Krisztián Hincz
Arch-supported tensile structures
with a special suspension system
CONTENTS






Existing arch-supported tensile structures
The block and tackle suspension system
Main steps of the numerical analysis
Dynamic relaxation method
Numerical examples
Future plans
BoA Pavilion, MA
BoA Pavilion, MA
BLOCK AND TACKLE SUSPENSION
SYSTEM
Árpád KOLOZSVÁRY, Roof Arches Without Bending Moments, 2006.
THE ARCH LOADS
Conventional
suspension system
Block and tackle
suspension system
In practice, how much can the bending moment of the
arches (due to tipical external loads) be decreased?
THE ANALYSED STRUCTURES
STRUCTURAL UNITS OF THE ANALYSED
STRUCTURES




Cable net
Suspension system
Truss arches
Safety cables
MODELLING OF THE BLOCK AND TACKLE
SUSPENSION SYSTEM
MODELLING OF THE BLOCK AND TACKLE
SUSPENSION SYSTEM
MAIN STEPS OF THE ANALYSIS
1. Truss arch and cable net topology generation
(Initial shape)
2. Form finding of the cable net with constant
cable forces (Theoretical shape)
3. Calculation of the stress-free lengths of the
cables
4. Determination of the construction shape
(prestress+dead load)
5. Load analysis
(prestress, dead load, snow load, wind load)
DINAMIC RELAXATION METHOD
 Step-by-step
 Nonlinear, static problems, determination of
equilibrium positions of tensile structures
 Fictitious motion from the initial position to the
equilibrium shape
 Fictitious masses
 Unbalanced (resultant) nodal forces
(member forces + external forces)
 Newton’s II. law
 Kinetic damping
TOPOLOGY GENERATION, INITIAL SHAPE
Initial data:
 Geometrical data of the truss arches (radius, angle,
depth, width)
 Number of suspended points
 Initial (constant) distance of the upper and lower
suspension points
FORM FINDING OF THE CABLE NET
 Constant force in the snow and wind cables
 The breakpoints of the ridge cables are fixed
 Coordinates, cable forces  unbalanced nodal forces
 Calculation of the stress-free (cutting) lengths
CONSTRUCTION SHAPE
 Constant suspension force
 Current coordinates, stress-free lengths, stiffness
(+self weight)  unbalanced nodal forces
 Stress-free lengths of the suspension cables
LOAD ANALYSIS
Unbalanced nodal forces:
 Meteorological loads
 Member forces
 Self-weight
Loads:
 Total snow load
 Two types of partial snow load
 Wind load
(+Self-weight and prestress)
MOVEMENT OF THE PULLEYS
2R
2r
Si
Upper pulleys roll if:
Si
Si 1 R  r 
R  r

or

Si 1 R  r 
Si
R  r
Lower pulleys roll if:
Si
r
 cot(45  arcsin
or
Si 1
2R
Si 1
r
 cot(45  arcsin

Si
2R
S i+1
Si
2R
2r
S i+1
Displacement:
li , li 1 , li0 , li01 , EA  
EXAMPLE STRUCTURE I.
 Individual suspension cables ↔ Block and tackle
suspension system
 Idealised pulleys
 Covered area: 120m·120m
MEMBER FORCES IN CASE OF
PARTIAL SNOW LOAD TYPE 1
MAXIMUM OF THE INTERNAL FORCES
AND BENDING MOMENTS
Normal Force [kN]
Shear Force [kN]
Bending Moment [kNm]
Load
ISC
BTSS
BTSS/ISC
ISC
BTSS
BTSS/ISC
ISC
BTSS
BTSS/ISC
Construction
shape
-7289
-7289
1.00
69
69
1.00
265
265
1.00
Total snow
load
-14808
-17833
1.20
-1074
36
-0.03
31001
783
0.03
Partial snow
load 1
-12393
-14459
1.17
-1427
118
-0.08
42528
3523
0.08
Partial snow
load 2
-9384
-11973
1.28
-512
53
-0.10
15554
512
0.03
Wind load
-9264
-10216
1.10
736
92
0.12
-19248
-1317
0.07
EXAMPLE STRUCTURE II.
How does the friction affect the elimination
of bending moments?
INTERNAL FORCES IN CASE OF
WIND LOAD
3500
Bending
Moment
[kNm]
Internal forces
3000
2500
2000
Normal
Force
[kN]
1500
1000
Shear
Force
[kN]
500
0
0
0.05
0.1
0.15
Coefficient of friction
0.2
0.25
INTERNAL FORCES IN CASE OF
PARTIAL SNOW LOAD TYPE 1
6000
Bending
Moment
[kNm]
Internal forces
5000
4000
Normal
Force
[kN]
3000
2000
Shear
Force
[kN]
1000
0
0
0.05
0.1
0.15
Coefficient of friction
0.2
0.25
CONCLUSIONS
 By the help of the developed procedures, arch
supported tensile roofs with block and tackle
suspension system can be analysed. The developed
procedures converge in every step of the analysis.
 The numerical results show that the block and tackle
suspension system reduces radically the in-plane
bending moments of the supporting arches.
FUTURE PLANS
 Topology of the cable net
 Theoretical shape of the cable net
 Number of suspension points
 Experiments to validate the numerical results.
K. HINCZ: ARCH-SUPPORTED TENSILE STRUCTURES WITH VERY LONG
CLEAR SPANS, JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR
SHELL AND SPATIAL STRUCTURES, Vol. 48 No. 2, 2007
maximum compression force [kN]
PBSS_TSnow
ISC_TSnow
PBSS_PSnow1
ISC_PSnow1
PBSS_PSnow2
ISC_PSnow2
PBSS_Wind
ISC_Wind
Prestress
8000
7000
6000
5000
4000
3000
2000
1000
0
125
150
175
200
225
250
275
300
initial prestress in the suspension cables [kN]
325
PBSS_TSnow
ISC_TSnow
PBSS_PSnow1
ISC_PSnow1
PBSS_PSnow2
ISC_PSnow2
PBSS_Wind
ISC_Wind
maximum displacement [m]
2
1.5
1
0.5
0
125
150
175
200
225
250
275
initial prestress in the suspension cables [kN]
300
325
PBSS_TSnow
ISC_TSnow
PBSS_PSnow1
ISC_PSnow1
PBSS_Wind
ISC_Wind
Prestress
maximum compression force [kN]
8000
7000
6000
5000
4000
3000
2000
1000
0
0
1
2
3
4
initial suspension length [m]
5
6
7
PBSS_TSnow
ISC_TSnow
PBSS_Psnow1
ISC_Psnow1
PBSS_Wind
ISC_Wind
Prestress
maximum compression force [kN]
1600
1400
1200
1000
800
600
400
200
0
0
1
2
3
4
initial suspension length [m]
5
6
7
QUESTIONS
 How much can the bending moment of the arches
be decreased? How do the tangential and out-ofplane movements of the pulleys and the friction
affect the elimination of bending moments?
 Can the cable net be prestressed during
construction by tensioning the suspension cables
only?
 What effect does the prestress level have on the
behaviour of the structure?
 What effect does the distance of the upper and
lower pulleys have?
MOTION OF THE BLOCK AND TACKLE II.
l i,0
li
S=
(li  li 1 )  (lin,0  lin1,0 )
(l  l
n
i ,0
Si
li EA
l 
S  EA
  lin,0*  lin,0
n*
i ,0
R
r
S i+1
l i+1,0
l i+1

l l 
k
(e.g.   10)
n 1
i ,0
n
i ,0
n
i 1,0
)
EA
EXAMPLE STRUCTURE I.
 Individual suspension cables ↔ Block and tackle
suspension system
 Force in the suspension cables: 25kN - 300kN
 Suspension length: 1m - 6m
 Idealised pulleys
QUESTIONS
 How much can the bending moment of the arches
be decreased?
How do the tangential and out-of-plane
movements of the pulleys and the friction affect
the elimination of bending moments?
 What effect does the prestress level have on the
behaviour of the structure?
 What effect does the distance of the upper and
lower pulleys have?