When a force is applied to a beam and removed vibrations may occur. If a beam bends it experiences compressive and tensile stresses and there exists a point between them where there is no bending stress. In the Euler-Bernoulli beam theory this neutral plane will always be perpendicular to the outer edge of the beam. The Timoshenko beam is more complex and the neutral plane in the beam is not always perpendicular to the edge of the beam under bending. Instead the neutral plane is deformed by an angle from its undeformed position. For each method we can calculate the eigenvalue of smallest amplitude and from these we can find the natural frequency of the beam. Knowing this, engineers are able to apply these results to real life applications such as bridges. By considering the structures natural frequencies and the external forces acting on them it allows engineers to design and construct larger safer structures.
Defining a Beam and its Forces
Before we try to understand how a beam vibrates it is essential for us to understand what exactly a beam is and how it behaves. A beam is defined as a structural element that carries loads primarily in bending. The term structural element is used to separate complex structures into their simple elements. Some examples of these elements are rods, ties, beams and shells and it is the beam element that we will be investigating in this paper. The load, or bending force, applied to the beam is a result from all the different forces that the beam experiences. The most common forces applied to beams are its own weight due to gravity and external loads that can be placed on or hung from the beam. These two forces mentioned are considered vertical gravitational forces but horizontal forces can also be applied to the beam such as wind or by an earthquake. It is through these forces being applied and removed that vibrations occur and the magnitude of these vibrations depends on the how and when these forces are applied.
Beams come in many different shapes and forms and in general they are characterised by their profile which considers the shape of their cross section, their length and their material. When beams are subject to bending they experience both compressive and tensile stresses depending on how the load is applied. If we consider only loads due to gravity when both ends of the beam are supported (so that the beam is bent downwards in the middle) the top of the beam will be under compression and the bottom of the beam will be under tension. There is a third internal force present called the shear force which is parallel to the lateral loading. The shear force can be described as a force, or a component of a force, that acts parallel to a plane (Popov, E., 1968). The compression and tension in the beam form a moment because they are equal in magnitude and opposite in direction. We call this moment the bending moment. The bending moment causes the beam to sag and deform when forces are applied. It should also be noted that the maximum compressive stress and tensile stress is found at the upper edge and lowest edge of the beam respectively. Since these two opposing maxima vary linearly there exists a point between them where there is no bending stress.
The most common shape of beam is the I-beam as it reduces mass without compromising strength and this means large structures can be constructed due to its low mass and high strength. It is commonly used in the construction of bridges and buildings but it is important to note that the I-beam is much stiffer vertically than it is horizontally and it is also easily twisted.
Frequency of a Vibrating System Using a Vertical Beam
If a force is applied then removed from a beam it is possible for the beam to vibrate and how it vibrates depends on the beams stiffness. If a load is applied to a beam it will produce a change in potential energy of the beam and in turn this has the potential of forming a vibrational system (McLachlan, N., W, 1951). To understand the frequency of a vibrating system we will first look at a vertical beam suspended from the roof as seen in Fig 1.1.
The factors above need to be considered when constructing large structures to avoid weakness in the building that could lead to disaster. One of the most famous bridge disasters was the Tacoma Narrows Bridge in the United States. One hypothesis for the bridges collapse is called the resonance hypothesis (Billah, K.; R. Scanlan, 1991) which is caused by wind passing over and through the bridge. In its most basic form we can think of the resonance as a child being pushed on a swing. The swing will start at rest until pushed by a force and if the child is pushed at the right time the resultant swing will have larger amplitude than the last. If this is repeated over and over again the swing will end up with a significant amplitude. In this example the person pushing the swing produces the driving force.
Resonance and the Tacoma Narrows Bridge
Resonance is defined as The increase in amplitude of oscillation of an electric or mechanical system exposed to a periodic force whose frequency is equal or very close to the natural undamped frequency of the system. (http://www.thefreedictionary.com/resonance)
The resonance experienced by the Tacoma Bridge was mechanical resonance and the driving force was created by the wind. To explain the resonance mathematically we use the linear differential equation,
Where and are the mass, damping coefficient and stiffness, respectively, of a linear system of displacement. denotes the radian frequency andis the amplitude of the driving force as a function of time. For the system above resonance occurs when is approximately equal to. When we call the natural frequency of the system. Equation (1.6) can be used to explain the resonance of the Tacoma Bridge. It is believed that the wind produced a fluctuating resultant force in resonance with the natural frequency of the structure. Over time this produced a steadily increasing amplitude until the bridge was destroyed, similar to the swing analogy stated earlier.
2, Beams as Eigenvalue Problems
When looking at beams we can think of them as eigenvalue problems. It is known that all eigenvalues and eigenvectors, ? and x respectively, satisfy the equation,
From this we can calculate corresponding eigenvalues and eigenvectors for different matrices . When an eigenvector is associated with an eigenvalue they are called an eigenpair. To make use of these eigenpairs in real life applications we generally are looking for the dominant eigenpair. To find the dominant eigenpair we must first find the dominant eigenvector which is defined as the eigenvector corresponding to the eigenvalue of largest magnitude of the matrix. One method to find the dominant eigenvector involves using the power method (Braun, M., 1983). Once the dominant eigenvalue is found it can now be applied to real life applications.
Applying Eigenvalues to the Tacoma Bridge Disaster
In the Tacoma Bridge example explained above the natural frequency of the bridge is described by the eigenvalue of smallest magnitude. To find this eigenvalue of lowest magnitude simply take the reciprocal of the dominant eigenvalue and this gives the eigenvalue of lowest magnitude. Now that we know the natural frequency of the structure it is possible to apply it to (2.1) and in doing so the resonance can be calculated and examined. This ensures that the structure is correctly constructed and that it will be stable in winds or other resonance causing circumstances.
If the dominant eigenvalue had been calculated for the Tacoma Bridge then the engineers would have been able to prevent the disaster from ever occurring. One way to help prevent resonance and structure vibration is to include some sort of dampening device into the structure. We can define dampening as the restraining of vibratory motion, such as mechanical oscillations, noise, and alternating electric currents, by dissipation of energy. (http://www.britannica.com/EBchecked/topic/150616/damping) During the construction of the Tacoma Bridge the engineers implemented several mechanical dampening methods to try and reduce the bridges oscillation. The first method involved attaching large cables to heavy weights at the base of the bridge. This would reduce the amplitude of the vibrations but unfortunately they snapped soon after installation meaning the bridge was allowed to oscillate freely again. The bridge was also fitted with hydraulic buffers on the towers of the bridge. This would have reduced the longitudinal motion of the bridge as it swayed but the seals of these buffers were damaged during construction rendering them near useless. All these factors contributed to the collapse of the Tacoma Bridge and unfortunately the understanding of the mathematics behind the resonance frequency was not well enough understood to save it from its collapse.
Cantilever, Mixed and Simple-supported Beams
We can consider most bridges as being a simply supported beam (Fig. 3.1). Two other common ways that beams can be supported are cantilever and mixed.
The boundary conditions vary depending on how the beam is supported. Consider the cantilever supported beam. Let define the length of the beam along the axis and let be where the beam touches the wall. At the fixed end of the beam there will be no deflection and there will not be any sloping present either. At the free end of the beam (in this case the right hand side of the beam) there will not be a bending moment and there is also no shear force present as it is at the utmost edge of the beam (Ballarini, R., 2003). From this it can be assumed that:
- there is no deflection where the beam is in contact with the wall.
- the beam touching the wall is horizontal.
- there is no bending moment at.
- there is no shearing force at.
Now that I have these assumptions I need to find an equation that describes the relationship between the deflection of the beam and the applied load. As I have 4 boundary conditions it is safe to assume that a fourth-order differential equation is needed to describe these relations.
Euler-Bernoulli Beam Theory
The most commonly used equation for calculating beam deflection in solid mechanics is the Euler-Bernoulli beam equation. It is a simplified version of the linear theory of elasticity which is also used for calculating deflections and load bearing characteristics of beams. The Euler-Bernoulli beam equation is a combination of four different subsets of beam theory. These being the kinematics, constitutive, force resultant and equilibrium theories and are explained below (Thomson, W. T., 1981).
To understand how the Euler-Bernoulli beam equation is obtained it is first important understand how these four different theories arise and how they are significant to the formation of the equation.
Kinematics describes the motion of the beam and its deflection. To understand the kinematics of the beam we first need to understand how the normals behave and how the beams cross section rotates. This is done by looking at Kirchoffs Assumptions (Labidi S., 2008).
- The normals remain straight
- The normals remain unstreched
- The normals remain to the normal (they are always perpendicular to the neutral plane)
It is seen from these assumptions that the normals will remain straight and unstreched. It is also assumed that there will be negligible strain occurring in the direction. The normals remain perpendicular to the neutral plane therefore the and dependence in is made explicit by using the geometric expression, (4.1) Where ? is the angle from the normal of the deformed beam. Now thatis dependant on the direct strain equation given by, (4.2) is used to find the strain through the beam, where e is the strain, (4.3) Now still assuming the normals are always perpendicular consider the cross section rotation and the neutral plane rotation and from these I calculate the beams displacement. This gives us the kinematics equation. (4.4)
The next equation needed to explain the Euler-Bernoulli equation is the constitutive equation. Constitutive equations relate two physical quantities and in the case of beam theory the relation between the direct stress s and direct strain e within a beam is being investigated. The direct stress is found from the equation, (4.5) The term direct implies that the force is perpendicular to the cross section of the beam. Beam theory can be explained using the 1-dimensional Hookes equation (4.6) Here is the Youngs modulus of the beam and it measures the stiffness of the material.
Next I consider the force resultants of the beam. These force resultants allow us to study the important stresses within the beam. They are useful to us as they are only considered functions of, whereas the stresses found in a beam are functions of and. Consider a beam cut at a point It is possible to find the direct stress and shear stress, and respectively, of that point. This is due to the direct stress at the cut producing a moment around the neutral plane. By summing all these individual moments over the whole of the cross-section it is possible to find the moment resultant M given by the equation, (4.7) Where z is the axis pointing out of the page. If I sum the shear stresses at the point the definition of the shear resultant V is found to be, (4.8) Inverting equations (4.7) and (4.8) gives the equation for the direct stress distribution along the beam due to bending, (4.9) Where I is the integral. To find the maximum bending stress in the beam consider the point that is furthest away from the neutral axis, in this case c. (4.11) Now that the maximum bending stress in the beam has been found I will now investigate the equilibrium aspect of the Euler-Bernoulli beam theory.
To keep things simple it is sufficient to deal with the stresses resultants rather than the stresses themselves. The reason, explained previously, is that the resultants are functions of only. The equilibrium equations are used to describe how the internal stresses of the beam are affected by the external loads being applied.
Assume there is equilibrium in the y-direction, then the equation for the shear resultant is, (4.12) From the moment equilibrium we get the moment resultant given by, (4.13) Where is the pressure from the load. Now that I have investigated these four theories it is possible to construct the Euler-Bernoulli beam equation. After doing this it will be possible to investigate how different loads affect the beams displacement.
Deriving the Euler-Bernoulli Beam Equation
To derive the Euler-Bernoulli beam equation I combine the equilibrium equations (4.12) and (4.13). This is done to eliminate , (4.14) Next use equation (4.7), (4.15) Using equation (4.6) to eliminate , (4.16) Insert equation (4.3) to eliminate, (4.17) Equation (4.4) shows therefore, (4.18) It is known that the definition for a beams area moment of inertia is, (4.19) Combining (4.18) and (4.19) the Euler-Bernoulli beam equation is obtained, (4.20)
Euler-Bernoulli Equation Assumptions
Before the Euler-Bernoulli beam equation is used for practical applications it is important to consider what assumptions have to be made. These assumptions are used as the full theory of elasticity is too complex for routine design to work. There are six assumptions that must be made when using the Euler-Bernoulli beam equation.
- The beam must be long and slender such that the length is significantly greater than the width and depth. This assumption is needed so that the compressive and tensile stresses that are perpendicular to the beam are significantly smaller than the stresses running parallel with the beam.
- The cross section of the beam is constant.
- The beam must be loaded in its plane of symmetry so that the torque acting on the beam is equal to zero.
- Only small deformations are possible which implies there is no buckling present, no plasticity and no soft materials. This assumption simplifies the theory of elasticity and puts it into linear form.
- The material must be isotropic.
- The plane sections remain a plane meaning that there should be no shear deformation.
Although these assumptions are important to the workings of the Euler-Beam equation in real life application it is unlikely that beams will completely satisfy all six assumptions. Even if the beam does not exactly match the assumptions it is still possible to make useful predictions from the Euler-Bernoulli equation. In some cases it is even possible to use simplified versions of the Euler-Bernoulli equation to get satisfactory answers.
Euler-Bernoulli Beam Boundary Conditions
Before applying the beam equation to real life models I must first consider the boundary conditions as discussed in Fig. 3.1. Looking at the assumptions for the cantilever beam we see thatas there is no deflection where the beam is in contact with the wall. as at the wall the beam is horizontal thus the derivative of the deflection function will be zero. as there is no bending moment at ( being the free end of the beam). since there is no shearing force at.
If a force was applied at L then this would produce a shear force meaning that the fourth boundary condition would not be valid. If the force applied to the free end was due to a weight of mass then the boundary condition would become where is acceleration due to gravity. It makes sense to use this as our new boundary condition because if the equation will reduce back to the original boundary condition.
If investigating a simply supported beam the boundary conditions would have to change. Assume that the beam is resting on its support at and The first two boundary conditions areandsince the beam can not experience any deflection at these points. The third and fourth boundary conditions arise due to the beam being able to rotate and therefore no torque is experienced. This gives us the third and fourth boundary conditions and. Now that I have these assumptions it is possible to apply the equation to real life situations.
Euler-Bernoulli Eigenvalue Problems
By considering a beams bending moment, its displacement, its shearing force equations and the Euler-Bernoulli beam equation it is possible to manipulate differential eigenvalue problems and form the partial differential equation for a beams bending vibration.
To find this equation we must first look at (4.20) and consider the bending vibration for a bending element. Figure (4.21) illustrates the forces and moments applied to the element (Meirovitch, L., 2002)
By using this diagram and considering the motion in the vertical direction a new force equation is obtained. (4.22) This equation can then be manipulated into the differential eigenvalue problem, (4.23) Where is the frequency of vibrations and is the displacement configuration assumed by the beam during harmonic oscillation. Using equation (4.23) it is possible to calculate eigenvalue problems for the Euler-Bernoulli beam equation. One such example is shown below.
Example Using the Euler-Bernoulli Equation on a Cantilever Beam.
Example (Taken from Meirovitch, L., (2002). Fundamentals of Vibrations. McGraw-Hill Companies page 399)
Solve the eigenvalue problem for a cantilever beam clamped at and free atand plot the first three nodes.
These frequencies are very useful and they allow us to calculate how different materials vibrate with regards to their length and elasticity. By using these equations engineers will be able to avoid another Tacoma Bridge catastrophe.
Strengths and Weaknesses of Euler-Bernoulli Equation
The main advantage of the Euler-Bernoulli beam equation is its ability to be simplified and applied to a large variety of practical situations. A common form of the equation used in engineering is where represents the section modulus, is the maximum allowable bending moment, and is the stress. This simple equation allows engineers to select the appropriate material for the job. One of the earliest main building applications that used the Euler- Bernoulli equation was building of the Eiffel Tower in 1889. It was also influential in the design of the Ferris Wheel in the late 19th Centaury.
As well as being able to simplify the Euler-Bernoulli beam equation, it is also possible to use its kinematic assumptions for more advanced analysis. By using different constitutive equations than the one stated in equation (4.6) it is possible to consider plastic or viscoelastic beam deformation, meaning the equation can be applied to a large variety of materials with varying properties. The theory behind the Euler-Bernoulli beam equation can also be used to analyse the bending effects on beam buckling, curved beams and composite beams. Although the Euler-Bernoulli equation has many applications, it does not consider the transverse shear strain on the beam. This is its major weakness and it can lead to low predictions for deflections and high predictions for natural frequencies. These errors are not so noticeable when the beam is thin but if the equation is applied to a thick beam these errors can become significant. To overcome these errors we need to consider more advanced theories. The most commonly used beam theory that considers the shear effect on beams is the Timoshenko Beam theory which was developed in the 1920s. Timoshenkos theory considers the effect of the shear force and it allows us to investigate the bending effects on shorter, thicker beams.
Timoshenko Beam Theory
The Timoshenko Beam theory considers the rotational inertia and shear effects of beams meaning it can be used to investigate the behaviour of short beams or beams that have been excited by a high frequency, such as an aircraft wing. The theory is based on the assumption that the plane normal to the beam axis before deformation is not normal to the axis after deformation but it still remains a plane deformed by an angle from its undeformed position (Thomas, D.L, Wilson, J. M., and Wilson, R. R. 1973). This means that the neutral axis of the beam will have a different curvature from the edges of the beam thus creating angular momentum along the beam. If the ratio between the cross-section and length further increases then the assumption that the plane section remains a plane no longer holds and as a result the shear lag phenomenon starts to become significant. The Timoshenko beam theory is only valid for beams where the shear lag is insignificant implying that the theory can consider shear deformation but only in small quantities. We can define the shear lag of the beam to be the delay in developing shear flow reactions to applied loads (Dowswell, B., and Barber, S. 2007).
Deriving the Timoshenko Beam Equation
To derive the Timoshenko beam theory we first consider a section of beam in the interval As before represents the deflection of the beams cross section from its initial position and letdenote the transverse deformation of the beam. In the Euler-Bernoulli beam theory we assume that the plane perpendicular to the beam surface when the beam is not deformed will still be perpendicular to the beam surface when the beam is deformed. Hence the deflectionwill be the same as the slope of the beams neutral axis. Equation (5.1) denotes the shear strain It is used to measure how much the plane perpendicular to the boundary surface of the beam in the undeformed state varies from being perpendicular to the boundary surface of the beam while in its deformed state.
It is known that in the section of beam the angular velocity and angular acceleration areandrespectively. Using these it is possible to describe the flexural acceleration of the section with the equation, (5.2) Dividing byand taking limits so that gives the following equation, (5.3) The shear force is the only force driving the transverse motion and by applying Newtons Second Law to the transverse deformation of the section of the beam I get, (5.4) where denotes the distribution of load in the positive direction of w. Dividing byand taking limitsthe equation, (5.5) is obtained. To construct the Timoshenko equation I need to find expressions forand in terms ofand. The connection between and was found previously when investigating the Euler-Bernoulli equation. The equation that connects them is, (5.6) To relate the shear forcetoandconsider the shear modules of the fabric of the beam by using the formula, (5.7) whereis the modulus of elasticity in the shear andis the shear strain. Therefore, (5.8) Where is the Timoshenko shear coefficient which depends on the geometry of the material. The most commonly used coefficient iswhich represents a rectangular cross section although other shear coefficients are available for differently shaped beams. This gives the Timoshenko equations which are two coupled partial differential equations, and . (5.9)
Boundary Conditions for a Free-Free Beam
The Timoshenko equation has been derived and now I will consider some of its boundary conditions. In the case of the cantilever beam in Fig. (3.1) the free end of the beam will have no shearing forces and its moment will be zero. Now suppose both ends of the beam are free ends then the boundary conditions will be, (5.10) By using separation of variables the Timoshenko equations for andform a coupled system of two second order differential equations containing and, and (5.11) Notice thatis an eigenvalue parameter and from equations (5.10) the conditions onandimply thatandmust satisfy the free-free boundary conditions. Therefore, (5.12)
Notice that the boundary value problem forandis self-adjoint, implying that non-trivial solutions for this problem exist and the eigenvalues for this model are real. It can be seen that the collection of all eigenvalues for this problem form a discrete unbounded set of real non-negative numbers. Furthermore, if the natural frequency of the beam is then it can be shown thatis one of the beams eigenvalues, making it now possible to determine eigenvalues from natural frequencies through experiments (Geist, B. 1994).
Suppose a beam with unknown elastic moduli and unknown mass density but the natural frequencies of the beam have been found through experiment. It is possible to calculate the sequence of eigenvalues but what information can be extracted from these eigenvalues? It is not easy to see how the eigenvalues depend on and due to these coefficients being highly nonlinear but the first step to obtaining general eigenvalue formulas for nonconstant beams is to derive eigenvalue formulas for the uniform Timoshenko beam. This will produce asymptotic formulas for eigenvalues of a Timoshenko beam.
The Frequency Equation
I want to derive the frequency equation for the Timoshenko beam. This will allow me to investigate vibrations of different Timoshenko beams. First consider a uniform Timoshenko beam where and are constant. I can simplify the equations (5.11) and (5.12) by letting, (5.13) So now equations (5.11) are equivalent to the differential equations, and (5.14) The boundary conditions in (5.12) become, (5.15) I can eliminateorfrom (5.14) and in doing so I find that andsatisfy the following two fourth order differential equations, and (5.16) Notice that these equations are now decoupled but coupling between and can still occur through the boundary conditions from equation (5.7). Define and as, (5.17) It is possible to derive valid general solutions to the equations in (5.16) when is not equal to 0 or 1. These solutions are, and (5.18) To findin terms ofsubstitute the general solutions for andinto equations (5.14) and nowcan be expressed in terms of . For or it can be shown that the solutions to the boundary value problem in (5.14) and (5.15) exist if and only if where and, (5.19) The matrix equationhas non trivial solutions if and only if the determinant of the matrixvanishes. Letand after some standard calculation, (5.20) When the frequency parameterandthe matrix equation defines solutions for (5.14) and (5.15). The frequency parameter is nonnegative implying that whenandthe free-free Timoshenko boundary problem has nontrivial solutions if and only if. Remember that the eigenvalue parameter in equations (5.11) satisfies , so it follows thatis the square root of a nonzero eigenvalue if and only if
Example using Timoshenkos Equation on a Cantilever Beam.
Example I will solve the same eigenvalue problem as I did for the Euler-Bernoulli beam equation as this will allow comparison of the methods. The example above investigated a cantilever beam clamped at and free atand we will now solve this using the Timoshenko Beam model.
The discriminant is always greater than 0. If both values for the discriminant are positive then it will produce purely exponential solutions. If one of the solutions is positive and the other negative then two real and two oscillatory solutions will be obtained.
We need to consider the boundary conditions for the free end and the clamped end of the beam. The point is free when the bending moment and shear force are simultaneously zero, such that, (11.1) For the clamped end,, the boundary conditions will be, (11.2)
We have seen that beam theories are very important and are found in many different places in our everyday life. The understanding of these beam theories is a fairly new discovery but it has allowed mathematicians to work with engineers to design and construct larger, more impressive structures. With the understanding of these beam theories we are able to predict how bridges and large structures will behave under different forces thus allowing safer construction and in turn produce structures that will be stable and safe. It is even thought that the discovery of the beam theories was a significant enabler to the start of the Second Industrial Revolution (Beaudreau, B., C. 2006). The example of the Tacoma Narrows Bridge disaster emphasises the importance of understanding these theories and without them we would almost certainly have many more structural disasters.
The Timoshenko Beam Theory is the simpler of the two theories and although it will not give as accurate results as the Timoshenko Theory its results are still valid and considered acceptable in construction. It is often referred to as the Engineers Beam theory as it is widely used by engineers to calculate how materials will react to different forces. The Euler-Bernoulli theory is also very versatile in comparison to the Timoshenko Model (Hobsbawm, E. J. 1999): it can be manipulated to form a wide variety of equations and it can used to calculate a large variety of materials with varying properties. One example of these is the Euler-Lagrange beam equation which investigates how the beams potential energy and kinetic energy are related, (12.1) whereis the linear mass density of the beam, describes the deflection of the beam. The first term on the right hand side represents the kinetic energy. The second term is the potential energy of the beam due to its internal forces and the remaining term is the potential energy due to the external load
Both theories have their limitations. As discussed earlier the Timoshenko beam theory is only valid for beams where the shear lag is insignificant, implying that the theory can consider shear deformation but only in small quantities. Unlike the Timoshenko theory, the Euler-Bernoulli equation does not consider the transverse shear strain on the beam. This is its major flaw and it leads to low predictions for deflections and high predications for natural frequencies when a short thick beam is being used (AS SEEN IN THE EXAMPLE, LOOK AT AND COMPARE THE EIGENVALUES AND FREQUENCIES FOR TIMOSHENKO AND EULER).
It is hard to say where future research will go as we have already considered the two main equations that investigate beams vibrations. One possible investigation could look at these equations in more detail by looking at all their possible applications, such as how the beams perform in varied temperatures. Or we could investigate how weaknesses and fractures in the beam are affected by vibrations. We could also look at how non uniform beams behave under vibration. An area that other researches have looked at it is described as The second spectrum of Timoshenko beam theory (Stephen, N., G. 2005). It is known that the Timoshenko frequency equation factorises for hinged hinged boundary conditions but it is now thought that it also factorises for hinged guided and guided guided cases. By using a higher derivative Lagrangian we can construct the familiar fourth order Timoshenko beam equation. At the moment this research does not have many practical applications but if more work was done in this area it may be possible to apply the Timoshenko theory to more real life scenarios.
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