IGA Cable Structures

STATIC AND VIBRATION ANALYSIS OF CABLE
STRUCTURES USING ISOGEOMETRIC
APPROACH
Thai Son
Sejong University
Advanced Structural Engineering Laboratory
MASTER’S THESIS DEFENSE
November 20, 2014
Thai Son (Sejong University) IGA-based analysis of cable structures November 20, 2014 1 / 55

Contents
1 Introduction
Analyses of cable structures
Goals and Objectives
2 IGA-based analysis of cable structures
Isogeometric analysis
Cable formulation
NURBS-based discretization
Form-ﬁnding of cable structures
Finite element models
3 Numerical examples
Static analysis
Free vibration analysis
4 Conclusions and remarks
Conclusions
Remarks

Introduction

Cable structures
Distinctive characteristics:
Light weight
High strength to weigh ratio
Aesthetic appearance
Low-cost construction
Drawbacks:
Highly geometrical non-linearity
Unstable and sensitive to the effects of external loads

Analysis of cable structures
Up until the present, there are 2 prevailing approaches to analyze the
cable structures’ behavior:
Exact elastic catenary expression
Finite element analysis

Elastic catenary expression
Features
Firstly introduced by Obrien and Francis in 1964
Based on mathematical analysis
The geometries are assumed to be represented by parabolic
curves
Advantage
The nonlinear effects are considered accurately with a small
number of elements
Disadvantages
Only acceptable when small curvatures are considered
Having problems in cases extremely high pretension cables are
taken into account
Having some restrictions due to the limitation of mathematical
expression
For vibration analysis, the complexities arise due to various cases
in reality make the general explicit solutions cannot be obtained
Generally restrictive to apply in engineering practices

Traditional finite element approach
Features
Normally employed Lagrangian functions to represent the
geometry and the displacement fields
Advantage
Versatile with no limitation and restriction regarding geometries
and loading conditions
Disadvantages
Need a fine mesh to generate the smooth curve of cables
Difficult to obtained the solutions for cables having a relatively
large sag
Mesh generation is generally time-consuming and costly
Generally less attractive to investigators recently

Motivations
Propose an alternative approach, which can take advantage of the
versatility of FEA but also derive the precise geometry of the
structures

Motivations
Propose an alternative approach, which can take advantage of the
versatility of FEA but also derive the precise geometry of the
structures
=⇒ ISOGEOMETRIC ANALYSIS APPROACH

Recently Isogeometric approach
Bridge the gap between ﬁnite element analysis (FEA) and
conventional computer-aided design tools
Employ B-splines, NURBS or currently T-splines as the attractive
alternatives of interpolation functions
Inherit the isoparametric concept from FEA, in which both physical
geometry and dependent variables of problems are expressed by
the same basis
Have the ability to present exact geometry of object, which is
virtually approximated in traditional FEA
Provide advantages in computational cost concerning to accuracy
with the newly introduced k-reﬁnement scheme

Objectives
Develop the NURBS-based isogeometric cable element as an
alternative approach to deal with static and free vibration
problems of cable structures
Employ the NURBS basis functions to model exactly the geometry
of cable structures and the displacement field
Utilize the penalty method to handle the form-finding for
determining the initial equilibrium geometries
Perform the refinement tests and compare the converged
solutions with those obtained form previous research to
demonstrate the accuracy and the efficiency of proposed
approach

IGA-based analysis of cable
structures

Knot vector and Basis functions
Knot vector
Ξ = ξ1, ξ2, ξ3, ..., ξi, ..., ξn+p+1 , ξi ≤ ξi+1
B-spline basis functions
Ni,0 (ξ) =
1 ξi ≤ ξ < ξi+1
0 otherwise
Ni,p =
ξ − ξi
ξi+p − ξi
Ni,p−1 (ξ) +
ξi+p+1 − ξ
ξi+p+1 − ξi+1
Ni,p−1 (ξ)
0
0.25
0.5
0.75
1
0 0.25 0.5 0.75 1
1,3N
2,3N 3,3N 4,3N 5,3N
6,3N
7,3N 8,3N
9,3N
Ξ={0, 0, 0, 0, 0.25, 0.5, 0.75, 0.75, 0.75, 1, 1, 1, 1}

Knot vector and Basis functions
The characteristics of B-spline basis functions:
constituting the partition of unity, which means
n
1
Ni,p (ξ) = 1
linear independent, which means
n
1
αiNi,p (ξ) = 0 ⇔ αi = 0, i = 1, ..., n
having compact support
Compared to shape function in traditional FEA, the B-splines are also:
non-negative over the entire parameter space
Cp−1 continuity across the element boundaries when knot vector
is uniform.

Control points and B-spline curves
B-spline curves are constructed with respect to a set of control points
(Pi, i=1,...,n) by a linear combination of basis functions:
C (ξ) =


x
y
z

 (ξ) =
n
i=1
PiNi,p (ξ) =
n
i=1


xi
yi
zi

 Ni,p (ξ)
1C
2C
3C
4C
5C
6C
7C
8C

Reﬁnement schemes
h-reﬁnement
additional knots are inserted to split the knot span, thence
rendering more elements
both physical geometry and parameterization are not changed
the number of control point is increased; besides, the continuities
over the newly created knots are similar to the inner original ones
0
0.25
0.5
0.75
1
0 0.5 1
0
0.25
0.5
0.75
1
0 0.25 0.5 0.75 1
{0, 0, 0, 0.5, 1, 1, 1} ⇒ {0, 0, 0, 0.25, 0.5, 0.75, 1, 1, 1}

Reﬁnement schemes
p-reﬁnement
elevates polynomial order of basis functions by increasing the
multiplicity of each knot value by one
the geometry and parameterization of the physical curve are not
changed
the number of basis function and control points are increased
the number of elements and the continuity across inner knots
remains the same
0
0.25
0.5
0.75
1
0 0.25 0.5 0.75 1
0
0.25
0.5
0.75
1
0 0.25 0.5 0.75 1
{0, 0, 0.5, 1, 1} ⇒ {0, 0, 0, 0.5, 0.5, 1, 1, 1}

Refinement schemes
k-refinement
newly introduced refinement schemes compared to FEA
both order of basic functions and the continuity across the knot
values (element boundaries) are increased
change the control points but still maintain the original geometry
and the parameterization of physical domain
0
0.25
0.5
0.75
1
0 0.25 0.5 0.75 1
0
0.25
0.5
0.75
1
0 0.25 0.5 0.75 1
{0, 0, 0.5, 1, 1} ⇒ {0, 0, 0, 0.5, 1, 1, 1}

Non-Uniform Rational B-splines (NURBS)
NURBS is the generalized form of B-spline in the frame work of
partition of unity with weighted values wi
Rp
i (ξ) =
Ni,p (ξ) wi
W (ξ)
=
Ni,p (ξ) wi
n
i=1 Ni,p (ξ)wi
While B-splines only can represent conical sections or curves,
NURBS have the ability to describe arbitrary physical domain
The NURBS and B-splines basis functions share the same tensor
product nature and refinement schemes
The NURBS curves are defined by a linear combination of the
NURBS basis functions and coefficients (control points) over the
parametric space
C (ξ) =
n
i=1
PiRp
i (ξ)

Incremental equation
The initial undeformed configuration (C0), the last known deformed
configuration (C1), and the current deformed configuration (C2)
0
C0
C1
C2
0
ix 00
ii d xx +0
dS
1
0
0
ii ux + 1
0
0
ii ux + d x d ui i0
0 1
+ +1
dS
2
0
0
ii ux + 2
0
0
ii ux + d x d ui i0
0 2
+ +2
dS
0
x2 ,1
x2 ,2
x2
0
x1 ,1
x1 ,2
x1
0
x3 ,1
x3 ,2
x3
1
dS
2
= d0
xi + d1
0 ui · d0
xi + d1
0 ui
2
dS
2
= d0
xi + d2
0 ui · d0
xi + d2
0 ui

0ε =
1
2
2dS
2
− 1dS
2
(0dS)
2
=0e+0η
where 0e and 0η
0e =
d0xi
0dS
dui
0dS
+
dui
0dS
d1
0 ui
0dS
0η =
1
2
dui
0dS
dui
0dS
0S
Aρ¨uiδuidS +
0S
AE0eδ (0e) dS +
0S
A1
0Pδ (0η) dS
= δ 2
0R −
0S
A1
0Pδ (0e) dS

Geometry and displacement ﬁelds
xi =
ncp
α=1
Rα (ξ) (xc
i )α, (i = 1, 2, 3) or x = Rxc
ui =
ncp
α=1
Rα (ξ) (uc
i )α; (i = 1, 2, 3) or u = Ruc
in which
xc = x1
1 , x1
2 , x1
3 , ..., xncp
1 , xncp
2 , xncp
3
T
uc = u1
1, u1
2, u1
3, ..., uncp
1 , uncp
2 , uncp
3
T
R =


R1 0 0
0 R1 0
0 0 R1
...
Rncp 0 0
0 Rncp 0
0 0 Rncp



0e =
1
(J)2
xT
c R,T
ξ R,ξ +1
0 uT
c R,T
ξ R,ξ · uc = BLuc
δ (0e) =
1
(J)2
xT
c R,T
ξ R,ξ +1
0 uT
c R,T
ξ R,ξ · δuc = BLδuc
0η =
1
2(J)2
uT
c R,T
ξ R,ξ · uc =
1
2
uT
c BT
NLBNLuc
δ (0η) =
1
(J)2
R,T
ξ R,ξ · δuc = BT
NLBNLδuc
J =
dS
dξ
=
3
i=1
ncp
α=1
R,α
ξ (ξ) (xc
i )α
2

m¨u + (kL + kNL) u =2
0 f −1
0 f
where
m = A
ξi+1
ξi
ρRT
RJdξ
kL = EA
ξi+1
ξi
BT
L BLJdξ
kNL = A
ξi+1
ξi
1
0PBT
NLBNLJdξ
1
0f = A
ξi+1
ξi
1
0PBT
L Jdξ

The penalty method has been developed to solve the multi-point
constraints equations in FEA. For a global static problem:
kgug = fg
Global stiffness matrix
kg(∈ Ra×a
) =
ne
1
(kL + kNL)
Global force vector
fg =
ne
1
2
0f − 1
0f
Assume that the system has m constraint equations:
cug − ˆu = 0
in which
c(∈ Rm×a) is the constant matrix (m < a)
ˆu is the m-dimensional constant vector

The penalty equation is written as
p = cug − û
in which p = 0 imply all constraint equations are satisfied
The total potential energy of the system can be reformed as:
Π =
1
2
uT
g kug − uT
g f +
1
2
pT
∆p
where ∆ is the m × m diagonal matrix of chosen constant numbers or
so-called penalty numbers α, (∆ii = α).
δΠ = 0 ⇒ kg + cT
∆c uc = fg + cT
∆û

Starting configuration with
unstressed length 0
S
Initial equilibrium configuration
under self-weight
yû
xû
Prescribed
displacements
Firstly, the cable is modeled in the state of straight, stress-free
one.
Then, the prescribed displacements are applied to supports to
obtain the determined geometrical location
This procedure is carried out according to the incremental
scheme.

Finite element models
For static analysis
kg + cT
∆c uc = fg + cT
∆ˆu
For free vibration analysis
mg ¨uc + kg + cT
∆c uc = 0

Numerical examples

Isolated cable under concentrated load
121.92 m
152.4 m
304.8 m
30.48 m
x
P = 35.586 kN
z
L1 L2
1
2
3
Table: Properties of cable
Data Value
Cross-sectional area (A) 548.4 mm2
Elastic modulus (E) 131 kN/mm2
Unstressed length (L1) 125.88 m
Unstressed length (L2) 186.85 m
Self-weight (w) 46.12 kN/m

P = 35.586 kN
1
2
3
7.93
ξ=0 ξL ξ=1
C0
ξL = L1/(L1 + L2)
Ξ = {(0)p+1, 2ξL/ne, ..., 2kξL/ne, (ξL)p, 2(1 − ξL)/ne, ...
, 2k(1 − ξL)/ne, (1)p+1}; k = ne/2

Researcher Approach Displacements
Vertical Horizontal
Jayaraman and Knudson Elastic straight -5.4710 -0.8450
Jayaraman and Knudson Elastic catenary -5.6260 -0.8590
Yang and Tsay Elastic catenary -5.6250 -0.8590
Thai and Kim Elastic catenary -5.6260 -0.8590
Salehi Ahmad Abad et al. Discrete elastic catenary -5.5920 -0.8550
Continuous elastic catenary -5.6260 -0.8590
Discrete elastic catenary with point load -5.8300 -0.8730
This study Isogeometric cable element
Order No. of element
1 2 -5.8037 -0.8923
1 4 -5.6686 -0.8674
1 8 -5.6351 -0.8612
1 16 -5.6268 -0.8597
1 32 -5.6247 -0.8593
1 64 -5.6241 -0.8592
1 128 -5.6240 -0.8592
1 256 -5.6240 -0.8592
2 2 -5.6665 -0.8689
2 4 -5.6281 -0.8602
2 8 -5.6241 -0.8593
2 16 -5.6239 -0.8592
2 32 -5.6239 -0.8592
2 64 -5.6239 -0.8592
3 2 -5.6239 -0.8592
3 4 -5.6239 -0.8592
3 8 -5.6239 -0.8592
3 16 -5.6239 -0.8592
3 32 -5.6239 -0.8592
3 64 -5.6239 -0.8592

5-cables net
Table: Structure’s properties
Data Value
Axial stiffness (EA)
Elastic 5000 daN
Inelastic ∞
Cable length (L)
L1 1.2887 m
L2 1.2887 m
L3 0.5912 m
L4 1.8740 m
L5 2.0978 m
Self-weight (w) 2.0 daN/m

5-cables net
Firstly, each cable is
initially described as a
straight and free stressed
segment
Ξ = { (0)p+1, 1/ne, 2/ne,...,
(ne-1)/ne, (1)p+1 }
Nodes x y z
P3 0.0000 0.0000 0.0000
P4 1.0000 0.0000 0.0000
P5 0.0000 1.0000 0.0000
P6 1.0000 1.0000 1.0000

5-cables net
Researcher Node Unelastic Elastic
x y z x y z
P3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
P4 1.0000 0.0000 0.0000 1.0000 0.0000 0.0000
P5 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000
P6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Greco et al. P1 0.5000 0.2500 -1.1143 0.4999 0.2499 -1.1148
P2 0.5000 0.7500 -0.9954 0.4994 0.7500 -0.9963
Isogeometric cable element
1 1
P1 0.5000 0.2093 -1.1693 0.5000 0.2092 -1.1697
P2 0.5395 0.7838 -1.0355 0.5386 0.7839 -1.0363
1 2
P1 0.5007 0.2449 -1.1410 0.5007 0.2448 -1.1414
P2 0.5137 0.7577 -1.0163 0.5131 0.7577 -1.0171
1 4
P1 0.5003 0.2491 -1.1237 0.5003 0.2491 -1.1241
P2 0.5032 0.7525 -1.0036 0.5027 0.7524 -1.0043
1 8
P1 0.5001 0.2498 -1.1169 0.5000 0.2498 -1.1173
P2 0.5005 0.7507 -0.9980 0.5001 0.7507 -0.9987
1 16
P1 0.5000 0.2499 -1.1150 0.5000 0.2499 -1.1154
P2 0.5000 0.7502 -0.9963 0.4996 0.7502 -0.9970
1 32
P1 0.5000 0.2500 -1.1146 0.4999 0.2499 -1.1150
P2 0.4999 0.7501 -0.9958 0.4995 0.7501 -0.9965
2 1
P1 0.5003 0.2475 -1.1204 0.5001 0.2077 -1.1682
P2 0.5034 0.7527 -0.9996 0.5386 0.7814 -1.0339
2 2
P1 0.5001 0.2079 -1.1658 0.5002 0.2091 -1.1620
P2 0.5389 0.7801 -1.0309 0.5355 0.7789 -1.0280
2 4
P1 0.5005 0.2130 -1.1545 0.5007 0.2202 -1.1462
P2 0.5296 0.7769 -1.0220 0.5204 0.7726 -1.0165

5-cables net
Researcher Node Unelastic Elastic
x y z x y z
P3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
P4 1.0000 0.0000 0.0000 1.0000 0.0000 0.0000
P5 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000
P6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Greco et al. P1 0.5000 0.2500 -1.1143 0.4999 0.2499 -1.1148
P2 0.5000 0.7500 -0.9954 0.4994 0.7500 -0.9963
Isogeometric cable element
2 8
P1 0.5007 0.2316 -1.1356 0.5005 0.2417 -1.1272
P2 0.5129 0.7648 -1.0090 0.5067 0.7573 -1.0043
2 16
P1 0.5003 0.2475 -1.1204 0.5001 0.2493 -1.1175
P2 0.5034 0.7527 -0.9996 0.5010 0.7509 -0.9981
2 32
P1 0.5001 0.2499 -1.1154 0.5000 0.2499 -1.1151
P2 0.5004 0.7503 -0.9963 0.4996 0.7501 -0.9966
3 1
P1 0.5005 0.2491 -1.1331 0.5005 0.2491 -1.1333
P2 0.5087 0.7546 -1.0111 0.5083 0.7546 -1.0116
3 2
P1 0.5005 0.2462 -1.1329 0.5006 0.2468 -1.1322
P2 0.5078 0.7570 -1.0092 0.5094 0.7544 -1.0086
3 4
P1 0.5004 0.2493 -1.1238 0.5003 0.2492 -1.1221
P2 0.5052 0.7518 -1.0025 0.5036 0.7518 -1.0018
3 8
P1 0.5001 0.2499 -1.1162 0.5000 0.2499 -1.1159
P2 0.5005 0.7503 -0.9971 0.4999 0.7502 -0.9972
3 16
P1 0.5000 0.2500 -1.1146 0.4999 0.2499 -1.1149
P2 0.5000 0.7501 -0.9960 0.4995 0.7501 -0.9964
3 32 P1 0.5000 0.2500 -1.1145 0.4999 0.2499 -1.1148
P2 0.5000 0.7501 -0.9959 0.4995 0.7501 -0.9964

Single span cables
Table: Knot vectors for refinement strategies
Refinement strategy No. of element Order Knot vector
h-refinement ne p∗ (0)p+1, 1/ne, 2/ne, ..., (ne − 1)/ne, (1)p+1
p-refinement n∗
e p (0)p+1, (1/ne)p, (2/ne)p, ..., ((ne − 1)/ne)p, (1)p+1
k-refinement n∗
e p (0)p+1, 1/ne, 2/ne, ..., (ne − 1)/ne, (1)p+1
Note: * is a fixed value

Single span cables
Table: First 10 natural frequencies of single span cable (AE/mgl=1)obtained
from h-reﬁnement
Order No. of elements Mode number
(p) (ne) 1 2 3 4 5 6 7 8 9 10
1 1 - - - - - - - - - -
2 2.9072 3.4107 - - - - - - - -
4 2.2517 3.0646 4.4873 5.7506 8.8941 12.7687 - - - -
8 2.0817 2.9901 3.9515 5.4914 6.0603 7.8206 8.9666 9.5934 12.5640 13.1790
16 2.0697 2.9736 3.8558 5.3081 5.7973 7.2685 8.6271 9.0377 10.8416 12.3923
32 2.0664 2.9693 3.8302 5.2540 5.7351 7.0835 8.5256 8.7089 10.2809 11.8373
64 2.0656 2.9682 3.8237 5.2399 5.7198 7.0366 8.4908 8.6342 10.1366 11.6278
2 1 3.2676 3.7138 - - - - - - - -
2 2.4001 3.2194 5.8614 7.0817 - - - - - -
4 2.1768 3.0165 4.3005 5.7950 6.4284 9.2443 12.2898 14.6199 - -
8 2.0751 2.9718 3.8655 5.3252 5.8420 7.4705 8.5994 9.4951 11.3043 12.2998
16 2.0658 2.9680 3.8233 5.2392 5.7193 7.0398 8.4927 8.6436 10.1812 11.7278
32 2.0654 2.9679 3.8217 5.2353 5.7149 7.0217 8.4776 8.6128 10.0921 11.5631
64 2.0653 2.9679 3.8216 5.2351 5.7147 7.0209 8.4769 8.6116 10.0886 11.5568
3 1 2.4663 3.2292 5.8418 6.8394 - - - - - -
2 2.3233 3.0771 4.5029 6.1879 8.8315 10.6741 - - - -
4 2.1185 2.9908 4.0544 5.6260 6.8055 8.7888 8.9651 12.3486 16.6902 19.9130
8 2.0684 2.9689 3.8377 5.2755 5.7683 7.3018 8.5477 9.3457 11.3653 12.1999
16 2.0654 2.9679 3.8216 5.2354 5.7151 7.0231 8.4796 8.6164 10.1096 11.6083
32 2.0654 2.9679 3.8216 5.2351 5.7147 7.0209 8.4769 8.6116 10.0885 11.5567
64 2.0654 2.9679 3.8216 5.2351 5.7147 7.0209 8.4769 8.6115 10.0884 11.5565

Single span cables
Table: First 10 natural frequencies of single span cable (AE/mgl=1) obtained
from p-reﬁnement
No. of elements Order Mode number
(ne) (p) 1 2 3 4 5 6 7 8 9 10
1 1 - - - - - - - - - -
2 3.2676 3.7138 - - - - - - - -
3 2.4663 3.2292 5.8418 6.8394
2 1 2.9072 3.4107 - - - - - - - -
2 2.3302 3.0731 4.4788 6.0783 9.1840 12.9009 - - - -
3 2.1364 3.0006 4.1414 5.6124 6.5785 8.8102 9.8508 13.2553 15.4267 24.2254
4 1 2.2517 3.0646 4.4873 5.7506 8.8941 12.7687 - - - -
2 2.0802 2.9879 3.9932 5.4386 6.1106 7.8126 8.8206 10.5660 12.7856 13.5716
3 2.0709 2.9687 3.8364 5.2852 5.7756 7.2402 8.5462 9.0612 11.0815 12.2131
8 1 2.0817 2.9901 3.9515 5.4914 6.0603 7.8206 8.9666 9.5934 12.5640 13.1790
2 2.0688 2.9696 3.8371 5.2677 5.7529 7.1481 8.5391 8.8454 10.4923 12.1775
3 2.0655 2.9679 3.8222 5.2362 5.7160 7.0298 8.4851 8.6283 10.1365 11.6421
16 1 2.0697 2.9736 3.8558 5.3081 5.7973 7.2685 8.6271 9.0377 10.8416 12.3923
2 2.0656 2.9680 3.8228 5.2377 5.7176 7.0315 8.4858 8.6273 10.1305 11.6269
3 2.0654 2.9679 3.8216 5.2352 5.7148 7.0211 8.4770 8.6118 10.0895 11.5586
32 1 2.0664 2.9693 3.8302 5.2540 5.7351 7.0835 8.5256 8.7089 10.2809 11.8373
2 2.0654 2.9679 3.8216 5.2353 5.7149 7.0216 8.4775 8.6126 10.0914 11.5615
3 2.0653 2.9679 3.8216 5.2351 5.7147 7.0209 8.4769 8.6115 10.0884 11.5565
64 1 2.0656 2.9682 3.8237 5.2399 5.7198 7.0366 8.4908 8.6342 10.1366 11.6278
2 2.0653 2.9679 3.8216 5.2351 5.7147 7.0209 8.4769 8.6116 10.0886 11.5568
3 2.0653 2.9679 3.8216 5.2351 5.7147 7.0209 8.4769 8.6115 10.0884 11.5565

Single span cables
Table: First 10 natural frequencies of single span cable (AE/mgl=1) obtained
from k-reﬁnement
No. of elements Order Mode number
(ne) (p) 1 2 3 4 5 6 7 8 9 10
1 1 - - - - - - - - - -
2 3.2676 3.7138 - - - - - - - -
3 2.4663 3.2292 5.8418 6.8394 - - - - - -
2 1 2.9072 3.4107 - - - - - - - -
2 2.4001 3.2194 5.8614 7.0817 - - - - - -
3 2.3233 3.0771 4.5029 6.1879 8.8315 10.6741 - - - -
4 1 2.2517 3.0646 4.4873 5.7506 8.8941 12.7687 - - - -
2 2.1768 3.0165 4.3005 5.7950 6.4284 9.2443 12.2898 14.6199 - -
3 2.1185 2.9908 4.0544 5.6260 6.8055 8.7888 8.9651 12.3486 16.6902 19.9130
8 1 2.0817 2.9901 3.9515 5.4914 6.0603 7.8206 8.9666 9.5934 12.5640 13.1790
2 2.0751 2.9718 3.8655 5.3252 5.8420 7.4705 8.5994 9.4951 11.3043 12.2998
3 2.0684 2.9689 3.8377 5.2755 5.7683 7.3018 8.5477 9.3457 11.3653 12.1999
16 1 2.0697 2.9736 3.8558 5.3081 5.7973 7.2685 8.6271 9.0377 10.8416 12.3923
2 2.0658 2.9680 3.8233 5.2392 5.7193 7.0398 8.4927 8.6436 10.1812 11.7278
3 2.0654 2.9679 3.8216 5.2354 5.7151 7.0231 8.4796 8.6164 10.1096 11.6083
32 1 2.0664 2.9693 3.8302 5.2540 5.7351 7.0835 8.5256 8.7089 10.2809 11.8373
2 2.0654 2.9679 3.8217 5.2353 5.7149 7.0217 8.4776 8.6128 10.0921 11.5631
3 2.0654 2.9679 3.8216 5.2351 5.7147 7.0209 8.4769 8.6116 10.0885 11.5567
64 1 2.0656 2.9682 3.8237 5.2399 5.7198 7.0366 8.4908 8.6342 10.1366 11.6278
2 2.0653 2.9679 3.8216 5.2351 5.7147 7.0209 8.4769 8.6116 10.0886 11.5568
3 2.0654 2.9679 3.8216 5.2351 5.7147 7.0209 8.4769 8.6115 10.0884 11.5565

Single span cables
Table: First 10 natural frequencies of single span cable (AE/mgl=1000)
Researcher Mode number
1 2 3 4 5 6 7 8 9 10
Current study 4.8025 7.4751 10.4182 13.0157 15.8359 18.4249 21.2096 23.7867 26.5644 29.1087
Henghold and Russell 4.8782 7.7553

Single span cables
AE/mgl=1

Single span cables
AE/mgl=1000

Flat cable nets
L
L
L
L
L
LL
L
L
L
2´2 3´3 4´4
5´5 6´6

Flat cable nets
Data Value
Cross-sectional area (A) 322.56 mm2
Elastic modulus (E) 137.895 kN/mm2
Cable’s length (L) 3.048 m
Prestress force (P) 333.673 kN
Self-weight (w) 46.12 kN/m
L
L
L
L
2´2 3´3
C0
C0
C0
C0
C0
C0
C0
C0
C0
C0
C0
C0
C0

Flat cable nets
Table: The ﬁrst 10 natural frequencies of 2×2 ﬂat cable net
Order No. of elements Mode number
(p) (ne) 1 2 3 4 5 6 7 8 9 10
1 12 61.5943 97.3892 97.3892 137.7291 360.9097 360.9097 409.3435 409.3435 695.4976 695.4976
24 59.5562 90.6061 90.6061 123.1886 184.1342 184.1342 191.0658 191.0658 191.4927 191.4927
48 59.0497 88.8907 88.8907 119.1123 172.1924 172.1924 178.0680 178.0680 178.3798 178.3798
96 58.9235 88.4641 88.4641 118.0995 169.2512 169.2512 174.8143 174.8143 175.0977 175.0977
192 58.8920 88.3577 88.3577 117.8470 168.5216 168.5216 174.0059 174.0059 174.2824 174.2824
2 12 58.9280 88.6538 88.6538 119.0457 169.1005 169.1005 174.7843 174.7843 175.0537 175.0537
24 58.8847 88.3474 88.3474 117.8740 169.0818 169.0818 174.7605 174.7605 175.0514 175.0514
48 58.8817 88.3237 88.3237 117.7694 168.3162 168.3162 173.7839 173.7839 174.0588 174.0588
96 58.8815 88.3223 88.3223 117.7633 168.2812 168.2812 173.7396 173.7396 174.0139 174.0139
192 58.8815 88.3222 88.3223 117.7630 168.2792 168.2792 173.7371 173.7372 174.0113 174.0114
3 12 58.8818 88.3282 88.3282 117.8051 169.0808 169.0808 174.7593 174.7593 175.0513 175.0513
24 58.8815 88.3224 88.3224 117.7647 168.2956 168.2956 173.7576 173.7576 174.0320 174.0320
48 58.8815 88.3222 88.3222 117.7630 168.2796 168.2796 173.7377 173.7377 174.0119 174.0119
96 58.8815 88.3222 88.3222 117.7629 168.2791 168.2791 173.7370 173.7370 174.0112 174.0112
192 58.8815 88.3222 88.3222 117.7629 168.2790 168.2790 173.7370 173.7370 174.0112 174.0112

Flat cable nets
Table: The ﬁrst 10 natural frequencies of 2×2 to 6×6 ﬂat cable net
Reseacher Cable model Mode number
1 2 3 4 5 6 7 8 9 10
Current work 2×2 58.8818 88.3282 88.3282 117.8051 169.0808 169.0808 174.7593 174.7593 175.0513 175.0513
3×3 58.8815 90.6725 90.6725 117.7710 117.7710 117.7710 144.8874 144.8874 176.7675 213.9293
4×4 58.8815 91.6137 91.6137 117.7651 123.5273 123.5273 147.2137 147.2137 147.2137 147.2137
5×5 58.8815 92.0917 92.0917 117.7637 126.2845 126.2845 148.2328 148.2328 155.9521 155.9521
6×6 58.8815 92.3694 92.3694 117.7632 127.8269 127.8269 148.7847 148.7847 160.6813 160.6813
Dhoopar et al. 2×2 58.9100 88.3500 88.3500 117.8100 - - - - - -
3×3 58.9100 90.7000 90.7000 117.8100 - - - - - -
4×4 58.9100 91.6400 91.6400 117.8100 - - - - - -
5×5 58.9100 92.1200 92.1200 117.8100 - - - - - -
6×6 58.9100 92.4000 92.4000 117.8100 - - - - - -

Flat cable nets
2 × 2

Flat cable nets
3 × 3

Flat cable nets
4 × 4

Flat cable nets
5 × 5

Flat cable nets
6 × 6

Conclusions and remarks

Conclusions
The application of NURBS-based IGA cable elements to analyze
the cable structures has no restrictions and limitations
For a certain number of refinement cycles, the converged
solutions of displacements and natural frequencies are obtained
with a reasonable number of elements
No particular effort is required since all cables are firstly modeled
in the configuration of straights elements. Higher order basis
function could be applied systematically
Generally, the higher order basis functions can provider better
convergence rate in p-and k-refinement schemes.
The favorable characteristics of the present NURBS-based
isogeometric cable element can overcome drawbacks of the
traditional FEA (complex geometries in cable nets and large sad
cables)

Remarks
The successful application of the proposed NURBS-based IGA
cable element to the static and free vibration analysis of cable
structure once again validate the efﬁciency and the versatility of
the isogeometric analysis approach in solving problems related to
solid and structural mechanics
It is undeniable that cables are merely simple tension structures
having 1D mechanical behavior, hence the advanced features of
IGA approach are not fully employed.
This simple study, however, can be considered as an fundamental
basis for some further potential considerations
The proposed elements can be taken into consideration with other
structural elements to stimulate the whole structures like
suspended cable bridge, stay cable bridges or cable roof systems.

The End
THANK YOU FOR YOUR ATTENTION!

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