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Formulation and application of multi-node sliding cable element for the analysis of Suspen-Dome structures
ARTICLE IN PRESS Finite Elements in Analysis and Design 46 (2010) 743–750
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Formulation and application of multi-node sliding cable element for the analysis of Suspen-Dome structures Z.H. Chen a,b, Y.J. Wu a, Y. Yin a,b,, C. Shan a a b
Department of Civil Engineering, Tianjin University, Tianjin 300072, China Tianjin Key Laboratory of Civil Engineering Structures and New Materials, Tianjin 300072, China
a r t i c l e in fo
abstract
Article history: Received 30 May 2009 Received in revised form 2 April 2010 Accepted 24 April 2010 Available online 15 May 2010
Multi-node sliding cable element was formulated for the analysis of cable structures with cables threading through a number of joints and being able to slide inside them. Tangent stiffness matrix was determined for the element based on uniform strain assumption. The element was implemented in ABAQUS as user defined element and applied in the static and nonlinear stability analysis of a Suspen-Dome structure. The rationality and effectiveness of multi-node sliding cable element for both sliding and non-sliding situations were verified by considering symmetric and asymmetric loads on the Suspen-Dome. The effect of cable sliding on the behaviour of the Suspen-Dome was also discussed. & 2010 Elsevier B.V. All rights reserved.
Keywords: Multi-node sliding cable element Tangent stiffness matrix Suspen-Dome structure Nonlinear stability analysis Effect of cable sliding
1. Introduction Steel cables are widely adopted in long-span structures due to their high strength and light self-weight. By pre-stressing the steel cables, more effective structures can be achieved to span longer distance. In this kind of pre-stressed cable structures, a steel cable usually threads through a number of joints. The distance between adjacent joints is relatively short and the sliding of the cable is often fixed by the joints. In finite element analysis of the structures, the cables can then be simplified as 2-node straight bar elements separated by adjacent joints, subjecting to only tension forces [1]. However, in some cable structures, special type of joints is designed to allow the steel cables to slide freely inside the joints. Though a continuous steel cable may pass through several joints, the tension forces in all cable segments remain a constant. For this situation, the above simplification may cause significant error due to the neglect of cable sliding. Some techniques have been presented to consider the cable sliding based on finite element analysis with simplified separate cable elements [2–5]. They all need manual iteration intervened by engineers, which is time consuming especially for large scale cable structures. Formulating sliding cable element is a more convenient way to simulate sliding cables. Sliding cable elements have been developed and verified for the analysis of parachute
Corresponding author at: Department of Civil Engineering, Tianjin University, Tianjin 300072, China. Tel.: +86 22 13752506876; fax: + 86 22 27404465. E-mail address: [email protected] (Y. Yin).
0168-874X/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2010.04.003
systems [6]. However, the element is a 3-node element and only the intermediate nodes can slide, which hampers its application in the analysis of large scale cable structures. It is then desired to formulate multi-node sliding cable element for sliding cables in pre-stressed cable structures. In this paper, a new kind of multi-node sliding cable element was formulated for the static and nonlinear stability analysis of a Suspen-Dome structure. The element has arbitrary number of sliding nodes and satisfies uniform strain assumption. The principle of virtual work and total Lagrange formulation were used to derive the tangent stiffness matrix of the element. The element was then implemented in commercial finite element software ABAQUS [7] as a user defined element. Both symmetric and asymmetric loads were considered for the Suspen-Dome structure to illustrate the rationality and effectiveness of the presented multi-node sliding cable element.
2. Formulation of multi-node sliding cable element A multi-node sliding cable element is as shown in Fig. 1. The cable is separated to N-1 segments by N nodes. All nodes can slide along the cable except the two end nodes. The number of degree of freedom of the sliding cable element is 3N. The fundamental kinematic assumption of the multi-node sliding cable element states that (1) the cable remains straight between adjacent nodes; (2) the self-weight of the cable can be ignored and all the loads act directly on the nodes; (3) the strain is uniform along the element,
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N-3
i
3
1
If x and x0 are node coordinates in present and initial configurations, respectively, then
N-1
2
x ¼ x0 þ u and du ¼ dx:
N-2
1
i
3
N-1
N-3
So
4
2
N
N-2
i+1
B¼
Fig. 1. Multi-node sliding cable element.
i.e., the strain in all segments of the cable is the same at any time and (4) the sliding node cannot slide across the adjacent nodes along the cable, which means that the order of the nodes along the cable remains unchanged. The principle of virtual work for the sliding cable element in total Lagrange formulation (TL) can be expressed as Z S11 de11 A0 dL ¼ Pi due ð1Þ L
in which e11 is the Green–Lagrange strain, S11 is the second Piola–Kirchhoff stress, L is the total length of the cable at initial configuration, P and u are the node loads and displacements, respectively, and A0 is the cross-sectional area of the cable, which is a constant over the entire element length. In the TL formulation, the integration is performed over the initial configuration. Because the strain and stress are assumed to be constant along the element, the integration in Eq. (1) is performed analytically S11 de11 A0 L ¼ P du
l @l L2 @x
If li is the length of i-th segment of the cable at present configuration and the order of the nodes along the cable remains unchanged, then vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 1 N 1 u 3 X X uX t ð13Þ l¼ li ¼ D2i,j i¼1
ð3Þ
in which B is the vector of 3N. Substituting Eq. (3) into Eq. (2), gives S11 BA0 L du ¼ P du
ð4Þ
Considering the randomicity of the variation of node displacements, du, the equilibrium equation of the sliding cable element can be expressed as S11 BA0 L ¼ P
ð5Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P3 2 P @l @ð N1 @ðlm1 þ lm Þ @ j ¼ 1 Dm1,j i ¼ 1 li Þ ¼ ¼ ¼ @xm,n @xm,n @xm,n @xm,n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P3 2 @ Dm1,n Dm,n j ¼ 1 Dm,j þ ¼ lm1 lm @xm,n l Dm1,n Dm,n lm1 lm L2
@ðS11 BA0 LÞ @S11 @B ¼ A0 LBT þ A0 LS11 @ue @ue @ue
ð7Þ
The second Piola–Kirchhoff stress for the one-dimensional sliding cable element is given by S11 ¼ S011 þ Ee11
@B @B @ððl=L2 Þð@l=@xÞÞ 1 ¼ ¼ ¼ 2 @u @x @x L
@l @l @2 l þl 2 @x @x @x
! ð16Þ
ð@B=@uÞ, ð@B=@xÞ, ð@l=@xÞð@l=@xÞ and ð@2 l=@x2 Þ are all matrix of 3N 3N, and ! @ ðDm1,n =lm1 ÞðDm,n =lm Þ @2 l @ð@l=@xÞ ¼ ¼ @xp,q @x @x2 3ðm1Þ þ n,3ðp1Þ þ q
@ðDm1,n =lm1 Þ @ðDm,n =lm Þ @xp,q @xp,q
¼
ð17Þ
In which p ¼1, 2, y, N and q¼1, 2 and 3. Then the items of the matrix, ð@2 l=@x2 Þ, can be calculated as ( 8 2 2 p¼m > 1 1 Dm1,n Dm,n > > þ . . . . . . > > q¼n < lm1 lm l3m l3m1 @2 l ( ¼ > @x2 p¼m D D D D > > m1,n m1,q m,n m,q . . . . . . 3ðm1Þ þ n,3ðp1Þ þ q > > : qan l3m l3m1 !
O
in which is the initial stress in the cable and E is the Young’s modulus. Then @ðS011 þ Ee11 Þ @S11 @e11 ¼E ¼ ¼ EB @u @u @u For the one-dimensional sliding Green–Lagrange strain is given by [8]
ð18aÞ
ð8Þ
S011
i (x , x , x )
ð9Þ cable
l2 L2 2L2
element,
x2
@e11 @ððl2 L2 Þ=2L2 Þ 1 @l l @l ¼ ¼ 2 2l ¼ 2 @u @u @u 2L L @u
Li
the
(x
i+1 ,x ,x
)
(x
i+ 1 ,x ,x
)
i (x , x , x )
li
ð10Þ 1
in which l is the total length of the cable at present configuration. Then B¼
ð15Þ
ð6Þ
in which K is the tangent stiffness matrix of the sliding cable element, and
e11 ¼
ð14Þ
in which m¼1, 2, y, N and n ¼1, 2 and 3.
The incremental equilibrium equation of the sliding cable element can then be termed as K Du ¼ P
j¼1
in which Di,j ¼ xi þ 1,j xi,j is the length of the i-th (i¼ 1, 2, y, N 1) cable segment in the j-th (j ¼1, 2 and 3) direction, as shown in Fig. 2. Define ðD0,n =l0 Þ ¼ 0 and ðDN,n =lN Þ ¼ 0, then for the n-th coordinate of node m,
B3ðm1Þ þ n ¼
de11 ¼ B du
K¼
i¼1
ð2Þ
Assume
ð12Þ
3
ð11Þ
Fig. 2. The i-th segment of multi-node sliding cable element.
ARTICLE IN PRESS Z.H. Chen et al. / Finite Elements in Analysis and Design 46 (2010) 743–750
( 8 p ¼ m þ1 > D2m,n 1 > > ...... > þ 3 ! > q¼n < l l 2 m m @ l ( ¼ 2 > p ¼ mþ 1 @x > > Dm,n Dm,q . . . . . . 3ðm1Þ þ n,3ðp1Þ þ q > > : q an l3m ð18bÞ ( 8 p ¼ m1 > D2m1,n 1 > > þ . . . . . . > > 3 q¼n < lm1 lm1 @2 l ( ¼ > p ¼ m1 @x2 Dm1,n Dm1,q > > 3ðm1Þ þ n,3ðp1Þ þ q > ...... > : q an l3m1 !
745
For convenient cable arrangement, single-layer lattice dome with parallel grid is adopted in the Suspen-Dome. Two kinds of steel tubes, f127 4 and f133 8, are adopted as the members of the single-layer lattice dome. The members are connected with each other by welded hollow spherical joints, which can be considered as rigid joints. Steel of Grade Q235B [10] is adopted for all members and joints of the single-layer lattice dome. According to primary analysis, only two rings of latitudinal cables are needed structurally along the outer two grids of the dome. The sectional areas are 285 mm2 for the outer latitudinal cables and 112 mm2 for the inner latitudinal cables and radial cables, respectively. Steel tube f89 4 of Grade Q235B is adopted for the struts.
ð18cÞ @2 l @x2
3.2. Setup of finite element models for the Suspen-Dome structure
! ¼ 0 . . . . . . p a m1, m or m þ1
ð18dÞ
3ðm1Þ þ n,3ðp1Þ þ q
Substitute Eqs. (14) and (18) into Eq. (16), gives
Two finite element models were setup with commercial finite element software ABAQUS, as shown in Fig. 4. Multi-node sliding cable element presented in this paper was implemented in
!# ( 8 " 2 2 p¼m > 1 Dm1,n Dm,n Dp1,q Dp,q 1 1 Dm1,n Dm,n > > l ...... þ 3 3 > > 2 q¼n < l l l l l l l L l m p m m1 p1 m1 m m1 @B " !# ( ¼ @u 3ðm1Þ þ n,3ðp1Þ þ q > p¼m Dm1,n Dm1,q Dm,n Dm,q 1 Dm1,n Dm,n Dp1,q Dp,q > > > þl ...... þ > L2 : qa n lm1 lm lp1 lp l3m l3m1 !# ( 8 " 2 p ¼ m þ1 > 1 Dm1,n Dm,n Dp1,q Dp,q 1 Dm,n > > l . . . . . . > > 2 3 q¼n < l l l l l l L m p m m1 p1 m @B ( ¼ @u 3ðm1Þ þ n,3ðp1Þ þ q > p ¼ m þ1 Dm,n Dm,q 1 Dm1,n Dm,n Dp1,q Dp,q > > > 2 þl ...... > :L q an lm1 lm lp1 lp l3m !# ( 8 " p ¼ m1 > D2m1,n 1 Dm1,n Dm,n Dp1,q Dp,q 1 > > l . . . . . . > > 2 3 q¼n < l l l l l L l m p m1 p1 m1 m1 @B " !# ( ¼ @u 3ðm1Þ þ n,3ðp1Þ þ q > p ¼ m1 Dm1,n Dm1,q 1 Dm1,n Dm,n Dp1,q Dp,q > > > þl ...... > L2 3 : qa n lm1 lm lp1 lp lm1 @B ¼ 0 . . . . . . p am1, m or m þ 1 @u 3ðm1Þ þ n,3ðp1Þ þ q The tangent stiffness matrix can then be determined for the multi-node sliding cable element by substituting Eqs. (8), (9), (15) and (19) into Eq. (7).
3. The studied Suspen-Dome structure and its finite element models 3.1. Description of the studied Suspen-Dome structure Suspen-Dome structure [9] is a new kind of spatial structures, composed of a single-layer lattice dome and a tensegric network to take advantage of both structural systems. A Suspen-Dome used as the roof structure in a business center hall in Tianjin, China, is as shown in Fig. 3. The span and rise of the dome are 35.4 and 4.6 m, respectively. The Suspen-Dome is made of a singlelayer lattice dome, struts and radial and latitudinal cables with appropriate pre-stresses. The ends of the struts hanging from the same ring of joints of the single-layer lattice dome are connected with the next ring of joints by radial cables, and connected with each other by latitudinal cables. Struts, radial and latitudinal cables, being a tensegric network, sustain vertical loads together with the single-layer lattice dome and, therefore, enable a more efficient and economic way of constructing large-span domes.
ð19aÞ
ð19bÞ
ð19cÞ
ð19dÞ
ABAQUS as a user defined element and two 32-node elements were setup for the two rings of latitudinal cables of Model-A. For the sake of comparison, the latitudinal cables of Model-B were assumed to be separated by adjacent joints and 2-node linear displacement 3-D truss element, T3D2, was adopted for the cable segments. Element types for other members of the Suspen-Dome are all same for Model-A and Model-B, as shown in Table 1. The Suspen-Dome was restrained vertically at the 32 nodes of the outer ring of the single-layer lattice dome. Appropriate restrains were applied at the center node of the single-layer lattice dome to prevent the rigid body displacements. Materials for all the members and cables were assumed to be elastic. The Young’s moduli are E¼ 2.06 108 kN/m2 for the steel tubes and E¼1.95 108 kN/m2 for the steel cables [11]. The material for steel cables was set to be no compression material.
4. Static analysis of the Suspen-Dome structure with multi-node sliding cable element 4.1. Static analysis of the Suspen-Dome under the pre-tension of latitudinal cables The latitudinal cables were pre-tensioned before applying the loads to produce upward displacements of the dome and prevent
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jg1
0
1 jg1
jg1 2
B
1
strut strut Latitudinal Cables
jg9
Radial Cables
jg8
35400
jg7
O
jg6
hg11
jg5
hg9
hg6
jg4
hg5
hg4
hg3
hg2
jg3
hg12
jg2
hg10
hg1
Pa th
B AC
AC B
th Pa
jg1
hg8
A
hg7
1
Single-layer lattice dome
C
B
A
4600
35400
615
Vertical node displacement (mm)
Fig. 3. Structural layout of the Suspen-Dome in Tianjin, China.
Model-A: Under pre-tension
18 Model-A: Under asymmetric load
Model-B: Under pre-tension
Model-B: Under asymmetric load
9 0
-20
-15
-10
-5
0
5
10
15
20
-9 -18 -27 -36 Horizontal distance from center node O (m)
Fig. 5. Vertical node displacements of the Suspen-Dome under pre-tension and symmetric loads. Fig. 4. Finite element model of the Suspen-Dome structure. Table 2 Member forces (kN) of the tensegric system under the pre-tension of the latitudinal cables. Table 1 Element type for finite element models of the Suspen-Dome. Members
Latitudinal cables
Radial cables
Struts
Outer
Inner
Outer
Inner
Outer
Inner
61.65 61.68
34.13 34.15
7.34 7.36
3.92 3.92
2.78 2.78
2.03 2.03
Element type Model-A
Single-layer lattice dome Strut Radial cables Latitude cables
Member locations
Model-B
Model-A Model-B
B33 (2-node 3-D cubic beam element) T3D2 (2-node linear displacement 3-D truss element) T3D2 (no compression) Multi-node sliding cable T3D2 (no element compression)
the cables from sagging under the load action. The pre-tension forces were applied as initial stresses, 215 MPa for the outer latitudinal cables and 305 MPa for the inner ones. The upward
node displacements along the Section 1-1 of the single layer lattice dome and the member forces of the tensegric system are compared in Fig. 5 and Table 2 for Model-A and Model-B. No cable sliding would occur in Model-A, because the initial stress was set to be uniform for each ring of the latitudinal cables. The analysis based on the two models gave exactly the same results, which validated the multi-node sliding cable element for pre-tension analysis of the cable structures. It should also be noticed that the pre-tension of the latitudinal cables produced obvious
ARTICLE IN PRESS
The service loads on the Suspen-Dome structure are dead load of 2.0 kN/m2 and live load of 0.5 kN/m2, acting symmetrically on entire span of the dome. For Model-B, the stresses in all segments of each ring of latitudinal cables remained uniform under the service loads because of the symmetry of the structure and the loads. Correspondingly, no cable sliding would occurred in the sliding cable elements adopted in Model-A. The vertical node displacements along the Section 1-1 of the single layer lattice dome and the member forces of the tensegric system under the service loads are compared in Fig. 5 and Table 3 for Model-A and Model-B. The analysis results based on Model-A and Model-B agreed very well, which validated the multi-node sliding cable element for non-sliding situation. It can be seen that the maximum vertical displacement of the Suspen-Dome under symmetric loads reaches 32.8 mm. 4.3. Static analysis of the Suspen-Dome under asymmetric loads The Suspen-Dome may subject to asymmetric loads during construction and service periods. The worst asymmetric load distribution is dead load of 2.0 kN/m2 and live load of 0.5 kN/m2, acting on half span of the dome during the construction. This asymmetric load distribution was considered to illustrate the rationality and effectiveness of the multi-node sliding cable element for the situation of cable sliding. For the Suspen-Dome of Model-B under the asymmetric loads on the right half span, the axial forces in different segments of the latitudinal cables are different because the sliding of the cables are not allowed, as shown in Fig. 6. The difference between the axial forces is 47.0 kN for the outer latitudinal cable segments. Different axial forces in the latitudinal cables results in different axial forces in radial cables and struts, as shown in Figs. 7 and 8. The node displacements and member stresses along the Section 1-1 of the single-layer lattice dome of Model-B are as Table 3 Member forces (kN) of the tensegric system under symmetric loads. Member locations
Model-A Model-B
Latitudinal cables
Radial cables
Struts
Outer
Inner
Outer
Inner
Outer
Inner
137.17 137.34
36.88 36.83
16.37 16.36
4.23 4.23
6.39 6.32
2.11 2.09
747
20 16 12 Inner radial cable Outer radial cable
8 4 0 0
5
10
15
20
25
30
35
Segment number of radial cables along Path ACB Fig. 7. Axial force of radial cables of Model-B under asymmetric loads.
0 0 Axial force of the strut (kN)
4.2. Static analysis of the Suspen-Dome under symmetric loads
3
6
9
12
15
18
-2
-4
-6 Inner strut
Outer strut
-8 Number of struts along Path ACB Fig. 8. Axial force of struts of Model-B under asymmetric loads.
18 Vertical node displacement (mm)
anti-load effects and the maximum upward displacement of the Suspen-Dome is 7.45 mm.
Axial force of cable segment (kN)
Z.H. Chen et al. / Finite Elements in Analysis and Design 46 (2010) 743–750
9 0 -20
-15
-10
-5
0
5
10
15
20
-9 Model-A: Under pre-tension
-18
Model-B: Under pre-tension Model-A: Under asymmetric load Model-B: Under asymmetric load
-27
-36 Horizontal distance from center node O (m) Fig. 9. Vertical node displacements of the Suspen-Dome under pre-tension and asymmetric loads.
150
Worst member stress (MPa)
Axial force of cable segment (kN)
150
120 90
Inner latitudinal cable Outer latitudinal cable
60 30
Model A: latitudinal member hg# Model B: latitudinal member hg# Model A: radial member jg# Model B: radial member jg#
100 50 0 0
2
4
6
8
10
12
14
-50
0 0
3
6
9
12
15
18
Segment number of latitudinal cables along Path ACB Fig. 6. Axial force of latitudinal cables of Model-B under asymmetric loads.
-100 Number of members # Fig. 10. Worst member stresses of single-layer lattice dome under asymmetric loads.
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25
Table 4 Member forces (kN) of the tensegric system under asymmetric loads.
Model-A Model-B (average value)
20
Latitudinal cables
Radial cables
Struts
Outer
Inner
Outer
Inner
Outer
Inner
100.60 103.78
36.20 36.39
12.37 12.37
4.15 4.18
4.70 4.64
2.07 2.06
Load (KN/m2)
Member locations
15
Model-A Model-B
10
5
0
0
0.04 0.08 0.12 0.16 Vertical displacement of Node A (m)
0.2
Fig. 11. Load–displacement curves by nonlinear stability analysis for symmetric loads.
25 20 2
Load (kN/m )
shown in Figs. 9 and 10. It can be seen that the maximum vertical displacement of the Suspen-Dome under asymmetric loads is 31.7 mm, almost same as that under symmetric loads. For the Suspen-Dome of Model-A under the asymmetric loads on the right half span, the axial forces in different segments of each ring of latitudinal cables remain uniform because of the sliding of the cables. The axial forces in the same ring of radial cables and struts remain uniform too because of the equilibrium request. The member forces of the tensegric system are as shown in Table 4. Compared with Model-B, the maximum axial force was reduced by 26.2% for the outer latitudinal cables, which is helpful for improving the material efficiency of the cables. On the other hand, the minimum axial force was increased by 20.5% for the outer latitudinal cables, which is helpful for preventing the cables from sagging under asymmetric loads. It was also noticed that the axial forces in the latitudinal cables of Model-A agreed with the average values of the axial forces in cable segments of corresponding ring of latitudinal cables for Model-B as shown in Table 4. This validates the rationality and effectiveness of the presented multi-node sliding cable element for the situation of cable sliding. The vertical node displacements and worst member stresses of the single-layer lattice dome along the Section 1-1 of Model-A are compared with those of Model-B, as shown in Figs. 9 and 10. Little difference was found between the node displacements and the worst member stresses of Model-A and Model-B. It can then be concluded that whether the latitudinal cables can slide or not has little effect on the behaviour of the single layer lattice dome even under asymmetric loads.
15 10
Model-A: Inner latitudinal cable Model-B: Inner latitudinal cable
5
Model-A: Outer latitudinal cable Model-B: Outer latitudinal cable
0 0
50
100 150 Axial force in cable (kN)
200
Fig. 12. Axial forces in latitudinal cables by nonlinear stability analysis for symmetric loads.
5. Stability analysis of the Suspen-Dome structure with multi-node sliding cable element The design of single-layer lattice domes may be controlled by structural stability [12]. Though the studied Suspen-Dome structure is reinforced with the tensegric system, its stability still needs to be concerned. In design practice, the stability of lattice dome structures is often evaluated by tracing the nonlinear equilibrium path of the structure under proportional loading. Geometric nonlinearity should be considered and the arc-length method [13] is often adopted for equilibrium path tracing. The stability of the Suspen-Dome structure was first analyzed for the symmetric loading condition. Uniform loads were applied on the entire span of Model-A and Model-B, respectively, after the pre-tension of the latitudinal cables. With the increase of the uniform loads, the structure stiffness of the two models decreased due to the deflection of the Suspen-Dome. No cable sliding occurred in sliding cable elements adopted in Model-A and all segments of the same ring of latitudinal cables in Model-B had equal axial forces during the loading process. The load–displacement curves of Node A (as shown in Fig. 3) and the axial forces in the latitudinal cables obtained by nonlinear stability analysis were compared in Fig. 11 and Fig. 12, respectively. The results based on Model-A and Model-B agreed very well. The critical load for the Suspen-Dome structure can determine to be 22.2 kN/m2
Fig. 13. Deflection of the Suspen-Dome at critical load for symmetric loading condition.
for symmetric loading condition, corresponding to the limit point on the load–displacement curve. Note that the inner ring of the latitudinal cables sagged before the critical load. The deflection of the Suspen-Dome at the critical load is as shown in Fig. 13 for symmetric loading condition. The stability of the Suspen-Dome structure under asymmetric loads was analyzed by applying uniform loads on the right half span of Model-A and Model-B, respectively, after the pre-tension of the latitudinal cables. The stiffness of the structure was also decreased with the increased asymmetric loads due to the
ARTICLE IN PRESS Z.H. Chen et al. / Finite Elements in Analysis and Design 46 (2010) 743–750
18
2
Load (kN/m )
13.5
9
18
4.5
13.5
0
Model-A Model-B: Min. Model-B: Max.
40
2
Load (kN/m )
deflection of the Suspen-Dome. The load–displacement curves of Node B obtained by nonlinear stability analysis were compared for Model-A and Model-B in Fig. 14. Only slight difference was found between critical loads based on Model-A and Model-B, which were 16.7 and 17.1 kN/m2, respectively. The structural deflection of the two models at the critical load also agreed well for the asymmetric loading condition and is as shown in Fig. 15.
9
4.5
0.05 0.15 0.25 0.35 Vertical displacement of Node B (m)
65
90 115 Axial force in cable (kN)
140
Fig. 17. Axial forces in outer latitudinal cables by nonlinear stability analysis for asymmetric loads.
Model-A Model-B
0 -0.05
749
0.45
Fig. 14. Load–displacement curves by nonlinear stability analysis for asymmetric loads.
However, the axial forces in the latitudinal cables obtained for Model-A and Model-B were quite different. In Model-B, different segments of the same ring of the latitudinal cables had different axial forces under asymmetric loads. In Model-A, each ring of the latitudinal cables had uniform axial force during the loading process, because of the sliding ability of the sliding cable element. The uniform axial forces of Model-A and the maximum and minimum axial forces of Model-B are compared in Figs. 16 and 17 for the inner and outer ring of the latitudinal cables, respectively. The uniform axial forces in the latitudinal cables in Model-A are approximately the average of the axial forces in all segments of the corresponding ring of the latitudinal cables in Model-B.
6. Conclusions
Fig. 15. Deflection of the Suspen-Dome at critical load for asymmetric loading condition.
18
2
Load (kN/m )
13.5
A new kind of multi-node sliding cable element was formulated in this paper for the analysis of cable structures with sliding cables. The element has arbitrary number of sliding nodes and satisfies uniform strain assumption. After the derivation of the tangent stiffness matrix, the multi-node sliding cable element was implemented in commercial finite element software ABAQUS as a user defined element and applied in the static and nonlinear stability analysis of a Suspen-Dome structure under the action of both symmetric and asymmetric loads. The following conclusions were obtained from the analysis results: (1) the rationality and effectiveness of the presented nonlinear multi-node sliding cable element was verified for both sliding and non-sliding situations; (2) with the sliding of the latitudinal cables, uniform axial force distribution can be achieved for the tensegric system of the Suspen-Dome, which is helpful for improving the material efficiency of the cables and preventing the cables from sagging under asymmetric loads; (3) whether the latitudinal cables can slide or not has little effect on static behaviour and nonlinear stability of the single-layer lattice dome even under asymmetric loads.
9 References
Model-A Model-B: Min. Model-B: Max.
4.5
0 0
30
60 90 Axial force in cable (kN)
120
Fig. 16. Axial forces in inner latitudinal cables by nonlinear stability analysis for asymmetric loads.
[1] P. Krishna, Cable-Suspend Roofs, McGraw-hill Book Company, 1978. [2] C.X. Li, D.H. Li, L.F. He, A new method of analysis for the force of the external prestressed cables, China Civil Engineering Journal 38 (6) (2005) 54–58 (In Chinese). [3] X.Q. Cui, Y.L. Guo, Influence of gliding cable joint on mechanical behavior of suspend-dome structures, International Journal of Space Structures 19 (3) (2004) 149–154. [4] G.F. Zhang, S.L. Dong, X. Zhuo, et al., Research on sliding cable in construction of suspend-dome structures, Journal of Zhejiang University (Engineering Science) 42 (6) (2008) 1051–1057 (In Chinese).
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[5] M. Aufaure, A three-node cable element ensuring the continuity of the horizontal tension: a clamp cable element, Computers & Structures 74 (2) (2000) 243–252. [6] B. Zhou, M.L. Accorsi, J.W. Leonard, Finite element formulation for modeling sliding cable elements, Computers & Structures 82 (2–3) (2004) 271–280. [7] ABAQUS/Standard user’s manual, Hibbitt, Karlsson & Sorensen Inc, Pawtucket (RI), 2002. [8] M.A. Crisfield, Non-Linear Finite Element Analysis of Solids and Structures, Volume 1: EssentialsJohn Wiley & Sons, 2000. [9] M. Kawaguchi, M. Abe, I. Tatemichi, Design, test and realization of ‘‘SuspenDome’’ system, Journal of the International Association for Shell and Spatial Structures 40 (131) (1999) 179–192.
[10] National Standard of the People’s Republic of China. Code for design of steel structures GB 500017-2003. Beijing; 2003. (in Chinese). [11] Technical specification for pre-stressed steel structures CECS212:2006, China Plan Press, Beijing; 2006. (in Chinese). [12] V. Gioncu, Buckling of reticulated shells: state-of-the-art, International Journal of Space Structures 10 (1) (1995) 1–46. [13] E. Riks, An incremental approach to the solution of the snapping and buckling problems, International Journal of Solids and Structures 15 (7) (1979) 529–551.
Contents lists available at ScienceDirect
Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel
Formulation and application of multi-node sliding cable element for the analysis of Suspen-Dome structures Z.H. Chen a,b, Y.J. Wu a, Y. Yin a,b,, C. Shan a a b
Department of Civil Engineering, Tianjin University, Tianjin 300072, China Tianjin Key Laboratory of Civil Engineering Structures and New Materials, Tianjin 300072, China
a r t i c l e in fo
abstract
Article history: Received 30 May 2009 Received in revised form 2 April 2010 Accepted 24 April 2010 Available online 15 May 2010
Multi-node sliding cable element was formulated for the analysis of cable structures with cables threading through a number of joints and being able to slide inside them. Tangent stiffness matrix was determined for the element based on uniform strain assumption. The element was implemented in ABAQUS as user defined element and applied in the static and nonlinear stability analysis of a Suspen-Dome structure. The rationality and effectiveness of multi-node sliding cable element for both sliding and non-sliding situations were verified by considering symmetric and asymmetric loads on the Suspen-Dome. The effect of cable sliding on the behaviour of the Suspen-Dome was also discussed. & 2010 Elsevier B.V. All rights reserved.
Keywords: Multi-node sliding cable element Tangent stiffness matrix Suspen-Dome structure Nonlinear stability analysis Effect of cable sliding
1. Introduction Steel cables are widely adopted in long-span structures due to their high strength and light self-weight. By pre-stressing the steel cables, more effective structures can be achieved to span longer distance. In this kind of pre-stressed cable structures, a steel cable usually threads through a number of joints. The distance between adjacent joints is relatively short and the sliding of the cable is often fixed by the joints. In finite element analysis of the structures, the cables can then be simplified as 2-node straight bar elements separated by adjacent joints, subjecting to only tension forces [1]. However, in some cable structures, special type of joints is designed to allow the steel cables to slide freely inside the joints. Though a continuous steel cable may pass through several joints, the tension forces in all cable segments remain a constant. For this situation, the above simplification may cause significant error due to the neglect of cable sliding. Some techniques have been presented to consider the cable sliding based on finite element analysis with simplified separate cable elements [2–5]. They all need manual iteration intervened by engineers, which is time consuming especially for large scale cable structures. Formulating sliding cable element is a more convenient way to simulate sliding cables. Sliding cable elements have been developed and verified for the analysis of parachute
Corresponding author at: Department of Civil Engineering, Tianjin University, Tianjin 300072, China. Tel.: +86 22 13752506876; fax: + 86 22 27404465. E-mail address: [email protected] (Y. Yin).
0168-874X/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2010.04.003
systems [6]. However, the element is a 3-node element and only the intermediate nodes can slide, which hampers its application in the analysis of large scale cable structures. It is then desired to formulate multi-node sliding cable element for sliding cables in pre-stressed cable structures. In this paper, a new kind of multi-node sliding cable element was formulated for the static and nonlinear stability analysis of a Suspen-Dome structure. The element has arbitrary number of sliding nodes and satisfies uniform strain assumption. The principle of virtual work and total Lagrange formulation were used to derive the tangent stiffness matrix of the element. The element was then implemented in commercial finite element software ABAQUS [7] as a user defined element. Both symmetric and asymmetric loads were considered for the Suspen-Dome structure to illustrate the rationality and effectiveness of the presented multi-node sliding cable element.
2. Formulation of multi-node sliding cable element A multi-node sliding cable element is as shown in Fig. 1. The cable is separated to N-1 segments by N nodes. All nodes can slide along the cable except the two end nodes. The number of degree of freedom of the sliding cable element is 3N. The fundamental kinematic assumption of the multi-node sliding cable element states that (1) the cable remains straight between adjacent nodes; (2) the self-weight of the cable can be ignored and all the loads act directly on the nodes; (3) the strain is uniform along the element,
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N-3
i
3
1
If x and x0 are node coordinates in present and initial configurations, respectively, then
N-1
2
x ¼ x0 þ u and du ¼ dx:
N-2
1
i
3
N-1
N-3
So
4
2
N
N-2
i+1
B¼
Fig. 1. Multi-node sliding cable element.
i.e., the strain in all segments of the cable is the same at any time and (4) the sliding node cannot slide across the adjacent nodes along the cable, which means that the order of the nodes along the cable remains unchanged. The principle of virtual work for the sliding cable element in total Lagrange formulation (TL) can be expressed as Z S11 de11 A0 dL ¼ Pi due ð1Þ L
in which e11 is the Green–Lagrange strain, S11 is the second Piola–Kirchhoff stress, L is the total length of the cable at initial configuration, P and u are the node loads and displacements, respectively, and A0 is the cross-sectional area of the cable, which is a constant over the entire element length. In the TL formulation, the integration is performed over the initial configuration. Because the strain and stress are assumed to be constant along the element, the integration in Eq. (1) is performed analytically S11 de11 A0 L ¼ P du
l @l L2 @x
If li is the length of i-th segment of the cable at present configuration and the order of the nodes along the cable remains unchanged, then vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 1 N 1 u 3 X X uX t ð13Þ l¼ li ¼ D2i,j i¼1
ð3Þ
in which B is the vector of 3N. Substituting Eq. (3) into Eq. (2), gives S11 BA0 L du ¼ P du
ð4Þ
Considering the randomicity of the variation of node displacements, du, the equilibrium equation of the sliding cable element can be expressed as S11 BA0 L ¼ P
ð5Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P3 2 P @l @ð N1 @ðlm1 þ lm Þ @ j ¼ 1 Dm1,j i ¼ 1 li Þ ¼ ¼ ¼ @xm,n @xm,n @xm,n @xm,n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P3 2 @ Dm1,n Dm,n j ¼ 1 Dm,j þ ¼ lm1 lm @xm,n l Dm1,n Dm,n lm1 lm L2
@ðS11 BA0 LÞ @S11 @B ¼ A0 LBT þ A0 LS11 @ue @ue @ue
ð7Þ
The second Piola–Kirchhoff stress for the one-dimensional sliding cable element is given by S11 ¼ S011 þ Ee11
@B @B @ððl=L2 Þð@l=@xÞÞ 1 ¼ ¼ ¼ 2 @u @x @x L
@l @l @2 l þl 2 @x @x @x
! ð16Þ
ð@B=@uÞ, ð@B=@xÞ, ð@l=@xÞð@l=@xÞ and ð@2 l=@x2 Þ are all matrix of 3N 3N, and ! @ ðDm1,n =lm1 ÞðDm,n =lm Þ @2 l @ð@l=@xÞ ¼ ¼ @xp,q @x @x2 3ðm1Þ þ n,3ðp1Þ þ q
@ðDm1,n =lm1 Þ @ðDm,n =lm Þ @xp,q @xp,q
¼
ð17Þ
In which p ¼1, 2, y, N and q¼1, 2 and 3. Then the items of the matrix, ð@2 l=@x2 Þ, can be calculated as ( 8 2 2 p¼m > 1 1 Dm1,n Dm,n > > þ . . . . . . > > q¼n < lm1 lm l3m l3m1 @2 l ( ¼ > @x2 p¼m D D D D > > m1,n m1,q m,n m,q . . . . . . 3ðm1Þ þ n,3ðp1Þ þ q > > : qan l3m l3m1 !
O
in which is the initial stress in the cable and E is the Young’s modulus. Then @ðS011 þ Ee11 Þ @S11 @e11 ¼E ¼ ¼ EB @u @u @u For the one-dimensional sliding Green–Lagrange strain is given by [8]
ð18aÞ
ð8Þ
S011
i (x , x , x )
ð9Þ cable
l2 L2 2L2
element,
x2
@e11 @ððl2 L2 Þ=2L2 Þ 1 @l l @l ¼ ¼ 2 2l ¼ 2 @u @u @u 2L L @u
Li
the
(x
i+1 ,x ,x
)
(x
i+ 1 ,x ,x
)
i (x , x , x )
li
ð10Þ 1
in which l is the total length of the cable at present configuration. Then B¼
ð15Þ
ð6Þ
in which K is the tangent stiffness matrix of the sliding cable element, and
e11 ¼
ð14Þ
in which m¼1, 2, y, N and n ¼1, 2 and 3.
The incremental equilibrium equation of the sliding cable element can then be termed as K Du ¼ P
j¼1
in which Di,j ¼ xi þ 1,j xi,j is the length of the i-th (i¼ 1, 2, y, N 1) cable segment in the j-th (j ¼1, 2 and 3) direction, as shown in Fig. 2. Define ðD0,n =l0 Þ ¼ 0 and ðDN,n =lN Þ ¼ 0, then for the n-th coordinate of node m,
B3ðm1Þ þ n ¼
de11 ¼ B du
K¼
i¼1
ð2Þ
Assume
ð12Þ
3
ð11Þ
Fig. 2. The i-th segment of multi-node sliding cable element.
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( 8 p ¼ m þ1 > D2m,n 1 > > ...... > þ 3 ! > q¼n < l l 2 m m @ l ( ¼ 2 > p ¼ mþ 1 @x > > Dm,n Dm,q . . . . . . 3ðm1Þ þ n,3ðp1Þ þ q > > : q an l3m ð18bÞ ( 8 p ¼ m1 > D2m1,n 1 > > þ . . . . . . > > 3 q¼n < lm1 lm1 @2 l ( ¼ > p ¼ m1 @x2 Dm1,n Dm1,q > > 3ðm1Þ þ n,3ðp1Þ þ q > ...... > : q an l3m1 !
745
For convenient cable arrangement, single-layer lattice dome with parallel grid is adopted in the Suspen-Dome. Two kinds of steel tubes, f127 4 and f133 8, are adopted as the members of the single-layer lattice dome. The members are connected with each other by welded hollow spherical joints, which can be considered as rigid joints. Steel of Grade Q235B [10] is adopted for all members and joints of the single-layer lattice dome. According to primary analysis, only two rings of latitudinal cables are needed structurally along the outer two grids of the dome. The sectional areas are 285 mm2 for the outer latitudinal cables and 112 mm2 for the inner latitudinal cables and radial cables, respectively. Steel tube f89 4 of Grade Q235B is adopted for the struts.
ð18cÞ @2 l @x2
3.2. Setup of finite element models for the Suspen-Dome structure
! ¼ 0 . . . . . . p a m1, m or m þ1
ð18dÞ
3ðm1Þ þ n,3ðp1Þ þ q
Substitute Eqs. (14) and (18) into Eq. (16), gives
Two finite element models were setup with commercial finite element software ABAQUS, as shown in Fig. 4. Multi-node sliding cable element presented in this paper was implemented in
!# ( 8 " 2 2 p¼m > 1 Dm1,n Dm,n Dp1,q Dp,q 1 1 Dm1,n Dm,n > > l ...... þ 3 3 > > 2 q¼n < l l l l l l l L l m p m m1 p1 m1 m m1 @B " !# ( ¼ @u 3ðm1Þ þ n,3ðp1Þ þ q > p¼m Dm1,n Dm1,q Dm,n Dm,q 1 Dm1,n Dm,n Dp1,q Dp,q > > > þl ...... þ > L2 : qa n lm1 lm lp1 lp l3m l3m1 !# ( 8 " 2 p ¼ m þ1 > 1 Dm1,n Dm,n Dp1,q Dp,q 1 Dm,n > > l . . . . . . > > 2 3 q¼n < l l l l l l L m p m m1 p1 m @B ( ¼ @u 3ðm1Þ þ n,3ðp1Þ þ q > p ¼ m þ1 Dm,n Dm,q 1 Dm1,n Dm,n Dp1,q Dp,q > > > 2 þl ...... > :L q an lm1 lm lp1 lp l3m !# ( 8 " p ¼ m1 > D2m1,n 1 Dm1,n Dm,n Dp1,q Dp,q 1 > > l . . . . . . > > 2 3 q¼n < l l l l l L l m p m1 p1 m1 m1 @B " !# ( ¼ @u 3ðm1Þ þ n,3ðp1Þ þ q > p ¼ m1 Dm1,n Dm1,q 1 Dm1,n Dm,n Dp1,q Dp,q > > > þl ...... > L2 3 : qa n lm1 lm lp1 lp lm1 @B ¼ 0 . . . . . . p am1, m or m þ 1 @u 3ðm1Þ þ n,3ðp1Þ þ q The tangent stiffness matrix can then be determined for the multi-node sliding cable element by substituting Eqs. (8), (9), (15) and (19) into Eq. (7).
3. The studied Suspen-Dome structure and its finite element models 3.1. Description of the studied Suspen-Dome structure Suspen-Dome structure [9] is a new kind of spatial structures, composed of a single-layer lattice dome and a tensegric network to take advantage of both structural systems. A Suspen-Dome used as the roof structure in a business center hall in Tianjin, China, is as shown in Fig. 3. The span and rise of the dome are 35.4 and 4.6 m, respectively. The Suspen-Dome is made of a singlelayer lattice dome, struts and radial and latitudinal cables with appropriate pre-stresses. The ends of the struts hanging from the same ring of joints of the single-layer lattice dome are connected with the next ring of joints by radial cables, and connected with each other by latitudinal cables. Struts, radial and latitudinal cables, being a tensegric network, sustain vertical loads together with the single-layer lattice dome and, therefore, enable a more efficient and economic way of constructing large-span domes.
ð19aÞ
ð19bÞ
ð19cÞ
ð19dÞ
ABAQUS as a user defined element and two 32-node elements were setup for the two rings of latitudinal cables of Model-A. For the sake of comparison, the latitudinal cables of Model-B were assumed to be separated by adjacent joints and 2-node linear displacement 3-D truss element, T3D2, was adopted for the cable segments. Element types for other members of the Suspen-Dome are all same for Model-A and Model-B, as shown in Table 1. The Suspen-Dome was restrained vertically at the 32 nodes of the outer ring of the single-layer lattice dome. Appropriate restrains were applied at the center node of the single-layer lattice dome to prevent the rigid body displacements. Materials for all the members and cables were assumed to be elastic. The Young’s moduli are E¼ 2.06 108 kN/m2 for the steel tubes and E¼1.95 108 kN/m2 for the steel cables [11]. The material for steel cables was set to be no compression material.
4. Static analysis of the Suspen-Dome structure with multi-node sliding cable element 4.1. Static analysis of the Suspen-Dome under the pre-tension of latitudinal cables The latitudinal cables were pre-tensioned before applying the loads to produce upward displacements of the dome and prevent
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jg1
0
1 jg1
jg1 2
B
1
strut strut Latitudinal Cables
jg9
Radial Cables
jg8
35400
jg7
O
jg6
hg11
jg5
hg9
hg6
jg4
hg5
hg4
hg3
hg2
jg3
hg12
jg2
hg10
hg1
Pa th
B AC
AC B
th Pa
jg1
hg8
A
hg7
1
Single-layer lattice dome
C
B
A
4600
35400
615
Vertical node displacement (mm)
Fig. 3. Structural layout of the Suspen-Dome in Tianjin, China.
Model-A: Under pre-tension
18 Model-A: Under asymmetric load
Model-B: Under pre-tension
Model-B: Under asymmetric load
9 0
-20
-15
-10
-5
0
5
10
15
20
-9 -18 -27 -36 Horizontal distance from center node O (m)
Fig. 5. Vertical node displacements of the Suspen-Dome under pre-tension and symmetric loads. Fig. 4. Finite element model of the Suspen-Dome structure. Table 2 Member forces (kN) of the tensegric system under the pre-tension of the latitudinal cables. Table 1 Element type for finite element models of the Suspen-Dome. Members
Latitudinal cables
Radial cables
Struts
Outer
Inner
Outer
Inner
Outer
Inner
61.65 61.68
34.13 34.15
7.34 7.36
3.92 3.92
2.78 2.78
2.03 2.03
Element type Model-A
Single-layer lattice dome Strut Radial cables Latitude cables
Member locations
Model-B
Model-A Model-B
B33 (2-node 3-D cubic beam element) T3D2 (2-node linear displacement 3-D truss element) T3D2 (no compression) Multi-node sliding cable T3D2 (no element compression)
the cables from sagging under the load action. The pre-tension forces were applied as initial stresses, 215 MPa for the outer latitudinal cables and 305 MPa for the inner ones. The upward
node displacements along the Section 1-1 of the single layer lattice dome and the member forces of the tensegric system are compared in Fig. 5 and Table 2 for Model-A and Model-B. No cable sliding would occur in Model-A, because the initial stress was set to be uniform for each ring of the latitudinal cables. The analysis based on the two models gave exactly the same results, which validated the multi-node sliding cable element for pre-tension analysis of the cable structures. It should also be noticed that the pre-tension of the latitudinal cables produced obvious
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The service loads on the Suspen-Dome structure are dead load of 2.0 kN/m2 and live load of 0.5 kN/m2, acting symmetrically on entire span of the dome. For Model-B, the stresses in all segments of each ring of latitudinal cables remained uniform under the service loads because of the symmetry of the structure and the loads. Correspondingly, no cable sliding would occurred in the sliding cable elements adopted in Model-A. The vertical node displacements along the Section 1-1 of the single layer lattice dome and the member forces of the tensegric system under the service loads are compared in Fig. 5 and Table 3 for Model-A and Model-B. The analysis results based on Model-A and Model-B agreed very well, which validated the multi-node sliding cable element for non-sliding situation. It can be seen that the maximum vertical displacement of the Suspen-Dome under symmetric loads reaches 32.8 mm. 4.3. Static analysis of the Suspen-Dome under asymmetric loads The Suspen-Dome may subject to asymmetric loads during construction and service periods. The worst asymmetric load distribution is dead load of 2.0 kN/m2 and live load of 0.5 kN/m2, acting on half span of the dome during the construction. This asymmetric load distribution was considered to illustrate the rationality and effectiveness of the multi-node sliding cable element for the situation of cable sliding. For the Suspen-Dome of Model-B under the asymmetric loads on the right half span, the axial forces in different segments of the latitudinal cables are different because the sliding of the cables are not allowed, as shown in Fig. 6. The difference between the axial forces is 47.0 kN for the outer latitudinal cable segments. Different axial forces in the latitudinal cables results in different axial forces in radial cables and struts, as shown in Figs. 7 and 8. The node displacements and member stresses along the Section 1-1 of the single-layer lattice dome of Model-B are as Table 3 Member forces (kN) of the tensegric system under symmetric loads. Member locations
Model-A Model-B
Latitudinal cables
Radial cables
Struts
Outer
Inner
Outer
Inner
Outer
Inner
137.17 137.34
36.88 36.83
16.37 16.36
4.23 4.23
6.39 6.32
2.11 2.09
747
20 16 12 Inner radial cable Outer radial cable
8 4 0 0
5
10
15
20
25
30
35
Segment number of radial cables along Path ACB Fig. 7. Axial force of radial cables of Model-B under asymmetric loads.
0 0 Axial force of the strut (kN)
4.2. Static analysis of the Suspen-Dome under symmetric loads
3
6
9
12
15
18
-2
-4
-6 Inner strut
Outer strut
-8 Number of struts along Path ACB Fig. 8. Axial force of struts of Model-B under asymmetric loads.
18 Vertical node displacement (mm)
anti-load effects and the maximum upward displacement of the Suspen-Dome is 7.45 mm.
Axial force of cable segment (kN)
Z.H. Chen et al. / Finite Elements in Analysis and Design 46 (2010) 743–750
9 0 -20
-15
-10
-5
0
5
10
15
20
-9 Model-A: Under pre-tension
-18
Model-B: Under pre-tension Model-A: Under asymmetric load Model-B: Under asymmetric load
-27
-36 Horizontal distance from center node O (m) Fig. 9. Vertical node displacements of the Suspen-Dome under pre-tension and asymmetric loads.
150
Worst member stress (MPa)
Axial force of cable segment (kN)
150
120 90
Inner latitudinal cable Outer latitudinal cable
60 30
Model A: latitudinal member hg# Model B: latitudinal member hg# Model A: radial member jg# Model B: radial member jg#
100 50 0 0
2
4
6
8
10
12
14
-50
0 0
3
6
9
12
15
18
Segment number of latitudinal cables along Path ACB Fig. 6. Axial force of latitudinal cables of Model-B under asymmetric loads.
-100 Number of members # Fig. 10. Worst member stresses of single-layer lattice dome under asymmetric loads.
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25
Table 4 Member forces (kN) of the tensegric system under asymmetric loads.
Model-A Model-B (average value)
20
Latitudinal cables
Radial cables
Struts
Outer
Inner
Outer
Inner
Outer
Inner
100.60 103.78
36.20 36.39
12.37 12.37
4.15 4.18
4.70 4.64
2.07 2.06
Load (KN/m2)
Member locations
15
Model-A Model-B
10
5
0
0
0.04 0.08 0.12 0.16 Vertical displacement of Node A (m)
0.2
Fig. 11. Load–displacement curves by nonlinear stability analysis for symmetric loads.
25 20 2
Load (kN/m )
shown in Figs. 9 and 10. It can be seen that the maximum vertical displacement of the Suspen-Dome under asymmetric loads is 31.7 mm, almost same as that under symmetric loads. For the Suspen-Dome of Model-A under the asymmetric loads on the right half span, the axial forces in different segments of each ring of latitudinal cables remain uniform because of the sliding of the cables. The axial forces in the same ring of radial cables and struts remain uniform too because of the equilibrium request. The member forces of the tensegric system are as shown in Table 4. Compared with Model-B, the maximum axial force was reduced by 26.2% for the outer latitudinal cables, which is helpful for improving the material efficiency of the cables. On the other hand, the minimum axial force was increased by 20.5% for the outer latitudinal cables, which is helpful for preventing the cables from sagging under asymmetric loads. It was also noticed that the axial forces in the latitudinal cables of Model-A agreed with the average values of the axial forces in cable segments of corresponding ring of latitudinal cables for Model-B as shown in Table 4. This validates the rationality and effectiveness of the presented multi-node sliding cable element for the situation of cable sliding. The vertical node displacements and worst member stresses of the single-layer lattice dome along the Section 1-1 of Model-A are compared with those of Model-B, as shown in Figs. 9 and 10. Little difference was found between the node displacements and the worst member stresses of Model-A and Model-B. It can then be concluded that whether the latitudinal cables can slide or not has little effect on the behaviour of the single layer lattice dome even under asymmetric loads.
15 10
Model-A: Inner latitudinal cable Model-B: Inner latitudinal cable
5
Model-A: Outer latitudinal cable Model-B: Outer latitudinal cable
0 0
50
100 150 Axial force in cable (kN)
200
Fig. 12. Axial forces in latitudinal cables by nonlinear stability analysis for symmetric loads.
5. Stability analysis of the Suspen-Dome structure with multi-node sliding cable element The design of single-layer lattice domes may be controlled by structural stability [12]. Though the studied Suspen-Dome structure is reinforced with the tensegric system, its stability still needs to be concerned. In design practice, the stability of lattice dome structures is often evaluated by tracing the nonlinear equilibrium path of the structure under proportional loading. Geometric nonlinearity should be considered and the arc-length method [13] is often adopted for equilibrium path tracing. The stability of the Suspen-Dome structure was first analyzed for the symmetric loading condition. Uniform loads were applied on the entire span of Model-A and Model-B, respectively, after the pre-tension of the latitudinal cables. With the increase of the uniform loads, the structure stiffness of the two models decreased due to the deflection of the Suspen-Dome. No cable sliding occurred in sliding cable elements adopted in Model-A and all segments of the same ring of latitudinal cables in Model-B had equal axial forces during the loading process. The load–displacement curves of Node A (as shown in Fig. 3) and the axial forces in the latitudinal cables obtained by nonlinear stability analysis were compared in Fig. 11 and Fig. 12, respectively. The results based on Model-A and Model-B agreed very well. The critical load for the Suspen-Dome structure can determine to be 22.2 kN/m2
Fig. 13. Deflection of the Suspen-Dome at critical load for symmetric loading condition.
for symmetric loading condition, corresponding to the limit point on the load–displacement curve. Note that the inner ring of the latitudinal cables sagged before the critical load. The deflection of the Suspen-Dome at the critical load is as shown in Fig. 13 for symmetric loading condition. The stability of the Suspen-Dome structure under asymmetric loads was analyzed by applying uniform loads on the right half span of Model-A and Model-B, respectively, after the pre-tension of the latitudinal cables. The stiffness of the structure was also decreased with the increased asymmetric loads due to the
ARTICLE IN PRESS Z.H. Chen et al. / Finite Elements in Analysis and Design 46 (2010) 743–750
18
2
Load (kN/m )
13.5
9
18
4.5
13.5
0
Model-A Model-B: Min. Model-B: Max.
40
2
Load (kN/m )
deflection of the Suspen-Dome. The load–displacement curves of Node B obtained by nonlinear stability analysis were compared for Model-A and Model-B in Fig. 14. Only slight difference was found between critical loads based on Model-A and Model-B, which were 16.7 and 17.1 kN/m2, respectively. The structural deflection of the two models at the critical load also agreed well for the asymmetric loading condition and is as shown in Fig. 15.
9
4.5
0.05 0.15 0.25 0.35 Vertical displacement of Node B (m)
65
90 115 Axial force in cable (kN)
140
Fig. 17. Axial forces in outer latitudinal cables by nonlinear stability analysis for asymmetric loads.
Model-A Model-B
0 -0.05
749
0.45
Fig. 14. Load–displacement curves by nonlinear stability analysis for asymmetric loads.
However, the axial forces in the latitudinal cables obtained for Model-A and Model-B were quite different. In Model-B, different segments of the same ring of the latitudinal cables had different axial forces under asymmetric loads. In Model-A, each ring of the latitudinal cables had uniform axial force during the loading process, because of the sliding ability of the sliding cable element. The uniform axial forces of Model-A and the maximum and minimum axial forces of Model-B are compared in Figs. 16 and 17 for the inner and outer ring of the latitudinal cables, respectively. The uniform axial forces in the latitudinal cables in Model-A are approximately the average of the axial forces in all segments of the corresponding ring of the latitudinal cables in Model-B.
6. Conclusions
Fig. 15. Deflection of the Suspen-Dome at critical load for asymmetric loading condition.
18
2
Load (kN/m )
13.5
A new kind of multi-node sliding cable element was formulated in this paper for the analysis of cable structures with sliding cables. The element has arbitrary number of sliding nodes and satisfies uniform strain assumption. After the derivation of the tangent stiffness matrix, the multi-node sliding cable element was implemented in commercial finite element software ABAQUS as a user defined element and applied in the static and nonlinear stability analysis of a Suspen-Dome structure under the action of both symmetric and asymmetric loads. The following conclusions were obtained from the analysis results: (1) the rationality and effectiveness of the presented nonlinear multi-node sliding cable element was verified for both sliding and non-sliding situations; (2) with the sliding of the latitudinal cables, uniform axial force distribution can be achieved for the tensegric system of the Suspen-Dome, which is helpful for improving the material efficiency of the cables and preventing the cables from sagging under asymmetric loads; (3) whether the latitudinal cables can slide or not has little effect on static behaviour and nonlinear stability of the single-layer lattice dome even under asymmetric loads.
9 References
Model-A Model-B: Min. Model-B: Max.
4.5
0 0
30
60 90 Axial force in cable (kN)
120
Fig. 16. Axial forces in inner latitudinal cables by nonlinear stability analysis for asymmetric loads.
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[10] National Standard of the People’s Republic of China. Code for design of steel structures GB 500017-2003. Beijing; 2003. (in Chinese). [11] Technical specification for pre-stressed steel structures CECS212:2006, China Plan Press, Beijing; 2006. (in Chinese). [12] V. Gioncu, Buckling of reticulated shells: state-of-the-art, International Journal of Space Structures 10 (1) (1995) 1–46. [13] E. Riks, An incremental approach to the solution of the snapping and buckling problems, International Journal of Solids and Structures 15 (7) (1979) 529–551.