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Modeling Error in the Two-Node Timoshenko Beam Finite Element

JOÃO ELIAS ABDALLA FILHO and MICHELLE SCHUNEMANN PEREIRA

Programa de Pós-Graduação em Engenharia Mecânica

Pontifícia Universidade Católica do Paraná

Rua Imaculada Conceição, 1155 Prado Velho, Curitiba, Paraná

BRASIL

Abstract: - A displacement-based finite element for the analysis of beams is formulated using Timoshenko´s beam theory and strain gradient notation. The element is first-order shear deformable as its longitudinal displacement possesses only the independent term and a term linear in the thickness coordinate z. Strain gradient notation allows for assessing the modeling capabilities and deficiencies of the finite element. A spurious term is found in the transverse shear strain polynomial and it is identified as being responsible for the artificial stiffening of the element. Removal of the spurious term renders a better element. Numerical analyses show that the corrected element converges faster than the original one to the analytical solutions of a given problem.

Key-Words: - Timoshenko beam, finite element, strain gradient notation, modeling error.

1 Introduction

The objective of this paper is to identify the source of a modeling error present in the two-node Timoshenko beam finite element which causes artificial stiffening. This is done in the context of laminated composites. The element is formulated according to a first-order shear deformation theory which is suitable for computing global responses such as displacements and natural frequencies [1]. Strain gradient notation is employed to facilitate the identification and removal of the spurious term from the finite element.

Errors can be introduced into the finite element model at the individual element level which make the element overly stiff. This artificial stiffening is thus a quantitative error, i.e. an error in magnitude. One source of artificial stiffening is known as parasitic shear which is caused by incorrect coupling between flexural and shear deformations [2]. Parasitic shear also introduces qualitative errors into laminated composite models. A qualitative error is defined as a misrepresentation of the nature of a deformation [3]. For instance, it can be an error in the form of the represented solution throughout the element. That is the case in shear deformable laminated beams such as the one formulated here where a constant or linearly varying shear force along the length of the structure is misrepresented by a parabolic distribution. Another form of qualitative error is the misrepresentation of the direction of a certain displacement or even the complete obliteration of that displacement. These have been shown to occur in the analysis of laminated composite plates [3, 4].

The source of parasitic shear is the use of incompatible polynomials as approximations for the field variables. Incompatibility of polynomials manifest itself in two different ways. The polynomial may be incomplete such as is the case in the displacement representation of a four-node quadrilateral element for modeling plane problems [2]. Further, as it occurs in beam and plate analysis, the polynomial may be inconsistent. That is, the polynomial is incomplete and its order is inconsistent with the order of the theory being modeled [2, 3]. The spurious terms present into the strains approximations that are associated to parasitic shear may be clearly identified and eliminated if a physically interpretable notation is employed. Strain gradient notation is used here to formulate a laminated composite beam element. In the following sections, a strain gradient-based beam finite element for laminated composites is formulated. A parasitic shear term is identified and easily eliminated. The effects of parasitic shear in the beam model are investigated through numerical analyses. As a qualitative error is identified, its convergence characteristics are studied through mesh refinement.

2 Beam Finite Element

In this section, a two dimensional beam finite element for the analysis of laminated composites is formulated. As shown in figure 1 below, the element is two-noded with three degrees-of-freedom at each node; namely, the longitudinal displacement u, the vertical displacement w, and the rotation q.

The assumptions for the macromechanical behavior of the laminated beam are as follows: (i) plane sections normal to the longitudinal surface remain plane, but not necessarily normal after bending. Therefore, there is transverse shear deformation of the beam; (ii) there is perfect bond between laminae such that slip or separation cannot be represented, and (iii) strains and stresses normal to the middle surface are negligible.

[pic]

Fig. 1– Two-dimensional beam finite element.

The beam´s displacement field is defined by:

[pic] (1)

[pic] (2)

where [pic] is the axial displacement, [pic] is the midsurface´s vertical displacement, which defines the element´s vertical displacement, and [pic] is the in-plane rotation. These independent fields are represented by the polynomial expansions in strain gradient notation below:

[pic] (3)

[pic] (4)

[pic] (5)

while the longitudinal displacement [pic] is:

[pic] (6)

The quantities in brackets are the polynomial coefficients which in regular notation are unknown at the formulation stage. In strain gradient notation, the physical meanings of the polynomial coefficients are identified at the early steps as revealed by equations 3, 4, 5 and 6. In these equations, [u]0, [w]0 e [q]0 are rigid body displacements; [(x]0 e [(xz]0 are constant normal and transverse shear strains, and [(x,z]0 is the flexural strain. In general, they are referred to as strain gradients. The subscript o in the notation is employed to indicate that the strain gradients result from the evaluation of the corresponding Taylor series expansions at a given origin.

The nodal degrees-of-freedom are related to the strain gradients through the following:

[pic] (7)

where the nodal degrees-of-freedom and strain gradients are arranged in the following vectors:

[pic] (8)

[pic] (9)

and matrix ( is given by:

[pic] (10)

The columns of this matrix are linearly independent vectors which means the strain gradients form a basis of linearly independent deformation quantities for the element, including the rigid body modes.

The strain field is given by the derivatives of the displacements:

[pic] (11)

[pic] (12)

Inspection of equation 11, the definition of normal strain, shows that the coefficients are related to this strain. Therefore, they are both legitimate. However, the expansion for the transverse shear strain, equation 12, shows the presence of a shear strain term, but also of the flexural strain term [pic]. This equation indicates that when the beam undergoes a flexural deformation, there is an increase in shear strain. As this is physically impossible since there are no couplings between transverse shear and flexural strains, the latter term is certainly spurious. The erroneous increase in shear strain energy due to longitudinal strains is known as parasitic shear and it is a source for artificial stiffening, which causes slow convergence. The parasitic shear term appears naturally during the formulation procedure, as it has just been demonstrated, and it must be removed as it may cause strong deleterious effects in the performance of the element. It is important to note that the spurious term has been precisely identified a-priori through strain gradient notation. The element is corrected by simply removing the spurious term from its transverse shear strain polynomial.

Strains are related to strain gradients through:

[pic] (13)

and the stiffness matrix is given symbolically by:

[pic] (14)

where [UM] is termed strain energy matrix and is given by:

[pic] (15)

where[pic] is the constitutive matrix (relates stresses to strains) of a typical lamina k in global coordinates. Each column of the strain energy matrix represents the strain energy associated to a different strain gradient of the basis set of the finite element. The main diagonal terms are the strain energy quantities associated to pure deformation modes or strain gradients. In turn, the off-diagonal terms are the strain energy quantities associated to the coupling of deformation modes or strain gradients.

3 Numerical Analysis

This section presents the solution of a laminated composite cantilever beam comprised of two laminae of graphite/epoxy. The mechanical properties of the graphite/epoxy are E11 = 138 GPa, E22 = E33 = 14,5 GPa, G12 = G23 = G31 = 5,86 GPa, (12 = (23 = (31 = 0,21. The beam is 2,0 m long and it is loaded with 2000 N/m distributed along its length as shown in figure 2. The laminae are of the same thickness, but the graphite fibers of the top laminae are directed 30( with respect to the axis of the beam while the fibers of the bottom lamina are directed along that axis. This is a non-symmetric laminate, thus, coupling between bending and extension will occur [5].

[pic]

Fig. 2 – Laminated composite cantilever beam with distributed load.

In order to validate the finite element model, analytical results should be available for comparison. Hence, an analytical solution for this beam problem is derived through the minimization of the total potential energy. The solutions for displacements are:

[pic] (16)

[pic] (17)

[pic] (18)

where Q0 is the distributed loading, and A11, B11 and D11 are stiffness properties of the laminated beam.

Further, the normal and transverse shear strain expressions for the beam are:

[pic] (19)

[pic] (20)

Five uniform meshes comprised of two, four, eight, sixteen and thirty-two elements, respectively, are employed. These results are compared to the analytical results provided by equations 16 through 20. Each mesh is analysed using the model containing parasitic shear first and the using the model corrected for parasitic shear. This allows for studying the influence of refinement and of the spurious term in the solutions. The results are shown in the figures below.

Figure 3 and figure 4 show, respectively, the solutions for the vertical displacement w along the length of the beam for the models with and without parasitic shear. Both solutions converge to the analytical solution provided by equation 17. However, the rate of convergence is greater for the model corrected for parasitic shear, as displayed in figure 4.

In turn, figure 5 and figure 6 show the variation of rotation q along the length of the beam. Again, both numerical solutions converge to the analytical one given by equation 18, and the rate of convergence presented by the corrected finite element model is much greater.

Further, figure 7 and figure 8 display the variation of the axial displacement u along the length of the beam. This displacement only exists because the laminated in non-symmetric, and, therefore, presents coupling between bending and axial deformation. As in the previous analyses, the corrected finite element model solution (figure 8) converges faster than the parasitic shear model solution (figure 7) to the analytic solution given by equation 16.

The next four plots show the convergence characteristics of the beam model for strains. Figure 9 and figure 10 show, respectively, the variation of normal strain (x along the length of the beam. Convergence to the solution provided by equation 19 occurs for both models, although due to artificial stiffening, the model with parasitic shear presents a slower convergence.

Figure 11 and figure 12 show, respectively, the variation of transverse shear strain (xz along the length of the beam. Convergence to the analytic solution provided by equations 20 occurs nicely when using the model corrected for parasitic shear, as displayed by figure 12. However, a rather erroneous representation of (xz is provided by the model containing parasitic shear, as shown by figure 11. According to equation 20, the transverse shear strain varies linearly with the length of the beam. The representation provided by the model containing parasitic shear is parabolic, which is then obviously erroneous. Such representation reveals not only a quantitative error as in the previous solutions, but also an error of the qualitative kind. In the present case, it is an error in the shape of the solution. Further, this deleterious effect is augmented by mesh refinement. As shown in figure 11, the numerical solution gets farther and farther away from the correct solution as the mesh is refined, revealing a situation where refinement of the model does not attenuate the erroneous representation due to parasitic shear. Therefore, the only remedy is the complete obliteration of the effects of parasitic shear, which here is attained by removing the spurious term from the transverse shear strain expression.

Inspection of the formulation presented above explains the parabolic shape of the plots in figure 11. The analytic expression for the beam´s normal strain given by equation 19 shows that (x depends on z, x and x2. Thus, (x,z, which is the spurious term in the expression for (xz, depends on x and x2. Hence, when (x,z multiplies x in the transverse shear strain (xz expression of the finite element (equation 12), it generates a term which is cubic in x, responsible then for the erroneous parabolic distribution of (xz along the beam´s length. On the other hand, after elimination of the spurious term, (xz becomes only dependent on the variable x, conforming to the analytic expression for the transverse shear strain provided by equation 20, thus displaying the correct linear variation along the beam´s length.

[pic]

Fig. 3 – Displacement w using the parasitic shear model.

[pic]

Fig. 4 – Displacement w using the corrected model.

[pic]

Fig. 5 – Rotation q using the parasitic shear model.

[pic]

Fig. 6 – Rotation q using the corrected model.

[pic]

Fig. 7 – Displacement u using the parasitic shear model.

[pic]

Fig. 8 – Displacement u using the corrected model.

[pic]

Fig. 9 – Strain ((x) using the parasitic shear model.

[pic]

Fig. 10 – Strain ((x) using the corrected model.

[pic]

Fig. 11 – Strain (xz(x) using the parasitic shear model.

[pic]

Fig. 12 – Strain (xz(x) using the parasitic shear model.

4 Conclusion

This paper presented the formulation in strain gradient notation of a two-node beam finite element for the analysis of laminated composites. Inspection of the beam element´s transverse shear strain expansion revealed the presence of the flexural term (x,z, identified as a spurious terms which is responsible for parasitic shear. The numerical analyses performed here demonstrated that this spurious term is the source of severe erroneous numerical solution representations. For coarser meshes, parasitic shear caused significant artificial stiffening in the beam model. Refinement attenuated this parasitic shear effect, leading to convergence. Also, it was demonstrated that parasitic shear causes an error of the qualitative type in the transverse shear strain solution. In the example problem, as the loading is uniformly distributed along the beam´s length, solution for transverse shear strains and stresses must vary linearly. However, parasitic shear rendered such solutions parabolic, an error which was completely removed when the element was corrected. On the contrary to what was expected, refinement did not attenuate such qualitative error. Instead, refinement strengthened the spurious term, which is a coefficient to a linear term in x (longitudinal coordinate), increasing the solutions error proportionally. The conclusions that can be drawn are the following. The beam element is efficient as the numerical solutions converge to the analytic solution. Strain gradient notation provides an efficient means for detecting spurious terms responsible for modeling deficiencies, which are inherent to the formulation process. Elimination of the spurious term present in the transverse shear strain expression corrected the element for parasitic shear, providing solutions which converge rapidly to the analytic one. Strain gradient notation presents a clear and effective way of removing the spurious term from the solution a-priori.

References:

[1] Vinson JR, Sierakowski RL., The Behavior of Structures Composed of Composite Materials, Boston: Martinus Nijhoff Publishers, 1986.

[2] Byrd DE., Identification and Elimination of Errors in Finite Element Analysis, Ph.D Dissertation, University of Colorado at Boulder, 1988.

[3] Abdalla F( JE., Qualitative and Discretization Error Analysis of Laminated Plate Models, Ph.D Dissertation, University of Colorado at Boulder, 1992.

[4] Dow JO, Abdalla F( JE., Qualitative Errors in Laminated Composite Plate Models. International Journal for Numerical Methods in Engineering, Vol. 37, 1994, pp 1215-1230.

[5] Jones RM., Mechanics of Composite Materials, New York: Hemisphere Publishing Corporation, 1975.

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