generation of shape functions for straight beam elements

7/28/2019 Generation of shape functions for straight beam elements

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Generation of shape functions for straight beam elements

Charles E. Augarde *

Department of Engineering Science, University of Oxford, Parks Road, Oxford, U.K.

Received 17 June 1997; received in revised form 14 February 1998

Abstract

Straight beam nite elements with greater than two nodes are used for edge stiening in plane stress analyses and

elsewhere. It is often necessary to match the number of nodes on the edge stiener 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

nite 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

element. # 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Finite elements; Beams; Hermitian interpolation; Shape functions

1. Introduction

Analysis of structures using the nite element

method is well established. Many formulations exist

for complex elements but simple elements remain pop-

ular since they are usually well-tested and easy to im-

plement into an analysis program. Two-dimensional

plane stress analysis, for thin structures subject to in-

plane loading, may employ continuum elements, such

as the fteen-node triangle, having a large number of

nodes along a side. Where edge stiening 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 dis-

placement and one rotation. Unless the structure is

loaded entirely laterally, axial stiness must also be in-

corporated, by an additional degree of freedom at each

node. With this amendment, beam elements are usually

referred to as frame elements. The axial eects are

uncoupled for straight beams and the determination of

suitable shape functions is straightforward [1].

The author has recently contributed to the develop-

ment of a complex three-dimensional nite 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 nite el-

ement code (OXFEM).

In this note we are concerned with the generation of

the shape functions which interpolate the lateral displa-

cements along beam elements having more than two

nodes. Bernoulli-Euler beam theory is assumed where

transverse shear deformation is zero. While most nite

element texts describe the simple two-noded beam [1, 4]

few explain how more complex elements may be

formulated [3].

Computers and Structures 68 (1998) 555560

0045-7949/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved.P I I : S 0 0 4 5 -7 9 4 9 (9 8 )0 0 0 7 1 -6

PERGAMON

* Author to whom correspondence should be addressed.


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2. Standard Procedure

The beam element of Fig. 1(a) is dened 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

lY 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 Fig. 1(a) is

w"x Nd 2

where

N fN1 N2 N3 N4g 8 dT

fw1 y1 w2 y2gX

3

Nj,(j= 1,4) are the bending shape functions and wi, yi,

(i= 1,2) are the displacements and rotations at the

nodes. Determination of the former, using the method

to be found in many nite element texts [4, 5], proceeds

by rst writing w(x) as an n-termed polynomial with

unknown coecients, n being the number of degrees of

freedom in the element

w"x Xaaa 4

where,

X f1 "x "x2 "x3g 8 aaaT fa1 a2la3l2a4l

3gX 5

Taking Eq. (4) and its rst derivative with respect to x,four further equations can be formed, one for each

degree of freedom:

d Aaaa 6

where

A

1 0 0 0

0 1 0 0

1 1 1 1

0 1 2 3

PTTR

QUUS 7

Solving for aaa in Eq. (6) and substituting into Eqs. (4)

and (2) gives

N XA1 8

The three-node beam of Fig. 1(b), of overall length 2 l,

has nodes at x= 1, 0, +1. The same procedure as

above yields

N fN1 N2 N3 N4 N5 N6gY 9

dT fw1 y1 w2 y2 w3 y3gY 10

X f1 "x "x2 "x3 "x4 "x5g 11

aaaT fa1 a2la3l

2a4l

3a5l

4a6l

5g 12

and to solve for N requires the inversion of

A

1 1 1 1 1 1

0 1 2 3 4 5

1 0 0 0 0 0

0 1 0 0 0 0

1 1 1 1 1 1

0 1 2 3 4 5

PTTTTTTR

QUUUUUUS

X 13

3. Hermitian Interpolation

The shape functions in Eq. (2) are Hermitian poly-

nomials 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

eects in frame elements. Considering small displace-

ments, the nodal rotations are the rst derivatives of

the unknown real displacement function at the nodes

thus fullling the denition of Hermitian interpolation.

This property allows an alternative procedure to be

used to determine the shape functions to that outlined

above.Fig. 1. Beam elements showing bending degrees of freedom.

C.E. Augarde / Computers and Structures 68 (1998) 555560556


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The generation of Hermitian (or Hermite) poly-

nomials from Lagrangian interpolation polynomials is

described in many mathematical texts [68]. Despite

the clear understanding that bending shape functionsare equivalent to Hermitian polynomials the technique

described below has not, to the author's knowledge,

been linked to the generation of those shape functions.

One-dimensional interpolation is required for

straight beam elements. The single dimension is along

the element centreline, dened as the x-axis. One-

dimensional Hermitian interpolation for an unknown,

w(x) proceeds as [6, 7]

wx nnodi1

Hr0iwi Hr1i

dw

dx

i

F F F Hrridrw

dxr

i

!14

where Hjir 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 poly-

nomial indicates the highest order derivative used in

the interpolation.

Comparison of Eqs. (2) and (14) reveals that the

bending shape functions are level one Hermitian poly-

nomials as follows:

N1 H101 N2 H

111 N3 H

102 N4 H

112X 15

Level one Hermitian polynomials are derived from

Lagrangian polynomials by the following formulae:

H

1

0i 1 2x xiL

H

ixiLix

2

16

H11i x xiLix2 17

where Li(x) is the one-dimensional Langrangian poly-

nomial of degree (nnod 1) calculated at node i, given

by

Lix nnod

j1YjTi

x xj

xi xj18

and L'i(x) is its rst derivative with respect to x. A

polynomial of order (nnod 2 1) is required to in-

terpolate over nnod points, each contributing two

values. Inspection of Eqs. (16) and (17) shows that theHermite polynomials are of the correct order for in-

terpolation.

Re-writing Eqs. (16) and (17) in terms of the non-

dimensional coordinate, x

H10i 1 2l"x "xiLHi"xiLi"x

2 19

H11i l"x "xiLi"x2X 20

The advantage of a derivation based on the

Lagrangian polynomials and their rst 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 el-

ements. The use of this procedure also provides a sys-

tematic approach to allow simpler coding.

4. Examples

4.1. Three-node beam

A three-node element with six bending degrees of

freedom and a total length of 2 l is shown in Fig. 1(b).

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 bend-

ing shape functions.

The Lagrangian polynomials are

L1x x x2x x3

x1 x2x1 x321

L2x x x1x x2

x2 x3x2 x122

L3x x x1x x2

x3 x1x3 x2X 23

Rewriting in terms of the non-dimensional coordinate

x and substituting for values of x at nodes [i.e. (x1, x2,x3) = (1, 0, 1)], gives

L1"x "x

2"x 1 24

L2"x "x"x 1 25

L3"x "x

2"x 1X 26

The derivatives are

L H1"x 1

2l2"x 1 27

L H2"x 1

l2"x 1 28

L H3"x 1

2l2"x 1X 29

The bending shape functions are equivalent to the fol-

lowing Hermite polynomials:

N2 H101 N3 H

111 N5 H

102 30

N6 H112 N8 H

103 N9 H

113X 31

C.E. Augarde / Computers and Structures 68 (1998) 555560 557


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Note that the rst four are not the same as those in

Eq. (15) since each set is based on dierent order

Lagrangian polynomials.

From Eqs. (24) and (27) and with substitution for x1we obtain

H101 1 2l"x 1 3

2l

!"x

2"x 1

h i2

"x2 5

4"x3

1

2"x4

3

4"x5X 32

Similarly,

H111 l"x 1"x

2

"x 1h i2

l

4

"x2 "x3 "x4 "x5 33

H102 1 2l"x 1

l

!"x"x 12 1 2"x2 "x4

34

H112 l"x"x"x 12 l"x 2"x3 "x5 35

Fig. 2Caption overleaf



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6/6

L2"x 27

16

1

3

"x "x2

3

"x3 ! 39

L3"x 27

16

1

3 "x

"x2

3 "x3

!40

L4"x 9

16

1

9

"x

9 "x2 "x3

!41

L H1"x 9

16l

1

9 2"x 3"x2

!42

L H2"x 27

16l1

2

3"x 3"x2

!43

L H3"x 27

16l1

2

3"x 3"x2

!44

L H4"x 9

16l

1

9 2"x 3"x2

!X 45

The rst two shape functions, relating to lateral displa-

cement and rotation at node 1, can then be derived as

N1 1

51213 15"x 243"x2 281"x3

1215"

x

4

1413"

x

5

729"

x

6

891"

x

7

46

N2 l

2561 "x 19"x2 19"x3

99"x4 99"x5 81"x6 81"x7X 47

Plotting these two functions [Fig. 3(a), (b)] shows that

they also satisfy the basic requirement of shape func-

tions as outlined above.

5. Conclusion

This work presents an alternative derivation of

bending shape functions for simple beam elements, forimplementation of many-noded straight beam elements

within a nite element analysis code. While the el-

ements described are simple, the theory will be of inter-

est to developers of other C1 continuous elements such

as rectangular plates.

Acknowledgements

This work was carried out as part of an EPSRC

funded project examining numerical modelling of tun-

nelling 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 RJ. Finite Elements in Solids and Structures.

London: Chapman & Hall, 1992.

[2] Augarde CE. Numerical modelling of tunnelling pro-

cesses for assessment of damage to buildings. D.Phil

thesis. University of Oxford, 1997.

[3] Dawe DJ. Matrix and Finite Element Displacement

Analysis of Structures. Oxford: Clarendon Press, 1984.

[4] Cook RD. Concepts and Applications of Finite Element

Analysis. 2nd ed. Chichester: John Wiley & Sons, 1981.[5] Mohr GA. Finite Elements for Solids, Fluids and

Optimisation. Oxford: Oxford University Press, 1992.

[6] Jacques I, Judd C. Numerical Analysis. London,

Chapman & Hall, 1987.

[7] Morris JL. Computational Methods in Elementary

Numerical Analysis. Chichester: John Wiley & Sons,

1983.

[8] Spanier J, Oldham KB. An Atlas of Functions. London:

Hemisphere, 1987.

Fig. 3. Shape functions for a four-node beam element.


generation of shape functions for straight beam elements

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