Raji, S. A. USEP: Journal of Research Information in Civil Engineering, Vol. 3, No. 1, 2006
STATIC ANALYSIS OF FREE-FORM SHELLS USING
AN IMPROVED NINE-NODE LAGRANGIAN SHELL
ELEMENT
S.A. Raji
Department of Civil Engineering, University of Ilorin,
P.M.B. 1515, Ilorin, Nigeria.
E-mail address : saraji@unilorin.edu.ng
Abstract
In this paper, static analysis was carried out using the finite element method
for free-form shell element developed earlier by the author (Raji,1998). The
numerical work involved in performing these analyses necessitated the use
of an electronic computer. A Fortran 77 program, DSHELL was developed
and used to analyse the finite elements. Finally, some benchmark examples
of uniform and variable thickness plates and free-form shells were
considered to test the reliability of the improved nine-node Lagrangian shell
element. Results showed that the new element predicted the stress behavior
in shell elements accurately.
Keywords
Benchmark tests, Static analysis, Free-form shells
1. Introduction
Structural analysis is a vital part of the overall design optimization task
because one has to be able to predict structural behavior for various trial
designs in order to guide the design improvement process. Structural
analysis of shells has become more feasible because of steady increase in
computer speed and power, giving discretisation method like the finite
element method preference over the explicit mathematical methods for
structural response.
Benchmarks should be chosen to satisfy the following requirements (Parish,
1997).

The geometries must be simple to define, so that the tests can be
constructed with minimum of effort.
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Raji, S. A. USEP: Journal of Research Information in Civil Engineering, Vol. 3, No. 1, 2006







The geometries must represent reasonably realistic problems.
The tests must cover all aspects of how the element might be used in
practice in an unambiguous manner.
The mesh and element boundaries should be defined unambigously and
with enough precision to guarantee that the problem specification has
no effect upon the results.
The chosen idealisation of mesh should not contain heavily distorted
elements with unrealistic shapes.
Solutions other than finite element solution should preferably exist for
the geometries and loadings used in the benchmark. This may be
relaxed where ‘exact’ or alternative solutions do not exist.
The tests must be based upon facilities that are likely to be availabe in
any general purpose system and must not, for example, use forms of
constraints or element assumptions that exist only in specialised
systems.
Where possible target stresses should be quoted in the benchmarks as
single displacements and stresses, rather than in graphical form.
Several papers (Raji 1998, Huang and Hinton 1986) and texts(Hinton
et.al.,1979) have presented many examples of the performance of the
Ahmad shell element, Huang-Hinton element and other elements. In this
paper, only a brief benchmark examples is used to check the formulation
and implementation of the proposed improved nine-node Lagrangian
element in predicting the displacements and stresses in plates and free-form
shells.
2. Program Structure for DSHELL
An analysis module, DSHELL was developed using Fortran 77 on a Sun
workstation and used to analyse the finite elements. Figure 1 summarises
the basic requirements of the computer program necessary for the complete
solution of a problem by the finite element method using program
DSHELL.
A modular approach is adopted for the program, with the various main
finite element operations being performed by separate subroutines. In the
program DSHELL, though the input files for static, sensitivity and
optimisation analysis are different, all the data sets have to be read in at the
begining of the program.
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Raji, S. A. USEP: Journal of Research Information in Civil Engineering, Vol. 3, No. 1, 2006
3. Numerical Examples for Static Analysis
The numerical experiments are based on showing how the proposed 9-node
Lagrangian element (Raji,1998) behaves in some applications. These
benchmark examples are used to test the accurary with which the bendings
and displacements are represented in plates and free-form shells.
INPUT
CHECK1
PROGRAM
SHELL1
ECHO
CHECK2
GAUSSQ
MODTS
STIFTS
BMATS
BMTT
THETA
JACOB3
SFR2
LOADTS
PROGRAM
SHELL2
-FRONT
PROGRAM
SHELL3
-REVERSE
INPUT
PROGRAM
SHELL4
-BACKSUB
OUTPUT
Fig. 1: Program DSHELL organisation
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Raji, S. A. USEP: Journal of Research Information in Civil Engineering, Vol. 3, No. 1, 2006
3.1 Plates and Shells
(a) Simply supported square plates with uniform loading
A uniformly loaded, simply supported square plate with hard simple
supports, that is, with the lateral displacements and tangential edge
rotations constrained to zero) is initially considered as shown in Figure 2.
y
x
0
1
20
40
w
60
80
( a) selective integration
(b) reduced integration
c) thin plate 'exact'
a
b,c
100
21
41
61
81
Nel
Fig. 3: Convergencce characteristics for thin square plates: edge
deflection w.r.t. 'exact' thin plate theory.
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Raji, S. A. USEP: Journal of Research Information in Civil Engineering, Vol. 3, No. 1, 2006
1.
2
1
a
(a) Selective integration
(b) Reduced integration
b,c
(c) Thin plate 'exact' [5
0.
8
No/Nx
0.
6
0.
4
0.
2
0
0
2
4
6
2x/a
8
1
0
Fig. 4:Shear force for square thin plate normalized w.r.t. thin plate theory.
The following values are assumed in the analysis to simplify the
presentation of the results in a non-dimensionalised form. The elastic
modulus E is taken to be 10.92t-3 , where t is the thickness of plate,
Poisson’s ratio  is assumed to be 0.3 so that the flexural rigidity D=1
where D=Et3/12(1-\2). A uniform load q of intensity 1.0 was applied to a
square plate of side length a. Two cases were considered; a moderately
thick } plate with t/a =0.1 and a thin} plate with t/a = 0.01 . From
considerations of symmetry only a quadrant of the plate is considered in the
analysis.
In this case, the boundary condition on the edges x=0 and x=a are simply
supported with zero moment. A closed form solution is not available but
the finite element values were compared with the plate theory solution
(Scordelis,1989). Figure 3 shows the rate of convergence of the edge
displacement using a regular mesh with both reduced and selective
integrations. The results demonstrate that the selective integration
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Raji, S. A. USEP: Journal of Research Information in Civil Engineering, Vol. 3, No. 1, 2006
technique converges rather slowly to the ‘exact’ thin plate solution. In
contrast, the reduced integration proves highly accurate for the deflection.
Figure 4 shows the distribution of the shear force for square thin plate
normalized with respect to the thin plate theory using both selective and
reduced integrations. The results are compared with the ‘exact’ thin plate
results. The reduced integration also gave very close values to the ‘exact’
thin plate values compared to the selective integration results.
Table 1 summarises the values of the strain energy, maximum displacement
and stress resultants respectively for the moderately thick and thin cases.
The deflections and stress resultants obtained are compared with the closed
form solutions. A very good comparison is obtained.
Table 1 : Strain energy, maximum displacements and stress resultants for
square plate under uniform load with hard simple support.
Plate
Type
t/a=0.1
Ndof
(active)
121
605
2006
‘exact’
t/a=0.01
‘exact’
121
605
3390
^
w
x qa 2 w x qa 4 / D
M x x qa 2
M xy x qa 2
x101
x101
S y x qa
x 103
x10 2
1.802747
1.802909
1.802911
0.42783
0.42793
0.42726
0.4929
0.4831
0.4793
0.3389
0.3301
0.3269
0.3321
0.3353
0.3368
-
0.42725
0.4787
0.3247
0.3376
1.703059
1.703509
1.703514
0.40717
0.40645
0.40643
0.4939
0.4819
0.4791
0.3383
0.3307
0.3260
0.3348
0.3359
0.3369
-
0.42727
0.4787
0.3249
0.3378
(b) Variable thickness plates
Secondly, variable thickness rectangular plates subjected to uniformly
distributed load with different thickness ratios, loading and support
conditions were considered. The plates have side lengths a and b in the xand y-directions respectively with an aspect ratio a/b = 2. The thicknesses
at x=0 and x=a are designated as t1 and t2 respectively with the flexural
rigidities at x=0 and x=a similarly designated as D1 and D2 respectively.
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Raji, S. A. USEP: Journal of Research Information in Civil Engineering, Vol. 3, No. 1, 2006
The material of the plate has a Poisson’s ratio  = 0.3. A hard simply
supported edge is represented by Sh, a clamped edge by C and a free edge
is represented by F.
The plates are analysed using an 8x8 mesh in program HSHELL. Table 2
displays the values of the maximum deflection at y=b/2 and strain energy.
The results of the analyses are found to be in excellent agreement with the
results presented using the Huang-Hinton element (Raji,1998).
Table 2 :Maximum Deflections and SE values for rectangular variable
thickness plates
Support
t / b t / b w2 x 10 4
w x10 4
1
Sh / Sh / Sh / Sh
0.01
S h / S h / S h / S h 0.1
2
0.03
PE
357.73
H-H
357.16
PE
487.19
H-H
487.4
0.3
393.1
392.4
523.8
523.9
Sh / C / Sh / F
0.01
0.02
43.69
43.69
4.589
4.598
Sh / C / Sh / F
0.1
0.2
49.09
49.01
4.792
4.783
*A hard simply supported edge is represented by Sh, a clamped edge by C
and a free edge is represented by F.
( c ) Cylindrical shell roof
The cylindrical shell roof shown in Figure 5 is now considered. The shell is
simply supported on diaphragms along the two curved edges and is free
along the other two edges. It is subjected to a self weight loading per unit
area of 4.31 kN/m^2, and the following material properties and dimensions
are assumed:
elastic modulus E = 2.07 x 1010 N/m2, Poisson’s ratio ν= 0.25, radius R =
7.62m, span L = 5.24 m. and thickness t =76 mm.
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Raji, S. A. USEP: Journal of Research Information in Civil Engineering, Vol. 3, No. 1, 2006
The ‘exact’ solution(Huang,1989) is obtained by solving the plate equation
with the aid of computer.
In Figure 6, the effects of normal 3 x3 Gauss point integration and
the reduced 2 x 2 integration using parabolic element is shown on the axial
displacements. Both types of integrations give convergence solutions as
expected but the reduced integration gives deflections which are almost
the same as the ‘exact’ ones(Huang,1989) .
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Raji, S. A. USEP: Journal of Research Information in Civil Engineering, Vol. 3, No. 1, 2006
The mathematical explanation for the improvement was explained by
Parish,1979 that the coordinates of the 2x2 integration points η,ζ= 1/√3
with the shear expressions are exactly evaluated.
The transverse moment at the mid-span of the cylindrical vault using 2x2
reduced integration with the substended angle is shown in Figure 8. The
values were compared with the ‘exact’ solution which quite agree. In this
case, the moment depends on the position at the mid-span. The moment is
minimum at the centre and increases to almost zero value at the outer edge.
Angles(Degrees)
0
-1
-2
-3
Nxo-4
-5
-6
-7
-8
-9
-40
-30
-15
15
0
30
40
reduced
selective
Fig. 7: Shear force (Nxo) at end of support of cylindrical vault
0
-40
-0.5
-1
w(mm)
-1.5
-30
-20
-10
0
10
20
30
40
Angles(Deg.)
reduced
selective
-2
-2.5
Fig. 8 : Transverse moment (Mo) at mid-span of cylindrical vault
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Raji, S. A. USEP: Journal of Research Information in Civil Engineering, Vol. 3, No. 1, 2006
Table 3 presents the maximum deflection, strain energy value and the
contribution of the membrane and bending energies to the total strain
energy using the proposed nine-node Lagrangian element. The results
compare well with those obtained by Scordelis and Lo (Huang,1989) .
The performance of the improved nine-node Lagrangian element in these
standard examples shows that the element is reliable and robust, and could
be confidently used for structural analysis, sensitivity and structural shape
optimisation of free-form shells.
Table 3 : Strain energy and maximum displacement values for cylindrical
shell roof under self weight.
Mesh
w(mm)
w
2
Size
Percentage
Contribution to SE
Membrane
Bending
2x2
94.441
8997.2
53.0
47.0
4x4
91.211
9609.6
47.7
52.3
6x6
91.593
9649.2
47.6
52.3
8x8
91.685
9656.8
47.6
52.3
10x10
91.721
9660.0
47.6
52.3
Shear contribution to SE is negligible.
4. Conclusion
The performance of the improved nine-node Lagrangian element in these
standard examples shows that the element is reliable and robust, and could
be confidently used for structural analysis, sensitivity and structural shape
optimization of free-form shells.
5. Acknowledgement
The author wishes to thank Prof. E. Hinton of the Civil Engineering
Department, University of Wales, Swansea for his useful discussions and
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Raji, S. A. USEP: Journal of Research Information in Civil Engineering, Vol. 3, No. 1, 2006
for making the use of his library available to him while he was carrying out
his Ph.D. bench work in Swansea.
6. References
NAFEMS, (2005), The International Association for the Engineering
Analysis Community, http://www.nafems.org, 2005.
Hinton, E., Schrefler, B. and Natali, A.(1979), A Finite Element Method for
Variable Thickness Plates, 22nd ICES User Group Int. Conf., McGill
University, Montreal, Canada.
Huang, H. C. and Hinton, E.(1986), A New Nine-Node Degenerated Shell
Element with Enhanced Membrane And Shear Interpolation”, Int. J. Num.
Meth. Engng., No. 22, pp.73-92.
Huang, H.C.(1989), Static and Dynamic Analyses of Plates and Shells,
Springer-Verlag, Berlin, Heidelberg.
Parish,H.(1979), A Critical Survey of the 9-Node Degenerated Shell
Element with Special Emphasis on Thin Shell Application and
Reduced Integration, Computer Methods”, Int. J. Applied Mechanics And
Engineering, No. 20, pp.323-350.
Raji, S.A. (1998), A Nine-node Lagrangian Shell Element for the Analysis
and Optimization of Shell Structures, Ph.D thesis, University of Lagos,
Lagos,Nigeria
Raji, S.A. (1997), Innovative Finite Element for Shells, 4th International
Conference on Structural Engineering Analysis of Modelling, SEAM4,
University of Science and Technology, Kumasi, Ghana, pp.386-399.
Rao, N.V.R. (1992), Analysis and Optimisation of Free-form Shell
Structures, Ph.D Thesis , University of Wales, Swansea.
Scordelis, A.C. and Lo, K.S.(1969), Computer Analysis of Cylindrical
Shells, ACI Journal, No. 61, pp.539-561.
Timoshenko,S. and Woinowsky-Krieger,S.(1959), Theory of Plates and
Shells, 2nd ed., Mc. Graw -Hill, NY.
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