Abstract. keywords: finite elements, beams, Hermitian interpolation, shape functions

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1 Generation of shape functions for straight beam elements Charles E. Augarde Originally published in Computers and Structures, 68 (1998) Corrections highlighted. Abstract Straight beam finite elements with greater than two nodes are used for edge stiffening in plane stress analyses and elsewhere. It is often necessary to match the number of nodes on the edge stiffener to the number on a whole plane stress element side. Beam elements employ shape functions which are recognised to be level one Hermitian polynomials. An alternative to the commonly adopted method for determining these shape functions is given in this note, using a formula widely reported in mathematical texts which has hitherto not been applied to this task in the finite element literature. The procedure derives shape functions for beams entirely from the set of Lagrangian interpolating polynomials. Examples are given for the derivation of functions for a three and four-noded beam elements. keywords: finite elements, beams, Hermitian interpolation, shape functions 1

2 Introduction Analysis of structures using the finite element method is well-established. Many formulations exist for complex elements but simple elements remain popular since they are usually well-tested and easy to implement into an analysis program. Two-dimensional plane stress analysis, for thin structures subject to in-plane loading, may employ continuum elements, such as the fifteen-node triangle, having a large number of nodes along a side. Where edge stiffening is required, beam elements can be connected to continua edge nodes. There is then a requirement for formulations of beam elements having more than two nodes. Conventional two-dimensional beam elements have two degrees of freedom at each node: one lateral displacement and one rotation. Unless the structure is loaded entirely laterally, axial stiffness must also be incorporated, by an additional degree of freedom at each node. With this amendment, beam elements are usually referred to as frame elements. The axial effects are uncoupled for straight beams and the determination of suitable shape functions is straightforward [1]. The author has recently contributed to the development of a complex threedimensional finite element model at the Department of Engineering Science, Oxford University to study the damage accruing to surface structures from adjacent tunnelling [2]. Modelling was also undertaken in two-dimensions (to validate the complex model) where tunnel linings were represented with simple beam elements. The procedure outlined in this note was used in the implementation of these beam elements into an existing in-house finite element code (OXFEM). In this note we are concerned with the generation of the shape functions which interpolate the lateral displacements along beam elements having more than two nodes. Bernoulli-Euler beam theory is assumed where transverse shear deformation is zero. While most finite element texts describe the simple twonoded beam [1,4] few explain how more complex elements may be formulated [3]. 2

3 Standard procedure The beam element of Figure 1a is defined by two nodes, a distance l apart along the element local x axis. A common convention, and that adopted here, is to make the nodal coordinates non-dimensional that is x = x l (1) so that the two-noded beam has nodes at x = 0, +1. Using shape functions, lateral displacement w( x) of the two-noded beam element of Figure 1a is w( x) = Nd (2) where N = { N 1 N 2 N 3 N 4 } & d T = { w 1 θ 1 w 2 θ 2 } (3) N j, (j = 1, 4) are the bending shape functions and w i, θ i, (i = 1, 2) are the displacements and rotations at the nodes. Determination of the former, using the method to be found in many finite element texts [4,5], proceeds by first writing w( x) as an n-termed polynomial with unknown coefficients, n being the number of degrees of freedom in the element where, w( x) = Xα (4) X = { 1 x x 2 x 3 } & α T = { α 1 α 2 l α 3 l 2 α 4 l 3 } (5) Taking equation (4) and its first derivative with respect to x, four further equations can be formed, one for each degree of freedom: d = Aα (6) where A = (7)

4 Solving for α in Equation (6) and substituting into Equations (4) and (2) gives N = XA 1 (8) The three-node beam of Figure 1b, of overall length 2l, has nodes at x = 1, 0, +1. The same procedure as above yields N = { N 1 N 2 N 3 N 4 N 5 N 6 }, (9) d T = { w 1 θ 1 w 2 θ 2 w 3 θ 3 }, (10) X = { 1 x x 2 x 3 x 4 x 5 } (11) α T = { α 1 α 2 l α 3 l 2 α 4 l 3 α 5 l 4 α 6 l 5 } (12) and to solve for N requires the inversion of A = (13) Hermitian interpolation The shape functions in Equation (2) are Hermitian polynomials since the displacement w(x) is interpolated from nodal rotations as well as nodal displacements. This contrasts with Lagrangian interpolation, used for continuum elements shape functions and for the axial effects in frame elements. Considering small displacements, the nodal rotations are the first derivatives of the unknown real displacement function at the nodes thus fulfilling the definition of Hermitian interpolation. This property allows an alternative procedure to be used to determine the shape functions to that outlined above The generation of Hermitian (or Hermite) polynomials from Lagrangian interpolation polynomials is described in many mathematical texts [6,7,8]. Despite the clear understanding that bending shape functions are equivalent to Hermitian polynomials the technique described below has not, to the author s knowledge, been linked to the generation of those shape functions. 4

5 One-dimensional interpolation is required for straight beam elements. The single dimension is along the element centreline, defined as the x-axis. Onedimensional Hermitian interpolation for an unknown, w(x) proceeds as [6,7] w(x) = nnod i=1 [ H r 0iw i + H r 1i ( ) dw +... Hri r dx i ( d r ) ] w dx r i (14) where H r ji is a Hermite polynomial of level r, relating to node i and to derivative order j of w. The sum is over the number of nodes, nnod where values of w and its derivatives are available. The level of the polynomial indicates the highest order derivative used in the interpolation. Comparison of Equations (2) and (14) reveals that the bending shape functions are level one Hermitian polynomials as follows: N 1 = H 1 01 N 2 = H 1 11 N 3 = H 1 02 N 4 = H 1 12 (15) Level one Hermitian polynomials are derived from Lagrangian polynomials by the following formulae: H 1 0i = [1 2(x x i )L i(x i )][L i (x)] 2 (16) H 1 1i = (x x i )[L i (x)] 2 (17) where L i (x) is the one-dimensional Langrangian polynomial of degree (nnod 1) calculated at node i, given by L i (x) = nnod j=1,j i x x j x i x j (18) and L i(x) is its first derivative with respect to x. A polynomial of order (nnod 2 1) is required to interpolate over nnod points, each contributing two values. Inspection of Equations (16) and (17) shows that the Hermite polynomials are of the correct order for interpolation. Re-writing Equations (16) and (17) in terms of the non-dimensional coordinate, x H 1 0i = [1 2l( x x i )L i( x i )][L i ( x)] 2 (19) H 1 1i = l( x x i )[L i ( x)] 2 (20) 5

6 The advantage of a derivation based on the Lagrangian polynomials and their first derivatives is that these are already likely to be present in a program code. The former are required for continuum elements and the derivatives are required for isoparametric elements. The use of this procedure also provides a systematic approach to allow simpler coding. Examples Three-node beam A three-node element with six bending degrees of freedom and total length 2l is shown in Figure 1b. This element has nodes at x = ( 1, 0, 1). The axial degrees of freedom are omitted from this element as in the derivations above. Shape functions for these degrees of freedom are the Lagrangian polynomials of order 2. From the preceding section, it is clear that these are also required for the derivation of the bending shape functions. The Lagrangian polynomials are L 1 (x) = (x x 2)(x x 3 ) (x 1 x 2 )(x 1 x 3 ) L 2 (x) = (x x 1)(x x 3 ) (x 2 x 1 )(x 2 x 3 ) L 3 (x) = (x x 1)(x x 2 ) (x 3 x 1 )(x 3 x 2 ) (21) (22) (23) Rewriting in terms of the non-dimensional coordinate x and substituting for values of x at nodes (i.e. ( x 1, x 2, x 3 ) = ( 1, 0, 1) ), gives The derivatives are L 1 ( x) = x ( x 1) 2 (24) L 2 ( x) = 1 x 2 (25) L 3 ( x) = x ( x + 1) (26) 2 L 1( x) = 1 (2 x 1) (27) 2l 6

7 L 2( x) = 2 ( x) (28) l L 3( x) = 1 (2 x + 1) (29) 2l The bending shape functions are equivalent to the following Hermite polynomials: N 1 = H 1 01 N 2 = H 1 11 N 3 = H 1 02 (30) N 4 = H 1 12 N 5 = H 1 03 N 6 = H 1 13 (31) Note that the first four are not the same as those in Equation (15) since each set is based on different order Lagrangian polynomials. From (24) and (27) and with substitution for x 1 we obtain [ ( H01 1 = 1 2l( x + 1) 3 )] [ ] x 2 2l 2 ( x 1) = x x3 1 2 x x5 (32) Similarly, [ ] x 2 H11 1 = l( x + 1) 2 ( x 1) = l [ x 2 x 3 x 4 + x 5] (33) 4 H 1 02 = [1] [ (1 x 2 ) ] 2 = 1 2 x 2 + x 4 (34) H12 1 = l x [ x( x + 1)] 2 = l [ x 2 x 3 + x 5] (35) [ ( )] [ ] 3 x 2 H03 1 = 1 2l( x 1) 2l 2 ( x + 1) = x x3 1 2 x4 3 (36) 4 x5 [ ] x 2 H13 1 = l( x 1) 2 ( x + 1) = l [ x 2 x 3 + x 4 + x 5] (37) 4 The suitability of the above as shape functions can be demonstrated by plotting each polynomial (Figure 2). The polynomials related to displacements have value 1 at the associated node and zero elsewhere. Those related to derivatives have slopes of 1 at the associated node. Four-node beam Figure 1c shows a beam with four nodes having the same overall length as the three-node beam. The non-dimensional coordinates ( x) of the nodes are 7

8 1, 1/3, +1/3 and +1. Eight shape functions are required, derived from the following Lagrangian polynomials and derivatives: L 1 ( x) = 9 [ x ] 9 + x2 x 3 (38) L 2 ( x) = 27 [ ] 1 x2 x x3 (39) L 3 ( x) = 27 [ ] 1 x2 + x x3 (40) L 4 ( x) = 9 [ x ] 9 + x2 + x 3 (41) L 1( x) = 9 [ ] 1 16l x 3 x2 (42) L 2( x) = 27 [ 1 2 ] 16l 3 x + 3 x2 (43) L 3( x) = 27 [1 2 ] 16l 3 x 3 x2 (44) L 4( x) = 9 [ 1 ] 16l x + 3 x2 (45) The first two shape functions, relating to lateral displacement and rotation at node 1, can then be derived as N 1 = 1 [ x 243 x x x x x x (46) 7] 512 N 2 = l [ 1 x 19 x x x 4 99 x 5 81 x x 7] (47) 256 Plotting these two functions (Figures 3a & b) shows that they also satisfy the basic requirement of shape functions as outlined above. Conclusion This work presents an alternative derivation of bending shape functions for simple beam elements, for implementation of many-noded straight beam elements within a finite element analysis code. While the elements described are simple the theory will be of interest to developers of other C 1 continuous elements such as rectangular plates. 8

9 Acknowledgements This work was carried out as part of an EPSRC funded project examining numerical modelling of tunnelling at the Department of Engineering Science, University of Oxford, UK under the guidance of Dr. H.J. Burd. The author would like to acknowledge the help and contribution of Dr. Burd to this work. References 1. Astley, R.J., Finite Elements in Solids and Structures, Chapman & Hall, London, Augarde, C.E., Numerical modelling of tunnelling processes for assessment of damage to buildings, D.Phil thesis, University of Oxford, Dawe, D.J., Matrix and Finite Element Displacement Analysis of Structures, Clarendon Press, Oxford, Cook, R.D., Concepts and Applications of Finite Element Analysis, 2nd edn., John Wiley & Sons, Chichester, Mohr, G.A., Finite Elements for Solids, Fluids and Optimisation, Oxford University Press, Oxford, Jacques, I. and Judd, C., Numerical Analysis, Chapman & Hall, London, Morris, J.L., Computational Methods in Elementary Numerical Analysis, John Wiley & Sons, Chichester, Spanier, J. and Oldham, K.B., An Atlas of Functions, Hemisphere, London,

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