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BUCKLING OF BEAMS

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- 1. TORSIONAL VIBRATIONS AND BUCKLING OF THIN- WALLED BEAMS ON ELASTIC FOUNDATION- DYNAMIC STIFFNESS METHOD Dr.N.V.Srinivasulu1, B.Suryanarayana2 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.com AbstractThe 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 elasticfoundation is discussed in this paper. An analytical method based on the dynamic stiffnessmatrix approach is developed including the effect of warping. A thin walled beam clamped atone end and simply supported at the other is considered. The different mode shapes and naturalfrequencies for the above beam is studied when it is resting on elastic foundation using dynamicstiffness matrix method.Introduction The problem of free torsional vibration and buckling of doubly symmetric thin- walledbeams of open section, subjected to an axial compressive static load and resting on continuouselastic foundation is discussed in this paper. An analytical method based on the dynamic stiffnessmatrix approach is developed including the effect of warping. The resulting transcendentalequation is solved for thin-walled beams clamped at one end and simply supported at the other.The dynamic stiffness matrix can be used to compute the natural frequencies and mode shapes ofeither a single beam with various end conditions or an assembly of beams. When severalelements are to be used the over all dynamic stiffness matrix of the complete structure must beassembled. The associated natural frequencies and mode shapes are extracted using Wittrick-Williams algorithm[10]. The algorithm guarantees that no natural frequency and its associatedmode shape are missed. This is, of course, not possible in the conventional finite element methodNumerical results for natural frequencies and buckling load for various values of warping andelastic foundation parameter are obtained and presented. The vibrations and buckling of continuously – supported finite and infinite beams onelastic foundation have applications in the design of aircraft structures, base frames for rotatingmachinery, rail road tracks, etc. several studies have been conducted on this topic, and valuablepractical methods for the analysis of beams on elastic foundation have been suggested. Adiscussion of various foundation models was presented by Kerr[2]. While there are a number of publications on flexural vibrations of rectangular beams andplates on elastic foundation, the literature on torsional vibrations and buckling of beams onelastic foundation is rather limited. Free torsional vibrations and stability of doubly-symmetriclong thin-walled beams of open section were investigated by Gere[1], Krishna Murthy[3] andJoga Rao[3] and Christiano[5] and Salmela[8] , Kameswara Rao[4]., used a finite elementmethod to study the problem of torsional vibration of long thin-walled beams of open section 1
- 2. resting on elastic foundations. In another publication Kameswara Rao and Appala satyam[6]developed approximate expressions for torsional frequency and buckling loads for thin walledbeams resting on Winkler-type elastic foundation and subjected to a time invariant axialcompressive force. It is known that higher mode frequencies predicted by approximate methods artegenerally in considerable error. In order to improve the situation, a large number of elements orterms in the series have to be included in the computations to get values with acceptableaccuracy. In view of the same, more and more effort is being put into developing frequencydependent „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 developedincluding the effects of Winkler – type elastic foundation and warping torsion. The resultingtranscendental frequency equation is solved for a beam clamped at one end and simply supportedat the other. Numerical results for torsional natural frequencies and buckling loads for sometypical values of warping and foundation parameters are presented. The approach presented inthis 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 andresting on a Winkler –type elastic foundation of torsional stiffness Ks. The beam is subjected to aconstant static axial compressive force P and is undergoing free torsional vibrations. Thecorresponding differential equation of motion can be written as 4ECw / z4 – (GCs – Ip/A) 2 / z2 + Ks + Ip) 2 / 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; , the mass densityof the material of the beam; ,the angle of twist; Z, the distance along the length of the beam andt, the time.For the torsional vibrations, the angle of twist (z,t) can be expressed in the form (z,t) = x(z)e ipt (2)in which x(z) is the modal shape function corresponding to each beam torsion natural frequencyp. The expression for x(z) which satisfies equation(1) can be written as:x(z) = A Cos z + B Sin z +C Cosh z +D sin z (3)in which L. L = (1/ 2) { + (k2- 2) + [(k2- 2)2 +4( 2-4 2)]1/2}1/2 (4)K2 = L2 GCs/ECw. 2 = Ip L2 /AECw (5) 2And = Ip L4 p2n/ ECw 2 =Ks L4/4ECw (6)From equation (4), the following relation between L and L is obtained. 2
- 3. ( L)2 = ( L)2 + K2 - 2 (7) Knowing and . The frequency parameter can be evaluated using the following equation: 2 2 = ( L)( L) + 4 (8) The four arbitrary constants A, B, C, and D in equation (3) can be determined from the boundary equation of the beam. For any single span beam, there will be two boundary conditions at each end and these 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 of angle of twist with respect to z is denoted by (z); the flange bending moment and the total twisting moment are given by M(z) and T(z). Considering clockwise rotations and moments to be positive, we have, (z) = d /dz, (9) hM(z) = -ECw( d2 /dz2) and T(z) = -ECw ( d3 /dz3) + (GCs – Ip/A) d /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 (0) and (0) and end forces, hM(0) and T(0) of the beam at z = 0, can be expressed as : (0) 1 0 1 0 A (0) = 0 0 B 2 2 hM(0) ECw 0 -ECw 0 C 2 2 T(0) 0 ECw 0 - ECw D Equation (11) can be written in an abbreviated form as follows: (0) = V(0)U (11) in a similar manner , the end displacements , (L)and (L) and end forces hM(L) and T(L), of the beam where z = l can be expressed as follows: 3
- 4. (L) = V(L)U where { (L)}T ={ (L), (L), hM(L), T(L)} {U}T = {A,B,C,D} and c s C S - s c S C [V(L)] = ECw 2c ECw 2s -ECw 2C -ECw 2S 2 2 2 2 - ECw s ECw c - ECw S - ECw C in which c = Cos L; s = Sin L; C = Cosh L; S= Sinh L.By eliminating the integration constant vector U from equation (11) and (12), and designatingthe left end of the element as i and the right end as j. the equation relating the end forces anddisplacements can be written as: Ti j11 j12 j13 j14 i HMi j21 j22 j23 j24 i Tj = j31 j32 j33 j34 j HMj j41 j42 j43 j44 j Symbolically it is written {F} = [J] {U} (12) where {F}T = {Ti, hMi, Tj, hMj} {U}T = { i, j, j, j}In the above equations the matrix [J] is the „exact‟ element dynamic stiffness matrix, which isalso a square symmetric matrix.The elements of [J] are given by: 2 j11 = H[( + 2}( Cs+ Sc] 2 j12 = -H[( - 2}(1-Cc)+2 Ss] 4
- 5. 2 j13 = - H[( + 2}( s+ S)] 2 j14 = -H[( + 2}(C-c)] 2 j22 = -(H/ )[( + 2}( Sc- Cs)] (13) j24 = (H/ )[( 2 + 2}( S- s)] j23 = -j14 j33 = j11 j34 = -j12 j44 = j22 and H= ECw /[2 [ (1-Cc)+( 2- 2) Ss]using the element dynamic stiffness matrix defined by equation (12) and (13). One can easily setup the general equilibrium equations for multi-span thin-walled beams, adopting the usual finiteelement assembly methods. Introducing the boundary conditions, the final set of equations canbe solved for eigen values by setting up the determinant of their matrix to zero. For conveniencethe signs of end forces 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 thatDet |DS| =0 (14) Equation (14) yields the frequency equation of continuous thin-walled beams in torsionresting on continuous elastic foundation and subjected to a constant axial compressive force. Itcan be noted that equation (14) is highly transcendental in terms of eigen values . The roots ofthe equation (14) can, therefore, be obtained by applying the Regula-Falsi method and theWittrick –Williams algorithm on a high speed digital computer. Exact values of frequencyparameter for simply supported and built in thin-walled beams are obtained in this chapterusing an error factor = 10-6.Results and Discussions The approach developed in the present work can be applied to the calculation of naturaltorsional frequencies and mode shapes of multi –span doubly symmetric thin-walled beams ofopen section such as beams of I-section. Beams with non uniform cross sections also can behandled very easily as the present approach is almost similar to the finite element method ofanalysis but with exact displacement shape functions. All classical and non- classical (elasticrestraints) 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. Theboundary conditions for this problem can be written as: (0) = 0; (0) = 0 (15) (l) = 0; M(l) = 0 (16) 5
- 6. Considering a one element solution and applying the boundary conditions defined by equation(15) and (16) gives, j22=0 (17)This gives, j22 = -(H/ )[( 2 + 2}( Sc- Cs)] =0 (18)as H and ( 2 + 2) are ,in general, non-zero, the frequency equation for the clamped, simplysupported beam can , therefore, be written as tanh L = tan L (19)Equation (19)is solved for values of warping parameter k=1 and k=10 and for various values offoundation parameter in the range 0-12. Figures 2 and 3 shows the variation of fundamental frequency and buckling loadparameters with foundation parameter for values of k equal to 1 and 10 respectively. It can bestated that even for the beams with non-uniform sections, multiple spans and complicatedboundary conditions accurate estimates of natural frequencies can be obtained using theapproach presented in this paper. A close look at the results presented in figures clearly reveal that the effect of an increasein axial compressive load parameter is to drastically decrease the fundamental frequency (N=1). Further more, the limiting load where becomes zero is the buckling load of the beamfor a specified value of warping parameter, K and foundation parameter, one can easily readthe values of buckling load parameter cr from these figures for =0, as can be expected, theeffect of elastic foundation is found to increase the frequency of vibration especially for the firstfew modes. However, this influence is seen be quite negligible on the modes higher than thethird. 2.5 2 1.5 1 Wk=.01 Wk=0.1 0.5 Wk=1.0 Wk=10 0 1 2 3 4 5 6 7 8 Fig.1. First mode 6
- 7. 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 Fig. 2. Second mode9876 Wk=.015 Wk=0.14 Wk=1.03 Wk=102 Wk=10010 1 2 3 4 5 6 7 8 Buckling Parameter Fig. 3. Third mode 7
- 8. 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 Fig4. Fourth mode161412 1st mode10 2nd mode 8 3rd mode 6 4th mode 4 5th mode 2 0 1 2 3 4 5 6 7 Fig. 5 fifth mode 8
- 9. 161412 1st mode 2nd mode10 3rd mode 4th mode86 5th mode420 1 2 3 4 5 6 7 Fig.6 sixth mode for Wk=0.1 16 14 12 10 1st mode 8 2nd mode 6 3rd mode 4 4th mode 5th mode 2 0 1 2 3 4 5 6 7 Fig.7 sixth mode for Wk=1.0 9
- 10. Concluding Remarks A dynamic stiffness matrix approach has been developed for computing the naturaltorsion frequencies and buckling loads of long, thin-walled beams of open section resting oncontinuous Winkler-Type elastic foundation and subjected to an axial time –invariantcompressive load. The approach presented in this chapter is quite general and can be applied fortreating beams with non-uniform cross sections and also non-classical boundary conditions.Using Wittrick-Williams algorithm, the torsional buckling loads, frequencies and correspondingmodal shapes are determined. Results for a beam clamped at one end and simply supported at theother have been presented, showing the influence on elastic foundation, and compressive load.While an increase in the values of elastic foundation parameter resulted in an increase infrequency, the effect of an increase in axial load parameter is found to be drastically decreasingthe frequency to zero at the limit when the load equals the buckling load for the beam. Fig.1(a)References: 1. 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. 10
- 11. 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, , 1975,169-171.10. 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 constant support and guidance for publishing this research paper. 11

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