This article introduces the method for computing lateral deflections of plane beams undergoing symmetric bending. Reviewers should be acquainted with: integration of ordinary differential Equations, and statics of plane beams under symmetric bending.

Our primary objective is to apply the discontinuous beam bending differential Equation to different application obviously representing beam bending. One of the most common types of structural components is a beam, recommended more in Civil and Mechanical Engineering. A beam resembles as a bar-like structural that is used to support transverse loading and carry it to the supports. Beams resist against transverse loads through a bending action, which creates compressive longitudinal stresses on one side of a beam and tensile stress on the other side. With these two combinations between compressive longitudinal stress and tensile stress, an internal bending moment starts to occur. In the case of a discontinuous load, we begin applying beam-bending differential Equation for each part of the beam.

All models use some sort of approximation to the underlying physics because beams are three-dimensional bodies.

Transverse loading being resisted on a preferred longitudinal plane is known as a

plane beam. Because the classical beam theory is the simplest and most associated

model for plane beams, it presents assumptions such as:

1) Planar symmetry: The longitudinal axis appears to be straight with a cross section of the beam being longitudinal plane of symmetry. Each sections that lie on the plane, both resultant of the transverse loads. The resultant of the transverse loads acting on each section lies on the plane.

2) Cross sectional variation: The cross section remains constant or varies.

3) Normality: The plane sections are originally normal to the longitudinal axis of the beam remain plane and normal to the reformed longitudinal axis upon bending.

4) Strain energy: Transverse shear and axial forces are ignored, while only internal strain energy from another object accounts for the bending moment deformations.

5) Linearization: The infinitesimal deformation of the beam is brought into the mix due to the consideration of transverse deflections, rotations and deformations.

6) Material model: The heterogeneous beams are fabricated with several elastic and isotropic materials, such as reinforced concrete.

Transverse shear and axial force are ignored, while only internal strain energy from another object accounts for the bending moment deformations. The assumption of infinitesimal deformation is brought into the mix due to the consideration of transverse deflections, rotations and deformations. The assumption is made that heterogeneous beams are fabricated with several elastic and isotropic materials.

Now we will begin our discussion on classical beam theory, also known as The Euler- Bernoulli Beam Theory.

・ Beam coordinates system:

The coordinate system throughout the beam undergoes a transverse loading at a point on the top surface will shorten. As for the other, it will elongate. This causes a neutral surface between the top and the bottom.

・ Beam motion:

We associate beam motion as the loading on a x, y plane beam is structured in to two dimensional displacement field and u and v are respected as the axial and transverse displacement components with respect to a beam point.

・ Beam loading:

is denoted as the transverse force per unit length occurs on the plane beam in a positive y direction. We can determine the strong and the weak loading points based on what beam is being used. For instance, support on a simply supported beam is found on the end points that prohibit transverse displacements. In contrast, one side of a cantilever beam does not have an end support, resulting with one being clamped on and the other being free. Airplane wings, diving boards, and stabilizers are prime examples of cantilever beams.

This section will explain the equivalent distributed force for a family of singular loading condition by using Schwartz’s distribution theory.

Definition 1. Let be a distributed force. The n^th order moment of that is given by

Definition 2. Let be a distributed force in the small open segment (interval). Also consider

Then

In 1959, Timoshenko and Woinowsky-Krieger [1] studied the concentrated double moment of the beam, which is the limiting situation of two opposite movements acting on two different separated points. They proved the result in a deflection with a discontinuous slope at the point of the concentrated double movement.

In this article, we want to study the equivalent distributed force in the loading function of a point moment of order n by using the distribution theory, refer [2] . We will show that the loading function for this loading condition is expressed by

where is the value of double movement and is the second distributional derivative of.

Theorem 1. The equivalent distributed force of a unit moment of order n applied at is

where is the nth distributional derivative of the Dirac Delta function.

Corollary 1. The equivalent distributed force for an upward concentrated force of magnitude p is

This was obtained by Timoshenko (1976) [3] and Shames (1989) [4] , where the limiting case of a load distributed over a very short portion of a beam. The shearing forces of an Euler-Bernoulli beam can be applied to act as another proof of this representation by using the discontinuity of a concentrated force, appears in Section 3.4.

Corollary 2. The equivalent distributed force of a clockwise concentrated moment of magnitude M is

This result was founded by Shames in (1989) [4] , in which this loading is considered to be the limiting case of two concentrated forces, apart, when goes to zero. The bending moment of an Bernoulli beam can be applied to act as another proof of this representation by using the discontinuity a concentrated moment introduces. It appears in Section 3.3.

Corollary 3. The equivalent distributed force of a concentrated double moment is given by Equation (4). As Timoshenko and Woinowsky-Krieger (1959) [1] mention, this loading results in a deflection with a discontinuous slope at the point of double moment. We see later that, in an Euler-Bernoulli beam with a jump discontinuity in slope, this forcing function appears.

In this section we obtain the equivalent distributed force of a distributed moment. We then give a mathematical explanation for corner condition in classical plate theory.

It can be shown that the force function of a distributed moment, , can be expressed in terms of m and

But distribution theory shows that for any function f

Hence,

Accordingly, for a distributed moment the first distributional derivative of, is the forcing function. Imagine a beam that has a length of L under a distributed moment, (see Figure 1(a)). The moment can be written as

Hence,

where H is Heaviside’s function. When substituting Equation (12) into Equation (10), the output is

The distributed moment is equivalent to the distributed force on both end points of the beam given by. Also distributed moment is equivalent to two

concentrated forces and at and, respectively, as shown in Figure 1(b). Similarly, if is a partially distributed moment in is equivalent to a distributed force in this interval and two concentrated forces at and.

Corner condition: Timoshenko and Woinowky-Krieger (1959) [1] were unable to explain the corner condition mathematically, although they mentioned it physically as below:

・ Corner conditions definition―physically: The polygonal loaded plates will usually produce concentrated reaction at corner points with the distributed reaction along the edges.

・ Corner conditions definition―mathematically: In classical plate theory Equation (14) consists of a system of distributed forces and two concentrated forces as the corner points.

Note: Corner conditions phenomenon does not appear in sheer deformation theo- ries.

The classical method of solving the differential Equation of Euler-Bernoulli beam with jump discontinuity in slope, deflection and flexural stiffness, is to solve the problem on both sides of the discontinuities and then apply boundary and continuity conditions. Here we will solve a problem of differential Equation in the space of generalized functions we solve the problem as a single beam using generalized functions therefore we will consider only one point of jump discontinuity and then generalize this idea with n singular points of an Euler-Bernoulli beam.

Euler-Bernoulli beam theory provides the following displacement field assumptions:

where, , are displacement components along the x, y, and z axes respectively. The beam lies along the x-axis and the loads are applied vertically along the z-axis. We can use Equation (15) and the foundation of virtual work, the governing equilibrium Equation can be declared as:

where EI is the flexural stiffness and q is a distributed force and is called the loading function.

The first three derivatives of w are continuous and the fourth derivative is piecewise continuous, only when q is a piecewise continuous function. On the other hand, there are some equivalent distributed conditions for which the loading function cannot be declared as a classical function. A general case of these conditions were studied in Section 2. However, displacement of the beam or its derivatives can sometimes have discontinuities that are separate from the loading condition. That introduces the focus of this section.

The beam shown in Figure 2 is of length L and the boundary conditions are arbitrary at. The flexural stiffness of the beam is changed periodically at. This point will also house discontinuities in slope and deflection. The most general case uses a combination of an internal hinge with a rotational spring along with a shear-free connection with a translational spring. The constants of the rotational and translation springs are and, respectively. Now we take

The beam has two components to it, segments AB and BC. Hence, why the Heaviside’s function is applied.

where w is the deflection of the beam, is the deflection of the segment AB, and is the deflection of the segment BC. After the calculation given in the appendix, the governing differential Equation of the beam is

where the distributional differentiation is denoted by the bar. As can be seen having jump discontinuities in slope and deflection is equivalent to having double and triple point moments, at the point of jump discontinuities. Also, continuity conditions can be written as

when you apply the four boundary conditions at as well as the continuity conditions from Equation (21), you are able to gather the deflection w. As can be seen only the force term changes; the form of the operator of the differential Equation is the same as that of Equation (16).

Much like the work introduced on Euler-Bernoulli beams; we can also acquire jump discontinuities in slope, deflection, flexural stiffness, and shear stiffness to a system of differential Equations of a Timoshenko beam [6] . Differentiation between Euler- Bernoulli beams and Timoshenko beams are the distant shear deformation in the Timoshenko beams. Shear deformation is where a force is being applied on one part and another force is being applied on another part, but in the opposite direction. Along the x, y, and z-axes, are displacement components, , and. The displacement field is provided by the Equations

, , and are displacement components in the plane. Also, is the rotation of the Timoshenko beam about the y-axis and the superscript T shows the deflection of the Timoshenko beam. The Governing system of differential Equations can be written as,

the shear modulus, G, is a distinguished ratio of the shear stress over the shear strain. The shear stress is the force applied to a certain amount of area it is applied to. The shear strain on the other hand, is the rate of change in displacement of the strain combine with,

the ratio of shear and flexural stiffness has now been defined.

We can substitute Equation (88) into Equation (87) yields,

Next, ponder a Timoshenko beam has jump discontinuities in the slope, deflection, shear stiffness, and flexural stiffness at on its length L. We refer to the heavy- side function as used for the Euler-Bernoulli beam to initiate the two Equations of deflection and rotation of the Timoshenko beam, we can write

Because deflection and rotation both have jump discontinuities at, we write

It is known that

and

Also, for an infinitesimal element including the discontinuity point at equilibrium implies

As we can see from Equation (94), and are the stiffness of the translational and rotational springs at. So after reviewing and comparing Equation (92), Equation (93) and Equation (94),

where

Differentiating Equation (90), we obtain

and

The deflection and rotation of each beam segment have continuous derivatives and hence they are governed by Equation (84), thus

and

Now if we let and we can obtain the governing system of equi-

librium Equations for the beam from Equation (97) through Equation (104),

Structural analysis of multi-cracked beams is of greater engineering interest. Research in this area has been mainly concentrated on two classes of problems:

・ definition of appropriate linear and non-linear models for representing the effects of cracks under static and dynamical loadings and

・ detection of position and severity of the damage by using either static or dynamic tests.

Here we will study an effective and physically based linear modeling of multi-cracked beams subject to the static loading. This result is useful for treating any type of concentrated damage occurring in slender and short beams, e.g. corrosion of steel bars in reinforced concrete members, defects of material and attacks of biotic agents in timber elements etc.

The idea of treating multi-cracked beams with equivalent linear springs at the crack’s position is based on the portion of each member into undamaged pieces between two consecutive cracks. For slender Euler-Bernoulli beam, the governing 4^th order differential Equation of bending can be written for each subsystem, but it is necessary to impose the pertinent continuity conditions between adjacent subsystems to obtain the static response of the whole beam.

As a result, the computational effort increases with the number of cracks, i.e for n cracks along the beam, algebraic Equations have to be solved to compute integration constants. But this method is inefficient for identifications purposes, when analysis are repeated until position and severity of the damage are found. So one can use finite element method (FEM) in which stiffness matrix and load vector of the non-cracked Euler-Bernoulli beam are modified with some dimensionless coefficients with effects of internal cracks.

Here we will be using generalized functions to handle static and kinematical discontinuities along the beam. Here we require the enforcement of continuity conditions at each jump, and hence additional integration constants are needed. This issue can be tracked in the formulation of “rigidity modeling”, which consists of singularities in the flectual stiffness represented by Dirac’s delta functions, which in turn are equivalent to internal hinges with rotational linear-elastic springs.

In this work, a non-trivial generalization to multiple discontinuities in the curvature and the slope functions of the integration procedure is presented. The case of Euler-Bernoulli beams under static loads is treated; discontinuities in the curvature and the slope function are modeled as unit step distributions and Dirac’s delta, respectively, in the flectural stiffness of the beam. Moreover, the presented procedure is also extended to cases of discontinuities in the axial displacement and in the vertical deflection modeled as Dirac’s deltas in the axial stiffness and the shear stiffness, respectively.

For the case of multiple slope discontinuities, we adopt a new technique of integration.

In the case of flexural stiffness provided by Equation (131), the governing Equation (129) takes the following form:

To solve the product of two Dirac’s delta distributions, we will rely on the technique proposed by Bagarello (1995; 2000) [9] . Bagarello indicates that the product of two Dirac’s deltas both centered at can be reduced to a singe Dirac’s delta multiplied by a constant A as follows:

where; and is a test function.

Hence,

Substituting Equation (142) into Equation (140) given the following explicit expression of the curvature for the considered beam model

Integration of Equation (143) provides the following slope function showing discontinuities at the abscissa

Further integration of Equation (144) provides the following closed form expression for the deflection function of the beam

The bending moment and shear force function formally coincide with Equation (126) and Equation (128), respectively. In fact, for statically determine beams, and should not depend on the adopted flexural stiffness. On the contrary, for statically indeterminate beams, the adopted flexural stiffness model will affect the expressions of the constants,.

The slope function defined in Equation (144) presents jump discontinuities at that are explicitly evaluated as follows:

and comparison of Equation (146) with the bending moment given by Equation (126) evaluated at leads to

Equation (147) corresponds to the presence of internal hinges at;, endowed with rotational springs with stiffness as shown in Figure 8(a), given as

Since rotational spring stiffness can take values from zero (no rotational spring) up to infinity (continuous beam at, with no internal hinges), discontinuity values, in view of Equation (148), can take values from 0 up to 1.

However, if rotational spring stiffness, are assigned to the related value have to be obtained by Equation (148) for a value of the quantity A among those proposed by Bagereuo (1995) [9] .

Chalishajar, D., States, A. and Lipscomb, B. (2016) On Applications of Generalized Functions in the Discontinuous Beam Bending Differential Equations. Applied Mathematics, 7, 1943- 1970. http://dx.doi.org/10.4236/am.2016.716160