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Full Paper Proc. of Int. Conf. on Recent Trends in Transportation, Environmental and Civil Engineering 2012

Analysis of Non-prismatic Beams with Nonlinear Material Using Basic Displacement Functions S.M. Bagheri1, R. Attarnejad2 1

School of Civil Engineering, University College of Engineering, University of Tehran, Iran. Email: 2 School of Civil Engineering, University College of Engineering, University of Tehran, Iran. Email: of solids and structures. Some of the books which are partially devoted to nonlinear analysis are Belytschko and Hughes(1983), Zienkiewicz and Taylor(1991), Bathe(1995) and Cook, Plesha and Malkus(1989). These books provide useful introductions to nonlinear finite element analysis.

Abstract— In this paper, analysis of non-prismatic EulerBernoulli beam elements with nonlinear materials is studied from a mechanical point of view. The method utilizes Basic Displacement functions (BDFs) which are derived using basic principles of structural mechanics to obtain new shape functions for non-uniform beams. The main virtue of these shape functions is their susceptibility to the variations in cross sectional area, moment of inertia and even modulus of elasticity along the axis of the beam element. This flexibilitybased formulation could be easily incorporated into standard displacement based finite element programs. Competence of the present method in nonlinear static analysis has been verified through a simple numerical example and the results are highly competing with those of other methods in the literature.

B. Non-prismatic Beams Most of the literatures on non-prismatic beams have displacement-based formulations established on an assumed or prescribed displacement field. This makes an extra hypothesis usually imposed in addition to the three essential relations: equilibrium of forces, compatibility of displacements and/or strains and the constitutional law of the material behavior. Due to this extra hypothesis, one of the three essential relations, generally the equilibrium equations, is satisfied only in certain interior points such as integration points; thus the formulation is approximate in nature. Unlike the stiffness method, application of flexibility-based methods ensures the exact satisfaction of the equilibrium equations at any interior point of the element.

Index Terms— Material nonlinearity, basic displacement function, non-prismatic beam

I. INTRODUCTION A. Geometrical Nonlinear Analysis The main deference between linear analysis and nonlinear ones is that in linear analysis we often use superposition, knowing that for a linearly elastic structure it is valid. However, an elastic analysis does not give information about the loads that will actually collapse a structure. Non-linear behavior of solids has two aspects: geometric non-linearity and material non-linearity. Material non-linearity will be discussed in this paper. The simplest form of a nonlinear material behavior is that of elasticity for which the stress is not linearly proportional to the strain. The pioneer studies in finite-element analysis of elasticplastic continua were made by Argyris [1], Pope [2] , Swedlow et al. [3], and Marcaf and King [4] within the geometrically linear (or “small strain”) approximation. Oden [5] reviews FE formulations in the non-linear elasticity context for arbitrary strains. The first elastic-plastic formulation appropriate to deformations of arbitrary magnitude was given by Hibbitt, Marcal and Rice [6]. Several excellent texts and monographs dedicated either around nonlinear finite element analysis have already been published. Books dealing only with nonlinear finite element analysis include Oden(1972), Crisfield(1991), Kleiber(1989), and Zhong(1993). Oden’s work is particularly noteworthy since it pioneered the field of nonlinear finite element analysis © 2012 ACEE DOI: 02.TECE.2012.2. 3

C. Innovate Formulation In the present paper a simple flexibility-based formulation is used to compute the deformation of general non-prismatic beam elements with material nonlinearity. This is an extension of the formulation presented earlier by Attarnejad in 2000 and 2010 [7,8]. The method does not involve any cumbersome mathematical/numerical operations and is capable of modeling non-prismatic beams with any type of cross-section and variation of cross sectional dimension, moment of inertia along the axis of the beam element. II. STRUCTURAL FORMULATION A. Basic Displacement Functions BDFs are mathematical functions with mechanical concepts. These functions represent displacement of the free end of a cantilever beam due to a unit load at distance x, Fig. 1. It is worth noting that BDFs have an equivalent interpretation regarding the reciprocal theorem, Fig. 2, which is stated as displacement of a point at distance x due to a unit load at the free end of a cantilever beam, With this interpretation, BDFs can be calculated from the relations shown in table I.


Full Paper Proc. of Int. Conf. on Recent Trends in Transportation, Environmental and Civil Engineering 2012 TABLE I. FIRST AND SECOND DERIVATIVES



The first and second derivatives of BDFs with respect to x, are obtained applying the Leibniz rule for differentiating integrals. These derivative relations are shown in table II. For more details see Reference [8]. TABLE II. BDFS RELATION

B. Shape Functions As described in Reference [8], the nodal Softness matrices of the element can be obtained using the BDFs as follows:  bu1 (0)  F11    0   0 

Figure 1. Definitions of BDFs.[10]

 bu 2 ( L)  F22    0   0 

 0   b 1 ( 0)  db 1   dx x 0 

0 bw1 (0) dbw1 dx x0


   b 2 ( L)  db 2   dx x L 



bw 2 ( L) dbw 2 dx x L


Shape functions derive out from the following relation: u ( x)  N u d   ba  G d .


w( x )  N w d   b f  G d  .




b 

b   0

0 bw 2



Figure 2. Equivalent definitions of BDFs[10]




b 1

 G   F   0


G  is © 2012 ACEE DOI: 02.TECE.2012.2. 3


 bu 1

In which the


0 F 



0 0 bu 2

0 0

a nd

b 2  and  . 


Full Paper Proc. of Int. Conf. on Recent Trends in Transportation, Environmental and Civil Engineering 2012 Considering linear behavior at beginning of loading, the initial stiffness matrix for a non-prismatic element can be calculated from the equation below: L

k   G (  b E 0



A( x)bu  dx T



  bwE 0 I ( x )bw dx )G  T




C. Fundamental Formulations The principle of virtual work is probably the most fundamental concept in finite element analysis. This principle provides a convenient way for developing a force- deflection relationship for a structural element. Based on the principle, the equation of equilibrium in the deformed body is:

d   f     . ( ) dV . T



In which  f  and d T are equivalent nodal force vector and nodal displacement vector.  ( ) is the axial stress that can be obtained from a given stress- strain relationship.  is the strain developing during loading which is calculated by the following equation:

 a  du( x)

2  y (d w( x)

). dx dx 2 Substituting Eqs. (3) and (4) into Eq. (7), leads to

 a  ba   y bf  T


 Gd   Bd .

Figure 3. Iteration procedure for direct method [11].

assemble them, K 0  4.Specify load FA  5.Initialize i: i = 0


6.Let F 1  FA  7.Increment i: i = i + 1


8.Solve K 0 D i  F i to obtain D i

This allows Eq. (6) to be rewritten as:

d   f    d B . ( ) dV . T


9. Dtotal  D total  Di




10.Use Dtotal to gain d  for each element, then form  a

So that

11.Obtain  f real from the Eq. () for each element

 f   G    b  y b. ( ) dV  . V



12.Transform  f real from local to global coordinates and


assemble them, F real

This Eq. will be used for calculating the unbalanced force vector in solution procedure.

13.Compute the unbalance force F res  FA   F real 14.Let F i1  F res


15.Repeat steps 7-15 until F res  0 16.Return to step 5 for the next set of loads

Among the procedures introduced by past researchers, the “initial stiffness method” has been chosen which is much used in material non-linearity problem [9] this method uses the initial stiffness matrix and since the real stiffness matrix is a function of unknown displacement, the modified NewtonRaphson procedure is used to achieve the deflection of an element, Fig. (3). The initial stiffness method steps are briefly outlined here:

IV. NUMERICAL EXAMPLE To check the competency of the present method, a simple cantilever beam with constant cross section has been used as a validation test. Section properties of the cantilever beam are shown in the Fig. (4). A bilinear behavior has been considered for the material properties of the beam, Fig. (5).

1.Form k0  for each element 2.Use initial undeformed geometry to form transformation matrix, T  3.Transform k0  from local to global coordinates and

© 2012 ACEE DOI: 02.TECE.2012.2. 3


Full Paper Proc. of Int. Conf. on Recent Trends in Transportation, Environmental and Civil Engineering 2012 CONCLUSIONS According to the previews section, the presented flexibility- based method provides similar results to exact solution in constant area section problems. We believe that in non-prismatic beams, the flexibility- based method results much better than the displacement-based methods; due to the exact shape functions derived using BDFs. The significant property of these shape functions is their susceptibility to the variations in cross sectional area, moment of inertia and even modulus of elasticity along the axis of the beam element. REFERENCES

Figure 4. Section properties

[1] J. H. Argyris, “Continua and discontinua”, 1st Conference on Matrix Methods of Structural Mechanics, 1965 (WrightPatterson Air Force Base, 1967) l-198. [2] G. G. Pope, “A discrete method for analysis of plane elastoplastic strain problems,” Aeron. Q., vol.17, pp. 83,1966. [3] J. L. Swedlow, M.L. Williams and W.H. Yang, “Elasto-plastic stresses and strains in cracked plates,” in: T. Yokobori et al. (eds.), Proceedings 1st International Conference on Fracture, Sendai 1965 (Japanese Sot. for Strength and Fracture, Tokyo, pp. 259-282, 1966) [4] P. V. Marcal, I. P. King, “Elastic-plastic analysis of twodimensional stress systems by the finite-element method,” lnt. J. Mech. Sci., vol. 9, pp. 143-155, 1967. [5] J. T. Oden, “Finite elements of non-linear continua,” McGrawHill, New York, 1972. [6] H. D. Hibbitt, P. V. Marcal, J. R. Rice, “A finite-element formulation for problems of large strain and large displacement,” Int. J. Solids Structures, vol. 6, pp. 1069-1086, 1970. [7] R. Attarnejad, “On the derivation of the geometric stiffness and consistent mass matrices for non-prismatic Euler– Bernoulli beam elements,” European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, July 2000. [8] R. Attarnejad, “Basic displacement functions in analysis of nonprismatic beams,” J. Engng. Comp., vol. 27-6, pp. 733745, 2010. [9] O. C. Zienkiewicz, S. Valliapan, I. P. King, “Elasto-platic solutions of engineering problems. Initial stress, finite element approach,” Int. J. Num. Meth. Engng., vol. 1, pp. 75-100, 1969. [10] M. A. Crisfield, “Non-linear Finite Element Analysis of Solids and Structures,” JOHN WILEY & SONS, London, 1991.

Figure 5. Material properties

The results of the presented method have been compared by the exact solution. As shown in Fig. (6). The comparison verifies the method results.

Figure 6. Comparison the results

© 2012 ACEE DOI: 02.TECE.2012.2. 3