Essay: Computer program for static and free vibration of a beam

Beam is a part of a structure placed horizontally which is capable of taking the load by resisting bending which are used in multiple industrial application, automobile application, architectural application, for supporting the loads coming on to them and also be reliable in a long run. So it is very much essential to know material property of beam as well as response of beam in various cases. In this thesis we studied some of the response of beam by using finite element method (FEM) and MATLAB. In the present scenario of digital computing it is becoming common to model a certain system to get a deep insight of the existing conditions. In this thesis the Euler-Bernoulli beam theory is considered to study linear static as well as the free vibrational charecteristic behavior of a beam. Based on the assumption of the theory a finite element computer program is developed in MATLAB for static analysis as well as predicting the mode shapes and the natural frequency using the approximation technique to get the required results. The Galerkin’s method also known as the weighted residual method is adopted for the calculation of approximate solutions which uses the test functions for the solution of the differential equation and we conquer the nearest possible solution of our problem. The thesis is mainly concerned in validating the numerical results obtained from the program. Hence the result is compared with exact solution of the analysis procedure and with a general finite element program called ABAQUS for a specific problem. The experimental were also carried out during the course for the calculation of the peak natural frequency and the results obtained are also compared with the present program.
Every physical structure needs to be studied in this world of superior engineering whenever it is subjected to any kind of force. The force could be in any form depending upon the nature of the force and its resulting characteristics. The force could arise due to any of the physical as well as any other structural changes in a system, it can be force due to some obstruction or some other surrounding phenomena which would be varied throughout the wide spectrum. As a result these certain forces are characterized as static and dynamic depending upon their nature and need to be analyzed to predict the structures behavior and response using certain mathematical equations and laws. The main objective of structural analysis is to determine internal forces, stresses and deformations of structures under various load effects.
Structural design is the technique used to perform the task within the specified limits of structures performances and its effects on the life of the structure. Therefore it should be designed accordingly using the given parameters and the designer having a sound knowledge of the happenings that may occur during the course of assemblage as well as during its life. The better the structure design the best will be the results or say the structure will behave according to the needs of the user. For this thorough study of the structural part of the system should be done as per the various forces statics and dynamics and the material behavior under various loading conditions so that the structure behavesaccordingly and serves for the purpose without much destruction and should be safe and economical.
Civil engineering structures are always designed to carry their own dead weight, superimposed loads and environmental loads such as wind or waves. These loads are usually treated as maximum loads not varying with time and hence as static loads. In some cases, the applied load involves not only static components but also contains a component varying with time which is a dynamic load. In the past, the effects of dynamic loading have often been evaluated by use of an equivalent static load, or by an impact factor, or by a modification of the factor of safety. Many developments have been carried out in order to try to quantify the effects produced by dynamic loading. Examples of structures where it is particularly important to consider dynamic loading effects are the construction of tall buildings, long bridges under wind-loading conditions and buildings in earthquake zones, etc.
Typical situations where it is necessary to consider more precisely the response produced by dynamic loading are vibrations due to equipment or machinery, impact load produced by traffic, snatch loading of cranes, impulsive load produced by blasts, earthquakes or explosions. So it is very important to study the vibrations in a structures. Vibration characteristics of a damaged and undamaged body are, as a rule, different. This difference is caused by a change in stiffness and can be used for the detection of damage and for the determination of its parameters.
The behavior of a structure is of very much importance for which it is necessary to accurately anticipate the characteristics of a structure. Hence the Finite Element Method (FEM) has been extensively used in the structural analysis. The Finite Element modelmay provide exact characteristics of a structure if the wavelength is large as compared to the mesh size. However the finite element solutions become increasingly inaccurate as the frequency increases. As a result we can improve the accuracy by refining the mesh which sometimes proves to be expensive and time taking.
1.2 Objective of the study :
ï,· The current investigation aims to develop a computer program for static as well as for the free vibration of a beam using finite element method.
ï,· To determine the static characteristics i.e slope, deflection for the simply supported beam.
ï,· To determine the free vibration characteristics i.e natural frequency and mode shapes for a cantilever beam and reaching the accurate results by FEM.
ï,· To validate the finite element program of beam with exact solutions of the Euler-Bernoulli beam theory and ABAQUS solutions.
ï,· To experimentally find the peak response of a cantilever beam by impact hammer test using accelerometer and FFT analyser.
1.3 Scope of Work
The beam was analysed as a single element but we can use the FEM for the purpose of more accurate results by dividing the single elements into number of elements i.e by providing the mesh and then by refining the mesh for getting more accurate results. The present work involves the study of FEM and using the same for the analysis of beam so as to achieve greater accuracy in the results. The experimental results have also been
found in accordance with the above procedures of design and the same has been validated with the developed program. The various elements are modelled by varying their lengths and also by varying their width and the peak natural frequency are computed using the experimental procedures.
S H Shin [4] Vibration analysis of a rotating cantilever beam is an important and peculiar subject of study in engineering. There are many engineering examples which can be idealized as rotating cantilever beams such as turbine blades or turbo engine blades and helicopter blades. For the proper design of the structures their vibration characteristics which are natural frequencies and mode shapes should be well identified. Compared to the vibration characteristics of non rotating structures those of rotating structures often vary significantly. The variation results from the stretching induced by the centrifugal inertia force due to the rotational motion. The stretching causes the increment of the bending stiffness of the structure which naturally results in the variation of natural frequencies and mode shapes. The equations of motion of a rotating cantilever beam are derived based on a new dynamic modeling method. With the coupling effect ignored the analysis results are consistent with the results obtained by the conventional modelling method. A modal formulation method is also introduced in this study to calculate the tuned angular speed of a rotating beam at which resonance occurs.
 Mousa Rezaee [14] derived a new analytical method for vibration analysis of a cracked simply supported beam is investigated. By considering a non linear model for the fatigue crack, the governing equation of motion of the cracked beam is solved usingperturbation method. The solution of the governing equation reveals the super harmonics of the fundamental frequency due to the nonlinear effects in the dynamic response of the cracked beam. Furthermore, considering such a solution, an explicit expression is also derived for the system damping changes due to the changes in the crack parameters, geometric dimensions and mechanical properties of the cracked beam. The results show that an increase in the crack severity and approaching the crack location to the middle of the beam increase the system damping. In order to validate the results, changes in the fundamental frequency ratios against the fatigue crack severities are compared with those of experimental results available in the literature. Also, a comparison is made between the free response of the cracked beam with a given crack depth and location obtained by the proposed analytical solution and that of the numerical method. The results of the proposed method agree with the experimental and numerical results.
 Chih Ling Huang [7] Rotating beams are often used as a simple model for propellers, turbine blades, and satellite booms. The free vibration frequencies of rotating beams have been extensively studied. Rotating beam differs from a non-rotating beam in having additional centrifugal force. The natural frequency of the flap wise bending vibration, and coupled lagwise bending and axial vibrations investigated for the rotating beam. A method based on the power series solution is proposed to solve the natural frequency of very slender rotating beam at high angular velocity. The rotating beam is subdivided into several equal segments. The governing equations of each segment are solved by a power series. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method. The effect of angular velocity, and slendernessratio on the natural frequency of rotating beams is investigated. The Free vibration of the beam is measured from the position of the steady state axial deformation.
 H.Ding,G.C [16] The axially moving beams has several applications, including robot arms, conveyor belts, high-speed magnetic tapes, and automobile engine belt. Understanding the vibrations of axially moving beams are important for the design of the devices. Recent developments in research on axially moving structures have been reviewed. Natural frequencies of nonlinear coupled planar vibration are investigated for axially moving beams in the supercritical transport speed ranges. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. The finite difference scheme is developed to calculate the non-trivial static equilibrium. The equations are cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. Under fixed boundary conditions, time series are calculated via the finite difference method. Based on the time series, the natural frequencies of nonlinear planar vibration, which are determined via discrete Fourier transform (DFT), are compared with the results of the Galerkin method for the corresponding governing equations without nonlinear parts. The effects of material parameters and vibration amplitude on the natural frequencies are investigated through parametric studies.
 Yang Xiang [13] Free vibration and elastic buckling of beams made of functionally graded materials (FGMs) containing open edge cracks are studied in this paper based on Timoshenko beam theory. The crack is modeled by a massless elastic rotational spring. It is assumed that the material properties follow exponential distributions along beam thickness direction. Analytical solutions of natural frequencies and critical buckling loadare obtained for cracked FGM beams with clamped-free, hinged-hinged, and clamped-clamped end supports. A detailed parametric study is conducted to study the influences of crack depth, crack location, total number of cracks, material properties, beam slenderness ratio, and end supports on the free vibration and buckling characteristics of cracked FGM beams.
 M.Shavezipur [10] A set of differential equations governing triply coupled vibrations of centrifugally stiffened beams, a refined dynamic finite element (RDFE) method is developed. The application of the proposed method is demonstrated to obtain numerical results for several examples. Some of these results are compared with those obtained from classical FEM and published results. As it was confirmed by numerical results, the RDFE method is a reliable solution method with drastically higher convergence rates compared to other numerical methods. The RDFE can be advantageously used when multiple natural frequencies and/or higher modes of the beam structures are of interest. It is important to note that the method is not limited to the equations introduced in this paper and can be extended to more advanced models which may include more geometric and material coupling terms. Many aerospace and terrestrial structures, such as aircraft wings, propeller blades, solar panels and satellite antenna, compressor, turbine and helicopter rotor blades, space structures, bridges, etc. can be modelled as a combination of beam elements with two or three coupled governing differential equations.
 Michael I Frisswell [19] A Method is proposed for the replacement of unknown stiffness with rigid connections in two systems of equation from a finite element model and from measured response functions. The frequency response can be determined from
standard modal tests and no special forcing arrangements or physical constraints are needed. The only use of constraints in mathematics where the physical behaviour of constrained system is inferred from the unconstrained measurements and predictions are obtained from constrained finite element equations since stiffness which are replaced by rigid connections can’t experience any elastic strain they can have no effect on the inferred measurements from an elastic structure. The method can be used to determine erroneous connections in a finite element model or to locate discrete nonlinearities. Simulated examples are used to illustrate the application of the technique.
 Hamid Zabihi [15] This paper presents an analytical investigation of the free vibrations of a cracked Timoshenko beam made up of functionally graded materials (FGMs). It is assumed that the beam is constructed of FGM materials with a power law variation of metal-ceramic volume fraction. The perspective of wave method is adopted for the analysis. The method considers the nature of the propagation and reflection of the waves along the beam. Consequently, the propagation, transmission and reflection matrices for various discontinuities located on the beam are derived. Such discontinuities may include crack, boundaries or change in section. By combining these matrices a global frequency matrix is formed. In order to investigate the effect of the beam’s structural synthesis, different natural frequencies are obtained and studied.
 R. Lassoued [8] An accurate procedure to determine free vibrations of beams and plates is presented. The natural frequencies are exact solutions of governing vibration equations witch load to a nonlinear homogeny system. The bilinear and linear structures considered simulate a bridge. The dynamic behavior of this one is analyzed by using the theory of the orthotropic plate simply supported on two sides and free on the two others.The plate can be excited by a convoy of constant or harmonic loads. The determination of the dynamic response of the structures considered requires knowledge of the free frequencies and the shape modes of vibrations. The formulation is based on the determination of the solution of the differential equations of vibrations. The boundary conditions corresponding to the shape modes permit to lead to a homogeneous system.
 Metin O Kaya [9] There has been a growing interest in the analysis of the free vibration characteristics of elastic structures that rotate with constant angular velocity. Numerous structural configurations such as turbine, compressor and helicopter blades, spinning spacecraft and satellite booms fall into this category. A simple equation (known as the Southwell equation), which is based on the Rayleigh energy theorem to estimate the natural frequencies of rotating cantilever beams. Earlier studies mainly focused on Euler Bernoulli beams. However, due to the inclusion of shear deformation and rotary inertia effects, Timoshenko beam theory is more accurate than Euler Bernoulli beam theory. Therefore, considerable research has been carried out on the free vibrations of rotating Timoshenko beams, recently. Recently, the Dynamic Stiffness Method for a rotating cantilever Timoshenko beam that is based on Fresenius series expansion and claims its superiority of finding more correct results. On the other hand, the advantage of the DTM is its simplicity and high accuracy.
 M.Shahidi [17] In this paper, the nonlinear governing equation of tapered beams, attempt has been made to analyze the nonlinear behavior of tapered beams analytically. The nonlinear governing equation is solved by employing the variational approach method (VAM) and Improved Amplitude-Formulation (IAFF). Despite the increasing expenses of building structures to maintain their linear behavior, nonlinearity has been
inevitable and therefore, nonlinear analysis has been of great importance to the scientists in the field. The major concern is to assess excellent approximations to the exact solutions for the whole range of the oscillation amplitude, reducing the respective error of angular frequency in comparison with the VAM and IAFF. The effect of vibration amplitude on the nonlinear frequency is discussed. It is predicted that there can be wide application of VAM and IAFF in engineering problems, as indicated in this paper.
 Sabah Mohammed Jamel Ali [21] A numerical solution to the frequency equation for the transverse vibration of a beam (Simply Supported with symmetric overhang) is done. It is proposed two limiting cases of a beam with no overhang, and no span. This agrees with the cases in which the supports are at the nodal Points of a freely vibrating beam. Also the numerical results compared with the analytical solutions for this study are coincident. An approximation to the solution of the frequency equation for beams with small overhang is presented and compared with the numerical solution. This approximation is quite useful to determine a beam’s flexural stiffness (EI), or modulus of elasticity (E), by free vibrating of a simply supported beam.
 W.L. LI [22] A simple and unified approach is presented for the vibration analysis of a generally supported beam. The flexural displacement of the beam is sought as the linear combination of a Fourier series and an auxiliary polynomial function. The polynomial function is introduced to take all the relevant discontinuities with the original displacement and its derivatives at the boundaries and the Fourier series now simply represents a residual or conditioned displacement that has at least three continuous derivatives. As a result, not only is it always possible to expand the displacement in a Fourier series for beams with any boundary conditions, but also the solution converges ata much faster speed. The reliability and robustness of the proposed technique are demonstrated through numerical examples.
 Gurgoze [5] The frequency response function is obtained through a formula, which was established for the reacceptance matrix of discrete systems subjected to linear constraint equations. The comparison of the numerical results obtained with those via a boundary value problem formulation justifies the approach used here. Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. In simplest terms, if a sine wave is injected into a system at a given frequency, a linear system will respond at that same frequency with a certain magnitude and a certain phase angle relative to the input. Also for a linear system, doubling the amplitude of the input will double the amplitude of the output.
 JinsuoNie [18] This paper is aimed at determining how material dependent damping can be specified conveniently in ANSYS in a mode superposition transient dynamic analysis. A simple cantilever beam is analyzed using various damping options in ANSYS. The mode superposition method is often used for dynamic analysis of complex structures, such as the seismic Category I structures in nuclear power plants, in place of the less efficient full method, which uses the full system matrices for calculation of the transient responses. In such applications, specification of material-dependent damping is usually desirable because complex structures can consist of multiple types of materials that may have different energy dissipation capabilities. A recent review of the ANSYS manual for several releases found that the use of material-dependent damping is notclearly explained for performing a mode superposition transient dynamic analysis. This paper includes several mode superposition transient dynamic analyses using different ways to specify damping in ANSYS, in order to determine how material-dependent damping can be specified conveniently in a mode superposition transient dynamic analysis.
 Shibabrat Naik [20] Studied of substantial importance in complaint structures, now days ,are the dynamic parameters such as the modal frequencies and damping constant of their components. These parameters are the essential technical information required in engineering analysis and design. In addition this information is needed for numerical simulations and finite element modeling to predict the response of structures to a variety of dynamic loadings. In this work, experimental modal testing of a cantilever beam has been performed to obtain the mode shapes, modal frequencies and the damping parameters. A fast fourier transform analyzer, PULSE lab shop was used to obtain the frequency response functions and subsequent extraction of modal data was performed using ME’s scope. These modal parameters were then checked using finite element analysis software, ANSYS which were found to comply with the experimental results. The range of applications for modal data is vast and includes checking modal frequencies, forming qualitative descriptions of the mode shapes as an aid to understanding dynamic structural behaviour for trouble shooting, verifying and improving analytical models. It is with this objective that the experimental method was standardized and thus the mathematical model can be updated further. The experimental methods include obtaining the FRF plots from a cantilever beam and then using the ME’s scope to obtain the various parts of the FRF plots like the magnitude, phase, real,
imaginary. Then the modal data was analyzed to obtain different parameters which were further compared with the model developed in ANSYS.
 D.Ravi Prasad [11] Modal analysis is a process of describing a structure in terms of its natural characteristics which are the frequency, damping and mode shapes —its dynamic properties. The change of modal characteristics directly provides an indication of structural condition based on changes in frequencies and mode shapes of vibration. This paper presents results of an experimental modal analysis of beams with different materials such as steel, brass, copper and aluminum. The beams were excited using an impact hammer excitation technique over the frequency range of interest, 0-2000 Hz. Response functions were obtained using vibration analyzer. The FRFs were processed using NV solutions modal analysis package to identify natural frequencies, damping and the corresponding mode shapes of the beam.
 Y.V Mohan Reddy [23] The paper mainly deals with the polynomial regression method which focuses on dynamic reanalysis of a simple beam structure. The method can be used to generate finite element system that deals with the calculations of stiffness and mass matrices of the structures. The method is applied to a simple beam and a T-Beam and the results were found to indicate high quality approximations of the frequencies. The final results of the regression and the finite element method were compared. Basically reanalysis methods are intended for the purpose of analysing the new designs using the information acquired from the older ones.2.2 Critique
From above literature review we have seen that so many works has been done on the analysis of the beams and structural members but not much work has been done on development of finite element program for static and free vibration analysis of various structural members. As a result of these we have seen that a program can be developed based on finite element analysis using the weighted residual method i.e the Galerkin’s Method which uses the test function for the solution of differential equations obtained during the analysis of simple as well as complex structure and the same method can be applied to model the beam using various beam conditions and the various characteristics such as slope and deflection and free vibration characteristics of a beam can be computed.
A beam is a horizontal or vertical structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending movement. Beams are traditionally descriptions of building or civil engineering structural elements, but smaller structures such as truck or automobile frames, machine frames, and other mechanical or structural systems contain beam structures that are designed and analyzed in a similar fashion. A beam is a three-dimensional object. Even the three-dimensional theory of elasticity is not perfect and further assumptions to simplify the theory must erode the extent to which it is applicable.
3.2 Types of Beams
Beams are characterized by their profile (the shape of their cross-section), their length, and their material. In contemporary constructions, beams are typically made of steel, reinforced concrete or wood. One of the most common types of steel beam is the I-beam or wide – flange beam (also known as a “universal beam” or, for stouter sections, a “universal column”). This is commonly used in steel-frame buildings and bridges. Other common beam profiles are the C- channels, the hollow structural section beam, the pipe,and the angle. Beams are also described by how they are supported. Supports restrict lateral or rotational movements so as to satisfy stability conditions as well as to limit the deformations to a certain allowance. A simple beam is supported by a pin support at one end and a roller support at the other end. A beam with a laterally and rotationally fixed support at one end with no support at the other end is called a cantilever beam. A beam simply supported at two points and having one end or both ends extended beyond the supports is called an overhanging beam. Fig.1 showing the different types of beams.An exact formulation of the beam problem was first investigated in terms of general elasticity equations by Pochhammer (1876) and Chree (1889) . They derived the equations that describe a vibrating solid cylinder. However, it is not practical to solve the full problem because it yields more information than usually needed in applications. Therefore, approximate solutions for transverse displacement are sufficient. The beamtheories under consideration all yield the transverse displacement as a solution. It was recognized by the early researchers that the bending effect is the single most important factor in a transversely vibrating beam. The Euler Bernoulli model includes the strain energy due to the bending and the kinetic energy due to the lateral displacement. The Euler Bernoulli model dates back to the 18th century. Jacob Bernoulli (1654-1705) first discovered that the curvature of an elastic beam at any point is proportional to the bending moment at that point. Daniel Bernoulli (1700-1782), nephew of Jacob, was the first one who formulated the differential equation of motion of a vibrating beam. Later, Jacob Bernoulli’s theory was accepted by Leonhard Euler (1707-1783) in his investigation of the shape of elastic beams under various loading conditions. Many advances on the elastic curves were made by Euler . The Euler-Bernoulli beam theory, sometimes called the classical beam theory, Euler beam theory, Bernoulli beam theory, or Bernoulli and Euler beam theory, is the most commonly used because it is simple and provides reasonable engineering approximations for many problems. However, the Euler Bernoulli model tends to slightly overestimate the natural frequencies. This problem is exacerbated for the natural frequencies of the higher modes. Also, the prediction is better for slender beams than non-slender beams.
Timoshenko (1921, 1922) proposed a beam theory which adds the effect of shear as well as the effect of rotation to the Euler-Bernoulli beam. The Timoshenko model is a major improvement for non-slender beams and for high-frequency responses where shear or rotary effects are not negligible. Following Timoshenko, several authors have obtained the frequency equations and the mode shapes for various boundary conditions. Some are Kruszewski (1949), Traill-Nash and Collar (1953), Dolph (1954), and Huang (1961).
The finite element method originated from the need of solving complex elasticity and structural analysis problem in civil and aeronautical engineering. Its development could be traced back to the work by Alexander Hrennikoff (1941) and Richard Courant (1942). While the approach used by these pioneers are different, they all share one essential characteristic: mesh discretization of a continuous domain into a set of discrete subdomains, usually called elements. Starting in 1947, Olgierd Zienkiewicz from Imperial College gathered those methods together into what is called the Finite Element Method, building the pioneering mathematical formalism of the method.
Hrennikofs work discretizes the domain by using a lattice analogy, while Courant’s approach divides the domain into finite triangular subregions to solve second order elliptic partial differential equations (PDEs) that arise from the problem of torsion of a cylinder. Courant’s contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by Rayleigh, Ritz, and Galerkin. Development of the finite element method began in the middle to late 1950s for airframe and structural analysis and gathered momentum at the University of Stuttgartthrough the work of John Argyris and at Berkeley through the work of Ray W. Clough in the 1960’s for use in civil engineering. By late 1950s, the key concepts of stiffness matrix and element assembly existed essentially in the form used today. NASA issued a request for proposals for the development of the finite element software NASTRAN in 1965. The method was again provided with a rigorous mathematical foundation in 1973 with the publication of Strang and Fix An Analysis of The Finite Element Method, and has since been generalized into a branch of applied mathematics for numerical modelling of physical systems in a wide variety of engineering disciplines, e.g., electromagnetism and fluid dynamics.The Timoshenko beam theory was developed by Ukrainian-born scientist Stephen Timoshenko in the beginning of the 20th century. The model takes into account shear deformation and rotational inertia effects, making it suitable for describing the behaviour of short beams, sandwich composite beams or beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of 4th order, but unlike ordinary beam theory – i.e. Bernoulli-Euler theory – there is also a second order spatial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted frequencies for a given set of boundary conditions. The latter effect is more noticeable frequencies as the wavelength becomes shorter, and thus the distance between opposing shear forces decreases.
If the shear modulus of the beam material approaches infinity and thus the beam becomes rigid in shear – and if rotational inertia effects are neglected, Timoshenko beam theory converges towards ordinary beam theory.
The normal rotates by an amount  x  (x) which is not equal to
x
ï,¶ï·
ï,¶
In static Timoshenko beam theory without axial effects, the displacements of the beam
are assumed to be given by
( , , ) ( ) x u x y z  z x …(1)
( , , ) 0 x u x y z  …(2)
( , ) ( ) x u x y  x
…(3)
where (x,y,z) are the coordinates of a point in the beam, u(x),u(y) ,u(z) are displacement
vector in the three coordinate directions, Ψ is the angle of rotation of the normal to the
mid-surface of the beam, and ω is the displacement of the mid surface in the z-direction.
The governing equations are the following uncoupled system of ordinary differential
equations:
2
2 ( , )
d d
EI q x t
dx dx
  
  
  …(4)
d 1 d d
EI
dx KAG dx dx
 

 
   
  …(5)
The Timoshenko beam theory for the static case is equivalent to the Euler Bernoulli
theory when the last term above is neglected, an approximation that is valid when
where L is the length of the beam.
Combining the two equations gives, for a homogeneous beam of constant cross-section,
4 2
4 2 ( )
d x EI d q
EI q x
dx KAG dx
 
…(7)
In Timoshenko beam theory without axial effects, the displacements of the beam are
assumed to be given by:-
( , , , ) ( , ); ( , , , ) 0; ( , , , ) ( , ) x y z u x y z t  z x t u x y z t  u x y z t  x t
…(8)
Starting from the above assumption, the Timoshenko beam theory, allowing for
vibrations, may be described with the coupled linear partial differential equations.
2
2 A q(x, t) KAG( )
t x x
 
 
ï,¶ ï,¶  ï,¶ 
     ï,¶ ï,¶  ï,¶ 
…(9)
2
2 I (EI ) KAG( )
t x x x
  
 
ï,¶ ï,¶ ï,¶ ï,¶
  
ï,¶ ï,¶ ï,¶ ï,¶ …(10)
where the dependent variables are ω(x, t) , the translational displacement of the beam,
and Ψ(x,t) , the angular displacement. Note that unlike the Euler Bernoulli theory, the
angular deflection is another variable and not approximated by the slope of the deflection.
Also,
ρ is the density of the beam material (but not the linear density).
A is the cross section area.
E is the elastic modulus
G is the shear modulus.
I is the second moment of area.
K , called the Timoshenko shear coefficient, depends on the geometry. Normally,
K =5/6 for a rectangular section.
q(x,t) is a distributed load (force per length).
These parameters are not necessarily constants.
For a linear elastic, isotropic, homogeneous beam of constant cross-section these two
equations can be combined to give
4 2 4 4 2 2
4 2 2 2 4 2 2
Im
( ) ( , )
E Jm I q EI q
EI m I q x t
x t KAG x t KAG t KAG t KAG x
    

ï,¶ ï,¶ ï,¶ ï,¶ ï,¶ ï,¶
      
ï,¶ ï,¶ ï,¶ ï,¶ ï,¶ ï,¶ ï,¶ …(11)
Further the above equation can be solved by applying the boundary conditions with the
help of numerical methods.
3.3.2 Euler Bernoulli Beam Theory
Euler Bernoulli’s Beam Theory also known as engineer’s beam theory or classical beam
theory is a simplification of the linear theory of elasticity which provides a means of
calculating the load carrying and deflection characteristics of beams . It covers the case
for small deflections of a beam which is subjected to lateral loads only. It is thus a special
case of Timoshenko beam theory which accounts for shear deformation and is applicable
for thick beams. It was first enunciated circa 1750, but was not applied on a large scale
until the development of the Eiffel tower and the Ferris wheel in the late 19th century.
Following these successful demonstrations, it quickly became a cornerstone of
engineering and an enabler of the second industrial revolution. Additional analysis tools
have been developed such as plate theory and finite element analysis, but the simplicity
of beam theory makes it an important tool in the sciences, especially structural and
mechanical engineering.
The Euler-Bernoulli equation describes the relationship between the beam’s deflection
and the applied load