COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Stretching Control of the Long-Span Roof Structure of Shanghai South
Railway Station
Y. Huang *, Y. F. Luo, R. Yu
College of Civil Engineering, Tongji University, Shanghai, 200092 China
Email: [email protected], [email protected]
Abstract: The roof of Shanghai South railway station is a long-span beam-cable hybrid steel structure. The essential
role of stretching cables between and under steel beams is to keep whole roof stable under possible load conditions.
Using finite element analysis computer program ANSYS, the time-history analysis of seismic response and the
stability behavior of the steel roof is calculated. Based on the numerical results, the optimal tension force of cables
and reasonable stretching sequence are obtained.
Combined with a time-history analysis and a buckling analysis, the tension forces of stretching steel cable are
optimized on the basis of guaranteeing no cable slack. Then four kinds of cable stretching schemes are proposed. An
optimal one is chosen to ensure cable stress value uniformity and construction convenience. Subsequently,
beginning with the ideal state of completed roof, the counter-stretching procedure, in which cables are disassembled
in the opposite direction of stretching, is modeled stepwise, and the initial cable force values are obtained to insure a
successful stretching construction.
Keywords: counter-stretching, FEA, buckling analysis, optimum design
INTRODUCTION
The Shanghai South Railway Station will be constructed and put into use this year. Shaped like a wheel and being the
world first round railway station, the station will serve as a landmark of Shanghai’s south gate and an important
interchange of many transportation means. The roof of Shanghai South Railway Station is a long-span beam-cable
hybrid steel structure. It is composed of three components: outer columns and inner columns, 18 main beams with
thin-walled olive cross-sections, and the peak pressure annulus at the ends of cantilever beams (Fig. 1). The main
beams are supported by the inner columns and outer columns. The peak pressure annulus at the beam’s inner ends
connects all the steel main beams together. There are X steel braced cables laid out between beams. The steel cables
provide pre-stress to keep roof stable under possible load conditions. The whole roof stiffness and integrity also are
enhanced by the tension force provided by steel cables.
The purlines between main beams are curvy bars laid out in the direction of a circle. They can’t provide good bracing
to resist the lateral force because of their no straight-line shape. So, there should be some measures to be taken to
enhance the connection between main beams. Fortunately, there are stretching steel cables to take this role. The cables
are laid out X-crossed between main beams but not through the center of circle of the roof. Combined with the
purlines, the stretching cables can provide tension force to resist the lateral force and moment of torque induced by
earthquake.
NUMERICAL SIMULATION MODEL
Composed of beams, purlines, columns and cables, the roof structure is a half-rigid beam-cable hybrid steel structure
system. All member steels are Q345 except stretching cable. The cable steels are M50. The space beam element is
adopted in analysis. The elastic support at the connection point of column and main beams are used to decrease
temperature stress. It is modeled by COMBIN14 element in ANSYS element library. The other supports are hinged
supports. The nodes between cables and beams are hinged connection. The several kinds of other elements are
adopted, such as beam element for main beams, columns, purlines, peak pressure annulus, link element for steel cable,
mass element for the gravity of the roof. The finite model is shown in Fig. 2.
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Figure 1: The steel roof of Shanghai South Railway Station
Figure 2: The finite element model
Table 1 gives the main parameters of materials and element type corresponding different parts of the roof of Shanghai
South Railway Station.
Table 1 The main parameters and element type for different parts
Main beam
Yielding stress
(MPa)
235
Column, purline
Cable
Structural parts
Elastic support
BEAM44
Elastic module
(MPa)
2.06E+5
235
BEAM44
2.06E+5
345
LINK10
2.06E+5
Element type
COMBIN14
THE DETERMINATION OF OPTIMAL FORCES OF STRETCHING CABLE
Just as the former discussion, the role of cables is providing tension force to ensure the whole roof an entity under
possible loads, and it is important to guarantee no cable-slack occurrence. It is obvious that a higher force of cable can
provide greater integral rigidity and torsional stiffness for the roof. But the difficulty for construction and the
additional stress of some beams and purlines induced by the additional cable force are added at the same time. So it is
essential to determinate a set of optimal cable forces. Four rules should to be observed as following:
(1) The cables don’t quit work under the most unfavorable load combinations of dead loads (DL), live loads (LL),
wind loads (WL) and temperature load.
(2) All the cables don’t quit work under the frequently occurred earthquake.
(3) Based on the rule 1 and rule 2, the force of stretched cable should be small as possible as it could be. The smaller
the force of cable is, the easier the construction can be conducted.
(4) The force of cables under possible load combinations should be closer as possible as it could be.
According to the structure characters and the design conditions, 14 kinds of load combinations of DL, LL, WL and
temperature loads are calculated to define an initial force of steel cables. Before the calculation, a force of cable is
presumed on the basis of designer’s project experience. If the force is not feasible, a new force is consumed. And if the
new force is still not does the work, another available force is tested until a reasonable force (N1) is obtained. We call
this repeat calculation method the trial method. Using trial method, the minimal forces of cables (N2) are achieved
under the condition of gravity loads. After the initial force of cables is defined, the elastic dynamic time-history
analysis under frequently occurred earthquake is conducted to obtain the maximal range of all cables (dN). The
optimal forces of steel cables are max(N1, N2+dN).
1. The determination of initial stretching force Considering the calculation time and the computer memory
capacity, the finite model without purlines is adopted in the analysis of elastic dynamic time-history. 14 kinds of load
combinations of DL, LL, WL and temperature loads and the gravity typical value loads condition are adopted in the
static analysis to determinate the value of N1 and N2. The nominal value of dead loads on roof is 1.3 kN/m2, the
nominal value of live loads on roof is 0.3kN/m2. The fundamental wind pressure is 0.55kN/m2. The shape coefficients
are determined according to wind tunnel experiments. +30o and -30o temperature loads change are considered. The
representative values of gravity load are “1.0DL+0.5LL”. According to the code of seismic design of buildings, four
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seismic waves (recommended by technical specification for Shanghai seismic design of buildings: SHW1 seismic
wave, SHW2 seismic wave, SHW3 seismic wave and SHW4 seismic wave) are selected in this paper. In the
time-history analysis, the peak acceleration is adjusted as 0.35m/s2 and the damping ratio is taken as 0.02 under 7
degree frequently occurred earthquake (FOE). Considering effect of vertical earthquake, the 1.0 time of horizontal
earthquake acceleration spectrum and 0.85 times of vertical earthquake acceleration spectrum are input at the same
time. The NewMark-beta time integration method is adopted in the time-history analysis. The coefficient for
representative values of gravity is 1.2. The coefficient for earthquake action is 1.3. The stress time-history curve for
the innermost circle X-cables are showed in Fig. 3.
50
50
40
40
30
30
20
20
10
10
0
0
0
500
1000
1500
2000
2500
3000
3500
4000
0
(a) The curve for the innermost circle X-cables
500
1000
1500
2000
2500
3000
3500
4000
(b) The curve for the innert circle X-cables
70
60
50
40
30
20
10
0
120
100
80
60
40
20
0
0
500
1000
1500
2000
2500
3000
3500
4000
(c) The curve for the outer circle X-cables
0
500
1000
1500
2000
2500
3000
3500
4000
(d) The curve for the outmost circle X-cables
Figure 3: Stress time-history curves for X-cables (horizonal axis unit:0.01s; vertical axis unit: MPa, SHW1)
The forces of cables for static calculation are showed in Table 2. DL is short for dead loads, LL is short for live loads,
WD is short for wind loads. There are four kinds of wind load conditions. IT is short for increasing temperature
condition, and DT is short for decreasing temperature condition. RG is short for representative values of gravity load.
EA is short for earthquake action.
Table 2 The forces of cables for static analysis
No.
The force of cable(MPa)
Combinations of loads
outmost
outer
inner innermost
01
1.35*DL+1.4*0.7*IT+1.4*0.7*LL
0.0
50.0
70.0
140.0
02
1.35*DL+1.4*0.7*DT+1.4*0.7*LL
0.0
50.0
70.0
130.0
03
1.2*DL+1.4*IT+1.4*0.7*LL
0.0
50.0
70.0
140.0
04
1.2*DL+1.4* DT +1.4*0.7*LL
0.0
50.0
60.0
130.0
05
1.0*DL+1.4* IT +1.4*0.7*LL
0.0
50.0
60.0
110.0
06
1.0*DL+1.4* DT +1.4*0.7*LL
0.0
40.0
60.0
100.0
07
1.2*DL+1.4*WL①+1.4*0.7* IT +1.4*0.7*LL
0.0
60.0
70.0
130.0
08
1.2*DL+1.4*WL②+1.4*0.7* IT +1.4*0.7*LL
0.0
60.0
70.0
110.0
09
1.2*DL+1.4*WL③+1.4*0.7* IT +1.4*0.7*LL
0.0
60.0
70.0
120.0
10
1.2*DL+1.4*WL④+1.4*0.7* IT +1.4*0.7*LL
0.0
60.0
70.0
120.0
11
1.0*DL+1.4*WL①+1.4*0.7* DT +1.4*0.7*LL
0.0
60.0
60.0
100.0
12
1.0*DL+1.4*WL②+1.4*0.7* DT +1.4*0.7*LL
0.0
50.0
60.0
90.0
13
1.0*DL+1.4*WL③+1.4*0.7* DT +1.4*0.7*LL
0.0
50.0
60.0
100.0
14
1.0*DL+1.4*WL④+1.4*0.7* DT +1.4*0.7*LL
0.0
50.0
60.0
100.0
15
1.2*RG
0.0
50.0
60.0
120.0
0.0
60.0
70.0
140.0
MAX(1~15)
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Table 2 shows us the minimal forces of cables on the base of no cable quitting work. From the outmost circle cables on
roof to the innermost cables, the forces increase gradually. The outmost cables need no stretching. The innermost
cables need the maximal stretching forces to guarantee not slackening. The analysis illustrates that the nonlinear
behavior is not strong. The final stresses of cables depend on the level of stretching. The initial stretching force is σ 0 ,
σ 1 is the final force of cable under combinations of loads, the increment of stretching force is Δσ , Δσ = σ 1 − σ 0 .
The analysis results show that the force of initial stretching has little influence on the increment Δσ . The MAX in
Table 2 can be taken as the initial force of cable.
The maximal range of all cables are showed in Table 3, the optimal stretching forces of cables is the larger one
between SUM(15+16) and MAX(1-15).
Table 3 The optimal stretching forces of cables
No.
Combinations of loads
outmost
outer
inner
innermost
15
1.2*RG
0.0
50.0
60.0
120.0
16
1.2*RG+1.3*EA
40.0
40.0
40.0
40.0
SUM（15+16）
40.0
90.0
100.0
160.0
MAX（1-15）
0.0
60.0
70.0
140.0
The optimal force
40.0
90.0
100.0
160.0
2. Overall stability analysis of the roof Since the main questions in structural design of long-span steel structures are
stability and deformation problems (Li, 1998), we can use a possible instability mode of the roof as the most
unfavorable distribution to give a conservative estimation of the structure’s performance under combinations of load.
Such method, named as the conformable imperfect mode method, is often used for sensitivity analysis of imperfects,
and has been proved effective. This method is presented as follows:
(1) The linear buckling analysis is carried out to get the eigenvalue λ . The eigenvalue equation is formulated:
([ K ] + λ [ S ]){ψ } = {0}
i
i
Where, [ K ] = stiffness matrix, [ S ] = stress stiffness matrix, λi = ith eigenvalue (used to multiply the loads which
generate [ S ] ), ψ i = ith eigenvector of displacements. Supposing during the stability tracing analysis under a load
combination, [ K ]NL becomes non-positive at the (i+1)-th incremental step, which means a limit or bifurcation point
was occurred. Then, the calculation goes back to the initial state of this step, and an eigenvalue analysis is carried out
to obtain the current possible instability modes.
(2) Considering the initial imperfect. According to the conformable imperfect mode method: Since the assemble error
are really a random distribution, the possible instability mode, usually the first mode, is used as the most unfavorable
estimation of the structure’s nonlinear behavior. According to the technical specification for latticed shells, the initial
imperfect is taken as 1/300 of shorter span of the structure.
(3) Nonlinear finite tracing analysis. The nonlinear equation is formulated:
([ K ]L + [ K ]NL ) {Δu} = {ΔP}
Where, [ K ]L is the elastic tangent stiffness matrix, [ K ]NL is geometrical nonlinear tangent stiffness matrix, Δu is
the displacement increment vector, ΔP is the node load vector. The Newton-Raphson solution method and the
arc-length method are applied during the analysis.
3. The mode analysis In order to obtain the most unfavorable loads combination, the mode analysis is carried out.
Four kinds of stretching forces are considered: 80MPa, 100MPa, 120MPa and the optimal forces obtained from above
time-history analysis. Five load combinations are adopted for the linear mode analysis: load combination 1
(LC1=1.0*dead load + 1.0*live load), load combination 2 (LC2=1.0*dead load +1.0* wind load1), load combination
3(LC3=1.0*dead load +1.0* wind load2), load combination 4(LC4=1.0*dead load +1.0* wind load3), load
combination 5(LC5=1.0*dead load +1.0* wind load4). The finite element analysis software ANSYS is adopted for
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numerical analysis. In order to obtain the buckling characters and buckling coefficients of the roof, 36 linear buckling
analyses are carried out.
The analysis shows that the first two buckling coefficients are very close under load combination 1, and the roof has
two similar modes: main beam vertical buckling nearby the roof center. From the third mode, the form of the roof
buckling has changed to main purlines buckling on the innermost circle of the roof. Under the load combination 2, 3
and 4, the roof shows some similar buckling characters: the main purlines’ buckling happens on the innermost circle of
the roof before the 19th mode, the main beams buckling happens after the 19th mode. The first 3 buckling modes and
a first purline buckling mode are showed in Fig. 4. The buckling coefficients of the roof under five load combinations
are showed in Table 4. For shortening the paper, only first five buckling coefficients are listed.
(a) The first buckling mode under LC1
(b) The second buckling mode under LC1
(c) The third buckling mode under LC1
(d) The 19th buckling mode under LC1
Figure 4: Four modes of the roof under load combination1 and the optimal stretching force
Table 4 The linear buckling coefficients λ
The value of stretching force(MPa)
Load
combinations
The optimal stretching force
80
100
120
λ
λ
λ
λ
1st
2nd
mode mode
3th
mode
4th
mode
1st
mode
2nd
mode
3th
mode
4th
mode
1st
mode
2nd
mode
3th
mode
4th
1st
2nd
3th
4th
mode mode mode mode mode
LC1
4.67
4.67
5.39
5.42
4.64
4.64
5.03
5.06
4.68
4.70
4.70
4.71
4.72
4.72
4.75
4.77
LC2
6.08
6.11
6.22
6.32
5.43
5.60
5.70
5.79
4.96
5.10
5.18
5.26
5.03
5.18
5.27
5.35
LC3
5.88
6.08
6.22
6.32
5.32
5.50
5.63
5.73
4.87
5.02
5.14
5.12
4.94
5.10
5.21
5.30
LC4
5.80
5.98
6.13
6.25
5.23
5.40
5.55
5.67
4.79
4.94
5.07
5.17
4.86
5.01
5.15
5.25
LC5
5.95
6.09
6.18
6.28
5.34
5.49
5.61
5.72
4.88
5.10
5.12
5.20
4.96
5.09
5.20
5.29
The data show that the load combination1 (LC1=1.0*dead load + 1.0*live load) is the most unfavorable load
combination and the optimal stretching force is the best stretching force.
4. The nonlinear analysis Two models, that is, the perfect model and the imperfect model are calculated to get the
nonlinear behavior of the roof. The first buckling model is taken as the initial imperfect model. According to the
technical specification for latticed shells, the initial imperfect is taken as 1/300 of shorter span of the structure. The
imperfect value L/300 = 152.0/300 = 0.5067m. The nonlinear overall stability coefficients are showed in Table 5. The
load-deflection curves of the imperfect structures under load combination1 are shown in Fig. 5.
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Table 5
Load combinations
The nonlinear buckling coefficients λ
The
λ of perfect model
The
λ of imperfect model
LC1
4.011
4.002
LC2
4.593
4.502
LC3
5.102
4.956
LC4
4.835
4.735
LC5
4.687
4.580
4.5
4
3.5
3
2.5
2
1.5
UX_18744
UY_18744
UZ_18744
1
0.5
0
-2
-1.5
-1
-0.5
0
0.5
Figure 5: The node (18744) load-deflection curves of the imperfect structures under load combination1
OPTIMIZATION OF CONSTRUCTION SCHEMES FOR STEEL CABLES STRETCHING
Using the optimal pre-stress forces got from above analysis as the final pre-stress force for the cables, the initial
stretching forces in construction process are obtained through a counter-stretching method, that is, beginning with
the ideal state of completed roof, the counter-stretching procedure, in which steel cables are disassembled in the
opposite direction of stretching, is modeled using the element live-and-death function in ANSYS. Then four
schemes of steel rods stretching are proposed: the clock-wise symmetry stretching scheme; the clock-wise
asymmetry stretching scheme; the anti-clockwise symmetry stretching scheme and the anti-clockwise asymmetry
stretching scheme. An optimal one is chosen to keep steel cables stress uniformity and a lower loss of the pre-stress.
Subsequently, an optimal stretching construction schemes is achieved. The initial stretching forces are used to guide
the construction.
The first set of stretching cables positions are showed in Fig. 6.
Figure 6: The positions of the first set of stretching cables
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The stresses of the cables in construction are showed in Fig. 7.
(a) The curve (ELEMENT 370) of the outmost circle
(c) The curve (ELEMENT 368) of the inner circle
(b) The curve (ELEMENT 361) of the outer circle
(d) The curve (ELEMENT 371) of the innermost circle
Figure 7: Stress curves of the cables
All the stresses of the outmost circle cables are higher than the optimal stretching force (40MPa), but the change trend
is very slow, almost equal to 50MPa during the whole analysis process. Considering the nonlinear behavior has little
influence on the stretching force of cables, 35 MPa is the recommended value for the initial force of the outmost circle
cables in the construction process. The stress change of the Fig. 7(b) is similar to Fig. 7(a), but the stress value is lower
than the optimal force. So the initial stress value of the outer circle cables can be taken the same value of the optimal
force (90MPa). Considering the inner circle cables and the innermost cable have a similar change trend: they have a
considerable increment during the process of construction, especially for the innermost cables. 100MPa is the
recommended value for the inner cables. 80 MPa is the recommended value for the innermost cables.
Considering the innermost circle cables stress have substantial increments during stretching schemes 3 (the direction
from outer circle to inner circle), the stretching scheme 3 is the best one for construction.
CONCLUSION
The optimal force value obtaining from time-history analysis and nonlinear stability analysis is the upper limit for the
cable stretching force. It can guarantee no cable quitting work under possible load combinations. But it is just an ideal
final value, that is, the value is obtained when the structure has been constructed. So the initial force of cables (can be
used to guide construction) should be achieved from a counter-construction procedure analysis.
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Journal of Southeast University, 2003; 33(5): 583-587.
2. Huang Y, Luo YF. Optimal design and buckling analysis of a long-span steel truss. ICASS 05, vol. 2,
pp.1329-1333
3. Structural Analysis Guide (ANSYS Release 5.7). ANSYS Inc.
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