Torsional vibrations and buckling of thin-walled beams on elastic foundation

Thin-Walled Structures 7 (1989) 73--82

Torsional

Vibrations and Buckling of Thin-Walled on Elastic Foundation

Beams

C. Kameswara Rao* & S. M i r z a Department of Mechanical Engineering. University of Ottawa. Ottawa. Canada KIN 6N5 (Received 23 November 1987; revised version received 15 March 1988; accepted 5 May 1988)

A BSTRA CT The problem of free torsional vibration and buckling of doubly symmetric thin-walled beams of open section, subjected to an axial compressive static load and resting on continuous elastic foundation, is discussed in this paper. An analytical method based on the dynamic stiffness matrix approach is developed, including the effects of warping. The resulting transcendental equation is solved for thin-walled beams clamped at one end and simply supported at the other. A computer program is developed, based on the dynamic stiffness matrix approach. The software consists of a master program to set up the dynamics stiffness matrix and to call specific subroutines to perform various system calculations. Numerical results for natural frequencies and buckling load for various vahtes of warping and elastic foundation parameter are obtained attd presented.

1 INTRODUCTION

The vibrations and buckling of continuously-supported finite and infinite beams on elastic foundation have applications in the design of aircraft structures, base frames for rotating machinery, railroad tracks, etc. Several studies have been conducted on this topic, and valuable practical methods for the analysis of beams on elastic foundation have been suggested.~-3 A discussion of various foundation models was presented by Kerr? While there are a number of publications on flexural vibrations of Correspondence address: Group Co-ordinator (SFA). Bharat Heavy Electricals Ltd, Corporate R & D Division, Vikas Nagar, Hyderabad-500593, India. 73 Thin-Walled Structures 02_63-8231/89/$03-50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain

74

C. Kameswara Rao, S. Mirza

rectangular beams and plates on elastic foundation, the literature on torsional vibrations and buckling of beams on elastic foundation is rather limited. Free torsional vibrations and stability of doubly-symmetric long thin-walled beams of open section were investigated by Gere.' Krishna Murthy and Joga Rao ~ and Christiano and Salmela. 7 Kameswara Rao et a l ? used a finite element method to study the problem of torsional vibration of long, thin-walled beams of open sections resting on elastic foundations. In another publication 9 Kameswara Rao and Appala Satyam developed approximate expressions for torsional frequency and buckling loads for thin-walled beams resting on Winkler-type elastic foundation and subjected to a time-invariant axial compressive force. It is known that higher mode frequencies predicted by approximate methods are generally in considerable error. In order to improve this situation, a large number of elements or terms in the series have to be included in the computations to get values with acceptable accuracy. In view of the same, more and more effort is being put into developing frequency dependent 'exact' finite elements or dynamic stiffness and mass matrices. In the present paper, an improved analytical method based on the dynamic stiffness matrix approach is developed including the effects of Winkler-type elastic foundation and warping torsion. The resulting transcendental frequency equation is solved for a beam clamped at one end and simply supported at the other. Numerical results for torsional natural frequencies and buckling loads for some typical values of warping and foundation parameters are presented. The approach presented in this paper is quite general and can be utilized in analyzing continuous thin-walled beams also.

2 FORMULATION AND ANALYSIS Consider a long doubly-symmetric thin-walled beam of open section of length L and resting on a Winkler-type elastic foundation of torsional stiffness K~. The beam is subjected to a constant static axial compressive force P and is undergoing free torsional vibrations. The corresponding differential equation of motion can be written as: 9 ECw 0 4 ¢k/Oz' - ( GC~ - PIp~A) 0"-~blOz2 + Ks ¢k + pip 0 2 4J/OF = 0

(1)

in which E is the modulus of elasticity; Cw, the warping constant; G, the shear modulus; Cs, the torsion constant; lp, the polar moment of inertia; A, the area of cross-section; p, the mass density of the material of the beam; ~b, the angle of twist; z, the distance along the length of the beam and t, the time.

Torsional vibrations and buckling of thin-walled beams

75

For free torsional vibrations, the angle of twist 4,(z, t) can be expressed in the form. tb(z,t) = x(z) eipt

(2)

in which x ( z ) is the modal shape function corresponding to each beam torsional natural frequency p. T h e expression for x ( z ) which satisfies eqn (1) can be written as: x(z) = A cosc~z + Bsinaz + CcoshBz + Dsinh/3z

(3)

in which, aL,/3L = (I/x/2) {-T-(K" - A-') + [(K" - .%:)-" + 4(h-" - 4-f')]'/2} 'l" K'- = L" G C d E G , A: = Pip L " / A E G

(4) (5)

and h z = plv L4p'/ECw, y: = Ks L4/4ECw

(6)

F r o m eqn (4), we have the following relation between a L and/3L: (/3L) z = (aL) 2 + K z - Az

(7)

K n o w i n g a a n d / 3 , the frequency parameter h can be evaluated using the following equation: h: = (ctL)(flL) + 4y z

(8)

T h e four arbitrary constants A, B, C and D in eqn (3) can be determined f r o m the b o u n d a r y conditions of the beam. For any single-span beam, there will be two b o u n d a r y conditions at each end and these four conditions then d e t e r m i n e the corresponding frequency and m o d e shape expressions.

3 D Y N A M I C STIFFNESS M A T R I X In o r d e r to p r o c e e d further, we must first introduce the following nomenclature: the variation of angle of twist ~bwith respect to z is d e n o t e d by O(z); the flange bending m o m e n t and the total twisting m o m e n t are given by M ( z ) a n d T ( z ) . Considering clockwise rotations and m o m e n t s to be positive, we have, O(z) = drMdz, hM(z) = - E C , ( d z 4~ldzz)

(9)

76

C. Kameswara Rao, S. Mirza

and (lO)

7"( Z ) -= -- ECw ( d 3 cb/dz -~) + ( G C, - P I p / A ) dch/dz

where ECw = If h:/2. If being the flange moment of inertia and h the distance between the center lines of the flanges of a thin-walled I-beam. Consider a uniform thin-walled I-beam element of length / as shown in Fig. 1. By combining eqns (3) and (9), the end displacements, ~b(0) and 0(0) and end forces, hM(O) and T(O), of the beam at z = 0, can be expressed as:

o(o)~ =

hM(0) /

T(O)J

o

¢~

o

EC,,a 2

0

-EC~,fl2

0

ECwafl2

0

(lla)

-ECwaZ fl ]

Equation (1 la) can be written in an abbreviated form as follows: (llb)

~(o) = v ( o ) u

In a similar manner, the end displacements, &(L) and O(L) and end forces hM(L) and T(L), of the beam where z = I can be expressed as follows: 8(L) = V(L) U

(12a)

where {8(L)}T = {d~(L),O(L).hM(L).T(L)}

(12b)

{U}T= {A,B,C,D}

(12c)

and I [V(/)] =

Cos ECwc~'-c

1-ECwaBZ s

sac

C ~S

ECwa2s

-ECw~C

ECwa~ c

-ECwo~:BS

S 13C

1

-ECwfl:S

(lZd)

-ECwct: ~C _l

in which c = cos aL; s = sin aL; C = cos h flL; S = sin h ilL. By eliminating the integration constant vector U from eqns (11) and (12), and designating the left end of element as i and the right end as j, the equation relating the end forces and displacements can be written as:

Torsional vibrations and buckling of thin-walled beams

hMi

J22

Ti / = hMj

J23

J'-~

0i

J33

J34

~j

J~4

0j

L.Sym

i ]f it

77

(13a)

S y m b o l i c a l l y it is w r i t t e n

{F} = [S]{U}

(tab)

where {F} v = {T~,hM~,Tj,hMi}

{u} ~ =

{4,~,o~,4,j,oj}

(13c)

(13d)

In e q n s (13a) a n d (13b) the matrix [J] is the 'exact" e l e m e n t d y n a m i c stiffness m a t r i x , w h i c h is also a s q u a r e s y m m e t r i c matrix. T h e e l e m e n t s o f [J] are given:

J,, = H(o;- + t~)(o, Cs + ~Sc) J,z = - H[(ot: - ~ ) ( 1 - Cc) + 2a#Ss] Jl3 = - H ( a z + 13z)(o~ + flS)

J14

- H ( ~ 2 + f f ) ( C - c)

J22 = - ( H / a # ) ( d

+ t ~ ) ( ~ S c - #Cs)

(14) J24 = ( H l a f l ) ( a 2 + ~ ) ( ~ S - #s) Jzs = ~J14 -/33 = Jtl

-J12

L,= and H = ECwafl/[2ctB(1 - Cc) + (132-a")Ss]

78

C. Kameswara Rao, S. Mirza

Fig. 1. Differential element of thin-walled I-beam.

Using the element dynamic stiffness matrix defined by eqns (13) and (14), one can easily set up the general equilibrium equations for multi-span thin-walled beams, adopting the usual finite element assembly methods. Introducing the boundary conditions, the final set of equations can be solved for eigenvalues by setting up the determinant of their matrix to zero. For convenience in programming, the signs of end forces and end displacements used in eqns (13) and (14) are all positive and are defined as shown in Fig. 1.

4 M E T H O D OF SOLUTION Denoting the assembled and modified dynamic stiffness matrix as [DS], we state that detlDS I = 0

(15)

Equation (15) yields the frequency equation of continuous thin-walled beams in torsion resting on continuous elastic foundation and subjected to a constant axial compressive force. It can be noted that eqn (15) is highly transcendental in terms of the eigenvalues h. The roots of eqn (15) can, therefore, be obtained by applying the Regula-Falsi method 1° and the Wittrick-Williams algorithm" on a high-speed digital computer. Exact values of the frequency parameter h for simply supported and built-in thin-walled beams are obtained in this paper using an error factor ~ = 10 -6. A computer program was developed based on the above-discussed approach. The software includes a master program to evaluate and assemble

Torsional vibrations and buckling Of thin-walled beams

79

the various element stiffness matrices and to finally solve the frequency equations. Various checks are incorporated in order to identify, and eliminate any false frequencies. Care is also taken to suitably adjust the required number of iterations to avoid missing any frequency in the specified range.

5 RESULTS AND DISCUSSION The approach developed in this paper can be applied to the calculation of natural torsional frequencies and mode shapes of multi-span doubly symmetric thin-walled beams of open section such as beams of I-section. Beams with non-uniform cross-sections also can be handled very easily as the present approach is almost similar to the finite element method of analysis but with exact displacement shape functions. All classical and non-classical (elastic restraints) boundary conditions can be incorporated in the present model without any difficulty. To demonstrate the effectiveness of the present approach, a single-span, thin-walled I-beam, clamped at one end (z = 0) and simply supported at the other end (z = l), is chosen. The boundary conditions for this problem can be written as: ~(0) = O: o(o) = o

(16)

tk(l) = 0 and M(/) = 0

(17)

Considering a one element solution and applying the boundary conditions defined by eqns (16) and (17) gives, J22 = 0

(18a)

This gives,

(n/,,/3)(w" + ~ ) ( ~ S c - /3Cs) = 0

08b)

As H and (a: +/3 2) are, in general, non-zero, the frequency equation for the clamped, simply supported beam can, therefore, be written as, a tanh/3L =/3 tan o~L

(18c)

Equation (18c) is solved for values of warping parameter K = 1 and K = 10 and for various values of foundation parameter y in the range 0-12. Figures 2 and 3 show the variation of fundamental frequency and buckling

80

C. Kameswara Rao, S. .~lirza i

i

i

J

a O 0 ~

300"

;

i

12

i

i

N :I

4

2OO

2

I00

2

I

3

4

5

~ZX

6

7

8

9

Fig. 2. Effect of elastic foundation on torsional frequencies and buckling loads of l-beams ( N = I . K = 1).

I

I

I

I

I

0

2

4

6

8

I0

12

Fig. 3. Effect of elastic foundation on torsional frequencies and buckling loads of I-beams (N = 1, K = 10).

Torsional vibrations and buckling o f thin-walled beams

81

TABLE 1 Comparison between Exact and Approximate Values of k" for the First Mode of Vibration of a Fixed, Simply Supported Beam Resting on Elastic Foundation Values o f h e from exact and approximate analyses ~ 7 = 4.0 K

1.0

10.0

3, = 8.0

3t = 12.0

A

Exact

Approx.

Exact

Approx.

Exact

Approx.

0-0 2.0 4.0 0-0 4.0 8.0

314.265 268.632 132.226 1 439.762 1 257.879 712.010

325.950 276.602 128.557 1 547.319 1 349.926 757.747

506.302 460.548 324.676 1 631.753 1 449.536 904.893

517.894 468-596 320.624 1 739.526 1 541-898 949-692

826-253 780.735 644-378 1 951-865 1 769.758 1 224.926

837.892 788-735 640.655 2 059-296 1 861.886 1 269.686

aResults from Galerkin's technique. 9

load parameters with foundation parameter for values of K equal to I and 10 respectively. To give an idea about the accuracy of the results obtained, comparison is made with results obtained using approximate methods such as Galerkin's technique. 9 Table 1 shows the comparison between the approximate values 9 and those from the present analysis. It can be stated that even for beams with non-uniform sections, multiple spans and complicated boundary conditions, accurate estimates of natural frequencies can be obtained using the approach presented in this paper. A close look at the results presented in Figs 2 and 3 clearly reveals that the effect of an increase in axial compressive load parameter A is to drastically decrease the fundamental frequency h(N = 1). Furthermore, the limiting load where h becomes zero is the buckling load of the beam for a specified value of warping parameter K and foundation parameter y. One can easily read the values of buckling load parameter AcTfrom these figures for h = 0. As can be expected, the effect of elastic foundation is found to increase the frequency of vibration especially for the first few modes. However, this influence is seen to be quite negligible on the modes higher than the third.

6 C O N C L U D I N G REMARKS A dynamic stiffness matrix approach has been developed for computing the natural torsion frequencies and buckling loads of long, thin-walled beams of open section resting on continuous Winkler-type elastic foundation and subjected to an axial time-invariant compressive load. The approach

82

c. KameswaraRao. S. Mirza

presented in this paper is quite general and can be applied for treating beams with non-uniform cross-sections and also non-classical boundary conditions. Using Wittrick and Williams" algorithm t~, a computer program has been developed to accurately determine the torsional buckling loads, frequencies and corresponding modal shapes. Results for a beam clamped at one end and simply supported at the other have been presented, showing the influence of elastic foundation, axial compressive load and warping. While an increase in the values of elastic foundation parameter resulted in an increase in frequency, the effect of an increase in axial load parameter is found to be to drastically decrease the frequency to zero at the limit when the load equals the buckling load for the beam.

REFERENCES 1. Hetenyi, M., Beams on elastic foundation. University of Michigan Press, Ann Arbor, 1946. 2. Vlasov, V. Z. & Leont'ev, U. N., Beams, plates and shells on elastic foundations. Translation from Russian, Israel program for scientific translations, Jerusalem, 196l. 3. Timoshenko, S. P. & Gere, J. M., Theory of elastic stability (2nd Edition). McGraw-Hill, New York, 1961. 4. Kerr, A. D., Elastic and viscoelastic foundation models. Journal of Applied Mechanics, 31 (1964) 491-8. 5. Gere, J. M., Torsional vibrations of beams of thin walled open section. Journal of Applied Mechanics, 21 (1954) 381-7. 6. Krishna Murthy, A. V. & Joga Rao, C. V., General theory of vibrations of cylindrical tubes. Journal of Aeronautical Society of India, 20 (1968) 1-38, 235-58. 7. Christiano, P. & Salmela, L., Frequencies of beams with elastic warping restraint. Journal of the Structures Division, 97 (1971) 1835--40. 8. Kameswara Rao, C., Gupta, B. V. R. & Rao, D. L. N., Torsional vibrations of thin walled beams on continuous elastic foundation using finite element method. Proceedings of International Conference on Finite Element Methods in Engineering, Coimbatore, 1974, pp. 231-48. 9. Kameswara Rao, C. & Appala Satyam, A., Torsional vibration and stability of thin walled beams on continuous elastic foundation. AIAA Journal, 13 (1975) 232-4. I0. Wendroff, B., Theoretical numerical analysis. Academic Press, New York, 1966. 11. Wittrick, W. H. & Williams, F. W., A general algorithm for computing natural frequencies of elastic structures. Quarterly Journal of Mechanics and Applied Mathematics, 24 (1971) 263-84.