The Use of Trigonometric Series for the Study of Isotropic Beam Deflection

Abstract

The average deformed fiber is a continuous and smooth function of the fourth order. The deflection and rotation of beams can be determined by various methods available in the literature. Thus, in this paper, the expression of the average deformed fiber is defined in advance, then it is considered an infinite sum of sinusoidal functions that are successively evaluated using sinusoidal trigonometric series if it is considered a periodic function, or using Fourier series if it is considered a non-periodic function. The method is examined by solving several beam problems. The results indicate that the method can be used with confidence for solving any bending beam problem.

1. Introduction

Bars subjected to bending are structural elements that take loads applied in the transverse plane. These loads act on a unit of reduced length (forces and/or moments of nodal forces), or on the entire length of the bar (loads whose distribution is governed by a constant, linear, or parabolic function). After the operating time, the loads can be static loads or dynamic loads. In the case of static loads, it is assumed that the velocities and accelerations are zero. Dynamic loads are the result of the action of masses in motion or the variation over time—according to some laws—of the loads applied directly to the bodies. Here, we can discuss inertia forces (high, constant, or continuously variable accelerations), shock loads (sudden–discontinuous–variations of speeds and accelerations) and periodic (sometimes random) variable loads of a very large number of repeated tasks times.

Under the action of these loads, the bar deforms, according to Figure 1. The deformed axis of the bar is called the deformed average fiber or elastic line because the study of beam deformations is generally carried out within the limit of small deformations, which do not exceed the elastic stress limit of the material from which this bar is made. The size of the deformations of the straight bars, under simple bending stress, is estimated using the following quantities: the displacement of the center of mass of the considered cross-section, which represents the ordinate of the cross-section, denoted by y; the angle of rotation of the cross-section φ_x; and the radius of curvature ρ_x or the curvature 1/ρ_x next to the section corresponding to the elevation x [1,2,3].

In the case of the periodic variation of the stress, a periodic variation of the deformations is obtained, giving rise to vibrations.

Thus, both in the case of static and dynamic stresses, the study of the displacement of the center of gravity of the cross-section of the bar is the central parameter because the rotation of the cross-section of the bar, in the field of elastic deformations, has low values.

For static loads deflection of beams can be determined by various approaches, some of which are: the direct integration method, Clebsch method for calculating integration constants, moment-area approach (Mohr method), energy method, which includes Castigliano’s theorems and unit load technique, stiffness and flexibility methods, principal superposition, three bending moments technique and singularity functions method, and Taylor’s expansion series [1,2,3,4,5,6,7].

An interesting radiograph of the different methods used to calculate the deformations of straight bars statically stressed in bending can be found in the reference [8].

This study aims to analyze the average deformed fiber in the case of a bar, simply supported, statically stressed in simple bending, using trigonometric series (Fourier series), appreciating the fact that the average deformed fiber is a periodic function that can be represented by a sum infinity of sine and cosine terms. The first application of the Fourier method to the solution of isotropic beam problems was given by Filon (a sinusoidal series) [9] and Ribiere (a cosinusoidal series) in 1889 [10].

The method of ordinary Fourier series in its classical form makes it possible to satisfy the boundary conditions on two opposite faces, and on the other two, up to self-balanced loads. The superposition of Fourier series on the other variable to remove the loads that arise reduces to an infinite system of algebraic equations to find the coefficients of the Fourier series of the required solutions. The exact solution of such a system has been obtained for only one case: when two of the opposite faces of the beam are clamped and the other two are subject to a prescribed load [11].

Thus, the Fourier series, for such applications, can be used to obtain optimal shape solution for a non-prismatic planar beam [12,13].

In the case of beams (straight or curved) made of composite materials the deflection functions, the Lagrange multiplier functions, and the loads are expressed as trigonometric series to satisfy the governing equations and the simply supported constraints at both ends [14,15,16].

Other studies on this topic can be found in the references [17,18,19,20,21].

In this paper, the expression of the average deformed fiber (the continuous and smooth function of the fourth order) is defined in advance for a straight bar subjected to plane bending because the plane of loads coincides with the main axes of inertia. The deformed average fiber is considered to represent an infinite sum of sinusoidal functions, which are successively evaluated through sinusoidal trigonometric series (if this is considered to be a periodic function) or using the Fourier series if this is considered to be a non-periodic function.

Thus, the displacement in the vertical plane of the center of gravity of the cross-section for a straight bar, simply supported, is studied for two loading cases: in the first case, a concentrated force acting in the vertical plane (in the direction of the y axis, according to Figure 1), positioned asymmetrically or symmetrically to the support points; in the second case of loading, a uniformly distributed load is considered over the entire length unit and which also acts in the direction of the y axis.

The calculation relations of the displacement obtained with the help of the trigonometric series and the Fourier series are compared with those offered by the specialized literature (obtained by direct integration of the differential equation of the deformed average fiber) to check the convergence of the obtained results [1,2,3,4,5,6].

In the final part of this paper, a case study is made for a simply supported bar loaded on the entire length unit through a distributed load, whose variation law is a linear function.

2. The Deformed Average Fiber of the Beam

The beam is subjected to simple bending using a system of external forces (nodal forces F₁ and F₂ and the uniformly distributed load q), perpendicular to the longitudinal axis, which act in a main plane of inertia. Under the action of these loads, the bar bends, its longitudinal axis deforming in the same plane as that of the external loads, producing plane bending.

Thus, according to Figure 1, in node ‘’a’’ located at a distance x from the articulated support (from node 1) a linear deformation will be recorded, represented by the arrow aa₁ = y and an angular deformation represented by the rotation angle (or slope) of the cross-section φ_x. In Figure 1, the points (nodes) where either the supports are defined, or the different types of loads act are marked with 1–6.

Nodes ‘’a’’ and a₁ define the position of the center of gravity of the cross-section in the undeformed position, respectively, in the deformed position of the straight bar before and after bending. In other words, the deformation aa₁ represents the displacement of the center of gravity of the cross-section in a direction perpendicular (normal) to the axis of the bar. After bending, the axis of the bar will not change its length, and the center of gravity of the cross-section will move laterally about the normal position on the undeformed axis, going through arc a₁a₂. This displacement represents an infinitely small higher-order about the length l of the bar. Consequently, this displacement can be neglected and the center of gravity of the cross-section, from the deformed position of the bar, will be in the direction of the normal to the undeformed axis.

The rotation angle or slope φ_x is an angular deformation and represents the angle by which the cross section corresponding to the x-axis rotates to its initial position.

The radius of curvature ρ_x, line O₁a₁, is normal to the tangent to the deformed axis next to the section corresponding to the x-axis.

The average deformed fiber, in the orthogonal system of axes xOy, is a shape function:

$$y=f\left(x\right)$$

(1)

If this function is assumed to be known, displacements, rotations, and curves can be determined. The rotations are obtained from the first derivative of the function f(x):

$$\frac{dy}{dx}=tg{\phi }_x\cong {\phi }_x$$

(2)

The curves are given by the expression:

$$\frac{1}{{\rho }_x}=\pm \frac{\frac{d^{2}y}{d{x}^2}}{{\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^{\frac{3}{2}}}$$

(3)

In the case of bars subjected to simple bending in the elastic domain, the rotations are small, and the term (dy/dx)² is neglected, relation (3) becoming:

$$\frac{1}{{\rho }_x}=\pm \frac{d^{2}y}{d{x}^2}$$

(4)

In conclusion, to calculate the curvature of the average deformed fiber, in a certain cross section, the rotation expression is derived from the respective section, or the second derivative of the displacement will be performed.

The problem can be solved the other way around. If the expression of the curves along the axis of the beam is known:

$$\pm \frac{d^{2}y}{d{x}^2}=\frac{1}{{\rho }_x}$$

(5)

the expression of rotations φ_x = dy/dx and displacements y = f(x) can be determined by successive integration.

The resulting integration constants will be determined from the condition of continuity and smoothness of the curve of the average deformed fiber along its entire length, because, by its physical nature, the deformed axis of the bar is a continuous and smooth curve.

The curve y = f(x) is continuous when a given value of the abscissa x corresponds to a single value of the ordinate y.

The term smoothness defines a curve that does not present breaks along its length. In other words, only one tangent can be taken at each point of the curve.

In conclusion, for a curve to be continuous and smooth (in the case of the average deformed fiber) the function y = f(x), and it is first derivative dy/dx = φx = f’(x) = y’, must be continuous throughout the length of the beam.

The relation:

$$\frac{1}{{\rho }_x}=\frac{M_i}{E·I_z}$$

(6)

expresses the connection between the curvature of the bar and the bending moment function.

Relation (4) expresses the mathematical formula of curvature:

$$\frac{1}{{\rho }_x}=\pm \frac{d^{2}y}{d{x}^2}$$

From the two expressions that define the curvature, the following relation is obtained:

$$\frac{d^{2}y}{d{x}^2}=\pm \frac{M_i}{E·I_z}$$

(7)

which is called the differential equation of the deformed average fiber.

From the analysis of this expression, the following observations can be highlighted: The sign of relation (7) is established by choosing an orthogonal system of reference axes. In particular, the origin of the orthogonal system of axes is defined in the center of gravity of a corresponding transverse section whose displacement is zero (in simple support, articulated or embedment); but, in general, it can be chosen in the center of gravity of any cross-section. The longitudinal axis of the bar before deformation coincides with the x-axis, with the positive sign oriented to the right. The axis of the ordinates y, with the positive direction down, will be perpendicular to the abscissa x, according to Figure 2, where the curve Oab represents the average deformed fiber. Points a and b are points of inflection of the deformed average fiber. On the portion Oa the deformed average fiber shows a positive curvature because its concavity is directed in the direction of the positive axis Oy, and on the portion ab, the curvature is negative. The bending moments corresponding to these curves have opposite signs.

According to Figure 2, where the y-axis is oriented downwards, relation (7) will be of the form:

$$\frac{d^{2}y}{d{x}^2}=-\frac{M_i}{E·I_z}$$

(8)

If the y-axis is oriented upwards, expression (7) will be of the form:

$$\frac{d^{2}y}{d{x}^2}=+\frac{M_i}{E·I_z}$$

(9)

If the beam is of the constant section, it follows that the bending stiffness modulus E·I_z is constant. By deriving expression (8) and considering the differential relations between efforts:

$$\frac{d{M}_i}{dx}=T_y şi \frac{d{T}_y}{dx}=-q\left(x\right)$$

(10)

the expression of the average deformed fiber can also be written in the following forms:

$$\frac{d^{3}y}{d{x}^3}=-\frac{T_y}{E·I_z} şi \frac{d^{4}y}{d{x}^4}=\frac{q\left(x\right)}{E·I_z}$$

(11)

In the node where the bending moment M_i is zero, the deformed average fiber presents an inflection point.

3. Trigonometric Series Applied in the Study of the Average Deformed Fiber

It is considered that the average deformed fiber can have a sinusoidal variation, and it is expressed mathematically in the following form:

$$y\left(x\right)=A·sin\left(\frac{2·\pi }{T}·x+\phi \right)$$

(12)

where A is the oscillation amplitude, φ represents the initial phase, and T is the oscillation period.

The deformed mean fiber is a function that can be expressed as an infinite sum of sinusoidal functions, according to the example in Figure 3. In Figure 3, the points (nodes) where either the supports are defined, or the different types of loads act are marked with 1–3.

The relation (12) can be expressed, in the most general case (considering the example in Figure 3), as follows:

$$y\left(x\right)=v_n·sin\frac{n·\pi ·x}{l}$$

(13)

where: n = 1, 2, 3, …

For case (a) from Figure 3, relation (13) has the following form:

$$y{\left(x\right)}_{\left(a\right)}=v_1·sin\frac{1·\pi ·x}{l}$$

(14)

For case (b) from Figure 3, relation (13) has the following form:

$$y{\left(x\right)}_{\left(b\right)}=v_2·sin\frac{2·\pi ·x}{l}$$

(15)

For case (c) from Figure 3, relation (13) has the following form:

$$y{\left(x\right)}_{\left(c\right)}=v_3·sin\frac{3·\pi ·x}{l}$$

(16)

By summing relations (14)–(16) we obtain:

$$y\left(x\right)={\displaystyle \sum }_{n=1}^{\infty }{v}_n·sin\frac{n·\pi ·x}{l}=v_1·sin\frac{\pi ·x}{l}+v_2·sin\frac{2·\pi ·x}{l}+v_3·sin\frac{3·\pi ·x}{l}+\dots$$

(17)

The coefficients v₁, v₂, v₃, …, represent the maximum ordinates of the sinusoids (oscillation amplitude), and indices (1,2,3, …) represent the number of half-waves on the length l of the bar (oscillation period).

The expression of the potential energy of elastic deformation for the bar subjected to bending is of the form:

$$U_d={{\displaystyle \int }}_0^l\frac{M_i^2}{2·E·I_z}·dx=\frac{E·I_z}{2}·{{\displaystyle \int }}_0^l{\left(\frac{d^{2}y}{d{x}^2}\right)}^2·dx$$

(18)

where:

$$M_i=-E·I_z·\frac{d^{2}y}{d{x}^2}$$

(19)

To calculate the potential energy of elastic deformation depending on the coefficients v₁, v₂, v₃, …, the trigonometric series (17) is derived twice, and the obtained expression is inserted into the relation (18):

$$\frac{dy}{dx}=\frac{\pi ·v_1}{l}·cos\frac{\pi ·x}{l}+\frac{2·\pi ·v_2}{l}·cos\frac{2·\pi ·x}{l}+\frac{3·\pi ·v_3}{l}·cos\frac{3·\pi ·x}{l}+\dots$$

(20)

$$\frac{d^{2}y}{d{x}^2}=-\frac{1^2·{\pi }^2·v_1}{l^2}·sin\frac{\pi ·x}{l}-\frac{2^2·{\pi }^2·v_2}{l^2}·sin\frac{2·\pi ·x}{l}-\frac{3^2·{\pi }^2·v_3}{l^2}·sin\frac{3·\pi ·x}{l}-\dots$$

(21)

$$\begin{gathered} {\left(\frac{d^{2}y}{d{x}^2}\right)}^2={\left(\frac{1·\pi }{l}\right)}^4·v_1^2·si{n}^2\frac{\pi ·x}{l}+{\left(\frac{2·\pi }{l}\right)}^4·v_2^2·si{n}^2\frac{2·\pi ·x}{l}+{\left(\frac{3·\pi }{l}\right)}^4·v_3^2·si{n}^2\frac{3·\pi ·x}{l}+\dots + \\ +2·v_1·v_2·\frac{1^2·2^2·{\pi }^4}{l^4}·sin\frac{\pi ·x}{l}·sin\frac{2·\pi ·x}{l}+2·v_2·v_3·\frac{2^2·3^2·{\pi }^4}{l^4}·sin\frac{2·\pi ·x}{l}·sin\frac{3·\pi ·x}{l}+ \\ +2·v_1·v_3·\frac{1^2·3^2·{\pi }^4}{l^4}·sin\frac{\pi ·x}{l}·sin\frac{3·\pi ·x}{l}+\dots \end{gathered}$$

(22)

The expression of the potential energy of elastic deformation becomes a sum of terms, generalized in the following form:

$${\left(\frac{n·\pi }{l}\right)}^4·v^2·si{n}^2\frac{n·\pi ·x}{l}$$

(23)

and

$$2·v_n·v_m·\frac{n^2·m^2·{\pi }^4}{l^4}·sin\frac{n·\pi ·x}{l}·sin\frac{m·\pi ·x}{l}$$

(24)

where indices n and m are integers.

By integrating these terms, it results:

$${{\displaystyle \int }}_0^{l}si{n}^2\frac{n·\pi ·x}{l}·dx=\frac{l}{n·\pi }·{\left[-\frac{1}{4}·sin\frac{2·n·\pi ·x}{l}+\frac{1}{2}·\frac{n·\pi ·x}{l}\right]}_0^l=\frac{l}{2}$$

(25)

$${{\displaystyle \int }}_0^{l}sin\frac{n·\pi ·x}{l}·sin\frac{m·\pi ·x}{l}·dx=\frac{l}{\pi }·{\left[\frac{1}{2·\left(n-m\right)}·sin\left(n-m\right)·\frac{\pi ·x}{l}-\frac{1}{2·\left(n+m\right)}·sin\left(n+m\right)·\frac{\pi ·x}{l}\right]}_0^l=0$$

(26)

Thus, relation (18) will be of the form:

$$U_d=\frac{E·I_z}{2}·\frac{{\pi }^4}{l^4}·\frac{l}{2}·\left(1^4·v_1^2+2^4·v_2^2+3^4·v_3^2+\dots \right)=\frac{E·I_z·{\pi }^4}{4·l^3}·{\displaystyle \sum }_{n=1}^{\infty }{n}^4·v_n^4$$

(27)

It is known that, at a reduced deformation of an elastic system, starting from the equilibrium position, the deformation potential energy is equal to the mechanical work produced by the external forces during this deformation. In other words, according to relation (17), a small deformation is equivalent to a small change in the coefficients v₁, v₂, v₃, which represent the arrows of the beam in different nodes. Thus, dv_n is considered the variation of the coefficient v_n. With this variation, the term:

$$v_n·sin\frac{n·\pi ·x}{l}$$

(28)

turns into:

$$\left(v_n+d{v}_n\right)·sin\frac{n·\pi ·x}{l}$$

(29)

This increase is equivalent to the introduction of a new sinusoid in the trigonometric series (17), of the form:

$$d{v}_n·sin\frac{n·\pi ·x}{l}$$

(30)

Through this additional deformation, the point of application of force F, located at an elevation a with the origin of the orthogonal system of reference axes (node 1 in Figure 3), moves with:

$$d{v}_n·sin\frac{n·\pi ·x}{l}$$

To determine the amplitude v_n, the condition is imposed that the external mechanical work (dL_ext) produced by the loads applied to the bar is transformed into potential energy of deformation (dL_int) as a result of moving to a new equilibrium position (dv_n). The magnitude dv_n is produced by a small static increase dF of the applied concentrated force (according to Figure 4).

In Figure 4, the straight-line OA represents the characteristic curve of the material from which the bar is made for the range of elastic deformations. The external mechanical work represents the area of the surface abcde under the characteristic curve, and is expressed as follows:

$$d{L}_{ext}=A_{abcde}=A_{abcd}+A_{ade}$$

or

$$d{L}_{ext}=F·d{v}_n+\frac{dF·d{v}_n}{2}\cong F·d{v}_n$$

The $\frac{dF·d{v}_n}{2}$ term is neglected considering that it is an infinitesimal of higher order.

The external mechanical work produced by the force F is of the form:

$$d{L}_{ext}=F·d{v}_n·sin\frac{n·\pi ·a}{l}$$

(31)

The internal mechanical work, expressed as a function of v₁, v₂, v₃, …, through the variation of a single coefficient, will present an increase in the form:

$$\begin{gathered} d{L}_{int}=\frac{\partial U_d}{\partial v_n}·d{v}_n=\frac{\partial }{\partial v_n}·\left(\frac{E·I_z·{\pi }^4}{4·l^3}·{\displaystyle \sum }_{n=1}^{\infty }{n}^4·v_n^4\right)·d{v}_n\underset{}{\Rightarrow } \\ d{L}_{int}=\frac{E·I_z·{\pi }^4}{2·l^3}·n^4·v_n·d{v}_n \end{gathered}$$

(32)

Applying the principle of conservation of energy, from the equality of the internal mechanical work, relation (32) with the external mechanical work, relation (31), results:

$$F·d{v}_n·sin\frac{n·\pi ·a}{l}=\frac{E·I_z·{\pi }^4}{2·l^3}·n^4·v_n·d{v}_n$$

(33)

From relation (33), we obtain:

$$v_n=\frac{2·F·l^3·sin\frac{n·\pi ·a}{l}}{E·I_z·{\pi }^4·n^4}$$

(34)

By replacing the v_n coefficients in relation (17), it results:

$$y\left(x\right)=\frac{2·F·l^3}{E·I_z·{\pi }^4}·\left(\frac{1}{1^4}·sin\frac{\pi ·a}{l}·sin\frac{\pi ·x}{l}+\frac{1}{2^4}·sin\frac{2·\pi ·a}{l}·sin\frac{2·\pi ·x}{l}+\frac{1}{3^4}·sin\frac{3·\pi ·a}{l}·sin\frac{3·\pi ·x}{l}\right)$$

(35)

or

$$y\left(x\right)=\frac{2·F·l^3}{E·I_z·{\pi }^4}·{\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n^4}·sin\frac{n·\pi ·a}{l}·sin\frac{n·\pi ·x}{l}$$

(36)

Relation (36) allows the calculation of the arrow y in any section of the considered beam. Thus, considering a = l/2 (according to Figure 5), the displacement of point 3 is evaluated as follows:

From relation (36), it follows:

$$\begin{gathered} v=\frac{2·F·l^3}{E·I_z·{\pi }^4}·(\frac{1}{1^4}·sin\frac{1·\pi ·\frac{\mathrm{l}}{2}}{l}·sin\frac{1·\pi ·\frac{\mathrm{l}}{2}}{l}+\frac{1}{2^4}·sin\frac{2·\pi ·\frac{\mathrm{l}}{2}}{2}·sin\frac{2·\pi ·\frac{\mathrm{l}}{2}}{l}+\frac{1}{3^4}·sin\frac{3·\pi ·\frac{\mathrm{l}}{2}}{l}·sin\frac{3·\pi ·\frac{\mathrm{l}}{2}}{l}+\dots )\underset{}{\Rightarrow } \\ v=\frac{2·F·l^3}{E·I_z·{\pi }^4}·\left(\frac{1}{1^4}·si{n}^2\frac{\pi }{2}+\frac{1}{2^4}·si{n}^2\pi +\frac{1}{3^4}·si{n}^2\frac{3·\pi }{2}+\dots \right)\underset{}{\Rightarrow } \\ v=\frac{2·F·l^3}{E·I_z·{\pi }^4}·\left(\frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+\dots \right) \end{gathered}$$

(37)

In Figure 5, the points (nodes), where either the supports are defined or the different types of loads act, are marked with 1–3.

If we consider, sequentially, the first term, then the first two terms, and finally the first three terms in the parenthesis, we get:

$$v=\frac{2·F·l^3}{E·I_z·{\pi }^4}·\frac{1}{1^4}=\frac{2·F·l^3}{97.211·E·I_z}=\frac{F·l^3}{48.605·E·I_z}=0.020574·\frac{F·l^3}{E·I_z}$$

(38)

$$v=\frac{F·l^3}{48.013·E·I_z}=0.020827·\frac{F·l^3}{E·I_z}$$

(39)

$$v=\frac{F·l^3}{47.937·E·I_z}=0.02086·\frac{F·l^3}{E·I_z}$$

(40)

The exact relation of the maximum displacement is expressed by the relation [1,2,3,4,5,6]:

$$v=\frac{F·l^3}{48·E·I_z}=0.020833\frac{F·l^3}{E·I_z}$$

(41)

The relative deviation between the data obtained from relations (38) and (41) is 1.243%, the data obtained from relations (39) and (41) is 0.028%, and the data obtained from relations (40) and (41) is 0.129%.

For the case where the bar is supported and subjected to bending using a uniformly distributed load, as in Figure 6, the concentrated force F, from relation (36), can be replaced using the expression:

$$F={{\displaystyle \int }}_0^{l}q·dx$$

(42)

In Figure 6, the points (nodes) where either the supports are defined or the different types of loads act are marked with 1–2.

Thus, the variation of the external mechanical work becomes:

$$\frac{d{L}_{ext}}{d{v}_n}={{\displaystyle \int }}_0^{l}q·dx·sin\frac{n·\pi ·x}{l}={\left[q·\left(-\frac{l}{n·\pi }\right)·cos\frac{n·\pi ·x}{l}\right]}_0^l=-\frac{q·l}{n·\pi }·\left(cosn·\pi -1\right)$$

(43)

Equating this expression with the increase in internal mechanical work results:

$$\frac{E·I_z·{\pi }^4}{2·l^3}·n^4·v_n=-\frac{q·l}{n·\pi }·\left(cosn·\pi -1\right)\underset{}{\Rightarrow }{v}_n=\frac{-2·q·l^4}{E·I_z·{\pi }^5·n^5}·\left(cosn·\pi -1\right)$$

(44)

Thus, the equation of the average deformed fiber will be of the form:

$$y\left(x\right)=-{\displaystyle \sum }_{n=1}^{\infty }\frac{2·q·l^4}{E·I_z·{\pi }^5·n^5}·\left(cosn·\pi -1\right)·sin\frac{n·\pi ·x}{l}$$

(45)

For n = 1, 3, 5, …

$$v_n=\frac{4·q·l^4}{E·I_z·{\pi }^5·n^5}$$

(46)

and for n = 2, 4, 6, …

$$v_n=0$$

(47)

In this situation, the expression of the average deformed fiber will be of the form:

$$y\left(x\right)=\frac{4·q·l^4}{E·I_z·{\pi }^5}·\left(\frac{1}{1^5}·sin\frac{1·\pi ·x}{l}+\frac{1}{3^5}·sin\frac{3·\pi ·x}{l}+\frac{1}{5^5}·sin\frac{5·\pi ·x}{l}+\dots \right)$$

(48)

The maximum displacement will be recorded in the middle of the distance between the two supports, for which x = l/2:

$$v=\frac{4·q·l^4}{E·I_z·{\pi }^5}·\left(\frac{1}{1^5}-\frac{1}{3^5}+\frac{1}{5^5}-\frac{1}{7^5}+\dots \right)$$

(49)

If we consider, sequentially, the first term, then the first two terms, and finally, the first three terms in the parenthesis, we receive:

$$v=\frac{4·q·l^4}{306.019·E·I_z}=0.013071·\frac{q·l^4}{E·I_z}$$

(50)

$$v=0.0130172·\frac{q·l^4}{E·I_z}$$

(51)

$$v=0.0130214·\frac{q·l^4}{E·I_z}$$

(52)

The exact relation of the maximum displacement is expressed by the relation [1,2,3,4,5,6]:

$$v=\frac{5·q·l^4}{384·E·I_z}=0.0130208·\frac{q·l^4}{E·I_z}$$

(53)

The relative deviation between the data obtained from relations (49) and (52) is 0.384%, the data obtained from relations (50) and (52) is 0.027%, and the data obtained from relations (51) and (52) is 0.004%.

4. Fourier Series Applied to the Study of the Average Deformed Fiber

Fourier series are expressed through the following mathematical relationship:

$$y\left(x\right)=\frac{a_o}{2}+{\displaystyle \sum }_{n=1}^{\infty }\left[a_n·cos\left(\frac{n·\pi ·x}{l}\right)+b_n·sin\left(\frac{n·\pi ·x}{l}\right)\right]$$

(54)

where a_o, a_n, and b_n are the Fourier coefficients of the function f(x), and are expressed through the following relations (Euler’s formulas):

$$a_o=\frac{1}{l}·\underset{-l}{\overset{l}{{\displaystyle \int }}}y\left(x\right)·dx$$

(55)

$$a_n=\frac{1}{l}·\underset{-l}{\overset{l}{{\displaystyle \int }}}y\left(x\right)·cos\frac{n·\pi ·x}{l}·dx$$

(56)

$$b_n=\frac{1}{l}·\underset{-l}{\overset{l}{{\displaystyle \int }}}y\left(x\right)·sin\frac{n·\pi ·x}{l}·dx$$

(57)

Since the average deformed fiber is a continuous and odd function, the terms a_o and a_n are zero.

Thus:

$$b_n=\frac{2}{l}·\underset{0}{\overset{l}{{\displaystyle \int }}}y\left(x\right)·sin\frac{n·\pi ·x}{l}·dx$$

(58)

Therefore, the Fourier series of an odd function, defined on the interval (0, l), is a sinusoidal series and relation (53) will be of the form:

$$y\left(x\right)={\displaystyle \sum }_{n=1}^{\infty }\left[b_n·sin\left(\frac{n·\pi ·x}{l}\right)\right]$$

(59)

For the case of support and request in Figure 6, a function of the form is defined:

$$y\left(x\right)={\displaystyle \sum }_{n=1}^{\infty }{C}_n·sin\frac{n·\pi ·x}{l}$$

(60)

which satisfies the condition y(x) = 0 for x = 0 and x = l.

Thus:

$$x=0\to y\left(0\right)={\displaystyle \sum }_{n=1}^{\infty }{C}_n·sin\frac{n·\pi ·\left(0\right)}{l}=0$$

(61)

$$x=l\to y\left(l\right)={\displaystyle \sum }_{n=1}^{\infty }{C}_n·sin\frac{n·\pi ·\left(l\right)}{l}=0$$

(62)

If, from Figure 6, we consider q = q(x) = y(x), from relation (53) it follows:

$$q\left(x\right)={\displaystyle \sum }_{n=1}^{\infty }\left[b_n·sin\left(\frac{n·\pi ·x}{l}\right)\right]$$

(63)

From relation (62), we obtain:

$$b_n=\frac{2}{l}·\underset{0}{\overset{l}{{\displaystyle \int }}}q\left(x\right)·sin\frac{n·\pi ·x}{l}·dx$$

(64)

From relations (11) and (62), it follows:

$$\frac{d^{4}y\left(x\right)}{d{x}^4}=\frac{{{\displaystyle \sum }}_{n=1}^{\infty }\left[b_n·sin\left(\frac{n·\pi ·x}{l}\right)\right] }{E·I_z}$$

(65)

Successively deriving relation (59) results in:

$$\frac{d^{4}y\left(x\right)}{d{x}^4}={\displaystyle \sum }_{n=1}^{\infty }{C}_n·{\left(\frac{n·\pi }{l}\right)}^4·sin\frac{n·\pi ·x}{l}$$

(66)

From relations (64) and (65), we obtain:

$$\begin{gathered} {\displaystyle \sum }_{n=1}^{\infty }{C}_n·{\left(\frac{n·\pi }{l}\right)}^4·sin\frac{n·\pi ·x}{l}=\frac{{{\displaystyle \sum }}_{n=1}^{\infty }\left[b_n·sin\left(\frac{n·\pi ·x}{l}\right)\right] }{E·I_z}\underset{}{\Rightarrow }{C}_n·{\left(\frac{n·\pi }{l}\right)}^4=\frac{b_n}{E·I}\underset{}{\Rightarrow } \\ {C}_n=\frac{b_n}{E·I}·{\left(\frac{l}{n·\pi }\right)}^4 \end{gathered}$$

(67)

Thus, from relation (63), it follows:

$$\begin{gathered} b_n=\frac{2·q\left(x\right)}{l}·\underset{0}{\overset{l}{{\displaystyle \int }}}sin\frac{n·\pi ·x}{l}·dx=\frac{2·q\left(x\right)}{l}·\left(-\frac{l}{n·\pi }·cos\frac{n·\pi ·x}{l}\right)|\begin{array}{c}l\\ 0\end{array}= \\ =-\frac{2·q\left(x\right)}{n·\pi }·\left(cos\frac{n·\pi ·l}{l}-cos\frac{n·\pi ·0}{l}\right)=-\frac{2·q\left(x\right)}{n·\pi }·\left(cosn·\pi -cos0\right) \\ =\frac{2·q\left(x\right)}{n·\pi }·\left(1-cosn·\pi \right)=\frac{2·q\left(x\right)}{n·\pi }·\left[1-\left(-1\right)\right]\underset{}{\Rightarrow } \\ {b}_n=\frac{4·q\left(x\right)}{n·\pi } \end{gathered}$$

(68)

where: n = 1, 3, 5, 7, 9, …, ∞—odd function.

From relation (66), it follows:

$$C_n=\frac{4·q\left(x\right)}{E·I}·\frac{l^4}{{\left(n·\pi \right)}^5}$$

(69)

Substituting relation (68) into relation (59) results in:

$$y\left(x\right)=\frac{4·q\left(x\right)·l^4}{{\pi }^5·E·I}{\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n^5}·sin\frac{n·\pi ·x}{l}$$

(70)

For x = l/2, from relation (69), we obtain:

$$y\left(x\right)=v=\frac{4·q\left(x\right)·l^4}{{\pi }^5·E·I}{\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n^5}·sin\frac{n·\pi ·\left(\frac{l}{2}\right)}{l}=\frac{4·q\left(x\right)·l^4}{{\pi }^5·E·I}{\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n^5}·sin\frac{n·\pi }{2}$$

(71)

For n = 1 from relation (70), it follows:

$$v=\frac{4·q\left(x\right)·l^4}{{\pi }^5·E·I}·\frac{1}{1^5}·sin\frac{1·\pi }{2}=\frac{4·q\left(x\right)·l^4}{{\pi }^5·E·I}=0.013071·\frac{q·l^4}{E·I_z}$$

(72)

It is observed that relation (71) is identically equal to relation (49).

For the case of support and load in Figure 7a, relation (57) is of the form:

$$\begin{gathered} b_n=\frac{2}{l}·\underset{a}{\overset{a+c}{{\displaystyle \int }}}\frac{F}{c}·sin\frac{n·\pi ·x}{l}·dx=\frac{2·F}{l}·\left(-\frac{l}{n·\pi ·c}·cos\frac{n·\pi ·x}{l}\right)|\begin{array}{c}a+c\\ a\end{array} \\ =-\frac{2·F}{n·\pi ·c}·\left[cos\frac{n·\pi ·\left(a+c\right)}{l}-cos\frac{n·\pi ·a}{l}\right] \\ {b}_n=-\frac{2·F}{n·\pi ·c}·\left(cos\frac{n·\pi ·a}{l}·cos\frac{n·\pi ·c}{l}-sin\frac{n·\pi ·a}{l}·sin\frac{n·\pi ·c}{l}-cos\frac{n·\pi ·a}{l}\right) \end{gathered}$$

(73)

If c tends to zero [according to Figure 7b], it follows:

$$cos\frac{n·\pi ·c}{l}=1;sin\frac{n·\pi ·c}{l}=\frac{n·\pi ·c}{l}$$

Thus, from relation (72), we obtain:

$$\begin{gathered} b_n=-\frac{2·F}{n·\pi ·c}·\left(cos\frac{n·\pi ·a}{l}-\frac{n·\pi ·c}{l}·sin\frac{n·\pi ·a}{l}-cos\frac{n·\pi ·a}{l}\right)=\frac{2·F}{n·\pi ·c}·\left(-\frac{n·\pi ·c}{l}·sin\frac{n·\pi ·a}{l}\right)\underset{}{\Rightarrow } \\ {b}_n=\frac{2·F}{l}·sin\frac{n·\pi ·a}{l} \end{gathered}$$

(74)

Substituting relation (73) into relation (66) yields:

$$C_n=\frac{b_n}{E·I}·{\left(\frac{l}{n·\pi }\right)}^4=\frac{2·F}{E·I}·\frac{l^3}{{\left(n·\pi \right)}^4}·sin\frac{n·\pi ·a}{l}$$

(75)

Substituting relation (74) into relation (59) results in:

$$y\left(x\right)={\displaystyle \sum }_{n=1}^{\infty }{C}_n·sin\frac{n·\pi ·x}{l}=\frac{2·F·l^3}{{\pi }^4·E·I}·{\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n^4}·sin\frac{n·\pi ·a}{l}·sin\frac{n·\pi ·x}{l}$$

(76)

For x = l/2, from relation (75), it follows:

$$y\left(x\right)=v=\frac{2·F·l^3}{{\pi }^4·E·I}·{\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n^4}·sin\frac{n·\pi ·\frac{l}{2}}{l}·sin\frac{n·\pi ·\frac{l}{2}}{l}=\frac{2·F·l^3}{{\pi }^4·E·I}·{\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n^4}·si{n}^2\frac{n·\pi }{2}$$

(77)

For n = 1, from relation (76), we obtain:

$$v=\frac{2·F·l^3}{{\pi }^4·E·I}·\frac{1}{1^4}·si{n}^2\frac{1·\pi }{2}=0.020531·\frac{F·l^3}{E·I}$$

(78)

The relative deviation between the data obtained from relations (41) and (77) is 1.449%.

5. Case Study Using the Fourier Series

It is defined as a simply supported bar loaded on the entire length unit (l) through a distributed load whose variation law is a linear function (q), according to Figure 8.

In a certain cross-section x, the magnitude of the distributed load q(x) is expressed through the following relationship:

$$q\left(x\right)=q·\frac{x}{l}$$

(79)

For the case of support and request in Figure 8, a function of the form is defined:

$$y\left(x\right)={\displaystyle \sum }_{n=1}^{\infty }{C}_n·sin\frac{n·\pi ·x}{l}$$

where:

$$C_n=\frac{b_n}{E·I}·{\left(\frac{l}{n·\pi }\right)}^4$$

If, from Figure 8, we consider q = q(x) = y(x), from relation (53), it follows:

$$q\left(x\right)={\displaystyle \sum }_{n=1}^{\infty }\left[b_n·sin\left(\frac{n·\pi ·x}{l}\right)\right]$$

(80)

From relation (79), we obtain:

$$b_n=\frac{2}{l}·\underset{0}{\overset{l}{{\displaystyle \int }}}q\left(x\right)·sin\frac{n·\pi ·x}{l}·dx$$

(81)

In relation (80), replacing q(x) with the expression from formula (78) results:

$$\begin{gathered} b_n=\frac{2}{l}·\underset{0}{\overset{l}{{\displaystyle \int }}}q·\frac{x}{l}·sin\frac{n·\pi ·x}{l}·dx=\frac{2·q}{l^2}·\underset{0}{\overset{l}{{\displaystyle \int }}}x·sin\frac{n·\pi ·x}{l}·dx= \\ =\frac{2·q}{l^2}·\left[{\left(\frac{l}{n·\pi }\right)}^2·sin\frac{n·\pi ·x}{l}-\frac{l}{n·\pi }·x·cos\frac{n·\pi ·x}{l}\right]|\begin{array}{c}l\\ \\ 0\end{array}=2·q·\left(-n·\pi ·cosn·\pi \right) \end{gathered}$$

Thus:

$$b_n=\frac{2·q}{n·\pi }·{\left(-1\right)}^{n+1}$$

(82)

for n = 1, 2, 3, 4, 5, …, ∞.

$$C_n=\frac{b_n}{E·I_z}·{\left(\frac{l}{n·\pi }\right)}^4=\frac{2·q·l^4}{{\pi }^5·E·I_z}·\frac{{\left(-1\right)}^{n+1}}{n^5}$$

(83)

Thus:

$$y\left(x\right)={\displaystyle \sum }_{n=1}^{\infty }{C}_n·sin\frac{n·\pi ·x}{l}=\frac{2·q·l^4}{{\pi }^5·E·I_z}·{\displaystyle \sum }_{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{n^5}·sin\frac{n·\pi ·x}{l}$$

(84)

Since for this case of support and loading y(x) for x = l/2, the calculation relation is of the form [1,2,3,4,5,6]:

$$y\left(x\right)=\frac{5·q·l^4}{768·E·I_z}=0.00651·\frac{q·l^4}{E·I_z}$$

(85)

For n = 1 from relation (83), it follows:

$$\begin{gathered} y\left(x\right)=\frac{2·q·l^4}{{\pi }^5·E·I_z}·{\displaystyle \sum }_{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{n^5}·sin\frac{n·\pi ·\left(\frac{l}{2}\right)}{l}=\frac{2·q·l^4}{{\pi }^5·E·I_z}·{\displaystyle \sum }_{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{n^5}·sin\frac{n·\pi }{2}\underset{}{\Rightarrow } \\ y\left(x\right)=0.00653·\frac{q·l^4}{E·I_z} \end{gathered}$$

(86)

The relative deviation between the data obtained from relations (84) and (85) is 0.306%.

6. Conclusions

The Fourier series method is used to obtain the solutions for the unknown transverse displacement y(x) as infinite series, both in the case of static and dynamic loads. In this work, the concept of the average deformed fiber was defined in advance, highlighting the fact that this is a continuous periodic or non-periodic function. In this work, the concept of the average deformed fiber was defined in advance, highlighting the fact that this is a continuous periodic or non-periodic function. This can be expressed through trigonometric series and Fourier series.

The following cases were defined: simple right beam, supported on its ends, loaded in the transverse plane using a concentrated force (F); in the second case, it is considered a simple right beam, supported on its ends, loaded in the transverse plane using a uniformly distributed load (q); and in the third case, it is considered a simple right beam, supported on the ends, loaded in the transverse plane using a linear load distributed q(x). All these types of loads are applied statically.

The determination of the displacement of the center of gravity of the cross-section is established using trigonometric series and Fourier series and the results obtained are compared with those provided by the specialized literature (the displacement of the center of gravity being determined by direct integration of the differential equation of the average deformed fiber).

For the first case of support and loading, using trigonometric series, the relative deviation between the data obtained from relations (38) and (41) is 1.243%, the data obtained from relations (39) and (41) is 0.028%, and the data obtained from relations (40) and (41) is 0.129%.

In the second case of support and loading, using trigonometric series, the relative deviation between the data obtained from relations (49) and (52) is 0.384%, the data obtained from relations (50) and (52) is 0.027%, and the data obtained from relations (51) and (52) is 0.004%.

For the first case of support and loading using Fourier series, it is observed that relation (71) is identically equal to relation (49).

In the second case of support and loading using Fourier series, the relative deviation between the data obtained from relations (41) and (77) is 1.449%.

In the third case of support and loading using Fourier series, the relative deviation between the data obtained from relations (84) and (85) is 0.306%.

The displacements are found as single series of infinite terms with good convergence.

The trigonometric series and Fourier series method has been successfully applied in this work to solve flexural problems.

Even though the method of trigonometric series and Fourier series is known and applied in many fields of science in the current applications of mechanics of materials, it is not currently used. For this reason, this work intends to popularize these methods by presenting them in an exhaustive and comprehensive form, starting from the presentation of the concept of medium deformed fiber to the solution of some current applications.

Thus, it can be concluded that the use of the trigonometric series and the Fourier series presents a series of advantages: they are well studied and have a solid theory, there are a variety of theorems and properties that can be used to understand the behavior of Fourier series and to perform calculations with them; they have a wide range of practical applications; offers an elegant and simple method that leads to fast and accurate solutions that can be easily implemented in a wide range of programs, such as MATLAB or Mathematica.