COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China
©2006 Tsinghua University Press & Springer
Parametric Analysis and Optimization of Partial Double-Layer Reticulated
Shells Using Uniform Design and Rotational Second Order Design
J. C. Xiao 1*, T. Liang 1, Y. Liu 2
1
2
School of Civil Engineering & Architecture, Guizhou University, Guiyang, 550003 China
Department of Civil Engineering, Dalhousie University, Halifax NS B3J 1Z1, Canada
Email: [email protected]
Abstract The determination of optimum design variables of partial double-layer reticulated shells is of a multistage
optimization problem, which is difficult to solve by normal methods. To fulfill the requirement of preliminary design,
uniform design and rotational second order design are proposed to make a quick optimization. This is a kind of
response surface method, which requires fewer times of computational experiment. In wide range of design variables,
approximate regression relations between the objective values (the amount of material consumption, maximum
downward/upward deflections, basic frequency and minimum geometrically nonlinear buckling load) and the main
design variables (high rise of shell, the single-layer area, and material design stress ratio for single-layer members) are
obtained using uniform design. By solving a simple problem of nonlinear programming, the approximate optimum
combination of design variables that meet the requirements of design specifications is decided. Within the local range
of the approximate results, rotational second order design is used to improve the precision. An example of
multipoint-supported hexagonal partial double-layer reticulated shell is taken to explain the process of analysis.
Design verification shows the reliability of the results.
Key words: parametric analysis, optimization, uniform design, rotational second order design, partial double-layer
reticulated shells
INTRODUCTION
The configuration of partial double-layer reticulated shells is based on the theory of uniform strength shells. In a partial
double-layer reticulated shell, the single-layer area corresponds to the thin part of uniform strength shell. The
double-layer area, the curvatures of its top and lower chords being equal, is the mapping of the thick part. Between the
single-layer and double-layer areas, stiffness transition region is established (Fig. 1). The orderly configuration of
partial double-layer reticulated shells provides well impression of indoor view. This economy kind of structures has
been used in practical engineering.
Figure 1: Section of uniform-strength spherical shell and section of partial double-layer reticulated shell
The design variables of a partial double-layer reticulated shell are with different natures. The determination of
optimum design variables of is of a multistage optimization problem. The structure is subjected to stress, local
buckling, overall buckling and displacement constraints. Due to the complexity of the problem, current approaches
and software packages can not efficiently meet the needs of the optimization of space trusses under several kinds of
loading combination conditions.
Sake and Kameshki employed optimality criteria approach and nonlinear finite element method for the optimization of
elastic framed domes. Gil and Andreu proposed a methodology combining a full stressed design optimization with a
conjugates gradient optimization for the identification of the optimum shape and cross-sections of a plane truss
⎯ 1046 ⎯
structure. Wang, etc presented an evolutionary optimization method for weight minimum problem of a space truss
structure, where nodal coordinates and element cross-sectional areas being optimized by using full stressed design and
evolutionary node shift method.
At the stage of preliminary design, simple and efficiency approaches to determinate basic design variables seem to be
very important. In the paper, uniform design and rotational second order design are proposed to make a quick
optimization. This is a kind of response surface method, which requires fewer times of computational experiment.
ANALYSIS PROCESS
Structural optimization can be divided into cross-sectional optimization, configurational optimization, topological
optimization and structural type optimization in terms of optimization stage. The benefits from optimization increase
with the enhancement of optimization stage. Due to the different magnitude or dimension of design variables, and the
interaction of design variables, the degree of difficulty of multistage optimization is relatively greater.
One of the aims of experimental design is to find the best combination of variables under certain constraint conditions.
In experimental design, there are not demands for the magnitude or dimension of design variables. The interactions
among variables can also be considered.
Uniform design, one of experimental design approaches, was put forward by Fang in 1985. It has many advantages,
such as fewer times of experiment, more variables and levels that can be considered. And the experimental points are
uniformly distributed in design space. By regression to the experimental results, the internal relationship between
objective values and design variables, as well as the approximate solutions, can be obtained. This method is suitable
for the multivariable and multilevel experimental design specially.
Current software packages are relatively much more perfect. Many of them have the function of full stressed
optimization, and data exchange modules between different software packages. PLAScad, developed by us, is one of
such software packages for the design of space trusses. PLAScad has many inline procedures, such as modeling,
structural analysis, cross-section full stressed optimization and nonlinear stability analysis.
Professional software packages for the design of reticulated shells can not undertake the optimization that meets the
requirements of strength, stiffness and stability at the same time.
For preliminary design, the efficiency of optimization analysis has gain more attention. To quickly determinate the
design variables, uniform design and rotational second design are used in different optimization range of variables. In
wide range of design variables, uniform design is adopted to make parameter study and approximate optimization.
Within the local range of the approximate results, rotational second order design is used to improve the precision.
In the analysis the calculation experiments are completed by professional reticulated shell design software packages.
The results are checked by common structural software. Regression computation uses MS-Excel. Nonlinear minimum
problem with constraint conditions is solved by Matlab. This approach leaves out much of complex programming
tasks.
In the process of calculation experiments, repetition works of modeling are necessary. Professional software packages
build the models at a fast rate by utilizing the grid formation rules, or through changing initial parameters and node
coordinates, or through adding/erasing members.
Analysis considerations are as follows:
(1) The single-layer area R is the only topological design variable considered. Configurational design variables include
high rise h、grid dimensions D and the thickness of double-layer reticulated shell t.
(2) Structural optimization is a problem considering the least cost of a structure that satisfies various constraint
conditions, such as strength, stiffness, frequency, stability, and etc. Most of current professional software packages can
undertake full stressed optimization only. The results meet the requirements of member strength and stability.
However, they do not always satisfy the needs of structural stiffness and ultimate load bearing capability. In
calculation experiments, thus, objective values include cost, displacements, ultimate load, basic frequency, and etc.
Referenced structures used to compute objective values are chosen those full-stressed optimization models made by
professional software packages. When deciding the optimum combination of design variables, cost is considered as the
only criterion, stiffness and stability being constraint conditions, basic frequency being design reference index.
(3) For totally double-layered reticulated shells, structural design is dominated by strength mainly. In the design of
totally single-layered reticulated shells, stability is the main dominated condition.
For partial double-layer reticulated shells, the single-layer area and material design stress ratio for single-layer
members significantly influence the objective values. To simplify the analysis, the single-layer members use the same
⎯ 1047 ⎯
material design stress ratio in the process of full-stressed optimization. While, the material design stresses for
double-layer members do not discount.
(4) To regularize the experimental objective values, the discrete cross sectional properties of members, discrete gages
of bolts, and discrete dimensions of joints are made to be continuous. These relationships are obtained by carrying out
the least square approximation.
Calculation steps are given below:
(1) Choose the main design variables, such as h, D, t, R, and material design stress ratio for single-layer members ρ.
(2) Determinate the wide change range of design variables.
(3) In wide range of design variables, use uniform design to make approximate optimization. (i) Decide the level
number of each design variable. (ii) Select an appropriate uniform design table and make the experimental scheme. (iii)
Do calculation experiments. (iv) Make the regression of the experimental results, and obtain the approximate relations
between the objective values and the main design variables. (v) By solving a simple problem of nonlinear
programming, get the approximate optimum results that meet the requirements of design specifications.
(4) Within the local range of the approximate results, use rotational second order design to improve the precision. (i)
Select an appropriate rotational second design table and make the experimental scheme. (iii) Do calculation
experiments. (iv) Make the regression of the experimental results. (v) Get the optimum results.
POSSIBLE POSITIONS OF SINGLE-LAYER REGIONS
The possible positions of single-layer regions can be determined according to the stress distribution of the
corresponding continuum shell or the corresponding single-layer reticulated shell. Fig. 1 shows the stress distribution
details of 4 kinds of point-supported single-layer reticulated shells under vertical uniformed load. In the figure, line
thickness represents the magnitude of member design stress. The regions of less stress, where are the possible positions
of single-layer regions, are enclosed by broken line.
(a) Triangular plane single-layer reticulated shell
(c) 4-block combined hyperboloid reticulated shell
(b) Hexagonal plane reticulated shell
(d) Combined double-spherical-center reticulated shells
Figure 2: Possible positions of single-layer regions
OTHER STEPS AND EXAMPLE
An engineering example is taken to explain other steps in detail.
1. Design conditions Fig. 2 is a 6-point pin-joint supported reticulated shell with projected equilateral hexagonal
plane. Its lower support points are 10 m above the ground. The length of one side of the projected plane is 40 m. The
⎯ 1048 ⎯
maximum diagonal length of the projected plane L is 80 m. The equation of top-chord surface of the reticulated shell is
z=−
x2 + y2
,
250
(1)
Where z is the vertical coordinate value. The origin of the space rectangular coordinates is at the center of the top-chord
surface.
The grids of reticulated shell are three-way. One side of the projected plane is divided into 12 segments. The thickness
of the double-layer region t is 2.5m.
Assume that that supporting structures are identical and the supporting structures satisfy the design requirements of all
calculation models. During comparison, thus, substitute the cost of a reticulated shell for the engineering cost.
Considering steel consumption is the dominating factor in the cost of a reticulated shell, use steel consumption as the
construction cost index.
Standard design loads considered are: (a) deck load 0.2kN/m2, (b) live load 0.5kN/m2, and (c) basic wind pressure
0.35kN/m2. The weight of structure is automatically calculated by PLAScad. Uniformed live load is exerted vertically
on the top-chord surface of the shell, and wind load being perpendicular to the top-chord tangent surface. Loads are
equivalently acted on the nodes. Ignore the effect of snow load, seeper load and earthquake effects. Load combinations
include: (a) 1.2 Dead Load, (b) 1.2 Dead Load + 1.4 Live Load (global), (c) 1.2 Dead Load + 1.4 Live Load (half span)
(d) 1.2 Dead Load +1.4 Wind Load, (e) 1.2 Dead Load + 1.4 Live Load (global) +0.8 Wind Load.
Figure 3: A calculation model
Elements are determined by using full stressed design. The stability of a compressive element is controlled by the
following formula:
Cf 0.85U1x M fx βU1y M fy
+
+
≤ 1.0
Cr
M rx
M ry
(2)
where Cf is compressive force under factored load. Mfx and Mfy are bending moment under factored load in two main
axils separately. Cr is factored compressive resistance. Mrx and Mry are factored moment resistances in two main axils
separately. U1x and U1y are amplification factors for stability analysis.
Members use round hollow steel Nodes are welded hollow spherical balls. During the full stressed design, members of
single-layer reticulated shell are treated as beam elements, members of double-layer shell being pin-joint bars.
Members linking fixed-joint and pin-joint are instead by transient members. Minimum geometrically nonlinear
buckling load is decided according to the approach (JGJ61-2003). In nonlinear elastic analysis, all members are
idealized as space beams.
2. Experimental design and results Design variables include ρ, R, and a, where R is represented by the number of
members at an single-layer outside k. The levels of design variables are given by Table 1.
Uniform design table use U15(53), which given by Professor Fang ‘s website http://www.math.hkbu.edu.hk/
UniformDesign. The deviation value of U15(53) is 0.013149.
The scheme and results of calculation experiments is shown as Table 2, where wup,max and wdown,max are the maximum
vertical displacement and the minimum vertical displacement separately. λ is the minimum geometrically nonlinear
buckling load factor. λ=1 corresponds to 1.2 Dead Load + 1.4 Live Load (half span). f is the basic frequency.
⎯ 1049 ⎯
Table 1 Levels of design variables
number
ρ
1
2
3
4
5
1.0
0.8
0.6
0.4
0.2
Design variables
k
2
4
6
8
10
a
60
80
100
120
140
Table 2 Scheme and results of calculation experiments
Number of
experiments
Combination of design variables
Results
ρ
k
a
1
1.0(1)
6(3)
140(5)
Steel
consumption
G/t
132.071
2
1.0(1)
2(1)
60(1)
3
0.6(3)
4(2)
4
0.6(3)
5
wup,max
mm
wdown,max
mm
λ
f
Hz
7.4
51.2
2.6546
1.6554
184.883
2.1
23.5
17.4270
1.2473
80(2)
158.558
1.1
27.9
8.5546
1.5303
10(5)
60(1)
97.808
3.4
23.8
7.0345
1.3798
0.8(2)
4(2)
100(3)
152.118
1.3
35.7
5.9919
1.6296
6
0.4(4)
8(4)
80(2)
121.217
0.9
25.9
9.2156
1.6480
7
0.2(5)
6(3)
60(1)
163.377
1.7
20.1
27.6010
1.2988
8
0.4(4)
4(2)
120(4)
152.199
0.2
37.3
9.6606
1.6553
9
0.8(2)
8(4)
80(2)
118.031
1.5
26.7
6.2624
1.6589
10
0.2(5)
10(5)
140(5)
137.214
3.9
51.5
10.6640
1.6364
11
0.4(4)
6(3)
120(4)
138.227
0.5
37.0
7.8579
1.6763
12
1.0(1)
10(5)
100(3)
83.565
5.1
45.0
3.6057
1.8431
13
0.8(2)
8(4)
120(4)
110.336
4.8
46.3
3.5253
1.7511
14
0.2(5)
2(1)
100(3)
164.187
0.4
32.3
15.9680
1.5946
15
0.6(3)
2(1)
140(5)
158.774
4.6
48.4
4.1095
1.6105
The value in parentheses corresponds to the number of design variable.
3. Regression and parametric analysis The regression of the results in Table 2 by using MS-Excel gives the
following approximate functional relations
⎧G = 178.6263329908 − 164.5071731726ρ + 12.0696096066k + 1.1232869916a + 328.4938318760ρ 2
⎪
⎪ − 3.1829977544k 2 − 0.0209088876a 2 − 3.8836716091ρk +0.0379648411ka − 0.4055356562ρa
⎪
3
3
3
⎪ − 155.3264756513ρ +0.1352773774k + 0.0000949046a
⎪
2
⎪ wup,max = 26.1093667139 − 8.9357153305ρ − 2.9866358280k − 0.3478698430a − 1.0091660810ρ
⎪
2
2
⎪ +0.2172011959k + 0.0009713169a + 0.3063825777ρk + 0.0089153409ka + 0.1517650331ρa
⎪
3
3
3
⎪ − 2.6489925895ρ − 0.0038250712k + 0.0000014337a
⎨
2
2
⎪ wdown,max = 50.5300116245 − 23.7724969965ρ − 4.4414468375k − 0.5521755067a + 15.1329999905ρ − 0.1603538414k
⎪
2
3
3
3
⎪ + 0.0061148410a + 0.0263746034ka + 0.3134421307ρa − 15.4419215572ρ + 0.0354763381k − 0.0000211505 a
⎪
2
2
⎪λ = 119.3189885923 − 93.7518414699ρ − 2.3103600457a + 107.3535944712ρ +0.0206006597a
⎪
3
3
3
⎪ + 0.0041231513ka − 0.0252634317ρa − 36.5108900975ρ − 0.0087977724k − 0.0000621412a
⎪
2
2
⎪ f = −1.3182803973 + 0.7042167440ρ − 0.0708726426k + 0.0697882012a − 1.3705781160ρ + 0.0167968361k
⎪
2
3
3
3
⎪⎩ − 0.0005393577a + 0.0156822812ρk − 0.0000886978ka + 0.7551478292ρ − 0.0009215711k + 0.0000013508a
⎯ 1050 ⎯
(3)
Here for convenience, k is looked as real number temporarily.
The basic data of regression analysis is shown as Table 3. According to JGJ61-2003 the minimum value of λ is 5. In
Table 3, the standard error of λ is about 36.62% of the minimum value, which overruns the design requirements of
JGJ61-2003. One of the main reasons leading to the errors is that there are more polynomial items of higher orders.
Another is that nonlinear calculation results in significant errors. The other is that R represented by k also affects the
results. Among the 15 times of calculation experiments, the maximum of λ is about 10.4 times of the minimum of λ.
The presence of big change extents of objective values is another reason. Notwithstanding, all the correlative
coefficients of regression equations are good enough. It shows that Eq. (3) gives correct relations between objective
values and design variables still. The interaction effects of design variables are include in the Eq. (3).
Table 3 Basic data of regression analysis
correlative
coefficients
Standard error
σ
F
Significance
F
Significance
order of
design
variables
G
wup,max
wdown,max
λ
f
0.99893
0.99988
0.99804
0.98622
0.99613
3.407
0.088
1.463
1.831
0.031
77.6903
687.5518
69.2973
19.7422
35.0633
0.0128
0.0015
0.0025
0.0022
0.0069
k 2 , ρ2 , ρ , ρ 3 ,
ρa ,k, ka , a ,
ρa , ka ,k, k 3 ,
k3 , ρ , a ,
a , a 2 , a3 , k 3 ,
k 3 , ρa , ρk , ka ,
k 2 , ρ , ρk , k 3 ,
ρ , a3 , a 2 , a ,
a 2 , a3 , ρ 2 ,
k 2 , ρ3 , ρ 2 , ρ ,
k, a3 , a 2 , a
a 2 , ρ3 , a3 , ρ 2
ρ3 , k 2 , ρ 2
ρ 3 , ka , ρa
k, ρk , ka
The relation graphs of objective values versus k and a when ρ is 1.0, 0.6 and 0.2 separately are shown in Fig. 3.
As a whole, the following conclusions can be drawn:
(1) All the response surfaces are curved, and are with significant waves.
(2) Under the same design conditions, the value of G when ρ= 0.6 is less than the value when ρ is 0.2 or 1.0.
(3) The wave amplitude of the response surface wup, max(k, a) enlarges with the increase of ρ.
(4) Under the same conditions, the value of wdown, max(k, a) when ρ= 0.2 is less than the value when ρ is 0.6 or 1.0.
(5) Under the same design conditions, the value of λ increases with the decrease of a.
(6) The value of f enlarges with the increase of k and a.
4. Optimization and verification For a certain k, the determination of the optimum combination of design variables
comes down to the following problem:
Minimize G ( ρ , a )
⎧wup, max ≤ L / 400 = 200
⎪
⎪wdown, max ≤ L / 400 = 200
⎪
s.t. ⎨λ ≥ 5
⎪0.2 ≤ ρ ≤ 1.0
⎪
⎪⎩60 ≤ a ≤ 140
(4)
where the constraint conditions of wup, max, wdown, max and λ are decided according to JGJ61-2003. By solving the
minimization problem when k is 4, 5 and 6 separately, the approximate condition solutions for preliminary design are
obtained (Table 4). The data of verification designs are also listed in Table 4. It can be seen that the condition solutions
are close to the values of verification design.
Within the local range of the approximate results, rotational second order design is used to improve the precision of the
results. In the response surface analysis, there are 6 additional calculation experiments. These data produce a new set
of regression equations. For k=5, the final optimization combination of design variables is given below:
⎧a = 120.521
⎨
⎩ ρ = 0.995
(5)
⎯ 1051 ⎯
□
175
G 吨150
125
100
140
120
100
2
□
140
120
100
2
a
□
180
160
G 吨140
120
100
G 吨160
140
120
100
2
a
4
4
k
a
4
80
6
140
120
80
6
k
8
80
6
k
8
8
10 60
10 60
ρ=1.0
10 60
ρ=0.6
ρ=0.2
(a) Response surface of G(k, a)
10
140
wu p,ma x
5
120
0
2
100
a
8
wu p,m a x 6
4
2
0
2
140
120
4
wu p,m a x 2
0
100
a
4
140
120
100
a
2
4
4
80
6
k
80
6
k
8
6
k
8
10 60
10 60
ρ=0.6
(b) Response surface of wup, max(k, a)
□
60
wdo w n,m ax mm 40
140
120
20
100a
2
4
□
60
50
wd o w n,m ax mm 40
30
20
2
ρ=0.2
□
50
140
120
100a
4
80
6
wdo w n,m ax mm 40
30
20
2
k
80
6
k
8
10 60
10 60
ρ=1.0
ρ=0.6
(c) Response surface of wdown, max(k, a)
15
140
120
100
l 10
5
0
140
120
2
a
100
4
ρ=0.2
25
l 20
140
15
120
2
a
100
a
4
4
80
k
100a
4
8
10 60
15
l 10
5
0
2
140
120
80
6
8
6
8
10 60
ρ=1.0
k
80
80
6
k
8
80
6
k
8
10 60
8
10 60
10 60
ρ=1.0
ρ=0.6
ρ=0.2
(d) Response surface of λ(k, a)
□
1.8
140
f Hz1.6
1.4
120
100 a
2
4
□
f Hz1.6
1.4
140
120
100 a
2
4
80
6
k
8
10 60
ρ=1.0
100 a
2
80
6
k
8
140
120
4
80
6
k
□
1.6
f Hz
1.4
8
10 60
ρ=0.6
(e) Response surface of f(k, a)
10 60
ρ=0.2
Figure 3: Relations of objective values vs. design variables k and a when ρ=1.0, 0.6 and 0.2 separately
⎯ 1052 ⎯
Table 4 Solution to programming problem and verification design
k
Solutions to
minimization
problems
a
4
5
6
ρ
115.583 0.605
120.401 1.000
118.959 1.000
comparison of results
Steel consumption G/t
wup,max/mm
wdown,max/mm
λ
Data given
Data given
Data given
Verificatio
Verificatio
Verification
Data given by Verification
by formulas
by formulas
by formulas
n design
n design
design
formulas (3)
design
(3)
(3)
(3)
147.791
137.365
127.392
148.632
139.659
130.201
1.489
3.405
3.621
1.310
3.209
3.571
39.528
43.529
43.043
40.130
44.551
44.379
5.60587
6.75618
6.5243
5.135
5.025
5.057
CONCLUSIONS
The structural design of a partial single-layer reticulated shell involves three categories of optimization: size, shape
and topology optimizations. To meet preliminary design requirements of computing efficiency and cost, a combined
experiment design method is adopted to make a quick optimization. This is a kind of response surface method, which
requires fewer times of computational experiment. In wide range of design variables, experimental tables use uniform
design tables. By means of regression of the calculation data, approximate regression relations between objective
values and design variables are obtained. By solving a simple problem of nonlinear programming, the approximate
optimum combination of design variables that meet the requirements of design specifications is decided. Within the
local range of the approximate results, rotational second order design is used to improve the precision. Example
analysis of a multipoint-supported hexagonal reticulated shell shows the reliability of the results.
Acknowledgements
The support of Guizhou Provence Excellent Young Scholars Encouragement Plan is gratefully acknowledged.
REFERENCES
1. Saka MP, Kameshki ES. Optimum design of nonlinear elastic framed domes. Advances in Engineering Software,
1998; 7-9: 519-528.
2. Gil L, Andreu A. Shape and cross-section optimization of a truss structure. Computers & Structures, 2001; 79:
681-689.
3. Wang D, Zhang WH, Jiang JS. Combined shape and sizing optimization of truss structures. Computational
mechanics, 2002; 29: 307-312.
4. Fang KT. Uniform Design and Uniform Design Table. Science press, Beijing, China, 1994 (in Chinese).
5. Chen WJ, He YL, Fu GY, etc. Stability analysis of the partial double-layer forms derived from the complete
double-layer reticulated dome. Journal of the International Association for Shell and Spatial Structures, 2001; 42
(137): 117-127.
6. Cochran WG, Cox GM. Experimental Design (second edition). John Wiley & Sons Inc, New York, USA, 1957.
7. JGJ61-2003. Chinese Specifications for Reticulated shells. 2003 (in Chinese).
⎯ 1053 ⎯