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Studies on performance of curved beam finite elements for analysis of thin arches
Computers & Srruc~ures Vol. 31, No. 6. pp. 997-1002. Printed in Great Britain.
1989 0
0045-7949/89 $3.00 + 0.00 I989 Pergamon Press plc
STUDIES ON PERFORMANCE OF CURVED BEAM FINITE ELEMENTS FOR ANALYSIS OF THIN ARCHES N. PANDIAN,T. V. S. R. AWA RAo and SATISHCHANDRA~ Structural Engineering Research Centre, Madras 600 113, India (Received 10 March 1988) Abstract-The finite element model with cubic polynomial shape functions to analyse thin arches has been modified to relieve membrane locking effects and thus to achieve better performance. Two different ways of improving the element stiffness have been studied, one by adopting the reduced integration technique and the other by adopting a least square fit to achieve field consistency for membrane strains. The curved cantilever beam of Kikuchi and two fixed circular arches of Dawe having different ratios of radius to thickness (one deep, the other shallow) were studied by using the improved models. The results obtained by the improved models are compared with those of others. The model with three point integration for membrane energy gives, as expected, identical results in comparison with field consistency. The model with two point reduced integration in membrane and four point integration for bending has been found to be superior in convergence.
I. INTRODU~ION
2. CUBIC-CUBICFINITE ELEMENT MODEL
Accurate and efficient analysis of thin curved beams and arches by employing the finite element method has been of considerable interest for quite some time, and it also serves as a forerunner for analysis of thin shells. A number of investigators [l-7] have formulated finite element models and studied their performance in solving a variety of arch problems including very thin arches. Ashwell and Gallagher [4,6] provide a very good comparison of the performance of different models. These studies indicate that the quintic-quintic finite element model has superior convergence. Recent investigators have concentrated on improving the earlier models and also in developing new elements incorporating rigid body modes or by relieving “locking” (membrane, or shear or both) by adopting reduced integration [5,8,9] or by adopting the field consistency[2] approach. The authors have been interested in studying the efficacy of field consistency and reduced integration techniques for analysis of arch problems using a finite element curved beam model with cubic polynomials for both axial and normal deformations, without taking into account shear deformations since this model has wide applications. In an earlier [l] investigation trigonometric displacement functions have been tried out to formulate the element. The aim of this paper is to present the application of reduced integration and field consistency techniques for arch problems to investigate their efficacy in solving the problem of membrane locking of the cubic-cubic finite element model. A description of the techniques adopted to improve the cubic finite element mode1 and also the results of case studies are presented in this paper. tPresent address: National Bangalore 560 017, India.
Aeronautical
Laboratory,
The displacement follows:
function empolyed herein is as
u = a, + u,a +
+a2 + a,a3
w = 6, + b,a +
&a2 +
(1)
b4a3.
The shape function in natural co-ordinate drawn in Fig. 1 is given by [lo]
system as
u = N,u, + N2u; + N,u2 + N4u;
(2)
where N,=1-3L:+2L:, N, = 3Li - 2L:
N,=IL:L,
and
N4 = -lL,L$.
The axial and bending strains for the curved beams are given by 1au E=i%a-R
w
(4)
Strain energy of the element is given by Li=~~(r2+;k2)d..
(5)
Element stiffness matrices and assembly of stiffness matrices were carried out by using a standard procedure. 997
998
N. PANDIAN et
al
4. TECHNIQUES
TO RELIEVE MEMBRANE
LOCKING
Field consistency
One way of correcting the problem is to assume a one degree higher polynomial for u than that for w, so that the state of inextensional bending can be represented. This kind of assumed displacement is termed as “field consistent” by Prathap [2]. Another approach as stated by Prathap is to force the field consistency by fitting a second degree polynomial by least square approximation for w displacement and considering such a fitted polynomial for evaluating membrane strain. This has been adopted in the present investigation and the function, smoothened by means of regression, is given in Table 1. L2=
a2
Reduced integration
a,
a -
-
-
a1
(Prime) indicates differential
with respect to a
Fig. 1. Curved beam element.
3. MEMBRANE
LOCKING OF CUBIC-CUBIC ELEMENT MODEL
FINITE
One of the major reasons identified for very slow convergence of certain curved finite element models is “membrane locking” that occurs due to the displacement functions of the same order being adopted for representing u and w displacements. It has been noted that rate of convergence of a finite element model also depends on the facility with which they can represent the state of inextensional bending of arch. This is examined in the following for the cubic-cubic model under consideration. Substituting eqn (1) in the axial strain equation (3)
+,a’.
(6)
Here, the first three terms of the right hand side of the strain equation are represented by both u and w terms while the last term [( - l/R)b4a3] depends only on the coefficient b4 of w displacement and therefore cannot represent inextensional bending unless LYtends to zero. Convergence of the solution is considerably delayed due to this term. This is known as “membrane locking”.
It is well-known that reduced integration has been widely adopted to relieve “shear locking” effects [5] in the case of elements formulated by taking into account shear deformations for computation of strain energy. Reduced integration below a certain order is also beset with problems of spurious mechanisms. Keeping this in mind reduced integration of different orders has been tried in this investigation to relieve membrane locking. The different finite element models studied and the order of reduced integration adopted in this investigation are given in Table 2. 5. NUMERICAL
STUDIES
Details of different arch problems herein are shown in Table 3.
Clamped arches with central point load [4]
Four arch geometries are considered and each arch has both ends clamped and carries a point load at the centre which acts normally to the arch centre line. The first two arches are semi-circular with the radius of the centre line taken to be 17in. and the arch thickness is to be 1 in. or 1/16in. (R/t = 17 and 272 respectively). The other two arches are of the same base length and thickness but subtend an angle of only 30” rather 180”. Making use of the symmetry of the problem, only half of the arch was analysed with the following boundary conditions: dW
Fixed end
a=w=-SO.0 aa
Centre point
u = - = 0.0. f3a
aw
Table 1. Rearession constants for smoothened shape functions Cubic functions (11 Shape functions to be used for calculating membrane strain
investigated
Smoothened functions (2)
N,=l-3L;+2L;
N, = 1.1- 1.2L,
N2 = IL:L, N, = 3L; - 2L; N4= -IL,L;
N2 = l(O.05 + 0.4L,-OSL:) N, = (-0.1 + 1.2LJ N4 = -Q-O.05 + 0.6L, -0SL;)
Studies on performance of curved beam finite elements
999
Table 2. Description of cubic-cubic models
Sk No.
Model
Gauss-Radau integration points* Membrane strain field
Description
1
cc
Cubic-cubic model
4
2
CCM
Cubic-cubic model with membrane strain field made consistent
4
3
CCR3
Cubic-fubic model by adopting reduced integration in the
3
4
CCR2
membrane strain field Cubic-cubic model by adopting reduced integration in the membrane strain field
2
* Four point integration is adopted for bending strain field throughout the study.
6. DISCUSSION
Circularly curved cantilever of Kikuchi[3]
A curved cantilever beam of length equal to its radius (R) subjected to a ~on~ntrated load in the normal direction at the free end was analysed. Three different cases with
p
=&=
10-3, 10e5 and lo-’
were studied (where I is the moment of inertia and A is the area of cross-section). The results obtained for the above problems by employing reduced integration and field consistency techniques are presented in Figs 2-5 in comparison with those obtained earlier for the finite element cubic-cubic model.
AND CONCLUSIONS
The convergent study of deflections under point load for various models and different test example problems has been presented in Figs 2-5. This study shows that the performance of the finite element model with field consistency is the same as that of the reduced integration model with three point integration for membrane and four point integration for bending. This is because, as mentioned earlier, the values of coefficients in shape functions of the cubiccubic model at the locations of three point integration (reduced) are the same as those for the corresponding smoothened functions in the case of the field consistency approach. It is interesting to note here that if the shape functions of columns 1 and 2 of TabIe 1 are equated and solved for the locations L, the values
Table 3. Geometry and material properties of case studies Sl. No.
Arch type
1. Deep arch OT = 180” R=l7”
*a 1" 2. Shallow arch a = 30”
R=65.68
Thick/thin Thick AE= 1.0 EI= 0.0833 Thin AE = 0.~3906 EI=O.l27fL3-05 Thick
Defl/stress fig. Deft. under pt. load Fig. 2a
Pt. load P=l
Fig. 2b
P=l
Fig. 3a
P=l
Fig. 3b
P=l
Fig. 4a
P=l
AE= 1.0 EZ= 0.0833 Thin AE = 0.003906
EI=O.I27ID-OS 3. Cantilever arch a = 1 rad.
/9= IO-’ ; f ;;I:
R = 288.764
Fig. Fig. Fig. Stress for
4b 5a 5b 8 = 10m3
N. PANDIAN et al.
1000 of freedom
Degrees
of freedom
Degrees 7
15
23
\ \\ \ \
b k a? lo-‘-
zk m E 8 tl P
-
31
39
47
KCR3)
\ \ \ \ \ \ h
lo-2-
‘\\
‘OCCRP
lo-3:
10-31
(b) Thin section
(a) Thick section Fig. 2. Convergence Degrees 7
102k
15
1
of central
deflection,
deep clamped Degrees
of freedom 23
I
31
I
39
arch. of freedom
lo2-
I
CCM .,p3)
,,-al I
I
I
“”
10-J
I
(b) Thin section
(a) Thick section Fig. 3. Convergence
I
of central
deflection,
shallow
clamped
arch.
I
I
Studies on performance of curved beam finite elements Degrees of freedom
Degrees of freedom
(b) p= 1O-5
(a)p=10m3
Fig. 4. Convergence of deflection, circular curved cantilever Degrees of freedom 102
td)
9
25
Degrees of freedom
33
41
“\
b
‘o*,,,
cc
lo-'=
E 8 5 P
17
“0CCM F.xR3~
CT
\ 10-z* -1\ \
10-Z
10-a I_-
(a) p = 10B7 (deflection)
(b) p = 10m3 (moment)
Fig. 5. Convergence of deflection and moment, circular curved cantilever.
1001
N. PANDIANet al.
1002
obtained will be exactly the same as those of the well-known Gauss-Radau three point integration. The influence of this can be seen in the results obtained by the two different models discussed earlier. The present study also indicates that the performance of the model with reduced integration (two point for membrane and four point for bending), in general is superior to that of other models. To achieve faster convergence, two point integration may be adopted with cubic-cubic arch finite elements. One point integration for membrane energy was also tried, but spurious mechanisms were found. This study is a forerunner for further investigations on doubly curved shell elements to take advantage of reduced integration and/or field consistency approach. Acknowledgement-We
wish to thank Dr. M. Ramaiah, Director, Structural Engineering Research Centre, Madras, for his kind permission to publish this paper.
REFERENCES
I. T. V. S. R. Appa Rao, N. Pandian, Satish Chandra and
Nagesh R. Iyer, Performance of straight and curved finite elements in the analysis of curved beams and
arches. Proc. Int. Conf. on Finite Elements in Computational Mechanics, 2.
pp. 113-122 (1985). G. Prathap, The curved beam/deep arch/finite ring element revisited. NAL-TAM-ST-501/257/83
(1983).
3, F. Kikuchi and K. Tanizawa, Accuracy and locking-free
property of the beam element approximation for arch problems. Comput. Strut 19, 103-I 10 (1984). 4. D. Ashwell and R. W. Gallagher, Finite Elements for Thin Shells and Curved Members, pp. 131-153. John Wiley, New York (1976). 5. A. K. Noor and J. M. Peters, Mixed models and reduced/selective integration displacement models for non-linear analysis of curved beams. Inc. J. Numer. Meth. Engng 7, 615-631 (1981). 6. D. Ashwell and R. W. Gallagher, Finite Elements for Thin Shells and Curved Members, pp. 91-111. John
Wiley, New York (1976). 7. Y. Yamada and Y. Ezawa, On curved finite elements for the analysis of circular arches. Inc. J. Numer. Meth. Engng 11, 1635-1651 (1977). 8. Satish Chandra, Nagesh R. Iyer, N. Pandian and T. V.
S. R. Appa Rao, Reduced integration for curved beam finite elements. Report No. PPS-RR-86-3, Structural Engng Res. Centre, Madras (1986). 9. H. Stolarski and T. Belytschko, Membrane locking and reduced integration for curved elements. Trans. J. appl. Mech. 49, 172-176 (1982).
IO. T. V. S. R. Appa Rao, K. Loganathan and R. H. Gallagher, A discrete stiffener element for doubly curved shells. Int. Conf. on Computer Application in Civil Engng, Roorkee, India (1979).
1989 0
0045-7949/89 $3.00 + 0.00 I989 Pergamon Press plc
STUDIES ON PERFORMANCE OF CURVED BEAM FINITE ELEMENTS FOR ANALYSIS OF THIN ARCHES N. PANDIAN,T. V. S. R. AWA RAo and SATISHCHANDRA~ Structural Engineering Research Centre, Madras 600 113, India (Received 10 March 1988) Abstract-The finite element model with cubic polynomial shape functions to analyse thin arches has been modified to relieve membrane locking effects and thus to achieve better performance. Two different ways of improving the element stiffness have been studied, one by adopting the reduced integration technique and the other by adopting a least square fit to achieve field consistency for membrane strains. The curved cantilever beam of Kikuchi and two fixed circular arches of Dawe having different ratios of radius to thickness (one deep, the other shallow) were studied by using the improved models. The results obtained by the improved models are compared with those of others. The model with three point integration for membrane energy gives, as expected, identical results in comparison with field consistency. The model with two point reduced integration in membrane and four point integration for bending has been found to be superior in convergence.
I. INTRODU~ION
2. CUBIC-CUBICFINITE ELEMENT MODEL
Accurate and efficient analysis of thin curved beams and arches by employing the finite element method has been of considerable interest for quite some time, and it also serves as a forerunner for analysis of thin shells. A number of investigators [l-7] have formulated finite element models and studied their performance in solving a variety of arch problems including very thin arches. Ashwell and Gallagher [4,6] provide a very good comparison of the performance of different models. These studies indicate that the quintic-quintic finite element model has superior convergence. Recent investigators have concentrated on improving the earlier models and also in developing new elements incorporating rigid body modes or by relieving “locking” (membrane, or shear or both) by adopting reduced integration [5,8,9] or by adopting the field consistency[2] approach. The authors have been interested in studying the efficacy of field consistency and reduced integration techniques for analysis of arch problems using a finite element curved beam model with cubic polynomials for both axial and normal deformations, without taking into account shear deformations since this model has wide applications. In an earlier [l] investigation trigonometric displacement functions have been tried out to formulate the element. The aim of this paper is to present the application of reduced integration and field consistency techniques for arch problems to investigate their efficacy in solving the problem of membrane locking of the cubic-cubic finite element model. A description of the techniques adopted to improve the cubic finite element mode1 and also the results of case studies are presented in this paper. tPresent address: National Bangalore 560 017, India.
Aeronautical
Laboratory,
The displacement follows:
function empolyed herein is as
u = a, + u,a +
+a2 + a,a3
w = 6, + b,a +
&a2 +
(1)
b4a3.
The shape function in natural co-ordinate drawn in Fig. 1 is given by [lo]
system as
u = N,u, + N2u; + N,u2 + N4u;
(2)
where N,=1-3L:+2L:, N, = 3Li - 2L:
N,=IL:L,
and
N4 = -lL,L$.
The axial and bending strains for the curved beams are given by 1au E=i%a-R
w
(4)
Strain energy of the element is given by Li=~~(r2+;k2)d..
(5)
Element stiffness matrices and assembly of stiffness matrices were carried out by using a standard procedure. 997
998
N. PANDIAN et
al
4. TECHNIQUES
TO RELIEVE MEMBRANE
LOCKING
Field consistency
One way of correcting the problem is to assume a one degree higher polynomial for u than that for w, so that the state of inextensional bending can be represented. This kind of assumed displacement is termed as “field consistent” by Prathap [2]. Another approach as stated by Prathap is to force the field consistency by fitting a second degree polynomial by least square approximation for w displacement and considering such a fitted polynomial for evaluating membrane strain. This has been adopted in the present investigation and the function, smoothened by means of regression, is given in Table 1. L2=
a2
Reduced integration
a,
a -
-
-
a1
(Prime) indicates differential
with respect to a
Fig. 1. Curved beam element.
3. MEMBRANE
LOCKING OF CUBIC-CUBIC ELEMENT MODEL
FINITE
One of the major reasons identified for very slow convergence of certain curved finite element models is “membrane locking” that occurs due to the displacement functions of the same order being adopted for representing u and w displacements. It has been noted that rate of convergence of a finite element model also depends on the facility with which they can represent the state of inextensional bending of arch. This is examined in the following for the cubic-cubic model under consideration. Substituting eqn (1) in the axial strain equation (3)
+,a’.
(6)
Here, the first three terms of the right hand side of the strain equation are represented by both u and w terms while the last term [( - l/R)b4a3] depends only on the coefficient b4 of w displacement and therefore cannot represent inextensional bending unless LYtends to zero. Convergence of the solution is considerably delayed due to this term. This is known as “membrane locking”.
It is well-known that reduced integration has been widely adopted to relieve “shear locking” effects [5] in the case of elements formulated by taking into account shear deformations for computation of strain energy. Reduced integration below a certain order is also beset with problems of spurious mechanisms. Keeping this in mind reduced integration of different orders has been tried in this investigation to relieve membrane locking. The different finite element models studied and the order of reduced integration adopted in this investigation are given in Table 2. 5. NUMERICAL
STUDIES
Details of different arch problems herein are shown in Table 3.
Clamped arches with central point load [4]
Four arch geometries are considered and each arch has both ends clamped and carries a point load at the centre which acts normally to the arch centre line. The first two arches are semi-circular with the radius of the centre line taken to be 17in. and the arch thickness is to be 1 in. or 1/16in. (R/t = 17 and 272 respectively). The other two arches are of the same base length and thickness but subtend an angle of only 30” rather 180”. Making use of the symmetry of the problem, only half of the arch was analysed with the following boundary conditions: dW
Fixed end
a=w=-SO.0 aa
Centre point
u = - = 0.0. f3a
aw
Table 1. Rearession constants for smoothened shape functions Cubic functions (11 Shape functions to be used for calculating membrane strain
investigated
Smoothened functions (2)
N,=l-3L;+2L;
N, = 1.1- 1.2L,
N2 = IL:L, N, = 3L; - 2L; N4= -IL,L;
N2 = l(O.05 + 0.4L,-OSL:) N, = (-0.1 + 1.2LJ N4 = -Q-O.05 + 0.6L, -0SL;)
Studies on performance of curved beam finite elements
999
Table 2. Description of cubic-cubic models
Sk No.
Model
Gauss-Radau integration points* Membrane strain field
Description
1
cc
Cubic-cubic model
4
2
CCM
Cubic-cubic model with membrane strain field made consistent
4
3
CCR3
Cubic-fubic model by adopting reduced integration in the
3
4
CCR2
membrane strain field Cubic-cubic model by adopting reduced integration in the membrane strain field
2
* Four point integration is adopted for bending strain field throughout the study.
6. DISCUSSION
Circularly curved cantilever of Kikuchi[3]
A curved cantilever beam of length equal to its radius (R) subjected to a ~on~ntrated load in the normal direction at the free end was analysed. Three different cases with
p
=&=
10-3, 10e5 and lo-’
were studied (where I is the moment of inertia and A is the area of cross-section). The results obtained for the above problems by employing reduced integration and field consistency techniques are presented in Figs 2-5 in comparison with those obtained earlier for the finite element cubic-cubic model.
AND CONCLUSIONS
The convergent study of deflections under point load for various models and different test example problems has been presented in Figs 2-5. This study shows that the performance of the finite element model with field consistency is the same as that of the reduced integration model with three point integration for membrane and four point integration for bending. This is because, as mentioned earlier, the values of coefficients in shape functions of the cubiccubic model at the locations of three point integration (reduced) are the same as those for the corresponding smoothened functions in the case of the field consistency approach. It is interesting to note here that if the shape functions of columns 1 and 2 of TabIe 1 are equated and solved for the locations L, the values
Table 3. Geometry and material properties of case studies Sl. No.
Arch type
1. Deep arch OT = 180” R=l7”
*a 1" 2. Shallow arch a = 30”
R=65.68
Thick/thin Thick AE= 1.0 EI= 0.0833 Thin AE = 0.~3906 EI=O.l27fL3-05 Thick
Defl/stress fig. Deft. under pt. load Fig. 2a
Pt. load P=l
Fig. 2b
P=l
Fig. 3a
P=l
Fig. 3b
P=l
Fig. 4a
P=l
AE= 1.0 EZ= 0.0833 Thin AE = 0.003906
EI=O.I27ID-OS 3. Cantilever arch a = 1 rad.
/9= IO-’ ; f ;;I:
R = 288.764
Fig. Fig. Fig. Stress for
4b 5a 5b 8 = 10m3
N. PANDIAN et al.
1000 of freedom
Degrees
of freedom
Degrees 7
15
23
\ \\ \ \
b k a? lo-‘-
zk m E 8 tl P
-
31
39
47
KCR3)
\ \ \ \ \ \ h
lo-2-
‘\\
‘OCCRP
lo-3:
10-31
(b) Thin section
(a) Thick section Fig. 2. Convergence Degrees 7
102k
15
1
of central
deflection,
deep clamped Degrees
of freedom 23
I
31
I
39
arch. of freedom
lo2-
I
CCM .,p3)
,,-al I
I
I
“”
10-J
I
(b) Thin section
(a) Thick section Fig. 3. Convergence
I
of central
deflection,
shallow
clamped
arch.
I
I
Studies on performance of curved beam finite elements Degrees of freedom
Degrees of freedom
(b) p= 1O-5
(a)p=10m3
Fig. 4. Convergence of deflection, circular curved cantilever Degrees of freedom 102
td)
9
25
Degrees of freedom
33
41
“\
b
‘o*,,,
cc
lo-'=
E 8 5 P
17
“0CCM F.xR3~
CT
\ 10-z* -1\ \
10-Z
10-a I_-
(a) p = 10B7 (deflection)
(b) p = 10m3 (moment)
Fig. 5. Convergence of deflection and moment, circular curved cantilever.
1001
N. PANDIANet al.
1002
obtained will be exactly the same as those of the well-known Gauss-Radau three point integration. The influence of this can be seen in the results obtained by the two different models discussed earlier. The present study also indicates that the performance of the model with reduced integration (two point for membrane and four point for bending), in general is superior to that of other models. To achieve faster convergence, two point integration may be adopted with cubic-cubic arch finite elements. One point integration for membrane energy was also tried, but spurious mechanisms were found. This study is a forerunner for further investigations on doubly curved shell elements to take advantage of reduced integration and/or field consistency approach. Acknowledgement-We
wish to thank Dr. M. Ramaiah, Director, Structural Engineering Research Centre, Madras, for his kind permission to publish this paper.
REFERENCES
I. T. V. S. R. Appa Rao, N. Pandian, Satish Chandra and
Nagesh R. Iyer, Performance of straight and curved finite elements in the analysis of curved beams and
arches. Proc. Int. Conf. on Finite Elements in Computational Mechanics, 2.
pp. 113-122 (1985). G. Prathap, The curved beam/deep arch/finite ring element revisited. NAL-TAM-ST-501/257/83
(1983).
3, F. Kikuchi and K. Tanizawa, Accuracy and locking-free
property of the beam element approximation for arch problems. Comput. Strut 19, 103-I 10 (1984). 4. D. Ashwell and R. W. Gallagher, Finite Elements for Thin Shells and Curved Members, pp. 131-153. John Wiley, New York (1976). 5. A. K. Noor and J. M. Peters, Mixed models and reduced/selective integration displacement models for non-linear analysis of curved beams. Inc. J. Numer. Meth. Engng 7, 615-631 (1981). 6. D. Ashwell and R. W. Gallagher, Finite Elements for Thin Shells and Curved Members, pp. 91-111. John
Wiley, New York (1976). 7. Y. Yamada and Y. Ezawa, On curved finite elements for the analysis of circular arches. Inc. J. Numer. Meth. Engng 11, 1635-1651 (1977). 8. Satish Chandra, Nagesh R. Iyer, N. Pandian and T. V.
S. R. Appa Rao, Reduced integration for curved beam finite elements. Report No. PPS-RR-86-3, Structural Engng Res. Centre, Madras (1986). 9. H. Stolarski and T. Belytschko, Membrane locking and reduced integration for curved elements. Trans. J. appl. Mech. 49, 172-176 (1982).
IO. T. V. S. R. Appa Rao, K. Loganathan and R. H. Gallagher, A discrete stiffener element for doubly curved shells. Int. Conf. on Computer Application in Civil Engng, Roorkee, India (1979).