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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

                                            Abstract

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 effect of warping. A thin walled beam clamped at
one end and simply supported at the other is considered. The different mode shapes and natural
frequencies for the above beam is studied 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 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 effect of warping. The resulting transcendental
equation 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 of
either a single beam with various end conditions or an assembly of beams. When several
elements are to be used the over all dynamic stiffness matrix of the complete structure must be
assembled. The associated natural frequencies and mode shapes are extracted using Wittrick-
Williams algorithm[10]. The algorithm guarantees that no natural frequency and its associated
mode shape are missed. This is, of course, not possible in the conventional finite element method
Numerical results for natural frequencies and buckling load for various values of warping and
elastic foundation parameter are obtained and presented.
        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, rail road 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. A
discussion of various foundation models was presented by Kerr[2].
        While there are a number of publications on flexural vibrations of 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[1], Krishna Murthy[3] and
Joga Rao[3] and Christiano[5] and Salmela[8] , Kameswara Rao[4]., used a finite element
method to study the problem of torsional vibration of long thin-walled beams of open section
                                                1
resting on elastic foundations. In another publication Kameswara Rao and Appala satyam[6]
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 arte
generally in considerable error. In order to improve the 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 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 a Winkler –type elastic foundation of torsional stiffness Ks. 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

       4
ECw        / 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 , the
torsion constant; Ip, the polar moment of Inertia; A, the area of cross section; , the mass density
of the material of the beam; ,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 (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 frequency
p. 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)
           2
And            = Ip L4 p2n/ ECw          2
                                             =Ks L4/4ECw                              (6)

From equation (4), the following relation between L and L is obtained.
                                                     2
( 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
(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 designating
the left end of the element as i and the right end as j. the equation relating the end forces and
displacements 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 is
also 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
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 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 eigen values by setting up the determinant of their matrix to zero. For convenience
the 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 that
Det |DS| =0                                                               (14)

        Equation (14) 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 equation (14) is highly transcendental in terms of eigen values . The roots of
the equation (14) can, therefore, be obtained by applying the Regula-Falsi method and the
Wittrick –Williams algorithm on a high speed digital computer. Exact values of frequency
parameter for simply supported and built in thin-walled beams are obtained in this chapter
using an error factor = 10-6.

Results and Discussions
        The approach developed in the present work 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) = 0; (0) = 0                                                         (15)


         (l) = 0; M(l) = 0                                                        (16)


                                                    5
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, simply
supported 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 of
foundation parameter in the range 0-12.
        Figures 2 and 3 shows the variation of fundamental frequency and buckling load
parameters with foundation parameter for values of k equal to 1 and 10 respectively. It can be
stated that even for the 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 figures clearly reveal that the effect of an increase
in 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 beam
for a specified value of warping parameter, K and foundation parameter, one can easily read
the values of buckling load parameter cr from these figures for =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 be quite negligible on the modes higher than the
third.



          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
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 mode




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




                        Fig. 3. Third mode


                                            7
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 mode




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


                          Fig. 5 fifth mode

                                      8
16



14



12                                                                     1st mode         2nd mode



10


                                                                       3rd mode         4th mode
8



6

                                                                       5th mode

4



2



0

         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
Concluding 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 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 a beam clamped at one end and simply supported at the
other 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 in
frequency, the effect of an increase in axial load parameter is found to be drastically decreasing
the 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
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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Torsional vibrations and buckling of thin WALLED BEAMS

  • 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 Abstract 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 effect of warping. A thin walled beam clamped at one end and simply supported at the other is considered. The different mode shapes and natural frequencies for the above beam is studied 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 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 effect of warping. The resulting transcendental equation 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 of either a single beam with various end conditions or an assembly of beams. When several elements are to be used the over all dynamic stiffness matrix of the complete structure must be assembled. The associated natural frequencies and mode shapes are extracted using Wittrick- Williams algorithm[10]. The algorithm guarantees that no natural frequency and its associated mode shape are missed. This is, of course, not possible in the conventional finite element method Numerical results for natural frequencies and buckling load for various values of warping and elastic foundation parameter are obtained and presented. 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, rail road 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. A discussion of various foundation models was presented by Kerr[2]. While there are a number of publications on flexural vibrations of 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[1], Krishna Murthy[3] and Joga Rao[3] and Christiano[5] and Salmela[8] , Kameswara Rao[4]., used a finite element method 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 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 arte generally in considerable error. In order to improve the 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 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 a Winkler –type elastic foundation of torsional stiffness Ks. 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 4 ECw / 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 , the torsion constant; Ip, the polar moment of Inertia; A, the area of cross section; , the mass density of the material of the beam; ,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 (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 frequency p. 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) 2 And = 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 designating the left end of the element as i and the right end as j. the equation relating the end forces and displacements 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 is also 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 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 eigen values by setting up the determinant of their matrix to zero. For convenience the 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 that Det |DS| =0 (14) Equation (14) 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 equation (14) is highly transcendental in terms of eigen values . The roots of the equation (14) can, therefore, be obtained by applying the Regula-Falsi method and the Wittrick –Williams algorithm on a high speed digital computer. Exact values of frequency parameter for simply supported and built in thin-walled beams are obtained in this chapter using an error factor = 10-6. Results and Discussions The approach developed in the present work 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) = 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, simply supported 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 of foundation parameter in the range 0-12. Figures 2 and 3 shows the variation of fundamental frequency and buckling load parameters with foundation parameter for values of k equal to 1 and 10 respectively. It can be stated that even for the 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 figures clearly reveal that the effect of an increase in 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 beam for a specified value of warping parameter, K and foundation parameter, one can easily read the values of buckling load parameter cr from these figures for =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 be quite negligible on the modes higher than the third. 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 mode 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 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 mode 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 Fig. 5 fifth mode 8
  • 9. 16 14 12 1st mode 2nd mode 10 3rd mode 4th mode 8 6 5th mode 4 2 0 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 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 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 a beam clamped at one end and simply supported at the other 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 in frequency, the effect of an increase in axial load parameter is found to be drastically decreasing the 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