COMPUTATIONAL ASPECTS OF CONFIGURATION DEPENDENT
INTERPOLATION IN NON–LINEAR HIGHER-ORDER 2D BEAM FINITE
ELEMENTS
Edita Papa, Gordan Jelenić
Faculty of Civil Engineering, University of Rijeka,
Radmile Matejčić 3, 51 000 Rijeka, Croatia, edita.papa@gradri.hr, gordan.jelenic@gradri.hr
1. Introduction
In non-linear 3D beam theory with rotational
degrees of freedom (Simo, 1985) configurationdependent interpolation may be utilized to
provide a result invariant to the choice of the
beam reference axis (Borri and Bottasso, 1994)
or invariant to a rigid-body rotation (Crisfield
and Jelenić, 1999). For 2D beam elements, the
latter issue vanishes, and such elements are
more illustrative for the study of accuracy of
the configuration-dependent interpolation in
higher-order elements.
Since the approximate character of the
finite-element method stems from the
introduction of interpolation functions for the
field variables and their variations, the actual
choice of the interpolation functions is of a
great importance for the accuracy of the finite
element method. The most commonly used
standard Lagrangian interpolation procedure
(with reduced integration used to eliminate the
shear locking effect) is satisfactory if our
attention is limited to finding the results for the
field variables at nodal points in linear analysis.
However, if we want to obtain the exact field
distribution this kind of interpolation is unable
to do so, even in linear analysis. To obtain the
exact solutions in linear analysis simply using
the right interpolation function and without
applying reduced integration, a different kind of
interpolation, called the linked interpolation is
needed, in which the unknown displacement
function do not depend only on the nodal
displacements, but also on the nodal rotations
(Jelenić and Papa, 2011). Marco Borri and
Carlo Bottasso have developed a so-called
fixed-pole formulation which effectively
generalizes the idea of linked interpolation to
non-linear two-noded beam elements. This
interpolation is called the helicoidal
interpolation by the authors and is based on the
fact that the tangent to the beam centroidal axis
and the normal of the cross section follow the
same transformation rule (Borri and Bottasso,
1994). This kind of interpolation can also be
obtained starting from the condition of constant
Reissner’s strain measures (Reissner, 1972).
2. Reissner’s beam theory
In this paper we analyze geometrically exact
Reissner’s beam theory (Reissner, 1972). In this
beam theory, normal strain  , shear strain 
and the change of the sectional orientation
along the length of the beam  are given by the
following expressions:
  cos  u' cos     sin   w' sin      1
(1)
   cos  u' sin      sin   w' cos   
(2)
  ',
(3)
where  is the slope of the beam in the initial
configuration,  is the function describing the
slope of the cross sections in the deformed
configuration as a function of the arc-length
coordinate x , u (x) and w(x ) are the horizontal
and vertical displacements that can be written in
terms of the components of the initial and the
deformed position vectors R x (x) , R y (x ) , rx (x)
strain energy  and the work of the applied
loading U ,  (  U )  0 , where
and ry (x) , respectively, and a dash denotes a
differentiation with respect to the arc-length
coordinate x (see Figure 1).
L
     N    T    M  dx
(4)
0
deformed
state
U   u p x  w p y   mz  dx  p  S
L
(5)
0
w(x)
u(x)
r(x)
ry(x)
x
Ry(x)
and N,T and M are the normal stress, shear
stress and stress-couple resultants, p x , p y and
reference
state
are distributed horizontal, vertical and
moment loading respectively, while p is a
vector of nodal displacements and rotations and
S is a vector of corresponding nodal loading.
Taking (1)-(3) and writing minimization of the
energy in a matrix form becomes
mz
R(x)
Rx(x)
u(x)
rx(x)
Figure 1 Kinematics of the problem.
Equilibrium of a beam of length L follows from
minimization of the difference between the
L
   U    N T
0

u ' cos 
u '  
L



 
M  Λ T  v' sin    Λ T  v'  dx   p x

0
 ' 
' 


 

L
 N T
 0
where D1 is a matrix of differential operators
given by
d
 dx

D1   0

0

0
d
dx
0

v' sin  

 u ' cos  .

d


dx
(9)
(6)
interpolation matrix, expression (6) becomes
(7)
L
M ΛT D1 ψ  dx    p x
0
u 
 
mz wdx  p  S  0,

 
After introducing interpolation u  ψ p , where
u is a vector of the unknown displacement and
rotation functions uT  u w  and ψ is the
where Λ is the rotation matrix of a crosssection given by
cos(   )  sin(    ) 0
Λ   sin(    ) cos(   ) 0 .

0
0
1
py
py

mz ψ dx  ST p  0

(8)
The first term within brackets in (8) is the
internal force vector, while the term within the
parentheses is the external force vector.
Depending on the interpolation ψ the results
will be more or less accurate.
Standard finite element procedure employs
Lagrangian polynomials as interpolation
functions (Simo and Vu-Quoc, 1986). This
interpolation implies independence between the
displacements and the rotation of the cross
section which is physically incorrect. Also, this
kind of independent interpolation results in
‘shear locking’ effect for certain aspect ratios
when full integration is used. Additionally, the
choice of the beam reference axis is an open
question for the cases of more complex nonhomogeneous or anisotropic beams, or beams
with twisted or curved geometry. In these
situations the axis of geometric, mass or shear
centers of the cross-section do not necessarily
coincide and choosing one of them for the beam
reference axis instead of another may lead to
different finite-element solutions. To overcome
these problems, it makes sense to re-consider
the choice of the interpolation functions that
provide the actual interdependence between the
displacement and the rotation fields. In linear
analysis, such interpolation is called the linked
interpolation, while in the non-linear analysis
we call it the configuration-dependent
interpolation.


  x   1 
As stated above, interpolation that provides
the interdependence between the displacement
and the rotation fields in non-linear analysis is
called
the
configuration-dependent
interpolation. In this section a brief overview of
the known studies will be given.
Borri and Bottaso (Borri and Bottasso, 1994)
introduce the so-called fixed pole interpolation
in which the equilibrium of cross-sectional
moments is stated with respect to a point which
the authors call the fixed pole. This point has all
the properties of the origin of an inertial
reference frame. The fixed-pole strain measures
are obtained by statically reducing all stress and
stress-couple resultants to the fixed pole.
Assuming a constant curvature in the deformed
state and considering that the tangent to the
beam centroidal axis and the normal of the
cross-section follow the same transformation
rule, for a two-noded element the following
interpolation is obtained:
x
x
 1   2
L
L
(10)
rx1 
rx   1 0
rx 2 
 N    N 
    

ry   0 1
ry 2 
ry1 
(11)
   1  
   1 
   1  
sin  2
I 2   cos 2
I1  sin  2
I1  
 2

 2

 2
 ,
N
  2  1 
  2  1 
  2  1  
I1  cos
I1 
sin 
  sin 
 2

 2

 2
 
(12)
where rx 0  rx1 , rx L   rx 2 , ry 0  ry1 , ry L   ry 2
 0  1 ,  L  2 while I i are the Lagrangian
polynomials for a two noded element:
I1  1 
3. Summary of the configurationdependent interpolation (Borri and
Bottasso, 1994)
x
x
; I2 
L
L
(13)
The same interpolation as in (10)-(12) can be
obtained by assuming constant strain measures
 ,  ,   in (1)-(3). This interpolation is nonlinear in the field variables (configurationdependent) and has the following limiting value
in the case of the non-linear analysis becoming
linear:
    i  0  Rxi  
rx  2  rxi 
 0
  1 
 
 
0
0  R yi   .
ry    I i  ryi      i 
  i 1    2  0
0
0   i  
 

 i 
This limiting case is the same as the linked
interpolation in (Jelenić and Papa, 2011) known
to be able of reproducing the exact solution of a
linear problem thus eliminating the shear-
(14)
locking completely. The latter, however, may
be written for an element with arbitrary number
of nodes:
    i  0  Rxi  
rx  n  rxi 
 0
  1 
 
 
0
0  R yi   .
ry    I i  ryi      i 
  i 1    n  0
0
0   i  
 

 i 
4. Configuration-dependent higherorder interpolation
5. Numerical examples
In order to generalize the non-linear BorriBottasso result for two-noded elements to
higher-order elements we may take the strain
measures to be linearly distributed rather than
constant, which however complicates the
procedure considerably. This is so because the
interpolation turns out to be related with
Fresnel’s integrals, for which analytical
solutions are not available (Jelenić and Papa,
2009). Alternatively, we may attempt to
generalize the linear higher-order result
obtained by Jelenić and Papa to non-linear
analysis. The simplest idea of how to perform
this generalization and come up with a
configuration-dependent interpolation is to
substitute the initial positions of the nodal
points in the linked interpolation with their
current positions:
I


Ii
 i    i  0 

n
r
rx  n 
  xi 
Ii
 


   i 
Ii
0 ryi 
ry   
 
  i 1  n
0
0
I i   i 
 



(15)
(16)
As it can be seen, the unknown position
vectors in the deformed state depend on the
rotations thus making this interpolation nonlinear in the unknown functions, i.e.
configuration-dependent.
In this section two numerical examples are
given. Both of them are calculated using the
standard Lagrangian interpolation and the
linked interpolation in non-linear analysis for
reduced integration as well as the full
integration.
5.1
Mattiasson cantilever beam
P
L
Figure 2 Mattiasson cantilever beam problem.
Mattiasson cantilever beam (Mattiasson,
1981) with a transversal point load (Figure 2) is
a large deflection problem most commonly
used in testing the behavior of geometrically
nonlinear beam finite element analysis. The
value of Young’s modulus is E=20000, the area
of the cross-section A=0.02, the shear area of
the cross-section As=A/1.2, the inertia of the
cross-section I=0.00025, the length of the beam
L=1, shear modulus is given with respect to
Young’s modulus G=10000 E and the applied
loading P=1. The results are given in Tables 1
and 2 for the reduced and the full integration.
Ten linear elements were used to solve this
problem and the results are compared with
those from (Mattiasson, 1981).
Standard
interpolation
Linked
interpolation
Results from
(Mattiasson, 1981)
u
0,0024662
0,0024662
0,00265
w
0,0662269
0,0662269
0,06636
ϕ
0,099659
0,099659
0,09964
Table 1 Reduced integration solutions.
Standard
interpolation
Linked
interpolation
8,04E-9
0,0025469
0,00265
w
0,00011978
0,0659063
0,06636
ϕ
0,00017967
0,0990763
0,09964
Table 2 Full integration solutions.
As it can be seen from the results shown in
the tables above, reduced integration gives the
same results for both the standard (Lagrangian)
and the linked interpolation. However, looking
at the results of the full integration it can be
seen that the linked interpolation gives much
better results than the standard interpolation.
Therefore we can conclude that the linked
interpolation is free from the locking effect even
when it is used in non-linear analysis.
u
6,46073
6,46073
w
22,4863
22,4863
Standard
interpolation
Linked
interpolation
u
0,0032633
0,421699
w
0,228125
4,56821
Table 4 Full integration solutions for the mesh
consisting of ten linear elements.
Taking quadratic elements to solve the same
problem increases the accuracy of the solution.
Those results are shown in Tables 5 and 6.
Standard
interpolation
Linked
interpolation
Results from
the literature
u
8,01638
8,01638
8,235
w
25,8625
25,8625
25,8823
Table 5 Reduced integration solutions for the
mesh of ten quadratic elements.
Hinged right-angle frame under fixed
point load
P
Linked
interpolation
Table 3 Reduced integration solutions for the
mesh consisting of ten linear elements.
Results from
(Mattiasson, 1981)
u
5.2
Standard
interpolation
Standard
interpolation
Linked
interpolation
u
3,34
5,42416
w
14,3643
19,4069
Table 6 Full integration solutions for the mesh of
ten quadratic elements.
L
L/5
6. Concluding remarks
L
Figure 3 Hinged right-angle frame under point
load.
This problem and its results are given in
(Simo and Vu-Quoc, 1986). The data are:
length L=120, inertia I=2, area of the cross
section A=6, Young’s modulus E=7.2E06,
Poisson’s ratio υ=0.3 and the vertical point load
P=15000. This problem has been solved using
ten linear and ten quadratic elements (five per
leg) and the results are given in Tables 3 and 4.
Results in (Simo and Vu-Quoc, 1986) are given
for ten quadratic elements.
In this paper the theory of higher-order
configuration-dependent interpolation (i.e.
nonlinear in the field variables) has been
introduced. Linked interpolation that gives
exact solutions in linear analysis has been used
to calculate the numerical examples in
nonlinear analysis. It is shown that, in the case
of reduced integration the results are identical
to the ones obtained by using the standard
Lagrangian interpolation. However, when using
full integration the linked interpolation is
significantly less sensitive to shear locking.
7. Acknowledgements
The results shown here were obtained within
the scientific project No 114-0000000-3025:
„Improved accuracy in non-linear beam
elements with finite 3D rotations” financially
supported by the Ministry of Science,
Education and Sports of the Republic of
Croatia.
8. References
J.C. Simo. A finite strain beam formulation.
The three-dimensional dynamic problem,
Part I. Computer Methods in Applied
Mechanics and Engineering, 49: 55-70,
1985.
M. Borri, C. Bottasso. An intrinsic beam model
based on a helicoidal approximation --- Part
I: Formulation. International Journal for
Numerical Methods in Engineering, 37:
2267-2289, 1994.
M.A. Crisfield, G. Jelenić. Objectivity of strain
measures in the geometrically exact threedimensional beam theory and its finiteelement implementation. Proceedings of the
Royal Society London Series A, 455: 11251147, 1999.
G. Jelenić, E. Papa. Exact solution of 3D
Timoshenko beam problem using linked
interpolation of arbitrary order. Arch Appl
Mech (2011) 81: 171-183.
E. Reissner. On one-dimensional largedisplacement finite-strain beam theory.
Journal of App. Math. And Phys (ZAMP).
23: 795-804, 1972.
G. Jelenić, E. Papa. Configuration-dependent
interpolation in non-linear 2D beam finite
elements. Proceedings of the 6th
International Congress of Croatian Society
of Mechanics, 2009.
K. Mattiasson. Numerical results from large
deflection beam and frame problems
analysed by means of elliptic integrals.
Int.J.Num.Meth.Engng., 16, 145-153, 1981.
J.C. Simo, L. Vu-Quoc. A three-dimensional
finite-strain
rod
model.
Part
II:
Computational aspects. Computer methods
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(1986) 79-116
J.C. Simo, L. Vu-Quoc. On the dynamics of
flexible beams under large overall motions
– the plane case: Part I and Part II. ASME
Journal of Applied Mechanics, 53: 849854, 1986.