Upload
anonymous-qm973oab3
View
221
Download
0
Embed Size (px)
Citation preview
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
1/20
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
2/20
32
Impact Factor (JCC): 5.9234
Figure a: Grillage Fou
The analysis of structures res
foundation’s response to applied loads
[1]. The Winkler model (one paramete
very simple but does not accurately rep
Figure 1: Defle
In order to eliminate the defic
Winkler’s model, by visualizing vario
springs [1] (Filonenko-Borodich (194
have been attempted to find an applic
Winkler model shortcomings improve
foundation and the spring constants a
following figure shows the physical rep
The vibrations of continuousl
the design of aircraft structures, base f
exists on this topic, and valuable practi
11]. Kameswara Rao et al [12-14] stud
resting on Winkler-type elastic foundati
of a thin-walled beams and subjected to
A
ndation Figure b: Mat Foundation under Large
ting on elastic foundations is usually based on a r
. A simple representation of elastic foundation was i
model), which has been originally developed for th
resents the characteristics of many practical foundatio
ctions of Winkler Foundation under Uniform Pres
iency of Winkler model, improved theories have bee
us types of interconnections such as shear layers
) [2]; Hetényi (1946) [3]; Pasternak (1954) [4]; Ke
able and simple model of representation of foundati
versions [6] [7] have been developed. A shear laye
ove and below this layer is assumed to be differen
resentation of the Winkler-Pasternak model.
Figure 2: Winkler-Pasternak Model
-supported finite and infinite beams on elastic found
ames for rotating machinery, railroad tracks, etc. Qui
al methods for the analysis of beams on elastic found
ied the problem of torsional vibration of long, thin-
ons using exact, finite element and approximate expr
a time-invariant axial compressive force.
. Sai Kumar & K. Srinivasa Rao
NAAS Rating: 3.01
Storage Tanks
latively simple model of the
troduced by Winkler in 1867
analysis of railroad tracks, is
ns.
sure q
n introduced on refinement of
nd beams along the Winkler
rr (1964) [5]). These theories
on medium. To overcome the
r is introduced in the Winkler
t as per this formulation. The
ation has wide applications in
te a good amount of literature
ation have been suggested. [8-
alled beams of open sections
ssions for torsional frequency
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
3/20
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 33
Winkler-Pasternak Foundation Using Dynamic Matrix Method
www.tjprc.org [email protected]
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 aTimoshenko 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 torsionT
warping torsion
T Total Non-Uniform Torsionk Modulus Of Subgrade ReactionP Pressure
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
4/20
34 A. Sai Kumar & K. Srinivasa Rao
Impact Factor (JCC): 5.9234 NAAS Rating: 3.01
Table 1: Contd., Shear Modulus∅ Angle Of TwistG Modulus Of Rigidity
Shear ConstantM Twisting Moment In Each Flangeh Distance Between The Center Lines Of The Flanges Moment Of Inertia Of Flange About Its Strong Axisu Lateral Displacement Of The Flange Centerline Warping ConstantE Young’s Modulus Mass Density Of The Material Of The Beam Polar Moment Of Inertia Winkler Foundation Stiffness Pasternak Layer StiffnessZ Distance Along The Length Of The Beam
Torsional Natural FrequencyK 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 isundergoing free torsional vibrations. The corresponding differential equation of motion can be written as:
∅ − + !∅! + ∅ − !∅" ! # $ (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 + -./5 61 + 7 /345 61 (3)
In which 61 849 01 are the positive, real quantities given by01%61 # : ;(?!)=@ (?!)!=A(B!C>D!)E (4)E # F GHIJ!KHD L
# :
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
5/20
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 35
Winkler-Pasternak Foundation Using Dynamic Matrix Method
www.tjprc.org [email protected]
# :
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
6/20
36
Impact Factor (JCC): 5.9234
b(1) # c (1)d Where
c(1) # [ ∅115O1P1 \ edfg # e , 2 7 f hc1i # - /6/6E-
60E
/
6-6E/60
E
-In which- # -./ 01% / # /34 01% #
Figure 3
The equation relating the end
[ P $5O$P15O1\
# $ 60E6E60E/6E-
$60E-6E/ m hdi [∅$$
∅11\
A
c0c0E6E0c
00Ec6E0
./5 61% c # /345 61
: Differential Element of Thin wall I Section Beam
orces and displacements can be written as
$ 6E00E6E0c0E$6E00Ec
. Sai Kumar & K. Srinivasa Rao
NAAS Rating: 3.01
(13)
(14)
(15)
(16)
(17)
(18)
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
7/20
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 37
Winkler-Pasternak Foundation Using Dynamic Matrix Method
www.tjprc.org [email protected]
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
q P) S5Ou SPu S5Ou S v # ] yzz y zE y zR y zA yEz y EE y ER y EA yRz y RE y RR y RA yAz y AE y AR y AA_ [{)){uu \ (19)
Eq. (19) is symbolically written as
e|f # h yiedf (20)In the Eq. (20) the matrix h yiis the ‘exact’ element dynamic stiffness matrix (DSM), which is also a square matrix.
The elements of h yiare yzz # }6 E + 0 E6c- +0/
yzE # }h 6E 0 E^ - +~60c/i yzR # } 6E + 0 E6/ + 0c yzA # } 6E + 0 E - yEE # } 60S 6E + 0 E6c- 0/ yEA # } 60S 6E + 0 E6c 0/
yER # y zA
yRR # y zz yRA # y zE yAA # y EE and
} # 60 h ~60^ - + 0E 6 Ec/iS (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 boundaryconditions, 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• # $ (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.
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
8/20
38 A. Sai Kumar & K. Srinivasa Rao
Impact Factor (JCC): 5.9234 NAAS Rating: 3.01
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 ' +-./ € # $ ) highly-transcendentalequations 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
∅ # $O # $‚ 8& ' # $ ∅ # $O # $‚ 8& ' # 1 (23)
Figure 4: Simply-Supported Beam resting on Winkler-Pasternak Foundation
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
9/20
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 39
Winkler-Pasternak Foundation Using Dynamic Matrix Method
www.tjprc.org [email protected]
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
• y EE yAA − y EA yAE• # $ (24)This gives,
(} 60)(6E + 0 E)ES /34(01) m /345(61) # $ (25)As H and (6E + 0 E)E are in general, non-zero. The frequency equation for the simply supported beam can,
therefore, be written as,
/34(01) m /345(61) # $ (26)Fixed-end beam
The boundary conditions for this problem can be written as
∅ # $ # $‚ 8& ' # $ ∅ # $ # $‚ 8& ' # 1 (27)
Figure 5: Fixed-end Beam Resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
(^−-./5(61) -./(01)) + (ƒ!C„!…)Eƒ„ m/345(61)/34( β1) # $ (28)Beam free at both the ends
The boundary conditions for this problem can be written as
P # $O # $‚ 8& ' # $ P # $O # $‚ 8& ' # 1 (29)
Figure 6: Beam Free at Both ends and resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
• y• # $ (30)This gives,
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
10/20
40 A. Sai Kumar & K. Srinivasa Rao
Impact Factor (JCC): 5.9234 NAAS Rating: 3.01
^ -./561 -./01 ƒ†C„†EƒU„U m /345(61) /34(01) # $ (31)Beam fixed at one end, simply-supported at other end
The boundary conditions for this problem can be written as∅ # $ # $‚ 8& ' # $ ∅ # $O # $‚ 8& ' # 1 (32)
Figure 7: Beam Fixed at one end, Simply-Supported at Other End
and Resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
• y EE• # $ (33)This gives,
(} 60)(6E + 0 E)S (6c- − 0/) # $ (34)As H and (6E + 0 E) are in general non-zero. The frequency equation for the simply supported beam can,
therefore, be written as,
6&84(01) −0&845(61) # $ (35)Beam fixed at one end, free at other end
The boundary conditions for this problem can be written as
∅ # $ # $‚ 8& ' # $ P # $O # $‚ 8& ' # 1 (36)
Figure 8: Beam Fixed at One End, Free at Other End and Resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
• y RR yAA − y RA yAR• # (37)This gives,
(ƒ
=„
)ƒ!„! -./5 (61) -./(01) + (ƒ!
C„!
)ƒ„ /345(61)/34(01)+~ # $ (38)
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
11/20
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
12/20
42 A. Sai Kumar & K. Srinivasa Rao
Impact Factor (JCC): 5.9234 NAAS Rating: 3.01
# $P # $‚ 8& ' # $ P # $O # $‚ 8& ' # 1 (46)
Figure 11: Beam Guided at One End, Free at Other End and Resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
• yzz(yRR yAA − y RA yAR) − yzR(yRz yAA − y RA yAz) + yzA(yRz yAR − y RR yAz)• # $ (47)This gives,
6R &84(01) + 0R &845(61) # $ (48)Beam guided at one end, simply supported at one end
The boundary conditions for this problem can be written as
# $P # $‚ 8& ' # $ ∅ # $O # $‚ 8& ' # 1 (49)
Figure 12: Beam Guided at One End, Simply Supported at other
End and Resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
• y AA yzz − y zA yAz• # $ (50)This gives,
Eƒ!„!(ƒ=„) -./5(61) -./(01) + ^ # $ (51)Beam guided at one end, clamped at other end
The boundary conditions for this problem can be written as
# $P # $‚ 8& ' # $ ∅ # $ # $‚ 8& ' # 1 (52)
Figure 13: Beam Guided at One and, fixed at Other End and Resting on Winkler-Pasternak Foundation
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
13/20
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 43
Winkler-Pasternak Foundation Using Dynamic Matrix Method
www.tjprc.org [email protected]
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
• yzz• # $ (53)This gives,
0&84(01) +6&845(61) # $ (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 theEq. (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€ˆ.The equations for the fixed-end beam and simply supported beam are solved for values of warping parameter$ ‰ ‰ ~$ and for various values of Winkler foundation parameter $ ‰ ‰ ^$ $$$ $$$ and values of Pasternak
foundation parameter $ ‰ ‰ ~QŠ .Table 1: Numerical Comparisons Frequency Parameters for a
Simply Supported Beam Fully Supported on a Winkler
‹Œ% ‹ 0Approx. 0exact 0.5Approx. 0.5exact 1Approx. 1exact 2.5Approx. 2.5exact0 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 $ ‰ ‰ ~$ and forvarious values of Winkler foundation parameter $ ‰ ‰ ^Š and values of Pasternak foundation parameter $ ‰ ‰ ^ 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, forsimply supported beam, one end fixed and other end free beam.
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
14/20
44
Impact Factor (JCC): 5.9234
Figure 14: Plot fo
Parameter for v
The influences of the found
conditions is shown in the Figure 14,
the beam-foundation system increases.
support stiffness, the foundation stiffne
beam increases as increases. It is ofoundation system increases.
Figure 15 shows the variatio
fixed-beam, one end fixed and other en
A close look at the results
parameter K is to drastically decrease
foundation is found to increase the freseen to be quite negligible on the mode
Figure 15: Plot for Infl
of Pasternak Foundatio
Furthermore, the equations fo
and for various values of Winkler fo$ ‰ ‰ ^ are presented in graphical fThe influences of the founda
shown in the Figure 16, The figure i
foundation system increases. The over
A
r Influence of Winkler Foundation Parameter on F
lues of Pasternak Foundation Parameter for Vari
ation parameters 849 , on the stability para
he figures indicate that the stability parameter incre
The overall stiffness of the beam-foundation system i
ss and the flexural rigidity of the beam. It is known
vious that the frequency parameter increases as the
of frequency parameter with warping parameter K
free beam, one end fixed and other end simply supp
resented in Figure 15 clearly reveals that the effe
the fundamental frequency. Furthermore, can be
quency of vibration especially for the first few mods higher than the third.
uence of Warping Parameter on Frequency Param
Parameter and Winkler Foundation Parameter f
guided end condition are also solved for values of w
undation parameter $ ‰ ‰ ^Š and values of Pa
orm for the first three modes.
ion parameters 849 , on the stability paramete
dicate that the stability parameter increases as the
ll stiffness of the beam-foundation system is an inte
. Sai Kumar & K. Srinivasa Rao
NAAS Rating: 3.01
requency
us BC’s
eter for different supporting
ases as the overall stiffness of
s an integrated resultant of the
hat the flexural rigidity of the
overall stiffness of the beam
, for simply supported beam,
rted beam.
ct of an increase in warping
xpected, the effect of elastic
es. However, this influence is
eter for Values
r Various BC’s
arping parameter $ ‰ ‰ ~$
sternak foundation parameter
for guided-end conditions is
verall stiffness of the beam-
rated resultant of the support
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
15/20
Torsional Vibrations of Doubly-Symmetric
Winkler-Pasternak Foundation Using Dyn
www.tjprc.org
stiffness, the foundation stiffness and
parameter increases as the overall stiffn
The variation of frequency p
free beam, one end guided and other en
The plots clearly show that w
vibration for constant values of warpin
the effect of Pasternak foundation Par
vibration and for constant values of wa
Figure 16: Plot for Infl
for values of
A close look at the results
parameter K is to drastically decrease
foundation is found to increase the fre
seen to be quite negligible on the mode
Figure 17: Plot For infl
of Pasternak Foundation
It can be finally concluded tha
on continuous elastic foundation, it is
and Pasternak foundation stiffness valu
and hence cannot be ignored.
Thin-Walled I-Beams Resting on
amic Matrix Method
the flexural rigidity of the guided-end beam. It i
ess of the beam foundation system increases.
rameter with foundation parameters
849 , for
d simply supported beam are shown graphically.
ile the Winkler foundation independently increases t
and the Pasternak foundation parameters. Interestin
meter is to decrease the natural torsional frequency
ping and Winkler foundation parameter.
ence of Winkler Foundation Parameter on Freque
asternak Foundation Parameter for Guided-End
resented in Figure 17 clearly reveals that the effe
the fundamental frequency. Furthermore, it can bequency of vibration especially for the first few mod
s higher than the third.
uence of Warping Parameter on Frequency Param
arameter and Winkler Foundation Parameter for
t for an appropriately designing the thin-walled beams
ery much necessary to model the foundation appropr
s as their combined influence on the natural torsiona
45
s obvious that the frequency
uided-clamped beam, guided
he frequency for any mode of
ly we can clearly observethat
significantly for any mode of
ncy Parameter
BC’s
ct of an increase in warping
expected, the effect of elastic
es. However, this influence is
eter for Values
Guided-end BC’s
of open cross sections resting
iately considering the Winkler
l frequency is quite significant
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
16/20
46 A. Sai Kumar & K. Srinivasa Rao
Impact Factor (JCC): 5.9234 NAAS Rating: 3.01
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 $ ‰ Ž ‰ ^$ $$$ $$$ and Pasternakfoundation parameter $ ‰ Ž ‰ ~QŠ and are presented in Table 1. The results compare very well with those fromDe Rosa, M. A. and M. J. Maurizio [25] and Yokoyama [26]
The influences of the foundation parameters
Ž% Ž and warping parameter K, on the stability parameter
for
various supporting conditions are shown in the Figures (14-17). The second foundation parameter Ž , tends toincrease the fundamental frequency for the same Winkler constant Ž. The effect of Ž , may be interpreted in thefollowing way: A simply supported beam, which is the weakest as far as stability ( # ‘Q’’) is concerned,acquires the stability of a beam which is clamped at both ends ( # ŠQ~“), by increasing the shear parameter ofthe 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 parameter.It can be finally concluded that for an appropriately designing the thin-walled beams of open cross sections resting
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 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.
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
17/20
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 47
Winkler-Pasternak Foundation Using Dynamic Matrix Method
www.tjprc.org [email protected]
REFERENCES
1. Winkler, E. "Theory of elasticity and strength." Dominicus Prague, Czechoslovakia (1867).
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."
Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhitekture, Moscow (1954).
5. Kerr, Arnold D., "Elastic and viscoelastic foundation models." Journal of Applied Mechanics 31.3 (1964): 491-498.
6.
S. C. Dutta en, R. Rana, “A critical review on idealization and modeling for interaction among soil-foundation-structure
system”. Elsevier Science Ltd., pp. 1579-1594, April 2002.
7.
Y. H. Wang, L. G. Tham en Y. K. Cheung, “Beams and Plates on Elastic Foundations: a review,” Wiley Inter Science, pp. 174-
182, May 2005.
8. Timoshenko, Stephen P., "Theory of bending, torsion and buckling of thin-walled members of open cross section." Journal of
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.
10. Christiano, Paul, and Larry Salmela., "Frequencies of beams with elastic warping restraint." Journal of the Structural
Division 97.6 (1971): 1835-1840.
11. E. J. Sapountzakis, Bars under Torsional loading: a generalized beam approach, ISRN Civil Engineering (2013) 1-39.
12. Rao, C. Kameswara, and S. Mirza, "Torsional vibrations and buckling of thin-walled beams on elastic foundation." Thin-
walled structures 7.1 (1989): 73-82.
13. C. Kameswara Rao and Appala Satyam, "Torsional Vibrations and Stability of Thin-walled Beams on Continuous Elastic
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.
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
18/20
48 A. Sai Kumar & K. Srinivasa Rao
Impact Factor (JCC): 5.9234 NAAS Rating: 3.01
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.
21. J.R. Banerjee, “Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping”, Computers &
Structures 59(4) (1996) 613-621.
22. Jun, Li, et al. "Coupled bending and torsional vibration of axially loaded Bernoulli–Euler beams including warping effects."
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.
27. Hassan, Mohamed Taha, and Mohamed Nassar., "Static and Dynamic Behavior of Tapered Beams on Two-Parameter
Foundation." Vol. 14.(2013): 176-182.
28.
Ferreira, António JM. “MATLAB codes for finite element analysis”, solids and structures. Vol. 157. Springer Science &
Business Media, 2008.
29.
http://in.mathworks.com/matlabcentral/fileexchange/48107-roots-for-non-linear-highly-transcendental-equation
30.
MATLAB 7.14 (R2012a)
31. Kameswara Rao, C. and Appa Rao, K., “Effect of longitudinal Inertia and Shear Deformation on the torsional Frequency and
Normal Modes of thin-walled open section beams”, journal of aeronautical society of India,1974., pp. 32-41.
32. Maurizi, M. J., R. E. Rossi, and J. A. Reyes., "Comments on “a note of generally restrained beams”, Journal of sound and
vibration 147.1 (1991): 167-171.
APPENDICES
ANNEX A
MATLAB CODE FOR SOLVING TORSIONAL FREQUENCY EQUATIONS
Main programfunction 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 method is exercised %
% tol: is the solution tolerance %
% n: is the maximum number of iterations of the algorithm. %
% FCT(TOL): deceleration of the function whose solution has to be found. %
% %
% OUTPUT: % % value: is the approximate solution %
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
19/20
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 49
Winkler-Pasternak Foundation Using Dynamic Matrix Method
www.tjprc.org [email protected]
% %
% USAGE: %
% [value] = code(...) to display a solution of the function %
% %
%-----------------------------BY: Kumar Sai--------------------------------
if nargin < 2error('Incorrect input!!! provide at least two input arguments')
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
8/20/2019 4. IJCSEIERD - TORSIONAL VIBRATIONS OF DOUBLY.pdf
20/20
50 A. Sai Kumar & K. Srinivasa Rao
Impact Factor (JCC): 5.9234 NAAS Rating: 3.01
fprintf('------------------ %-10.10f-------------------\n',t*10^3);
fprintf('------The value of the Frequency parameter is-----\r\n');
fprintf('.................>>> %5.7f