Abstract
We investigate semianalytical solutions of Euler-Bernoulli beam equation by using Laplace transform and Adomian decomposition method (LADM). The deformation of a uniform flexible cantilever beam is formulated to initial value problems. We separate the problems into 2 cases: integer order for small deformation and fractional order for large deformation. The numerical results show the approximated solutions of deflection curve, moment diagram, and shear diagram of the presented method.
1. Introduction
The Euler-Bernoulli beam theory states that the action load produces the bending moment which is proportional to deflection characteristics of the beam. The equation of this law can be written as follows:where is deflection curve of a uniform beam, the modulus of elasticity , and the moment of inertia . We note that and are both constant and the product of and is called beam stiffness. In a case of small deformation, we assume that is infinitesimal. Equation (1) is reduced to the well-known fourth-order linear differential equation: In this study we consider the uniform flexible of cantilever beam; see Figure 1. The parameters and are undeformed length and horizontal displacement, respectively. The deformed length of beam is verified by the integral , where . It was shown in [1] that the slope of deflection curve represents the following equation , where for the known function . The deflection curve in Figure 1 corresponds to initial value problem of the geometric problem:where . If the slope is very small, the linear Euler-Bernoulli beam theory [2] governs the problemThe solution of the problem (4) isOur method proposes to reform problem (3) in a sense of fractional calculus without linearization. Because fractional calculus is the great idea to describe behavior in nature, for example, force response of viscoelastic material [3, 4], fluid flow [5], and fitting experimental data [6]. For convenience, we set and . We develop (1) to deal with fractional order ; system (3) becomesWe use Adomian polynomial to approximate a nonlinear term and derive a semianalytical solution by use of Laplace transform for the initial value problem (6).
2. Preliminaries
In this section we introduce some definitions and theory of fractional calculus and Laplace transform which are used in our method. See [7] for more details.
Definition 1. The Riemann-Liouville fractional integral operator of order is defined as
Definition 2. The Caputo fractional derivative of order is defined as
Theorem 3. If and , then .
Theorem 4. The Laplace transform of the Caputo fractional derivative is given bywhere and is a nonnegative integer. The inverse Laplace transform of power function is given by
3. Semianalytical Solution via LADM
To formulate the general solution of problem (6), we replace with in the equation of problem (6) and applied Theorem 3. We obtainTaking Laplace transform of (11) givesUsing Theorem 4 and replacing by , we can rewrite asWe take inverse Laplace transform of (13), and the following operator equation is obtained:We apply Adomian polynomial [8, 9] to approximate a nonlinear term of (14), by settingEquation (14) becomes Assume that the solution of (11) can be written as . From the combination of Laplace transform and Adomian decomposition method, we obtain the formulation of LADM for fractional order of cantilever beam deflection equation:In this work, we apply the first four terms of LADM which are provided from following recurrence:whereThe components of adomian polynomial are given by The iteration results of recurrence (18) are shown as follows: The first four terms’ approximate solution is : Since and , thus . This yields
3.1. Integer Order for Small Deformation
We simulate some deflection curve of the results in (24) and (5) which are solution of LADM and classical method, respectively. The parameters are in., in., kip in.2, , and kip. Figure 2 shows the deflection of cantilever beam, obtained by using classical method and LADM. LADM has accurate slope around the free end of the beam as well as classical method. Moreover LADM shows that the deflection curve remains nearly straight line in the case of small deformation.
The effects of various loads influencing maximum deflection at the free end and length of the deformed beam are presented in Table 1. We observe that the calculation of nonlinear integer order deformation via LADM fails at kips and kips. Since thus integral of bending moment is greater than stiffness of the beam. This yields large deformation, large rotation, and large strain, namely, some concept of large deformation. Not only does LADM fails, but also classical method does, because it contradicts the assumption that the lengths of deformed and undeformed beam are identical. This is the motivation for reform in fractional order model.
3.2. Fractional Order for Large Deformation
There are many methods for describing a large deformation of a beam, for example, the Laplace-Padé coupling with NDHPM and HPM [10], VIM [11], and pseudolinear system [1]. In this section, we introduce the process of investigating semianalytical solution via LADM and some numerical results of pseudolinear system in [1].
To illustrate the numerical results we suppose that horizontal displacement . The beam parameters are in. and kip in.2. The fact from our assumption is
In this case we fixed the concentrate load kip. We get the deformed length and initial condition . Fractional order is determined by simulating the results of slope at and deformed length. Table 2 shows that order is given, and , which is the most accurate for the fact in (25). We design order for this problem. Using the first four terms’ approximation, the deflection is shown in Figure 3(a). The slope deflection of psudolinear system and LADM are presented in Table 3. Moment diagram can be computed from the deflection curve as follows: and for shear diagram. The results of both moment and shear diagram are shown in Figures 3(b) and 3(c), respectively.
(a) Large deflection curve
(b) Moment diagram
(c) Shear diagram
4. Conclusion
In this study, we use Euler-Bernoulli beam equation for describing the uniform flexible cantilever beam with a concentrated load. The initial conditions are given by calculating slope of the beam. We use LADM to determine the semianalytical solution. LADM with integer order system can approximate solution without cancellation a nonlinear term in the case of small deformation. For the large deformation, we reform the problem to fractional order system and estimate fractional order which conserves the fact from the assumption. Finally, we show that LADM gives the solution as a polynomial expression which is the advantage for analyzing moment and shear diagrams. LADM may be a powerful and successive method for solving nonlinear science and engineering problem.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.