4. ijcseierd - torsional vibrations of doubly - copy

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TORSIONAL VIBRATIONS OF DOUBLY-SYMMETRIC THIN-WALLE D

I-BEAMS RESTING ON WINKLER-PASTERNAK FOUNDATION

USING DYNAMIC MATRIX METHOD

A. SAI KUMAR 1 & K. SRINIVASA RAO 2 1Research Scholar, Department of Mechanical Engineering, Vasavi College of Engineering (Autonomous),

Ibrahimbagh, Hyderabad, Telangana, India 2Associate Professor, Department of Mechanical Engineering, Vasavi College of Engineering (Autonomous),

Ibrahimbagh, Hyderabad, Telangana, India

ABSTRACT

The present work deals with Dynamic Stiffness Analysis of free torsional vibration of doubly

symmetric thin-walled beams of open section and resting on Winkler-Pasternak elastic foundation. A new

dynamic stiffness matrix is developed in this paper which includes the effects of warping and Winkler-

Pasternak foundation on its frequencies of vibration. The resulting transcendental frequency equations for

all classical and various special boundary conditions are solved for thin-walled beams of open cross section

for varying values of warping Winkler and Pasternak foundation parameters on its frequencies of

vibration.

A new MATLAB computer program has been developed based on the dynamic stiffness matrix

approach to solve the highly transcendental frequency equations and to accurately determine the torsional

natural frequencies for all classical and various special boundary conditions. The MATLAB code

developed consists of a master program based on modified BISECTION method and to call specific

subroutines to set up the dynamic stiffness matrix to perform various parametric calculations. Numerical

results for natural frequencies for various values of warping and Winkler and Pasternak foundation

parameters are obtained and presented in graphical form showing their parametric influence clearly.

KEYWORDS: Warping, Dynamic Stiffness Matrix, Winkler-Pasternak Foundation, MATLAB, Bisection

Method

Received: Dec 29, 2015; Accepted: Jan 07, 2016; Published: Jan 19, 2016; Paper Id.: IJCSEIERDFEB20164

INTRODUCTION

The problem of a beam (or plate) on an elastic foundation is important in both the civil and mechanical

engineering fields, since it constitutes a practical idealization for many problems (e.g. the behavior of a shaft

within a hydro-dynamically lubricated bearing, a floating body on the water, etc.). The concept of beams and slabs

on elastic foundations has been extensively used by geotechnical, pavement and railroad engineers for foundation

design and analysis.

Original A

rticle International Journal of Civil, Structural, Environmental and Infrastructure Engineering Research and Development(IJCSEIERD) ISSN(P): 2249-6866; ISSN(E): 2249-7978 Vol. 6, Issue 1, Feb 2016, 31-50 © TJPRC Pvt. Ltd.

32

Impact Factor (JCC): 5.9234

Figure a: Grillage Foundation

The analysis of structures resting on elastic foundations is usually based on a relatively simple model of the

foundation’s response to applied loads. A simple representation of elastic foundation was introduc

[1]. The Winkler model (one parameter model), which has been originally developed for the analysis of railroad tracks, is

very simple but does not accurately represents the characteristics of many practical foundations.

Figure 1: Deflections of Winkler

In order to eliminate the deficiency of Winkler model, improved theories have been introduced on refinement of

Winkler’s model, by visualizing various types of interconnectio

springs [1] (Filonenko-Borodich (1940) [2]; Hetényi (1946) [3]; Pasternak (1954) [4]; Kerr (1964) [5]). These theories

have been attempted to find an applicable and simple model of representation of found

Winkler model shortcomings improved versions [6] [7] have been developed. A shear layer is introduced in the Winkler

foundation and the spring constants above and below this layer is assumed to be different as per this formula

following figure shows the physical representation of the Winkler

The vibrations of continuously

the design of aircraft structures, base frames for rotating machinery, railroad tracks, etc. Quite a good amount of literatur

exists on this topic, and valuable practical methods for the analysis of beams on elastic foundation have been su

11]. Kameswara Rao et al [12-14] studied the problem of torsional vibration of long, thin

resting on Winkler-type elastic foundations using exact, finite element and approximate expressions for torsional frequency

of a thin-walled beams and subjected to a time

A. Sai Kumar & K. Srinivasa Rao

Foundation Figure b: Mat Foundation under Large Storage Tanks

structures resting on elastic foundations is usually based on a relatively simple model of the

foundation’s response to applied loads. A simple representation of elastic foundation was introduc



: Deflections of Winkler Foundation under Uniform Pressure


Winkler’s model, by visualizing various types of interconnections such as shear layers and beams along the Winkler

Borodich (1940) [2]; Hetényi (1946) [3]; Pasternak (1954) [4]; Kerr (1964) [5]). These theories

have been attempted to find an applicable and simple model of representation of foundation medium. To overcome the


foundation and the spring constants above and below this layer is assumed to be different as per this formula

following figure shows the physical representation of the Winkler-Pasternak model.

Figure 2: Winkler-Pasternak Model

The vibrations of continuously-supported finite and infinite beams on elastic foundation has wide

the design of aircraft structures, base frames for rotating machinery, railroad tracks, etc. Quite a good amount of literatur

exists on this topic, and valuable practical methods for the analysis of beams on elastic foundation have been su

14] studied the problem of torsional vibration of long, thin-walled beams of open sections

type elastic foundations using exact, finite element and approximate expressions for torsional frequency

walled beams and subjected to a time-invariant axial compressive force.


NAAS Rating: 3.01

Storage Tanks

structures resting on elastic foundations is usually based on a relatively simple model of the

foundation’s response to applied loads. A simple representation of elastic foundation was introduced by Winkler in 1867



Foundation under Uniform Pressure q


ns such as shear layers and beams along the Winkler

Borodich (1940) [2]; Hetényi (1946) [3]; Pasternak (1954) [4]; Kerr (1964) [5]). These theories

ation medium. To overcome the


foundation and the spring constants above and below this layer is assumed to be different as per this formulation. The

supported finite and infinite beams on elastic foundation has wide applications in

the design of aircraft structures, base frames for rotating machinery, railroad tracks, etc. Quite a good amount of literature

exists on this topic, and valuable practical methods for the analysis of beams on elastic foundation have been suggested. [8-

walled beams of open sections

type elastic foundations using exact, finite element and approximate expressions for torsional frequency

Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 33 Winkler-Pasternak Foundation Using Dynamic Matrix Method


It is well known that a dynamic stiffness matrix is mostly formed by frequency-dependent shape functions which

are exact solutions of the governing differential equations. It overcomes the discretization errors and is capable of

predicting an infinite number of natural modes by means of a finite number of degrees of freedom. This method has been

applied successfully for many dynamic problems including natural vibration. A general dynamic-stiffness matrix of a

Timoshenko beam for transverse vibrations was derived including the effects of rotary inertia of the mass, shear distortion,

structural damping, axial force, elastic spring and dashpot foundation [15]. Analytical expressions were derived for the

coupled bending-torsional dynamic stiffness matrix terms of an axially loaded uniform Timoshenko beam element [16-20]

and also a dynamic stiffness matrix is derived based on Bernoulli–Euler beam theory for determining natural frequencies

and mode shapes of the coupled bending-torsion vibration of axially loaded thin-walled beams with mono-symmetrical

cross sections, by using a general solution of the governing differential equations of motion including the effect of warping

stiffness and axial force [21] and [22]. Using the technical computing program Mathematica, a new dynamic stiffness

matrix was derived based on the power series method for the spatially coupled free vibration analysis of thin-walled curved

beam with non-symmetric cross-section on Winkler and also Pasternak types of elastic foundation [23] and [24]. The free

vibration frequencies of a beam were also derived with flexible ends resting on Pasternak soil, in the presence of a

concentrated mass at an arbitrary intermediate abscissa [25]. The static and dynamic behaviors of tapered beams were

studied using the differential quadrature method (DQM) [26] and also a finite element procedure was developed for

analyzing the flexural vibrations of a uniform Timoshenko beam-column on a two-parameter elastic foundation [27].

Though many interesting studies are reported in the literature [8-27], the case of doubly-symmetric thin-walled

open section beams resting on Winkler–Pasternak foundation is not dealt sufficiently in the available literature to the best

of the author’s knowledge.

In view of the above, the present work deals with dynamic stiffness analysis of free torsional vibration of doubly

symmetric thin-walled beams of open section and resting on Winkler-Pasternak elastic foundation. A new dynamic stiffness

matrix (DSM) is developed which includes the effects of warping and Winkler-Pasternak foundation on its frequencies of

vibration. The resulting transcendental frequency equations for all classical and various special boundary conditions are

solved for thin-walled beams of open cross section for varying values of warping and Winkler, Pasternak foundation

parameters on its frequencies of vibration.

A new MATLAB computer program is developed based on the dynamic stiffness matrix approach to solve the

highly transcendental frequency equations for all classical and various special boundary conditions. The MATLAB code

developed consists of a master program based on modified BISECTION method and to call specific subroutines to set up

the dynamic stiffness matrix to perform various parametric calculations. Numerical results for natural frequencies for

various values of warping and Winkler and Pasternak foundation parameters are obtained and presented in graphical form

showing their parametric influence clearly.

NOMENCLATURE

Table 1 �� St. Venant’s torsion T� warping torsion T� Total Non-Uniform Torsion k Modulus Of Subgrade Reaction P Pressure

34 A. Sai Kumar & K. Srinivasa Rao

Impact Factor (JCC): 5.9234 NAAS Rating: 3.01

Table 1: Contd., �� Shear Modulus ∅ Angle Of Twist G Modulus Of Rigidity Shear Constant M Twisting Moment In Each Flange h Distance Between The Center Lines Of The Flanges �� Moment Of Inertia Of Flange About Its Strong Axis u Lateral Displacement Of The Flange Centerline Warping Constant E Young’s Modulus � Mass Density Of The Material Of The Beam �� Polar Moment Of Inertia � Winkler Foundation Stiffness �� Pasternak Layer Stiffness Z Distance Along The Length Of The Beam � Torsional Natural Frequency K Non-Dimensional Warping Parameter �� Non-Dimensional Pasternak Foundation Parameter �� Non-Dimensional Winkler Foundation Parameter � Non-dimensional frequency parameter �(�) Variation Of Angle Of Twist ∅

FORMULATION AND ANALYSIS

Consider a long doubly-symmetric thin-walled beam of open-section of length L and resting on a Winkler-

Pasternak type elastic foundation of Winkler torsional stiffness (� )and Pasternak layer stiffness(��). The beam is

undergoing free torsional vibrations. The corresponding differential equation of motion can be written as:

� ��∅�� − �� + �� !∅��! + � ∅ − �� !∅�"! = 0 (1)

For free torsional vibrations, the angle of twist ∅(�, &) can be expressed in the form,

∅(�, &) = '(�)()*" (2)

In which '(�) is the modal shape function corresponding to each beam torsional natural frequency �.

The expression for '(�) which satisfies Eq. (1) can be written as

'(�) = , -./ 01 + 2 /34 01 + -./ℎ 61 + 7 /34ℎ 61 (3)

In which 61 849 01 are the positive, real quantities given by

01, 61

= :∓(<!=>?!)=@(<!=>?!)!=A(B!C>D!)E (4)

�E = FGHIJ!KHD L

�� = :<? J!KHD



�� = :<D J�KHD

�A = FMN?*!J�KHD L (5)

From Eq. (4) we have the following relation between 61 849 01

(01)E = (61)E + �E − ��E (6)

Knowing 6 849 0, the frequency parameter λ can be evaluated using the relation

�E = (61)(01) + ��E (7)

The four arbitrary constants A, B, C and D in Eq. (3) can be determined from the boundary conditions 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.

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 �(�). The flange bending moment and the total twisting moment are given by O(�) and P(�). Considering clockwise rotations and moments to be positive, we have

�(�) = 9∅ 9�, ℎO(�) = −� 9E∅ 9�R⁄⁄ (8)

P(T) = −�� UV�WU + �X �V�� (9)

Where E� is termed as the warping rigidity of the section,

� = NYZ!E (10)

Consider a uniform thin-walled I-beam element of length L as shown in the Figure. 3. By combining the Eq. (3)

and Eq. (8), the end displacements, ∅(0)and �(0) and end forces ℎO(0) and P(0), of the beam at � = 0, can be expressed

as

[ ∅(0)�(0)ℎO(0)P(0) \

= ] 1 0 1 00 6 0 0��6E0 0��60E ��0E0 0−��6E0_ `,27a (11)

Eq. (3.28) can be abbreviated as follows

b(0) = c(0)d (12)

In a similar manner, the end displacements, ∅(1)and �(1) and end forces ℎO(1) and P(1), of the beam at � = 1,

can be expressed as

36


b(1) = c(1)d

Where

c(1) = [ ∅�1��1�5O�1�P�1� \

edfg # e, 2 7f

hc�1�i # jkk

l - /�6/��6E-��60E/6-��6E/��60E-

In which

- # -./ 01, / # /34 01, # -./

Figure 3

The equation relating the end forces and displacements can be written as

[ P�0�5O�0�P�1�5O�1�\

# jkkkl 0 ��60E��6E��60E/��6E-

0��60E-��6E/ ∗ hdi [∅�0��0�∅�1��1�\


c0c��0E��6E0c0��0Ec��6E0noo

p

-./5 61, c # /345 61

3: Differential Element of Thin wall I Section Beam

The equation relating the end forces and displacements can be written as

0 ��6E0��0E��6E0c��0E0��6E0��0Ec noo

op


NAAS Rating: 3.01

(13)

(14)

(15)

(16)

(17)

(18)



By eliminating the integration constant vector U and designating the left end element as I and the right end as j,

the final equation relating the end forces and displacements can be written as

qrsrt P) ��⁄5Ou ��⁄Pu ��⁄5Ou ��⁄ vrw

rx # ]yzz yzE yzR yzAyEz yEE yER yEAyRz yRE yRR yRAyAz yAE yAR yAA_ [{)�){u�u

\ (19)

Eq. (19) is symbolically written as

e|f # hyiedf (20)

In the Eq. (20) the matrix hyiis the ‘exact’ element dynamic stiffness matrix (DSM), which is also a square matrix.

The elements of hyiare

yzz # }�6E + 0E��6c- + 0/�

yzE # �}h�6E � 0E��1 � -� + 260c/i yzR # �}�6E + 0E��6/ + 0c�

yzA # �}�6E + 0E�� -�

yEE # ��} 60⁄ ��6E + 0E��6c- � 0/�

yEA # �} 60⁄ ��6E + 0E��6c � 0/�

yER # �yzA

yRR # yzz

yRA # �yzE

yAA # yEE

and

} # 60 h260�1 � - + �0E � 6E�c/i⁄ (21)

Using the element dynamic stiffness matrix defined by Eq. (20), 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

METHOD OF SOLUTION

Denoting the modified dynamic stiffness matrix as [J], we state that

9(&|y| # 0 (22)

The above equation yields the frequency equation of continuous thin-walled beams in torsion resting on Winkler-

Pasternak type foundation. It can be noted that above equation is highly transcendental, the roots of equation can, therefore,

be obtained by applying the bisection method using MATLAB code on a high-speed digital computer.



A new MATLAB code [ANNEX A] was developed based on bisection method, which consists of master program

i.e. (code.m [ANNEX A]) and to call specific subroutines i.e. (FCT.m [ANNEX A]) to perform various parametric

calculations and was published in MATLAB Central official online library [29] which was cited and referred by few

researchers.

These are some of the key highlights of the new MATLAB code

• Primarily, it solves almost any given linear, non-linear & highly-transcendental equations.

• Additional key highlight of this code is, the equation whose roots are to be found, can be defined separately in an

“.m” file, which facilitates to solve multi-variable (example'E + �E + /34 ' + -./ � # 0) highly-transcendental

equations of any size, where the existing MATLAB codes fail to compute.

• This code is made robust in such a manner that it can automatically save and write the detailed informationsuch as

no. of iterations; the corresponding values of the variables and the computing time are automatically saved, into a

(.txt) file format, in a systematic tabular form.

• This code has been tested on MATLAB 7.14 (R2012a) [30] for all possible types of equations and proved to be

accurate.

Exact values of the frequency parameter λ for various boundary conditions of thin-walled open section beam are

obtained and the results are presented both in tabular and graphical form in this paper for varying values of warping,

Winkler foundation and Pasternak foundation parameters.

RESULTS AND DISCUSSIONS

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.

In the following, six common type and four new types of beams will be identified by a compound objective which

describes the end conditions at (z = 0 and z = L). They are

• Simply-supported beam

The boundary conditions for this problem can be written as

� ∅ = 0O = 0� 8& ' = 0 � ∅ = 0O = 0� 8& ' = 1 (23)

Figure 4: Simply-Supported Beam resting on Winkler-Pasternak Foundation



Considering a one element solution and applying these boundary conditions to Eq. (19) gives

| yEEyAA − yEAyAE| = 0 (24)

This gives,

(} 60)(6E + 0E)E⁄ /34(01) ∗ /34ℎ(61) = 0 (25)

As H and (6E + 0E)E are in general, non-zero. The frequency equation for the simply supported beam can,

therefore, be written as,

/34(01) ∗ /34ℎ(61) = 0 (26)

• Fixed-end beam


�∅ = 0� = 0� 8& ' = 0 �∅ = 0� = 0� 8& ' = 1 (27)

Figure 5: Fixed-end Beam Resting on Winkler-Pasternak Foundation


(1 − -./ℎ(61) -./(01)) + (�!C�!�)E�� ∗ /34ℎ(61) /34(β1) = 0 (28)

• Beam free at both the ends


� P = 0O = 0� 8& ' = 0 � P = 0O = 0� 8& ' = 1 (29)

Figure 6: Beam Free at Both ends and resting on Winkler-Pasternak Foundation


|y| = 0 (30)

This gives,



�1 � -./5�61� -./�01�� C��E��U�U� ∗ /34ℎ(61) /34(01) = 0 (31)

• Beam fixed at one end, simply-supported at other end


�∅ = 0� = 0� 8& ' = 0 � ∅ = 0O = 0� 8& ' = 1 (32)

Figure 7: Beam Fixed at one end, Simply-Supported at Other End and Resting on Winkler-Pasternak Foundation


| yEE| = 0 (33)

This gives,

(} 60)(6E + 0E)⁄ (6c- − 0/) = 0 (34)

As H and (6E + 0E) are in general non-zero. The frequency equation for the simply supported beam can,


6 &84(01) − 0 &84ℎ(61) = 0 (35)

• Beam fixed at one end, free at other end


�∅ = 0� = 0� 8& ' = 0 � P = 0O = 0� 8& ' = 1 (36)

Figure 8: Beam Fixed at One End, Free at Other End and Resting on Winkler-Pasternak Foundation


| yRRyAA − yRAyAR| = (37)

This gives,

(��=��)�!�! -./ℎ(61) -./(01) + (�!C�!)�� /34ℎ(61) /34(01) + 2 = 0 (38)



• Beam simply-supported at one end, free at other end


� ∅ = 0O = 0� 8& ' = 0 � P = 0O = 0� 8& ' = L (39)

Figure 9: Beam Simply-Supported at one end, Free at Other End and Resting on Winkler-Pasternak Foundation


|yEE(yRRyAA − yRAyAR) − yER(yREyAA − yRAyER) + yEA(yREyAR − yRRyAE)| = 0 (40)

This gives,

0R &84(01) − 6R &84ℎ(61) = 0 (41)

• Guided-end beam


�� = 0P = 0� 8& ' = 0 �� = 0P = 0� 8& ' = 1 (42)

Figure 10: Beam Guided at both the ends and resting on Winkler-Pasternak Foundation


| yRRyzz − yzRyRz| = 0 (43)

This gives,

(})(60)(/34(01) ∗ /34ℎ(61) = 0 (44)

As H and (6E + 0E)E are in general, non-zero. The frequency equation for the simply supported beam can,


(/34(01) ∗ /34ℎ(61) = 0 (45)

• Beam guided at one end, free at other end




�� # 0P = 0� 8& ' = 0 � P = 0O = 0� 8& ' = 1 (46)

Figure 11: Beam Guided at One End, Free at Other End and Resting on Winkler-Pasternak Foundation


|yzz(yRRyAA − yRAyAR) − yzR(yRzyAA − yRAyAz) + yzA(yRzyAR − yRRyAz)| = 0 (47)

This gives,

6R &84(01) + 0R &84ℎ(61) = 0 (48)

• Beam guided at one end, simply supported at one end


�� = 0P = 0� 8& ' = 0 � ∅ = 0O = 0� 8& ' = 1 (49)

Figure 12: Beam Guided at One End, Simply Supported at other End and Resting on Winkler-Pasternak Foundation


| yAAyzz − yzAyAz| = 0 (50)

This gives,

E�!�!(��=��) -./ℎ(61) -./(01) + 1 = 0 (51)

• Beam guided at one end, clamped at other end


�� = 0P = 0� 8& ' = 0 �∅ = 0� = 0� 8& ' = 1 (52)

Figure 13: Beam Guided at One and, fixed at Other End and Resting on Winkler-Pasternak Foundation




|yzz| # 0 (53)

This gives,

0 &84(01) + 6 &84ℎ(61) = 0 (54)

The general dynamic stiffness matrix defined by Eq. (33) and Eq. (47) of Ref. [15], for Euler-Bernoulli beam is

observed to be same as Eq. (19) and Eq. (21), but for only difference that the axial force was included in the Eq. (33) and

Eq. (47) of Ref. [15], in defining the roots 6 849 0 where as in the present paper the axial force was not included in the

Eq. (19) and Eq. (21).

The first order approximation equations (Eq. 11, Eq. 18, Eq. 22, Eq. 25, Eq. 27, and Eq. 30) of Ref. [31], for

torsional vibrations of uniform doubly symmetric thin walled open cross section are observed to be same as equations (Eq.

(25), Eq. (28), Eq. (31), Eq. (35), Eq. (38) and Eq. 41)).

The transcendental frequency equations (Eq. (4a), Eq. (4b), Eq. (4e) and Eq. (4f)) of Ref. [32], for generally

restrained beams are observed to be same as equations (Eq. (45), Eq. (48), Eq. (51) and Eq. (54)) of the present paper, but

for only difference that the roots 6 849 0 are considered to be equal and have same sign and defined as��.

The equations for the fixed-end beam and simply supported beam are solved for values of warping parameter 0 ≤ � ≤ 20 and for various values of Winkler foundation parameter 0 ≤ � ≤ 10 000 000 and values of Pasternak

foundation parameter 0 ≤ �� ≤ 2.5.

Table 1: Numerical Comparisons Frequency Parameters for a Simply Supported Beam Fully Supported on a Winkler

��, �� 0 Approx.

0 exact

0.5 Approx.

0.5 exact

1 Approx.

1 exact

2.5 Approx.

2.5 exact

0 3.14159 3.1415930 3.4767 3.180679 3.7360 3.2183142 4.2970 3.3240213 1 3.1496 3.1496248 3.48267 3.1883879 3.74078 3.2257872 4.30016 3.3308077

100 3.74836 3.7483635 3.9608 3.7715736 4.10437 3.7943618 4.58239 3.8603697 10000 10.0244 10.0242642 10.036 10.0255880 10.048 10.0267120 10.084 10.0303817

1000000 31.6235 31.6235460 31.6239 31.6235850 31.624 31.6236240 31.625 31.6237412 -Pasternak foundation between finite element method [Ref. 25, 26, 27] and available results

The above Table depicts a comparison between the finite element and exact results for the frequency parameters

of asimply-supported beam fully supported on a Winkler-Pasternak foundation. The results obtained by the dynamic

stiffness matrix approach agree very closely, with the solutions computed from the frequency equations reported in

References [25] [26] and [27].

Furthermore, the equations for all the BC’s are also solved for values of warping parameter 0 ≤ � ≤ 20 and for

various values of Winkler foundation parameter 0 ≤ � ≤ 15 and values of Pasternak foundation parameter 0 ≤ �� ≤ 1

and are presented in graphical form for the first three modes.

The following Figure 14 shows the variation of frequency parameter with foundation parameters� 849 γ�, for

simply supported beam, one end fixed and other end free beam.

44


Figure 14: Plot for Parameter for values of Pasternak

The influences of the foundation

conditions is shown in the Figure 14, The figures indicate that the stability parameter increases as the overall stiffness of

the beam-foundation system increases. The overall

support stiffness, the foundation stiffness and the flexural rigidity of the beam. It is known that the flexural rigidity of

beam increases as � increases. It is obvious that th

foundation system increases.

Figure 15 shows the variation of frequency parameter with warping parameter K, for simply supported beam,

fixed-beam, one end fixed and other end free beam, o

A close look at the results presented in Figure 15 clearly reveals that the effect of an increase in warping

parameter K is to drastically decrease the fundamental frequency

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.

Figure 15: Plot for Influence of Pasternak Foundation Parameter

Furthermore, the equations for guided end condition are also solved for values of warping parameter

and for various values of Winkler foundation parameter 0 � �� 1 are presented in graphical form for the first three modes.

The influences of the foundation parameters

shown in the Figure 16, The figure indicate that the stability parameter increases as the overall stiffness of the beam

foundation system increases. The overall stiffness of the beam


for Influence of Winkler Foundation Parameter on Frequency for values of Pasternak Foundation Parameter for Various B

The influences of the foundation parameters� 849 ��, on the stability parameter for different supporting


foundation system increases. The overall stiffness of the beam-foundation system is an integrated resultant of the

support stiffness, the foundation stiffness and the flexural rigidity of the beam. It is known that the flexural rigidity of

increases. It is obvious that the frequency parameter increases as the overall stiffness of the beam


beam, one end fixed and other end free beam, one end fixed and other end simply supported beam


parameter K is to drastically decrease the fundamental frequency�. Furthermore, can be expected,



Influence of Warping Parameter on Frequency Parameter Foundation Parameter and Winkler Foundation Parameter for

Furthermore, the equations for guided end condition are also solved for values of warping parameter

and for various values of Winkler foundation parameter 0 � � � 15 and values of Pasternak foundation parameter

are presented in graphical form for the first three modes.

The influences of the foundation parameters� 849 ��, on the stability parameter for guided

shown in the Figure 16, The figure indicate that the stability parameter increases as the overall stiffness of the beam

foundation system increases. The overall stiffness of the beam-foundation system is an integr


NAAS Rating: 3.01

Frequency Various BC’s

, on the stability parameter for different supporting


foundation system is an integrated resultant of the

support stiffness, the foundation stiffness and the flexural rigidity of the beam. It is known that the flexural rigidity of the

e frequency parameter increases as the overall stiffness of the beam


ne end fixed and other end simply supported beam.


. Furthermore, can be expected, the effect of elastic


Frequency Parameter for Values for Various BC’s

Furthermore, the equations for guided end condition are also solved for values of warping parameter 0 � � � 20

and values of Pasternak foundation parameter

ity parameter for guided-end conditions is

shown in the Figure 16, The figure indicate that the stability parameter increases as the overall stiffness of the beam-

foundation system is an integrated resultant of the support

Torsional Vibrations of Doubly-Symmetric Winkler-Pasternak Foundation Using Dynamic Matrix Method

www.tjprc.org

stiffness, the foundation stiffness and the flexural rigidity of the guided

parameter increases as the overall stiffness of the beam foundation system increases.

The variation of frequency parameter with foundation parameters

free beam, one end guided and other end simply supported beam are shown graphically.

The plots clearly show that while the Winkler foundation independently increase

vibration for constant values of warping and the Pasternak foundation parameters. Interestingly we can clearly observe

the effect of Pasternak foundation Parameter is to decrease the natural torsional frequency significa

vibration and for constant values of warping and Winkler foundation parameter.

Figure 16: Plot for Influence for values of Pasternak


parameter K is to drastically decrease the fundamental frequency



Figure 17: Plot For influof Pasternak Foundation Parameter

It can be finally concluded that for an appropriately designing the thin

on continuous elastic foundation, it is very much necessary to model the foundation appropriately considering the Winkler

and Pasternak foundation stiffness values as their combined influence on the natural torsional frequency is quite significa

and hence cannot be ignored.

Symmetric Thin-Walled I-Beams Resting on Dynamic Matrix Method

stiffness, the foundation stiffness and the flexural rigidity of the guided-end beam. It is obvious that the frequency

parameter increases as the overall stiffness of the beam foundation system increases.

uency parameter with foundation parameters� 849 ��, for guided

free beam, one end guided and other end simply supported beam are shown graphically.

The plots clearly show that while the Winkler foundation independently increases the frequency for any mode of

vibration for constant values of warping and the Pasternak foundation parameters. Interestingly we can clearly observe

the effect of Pasternak foundation Parameter is to decrease the natural torsional frequency significa

vibration and for constant values of warping and Winkler foundation parameter.

Influence of Winkler Foundation Parameter on Frequency Paramefor values of Pasternak Foundation Parameter for Guided-End BC’s


parameter K is to drastically decrease the fundamental frequency�. Furthermore, it can be expected, the



influ ence of Warping Parameter on Frequency Parameter Parameter and Winkler Foundation Parameter for

It can be finally concluded that for an appropriately designing the thin-walled beams of open cross sectio


and Pasternak foundation stiffness values as their combined influence on the natural torsional frequency is quite significa

45

[email protected]

end beam. It is obvious that the frequency

, for guided-clamped beam, guided

s the frequency for any mode of

vibration for constant values of warping and the Pasternak foundation parameters. Interestingly we can clearly observethat

the effect of Pasternak foundation Parameter is to decrease the natural torsional frequency significantly for any mode of

Frequency Parameter BC’s


. Furthermore, it can be expected, the effect of elastic


Frequency Parameter for Values for Guided-end BC’s

walled beams of open cross sections resting


and Pasternak foundation stiffness values as their combined influence on the natural torsional frequency is quite significant



CONCLUSIONS

In this paper, a dynamic stiffness matrix (DSM) approach has been developed for computing the natural torsion

frequencies of long, doubly-symmetric thin-walled beams of open section resting on continuous Winkler-Pasternak type

elastic foundation. The approach presented in this thesis is quite general and can be applied for treating beams with non-

uniform cross-sections and also non-classical boundary conditions. A new MATLAB computer program has been

developed based on the dynamic stiffness matrix approach to solve the highly transcendental frequency equations and to

accurately determine the torsional natural frequencies for all classical and various special boundary conditions. Numerical

results for natural frequencies for various values of warping and Winkler and Pasternak-foundation parameters are obtained

and presented in both tabular as well as graphical form showing their parametric influence clearly. From the results

obtained following conclusions are drawn.

• An attempt was made to validate the present formulation of the problem for various boundary conditions. There is

very good agreement between the results, the general dynamic stiffness matrix defined by Eq. (33) and Eq. (47) of

Yung-Hsiang Chen [15], for Euler-Bernoulli beam is observed to be same as Eq. (19) and Eq. (20) in this paper,

but for only difference that the axial force was not included in the present paper.

• Further validation of the model was done by comparing the results obtained for simply supported beams and are

solved for values of various values of Winkler foundation parameter 0 ≤ γ� ≤ 10 000 000 and Pasternak

foundation parameter 0 ≤ γ� ≤ 2.5 and are presented in Table 1. The results compare very well with those from

De Rosa, M. A. and M. J. Maurizio [25] and Yokoyama [26]

• The influences of the foundation parametersγ�, γ� and warping parameter K, on the stability parameter λ for

various supporting conditions are shown in the Figures (14-17). The second foundation parameter γ�, tends to

increase the fundamental frequency for the same Winkler constantγ�. The effect of γ�, may be interpreted in the

following way: A simply supported beam, which is the weakest as far as stability (λ = 3.66) is concerned,

acquires the stability of a beam which is clamped at both ends (λ = 5.24), by increasing the shear parameter of

the foundation especially for the first mode. However, this influence is seen to be quite negligible on the modes

higher than the first. Also, it is found that the effect of an increase in warping parameter K is to drastically

decrease the stability parameterλ.

It can be finally concluded that for an appropriately designing the thin-walled beams of open cross sections resting


and Pasternak foundation stiffness values as their combined influence on the natural torsional frequency is quite significant

and hence cannot be ignored.

Future Work

The Dynamic Stiffness Matrix (DSM) approach could be implemented to multi span beams of open section

resting on various possible types of elastic foundations, including the effects of longitudinal inertia, axial compressive load,

time varying loads and shear deformation. This Dynamic Stiffness Matrix (DSM) approach could also be implemented not

only to beams but also to pipes conveying fluid and carbon Nano-tubes conveying fluid resting on visco-elastic foundation

and three parameter foundation models.



REFERENCES

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2. Filonenko-Borodich M. M., "Some approximate theories of the elastic foundation." Uchenyie Zapiski Moskovskogo

Gosudarstvennogo Universiteta Mekhanica 46 (1940): 3-18.

3. Hetényi, Miklós, and Miklbos Imre Hetbenyi., “Beams on elastic foundation: theory with applications in the fields of civil and

mechanical engineering”. Vol. 16. University of Michigan Press, 1946.

4. Pasternak, P. L., "On a new method of analysis of an elastic foundation by means of two foundation constants."

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system”. Elsevier Science Ltd., pp. 1579-1594, April 2002.

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182, May 2005.

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the Franklin Institute 239.4 (1945): 249-268.

9. Gere, JMt., "Torsional vibrations of beams of thin-walled open section." Journal of Applied Mechanics-Transactions of the

ASME 21.4 (1954): 381-387.

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Foundation", AIAA Journal, Vol. 13, 1975, pp. 232- 234.

14. C. Kameswara Rao and S. Mirza., “Torsional vibrations and buckling of thin walled beams on Elastic foundation”, Thin-

Walled Structures (1989) 73-82.

15. Yung-Hsiang Chen., “General dynamic-stiffness matrix of a Timoshenko beam for transverse vibrations”, Earthquake

Engineering and Structural dynamics (1987) 391-402

16. J.R. Banerjee., and F.W. Williams., “Coupled bending-torsional dynamic stiffness matrix of an Axially loaded Timoshenko

beam element”, Journal of Solids Structures, Elsevier science publishers 31 (1994) 749-762

17. P.O. Friberg, “Coupled vibrations of beams-an exact dynamic element stiffness matrix”, International Journal for Numerical

Methods in Engineering 19 (1983) 479-493

18. J.R. Banerjee, “Coupled bending-torsional dynamic stiffness matrix for beam elements”, International Journal for Numerical

Methods in Engineering 28 (1989)1283-1298.

19. Zongfen Zhang and Suhuan Chen, “A new method for the vibration of thin-walled beams”, Computers & Structures 39(6)

(1991) 597-601.



20. J.R. Banerjee and F.W. Williams, “Coupled bending-torsional dynamic stiffness matrix for Timoshenko beam elements”,

Computers & Structures 42(3) (1992)301-310.

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Structures 59(4) (1996) 613-621.

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Applied Acoustics 65.2 (2004): 153-170.

23. Kim, Nam-Il, Chung C. Fu, and Moon-Young Kim. "Dynamic stiffness matrix of non-symmetric thin-walled curved beam on

Winkler and Pasternak type foundations." Advances in Engineering Software 38.3 (2007): 158-171.

24. Nam-Il Kim, Ji-Hun Lee, Moon-Young Kim, “Exact dynamic stiffness matrix of non-symmetric thin-walled beamson elastic

foundation using power series method”, Advances in Engineering Software 36 (2005) 518–532.

25. De Rosa, M. A., and M. J. Maurizi,. "The Influence of Concentrated Masses and Pasternak Soil on the Free Vibrations of Euler

Beams-Exact Solution." Journal of Sound and Vibration 212.4 (1998): 573-581.

26. Yokoyama, T., "Vibrations of Timoshenko beam‐columns on two‐parameter elastic foundations." Earthquake Engineering &

Structural Dynamics 20.4 (1991): 355-370.

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29. http://in.mathworks.com/matlabcentral/fileexchange/48107-roots-for-non-linear-highly-transcendental-equation

30. MATLAB 7.14 (R2012a)

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Normal Modes of thin-walled open section beams”, journal of aeronautical society of India,1974., pp. 32-41.

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vibration 147.1 (1991): 167-171.

APPENDICES

ANNEX A

MATLAB CODE FOR SOLVING TORSIONAL FREQUENCY EQUATIO NS

Main program function code(a,b) % This code finds a solution to f(x) = 0 % % % % it finds a root given in the continuous function on the interval [a,b], % % where f(a) and f(b) have opposite signs. % % % % INPUT: % % a,b: define the interval over which the met hod is exercised % % tol: is the solution tolerance % % n: is the maximum number of iterations of the algorithm. % % FCT(TOL): deceleration of the function whose sol ution has to be found. % % % % OUTPUT: % % value: is the approximate solution %



% % % USAGE: % % [value] = code(...) to display a solution of th e function % % % %-----------------------------BY: Kumar Sai-------- ------------------------ if nargin < 2 error( 'Incorrect input!!! provide at least two input argu ments' ) end tol=10^-6; n=100; if a > b X = a; a = b; b = X; end if a==b error( 'Input (a) cannot equal (b)' ) end TOL=a; fa = FCT(TOL); TOL=b; fb = FCT(TOL); if fa*fb > 0 error( 'f(a) and f(b) have the same sign' ) end if tol < 0 error( 'tolerance must be a positive number' ) end fileID = fopen( 'results.txt' , 'w' ); fprintf(fileID, 'Solution of Non-Linear Transcendental Freq. Eq.\r\ n' ); fprintf(fileID, '-----------------By: Kumar Sai-----------------\r\ n' ); fprintf(fileID, '---------Number of Iterations obtained---------\r\ n' ); fprintf(fileID, ' I a b c p \r\n ' ); fprintf(fileID, '----------------------------------------------\r\n ' ); tic; I = 1; while I <= n c = (b - a) / 2.0; p = a + c; TOL=p; fp = FCT(TOL); A = [I;a;b;c;p]; fprintf(fileID, '%3u: % 5.7f % 5.7f % 5.7f % 5.7f\r\n' ,A); if I == 1 Result{1} = '-------------------------------------------------- ' ; Result{2} = '--Solution of Non-Linear Transcendental Freq. Eq.- ' ; Result{3} = '-------------------------------------------------- ' ; Result{4} = '-----The value of the Frequency parameter is------ ' ; Result{5} = '-------------------------------------------------- ' ; Result{6} = '----------time elapsed in milliseconds------------ ' ; end if abs(fp) < 1.0e-20 || c < tol Result = char(Result); disp(Result) value = p; t=toc; fprintf(fileID, '--------------------------------------------\r\n' ); fprintf(fileID, '---The value of the Frequency parameter is--\r\n' ); fprintf(fileID, '..............>>> %5.7f <<<...........\r\n' ,value); fprintf(fileID, 'time elapsed in milliseconds: %-10.10f\r\n' ,t*10^3);



fprintf( '------------------ %-10.10f-------------------\n' ,t*10^3); fprintf( '------The value of the Frequency parameter is----- \r\n' ); fprintf( '.................>>> %5.7f <<<...............\r\n' ,value); fclose(fileID); msgbox( 'procedure completed successfully !!' ) break else I = I+1; if fa*fp > 0 a = p; fa = fp; else b = p; end end end end Sub program for simply supported beam

function [F] =FCT(TOL) k2=0; d2=0; G1=0; G2=0; C=(k2+d2+G2); B4=TOL^4; ALP=(sqrt(-C+sqrt((C^2)+4*(B4-G1)))/sqrt(2)); BTA=(sqrt(C+sqrt((C^2)+4*(B4-G1)))/sqrt(2)); F=sinh(BTA)*sin(ALP); end

4. ijcseierd - torsional vibrations of doubly - copy

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