267_MS10_ABS_1954

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4

1919

ABSTRACT: This paper compares the effectiveness of different schemes of dynamic relaxation method (DRM) for the analysis of cable and membrane structures. DRM is an iterative process that is used to find static equilibrium. DRM is not used for the dynamic analysis of structures; a dynamic solution is used for a fictitious damped structure to achieve a static solution. The stability of the method depends on the fictitious variables (i.e. mass a damping) and time step. The effect of mass distribution along the structure is also studied in the paper. Eight different schemes DRM will be used in this paper. Schemes A and B are based on the theory of viscous damping. Schemes C, D and E are based on the theory of kinetic damping (KD) with a peak in the middle of the time step and schemes F, G and H are based on the theory of KD with parabolic approximation.

A cable is approximated as a tension bar, a catenary (several tension bars) and a perfectly flexible element. For membrane structures a triangular element is considered. The chosen methods are applied to six constructions. The cable structures are analyzed in Examples 1 to 3, the membrane structures are analyzed in Examples 4 to 6.

The results imply that that it is impossible to determine the best scheme. In this context, it may be noticed that the methods based on kinetic damping appear more stable and faster. For bar element, catenary and cable elements the results confirm that it is beneficial to divide the same amount of mass into all nodes of the structure proportionally to the stiffest node of the solved structure (schemes C and F). For membrane element it is preferred to use the kinetic damping method with the approximation of the kinetic energy peak in the middle of the time step tΔ . KEY WORDS: Dynamic relaxation, Cable structures, Membrane structures, Kinetic damping.

1 INTRODUCTION The load analysis of cable-membrane structures is a geometrical nonlinear problem. For numerical modelling of cable structures can be used idealization of the structure into the elements and nodes. The surface of membrane structure is discretized into a system of joints and triangle membrane elements. The edges of the triangle form the connection between the joints and they are called links.

The joints can be divided into two groups – supported or unsupported ones. The equilibrium of position unsupported nodes loaded by the nodal loads can be searched iteratively. Large displacements of the structure and small deformation of elements are considered.

Several methods exist to solve these structures. The dynamic relaxation method (DRM) will be examined in this paper. The stability of the method depends on the fictitious variables (i.e. mass a damping) and time step. This paper compares the effectiveness of different schemes of DRM for the analysis of cable and membrane structures. The effect of mass distribution along the structure is also studied in the paper. This paper develops papers [1] and [2].

2 ELEMENTS A cable can be approximated as a tension bar, a catenary (several tension bars) and a perfectly flexible element (where bending moments to zero). Homogeneous material with a constant cross-section throughout its length is assumed in all cases. For membrane structures a triangular element is considered.

2.1 Tension bar The bar connects the endpoints and carries only positive normal force. The internal force T (normal force) in one bar element can be calculated according to the well-known Equation (1).

( )00

srsEAT −= , (1)

where:

E is the Young’s modulus of elasticity, A is the cross-sectional area, r is the distance between two end joints in the chord

direction (current length), 0s is the un-elongated length of element (slack length).

If the force T is negative, then it is equal to zero. The deadweight of strut has been assumed to be concentrated equally at its two end joints. The used bar element can be seen in Figure 1.

Figure 1. Bar element.

Analysis of cable – membrane structures using the Dynamic Relaxation Method

M. Hüttner1, J. Máca2, P. Fajman3

1Dep. of Mechanics, Faculty of Civil Eng., Czech Tech. University in Prague, Ing. Miloš Hüttner, Prague, Czech Rep. 2 Dep. of Mechanics, Faculty of Civil Eng., Czech Tech. University in Prague, Prof. Jiří Máca, CSc., Prague, Czech Rep.

3Dep. of Mechanics, Faculty of Civil Eng., Czech Tech. University in Prague, Doc. Petr Fajman, CSc., Prague, Czech Rep.

email: [email protected], maca@ fsv.cvut.cz, fajman@ fsv.cvut.cz

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

1920

2.2 Catenary The basic assumption of this theory is that the behaviour of a cable can be approximated by a few bars. These bars are interconnected by joints and sustain only positive normal force. The behaviour of individual bars is described in Chapter 2.1. As found in [3], five bars are well enough to describe correctly the characteristics of the cable.

2.3 Cable element The basic assumption of the analysis of a flexible elastic cable is that the cable is regarded to be perfectly flexible and is devoid of any flexural rigidity. Load on a cable, which must include at least self-weight, is distributed uniformly along the curve of the cable which is assumed to be a parabola. The detailed analysis can be found in [4] and [5].

It is necessary for the aim of the study to use an internal force T , which is always positive and the importance of which is shown in Figure 2.

Figure 2. Cable element.

Force T can be calculated iteratively from Equation (2).

( ) ( )

012

81

4

2ln2ln2

),,,,,(

2

222

0

2

0

=⎟⎟⎠

⎞⎜⎜⎝

⎛++−

−−++−+

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−−⎟

⎟⎠

⎞⎜⎜⎝

⎛++−=

=

TrQ

rc

rl

EAT

sbaT

barQc

lTa

lTrQ

lc

lTb

lTrQ

lc

rQTl

QsclrTg

(2)

where: E is the Young’s modulus of elasticity, A is the cross-sectional area, r is the distance between two end joints in the chord

direction (current length), 0s is the un-elongated length of element (slack length).

l is the horizontal distance between the two end joints, c is the vertical separation between joint j and joint i

(can be negative), Q is the resultant of the vertical uniform load q acting

vertically the entire length of parabolic curved cable, while 0qsQ = .

For reasons of clarity, Equation (2) introduces two more substitutions:

crQTTlTcrQa 444 222222 +++= , (3)

crQTTlTcrQb 444 222222 −++= . (4)

2.4 Membrane element For a membrane structures the natural stiffness element can be used for calculation of internal forces. The original formulation of the natural stiffness element is credited to Argyris [6] but the formulation here follows the work of Barnes [7] and Topping [8]. For the formulation of the natural stiffness element a triangular element is considered. This element has only in-plane stiffness so the element formulation is with respect to displacements in the local coordinate directions. Using equations of equilibrium, it is possible to convert the surface stress within the element into forces along the sides of the triangle. General application of this element is described e.g. in [8].

In this case the idealization of a typical element is as shown in Figure 3 where the local coordinate system is conveniently chosen such a way that the axis coincides with the first side. The stresses in the element with respect to x′ and y′ directions, with z′σ equal to zero, are the standard plane stress formulation for an isotropic material [9].

Figure 3. Membrane element.

In Figure 3 is:

1l , 2l , 3l is the length of the edge 1, 2, 3

2θ is the inclination of the edge 2 to the local x′ axis,

3θ is the inclination of the edge 3 to the local x′ axis. The initial forces 1T , 2T and 3T of sides 1, 2 and 3 are

defined for membrane element as:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

´

´

´

22

33

23323223

03

02

01

3

2

1

0

0

1

/1000/1000/1

xy

y

x

Qb

Qc

Qb

Qc

Qbaba

Qcaca

ll

lAd

TTT

τσσ

(5)

and

( ) ( )

( ) ( )( )

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔ

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

−−

−−

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

3

2

1

3

2

3

3

3

2332

2

2

2

3

2

3223

1

22

22

´

´

´

001

1200

011

011

lll

Qlb

Qlb

Qlbaba

Qlc

Qlc

Qlcaca

l

E

EE

EE

xy

y

x

ν

ννν

νν

ν

τσσ

(6)

where:


1921

A is the area of the membrane element, d is the thickness of the membrane, E is the Young’s modulus of elasticity, ν is the Poisson’s ratio,

01l is the length of the edge 1 of the unloaded element, 02l is the length of the edge 2 of the unloaded element, 03l is the length of the edge 3 of the unloaded element,

1lΔ , 2lΔ , 3lΔ is the elongation of the edge 1, 2 and 3 Furthermore, they are used substitutions (for h = 1, 2, 3):

hha θ2cos= , (7)

hhb θ2sin= , (8)

hhhc θθ cossin= , (9)

and

2332 cbcbQ −= . (10)

The direct stiffness hS of side h ( h =1, 2, 3) is defined for membrane element as:

( )

202

3

h

h

l

EAdS ⋅= . (11)

3 DYNAMIC RELAXATION The dynamic relaxation method (DRM) is an iterative process that is used for the static analysis of structures. DRM is not used for the dynamic analysis of structures; a dynamic solution is used for a fictitious damped structure to achieve a static solution.

The theory of this method was first described by Day [10]. During several years, the DRM have been improved progressively. The kinetic damping technique was suggested by Cundall [11]. Topping [8] and Lewis [12] also contributed to the kinetic damping method. Practical examples of the application can be seen in [1,2,3,12,13].

3.1 Principle The basic unknowns are nodal velocities, which are calculated from nodal displacements. The discretization from timeline with time step tΔ will be performed. During the step tΔ a linear change of velocity is assumed. The acceleration during the step tΔ is thus considered to be constant. By substituting the above assumptions the velocity for joint i in direction m ( x , y and z ) can be expressed in a new time point ( t + tΔ /2) thus:

2//2//

2//)2/()2/(

imim

tim

imim

imimttim

ttim CtM

RCtMCtM

vv+Δ

++Δ−Δ

= Δ−Δ+ ,(12)

where: timR is the residual force at the nodal point i , in the

direction m and at the time t , imM is the fictitious mass at the nodal point i and in

the direction m ,

imC is the fictitious damping factor for the nodal point i and in the direction m ,

)2/( ttimv Δ+ is the velocity at the nodal point i in the direction

m and at the time t . The current coordinates of the nodal point i at the time

instant ( t + tΔ ) may then be expressed as follows:

)2/()( ttix

tti vtx Δ+Δ+ ⋅Δ= . (13)

Similarly, equations may be written for the y and z coordinate directions. From the imbalance (between external and internal forces) in the node i , we may calculate the residual forces t

imR ( m = x , y , or z ) for the corresponding node at the time t .

∑

−−=

kt

k

tj

tit

kixtix r

xxTPR

∑

−−=

kt

k

tj

tit

kiytiy r

yyTPR

∑∑−

−+=k

tk

tj

tit

kk

kiz

tiz r

zzTQPR

2, (14)

where: k is the index of the link (element or edge)

entering the nodal point i . j is the second endpoint on the link k .

ixP is the external load at the nodal point i in the direction x ,

iyP is the external load at the nodal point i in the direction y ,

izP is the external load at the nodal point i in the direction z ,

kQ is the resultant of the vertical uniform load for each link k ,

tix , t

iy , tiz are the current coordinates of the nodal point i ,

tkr is the distance between two end joints for each

link k . The internal force t

kT for each link k can be calculated from Equation (2) – for the cable element; from Equation (1) – for the bar element; and from Equations (5) – for the edge of membrane element.

If the sum of the forces tkT in the link k is less than zero so

it must be set equal to zero. In order to start calculations, the velocity at the time point tΔ /2 must be calculated. Using the initial conditions for the

time t = 0 where 00 =imv , we obtain:

0)2/(

2 imim

tim R

Mtv Δ

=Δ , (15)

where 0imR are residual forces at the time t = 0.


1922

At each time instant, it is also possible to calculate the kinetic energy kinU throughout the structure:

( )2)2/()2/(kin

ttim

n

i

p

mim

tt vMU Δ+Δ+ ∑∑= , (16)

where n is the number of joints and p is the number of dimensions (3D or 2D).

3.2 Schemes Eight different schemes DRM will be used in this paper. Schemes A and B are based on the theory of viscous damping [10]. Schemes C, D and E are based on the theory of kinetic damping (KD) with a peak in the middle of the time step [8] and schemes F, G and H are based on the theory of KD with parabolic approximation [12].

3.2.1 Scheme A Discretized mass M is chosen in this scheme the same for each node and all directions from follows Equation (17):

( )imStM max2

2Δ= , (17)

where imS is the largest direct stiffness of the i -th joint in the m direction.

The stiffness kS of each link k (entering into joint i ) is for the element (cable or bar) represented by two components – namely, the geometric stiffness G

kS and the elastic stiffness EkS .

k

k

k

kkGk

Ekk r

Ts

AESSS +=+=0

. (18)

Stiffness kS 2D membrane element should be determined for each edge of the equation (11).

Hence, the following Equation (17) applies to the stiffness of the node imS :

∑=k

mkim SS , , (19)

where mkS , is the stiffness kS - from Equation (18), respectively from Equation (11) - distributed in the m direction.

Viscous damping coefficient C for the whole structure is calculated using the coefficient of critical damping [12]:

“The iterative algorithm converges fastest when using the critically damped mode. In an undamped mode, the structure will oscillate around its position of equilibrium, and the viscous damping coefficient, known as critical damping, may be found from Equation (20):

tN

MMSC im

imim Δ=⋅=

π42 , (20)

where N denotes the number of iterations required to complete one cycle of oscillation. It may now be seen that in order to obtain the value of the viscous damping coefficient, an additional computer run is necessary, with C set to zero.”

3.2.2 Scheme B In this scheme, the fictitious values iM and iC are calculated for each joint i separately.

The discretized mass iM for each joint i is calculated from Equation (21):

ii StM2

2Δ= , (21)

where

( )iziyixi SSSS ,,max= . (22)

The viscous damping coefficient iC for the nodal point i is calculated as follows:

t

MMSC iiii Δ

⋅=⋅= 82 (23)

3.2.3 Scheme C This scheme is based on the theory of KD with a peak in the middle of the time step [8].

“When the technique of kinetic damping is employed, the viscous damping coefficient is taken as zero. The system is brought to rest by following a process stopping the iterations, whenever a peak in the kinetic energy of the entire system is detected, and then restarting the computation from the current configuration, but with zero initial velocity [12].”

The coordinates are set to )2/( ttix Δ− when the peak is

assumed to have occurred. Then

)2/()2/(

2tt

ixti

tti vtxx Δ−Δ− ⋅

Δ−= . (24)

Similarly, equations may be written for the y and z coordinate directions.

The mass for whole structure is calculated from Equation (17).

3.2.4 Scheme D This scheme is similar to the Scheme C, but the discretized mass iM for each joint i is calculated separately from Equation (21).

3.2.5 Scheme E This scheme is similar to Scheme D but masses iM are recalculated after each restart of the kinetic energy.

3.2.6 Schemes F, G and H These schemes are similar to Scheme C (respectively D and E) but there is used the theory of KD with parabolic approximation [12].

The trace of kinetic energy near the peak can be approximated by a parabolic curve. The coordinates are set to

)( ttix Δ−β when the peak is assumed to have occurred. Then

)2/()( ttix

ti

tti vtxx Δ−Δ− ⋅Δ−= ββ , (25)

where


1923

123

23

2 KEKEKEKEKE+−

−=β , (26)

and where ( )2/kin3

ttUKE Δ+= is the kinetic energy at the time

point ( 2/tt Δ+ ), ( )2/kin2

ttUKE Δ−= , ( )2/3kin1

ttUKE Δ−= .

4 EXAMPLES The chosen methods are applied to six structures. The cable structures are analyzed in Examples 1 to 3, the membrane structures are analyzed in Examples 4 to 6. The initial geometry (no internal stress) is evident from individual figures.

The calculations were terminated when the kinetic energy of the structure was less than 1.10-6 kJ and while the residual forces of all the degrees of freedom were less than the limit value limR (defined individually for each example). The maximum number of iterations is chosen according to the elements – max. 500000 iterations for bar elements max. 5000 iterations for cable and membrane elements. The time step is chosen tΔ = 1 s in all calculations.

Self-created scripts in MATLAB 7.14.0.739 (2012a) was used for all calculations. The calculations were carried out on the computer ASUS processor AMD E-450 APU – 1.65 GHz, memory 4GB RAM.

The results of the calculations (the number of iterations and the CPU time) are presented in Tables 1 to 9.

4.1 Example 1 A suspended cable ring shown in Figure 4, which has been discussed in [3], is analysed here to show the accuracy and speed of the computations developed in this paper.

Figure 4. A plan view of Example 1.

The structure consists of 16 cables connected to 16 joints (1-8 are free, A-H are fixed) with an inner radius of 35 m and outer radius of 75 m. The structures have 8 radial cables and 8 tangential cables. All cables have the same cross-sectional area A = 1.96344·10-3 m2 and the same Young's modulus E = 170 GPa. The slack length 0s of all radial cables is 40 m, and that of all ring cables is 32 m. Any external node loads

acts on the structure. An uniform load q = 1.5105·10-1 kN/m acts on each cable. The coordinates z of the unsupported nodes were always set to zero.

The limit value of a residual force is limR = 0.01 kN. The accuracy of the calculation is approximately ± 1 cm.

Table 1. The number of iterations – Example 1.

scheme/element bar catenary cable viscous damping

A 1420 - 1266 B 128759 - -

kinetic damping Δt/2

C 278 39894 799 D 484 - 913 E 546 14874 1591

kinetic damping βΔt

F 270 114559 777 G 385 - 904 H 535 16680 1554

Table 2. Time of solution (CPU time in seconds) – Example 1.


A 0.72 - 306.20 B 52.80 - -


C 0.28 63.60 249.20 D 0.30 - 278.90 E 0.36 28.80 527.30


F 0.28 184.80 247.50 G 0.30 - 275.20 H 0.38 36.20 539.50

4.2 Example 2 – Hypar net The example is taken from Lewis [12]: “Two sets of straight

line cables generate a model of a hyperbolic paraboloid surface, shown in Figure 5.

Figure 5. Topology of Hypar net.

The structure has 36 degrees of freedom. The load izP of 0.0157 kN is applied at all internal nodes, except for nodes 17, 21, and 22. The cross-sectional area of the cables is 0.785 mm2 and Young’s modulus is 124.8 kN/mm2. The pre-tension force in all cables is 0.2 kN. The previous analysis and experimental measurements of the same structure are reported in [14].”

The limit value of a residual force is limR = 0.01 kN. The accuracy of the calculation is approximately ± 1 mm.


1924



A 177 1042 135 B 8889 - 5920


C 214 1176 149 D 197 - 139 E 376 3078 237


F 215 1161 145 G 209 - 165 H 252 2894 234



A 0.12 3.40 52.00 B 6.90 - 3246.70


C 0.15 4.10 55.80 D 0.14 - 52.80 E 0.32 12.10 88.40


F 0.15 3.73 55.00 G 0.15 - 66.90 H 0.32 11.39 86.90

4.3 Example 3 – Tram stop Furthermore, one real structure was tested, see Figure 6.

Figure 6. Barrandov tram stop.

It is a cable-membrane structure with a central symmetry, only one fourth of the structure was subjected to modelling. The canvas was represented by forces exerted by the structure’s own weight and prestressi in the model. The background for the creation of the model was the geodetic survey of the existing structure made in November 2013, see Figure 7. The measurement of forces in cables has been discussed in [15].

The anchoring of the cables into anchor blocks was modelled as a fixed support. The point where the cables were anchored into anchor blocks was modelled as a sliding joint allowing the joint’s motion only in the direction of the x and y axes. The structural scheme of the structure considered is evident from Figure 8.

The parameters of cables are as follows: cables 1 – 7: E = 140 GPa, A = 230 mm2, q = 0.021 kN/m cables 8 – 30: E = 140 GPa, A = 180 mm2, q = 0.017 kN/m.

The following cable lengths (selected cables) were considered: cable 1 – 11.221 m, cable 2 – 13.618 m, cable 3 – 14.340 m, cable 4 – 13.020 m, cable 5 – 14.530 m, cable 6 – 13.820 m.

The load is applied at joints 18 and 19 where the forces exerted by the canvas act. These forces amount to:

zP18 = 12 kN and 19P = 16 kN. The limit value of a residual force is limR = 0.1 kN. The

accuracy of the calculation is approximately ± 5 cm.

Figure 7. The perspective view of Barrandov tram stop.

Figure 8. A plan view of a tram stop.



A 488 11461 448 B 1353 - 1290


C 136 2767 120 D 115 - 121 E 309 5320 301


F 125 2880 118 G 130 - 135 H 332 4446 313


1925



A 0.36 38.60 175.40 B 1.11 - 475.10


C 0.12 9.72 59.60 D 0.11 - 60.40 E 0.45 23.56 130.20


F 0.10 10.00 58.20 G 0.12 - 64.20 H 0.33 19.81 132.30

4.4 Example 4 It is a membrane structure with 6 nodes (of which 2 are unsupported) interconnected with 4 membrane elements. This structure is shown in Figure 9. The load izP = 3 kN acts on both unsupported joints. The parameters of membranes are always E = 500 MPa, d = 1 mm and ν = 0.3.


Figure 9. Topology and initial geometry of Example 4.

Table 7. The number of iterations and time of solution (CPU time in seconds) – Example 4.

scheme number of iterations

time of solution

viscous damping

A 66 0.32 B 981 2.45


C 42 0.27 D 42 0.27 E 43 0.28


F 24 0.21 G 24 0.20 H 26 0.18

4.5 Example 5


It is a membrane structure with 15 nodes (of which 11 are unsupported) interconnected with 16 membrane elements. This structure is shown in Figure 10. The load P = 0.75 kN. The parameters of membranes are always E = 500 MPa, d = 1 mm and ν = 0.3.




time of solution

viscous damping

A 224 2.82 B 3285 34.20


C 231 2.35 D 113 1.14 E 189 1.92


F 249 2.81 G 114 1.30 H 199 2.34

4.6 Example 6 It is a membrane structure with 65 nodes (of which 57 are unsupported) interconnected with 96 membrane elements. The topology and initial geometry of this structure is shown in Figure 11.


The load izP = 10 kN for internal joints and izP = 5 kN for all external joints. The parameters of membranes are always E = 500 MPa, d = 1 mm and ν = 0.3.

The limit value of a residual force is limR = 1 kN. The accuracy of the calculation is approximately ± 2 cm.


scheme number of

iterations time of

solution viscous damping

A 617 33.90 B 7065 345.60


C 449 20.70 D 781 39.00 E 1310 62.80


F 1427 66.20 G 1641 76.10 H 1222 59.00


1926

5 FINAL COMMENTS The overall ranking of methods sorted by the number of errors (sum of all examples), the total number of iterations and the total CPU time are shown in Tables 10 to13.

The results imply that that it is impossible to determine clearly the best scheme. In this context, it may be noticed that the methods based on kinetic damping appear more stable and faster, which confirms the conclusions presented in [1,2,13]. For bar element, catenary and cable elements the results confirm that it is beneficial to divide the same amount of mass into all nodes of the structure proportionally to the stiffest node of the solved structure (schemes C and F). For membrane element it is preferred to use the kinetic damping method with the approximation of the kinetic energy peak in the middle of the time step tΔ .

6 CONCLUSIONS It may be concluded that the Scheme C based on kinetic damping with a peak in the middle of the time step and the equal mass divided into all nodes proportionally to the stiffest node has proved the most comprehensive results.

Table 10. The summary of results – bar element.


time of solution

errors rank

viscous damping

A 2085 1.20 0 7 B 139001 60.81 0 8


C 628 0.55 0 2 D 796 0.54 0 4 E 1231 1.13 0 6


F 610 0.53 0 1 G 724 0.57 0 3 H 1119 1.03 0 5

Table 11. The summary of results – catenary.


time of solution

errors rank

viscous damping

A 12503 41.95 1 5 B 0 0 3 8


C 43837 77.42 0 3 D 0 0 3 8 E 23272 64.46 0 1


F 118600 198.55 0 4 G 0 0 3 8 H 24020 67.40 0 2

Table 12. The summary of results – cable element.


time of solution

errors rank

viscous damping

A 1849 533.65 0 5 B 7210 3721.75 1 8


C 1068 364.60 0 2 D 1173 392.12 0 3 E 2129 745.88 0 7


F 1040 360.81 0 1 G 1204 406.23 0 4 H 2101 758.71 0 6

Table 13. The summary of results – membrane element.


time of solution

errors rank

viscous damping

A 907 37.04 0 2 B 11331 382.25 0 8


C 722 23.32 0 1 D 936 40.41 0 3 E 1542 65.00 0 5


F 1700 69.22 0 6 G 1779 77.60 0 7 H 1447 61.52 0 4

ACKNOWLEDGMENTS The results presented in this paper are outputs of the research project “P105/11/1529 - Cable - membrane structures analyses” supported by Czech Science Foundation and project “SGS14/029/OHK1/1T/11 - Advanced algorithms for numerical modelling in mechanics of structures and materials” supported by the Czech Technical University in Prague.

REFERENCES [1] M. Hüttner, J. Máca and P. Fajman, Analysis of Cable Structures using

the Dynamic Relaxation Method, in B.H.V. Topping, P. Iványi, (Editors), Proceedings of the Fourteenth International Conference on Civil, Structural and Environmental Engineering Computing, Civil-Comp Press, Stirlingshire, UK, Paper 145, 2013. doi:10.4203/ccp.102.145

[2] M. Hüttner, J. Máca, Membrane structures - dynamic relaxation, Proceedings of the 4th Conference Nano & Macro Mechanics. Prague, ČVUT, 2013, pp75-82. ISBN 978-80-01-05332-4.

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