The document presents an analytical method called the dynamic stiffness matrix approach to analyze the torsional vibrations and buckling of thin-walled beams of open section that are resting on an elastic foundation. The method is used to study a thin-walled beam that is clamped at one end and simply supported at the other. Numerical results for the natural frequencies and buckling loads are obtained for different values of warping and elastic foundation parameters.
An Asymptotic Approach of The Crack Extension In Linear PiezoelectricityIRJESJOURNAL
Abstract: As a result of a theoretical technique for elucidating the fracture mechanics of piezoelectric materials, this paper provides, on the basis of the three-dimensional model of thin plates, an asymptotic behavior in the Griffith’s criterion for a weakly anisotropic thin plate with symmetry of order six, through a mathematical analysis of perturbations due to the presence of a crack. It is particularly established, in this work, the effects of both electric field and singularity of the in-plane mechanical displacement on the piezoelectric energy
Structural engineering iii- Dr. Iftekhar Anam
Joint Displacements and Forces,Assembly of Stiffness Matrix and Load Vector of a Truss,Stiffness Matrix for 2-Dimensional Frame Members in the Local Axes System,Transformation of Stiffness Matrix from Local to Global Axes,Stiffness Method for 2-D Frame neglecting Axial Deformations,Problems on Stiffness Method for Beams/Frames,Assembly of Stiffness Matrix and Load Vector of a Three-Dimensional Truss,Calculation of Degree of Kinematic Indeterminacy (Doki)
Determine the doki (i.e., size of the stiffness matrix) for the structures shown below,Material Nonlinearity and Plastic Moment,
http://www.uap-bd.edu/ce/anam/
International Journal of Engineering Research and Development (IJERD)IJERD Editor
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An Asymptotic Approach of The Crack Extension In Linear PiezoelectricityIRJESJOURNAL
Abstract: As a result of a theoretical technique for elucidating the fracture mechanics of piezoelectric materials, this paper provides, on the basis of the three-dimensional model of thin plates, an asymptotic behavior in the Griffith’s criterion for a weakly anisotropic thin plate with symmetry of order six, through a mathematical analysis of perturbations due to the presence of a crack. It is particularly established, in this work, the effects of both electric field and singularity of the in-plane mechanical displacement on the piezoelectric energy
Structural engineering iii- Dr. Iftekhar Anam
Joint Displacements and Forces,Assembly of Stiffness Matrix and Load Vector of a Truss,Stiffness Matrix for 2-Dimensional Frame Members in the Local Axes System,Transformation of Stiffness Matrix from Local to Global Axes,Stiffness Method for 2-D Frame neglecting Axial Deformations,Problems on Stiffness Method for Beams/Frames,Assembly of Stiffness Matrix and Load Vector of a Three-Dimensional Truss,Calculation of Degree of Kinematic Indeterminacy (Doki)
Determine the doki (i.e., size of the stiffness matrix) for the structures shown below,Material Nonlinearity and Plastic Moment,
http://www.uap-bd.edu/ce/anam/
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
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Additional Conservation Laws for Two-Velocity Hydrodynamics Equations with th...inventy
A series of the differential identities connecting velocities, pressure and body force in the twovelocity hydrodynamics equations with equilibrium of pressure phases in reversible hydrodynamic approximation is obtaned.
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein SpacesSEENET-MTP
Balkan Workshop BW2013
Beyond the Standard Models
25 – 29 April, 2013, Vrnjačka Banja, Serbia
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein Spaces
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
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Additional Conservation Laws for Two-Velocity Hydrodynamics Equations with th...inventy
A series of the differential identities connecting velocities, pressure and body force in the twovelocity hydrodynamics equations with equilibrium of pressure phases in reversible hydrodynamic approximation is obtaned.
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein SpacesSEENET-MTP
Balkan Workshop BW2013
Beyond the Standard Models
25 – 29 April, 2013, Vrnjačka Banja, Serbia
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein Spaces
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
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Abstract
Unreinforced masonry in fills has long been known to affect the strength and stiffness of frame. Under the action of a lateral load the
principal compressive diagonal acts as a strut or bracing and increases the initial lateral stiffness of the framed structure. However,
in the presence of openings in walls, which is more practical, the behavior of infill changes. The primary objective of this paper is to
study the variation of lateral stiffness, principal compressive diagonal strut width with modulus of masonry infill, with and without
openings. In the study a non-linear (multi-linear) analysis is performed, since it is more realistic. From the analysis performed using
ANSYS Version 10.0 it is seen that linear analysis over-estimates the lateral stiffness of the infill frame. Further, it is observed that the
width of compressive diagonal generally decreases with increase in the modulus of masonry. It is also seen that incase of two frames
with equal area of openings, the frame with larger width of opening exhibits slightly more initial lateral stiffness due to possibility of
formation of single diagonal strut.
Key words: Infill frame, principal compressive diagonal, initial lateral stiffness, equivalent strut width, multi-linear
analysis.
Effect of pitch and nominal diameter on load distribution and efficiency in m...eSAT Journals
Abstract Lead screws are the devices which are used for power transmission or to have linear motion. It is theoretically assumed that applied load is evenly distributed among the thread pair in contact. However, practically it is observed that load is not uniformly distributed among threads. The first thread carries the maximum load and later the load on each thread reduces. Numerous studies have been carried out for analytical calculation of the load distribution using spring stiffness method. But these studies are for screw and nut combination. Not much study has been done to find load distribution on threads of a lead screw. The maximum load acting on one thread is an important parameter in lead screw design. The load decides the fatigue life of the screw and nut. To have better life of threads, the load distribution should be uniform to have fewer loads on single thread. The load is also important to know the deflection of thread which affects the positional accuracy of the lead screw drives. This paper focuses on analyzing mathematically the various thread parameters which affects the load distribution in threads and the corresponding effect on efficiency. The spring model method proposed in [1], [4] has different constant coefficient which are depending on thread geometry and material. If there are n numbers of threads in contact, there will be (n-1) number of equations in (n-1) unknowns. These are linear difference equations and can be solved by matrix elimination method. The results obtained from analytical solution are validated with the FEM (Finite Elements Method) results. Keywords: Lead Screws, Load Distribution, Thread Parameters, Efficiency, Linear Drives
A review on study of composite materials in presence of crackseSAT Journals
Abstract
Composites materials are commonly used in automobiles, aircraft structures etc. due to their high specific strength and stiffness. Composites ability to retain functionality in the presence of damage is a crucial, safety and economic issue. The fatigue failure mechanisms have been widely studied. Matrix cracks, fiber break, dis-bonding and de-laminations are the main causes for progressive failure of composites under fatigue loads. In this paper, detailed review on composite in presence of cracks under different types of failure mechanisms etc have been discussed.
Keywords: Fatigue Failure Mechanisms, cracks in composite
Periodic material-based vibration isolation for satellitesIJERA Editor
The vibration environment of a satellite is very severe during launch. Isolating the satellitevibrations during
launch will significantly enhance reliability and lifespan, and reduce the weight of satellite structure and
manufacturing cost. Guided by the recent advances in solid-state physics research, a new type of satellite
vibration isolator is proposed by usingperiodic material that is hence called periodic isolator. The periodic
isolator possesses a unique dynamic property, i.e., frequency band gaps. External vibrations with frequencies
falling in the frequency band gaps of the periodic isolator are to be isolated. Using the elastodynamics and the
Bloch-Floquet theorem, the frequency band gaps of periodic isolators are determined. A parametric study is
conducted to provide guidelines for the design of periodic isolators. Based on these analytical results, a finite
element model of a micro-satellite with a set of designed periodic isolators is built to show the feasibility of
vibration isolation. The periodic isolator is found to be a multi-directional isolator that provides vibration
isolation in the three directions.
Periodic material-based vibration isolation for satellitesIJERA Editor
The vibration environment of a satellite is very severe during launch. Isolating the satellitevibrations during
launch will significantly enhance reliability and lifespan, and reduce the weight of satellite structure and
manufacturing cost. Guided by the recent advances in solid-state physics research, a new type of satellite
vibration isolator is proposed by usingperiodic material that is hence called periodic isolator. The periodic
isolator possesses a unique dynamic property, i.e., frequency band gaps. External vibrations with frequencies
falling in the frequency band gaps of the periodic isolator are to be isolated. Using the elastodynamics and the
Bloch-Floquet theorem, the frequency band gaps of periodic isolators are determined. A parametric study is
conducted to provide guidelines for the design of periodic isolators. Based on these analytical results, a finite
element model of a micro-satellite with a set of designed periodic isolators is built to show the feasibility of
vibration isolation. The periodic isolator is found to be a multi-directional isolator that provides vibration
isolation in the three directions.
Comparative study of results obtained by analysis of structures using ANSYS, ...IOSR Journals
The analysis of complex structures like frames, trusses and beams is carried out using the Finite
Element Method (FEM) in software products like ANSYS and STAAD. The aim of this paper is to compare the
deformation results of simple and complex structures obtained using these products. The same structures are
also analyzed by a MATLAB program to provide a common reference for comparison. STAAD is used by civil
engineers to analyze structures like beams and columns while ANSYS is generally used by mechanical engineers
for structural analysis of machines, automobile roll cage, etc. Since both products employ the same fundamental
principle of FEM, there should be no difference in their results. Results however, prove contradictory to this for
complex structures. Since FEM is an approximate method, accuracy of the solutions cannot be a basis for their
comparison and hence, none of the varying results can be termed as better or worse. Their comparison may,
however, point to conservative results, significant digits and magnitude of difference so as to enable the analyst
to select the software best suited for the particular application of his or her structure.
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Bound State Solution of the Klein–Gordon Equation for the Modified Screened C...BRNSS Publication Hub
We present solution of the Klein–Gordon equation for the modified screened Coulomb potential (Yukawa) plus inversely quadratic Yukawa potential through formula method. The conventional formula method which constitutes a simple formula for finding bound state solution of any quantum mechanical wave equation, which is simplified to the form; 2122233()()''()'()()0(1)(1)kksAsBscsssskssks−++ψ+ψ+ψ=−−. The bound state energy eigenvalues and its corresponding wave function obtained with its efficiency in spectroscopy.
Key words: Bound state, inversely quadratic Yukawa, Klein–Gordon, modified screened coulomb (Yukawa), quantum wave equation
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A general approach is presented to describing nonlinear classical Maxwell electrodynamics with conformal symmetry. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on conformal-invariant functionals of the field strengths. This allows a characterization of Lagrangian and non-Lagrangian theories. We obtain a general formula for possible Lagrangian densities in nonlinear conformal-invariant electrodynamics. This generalizes the standard Lagrangian of classical linear electrodynamics so as to preserve the conformal symmetry.
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Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
2. N.V. Srinivasulu, B. Suryanarayana & S. Jaikrishna 2
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 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
ECw ∂4φ/∂z4 – (GCs –ρIp/A)∂2φ/∂z2 + Ksφ+ ρIp)∂2φ/∂t2 (1)
In which E, the modulus of elasticity; Cw ,the warping constant; G, the shear modulus; Cs , the
torsion constant; Ip, the polar moment of Inertia; A, the area of cross section; ρ, the mass density of the
material of the beam; φ,the angle of twist; Z, the distance along the length of the beam and t, the time.
For the torsional vibrations, the angle of twist φ(z,t) can be expressed in the form
φ(z,t) = x(z)e ipt (2)
in which x(z) is the modal shape function corresponding to each beam torsion natural frequency p. The
expression for x(z) which satisfies equation(1) can be written as:
x(z) = A Cos αz + B Sinαz +C Coshβz +D sinβz (3)
in which
αL.βL = (1/√2) {+ (k2-∆2) + [(k2-∆2)2 +4(λ2-4γ2)]1/2}1/2 (4)
3. 3 Torsional Vibrations and Buckling of Thin-Walled Beams on
Elastic Foundation-Dynamic Stiffness Method
K2 = L2 GCs/ECw.∆2 =ρ Ip L2 /AECw (5)
And λ2 = ρIp L4 p2n/ ECw γ2=Ks L4/4ECw (6)
From equation (4), the following relation between αL and βL is obtained.
(βL)2 = (αL)2 + K2 - ∆2 (7)
Knowing α and β. The frequency parameter λ can be evaluated using the following equation:
λ2 = (αL)(βL) + 4γ2 (8)
The four arbitrary constants A, B, C, and D in equation (3) can be determined from the 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 φ with respect to z is denoted by θ(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,
θ(z) = dφ/dz, (9)
hM(z) = -ECw( d2φ/dz2) and
T(z) = -ECw ( d3φ/dz3) + (GCs –ρIp/A) dφ/dz (10)
Where ECw = If h2/2,
If = the flange moment of inertia and
h= the distance between the centre lines of flanges of a thin-walled I-beam.
Consider a uniform thin-walled I-beam element of length L as shown in fig.1(a).
By combining the equation (3) and (9), the end displacements φ(0) and θ(0) and end forces, hM(0)
and T(0) of the beam at z = 0, can be expressed as :
φ(0) 1 0 1 0 A
θ(0) = 0 α 0 β B
hM(0) ECwα2 0 -ECwβ2 0 C
T(0) 0 ECwαβ2 0 - ECwα2 β D
4. N.V. Srinivasulu, B. Suryanarayana & S. Jaikrishna 4
Equation (11) can be written in an abbreviated form as follows:
δ(0) = V(0)U (11)
in a similar manner , the end displacements , φ(L)and θ(L) and end forces hM(L) and T(L), of the beam
where z = l can be expressed as follows:
δ(L) = V(L)U where
{δ(L)}T ={φ(L), θ(L), hM(L), T(L)}
{U}T = {A,B,C,D}
and
c s C S
-αs αc βS βC
[V(L)] = ECwα2c ECwα2s -ECwβ2C -ECwβ2S
- ECwαβ2s ECwαβ2c - ECwα2βS - ECwα2βC
in which c = CosαL; s = SinαL; C = CoshβL; S= SinhβL.
By eliminating the integration constant vector U from equation (11) and (12), and designating 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 ϕi
HMi j21 j22 j23 j24 θi
Tj = j31 j32 j33 j34 ϕj
HMj j41 j42 j43 j44 θj
Symbolically it is written
{F} = [J] {U} (12)
where
F}T = {Ti, hMi, Tj, hMj}
U}T = {φi, θj, φj, θj}
5. 5 Torsional Vibrations and Buckling of Thin-Walled Beams on
Elastic Foundation-Dynamic Stiffness Method
In the above equations the matrix [J] is the ‘exact’ element dynamic stiffness matrix, which is also a
square symmetric matrix.
The elements of [J] are given by:
j11 = H[(α2 +β2}(αCs+βSc]
j12 = -H[(α2 -β2}(1-Cc)+2αβSs]
j13 = - H[(α2 +β2}(αs+βS)]
j14 = -H[(α2 +β2}(C-c)]
j22 = -(H/αβ)[(α2 +β2}(αSc-βCs)] (13)
2 2
j24 = (H/αβ)[(α +β }(αS-βs)]
j23 = -j14
j33 = j11
j34 = -j12
j44 = j22
and H= ECw /[2αβ[ (1-Cc)+( β2-α2) Ss]
using the element dynamic stiffness matrix defined by equation (12) and (13). One can easily set up 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 λ. 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 λ for simply supported and built in
thin-walled beams are obtained in this chapter using an error factor ε= 10-6.
RESULTS AND DISCUSSIONS
The approach developed in the present work can be applied to the calculation of natural 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
6. N.V. Srinivasulu, B. Suryanarayana & S. Jaikrishna 6
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:
φ(0) = 0; θ(0) = 0 (15)
φ(l) = 0; M(l) = 0 (16)
Considering a one element solution and applying the boundary conditions defined by equation (15)
and (16) gives,
j22=0 (17)
This gives,
j22 = -(H/αβ)[(α2 +β2}(αSc-βCs)] = 0 (18)
as H and (α2 +β2) are ,in general, non-zero, the frequency equation for the clamped, simply supported
beam can , therefore, be written as
αtanhβL = βtanαL (19)
Equation (19)is solved for values of warping parameter k=1 and k=10 and for various values of
foundation parameter γ 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 λ(N=1). Further more,
the limiting load where λ becomes zero is the buckling load of the beam for a specified value of warping
parameter, K and foundation parameter, γ one can easily read the values of buckling load parameter ∆cr
from these figures for λ=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.
10. N.V. Srinivasulu, B. Suryanarayana & S. Jaikrishna 10
16
14
12
10
8 1st mode
2nd mode
6
3rd mode
4 4th mode
5th mode
2
0
1 2 3 4 5 6 7
Figure 7 : Sixth mode for Wk=1.0
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.
11. 11 Torsional Vibrations and Buckling of Thin-Walled Beams on
Elastic Foundation-Dynamic Stiffness Method
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.
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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.