Euler-Bernoulli and Timoshenko Beam Theories Analytical and Numerical Comprehensive Revision

DOI: http://dx.doi.org/10.24018/ejers.2021.6.7.2626 Vol 6 | Issue 7 | November 2021 20 Abstract — Obtaining reliable and efficient results of a specified problem solution depends upon understanding the strategy of the method of analysis, which is emanated from all related physical basics of the problem, formulated with master mathematical tools to give its governing mathematical model. These two categories require deep study in a wide range of references and literature in order not only to apply the method professionally, but also to look for improvements, developments, and contributions in the field of the method. Consequently, although Euler-Bernoulli and Timoshenko beam theories are the oldest ones, but surely, they represent a cornerstone for most modern methods in structural analysis; In what follows, a detailed revision of these theories and their applications analytically and in numerical style is presented in a proper and simplified entrance to be able to understand more advanced topics such as thin and thick plate theories. Illustrative examples will be used to show and discuss the methods.


I. INTRODUCTION
Most practical engineering structures consist of structural members such as beams, plates, and shells in which a beam member have a large ratio of length to lateral dimensions and is subjected to forces both along and transverse to its length as well as moments that tend to rotate the member about an axis perpendicular to its length. The corresponding unknowns depend only on the beam-axis position and its structural behavior governed by the well-known beam theories, which represent the cornerstone of more modern analysis theories for many other structural members. The beam theories, in case of small deflections, are classified into two main groups. The first ignores the shear deformations whereas the other takes this effect into account.
In this article a grasp simplified revision of the fundamentals for such important foundational beam theories is presented. These beam theories have great importance in plate and shell studies since beams are counterpart of plates. Furthermore, beam theories enhance great value for understanding the formulating process of the finite elements of thin and thick structural members.

II. EULER-BEMOULLI BEAM THEORY
Due to the nature of beams in structural engineering, it is more convenient to investigate a single spanned, prismatic constant cross sectional homogenous beam in Cartesian righthanded 3D-coordinate axes system, such that, the positive x is left to right direction and positive y is directed toward the viewer and positive z is in a downward direction. All terminologies and signed quantities in whole of this article will be referenced to this coordinate system which is also the most used one in analytical and numerical plate theories and in finite element methods. Moreover, in order to clarify the basic fundamentals of such beam analysis, it is better to consider as an illustrative entrance, the real deflection curve for a built-in beam subjected to uniform load as shown in Fig.1(a), in which two inflection points divided the beam into three parts. The outer parts concaved-down while the middle part concaved-up. Points 1,2,3 and 4 are general points at which the perpendiculars to the tangents (i.e., normals) are shown. These portions are rearranged as shown in Fig.1(b) for mathematical purposes. This deflection curve was obtained based on kinematical assumptions due to Euler/Bernoulli-Navier hypothesis, which in turn leads to the well-known Euler-Bemoulli Beam Theory EBBT. The relevant basic data and relations concerned with Euler-Bemoulli Beam EBB which are followed in many important references such as [1]- [5] are summarized in the Table I. The most often used sign convention in main textbooks for analysis quantities such as shown in Fig. 1(a), as well as the mathematical definition of positive orientation of surfaces were followed in this article.

III. SHEARING STRESSES & STRAINS IN BEAMS
Practically, in a non-uniform bending moment case, both shear and bending coexist at the beam section with ⁄ = ≠ 0 which is different from the case of pure bending. The vertical shear force acting tangentially upon the plane of a rectangular cross section having width , thickness , area = and moment of inertia = 3 /12, produces an average shear stress = / , which is constant throughout the entire section. However, the actual shear stress as introduced by Timoshenko [1] is given by: which follows a parabolic law and vanishes at free surfaces = ± /2 as it must be since the complementary shear stresses are zero. This stress has a maximum value at points that lies on the neutral axis (i.e., = 0) of the cross section; that is: . Such structural beam members, which account for the shearing effect are known as Timoshenko Beam and are governed by Timoshenko Beam theory (TBT). Such shear stress distribution causes the initially plane cross section to become warped (i.e., curved), but it remains orthogonal to the upper and lower surfaces, furthermore Timoshenko [1] showed that this warpage is the same as long as the shearing force remains constant or continuously varying along the beam, in which the adjacent sides of infinitesimal sectional element slides to each other [1], by same amount of displacement in either sides of bent portion, provided no elongation or shortening of the longitudinal beam fibers occur. In order to show the real distortion of such warped sections diagrammatically, it is more helpful to use the mathematical solutions based on theory of elasticity as given by Popov [4], Timoshenko [6] and later on by Reddy [7]. The 2D-Elasticity solution introduced by Timoshenko for a rectangular cross sectional cantilever beam of length subjected to point load applied at its free end leads to two expressions for longitudinal and transversal (i.e., deflection) displacements; ( , ) & ( , ) respectively. The first one leads to a deflection curve that exactly coincides with the one obtained by EBBT due to constraint condition ⁄ = 0 at the left fixed end, whereas the second expression is based on the constraint condition ⁄ = 0 at fixed end of the beam. These two expressions are: where = /2, modulus of elasticity and shear modulus, which show the contribution of shear effect to the displacements. Substitution of = ± /2 in these 2Dequations, instead of giving the deflected longitudinal fibers within the beam, it will draw only the deflection curve of the longitudinal beam axis (i.e., at = 0), which is: The 3D-Elasticity solution introduced by Reddy for the same cantilever beam leads to the same last deflection equation ( ) for small / ratio but with K s = 4/3.
Along thickness-wise (i.e., at any z), the longitudinal displacement function ( , ) at any along span length will draw the warped sections, these distorted sections are determined for = and plotted exactly using Maple package, in Fig. 2(a) & Fig.2(b) for both 2D-& 3D-solutions which in turn are shows a non-usual condition at built-in end since practically, the fixed support is not free to distort [6].  In case of pure shear, the angle between each of tangents to each of these warped curves and their corresponding initial normal Fig. 2(c) is equal to the shearing strain provided that its maximum value occurs at neutral axis as natural consequence of shearing stress distribution effect, which in turn was assigned by Timoshenko for the whole of each section although it must vary in the same manner as shearing stress. Consequently, this theory relaxes the Bernoulli-Navier normality hypothesis, but at the same time neglects the warpage shape of originally plane cross section, therefore this assumed constant shearing strain through the beam thickness should correspond to a constant shearing stress Therefore, the pertinent constitutive Hookean relationship gives: In general, = ⁄ , where the so-called shear correction factor or shear stiffness K s is introduced, to compensate for or correct the error due to the foregoing assumption. This factor as introduced in first time by Timoshenko in 1920 /1921; was 2/3 for static analysis [8] and 8/9 for dynamic analysis [9] has been the subject of many investigations and many other approximations now exist, such as Roark's values [10] which are based on the work of Newlin and Trayer [11] for rectangular cross-sectional beam as = 5/6. A more accurate value was given by Cowper [12] as: = 10(1 + ) (12 + 11 ) ⁄ , which is independent of the aspect ratio of the rectangular cross section, and also agrees with Timoshenko's and Roark's values for = 0 [13].
As a closure, since (TBT) assumes a deformed straight section and neglecting the warpage which may be described by a higher degree (order) polynomial of z, then recently this theory is also, well and wide known as First order Shear Deformation Theory (FSDT)

IV. TIMOSHENKO BEAM THEORY
The deformation of elements of Timoshenko beam undergoes a rotation of the beam cross-section due to bending, in which as will be seen later in illustrative examples this rotation is independent of shear effect for all statically determinate beams and also all indeterminate beams having full symmetric conditions whereas except these beam-types, this rotation will depend on shear stiffness . the bending moment will behave in the same way. Therefore, the rotation in general will not be the same as in the Euler-Bernoulli beam. That is; ≠ ≡ ( )⁄ . Moreover, the deformation of such elements also undergoes an additional angular rotation caused by shear. Thereby the total rotation of the Normal is the slope of the deflection curve that is; ( )⁄ and numerically is equal the sum of these two rotations as ( )⁄ = ( + ), where the superscripts E & T refer to Euler and Timoshenko theories respectively.
The previous mathematical elasticity solutions can also be utilized to emphasize such sum, which follows what was visualized by Timoshenko [8] and others [14]- [17], where the visualization appeared by the almost similar figures; 12.1, 3.4.1, 10.1.1 and 8.27 given in Reddy et al. [18]- [21] respectively, is redrawn as shown in Fig. 4: This figure shows that the total rotation of the normal denoted was utilized instead of ( )⁄ , as a sum of angular rotation ≡ and the rotation − 0 ( )⁄ in which the later one maintains Bernoulli-Navier normality hypothesis (i.e., section remains normal ) as seen in Reddyfigures (i.e., Fig. 4). Thus, one may understand that 0 ( ) ≡ , in which this 0 ( ) becomes an essential part in describing the shearing angular strain as: Consequently, Reddy included shear force in the governing equilibrium equations (2.5.33, [7]) & (2.5.34, [7]) and also equations (4.3.10 a & b, [19]) which are:  [19] showed that the deflection is the one which was obtained. These last two equations were also given in Wang [22], by his Eqs. (2.1.31) & (2.1.32) but at this time the superscript T denoting to Timoshenko is added. These two − expressions will be discussed later in the light of the obtained analytical solutions of incoming illustrative examples. Therewith the total rotation described by Timoshenko [8] as discussed earlier will be followed hereinafter. Consequently, in similar manner to elementary EBBT, the bending longitudinal displacement ( , ) = − ( ), i.e., not affected by shears, and then bending curvature is = − ⁄ , because of ( ) decreases vs. increases in arc length of deflected curve then the previously momentcurvature equation becomes as given by Timoshenko [8]: The equilibrium equation ( ) = ( )⁄ yields: Therefore, from the well-known definition of twodimensional strain field in terms of displacements, the engineering shearing strain is given by: where it is evident that a direct substitution for earlier ⁄ satisfies the aforementioned statement that; the angular rotation represents the shearing strain, thereby alternatively, the shear force (2) can be expressed as: Definitely in cases of statically determinate beams in which ( ) & ( ) can be expressed in terms of known applied loads, then ( ) and ( ) can be easily obtained by means of (1) & (2) respectively, while in other beam problems, the following derived relationships provide the way for solution: • Equality of (2) and (3) yields: • From (2) and (3), the equilibrium equation can be re-expressed as: • From which the following two equations can easily be concluded: • Equality of middle parts in (5) gives: • Furthermore, twice differentiation of (7) is: • Then from (6) into (9) which is the fourth order linear differential equation governing the deflection of Timoshenko beam, It is evident that the analytical solution of (10) requires the application of four boundary conditions (BC`s) which may give rise to some complexity, Therefore our proposed systemized routine is that, the solution of such problem can be simplified by solving (6) with its BC`s for , then a direct evaluation of (7) with its BC`s for , and the complete solution for & , can be obtained through determination of remaining arbitrary constants by using of automatic relation (4), as will be illustrated in several examples later. But in the following section a numerical solution for both Euler-Bernoulli as well as Timoshenko beams using finite element method with mixed formulation is also investigated for completion of beam analysis alternatives as compared with analytical solutions. Moreover, such investigation provides a simple foundational entrance for interested beginners in more advanced topics.

V. MIXED FINITE BEAM ELEMENT MODELS
Finite element formulations in elasticity provides wide strategies to tackle many kinds of problems, usually based on multi-field variational principles in the form of weightedresidual integrals as well as their weaken forms in which the mixing between these two strong and weak forms nowadays is often known as primal-mixed and Dual-mixed formulation. A valuable study of such topic is given in [23], in what follows the weak formulation will be followed.

A. Euler-Bernoulli beam Mixed Finite Model
In order to maintain the symmetry as it will be seen later, the equilibrium governing equations can be rewritten as: The mixed finite element model can be constructed by applying either of both equivalently: three steps procedure due to Reddy [21], or weak form approach described by Zienkiewicz in Ref. [15]. Over an element Ω = ( , ) ≡ (0, ℎ) whose fixed length is ℎ = − , the following weighted-residual integrals: or in augmented matrix form, (where the superscript T over matrix reads transpose), as: Substitution of such interpolations into weak form yields the following finite element model: 0 is the intensity of uniformly distributed load.
Observing that the first integral in (12) does not conform in symmetric condition with its corresponding one in (11), and for the purpose of retaining the symmetry, it is required to repeat the integration in the form: and after weakening of its first part, becomes: The corresponding basic relation: and its weighted-residual integral is: Weakening this first integral yields: Retaining the conformity with (13) in conditions of symmetry for the first integral as: Finally multiplying both sides of (13) by (-1) yielding: The weak forms (11), (12)*, (13)* and (14) provide the following mixed finite element augmented symmetrical matrix and loading vector for Timoshenko beam when all boundary terms are omitted:

VI. ILLUSTRATIVE ANALYTICAL EXAMPLES
In order to verify the aforementioned explained revision, a selected single span beam problem of length L, prismatic cross section with constant EI subjected to uniform distributed load UDL; q(x) of intensity q or concentrated load CONL; P at a specified point, will be resolved analytically using both theorems EBBT and TBT in subsequent paragraphs.

A. Determinate Euler-Bernoulli Beams 1) Simply Supported Beam with CONL at Mid-Span
In this case the deflection curve is symmetric about = 2 ⁄ , thereby it has a symmetry condition and then it can be treated as half of the given beam governed by a homogeneous differential equation of Euler-Bernoulli Beam. Observing that the symmetric condition now becomes as boundary conditions for half of the beam, then: The solution for /2 ≤ ≤ can be obtained by replacing each by ( − ) in the last expressions, and the two parts can be combined in terms of unit step (Heaviside) function for complete solution.

2) Simply Supported Beam with UDL
Similarly, but for non-homogenous differential equation:

3) Left-End Supported Cantilever Beam with CONL
For a beam loaded at its free end and applying the BC`s: which is identical, to what given in example (4.3.1) of Reddy [19], and it is evident that, ( ) is composed of two parts, the first one is the deflection due to bending and the other is due to shearing effect which will be discussed in more details in the next example. Then: which is a non-usual condition in elementary analysis for such beam case, whereas the section at such position is remaining undistorted as pointed out by Timoshenko [1] page 173.

1) Simply Supported Beam with UDL
Similarly, systemized routine for such Timoshenko Beam yields, for eq. Then Eq. (4) becomes: in which automatically yields: 3 = 3 24 ⁄ , therefore: Equivalently with dimensionless numerical coefficients: which are identical, (although is missing print), to what is given in table (4.3.1) by Reddy [19], and also to eq.(2.4.6) given by Wang in [22]; if the typographical mistake 3 3 ⁄ instead of 2 2 ⁄ is ignored in -part. Since = /2(1 + ) and = 3 12 ⁄ ≡ 2 12 ⁄ , then = 12 2 ⁄ and 1⁄ = 2(1 + ) 2 12 ⁄ , therefore ( ) after multiplying by 2 2 ⁄ can be written in an alternative form as: In which for some fixed values (except its thickness ) for such beam parameters including = 5/6, then as increases ( ) increases as shown in Fig. 5, therefore the deflection ( ) as well as total rotation ( )/ are depending on ( ⁄ ) ratio while the bending rotation ( ) as well as bending stresses in this case remains unaffected. which is identical to table 4.3.1 in Reddy [19], and: A noticeable difference between ( )/ and ( ) is the constant ⁄ so that the warpage remaining the same for constant shear as mentioned above, furthermore ( )/ for various ratios ( ⁄ ), clarifies the aforementioned nonusual distortion condition at built-in end, although this distortion approaches to ( ) ≡ 0 for the ratios ⁄ < 0.02, it increases as the ratio increases.
Moreover, if the right-end, instead of the left one is built in, the − ′′ (0) = − due to sign convention for shears, and the solution will be identical to the one obtained by replacing each by ( − ), in the solution of the left-end case above, that is: which is also identical to [28], and which is identical to eq.(2.4.23) in Wang [22], and which is identical to [19], except the misprint of , and ( ) = Rebooting of in the term ( ) ( and similarly: Remarks: • The ( )-expression is fully identical to the given solution for problem 4.11 entitled Hinged-Fixed beam in Reddy [19] page 236, but such entitled Reddy-problem has a different solution structure and the authors believe that the meant in the question is left to right Fixed-Hinged beam. • The bending rotation ( ) is structurally identical but in opposite sign as explained above and also depend on shear stiffness , provided that the bending moment do the same.
• As discussed in example;1(VI-C) above, the same conclusion can also be drawn for ( )⁄ − ( ). The deflection , rotation and bending moment of such a problem case are shown in Fig. 6, Fig. 7 & Fig. 8 for various ratios ( ⁄ ).   Since the obtained ( ) in this Timoshenko beam example differs from all foregoing ones because of its dependency on shear stiffness coefficient, while its corresponding Euler-Bernoulli beam case does not include such factor, moreover the explicit expression of ( ) for such beam-type problem according to Reddy formulation is given in Reddy [19] page 236, then it is possible to compare graphically the following quantities of such beam: • ( )⁄ obtained in the illustrative example;

3) Built-in Beam with UDL & Central Point Load
This problem although it can be solved by superposition technique, it will be treated in similar manner of earlier simply supported beam with concentrated load, then the half beam governed by the nonhomogeneous differential equation for UDL only: In which its first part is in opposite sign to what is given by Reddy in his example 4.3.2 by eq. 4.3.25 in Ref. [19], as it is must be due to reason explained in example 1(VI-C) but the second one doesn't do that although its similar to the case in example 3(VI-C) mentioned above, when P & L replaced by P 2 ⁄ & L/2 respectively, and accordingly the second part of Reddy solution should be also with minus sign. Therefore, in the absence of the reason to the authors for this contradicting case, obviously; in general, the typographical mistakes in foundational references are the major confusing difficulties encounters the interested beginners in advanced topics. The deflection ( ) in compact form is given by: ] + 2 ( ) which fortunately intact from such problems and is fully identical to the one given by indicated Reddy solution.
As a closure to these detailed discussed illustrative examples; the established relationships between deflection, rotation, bending moment, and shear-force of Timoshenko beam theory (TBT) and their corresponding quantities due to Euler-Bernoulli beam theory (EBBT) given by Wang [22] in the Table 2

VII. MIXED FINITE ELEMENT SOLUTIONS
To this end, it is quite evident that more complicated applied loading patterns to Timoshenko beam resulted in very complicated analytical solutions even through superposition technique. Therefore, the numerical solutions might be the better choice to tackle such problems, which for example can be carried out using the TBT explicit augmented matrices developed above which in turn can be implemented in any computer programming language (authors develop this implementation in a visual basic VB6 programing language to obtain the solutions of the foregoing illustrative examples as shown in next tables) Therefore, for fixed given beam parameters including = 5 6 ⁄ , = 3 10 ⁄ & / = 0.3, the Mixed Finite Element Method (MFEM) utilizing 40-mesh elements yields the results for the deflection and bending moment based on TBT, gives good agreement results as shown in tables [2]- [8], coupled with theoretical results based on EBT & TBT derived above, for comparing purposes.