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Journal of Applied Mathematics
Volume 2014, Article ID 686421, 7 pages
Research Article

Conservative Semidiscrete Difference Schemes for Timoshenko Systems

Department of Mathematics, Federal University of Pará, Augusto Corrêa Street 01, 66075-110 Belém, PA, Brazil

Received 2 December 2013; Accepted 5 February 2014; Published 7 May 2014

Academic Editor: Suh-Yuh Yang

Copyright © 2014 D. S. Almeida Júnior. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We present a parameterized family of finite-difference schemes to analyze the energy properties for linearly elastic constant-coefficient Timoshenko systems considering shear deformation and rotatory inertia. We derive numerical energies showing the positivity, and the energy conservation property and we show how to avoid a numerical anomaly known as locking phenomenon on shear force. Our method of proof relies on discrete multiplier techniques.

1. Introduction

The importance of beam theories is well known in the world of engineering and mathematics. The most commonly used beam theories of linear elastic structures are the Euler-Bernoulli beam, the Rayleigh beam, and the Timoshenko beam, with the last being a hyperbolic partial differential equation that has many applications in cutting-edge technologies for flexible structures, such as robotics. The governing equations of Timoshenko beams are considered an improvement over the Euler-Bernoulli and Rayleigh beams since shear deformation is taken into account. The work of Han et al. [1] offers an excellent description of these models for transversely vibrating uniform beams.

There is a large number of publications concerning the study of Timoshenko systems [2]. In general, this has been always a description through a system of second-order differential equations, in which the vibration amplitude and the angle due to pure bending were the searched functions. One instructive introduction in [3] contains important references to the early progress in investigating the solutions that represent the mechanical deformations of the Timoshenko model.

The 1D equations to the theory of Timoshenko for plane beams are given by where is the time, is the distance along the center line of beam, is the transverse displacement, is the rotation of the neutral axis due to bending, is the mass density of material, is the bending moment, is the transverse shear force, is the cross-section area, and is the second moment of cross-section area. The relationship between stress and stretch for elastic behavior of the elastic beam is given by where is Young's modulus, is the modulus of rigidity, and is the transverse shear factor.

Therefore, taking into account (2) and (1), Timoshenko [2] established the following differential equations that incorporate the effects of transverse shear in the cross section of the beam of length : where , and .

Now let us mention some numerical aspects concerning 1D Timoshenko systems. Space semidiscretizations are common in many works dealing with Timoshenko beams [48]. In general, they arise in the design of numerical schemes to compute approximated solutions of continuous models. For example, a common approach consists in choosing a spatial discretization taking into account the application of finite element methods on partial differential equations. The results are systems of ordinary differential equations in time .

It is well known that standard Galerkin finite element method using equal-order piecewise linear approximations for the rotations and displacement yields locking phenomenon. This means that this method produces unsatisfactory numerical results when the thickness is very small. From the point of view of the numerical analysis, the locking phenomenon usually shows itself in that the a priori error estimates for these methods depend on the thickness of the structure in such a way that it degenerates when this parameter becomes small. We recommend the notable works by Arnold in [9], Hughes et al. [10], Prathap and Bhashyam [11], Li [12], Loula et al. [13, 14], and Reddy [15] to a deep discussion of the subject. In particular, the locking phenomenon is characterized by an overestimation on rigidity coefficient given by in the case of standard finite element method where is the mesh-size (see [11]).

For a finite the overestimation is grossly exaggerated by the shear stiffness term and it modifies the bending rigidity. In particular, for plane beams of rectangular geometry with width and thickness we get and is rewritten as

We can see the locking on shear force in the limit case . To avoid it we can take the limit , but naturally there exist limitations to computational cost.

This paper is mainly concerned with theoretical analysis of finite-difference schemes applied to Timoshenko equations (3)-(4) aiming to identify the locking numbers. To identify them, we use the energy method adapted for our numerical approach. We do not attempt to converge questions for the semidiscrete schemes treated here. We refer the readers to the excellent survey of Zuazua [16] where semidiscrete hyperbolic problems have been considered.

The remainder of the paper is organized as follows. In Section 2 we established the energy conservation property to continuous model of Timoshenko beams. In Section 3 we introduce numerical schemes in spatial finite difference applied to Timoshenko equations and we show how the locking numbers appear. To do this, we use the -scheme. Finally, in Section 4 we conclude our work with future perspectives.

2. Energy Conservation Property of the Timoshenko Equations

In this section we describe the energy conservation property with respect to energy functional of the Timoshenko equations.

For our purposes we consider here the following boundary conditions of more practical interest that are given byfree-free: built in-free: hinged-hinged:built in-hinged:built in-built in:for all . Additionally we consider initial conditions given by

There exists an important nonlinear functional related to Timoshenko equations. It is an energy functional composed of potential and kinetic energy. Indeed, taking into consideration any boundary conditions (7)–(11) the energy of solutions for Timoshenko equations (3)-(4) is given by

As usual, we can find that if we multiply formally the first equation in (3) by we get and then

On the other hand, if we multiply formally the second equation in (4) by we get

If we add up (15) and (16) we get and then

Taking any boundary conditions (7)–(11) we obtain that

We call this the energy conservation property and the Timoshenko equations are purely conservative. Therefore, it is interesting and convenient that numerical schemes applied to Timoshenko equations preserve consistently the property (19). That is to say, the numerical energy must be nonlinear and positive and it will obey the energy conservation property.

3. Finite-Difference Semidiscretizations and Properties

In this section, we introduce numerical schemes in finite-difference semi-discretization applied to 1D Timoshenko equations.

Firstly, we present the equations used by Wright [8] and we obtain the associated energy (locking free) by using the energy method. This method consists in the use of the discrete multiplier techniques as described by Infante and Zuazua [17].

Secondly, we consider the -scheme applied to angle rotation and we determine the values of to achieve the positivity of the associated energy and to avoid the number in (6). Again, we use the discrete multiplier techniques.

For our purposes we consider an integer nonnegative and a spatial subdivision of the interval given by with each node of the mesh. We write the following finite-difference semidiscretization to (3)-(4) as where we use to discrete Laplacian in one spatial dimension. Here “” denotes derivation with respect to time. Moreover, we use for all and to denote the approximate values of the continuous solutions and on the mesh, respectively. The homogeneous boundary conditions (7)–(11) in semidiscrete setting are given by

(i) free-free:

(ii) built in-free:

(iii) hinged-hinged:

(iv) built in-hinged:

(v) built in-built in:

. The initial conditions are discretized as follows:

Equations (21)-(22) were the object of the study in the works of Wright [7, 8] and the references contained therein. In particular, these equations can be obtained by assembling the stiffness and (lumped) mass matrices of a two-noded beam element which has linear shape functions for rotation and displacement . Moreover, these equations are locking free; that is, the associated numerical energy is free of the overestimation (6). See Proposition 1 below.

The energy of system (21)–(29) is given by which is a discretization of the continuous energy given in (13). Note that is free of the overestimation (5).

Proposition 1 (conservation of energy). For any and solutions of (21)-(22) and for any boundary conditions (24)–(28) and initial conditions (29), the energy is given by and it satisfies

Proof. To prove our affirmation we use the energy method in the following manner: we multiply (21)-(22) by using the discrete multipliers and , respectively, and we take the sum over . Then we obtain
Some elementary calculations show that
and then we can write
Adding the functionals , and collecting the terms above we can write
We note that the boundary conditions (24)–(28) result naturally from this procedure. Therefore, for any boundary conditions (24)–(28) we get
and therefore the system (21)–(29) is conservative in agreement with continuous counterpart.

3.1. The Locking Number from -Scheme

In general, discrete models may not share the same qualitative behavior as their continuous counterparts. This is the case of locking phenomenon on shear force in Timoshenko beams characterized by overestimation (6).

Here we show for -scheme how locking number appears. We assume a convex linear combination to angle rotation in the following manner: we replace by

Then we establish a difference scheme for (3)-(4) as follows: where is a positive parameter. For simplicity we consider homogeneous Dirichlet boundary conditions

The energy of (40)-(41) is given by where we have that

It is clear that the positivity of holds for and only for we have that . To see how the locking number for any affects the numerical solution, again we consider a rectangular beam, where with width and thickness and then

Clearly for we do not have accurate numerical approximations when the beam thickness is small. This is the case of (equivalent to standard finite element method) such that and the same problem occurs with standard finite difference given by .

Proposition 2. For any and solutions of (40)-(41) we have

Proof. In what follows we use the results from Proposition 1. Then we have
On the other hand,
Now is rewritten as
On the other hand, having in mind the boundary conditions (41), we have that and then
Therefore, we obtain that and then we get where is given by (42).

Remark 3. Deriving the energy , collecting the common terms, and using the conservation law (53) we obtain the semidiscrete equations given by
Naturally, we can notice that (54) is inspired in a dispersive approximation of the Timoshenko equations given by
For any , (56) is a typical example of a modified PDE. In [18], LeVeque offers an excellent approach of numerical methods applied to PDE that modify the original PDE.

4. Conclusion

In this work, we have developed nonstandard finite-difference semidiscretizations applied to the Timoshenko equations (3)-(4) by using the -method. We showed by the discrete energy method that the numerical equations preserve the positivity and the conservation to numerical energy and we identified a numerical anomaly known as locking phenomenon for .

The results presented here are important from point of view of the theoretical numerical analysis. Actually, an important problem from stabilization theory of the Timoshenko equations (3)-(4) says respect to exponential decay by taking into account some dissipative mechanism. In particular, Muñoz Rivera and Racke [19, 20] showed that (3)-(4) on effect of an only damping of the type on angle rotation equations are exponentially stable if and only if . Therefore, it is interesting to know if the locking free scheme preserves the same qualitative behavior.

Important issues to be investigated concern observability and control of numerical solutions of Timoshenko equations. To the best of our knowledge, we did not find in the literature numerical problems concerning observability uniform to semidiscrete schemes applied to the Timoshenko equations.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.


The author would like to acknowledge the support of the following corporations: UFPA, PROPESP, and FADESP through PAPQ. The author thanks the anonymous referees for their valuable comments as well as reference suggestions that helped to improve the presentation of this paper.


  1. S. M. Han, H. Benaroya, and T. Wei, “Dynamics of transversely vibrating beams using four engineering theories,” Journal of Sound and Vibration, vol. 225, no. 5, pp. 935–988, 1999. View at Google Scholar · View at Scopus
  2. S. P. Timoshenko, “On the correction for shear of the differential equation for transverse vibrations of prismatic bars,” Philosophical Magazine, vol. 6, pp. 744–746, 1921. View at Google Scholar
  3. T. C. Huang, “The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions,” Journal of Applied Mechanics, vol. 28, pp. 579–584, 1961. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. D. Krieg, “On the behavior of a numerical approximation to the rotatory inertia and transverse shear plate,” Journal of Applied Mechanics, vol. 40, no. 4, pp. 977–982, 1973. View at Google Scholar · View at Scopus
  5. T. Belytschko and W. L. Mindle, “Flexural wave propagation behavior of lumped mass approximations,” Computers and Structures, vol. 12, no. 6, pp. 805–812, 1980. View at Google Scholar · View at Scopus
  6. W. L. Mindle and T. Belytschko, “A study of shear factors in reduced-selective integration mindlin beam elements,” Computers and Structures, vol. 17, no. 3, pp. 339–344, 1983. View at Google Scholar · View at Scopus
  7. J. P. Wright, “A mixed time integration method for Timoshenko and Mindlin type elements,” Communications in Applied Numerical Methods, vol. 3, no. 3, pp. 181–185, 1987. View at Google Scholar · View at Scopus
  8. J. P. Wright, “Numerical stability of a variable time step explicit method for Timoshenko and Mindlin type structures,” Communications in Numerical Methods in Engineering, vol. 14, no. 2, pp. 81–86, 1998. View at Google Scholar · View at Scopus
  9. D. N. Arnold, “Discretization by finite elements of a model parameter dependent problem,” Numerische Mathematik, vol. 37, no. 3, pp. 405–421, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T. J. R. Hughes, R. L. Taylor, and W. Kanoknukulchai, “A simple and efficient finite element method for plate bending,” International Journal for Numerical Methods in Engineering, vol. 11, no. 10, pp. 1529–1543, 1977. View at Google Scholar · View at Scopus
  11. G. Prathap and G. R. Bhashyam, “Reduced integration and the shear-flexible beam element,” International Journal for Numerical Methods in Engineering, vol. 18, no. 2, pp. 195–210, 1982. View at Google Scholar · View at Scopus
  12. L. K. Li, “Discretization of the Timoshenko beam problem by the p and the h-p versions of the finite element method,” Numerische Mathematik, vol. 57, no. 4, pp. 413–420, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  13. A. F. D. Loula, T. J. R. Hughes, and L. P. Franca, “Petrov-Galerkin formulations of the Timoshenko beam problem,” Computer Methods in Applied Mechanics and Engineering, vol. 63, no. 2, pp. 115–132, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. A. F. D. Loula, T. J. R. Hughes, L. P. Franca, and I. Miranda, “Mixed Petrov-Galerkin methods for the Timoshenko beam problem,” Computer Methods in Applied Mechanics and Engineering, vol. 63, no. 2, pp. 133–154, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. J. N. Reddy, “On locking-free shear deformable beam finite elements,” Computer Methods in Applied Mechanics and Engineering, vol. 149, no. 1–4, pp. 113–132, 1997. View at Google Scholar · View at Scopus
  16. E. Zuazua, “Propagation, observation, and control of waves approximated by finite difference methods,” SIAM Review, vol. 47, no. 2, pp. 197–243, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. A. Infante and E. Zuazua, “Boundary observability for the space semi-discretizations of the 1-D wave equation,” Mathematical Modelling and Numerical Analysis, vol. 33, no. 2, pp. 407–438, 1999. View at Google Scholar · View at MathSciNet
  18. R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J. E. Muñoz Rivera and R. Racke, “Mildly dissipative nonlinear Timoshenko systems—global existence and exponential stability,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 248–278, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. J. E. Muñoz Rivera and R. Racke, “Global stability for damped Timoshenko systems,” Discrete and Continuous Dynamical Systems B, vol. 9, no. 6, pp. 1625–1639, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet