 60.56514

65.220 67.835 72.374 75.424 79.430 109.696 178.454

22.21809 23.69566 24.73637 26.08573 27.63726 39.66281 70.62262

65.671 68.602 73.241 76.730 81.228 116.519 207.690

0.05

pnnpn

1 2 3 4 5 10 15

9.87363 10.38309 10.66476 10.96153 11.27514 13.15644 15.79006

39.541 39.604 42.560 43.774 45.039 52.588 63.144

9.87662 10.79554 11.34201 11.94456 12.61419 17.52398 28.68760

39.587 39.696 45.014 47.500 50.197 69.824 114.459

9.87712 10.86666 11.46249 12.12488 12.86787 18.54814 33.20228

39.594 39.711 45.433 48.167 51.156 73.823 132.292

0.05

clnfr

1 2 3 4 5 10 15

2.59549 2.66055 2.73012 2.80365 2.88135 3.34531 3.98757

23.354 24.043 24.599 25.244 25.936 30.102 35.884

2.69834 2.82484 2.96552 3.12130 3.29452 4.56098 7.41043

24.284 25.478 26.669 28.050 29.596 40.960 66.591

2.71607 2.85406 3.00860 3.18121 3.37500 4.85466 8.64662

24.444 25.720 27.039 28.570 30.300 43.564 77.626

0.1

clncl clnpn clnsl clnfr pnncl pnnsl

6

56.22845 29.68722 14.08971 3.62774 27.96392 3.42518

114.642 87.738 56.228 32.648 82.143 30.744

81.50164 45.58526 20.68347 5.59473 39.16847 4.82553

164.134 132.493 81.502 49.987 112.962 42.951

87.89926 49.97616 22.42045 6.14813 41.94566 5.17710

176.163 144.050 87.899 54.728 120.255 45.951

was expressed as (2n)th order determinants equated to zero. Frequencies of beams with up to 6 stiffeners were tabulated. In Ref. [13] the difficulty of finding roots of determinantal equations order greater than about 4 was mentioned. One surmises that the number of stiffeners in [14] was limited to 6 because of the performance of the method in Ref. [15] to calculate the roots. Ref. [16] used a family of C1 functions as the assumed modes. The frequencies tabulated were for beams with up to 6 stiffeners. Limitations or otherwise of calculations for beams with more stiffeners was not mentioned. In the present paper, the frequencies were recalculated by

S. Naguleswaran / Applied Mathematical Modelling 30 (2006) 261–277

275

Table 6 Same as Table 5 for stiffener arrangement (c) b

BC

n

Type 1

Type 2

Type 3

sc,1

sc,2

sc,1

sc,2

sc,1

sc,2

0.05

clncl

1 2 3 4 5 10 15

41.52430 40.34771 41.56771 42.79478 44.07662 51.73541 62.53697

80.833 89.877 85.238 87.629 90.184 105.737 127.844

43.17290 41.04989 43.24396 45.59744 48.19656 67.24188 110.98904

80.885 97.769 88.819 93.424 98.599 137.160 226.356

43.45688 41.17129 43.53355 46.09434 48.95006 70.74740 127.34571

80.893 99.172 89.432 94.449 100.136 144.235 259.455

0.05

clnpn

1 2 3 4 5 10 15

20.85011 21.29994 21.84941 22.44347 23.07716 26.91091 32.29719

61.044 63.098 64.518 66.254 68.123 79.473 95.432

21.36400 22.20256 23.26736 24.47305 25.82558 35.82505 58.63608

62.110 65.657 68.475 71.990 75.958 105.404 172.759

21.45115 22.35935 23.52023 24.84558 26.34615 37.91214 67.84010

62.291 66.077 69.162 73.021 77.417 111.403 199.543

0.05

pnnpn

1 2 3 4 5 10 15

10.38194 10.63717 10.91284 11.20550 11.51539 13.37046 15.94239

39.495 42.740 43.723 44.854 46.074 53.466 63.760

10.79333 11.28744 11.84368 12.46243 13.15181 18.20376 29.58734

39.506 45.365 47.390 49.793 52.508 72.605 118.088

10.86425 11.40279 12.01358 12.69884 13.46972 19.36843 34.50363

39.508 45.815 48.044 50.706 53.740 77.172 137.530

0.05

clnfr

1 2 3 4 5 10 15

2.53029 2.59680 2.66696 2.74105 2.81939 3.28949 3.94767

22.774 23.360 23.989 24.655 25.359 29.593 35.523

2.57906 2.70239 2.83834 2.98878 3.15612 4.38373 7.17479

23.216 24.285 25.496 26.842 28.342 39.366 64.471

2.58733 2.72076 2.86898 3.03439 3.22010 4.64076 8.30571

23.291 24.445 25.763 27.241 28.904 41.648 74.572

0.1

clncl clnpn clnsl clnfr pnncl pnnpn

6

54.67990 28.80344 13.71844 3.52373 28.80344 14.47418

111.232 84.894 54.680 31.674 84.894 57.880

76.10831 42.18872 19.30007 5.18343 42.18872 22.20424

153.257 122.570 76.108 46.288 122.570 87.962

81.30116 45.69853 20.68616 5.62348 45.69853 24.35887

163.299 132.046 81.301 50.089 132.046 96.063

the method described in the present paper of beams with up to 25 stiffeners. The results for a pnnpn beam and tie-bar are tabulated in Table 7 and compared with the results in Refs. [14,16]. In these references since the axial length of the stiffeners decrease with increase in n, the change in frequency with n is small. In Ref. [14] the buckling was treated as a problem in statics and the first critical axial force was tabulated for ring-stiffener arrangement (a), (b) and (c). In Ref. [16], C1 and C2 functions were used. In the present paper, the first and second critical axial force were calculated (up to

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Table 7 Frequencies of pnnpn beam with stiffener arrangement (c) calculated by present method compared with results in Refs. [14,16] n

s0

Present

Ref. [14]

Ref. [16]

X1

X2

X3

X1

X2

X3

X1

X2

X3

8.6210 8.6298 8.6253 8.6222

39.274 34.571 34.617 34.577

84.688 88.188 77.898 78.077

0.00

1 2 3 4 8 15 25

8.62103 8.62982 8.62533 8.62222 8.61746 8.61575 8.61523

39.2745 34.5708 34.6168 34.5768 34.5010 34.4734 34.4649

84.6871 88.1883 77.8981 78.0763 77.7466 77.6043 77.5612

8.621 8.629 8.625 8.622

39.27 34.57 34.62 34.58

84.69 88.19 77.90 78.08

7.896

1 2 3 4 5 15 25

4.76782 4.73312 4.72528 4.72234 4.71937 4.71864 4.71846

35.1440 31.4566 31.4525 31.4080 31.3361 31.3114 31.3040

81.3332 84.2020 74.8860 74.9966 74.6607 74.5243 74.4834

4.768 4.733 4.725 4.722

35.14 31.46 31.45 31.41

81.33 84.20 74.89 75.00

System parameters: EIR/EIB = mR/mB = 4.0, b = 0.2, a from Eq. (22c), b = ab, axial force s0 in first column. Table 8 Buckling axial force obtained by present method compared with results in Refs. [14,16] of the pnnpn system in Table 7 n 1 2 3 4 5 6 7 8 10 15 20

Present

Ref. [14] sc,1

sc,1

sc,2

11.36596 11.29171 11.28168 11.27937 11.27877 11.27864 11.27867 11.27875 11.27891 11.27918 11.27931

39.620 45.488 45.165 45.116 45.105 45.103 45.104 45.105 45.108 45.112 45.114

Ref. [16] C1 (5 terms)

sc,1 calculated by C2 (5 terms)

C2 (10 terms)

11.37 11.29 11.28 11.28

11.366 11.292 11.283 11.280

12.444 13.020 14.812 14.819

11.822 12.400 12.447 12.984

11.28

11.280

14.810

14.808

11.28

11.280

14.807

14.807

n = 20) by the method outlined and are tabulated in Table 8 along with the results in Refs. [14,16]. The accuracy of the first critical value obtained with C2 functions is poor. Problems (if any) encountered in obtaining the second critical axial force was not mentioned in Refs. [14,16].

4. Concluding remarks The frequency equations of multi-stepped Euler–Bernoulli tie-bars are expressed as 2nd order determinants equated to zero for 16 combinations of classical boundary conditions. Schemes to

S. Naguleswaran / Applied Mathematical Modelling 30 (2006) 261–277

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recursively calculate the elements of the determinant and then to calculate the roots are presented. The tie-bar considered in this paper consisted of a beam with equi-spaced ring-stiffeners. It was assumed that the stiffener and the beam behaved as a homogeneous unit. The first three frequency parameters are tabulated for three cases of stiffener arrangement. The scheme outlined satisfactorily handled any number of stiffeners and numerical problems were not encountered. Euler buckling occurs for certain compressive axial forces at which a frequency parameter is zero. The first two critical combinations are tabulated for the example sets of beam and stiffener parameters. The 16 combinations of the boundary conditions are grouped into sub-sets which have common buckling forces. The tables may be used to judge frequencies and buckling axial forces by numerical methods like Rayleigh–Ritz, finite element, finite difference, etc. Although results are presented for the three types of beams, the method developed is applicable for any type of step changes in crosssection.

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