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- 1. International Journal of Applied EngineeringResearch and Development (IJAERD)ISSN:2250–1584Vol.2, Issue 1 Mar 2012 1-12© TJPRC Pvt. Ltd., TORSIONAL VIBRATIONS AND BUCKLING OF THIN-WALLED BEAMS ON ELASTIC FOUNDATION-DYNAMIC STIFFNESS METHOD N.V. SRINIVASULU1, B. SURYANARAYANA2AND S. JAIKRISHNA3 1,2. Associate Professor, Department of mechanical Engg., CBIT, Hyderabad-500075.AP.India 3. Sr.Asst.Professor, Department of Mechanical Engg., MJCET., Banjara Hills, Hyderabad. Mail: vaastusrinivas@gmail.comABSTRACT The problem of free torsional vibration and buckling of doubly symmetric thin- walled beams ofopen section, subjected to an axial compressive static load and resting on continuous elastic foundation isdiscussed in this paper. An analytical method based on the dynamic stiffness matrix approach isdeveloped including the effect of warping. A thin walled beam clamped at one end and simply supportedat the other is considered. The different mode shapes and natural frequencies for the above beam isstudied when it is resting on elastic foundation using dynamic stiffness matrix method.INTRODUCTION The problem of free torsional vibration and buckling of doubly symmetric thin- walled beams ofopen section, subjected to an axial compressive static load and resting on continuous elastic foundation isdiscussed in this paper. An analytical method based on the dynamic stiffness matrix approach isdeveloped including the effect of warping. The resulting transcendental equation is solved for thin-walledbeams clamped at one end and simply supported at the other. The dynamic stiffness matrix can be usedto compute the natural frequencies and mode shapes of either a single beam with various end conditionsor an assembly of beams. When several elements are to be used the over all dynamic stiffness matrix ofthe complete structure must be assembled. The associated natural frequencies and mode shapes areextracted using Wittrick-Williams algorithm [10]. The algorithm guarantees that no natural frequencyand its associated mode shape are missed. This is, of course, not possible in the conventional finiteelement method Numerical results for natural frequencies and buckling load for various values ofwarping and elastic foundation parameter are obtained and presented. The vibrations and buckling of continuously – supported finite and infinite beams on elasticfoundation 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 theanalysis of beams on elastic foundation have been suggested. A discussion of various foundationmodels was presented by Kerr[2]. While there are a number of publications on flexural vibrations of rectangular beams and plates onelastic foundation, the literature on torsional vibrations and buckling of beams on elastic foundation israther limited. Free torsional vibrations and stability of doubly-symmetric long thin-walled beams of
- 2. N.V. Srinivasulu, B. Suryanarayana & S. Jaikrishna 2open section were investigated by Gere[1], Krishna Murthy[3] and Joga Rao[3] and Christiano[5] andSalmela[8] , Kameswara Rao[4]., used a finite element method to study the problem of torsionalvibration of long thin-walled beams of open section resting on elastic foundations. In another publicationKameswara Rao and Appala satyam[6] developed approximate expressions for torsional frequency andbuckling loads for thin walled beams resting on Winkler-type elastic foundation and subjected to a timeinvariant axial compressive force. It is known that higher mode frequencies predicted by approximate methods arte generally inconsiderable error. In order to improve the situation, a large number of elements or terms in the serieshave 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 ordynamic stiffness and mass matrices. In the present paper, an improved analytical method based on thedynamic stiffness matrix approach is developed including the effects of Winkler – type elastic foundationand warping torsion. The resulting transcendental frequency equation is solved for a beam clamped atone end and simply supported at the other. Numerical results for torsional natural frequencies andbuckling loads for some typical values of warping and foundation parameters are presented. Theapproach presented in this chapter is quite general and can be utilised in analyzing continuous thin –walled beams also.FORMULATION AND ANALYSIS Consider a long doubly-symmetric thin walled beam of open section of length L and resting on aWinkler –type elastic foundation of torsional stiffness Ks. The beam is subjected to a constant static axialcompressive force P and is undergoing free torsional vibrations. The corresponding differential equationof motion can be written as ECw ∂4φ/∂z4 – (GCs –ρIp/A)∂2φ/∂z2 + Ksφ+ ρIp)∂2φ/∂t2 (1) In which E, the modulus of elasticity; Cw ,the warping constant; G, the shear modulus; Cs , thetorsion constant; Ip, the polar moment of Inertia; A, the area of cross section; ρ, the mass density of thematerial of the beam; φ,the angle of twist; Z, the distance along the length of the beam and t, the time. For the torsional vibrations, the angle of twist φ(z,t) can be expressed in the form φ(z,t) = x(z)e ipt (2)in which x(z) is the modal shape function corresponding to each beam torsion natural frequency p. Theexpression for x(z) which satisfies equation(1) can be written as: x(z) = A Cos αz + B Sinαz +C Coshβz +D sinβz (3)in which αL.βL = (1/√2) {+ (k2-∆2) + [(k2-∆2)2 +4(λ2-4γ2)]1/2}1/2 (4)
- 3. 3 Torsional Vibrations and Buckling of Thin-Walled Beams on Elastic Foundation-Dynamic Stiffness Method K2 = L2 GCs/ECw.∆2 =ρ Ip L2 /AECw (5)And λ2 = ρIp L4 p2n/ ECw γ2=Ks L4/4ECw (6) From equation (4), the following relation between αL and βL is obtained. (βL)2 = (αL)2 + K2 - ∆2 (7)Knowing α and β. The frequency parameter λ can be evaluated using the following equation: λ2 = (αL)(βL) + 4γ2 (8) The four arbitrary constants A, B, C, and D in equation (3) can be determined from the boundaryequation of the beam. For any single span beam, there will be two boundary conditions at each end andthese four conditions then determine the corresponding frequency and mode shape expressions.3. DYNAMIC STIFFNESS MATRIX In order to proceed further, we must first introduce the following nomenclature: the variation ofangle of twist φ with respect to z is denoted by θ(z); the flange bending moment and the total twistingmoment are given by M(z) and T(z). Considering clockwise rotations and moments to be positive, wehave, θ(z) = dφ/dz, (9) hM(z) = -ECw( d2φ/dz2) and T(z) = -ECw ( d3φ/dz3) + (GCs –ρIp/A) dφ/dz (10)Where ECw = If h2/2, If = the flange moment of inertia and h= the distance between the centre lines of flanges of a thin-walled I-beam. Consider a uniform thin-walled I-beam element of length L as shown in fig.1(a). By combining the equation (3) and (9), the end displacements φ(0) and θ(0) and end forces, hM(0)and T(0) of the beam at z = 0, can be expressed as : φ(0) 1 0 1 0 A θ(0) = 0 α 0 β B hM(0) ECwα2 0 -ECwβ2 0 C T(0) 0 ECwαβ2 0 - ECwα2 β D
- 4. N.V. Srinivasulu, B. Suryanarayana & S. Jaikrishna 4 Equation (11) can be written in an abbreviated form as follows: δ(0) = V(0)U (11)in a similar manner , the end displacements , φ(L)and θ(L) and end forces hM(L) and T(L), of the beamwhere z = l can be expressed as follows: δ(L) = V(L)U where {δ(L)}T ={φ(L), θ(L), hM(L), T(L)} {U}T = {A,B,C,D} and c s C S -αs αc βS βC [V(L)] = ECwα2c ECwα2s -ECwβ2C -ECwβ2S - ECwαβ2s ECwαβ2c - ECwα2βS - ECwα2βC in which c = CosαL; s = SinαL; C = CoshβL; S= SinhβL. By eliminating the integration constant vector U from equation (11) and (12), and designating theleft end of the element as i and the right end as j. the equation relating the end forces and displacementscan be written as: Ti j11 j12 j13 j14 ϕi HMi j21 j22 j23 j24 θi Tj = j31 j32 j33 j34 ϕj HMj j41 j42 j43 j44 θj Symbolically it is written {F} = [J] {U} (12)where F}T = {Ti, hMi, Tj, hMj} U}T = {φi, θj, φj, θj}
- 5. 5 Torsional Vibrations and Buckling of Thin-Walled Beams on Elastic Foundation-Dynamic Stiffness Method In the above equations the matrix [J] is the ‘exact’ element dynamic stiffness matrix, which is also asquare symmetric matrix. The elements of [J] are given by: j11 = H[(α2 +β2}(αCs+βSc] j12 = -H[(α2 -β2}(1-Cc)+2αβSs] j13 = - H[(α2 +β2}(αs+βS)] j14 = -H[(α2 +β2}(C-c)] j22 = -(H/αβ)[(α2 +β2}(αSc-βCs)] (13) 2 2 j24 = (H/αβ)[(α +β }(αS-βs)] j23 = -j14 j33 = j11 j34 = -j12 j44 = j22 and H= ECw /[2αβ[ (1-Cc)+( β2-α2) Ss]using the element dynamic stiffness matrix defined by equation (12) and (13). One can easily set up thegeneral equilibrium equations for multi-span thin-walled beams, adopting the usual finite elementassembly methods. Introducing the boundary conditions, the final set of equations can be solved foreigen values by setting up the determinant of their matrix to zero. For convenience the signs of endforces and end displacements used in equation are all taken as positive.METHOD OF SOLUTION Denoting the assembled and modified dynamic stiffness matrix as [DS], we state that Det |DS| =0 (14) Equation (14) yields the frequency equation of continuous thin-walled beams in torsion resting oncontinuous elastic foundation and subjected to a constant axial compressive force. It can be noted thatequation (14) is highly transcendental in terms of eigen values λ. The roots of the equation (14) can,therefore, be obtained by applying the Regula-Falsi method and the Wittrick –Williams algorithm on ahigh speed digital computer. Exact values of frequency parameter λ for simply supported and built inthin-walled beams are obtained in this chapter using an error factor ε= 10-6.RESULTS AND DISCUSSIONS The approach developed in the present work can be applied to the calculation of natural torsionalfrequencies and mode shapes of multi –span doubly symmetric thin-walled beams of open section suchas beams of I-section. Beams with non uniform cross sections also can be handled very easily as thepresent approach is almost similar to the finite element method of analysis but with exact displacement
- 6. N.V. Srinivasulu, B. Suryanarayana & S. Jaikrishna 6shape functions. All classical and non- classical (elastic restraints) boundary conditions can beincorporated in the present model without any difficulty. To demonstrate the effectiveness of the present approach, a single span thin walled I-beam clampedat one end (z=0) and simply supported at the other end(z=l)is chosen. The boundary conditions for thisproblem can be written as: φ(0) = 0; θ(0) = 0 (15) φ(l) = 0; M(l) = 0 (16) Considering a one element solution and applying the boundary conditions defined by equation (15)and (16) gives, j22=0 (17)This gives, j22 = -(H/αβ)[(α2 +β2}(αSc-βCs)] = 0 (18)as H and (α2 +β2) are ,in general, non-zero, the frequency equation for the clamped, simply supportedbeam can , therefore, be written as αtanhβL = βtanαL (19) Equation (19)is solved for values of warping parameter k=1 and k=10 and for various values offoundation parameter γ in the range 0-12. Figures 2 and 3 shows the variation of fundamental frequency and buckling load parameters withfoundation parameter for values of k equal to 1 and 10 respectively. It can be stated that even for thebeams with non-uniform sections, multiple spans and complicated boundary conditions accurateestimates of natural frequencies can be obtained using the approach presented in this paper. A close look at the results presented in figures clearly reveal that the effect of an increase in axialcompressive load parameter ∆is to drastically decrease the fundamental frequency λ(N=1). Further more,the limiting load where λ becomes zero is the buckling load of the beam for a specified value of warpingparameter, K and foundation parameter, γ one can easily read the values of buckling load parameter ∆crfrom these figures for λ=0, as can be expected, the effect of elastic foundation is found to increase thefrequency of vibration especially for the first few modes. However, this influence is seen be quitenegligible on the modes higher than the third.
- 7. 7 Torsional Vibrations and Buckling of Thin-Walled Beams on Elastic Foundation-Dynamic Stiffness Method 2.5 2 1.5 1 Wk=.01 Wk=0.1 Wk=1.0 0.5 Wk=10 Wk=100 0 1 2 3 4 5 6 7 8 Figure 1 : First mode 6 5 4 Wk=.01 Wk=0.1 3 Wk=1.0 Wk=10 2 Wk=100 1 0 1 2 3 4 5 6 7 8 Figure 2 : Second mode
- 8. N.V. Srinivasulu, B. Suryanarayana & S. Jaikrishna 8 9 8 7 6 Wk=.01 5 Wk=0.1 4 Wk=1.0 3 Wk=10 2 Wk=100 1 0 1 2 3 4 5 6 7 8 Buckling Parameter Figure 3 : Third mode 12 10 Wk=.01 8 Wk=0.1 6 Wk=1.0 Wk=10 4 Wk=100 2 0 1 2 3 4 5 6 7 8 Figure 4 : Fourth mode
- 9. 9 Torsional Vibrations and Buckling of Thin-Walled Beams on Elastic Foundation-Dynamic Stiffness Method 16 14 12 1st mode 10 2nd mode 8 3rd mode 6 4th mode 4 5th mode 2 0 1 2 3 4 5 6 7 Figure 5 : Fifth mode 16 14 12 1st mode 2nd mode 3r d mode 10 8 6 4th mode 5th mode 4 2 0 1 2 3 4 5 6 7 Figure 6 : Sixth mode for Wk=0.1
- 10. N.V. Srinivasulu, B. Suryanarayana & S. Jaikrishna 10 16 14 12 10 8 1st mode 2nd mode 6 3rd mode 4 4th mode 5th mode 2 0 1 2 3 4 5 6 7 Figure 7 : Sixth mode for Wk=1.0CONCLUDING REMARKS A dynamic stiffness matrix approach has been developed for computing the natural torsionfrequencies and buckling loads of long, thin-walled beams of open section resting on continuousWinkler-Type elastic foundation and subjected to an axial time –invariant compressive load. Theapproach presented in this chapter is quite general and can be applied for treating beams with non-uniform cross sections and also non-classical boundary conditions. Using Wittrick-Williams algorithm,the torsional buckling loads, frequencies and corresponding modal shapes are determined. Results for abeam clamped at one end and simply supported at the other have been presented, showing the influenceon elastic foundation, and compressive load. While an increase in the values of elastic foundationparameter resulted in an increase in frequency, the effect of an increase in axial load parameter is foundto be drastically decreasing the frequency to zero at the limit when the load equals the buckling load forthe beam.
- 11. 11 Torsional Vibrations and Buckling of Thin-Walled Beams on Elastic Foundation-Dynamic Stiffness Method Fig. 1(a)REFERENCES1. Gere J.M, “Torsional Vibrations of Thin- Walled Open Sections, “Journal of Applied Mechanics”, 29(9), 1987, 381-387.2. Kerr.A.D., “Elastic And Viscoelastic Foundation Models”, Journal of Applied Mechanics,31,221-228.3. Krishnamurthy A.V and Joga Rao C.V., “General Theory of Vibrations Of Cylindrical Tubes”, Journal of Structures, 97, 1971, 1835-1840.4. Kameswara Rao.C, Gupta.B.V.R.and D.L.N., “Torsional Vibrations of Thin Walled Beams on Continuous Elastic Foundation Using Finite Element Methods In Engineering”, Coimbatore, 1974, 231-48.5. Christiano.P and Salmela.L. “Frequencies of Beams With Elastic Warping Restraint”, Journal of Aeronautical Society Of India, 20, 1968, 235-58.6. Kameswara Rao.C & Appala Satyam.A., “Torsional Vibration And Stability of Thin-Walled Beams on Continuous Elastic Foundation”, AIAA Journal, 13, 1975, 232-234.7. Wittrick.W.H. & Williams,E.W., “A General Algorithm For Computing Natural Frequencies of Elastic Structures”, Quarterly Journal of Mechanics and Applied Mathematics,24,1971,263-84.8. Kameswara Rao, C., Ph.D Thesis, Andhra University, Waltair, “Torsional Vibrations and Stability of Thin-Walled Beams of Open Sections Resting on Continuous Elastic Foundation”, 1975.9. Kameswara Rao,C., Sarma P.K, “The Fundamental Flexural Frequency Of Simply-Supported I- Beams With Uniform Taper”, Journal of The Aeronautical Society Of India ,27, , 1975,169-171.
- 12. N.V. Srinivasulu, B. Suryanarayana & S. Jaikrishna 1210. Hutton S.G and Anderson D.L, “Finite Element Method, A Galerkin Approach”, Journal of The Engineering Mechanics Division, Proceedings of The American Society of Civil Engineers 97, Em5, 1503-1520, 1971.11. Srinivasulu, N.V.,Ph.D Thesis, Osmania University, Hyderabad, “Vibrations of thin walled composite I-beams” 2010.ACKNOWLEDGEMENTS We thank Management, Principal, Head and staff of Mech Engg Dept of CBIT for their constantsupport and guidance for publishing this research paper.

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