River Crossing

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

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ABSTRACT: A challenging design of a transmission line (TL) is presently being carried out in Brazil: a double circuit 500kV TL in the Amazonian region. In addition to the significant length, over 1400 km in the rain forest, the design has to cope with large river crossings and severe environmental constrains. Some of these crossings demand single spans more than 2000 meters long. The crossing of the Trombetas River is one of the most important, having a total length of 5100 meters. The proposed design has located a tower on an island. Each of the two main spans is approximately 1600m long and demands two latticed steel suspension towers 190m high and a third 120m high. Moreover, these towers must be erected on concrete foundations that reach 10m above terrain level, to maintain the base of the towers above maximum river flood level. The present work describes the structural analysis of the special TL towers for this crossing. The focus is on numerical predictions of the dynamic response of the 190m high latticed steel structure subjected to cable rupture (conductors and shield wires). The entire crossing section is modeled including the two highest towers and all other elements: foundations, conductor cables, shield wires and insulator strings. To perform the analysis, direct explicit numerical integration of the equations of motion in the time domain is adopted, using the central finite differences scheme. The responses obtained by such approach are then compared to responses determined by standard practice. An additional feature of this study was the consideration of the dynamic structural interaction between the structure and the foundation (soil-structure interaction), which attained particular importance in view of the soft soil foundation at the site.

KEY WORDS: Dynamic analysis; Latticed steel towers; Crossing towers; Transmission line structures; Cable rupture; Finite central differences; Soil-structure interaction.

1 INTRODUCTION The 500kV Transmission Line (TL) from the Tucuruí hydroelectric power plant to Macapá and Manaus, within the brazilian amazonic region, presents great engineering challenges, such as large river crossings and very severe environmental constrains. In addition, during the design stage the available information on the foundation soil and local geology presented large uncertainties. In this context it was decided to resort to towers about 190m high in order to attain spans as long as 2000m. Obviously these structures demanded a detailed assessment, mainly about the structural response of the towers due to dynamic loads.

The present paper aims at describing the evaluation of the dynamic response of the main steel tower for the Trombetas River crossing for cable rupture, which is one of the loading cases considered in design. The studies were carried out through the analysis of an entire section of the transmission line (towers, cables and insulator strings), representing cables and structures by means of truss elements and solving the resulting equations of motion by direct explicit numerical integration. This methodology was programmed through software developed by the authors.

In summary, the paper describes in detail the determination of the dynamic response of tower GTS 01 subjected to cable rupture, i.e., the evolution with time of displacements at the top of the tower and axial forces in structural elements. Peak values are compared with the response obtained through conventional TL design methods.

2 DESCRIPTION OF THE CROSSING AND THE STRUCTURAL SYSTEM

2.1 Crossing over the Trombetas River The crossing TL over the Trombetas River is of the A-S-S-S-A type, in other words, it is composed of anchor towers at both ends (GTA 00 and GTA 01) and a central section with three suspension towers (GTS 00, GTS 01 and GTS 02), as shown in Figure 1.

Figure 1. View of the crossing over the Trombetas River.

Dynamic analysis due to cable rupture of a transmission line tower for large river crossing in Brazil considering the soil-structure interaction

João Kaminski Jr.1, Leandro F. Fadel Miguel2, Jorge Daniel Riera3, Ruy Carlos Ramos de Menezes3, Letícia F. Fadel Miguel3

1Department of Structures and Civil Construction, Federal University of Santa Maria, Santa Maria, Brazil 2 Department of Civil Eng., Federal University of Santa Catarina, Florianópolis, Brazil

3 Department of Civil Eng., Federal University of Rio Grande do Sul, Porto Alegre, Brazil email: [email protected], [email protected], [email protected], [email protected], [email protected]

Tower GTS 01 Tower GTS 00

Tower GTS 02

Towers GTA 01

Towers GTA 00

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

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The main spans of the Trombetas River crossing design are 1598m and 1590m long, while the suspension towers should have useful heights equal to 190m and 119m. Two crossing towers (GTS 00 and GTS 01) should have their foundations lifted up about 10m, due to the elevation of the river level during the flooding season.

2.2 General considerations Differently from static analysis, the mass distribution of a structure plays a fundamental role in its dynamic analysis. Therefore, especial attention was devoted to the correct determination of the masses in the computational model of tower GTS 01. For instance, the masses of main bars are automatically calculated and assigned by the program to the nodes of the model. The mass of secondary bars, which are often introduced just for bracing main bars but do not carry loads in a linear analysis and hence need not be included in the model, were calculated and distributed manually. Additional masses, for example applied loads due to bolts, steel plates, galvanization and equipment, were carefully calculated and lumped at the corresponding nodal points of the model.

Regarding structural damping, it is known that energy dissipation in steel lattice towers increases with the vibration amplitude. Limited experimental evidence suggests critical damping ratios around 10% for large response amplitudes [1]. In this paper, the suggested 10% value was adopted.

2.3 Considerations on the foundations During the first design stage the available information on local geology presented large uncertainties, indicating soils with low support capacity. Additionally, due to the elevation of the river level during the flooding season, the crossing towers (GTS 00 and GTS 01) should have their foundations lifted up about 10m. For these reasons, the traditional consideration of structures erected over a rigid base becomes unviable.

Within this context, the study was conducted through different stages. Initially, aiming to assess the influence of dynamic modeling on displacements at the top of the tower, axial forces in structural elements and support reactions, the obtained results were compared to those provided by a conventional static analysis, with a rigid foundation, i.e., with undisplaceable supports. After, the simulations were carried out considering a foundation with an expected flexibility for the analyzed structure and, finally, considering the possibility of a floating foundation. The stiffness coefficients were determined from similar TL structure solutions already executed, as it is explained.

The expected flexibility was determined considering that each tower leg is supported by 10m-high tubular concrete columns with an outside diameter of 2.50m and thickness of 0.10m. Assuming a Young Modulus of Ec = 25×109N/m2, the stiffness coefficients would be:

Horizontal in the direction normal to the LT: kx = 4.08×107N/m; Horizontal in the direction of the LT: kz = 4.08×107N/m; Vertical direction: ky = 4.08×108N/m.

Observing that the stiffness coefficient in vertical direction is equal to ky = 1.88×109N/m. Therefore, the adopted coefficient above represents an estimated lower limit, considering the connections with base and tower.

Assuming that the flexibility of the foundations would present the same stiffness as the tubular concrete columns, the total coefficients would be:


With regard to the floating foundation, if a floating base

area of 100m2 was admitted, this would result in a lower limit vertical stiffness equal to ky = 1.0×106N/m, which is considered extremely low. Consequently, stiffness coefficients to the floating foundation were taken:


Starting from these coefficients, the support reactions were

evaluated, through a full dynamic simulation. After the comparison to the static analysis, maximum values were used and the real foundations could be designed, providing real stiffness coefficients.

The design of the foundations determinates the use of a concrete block with 7.0 x 7.0 x 2.60m in each tower leg, which is supported by nine inclined tubular metallic piles with outside diameter of 1.0m and thickness of 12.5mm. Through force vs. displacements tables, showed in Table 1, the real stiffness coefficients were able to be evaluated.

Table 1. Force vs. displacements for real foundations.

Compression Tension

Force (kN)

Displacement (m)

Force (kN)

Displacement (m)

X 2119.76 0.06260 -1191.21 -0.04732

Y 15788.32 0.01117 -4978.54 -0.00125

Z -2124.98 -0.05132 1229.73 0.03266

The stiffness coefficients to the real foundations are: Horizontal in the direction normal to the LT: kx = 3.452×107N/m; Horizontal in the direction of the LT: kz = 3.452×107N/m; Vertical direction, for tension: ky = 3.983×109N/m; Vertical direction, for compression: ky = 1.413×109N/m.


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Note that the horizontal coefficients kx and kz were taken as the average between the values presented in Table 1, for tension and compression in both directions.

3 DESIGN PROCEDURES FOR CABLE RUPTURE Usual design procedures of TL structures consider all acting dynamic loads, such as wind or cable rupture, by means of equivalent static loads. Specifically in connection to cable rupture, in usual design practice the load is applied directly on the tower, in the longitudinal direction of the TL, with a magnitude equal to the residual static load subsequent to the cable failure. For conductor cables, this magnitude is around 80% to 85% of the EDS condition (Every Day Stress).

In the case of the crossing section on the Trombetas River, the conductor cables were designed for a tension equal to 22% of its capacity (UTS - Ultimate Tension Stress). Therefore, the magnitude of the load that should be applied on the GTS 01 tower, in the longitudinal direction, must be around 18% of its UTS, jointly with other relevant loads in the vertical direction due to dead weight of the tower, equipment, conductor cables that did not break and shield wires.

4 SOLUTION METHOD To perform the dynamic analysis, direct explicit numerical integration of the equations of motion in the time domain was adopted, using the central finite differences scheme, because it does not require assembling or updating the system global stiffness matrix. Integration is accomplished at element level, which constitutes an advantage in non-linear problems. When the system mass and damping matrices M and C are both diagonal, the method becomes explicit and the expression in central finite differences for the displacement at any node in either the x, y or z direction, at time t + Δt, may be written as:

( )

( )( ) ( ) ( ) ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡⋅Δ⋅−−⋅+

⋅⋅

⋅Δ⋅+

=

−

+

Δttmt

2t

mΔtt

q t/2 c1 q 2 m

Δt f

t/2 c1

1 q

(1)

in which q denotes the nodal coordinate in either the x, y or z direction, f(t) the resultant nodal force component in the corresponding direction at time t, cm = c/m is a constant, m the nodal mass and c the nodal damping coefficient, assumed proportional to mass m. The resultant nodal force f(t) consists of gravitational forces (dead weight and external nodal forces), and axial forces in the truss elements. It is important to quote that geometrical non-linearity is always considered, since the nodal coordinates are updated after each integration step Δt.

Convergence and accuracy of the solution depend basically on the integration time interval Δt. Since the method is only conditionally stable [2], it is necessary that Δt ≤ Δtcrit. For latticed structures, the critical time interval Δtcrit can be estimated by [3]:

ρE

L t Δt (0)min

crit =Δ≤ (2)

in which Lmin(0) is the initial length (in t = 0) of the smallest truss element, E is the elastic modulus and ρ is the material mass density. Additional details about the integration method applied to dynamic analysis of TL towers and cables can be found in [4], [5], [6] and [7].

5 MECHANICAL MODEL FOR THE DYNAMIC ANALYSIS

5.1 Crossing over the Trombetas River To evaluate the dynamic response of the GTS 01 tower subjected to cable rupture, a mechanical model with the entire crossing section over the Trombetas River was modeled, including the two highest towers (GTS 01 and GTS 02), conductor cables, shield wires, insulator strings as well as the foundations. Such model with all elements is presented in Figure 2.

Figure 2. Mechanical model of the crossing section on the Trombetas River.

The insulator strings for each conductor cables bundle in the

GTS towers are double, as showed in Figure 3. The length of all insulator strings in the GTS towers is 7.15m.

Figure 3. Detail of GTS 01 tower in the mechanical model.

The crossing section on the Trombetas River has a total length exceeding 5100m. The model presents the following spans: 1037.71m between the anchor GTA 00 towers and the GTS 00 suspension tower , 1598.0m between the GTS 00

Tower GTS 01

Tower GTS 02

Location of the Tower GTS 00

Location of Towers GTA 01

Location of Towers GTA 00


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suspension tower and the GTS 01 suspension tower, 1590.0m between the GTS 01 suspension tower and the GTS 02 suspension tower and finally 961.61m between the GTS 00 suspension tower and the GTA 01 anchor towers . The conductor cables used in the crossing section are bundles with four AACSR 535/240 cables (AACSR - Aluminum Alloy Conductor Steel Reinforced). Each cable has 775.06mm2 total cross sectional area (aluminum alloy + steel). The shield wires are OPGW type (OPGW - Optical Fiber Composite Overhead Ground Wire) with 349.14mm2 cross sectional area. Other properties of the conductor cable AACSR 535/240 and of the shield wire OPGW are presented in Table 2 and Table 3, respectively.

In the mechanical model, the bundles were replaced by a single cable element, with outside diameter, cross section area, tension capacity and unit weight equal to four times the values presented in Table 2.

Table 2. Properties of the AACSR 535/240 conductor cable.

External diameter of the conductor cable 36.21 mm 0.03621 m

Cross sectional area (aluminum alloy) 535.70 mm2 535.70 × 10-6 m2

Cross sectional area (steel) 239.36 mm2 239.36 × 10-6 m2

Total cross sectional area (aluminum alloy + steel)

775.06 mm2 775.06 × 10-6 m2

Tension capacity of the conductor cable 49950.0 daN 499500 N

Unit weight of the conductor cable 3.464 daN/m 34.64 N/m

Elastic modulus in tension 94.50 daN/mm2/100 9.45 × 1010 N/m2

Table 3. Properties of the OPGW shield wire.

External diameter of the shield wire 24.30 mm 0.0243 m

Cross sectional area 349.14 mm2 349.14 × 10-6 m2

Tension capacity of the shield wire 39768.76 daN 397687.6 N

Unit weight 2.2563 daN/m 22.563 N/m

Elastic modulus in tension 129.85 daN/mm2/100 12.98 × 1010 N/m2

5.2 Constitutive law of conductor cables and shield wires Cables are formed by the association of threads, able to carry only tensile forces. In this paper, a linear model is used to calculate cables sags, elongations and tensions, i.e., the cable stress-strain diagram, at constant temperature, is a straight line. The following constitutive law was adopted for conductor cables and shield wires in tension:

OCCCCC L / L A E F Δ= (3)

in which AC denotes the cross sectional area of the cable element (m2), equal for conductor cables to the total area (aluminum alloy + steel) and for shield wires to the steel area; EC the elastic modulus in tension (N/m2); FC the tension force in the cable element (N); ΔLC the elongation of the element (m) and LOC the unstressed length of the cable element (m).The values used in Equation (3) to calculate the tension forces in the cable elements are presented in Tables 1 and 2. Suspended cables in TL present the form of a catenary. In the condition EDS, the conductor cables AACSR 535/240 used in the crossing section were designed for a tension of 22% of its capacity (UTS - Ultimate Tension Stress). The shield wires were designed for maximum sag equal to 90% of the conductor cables maximum sag, resulting in a tension around 20% of the shield wires UTS.

When the suspension points of the cable have the same height, the catenary is symmetrical in relation to the center of the span (central axis), where the vertex is located, i.e. the point where the maximum sag occurs. In the case of supports with different heights, the catenary is not symmetrical and the maximum sag fe does not occur at the center of the span, as shown in Figure 4. The sag depends on the span length, on the temperature and on the tension in the cable when it is fixed at the supports.

Figure 4. Suspended cable between supports “1” and “2” with different heights (B ≠ 0).

At the beginning of the analysis (initial condition, t = 0s) the

cable should be in a position such that, after the application of dead loads, it is subjected to the design tension force, equivalent to a percentile of the tensile strength of the cable, with the theoretical catenary (ftheoretical) and the maximum sag (fe). The formulation used to determine the theoretical catenary, the maximum sag, the position of the maximum sag (x0) and the theoretical length of the cables is described by [6]. Additional details are given by [8].

5.3 Constitutive law of insulator strings The insulator strings were modeled with elements able to carry only tensile forces. In this paper, a linear model is used to describe the force-displacement behavior of these elements. As mentioned before, all the insulator strings in the GTS towers are double with 7.15m length. The following constitutive law was adopted for insulator strings in tension:

OIIIII L / L A E F Δ= (4)

in which AI denotes the cross section area of the insulator string element (m2); EI the elastic modulus in tension of the steel that joins the insulators (N/m2); FI the tension force in

1 L

Le

B

f e

y

x

2

ftheoretical x0


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the insulator string element (N); ΔLI the elongation of the element (m) and LIC the unstressed length of the element (m). The values used in the Equation (4) to calculate the tensile forces in the insulator strings are presented in Table 4.

Table 4. Properties of insulator strings.

Cross sectional area of two insulators strings 5.00 × 10-4 m2 Weight for meter of two insulators strings 466.5 N/m Elastic modulus in tension 2.00 × 1011 N/m2

5.4 Constitutive law of bars of the towers Towers GTS 01 and GTS 02 were designed for ASTM A572 steel, with elastic modulus E = 200GPa. The following linear model, both in tension as well as in compression, was adopted to describe the force-displacement behavior of the truss elements:

OBBBBB L / L A E F Δ= (5)

in which AB denotes the cross sectional area of the truss element (m2); EB the elastic modulus of ASTM A572 steel (N/m2); FB the tension or compression force in the element (N); ΔLB the elongation or shortening of the element (m) and LOB the unstressed length of the truss (m).

5.5 Load application The total duration of the dynamic analysis was limited to 40 s. The dead weight of cables, towers, insulators and additional masses was gradually applied during 5 s, allowing 15 s to damp out induced vibrations. Rupture of the cable occurs 20 s after beginning the integration process. The ensuing 20 s were used for the analysis after rupture. In this period, the evolution with time of axial forces in each truss element and of the displacements at the top of tower GTS 01 were determined, and the maximum values identified. The results are presented in Section 6. It should be underlined that it is assumed that rupture of a conductor cable bundle takes place, which implies that the fours cables of the bundle break at the same time. Four analysis of cable rupture were performed, one for rupture of a shield wire and one for each conductor cable bundle of one side of tower GTS 01. The cable elements (conductor cable bundle and shield wire) assumed to break are illustrated in Figure 5.

Figure 5. Cable elements assumed to break.

6 RESULTS AND DISCUSSIONS The dynamic response of tower GTS 01, was obtained using explicit numerical integration. To determine the response envelope, rupture of shield wire and all bundles was evaluated in sequence, for three support conditions: rigid foundation (undisplaceable supports), real foundation and floating foundation (very flexible supports). The tower peak response due to cable rupture, by means of equivalent static loads, with rigid foundation, was also determined using a FEM program. The results are presented below.

6.1 Dynamic response due to rupture of a conductor cable bundle

The variation with time of displacements of four nodes at the top of tower GTS 01, shown in Figure 10, in the longitudinal direction to TL (axis z), due to rupture of the conductor cable bundle 01, is shown in Figure 6. Figure 7 presents the evolution with time of axial forces in some selected diagonal elements of tower GTS 01, identified in Figures 9 and 10, due to the rupture of conductor cable bundle 01. Similarly, Figure 8 shows the axial forces in selected main members of tower GTS 01, also identified in Figures 9 and 10, due to rupture of the conductor cable bundle 01. All these results are from the model with floating foundation. Figure 6. Nodal displacements at top of tower GTS 01, in the

longitudinal direction to TL (axis z), due to the rupture of conductor cable bundle 01, from the model with

floating foundation. Figure 7. Axial forces in diagonal elements of tower GTS 01,

due to rupture of the conductor cable bundle 01, from the model with floating foundation.

Element 3488 Conductor cable

(Bundle 01)

Element 4008 Condutor cable

(Bundle 02)

Element 4528 Condutor cable

(Bundle 03)

Element 2448 Shield wire

Tower GTS 01

0 5 10 15 20 25 30 35 40

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Time (s)

Dis

plac

emen

t (m

)

Node 23Node 25Node 34Node 40

20 22 24 26 28 30 32 34 36 38 40-4

-2

0

2

4x 10

5

Time (s)

Axi

al f

orce

(N

)

Element 569Element 570Element 571Element 572Element 581Element 582Element 583Element 584

20 22 24 26 28 30 32 34 36 38 40-4

-2

0

2

4x 10

5

Time (s)

Axi

al f

orce

(N

)



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Figure 8. Axial forces in main members of tower GTS 01 due

to the rupture of a conductor cable bundle 01, from the model with floating foundation.

Figure 9. Selected diagonal and main members of the lower part of tower GTS 01.

Figure 10. Selected nodes, diagonals and main members of the

upper part of tower GTS 01.

Similar results, presenting however smaller amplitudes, were obtained when the rupture of the conductor cable bundles 02, 03 and shield wire were simulated.

The evolution over time of axial forces in some selected elements of tower GTS 01 (diagonal and main member), identified in Figure 10, due to the rupture of conductor cable bundle 01, for the three support conditions, are presented in Figure 11. Similarly, Figure 12 shows the support reactions of node 148 due to rupture of the conductor cable bundle 01.

Figure 11. Axial forces in main member 556 (left) and diagonal member 581 (right) of the tower GTS 01 due to the

rupture of a conductor cable bundle 01, for three support conditions.

Figure 12. Transversal (left) and vertical (right) support reactions in node 148 of the tower GTS 01 due to the rupture of a conductor cable bundle 01, for three support conditions.

The variation over time of displacements of nodes 25 and

148, which are located at the top and at the bottom of tower GTS 01 (shown in Figures 9 and 10), respectively, in the longitudinal direction to TL (axis z), due to rupture of the conductor cable bundle 01, for the three support conditions, are shown in Figure 13. Figure 13. Displacements at the top (Node 25 – left) and at the

bottom (Node 148 – right) of the tower GTS 01, in the longitudinal direction to TL (axis z), due to the rupture of a

conductor cable bundle 01, for three support conditions.

20 22 24 26 28 30 32 34 36 38 40-3

-2

-1

0

1x 10

6

Time (s)

Axi

al f

orce

(N

)


20 22 24 26 28 30 32 34 36 38 40-3

-2

-1

0

1x 10

6

Time (s)

Axi

al f

orce

(N

)


Diagonal - Element 570 Diagonal - Element 572

Diagonal - Element 569




Main member – Element 553 Main member – Element 554


Node 23 Node 25 Node 40 Node 34

Details in Figure 10

Main member – Element 1058Main member – Element 1061





Main member – Element 688 Main member – Element 687 Diagonal - Element 665





Node 148 Node 149

Node 150 Node 151

20 25 30 35 40-2

-1.5

-1

-0.5

0x 106

Time (s)

Axi

al F

orce

(N

)

FloatingRealRigid

20 25 30 35 40-4

-3

-2

-1

0

1x 105

Time (s)

Axi

al F

orce

(N

)0 10 20 30 40

-4

-3

-2

-1

0x 105

Time (s)

Tra

nsve

rsal

Rea

ctio

n (N

)

FloatingRealRigid

0 10 20 30 400

0.5

1

1.5

2x 106

Time (s)V

ertic

al R

eact

ion

(N)

0 10 20 30 40-0.8

-0.6

-0.4

-0.2

0

0.2

Time (s)

Dis

plac

emen

t (m

)

FloatingRealRigid

0 10 20 30 400

0.05

0.1

0.15

0.2

Time (s)

Dis

plac

emen

t (m

)


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Similar results, presenting smaller amplitudes, were obtained when the rupture of the conductor cable bundles 02, 03 and shield wire was simulated.

6.2 Static response due to rupture of a conductor cable bundle

The nodal displacements at the top of tower GTS 01, identified in Figure 10, in the longitudinal direction to TL (axis z), for the standard static analysis of rupture of the conductor cable bundle 01, are indicated in Table 5.

Table 5. Displacements (z direction) at the top of tower GTS 01 due to rupture of the conductor cable bundle 01

with rigid foundation.

Node Displacement in the longitudinal direction to TL - axis z (m)

23 - 0.933 25 - 0.660 34 - 0.350 40 - 0.605

Tables 6 and 7 present the axial forces in some diagonal and

main members of the tower, also identified in Figures 9 and 10, according to a static analysis of loads due to rupture of the conductor cable bundle 01, with rigid foundation, along with the peak and final state values of the dynamic analysis, for three support conditions.

Table 6. Axial forces in some main members of the tower GTS 01 determined through static and dynamic analysis due to rupture of the conductor cable bundle 01, for three support

conditions.

Main member

Static analysis

Axial force (kN)

Dynamic analysis

Rigid foundation

Real foundation

Floating foundation

Peak axial force (kN)

Final state axial force (kN)





553 288 660 299 633 298 492 279

554 40 518 53 489 52 322 32

555 - 1288 - 1670 - 1318 - 1644 - 1317 - 1511 - 1297

556 - 1121 - 1665 - 1160 - 1635 - 1159 - 1505 - 1138

685 127 720 142 656 140 339 93

686 50 747 68 681 65 324 18

687 - 1599 - 2222 - 1633 - 2160 - 1631 - 1845 - 1582

688 - 1532 - 2274 - 1568 - 2194 - 1566 - 1829 - 1518

Table 7. Axial forces in some diagonal members of the tower GTS 01 determined through static and dynamic analysis due to rupture of the conductor cable bundle 01, for three support

conditions.

Diagonalmember

Static analysis

Axial force (kN)

Dynamic analysis

Rigid foundation

Real foundation

Floating foundation







569 217 371 215 373 215 372 216

570 - 204 - 372 - 204 - 374 - 204 - 369 - 204

571 192 382 185 359 185 355 186

572 - 184 - 375 - 189 - 377 - 189 - 370 - 189

581 - 210 - 383 - 209 - 378 - 209 - 358 - 205

582 184 352 183 358 183 369 189

583 - 156 - 321 - 154 - 330 - 154 - 338 - 159

584 229 409 229 403 228 380 224

Observing Tables 6 and 7, it may be noticed that the peak

values of axial forces provided by dynamic analysis (for three support conditions) due to rupture of the conductor cable bundle 01 were, in average, two times higher than those obtained in the standard static analysis. However, after the responses became stable (final state), i.e., without the dynamic amplification effects, the results were quite similar.

This can lead to important conclusions. Firstly, it may be pointed out that rigid and real supports models, in the dynamic analysis, presented almost coincident peak and final state values for axial forces on the selected elements, while the responses for the floating foundation tended to be slightly lower.

Additionally, it can be observed that the close correlation between the final state in the dynamic analysis and the standard static predictions constitutes strong evidence of the robustness of the latter, which is normally adopted for engineering design. Within this context, an interesting conclusion is that the presence of the remaining cables after the rupture of conductor cable bundled 01 presented a marginal influence on the distribution of the internal forces and did not affect the results.

On the other hand, dynamic amplification may approach 1.50 for main members and significantly exceed that value in case of diagonal members. It is thus concluded that dynamic amplification effects are not negligible in TL crossings and may cause failure of the towers if not properly taken into account for design purposes.

The support reactions of static and dynamic analysis, due to the rupture of conductor cable bundle 01, can be analyzed from Table 8. In the same as for axial forces, the peak values provided by the dynamic analysis were higher than those


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obtained in the static analysis. Again, after the responses became stable (final state) the results were quite similar.

Table 8. Support reactions in node 151 of the tower GTS 01 determined through static and dynamic analysis due to rupture of the conductor cable bundle 01, for three support conditions.

Reaction (kN) Static

analysis (kN)

Dynamic analysis (kN)

Rigid Foundation

Real Foundation

Floating Foundation

Transversal (x) - 383 - 497 - 553 - 477

Vertical (y) 2530 3301 3224 2548

Longitudinal (z) 395 552 591 483

Concerning support flexibility, the highest vertical reaction

was obtained due to the rigid foundation while the highest horizontal reaction was determined from real foundations, considering that both conditions were quite similar. The results for the floating foundation were around 20% lower. This illustrates the importance of considering the flexibility of foundations in simulations.

Regarding the displacements in the longitudinal direction to TL (axis z) at the top and at the bottom of tower GTS 01, due to rupture of the conductor cable bundle 01, the values for rigid and real foundations were, again, quite similar. As it could be expected, the displacements for floating foundations were much higher, around three times more.

In fact, cable rupture tends to be more relevant in TL which adopting single cables, since the probability of failure of all cables of a bundle at the same instant is considerably low. Anyhow, the cable rupture hypothesis provides additional longitudinal strength for preventing cascading failures at a relatively low cost [9].

7 CONCLUSIONS This paper describes the dynamic analysis of a four spans section of a TL crossing over the Trombetas River, which includes two 190m-high TL steel towers, subjected to cable rupture. The entire TL segment was modeled, including the two highest towers and all other elements: foundations, conductors, shield cables and insulator strings. The computed dynamic response of the tower GTS 01 was then compared with the static response obtained by standard procedures.

Since the latter aim at determining axial forces, reactions and displacements after rupture has occurrred, the close correlation of the final state in the dynamic analysis with the standard static predictions constitutes strong evidence of the robustness of both models. In addition, an interesting conclusion is that the presence of remaining cables, after rupture of another conductor cable, presented a marginal influence on the distribution of the internal forces and did not affect the results.

On the other hand, dynamic amplification may approach 1.50 for main members and significantly exceed that value in case of diagonal members. It is thus concluded that dynamic amplification effects are not negligible in TL crossings and may cause failure of the towers if not properly taken into account for design purposes. Furthermore, the

influence of support flexibility was important on the determination of reaction support values, since the highest vertical reaction was obtained with the rigid foundation model while the highest horizontal reaction was determined from real foundations.

ACKNOWLEDGMENTS The authors acknowledge CNPq and CAPES by the financial support.

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[6] Kaminski Jr., J. Incertezas de modelo na análise de torres metálicas treliçadas de linhas de transmissão, Doctoral Thesis, PPGEC, UFRGS, Brazil, 2007.

[7] Kaminski Jr., J.; Riera, J. D.; Menezes, R. C. R. and Miguel, L. F. F. Model uncertainty in the assessment of transmission line towers subjected to cable rupture, Engineering Structures, Vol. 30, pp. 2935-2944, DOI: 10.1016/j.engstruct.2008.03.011, 2008.

[8] Irvine, H. M. and Caughey, T. K. The linear theory of free vibrations of a suspended cable, Proceedings of the Royal Society of London, No. A341, pp. 299-315, 1974.

[9] Fang, S. J.; Roy, S. and Kramer, J. Transmission Structures, Structural Engineering Handbook. Ed. Chen Wai-Fah, CRC Press LLC, 1999.

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River Crossing