Alfredo Camara Casado Apendicesa

Appendix A Cable-stayed bridges constructed in seismic prone areas

Relevant information about the project and construction of major cable-stayed bridge in seismic areas is collected in this appendix, presenting the most relevant dimensions followed by details about the seismic measures adopted in order to control the seismic response of the structure.

A.1 Rion-Antirion bridge (Greece) The Rion-Antirion bridge is located in a highly seismic area in western Greece. Opened in 2004, it is a ve span continuous cable-stayed bridge with semi-harp lateral cable arrangement and total length of 2252 m. The three main spans are 560 meters long each. Figure A.1 presents the general schematic elevation of the bridge.

Figure A.1. Schematic elevation of Rion-Antirion cable-stayed bridge (Greece). Measurements in meters.

The structure holds four trac lanes and the deck section is composite, 27.2 m total width, presenting a classical conguration with two longitudinal steel `I'-shaped

310

Appendix A. Constructed cable-stayed bridges

2.2 m high edge girders separated 22.1 m where cable anchors are directly attached. Transverse metallic beams of the deck are disposed between both longitudinal girders each 4.06 m. Upper concrete slab is 25 cm thick. Very high longitudinal stiness has been conferred to the towers in order to solve the characteristic problem of the reduced stiness along this direction in a multiplespan cable-supported bridge. Such stiening is obtained by means of four concrete legs joined with a special composite element on the top of the tower, which encloses the anchors of the cable-system and increases the stability when the structure is subjected to static or dynamic actions.

Anti-seismic measures in Rion-Antirion bridge Seismic action governed the project of this structure. The bridge was designed to resist an earthquake with moment magnitude equal to 7, 0.48 g in terms of PGA, and transverse tectonics movements between consecutive towers of 2 meters. The response spectrum employed in the design has an extreme spectral acceleration of 1.2 g and a return period of 2000 years. Both local subsoil eects and soil-structure interaction were taken into account. The deck is completely suspended by the cable-system in vertical direction, which together with the great stiness projected in the towers, makes it safe to rely on such monolithic elements in longitudinal direction, whilst Viscous uid Dampers (VD) control transverse movements (Virlogeux, [Virlogeux 2001]). Hence a partial isolation has been selected, combining dissipation devices with allowed seismic demand in the towers (section 2.3.). Due to the great importance of the bridge, many research works have been conducted about its seismic response; [Infanti 2004] [Morgenthal 1999] [Combault 2005] [Dobry 2003] [Lecinq 2003] [Pecker 2004], among others. The seismic control system is equipped with passive devices; viscous dampers (VD) plus fuse restrainers in parallel which establish the transverse connection between the deck and each tower. Fuse restrainers have been designed to behave like sti links dealing with moderate earthquakes or strong winds, limiting this way the serviceability displacements (SLS). However, if the structure is subjected to the design earthquake action (with an associated return period larger than 350 years) fuse restrainers fail, releasing the free movement of the dampers to maximize the energy dissipation, being necessary their substitution after the ground motion. The deck is not vertically linked to the towers as it was already mentioned; instead it is completely supported by the stays in order to maximize the eectiveness of the cable-system and the global seismic response (see section 2.2.6).

In

transverse direction, dissipation devices are disposed in deck-tower connection in order to restrict the relative movements of the deck under severe earthquakes; in longitudinal direction, the deck is free in its whole length (2252 meters) trying to accommodate tectonic and thermal movements, designing the neoprene expansion joints over the abutments to admit movements up to 5 meters under strong earthquakes [Combault 2005].

A.1. Rion-Antirion bridge

311

Each transverse deck-tower connection has 4 viscous dampers (VD) and one fuse restrainer. VD located in the towers have an extreme reaction capacity of

3500

±1750

0.15

mm maximum stroke and damping constant

Cd = 3000

kN/(m/s)

kN,

each,

whilst fuse restrainers fail when their force reach 10500 kN. Each connection of the deck and the transition piers in the approach viaduct isolate the deck with elastomeric bearings which increase the vibration periods and reduce the spectral accelerations, moreover, VD are incorporated in order to dissipate energy and a fuse restrainer is integrated inside one of the dampers. These are the technical data of the seismic devices in the transition piers: two VD (Fmax =3500 kN,

±2600

mm stroke), one fuse restrainer (Fmax =3400 kN) and elastomeric bear-

ings.

The controlling parameter of VD nonlinear response constitutive expression for convenience, is low (αd

αd ,

repeated below the

= 0.15) in order to obtain a great

dissipation of energy and a nearly constant force for a wide range of velocities. The work of Infanti et al [Infanti 2004] collects detailed information about the mechanical characteristics of VD and fuse restrainers employed in Rion-Antirion bridge, along with the results of the laboratory test conducted in such elements (fatigue tests).

Fd = Cd |x| ˙ αd Where

Fd

is the reaction of the damper,

sign (x) ˙

(A.1)

Cd

x˙

is the damping constant and

the

relative velocity. Fuse restrainers have spherical hinges in their ends in order to ensure the alignment of the load and the fuse axis (bending is not allowed in this element). These devices are designed to resist the same stroke than the dampers, trying to avoid their destruction during extreme ground motions. They admit a readjustment of its length to allow slow tectonic movements without introducing forces in the deck. Figure A.2, taken from the report of Infanti et al [Infanti 2004], illustrates the seismic devices installed in deck-tower and deck-transition pier connections. Other seismic devices in Rion-Antirion bridge are the dampers inside the anchorages of the cables, which increase their small damping to limit their vibrations [Lecinq 2003]. It was observed that, if the extreme design earthquake occurs during the construction of the towers, before joining the four legs, three of them could be subjected to tensile stresses and the fourth might resist all the compressive load. In light of these numerical results, it was decided to reinforce and prestress such legs strongly; furthermore, a powerful bracing system between these elements was set during the construction [Combault 2005]. The foundation has been established over exceptionally soft and deep soil, which has been improved by introducing metallic columns with 20 to 30 m high.

Such

columns are not connected to the concrete foundation of the towers (with a diameter of 90 m), they are separated by a layer of gravel (2.8 m thick) forming a sort of plastic hinge if the extreme design earthquake strikes [Dobry 2003] [Pecker 2004].

312

Appendix A. Constructed cable-stayed bridges

Figure A.2. Seismic control system in the connection of the deck with the towers (left) and transition piers (right). Taken from the report of Infanti et al [Infanti 2004].

A.2 Memorial Bill Emerson bridge (USA) Memorial Bill Emerson bridge (opened to trac in 2003) crosses over Mississippi river and has two lateral cable planes with semi-harp arrangement and `H'-shaped towers. The main span is 351 meters, both lateral spans are 143 m long and the length of the approach viaduct is 570 m. The deck is composite, 29.3 m wide and holds road trac. Figure A.3 shows the schematic bridge elevation and the crosssection of its deck.

Anti-seismic measures in Memorial Bill Emerson Bridge This bridge is close to an active fault and important seismic risk is present in the area, the structure has been designed to resist earthquakes of moment magnitude higher than 7.5. The bridge is equipped with 84 accelerometers which monitor its real-time response under any kind of action, being an important data source in the study of the seismic behaviour of cable-stayed bridges [Chen 2007] [Çelebi M. 2005]. A good agreement between the reality of the structure and nite element predictions was achieved in light of the response obtained when it was subjected to a signicant ground motion in 2005 [Chen 2007]. Each deck-tower connection employs 8 Shock Transmission Units (STU) in longitudinal direction (parallel to the trac), with a capacity of 6670 kN and

±180

mm

stroke each, disposing a sti link transversely. The deck is joined to the towers in vertical direction by means of two POT supports. The connection between the deck

A.2. Memorial Bill Emerson bridge

313

Figure A.3. Elevation scheme of Memorial Bill Emerson (Cape Girardeau) bridge and typical cross-section of its deck. Measurements in meters.

and the abutments has two retention devices, a xed transverse link and an expansion joint in longitudinal direction, allowing the movement along the latter axis and the rotation about transverse and vertical axes. Therefore, under service loads the connections release the deck of this bridge in longitudinal direction and x it transversely, whilst under important earthquake the deck is also restrained longitudinally by the towers. Figure A.4 summarizes the seismic strategies employed in the bridge.

A benchmark problem about the incorporation of passive, active, semi-active devices, or a combination of them (hybrid control) in this bridge was proposed. Many researchers have published works with dierent possibilities, here just some references are collected: [Caicedo 2003] [Dyke 2003] [Jung 2003] [Park 2003a] [Park 2003b] [Agrawal 2002] [He 2001], among others.

The interested reader is referred to the

website http://mase.wustl.edu/wusceel/quake/bridgebenchmark/index.htm for more information, including the description of the benchmark problem stages, dynamic models implemented in Matlab [Matlab 2011] and several papers with dierent proposals. The rst stage of the problem exclusively considers the longitudinal component of the earthquake, which is the most demanding one in this case, whilst in the second stage the three-directional seismic excitation and its spatial variability were considered (it was also included the dynamic eect of snow live load over the deck increasing the mass and hence the vibration period).

314

Appendix A. Constructed cable-stayed bridges

Stiff link (transv.) Expansion joint (longitudinal)

Stiff link (transv.) STU (longitudinal)

Stiff link (transv.) Expansion joint (longitudinal)

a. Stiff link in transverse direction

b

b aa

b. Shock Transmission Units (STU) in longitudinal direction

Figure A.4. Seismic devices equipped in Memorial Bill Emerson bridge.

A.3 Tsurumi Fairway bridge (Japan) The length of Tsurumi Fairway bridge is 1020 m being the main central span 510 m. One central cable plane in semi-harp arrangement has been disposed. It was opened to trac in 1994. Figure A.5 represents the schematic elevation of the structure.

Figure A.5. Schematic elevation of Tsurumi Fairway bridge (Japan). Measurements in meters.

The deck section is a metallic box with enough torsional stiness to resist the eccentric forces which the central cable-system is unable to withstand. The total width of the deck is 38 m and the height 4 m. It is designed to support road trac. The pylons are inverted `Y'-shaped towers with lower diamond, they are made

A.4. Yokohama Bay bridge

315

of steel above the vertical pier, which is composite.

The foundation is made of

reinforced concrete.

Anti-seismic measures in Tsurumi Fairway bridge The structure is located in an area with important seismicity; it is designed to resist earthquakes up to 8.2 moment magnitude with a return period of only 75 years. In the future, the construction of a twin bridge close to this one is foreseen, which makes the transverse space available for deck-tower connections very reduced (see gure A.6), being necessary the limitation of relative movements.

With this

objective in mind, horizontal cables are disposed linking the deck and the tower in longitudinal direction, parallel to the axis of the former and inside its section. Four cables are anchored to each tower, two along one side of the deck and the other two along the opposite, as it may be observed in gure A.6, where deck-tower connection and vertical pier are detailed.

The length of each cable is 117 m and

their total diameter 167 mm. Their initial prestress was projected in order to keep the tension below the failure level during extreme earthquakes [Enomoto 1994]. Two vane-type dampers (VD) are installed over the metallic part of the tower below the deck as it is observed in gure A.6, whilst gure A.7 depicts details of their shape and constitutive behaviour. Such dampers help to reduce the longitudinal oscillation of the deck, transforming this movement in a rotation about the transverse axis in the devices. The longitudinal reaction introduced in the deck by the dampers is proportional to the square of the relative velocity (αd limit value of the force (Fmax

= 2000

= 2)

up to a

kN), beyond this level the reaction is main-

tained constant as long as the velocity increases, releasing pressure through orices. The parabolic response of the damper causes low reactions under slow movements (wind or moderate earthquakes) and a strong increase under important seismic intensities (see gure A.7), acting as shock transmission units (STU). Seismic analysis concluded that horizontal displacements are reduced by 30 % if these dampers are incorporated [Enomoto 1994].

Cd = 104

The damping constant of the vane-type damper is

2

kN/(m/s) .

The deck is xed to the towers in vertical and transverse directions.

A.4 Yokohama Bay bridge (Japan) Yokohama Bay bridge (nished in 1989) is a cable-stayed structure with three spans, the central one is 460 m and laterals are 200 m long. In gure A.8 the elevation of this structure is schematically represented. The deck is a metallic sti truss which supports 6 road trac lanes in the upper part and two in the lower level. The `H'-shaped towers are metallic and the cablesystem is lateral with semi-harp arrangement.

316

Appendix A. Constructed cable-stayed bridges

Damper force; F

Figure A.6. Detail of deck-tower connection and vertical pier in Tsurumi Fairway bridge (taken from the work of Enomoto et al [Enomoto 1994]).

Control pressure unit

Relative velocity; v

Figure A.7. Left: Detail of vane-type dampers (VD) employed in Tsurumi Fairway bridge

(taken from the work of Enomoto et al [Enomoto 1994]). Right: Constitutive properties of equipped vane dampers.

A.4. Yokohama Bay bridge

317

Figure A.8. Elevation scheme of Yokohama Bay bridge. Measurements in meters.

Anti-seismic measures in Yokohama Bay bridge This structure belongs to a key infrastructure network between Tokyo and Yokohama, and it is located in a highly seismic area. It was designed and constructed before Great Hanshin (Kobe) earthquake in 1995, and therefore the eect of neareld records was not considered (this eect was introduced in the Japanese seismic code before such event), the design earthquake intensity was neither as strong as the one recorded in Kobe signals. A detailed seismic analysis considering the reduced stiness of the soil and the important height of the gravity center of the structure was performed. The link between the deck and the towers, and also between this element and the transition piers at the end of cable-stayed length, is executed by means of LinkBearing Connections (LBC). The purpose of these devices is to release longitudinally the deck during strong ground motions in order to reduce the inertia forces which may be introduced in the towers and the abutments by the deck, increasing the longitudinal vibration modes (hence reducing associated spectral acceleration) and magnifying displacements.

The length of LBC is limited to 2 meters in order to

prevent excessive seismic movements. This connection releases the deck under slow actions like thermal eects.

Figure A.9 includes the dimensions in millimeters of

LBC equipped in the towers, the points were they are located along the deck and one picture with their disposition in the tower. The behaviour of LBC is nonlinear due to the change of direction of the strong axial load generated when the deck is moved (geometric nonlinearity). In gure A.10, taken from the work of Yamaguchi and Furukawa [Yamaguchi 2004], the nonlinear real behaviour of LBC and its linearized response, which was employed in the design stage, may be observed. The vertical movement of the deck is free in the connections, suspended from the cable-system. Transversely, the deck is xed to the towers. The research of Fujino and Siringoringo [Fujino 2006], based on recorded response data of Yokohama Bay bridge under 14 earthquakes with dierent intensity

318

Appendix A. Constructed cable-stayed bridges

Figure A.9. Dimensions (in mm) of LBC disposed in the towers of Yokohama Bay bridge

(left). Points along the bridge where LBC are equipped (top right corner). Picture including the LBC joining the deck and the tower (bottom right corner).

Figure A.10. Left: Geometric eects associated with LBC response in Yokohama Bay.

Right: Nonlinear and linear (design) response of LBC. Taken from the work of Yamaguchi and Furukawa [Yamaguchi 2004].

A.5. Ting Kau bridge

319

levels, demonstrated that LBC releases the deck if the structure is subjected to strong records, however under moderate events the connections constrain the deck in one or both two transition piers, increasing considerably the bending moment produced in them, which is able to exceed even the value recorded under strong earthquakes. This failure is probably due to the friction, which leads to the following observed behaviour; under strong records LBC initially releases the deck as it was expected, but several seconds before it is xed when the intensity is reduced. Otsuka et al [Otsuka 2004] conducted a seismic analysis about this bridge with acceleration spectra adapted to the modern seismic Japanese code, proposing the strengthening of transition piers and upper transverse struts in the towers, among other considerations. Yamaguchi and Furukawa [Yamaguchi 2004] studied the seismic behaviour of Yokohama Bay structure by means of two-dimensional analysis, verifying the inadequate response when the acceleration design spectra compliant with modern Japanese seismic code is employed, presenting excessive longitudinal movements of the deck, the yielding of both towers at foundation level and transition piers besides the loss of tension in several cables. The installation of viscous dampers (VD) in all LBC was proposed, observing numerically the remarkable improvement of the seismic behaviour if care is taken in the design of VD properties in order to avoid excessive stresses in transition piers. The fundamental period of this bridge is longitudinal, approximately 8 s, due to the free movement of the deck with respect to the supports in this direction. This long period governs its dynamic response.

A.5 Ting Kau bridge (China) Ting Kau bridge is a multiple-span cable-stayed structure with 4 spans located in Hong-Kong and opened to trac in 1998. Main central spans are 448 and 475 m long respectively, whilst lateral spans are 127 m each, composing a total length of 1177 m. Its distinctive characteristic is the employment of diagonal stabilization cables to increase the longitudinal stiness of the bridge, constraining the movement of the upper level of the central tower (see section 2.2.8). Altogether, 8 longitudinal stabilization cables are disposed linking the top section of the central tower and the area close to the connection of the deck with both lateral towers. The length of such diagonal cables, 465 m, established a record [Yuen 2007]. The towers are single concrete masts which separate both sides of deck section (gure A.11). The selection of this kind of towers meets the requirement of a structure with reduced surface due to wind considerations. Transverse cable-system was introduced in order to stabilize the slender masts in transverse direction, joining the upper section with the sti lower area of the tower below the deck (gure A.11). The deck is a composite twin-box section; the boxes are separated 5.2 meters and connected through transverse metallic struts. Each side of the deck is 18.77 m

320

Appendix A. Constructed cable-stayed bridges

wide and holds three trac lanes plus one emergency lane. Longitudinal metallic girders of the deck are 1.75 m deep. Four lateral cable planes are disposed, anchoring two at both sides of the deck section. The semi-harp cable-arrangement presents a separation between cables of 13.5 m. Figure A.11 depicts the schematic elevation of the bridge and one of its towers.

Longitudinal stabilization cables (diagonals)

Transverse stabilization cables

Figure A.11. Left: Schematic elevation of Ting Kau bridge (measurements in meters). Right: Tower detail in transverse direction.

Anti-seismic measures in Ting Kau bridge Support conditions along the deck were carefully studied in order to resist the seismic action and the strong winds. The deck is transversely constrained by POT supports in both lateral sides of the cross-section at the connections with the extreme sides of the bridge and the towers. Longitudinally, the deck is only constrained by the central mast through vertical POT supports able to withstand up to 30000 kN service load.

Both lateral masts dispose bearings prepared to resist the tension imposed

by diagonal cables, with a tensile capacity of 20000 kN and extreme horizontal displacements of 1 m.

Both ends of the deck have longitudinal expansion joints

with maximum admissible movements of

±500

mm [Ayala 1999].

Ting Kau bridge is specially exible due to the slenderness of the towers, the twin section of the deck and the great length of diagonal cables, which cause remarkably strong modal couplings only observed in this structure. Diagonal longitudinal stabilization cables decisively aect its dynamic response [Ni 2000]. Ko et al [Ko 2001] studied the seismic behaviour of Ting Kau bridge, observing the importance of longitudinal and transverse stabilization cables; they result very eective in controlling the vertical seismic excitation. The modal analysis revealed that the results recorded in the central tower fall on the unsafe side if only one element per cable (OECS) is considered. Internal viscoelastic dampers (VE) are incorporated inside the anchorages of longitudinal and transverse stabilization cables to control their vibrations.

The

A.6. Stonecutters bridge

321

bridge is equipped with more than 200 sensors which give information of its response under wind, earthquake, live loads, etc.

A.6 Stonecutters bridge (China) Stonecutters bridge, located in Hong Kong and opened in 2009, is nowadays the second longest cable-stayed bridge in the world with a main span of 1018 m. Lateral spans present intermediate piers to retain the central one and are both 289 m long. Two lateral cable planes with semi-harp arrangement has been disposed. The distance between consecutive cables in central and lateral spans is 18 m and 10 m respectively. The towers are single masts with circular concrete section in the lower and intermediate levels, and composite with stainless steel cover in the upper area where anchorages are located.

The incorporation of concrete in this part of the tower

increase the damping in comparison with the solution exclusively metallic. Auxiliary piers in lateral spans are approximately 60 m high and their exibility is enough to prevent the use of sliding bearings in their connection with the deck, which is completely encastred.

Transition piers (in both extremes of the cable-

stayed length) are gantry structures to provide lateral stability and enough surface to install the support devices of approach viaducts. The deck section is a twin box; metallic in the central span and made of concrete in the lateral spans to counteract the weight of the main one. Altogether 6 trac lanes (3 per both sides of the twin section) are supported. In the main span the total width of the deck is 51 meters and the height of the section reaches 3.9 m in the transverse struts linking both sides of the girder. Figure A.12 presents the elevation scheme of Stonecutters bridge

Figure A.12. Elevation scheme of Stonecutters bridge. Measurements in meters.

Anti-seismic consideration in Stonecutters bridge Stonecutters bridge is located in a seismic prone area. The completely elastic response of the structure under moderate earthquakes was required in the design, with

322

Appendix A. Constructed cable-stayed bridges

120 years return period (Serviceability Limit State), and admitting some yielding in certain parts of the towers but requiring its repair without interrupting the service under strong seismic events, with 2400 years return period (Ultimate Limit State). If the structure is subjected to an extreme earthquake, with 6000 years return period, the structural integrity is required in order to support the access of health personnel (Extreme Limit State) [Hansen 2007]. Therefore, partial isolation (mitigation through seismic devices and structural dissipation) is applied to the seismic design. The eects of spatial variability of the seismic action were taken into account due to the great length of the bridge. The deck-tower connection is solved by means of sliding bearings (SB) in transverse direction, and longitudinally through 4 Shock Transmission Units (STU) with velocity exponent

αd = 2

[Colato 2008]. These STU act only under fast dynamic

solicitations (earthquake or wind) and release the deck under slow movements (thermal eects, creep, shrinkage, etc.). The relative movement between the deck and the towers is free in vertical direction [Hussain 2002].

Figure A.13 presents the

conguration of such connection.

Figure A.13. Detail of deck-tower connection in Stonecutters bridge. Internal dampers have been installed in the anchorage of the cables in order to increase their damping, obtaining a critical damping factor of 4 %. The risk of resonance in some cables due to wind or even earthquake excitations was observed, since their fundamental local frequency was close to the excitation of the anchorages (or half this value), this problem was prevented using Tuned Mass Dampers (TMD) to increase the excitation period in the towers.

Appendix B On the dimensions and characteristics of the proposed bridges

A large number of cable-stayed bridges have been studied in this thesis, among their characteristics dimensions there are variable parameters (like the tower shape), and also canonical proportions which are taken from constructed cable-stayed bridges and maintained constant since they are not the object of the study. The most signicant parameters of the proposed cable-stayed bridges are described in this appendix, including the bridge geometry, the spring stiness representing the tower foundations, and the calculation of the cable properties, among other valuable information.

B.1 Tower height and span distribution Both towers have the same dimensions in all the models, which are in turn completely symmetric about the vertical plane crossing the span center transversely (Y ) and also about the vertical plane along the longitudinal axis of the deck (X ). The classical canonical relationship between the height of the towers above the level of the deck (H ) and the main span (LP ), as well as the ratio between the main and the lateral spans (LS ) was suggested by Como et al [Como 1985] more than two decades ago:

H 1 LS 1 = ; = LP 5 LP 3

(B.1)

Nowadays, the trend is to increase the size of the towers as it is observed in tables B.1 and B.2 [Manterola 1994], where the mentioned ratios are obtained in several constructed cable-stayed bridges worldwide.

In light of these two tables,

it has been decided to modify the classical parameters proposed by Como et al, considering instead the followings:

H 1 LS 1 = ; = LP 4.8 LP 2.5

(B.2)

The height of the towers between the deck level and the foundation (Hi ) is determined by the local conditions and the trac which is expected to cross below the bridge, among other factors. It is very important in dynamic calculations, since

324

Appendix B. Dimensions and characteristics of the models

Ratios Bridge

LP

LS

Jacksonville∗ Diepoldsau Skarnsundet Helgeland Guadiana Sama Langreo∗ Arade Yobukoh C. Saone G. Isere∗ Centenario

396 97 530 425 324 130∗ 256 250 151.8 103∗ 265

198 40.5 190 177.5 135 65 107 121 44.8 67 102

Deck

Deck

Tower

height; h

width; B

height; H

1.52 0.55 2.15 1.36 2.50 1.20 1.63 2.20 1.03 1.90 2.70

32.0 14.5 11.3 12.0 18.0 14.2 17.0 10.9 15.5 11.8 22.0

90.0 28.7 105.0 89.3 77.0 48.0 52.6 62.6 33.3 38.0 60.0

H LP

h LP

LS LP

1/4.33

1/264

1/2.00

1/3.38

1/176

1/2.39

1/5.00

1/246

1/2.79

1/4.75

1/310

1/2.39

1/4.20

1/129

1/2.40

1/5.40

1/216

1/4.00

1/4.80

1/157

1/2.39

1/4.00

1/113

1/2.10

1/4.55

1/146

1/3.39

1/5.40

1/108

1/3.07

1/4.40

1/98

1/2.60

Table B.1. Summary of the dimensions and geometric ratios in Lateral Cable Plane (LCP) cable-stayed bridges. Measurements in meters. The symbol (∗ ) denotes that the bridge has only one tower and hence the main span has been doubled in the calculation of the ratios. The symbol (h ) indicates harp-cable assembly (the rest are semi-harp). Adapted from the work of Manterola [Manterola 1994].

Ratios Bridge

LP

LS

Tampa Aomori Coatzacoalcos Usui∗ Wandre∗ Elorn Chandoline Elba

363 240 288 111 168 400 140 123

164 128 112 111 144 100 72 61

Deck

Deck

Tower

height; h

width; B

height; H

4.47 2.5-3.5 3.30 2.50 3.30 3.47 2.50 2.50

28.8 25.0 18.0 21.4 21.8 23.1 27.0 32.3

73.7 64.3 61.3 61.0 79.0 83.0 30.4 28.0

H LP

h LP

LS LP

1/4.92

1/181

1/2.21

1/3.73

1/68-1/96

1/1.87

1/4.68

1/87

1/2.57

1/3.63

1/88

1/2.00

1/4.25

1/100

1/1.93

1/4.81

1/115

1/4.00

1/4.61

1/56

1/1.95

1/4.40

1/49

1/2.01

Table B.2. Summary of the dimensions and geometric ratios in Central Cable Plane (CCP)

cable-stayed bridges. Measurements in meters. The symbol (∗ ) denotes that the bridge has only one tower and hence the main span has been doubled in the calculation of the ratios. All the bridges have semi-harp cable congurations. Adapted from the work of Manterola [Manterola 1994].

B.2. Cross-section of the deck the height of the gravity center of the bridge is strongly inuenced by

325 Hi .

However,

this parameter is governed by the specic site conditions and, therefore, it is not considered in the study as a design variable. Instead, a value proportional to the height of the tower above the deck is xed so that when it changes, the proportions of the model remain unchanged. It was decided to consider a reasonable value of

Hi = 0.5H ,

being

H

the height of the towers above the deck level.

The length of the central part of the deck between the end cables corresponding to each tower is considered constant and equal to 20 m in all the models, which is twice the separation between the cable anchorages in the main span (∆D,p ), as it is represented in gure B.1.

Figure B.1. Schematic bridge elevation. Measurements in meters.

Intermediate piers in the side spans The current trend is to dispose intermediate supports in side spans which helps to reduce the longitudinal displacement of the upper part of the towers, thus increasing the eectiveness of the cable-system. These intermediate piers only constrain the vertical movement of the deck, releasing it in longitudinal and transverse direction, because in this latter direction the towers are very rigid and are able to resist the transverse forces (especially due to wind and earthquakes). Intermediate piers are located in this thesis separated from the abutment around 40 % of the side span (see gure B.1), in light of constructed cable-stayed bridges. The eect of these lateral supports in the static response and the vibration modes of cable-stayed bridges have been studied by the author in a previous research [Camara 2008], observing the improvement of the longitudinal response due to the stiening of the lateral spans and the reduction of the rst vibration periods.

B.2 Cross-section of the deck The deck holds four road trac lanes with a steel-concrete composite section, which is commonly used to cover all the considered main spans (from 200 to 600 m). The axis of the deck is horizontal and straight both in plan and elevation. width (B ) is constant and equal to 25 m.

The total

326

Appendix B. Dimensions and characteristics of the models

The design of the deck section in bridges with lateral cable planes (LCP) is in accordance with the current trend in constructed cable-stayed bridges, disposing two longitudinal edge steel girders and one upper concrete slab. On the other hand, the cross-section of the deck adopted in structures with central cable plane (CCP) is an `U'-shaped steel section below the concrete slab, which is able to withstand the torsion that is not resisted by the cable-system. Tables B.1 and B.2 included the ratio between the height of the deck (h) and the main span (LP ) in constructed cable-stayed LCP and CCP bridges.

As it

may be observed, the height of the deck is higher in structures with central cable arrangement (CCP) since the torsional resistance is entrusted to the deck.

The

averaged deck ratios from these tables are:

h 1 = → LP 180 1 h = → LP 90

LCP

(B.3)

CCP

(B.4)

However, expression (B.3) is questionable since, in real projects, the depth of the deck in LCP cable-stayed bridges is nearly independent of their main span, instead it is inuenced by the transverse and longitudinal distance between consecutive cable anchorages. Nonetheless, the depth of the deck slightly increases with the main span because of the increment in the length of the cables and the consequent increase of the exibility, which requires a stier deck due to wind considerations. The work of Astiz [Astiz 2001] suggests the variation of the depth of the deck with the main span in several composite LCP cable-stayed bridges:

h = 0.78 + 0.00302LP → Where

h

and

LP

LCP

(B.5)

are expressed in meters.

Static analysis conducted here have veried that the level of stresses in LCP and CCP composite decks are admissible considering the worst situation; imposing the self-weight and the live load in the central part of the bridge, where the distance between cable anchorages is larger (20 meters). Therefore, the relationship between the depth of the deck and the main span in LCP models proposed by Astiz (expression (B.5)) and the ratio presented in equation (B.4) for CCP models are accepted. Figure B.2 presents the considered sections of the deck in both LCP and CCP models (excluding the non-structural mass), being their principal dimensions constant or variable in terms of the main span. The dimensions of the longitudinal and transverse girders in LCP deck cross-sections, or the plate thickness and stieners in CCP models, are the result of the simplied elastic static analysis mentioned above (with 20 m span), preventing the extreme stresses from exceeding the design allowable values. The longitudinal separation of the cables at the level of the deck was expressed in gure B.1 and discussed further in section B.5.

B.3. Dimensions of the towers

*

327

every every

*

every

(a) Deck cross-section for LCP arrangement

Cables every 10 m in the main span and every in lateral spans

0,2

* *

Diaprhagms each 5 m in the main span, and every each 2 in the lateral spans Profiles of the diaphragm: Vertical: 2 UPN 200 Diagonal: 2 UPN 300

*

(b) Deck cross-section for CCP arrangement

Figure B.2. Composite deck cross-sections employed in lateral and central cable congu-

rations, parameterized in terms of the main span (LP ). Measurements in meters.(*) Plate thickness is larger at localized areas; this is a mean value for analysis.

B.3 Dimensions of the towers The dimensions of the towers in proposed models are obtained from the specic study of a set of constructed cable-stayed bridges, which is deemed representative of each studied tower shape and cable-system arrangement. The geometric ratios, which lead to the nal cross-sections and proportions of the towers, are presented

328

Appendix B. Dimensions and characteristics of the models

here in tables B.3 to B.8 and graphically in gures B.3 to B.7. The cable-system assembly is always semi-harp, trying to minimize the length of the area where the anchorages are located (HA ) in order to maximize the eectiveness of the cables. Therefore, the distance between anchorages in the tower is reduced to a minimum space which allows the construction. In these bridges, such distance is reasonably

∆T = 2

m, which xes the length of the anchorage area in

the tower:

HA = ∆T (NT − 1) Where

NT

(B.6)

is the number of stays per half main span and cable-plane (obtained

with the expression the main span

LP .

NT = (LP − 20)/20,

1

justied in section B.5) which depends

on

If the area of the tower where the anchorages are located is not vertical but forms an angle

α < 90◦

2

with the horizontal line

(being this value included in the

geometrical description of each tower in gures B.3 to B.7), the length

HA

HA = ∆T (NT − 1) sin(α) | {z }

is:

(B.7)

≈1

Due to the large number of cable-stayed bridges studied in this thesis, and the required parametrization, the design simplicity is strongly recommended.

Hence,

constant sections between dierent parts of the towers have been considered in the parametric denition. To facilitate the comparison of the results between models with dierent towers, the sections have been established as similar as possible. The proportions and sections of `H'-shaped towers in terms of the height above the deck, which in turn depends on the main span (H

= LP /4.8), have been obtained

from a set of bridges with the same type of towers, included in tables B.3 and B.4, along with the ratios proposed here (X ,

Y

and

Z

are the longitudinal, transverse and

vertical directions respectively). The nal version of the `H'-shaped tower employed in the thesis is illustrated in gure B.3. The proportions of the `Y-shaped' towers with and without lower diamond of

3

several cable-stayed bridges are summarized in tables B.5 and B.6 , including the proposed ratios used in the nal version of the towers employed in the thesis, which are presented in gures B.4 and B.5. Finally, the dimensions of the `A'-shaped towers with and without lower diamond in several cable-stayed bridges are presented in tables B.7 and B.8, besides the proposed parameters employed in this work which are illustrated graphically in gures B.6 and B.7.

1 The

number of cables is also inuenced by the vertical connection between the deck and the tower; if it is xed in vertical direction, the closest pair of stays at each side of the tower are eliminated, compared with the solution with free vertical connection (the one employed here). 2 The angle between the legs above the deck and the horizontal line, α, has been referred as β b in chapter 7. For convenience, the notation α is maintained in this appendix. 3 Again, (∗ ) denotes that the bridge has only one tower and hence the main span has been doubled in the calculation of the ratios.

B.3. Dimensions of the towers

329 Anchorage

Bridge

Barrios de Luna, 1983 Suez Canal, 2001 Bill Emerson, 2003 Yokohama Bay, 1989 Sidney Lanier, 2003

LP

Inclined

area

H

Lower

legs

piers

a (X )

b (Y )

a (X )

b (Y )

a (X )

b (Y )

90 80 75 105 87

H/24

H/20

H/20

H/18

H/15

H/16

H/18

H/32

H/13

H/16

H/11

H/11

H/11

H/27

H/11

H/23

H/11

H/10

H/20

H/26

H/17

H/26

H/15

H/18

H/13

H/31

H/13

H/23

H/13

H/13

Proposed

H/15

H/25

H/15

H/20

H/13

H/13

440 404 351 460 375

Table B.3. Dimensionless proportions of several cable-stayed bridges with `H'-shaped tow-

ers and proposed ratios in this study. `Inclined legs' refers to the part of the tower between the anchorage area and the lower strut, `lower piers' refers to the legs of the tower below the lower strut. Parameters detailed in gure B.3. Measurements in meters.

Bridge

Barrios de Luna, 1983 Suez Canal, 2001 Bill Emerson, 2003 Yokohama Bay, 1989 Sidney Lanier, 2003

LP

H

Upper

Intermediate

Lower

strut

strut

strut

a (X )

h (Z )

a (X )

h (Z )

90 80 75 105 87

H/24

H/25

H/22

H/22

-

-

-

-

H/19

H/19

H/17

H/17

H/14

H/17

H/15

H/15

H/26

H/21

H/21

H/17

H/15

H/19

H/15

H/16

Proposed

H/24

H/25

H/18

H/19

H/15

H/16

440 404 351 460 375

a (X )

h (Z )

Table B.4. Dimensionless proportions of cable-stayed bridges with `H'-shaped towers and proposed ratios (continued). Parameters detailed in gure B.3. Measurements in meters.

Bridge

LP

Rijeka Dubrovacka, 2002 Boine, 2003 Rhein Flehe, 1979 Wandre, 1989 Incheon, 2009

244 170∗ 335∗ 368∗ 168∗ 800

H

Anchorage

Inclined

Lower

area

legs

piers

a (X )

b (Y )

a (X )

b (Y )

a (X )

b (Y )

93 71 115 130 79 171

H/18.6

H/16.6

H/18.6

H/23.3

H/18.6

H/23.3

H/16.9

H/8.9

H/16.9

H/19.2

H/14.2

H/14.8

H/28.7

H/14.4

H/28.7

H/29.4

-

-

H/20.2

H/16.4

H/20.2

H/24.4

H/20.2

H/24.4

H/19

H/19.7

H/19

H/24.6

H/19

H/24.6

H/21

H/28.4

H/21

H/34

H/21

H/21

Proposed

H/18

H/13

H/18

H/23

H/18

H/20

∗

Table B.5. Dimensionless proportions of several cable-stayed bridges incorporating `Y'-

shaped towers with and without lower diamond, and proposed ratios in this study. `Inclined legs'; legs between the anchorage area and the lower strut. `Lower piers'; legs below the lower strut. Parameters detailed in gures B.4 and B.5. Measurements in meters.

330

Appendix B. Dimensions and characteristics of the models

Bridge

LP

Lower

Vertical pier

strut

(lower diamond)

H

a (X ) Rijeka Dubrovacka, 2002 Boine, 2003 Rhein Flehe, 1979 Wandre, 1989 Incheon, 2009

244∗ 170∗ 335∗ 368∗ 168∗ 800

h (Z )

a (X )

b (Y )

93 71 115 130 79 171

H/18.6

H/31

H/17.7

H/18.2

-

-

-

-

H/28.7

1.4B

H/20.2

H/23.5

-

-

H/21

H/29

-

-

Proposed

H/18

H/23

H/15

H/6

Table B.6. Dimensionless proportions of several cable-stayed bridges incorporating `Y'-

shaped towers with and without lower diamond, and proposed ratios in this study (continued). Parameters detailed in gures B.4 and B.5. Measurements in meters.

Bridge

William Natcher, 2002 Chao Phraya, 2006 Jindo, 1984 Helgeland, 1991

LP

366 500 344 425

Inclined

Lower

legs

piers

H

79.7 120 69 90

Proposed

a (X )

b (Y )

a (X )

b (Y )

H/16

H/29.5

H/12.3

H/17.7

H/18.4

H/27.3

H/15.5

H/13.6

H/18

H/34

H/16

H/12.5

H/17

H/36

H/15

H/22

H/18

H/23

H/18

H/20

Table B.7. Dimensionless proportions of several cable-stayed bridges incorporating `A'shaped towers with and without lower diamond, and proposed ratios in this study. `Inclined legs'; legs between the anchorage area and the lower strut. `Lower piers'; legs below the lower strut. Parameters detailed in gures B.6 and B.7. Measurements in meters.

Bridge

William Natcher, 2002 Chao Phraya, 2006 Jindo, 1984 Helgeland, 1991

LP

H

Lower

Vertical pier

strut

(lower diamond)

a (X )

h (Z )

a (X )

b (Y )

79.7 120 69 89

H/13.3

H/29.5

H/10

H/3.5

H/20

H/13.3

H/16

H/12.5

-

-

H/15

H/25

H/15

H/7.4

Proposed

H/18

H/23

H/15

H/6

366 500 344 425

Table B.8. Dimensionless proportions of several cable-stayed bridges incorporating `A'-

shaped towers with and without lower diamond, and proposed ratios in this study (continued). Parameters detailed in gures B.6 and B.7. Measurements in meters.

B.3. Dimensions of the towers

331

ΔT = 2

bT ,A

F

aT ,A =

HA

H =

bT ,A =

A

LP 4, 8

Zona de Atirantamiento “HA”

B = 25

H 25

H 15

H = 25

H 13

bT ,bi

B

D

hT ,vtm = C

bT ,pi =

C

H 25

H 19

H 24

tc

x (Tráfico)

y

Sección E – E: H aT ,vtm = 18

D

H i = 0, 5H

aT ,vts =

z

x (Tráfico) y z

H = 20

tc Sección F – F:

H 13

hT ,vts =

tc

H 20

y

z Sección C – C:

A

aT ,pi =

H 15

x

bT ,pi = E

bT ,bi =

aT ,bi =

tc

E

B

Sección F – F:

aT ,vti = hT ,vti =

tc

z

H 13

y

H 16

H 15

tc

x (tráfico)

En esta torre solo es posible el atirantamiento lateral

z

≈ bT ,bi + 2 + B

y

x (tráfico)

Atirantamiento lateral: z

≈

Sección B – B:

Sección A – A:

CL F

α

bT ,bi

≈

2

ddvi = ZTgi + 1 +

2

y

x (tráfico)

2

B = 25 1

bT ,A

bT ,bi

ΔT = 2

hT ,vti 2

1

bT ,A =

⎛ ⎜ H − H A − ΔT α ≈ tan ⎜ bT ,bi ⎜⎜ 1 + 2 ⎝

⎞ ⎟ ⎟ ⎟⎟ ⎠

z

aT =

H 15

H 25

aT − tc 2

y x (tráfico)

tc

Figure B.3. Parameterized dimensions of the `H'-shaped tower in terms of the main span LP , since H = LP /4.8. H-LCP models.

332

Appendix B. Dimensions and characteristics of the models

A HA

H =

bT ,A =

aT ,A =

H 18

Zona de Atirantamiento “HA”

tc

bT ,bi =

H aT ,bi = 18 x z

ΔT

LP 4, 8

H 13

bT ,A =

A

H 13

Sección B – B:

Sección A – A:

CL

ΔT

bT ,pi =

B

bT ,bi =

B

H 18

aT ,vti =

H 23

H 18

tc

z

x

H 23

Sección D – D:

H 20

hT ,vti =

tc

tc y

Sección C – C:

aT ,pi =

H 23

z

y

y

x (tráfico)

D D

H i = 0, 5H

bT ,pi

C

C

⎛ ⎜ H − H A − 2ΔT α ≈ tan ⎜ b ⎜⎜ B + 1 + T ,bi 2 ⎝ 2

H = 20

α

⎞ ⎟ ⎟ ⎟⎟ ⎠

z ≈B N +2+2

y

x (tráfico)

25

≈

bT ,bi

≈

2

Hi + bT ,pi tan α

Atirantamiento lateral:

bT ,bi

bT ,A − 2tc − 0, 8

1

hT ,vti 2

2

y

− tc

ΔT

tc

tc

1

z y

x

bT ,A

ΔT

0, 4

bT ,A =

x

z

2 ZTgi + 1 +

B = 25

Atirantamiento central:

aT ,A =

H 18

H 15

bT ,A 2 aT ,A 2

− tc

z y x (tráfico)

bT ,A =

H 15

aT ,A − 0, 4 − tc

aT ,A =

H 18

4 aT ,A 2

− tc

tc

tc bT ,A 4

Figure B.4. Parameterized dimensions of the inverted `Y'-shaped tower without lower diamond in terms of the main span LP , since H = LP /4.8. Y-LCP and Y-CCP models.

B.3. Dimensions of the towers

333

Figure B.5. Parameterized dimensions of the inverted `Y'-shaped tower with lower diamond in terms of the main span LP , since H = LP /4.8. YD-LCP and YD-CCP models.

334

Appendix B. Dimensions and characteristics of the models

A H A − 2ΔT

Sección A – A:

CL

ΔT

A

HA

aT ,A

H = 18

bT ,bi = aT ,bi =

tc

Zona de Atirantamiento “HA”

x

H 18

z

2ΔT

H =

Sección B – B:

Variable

bT ,pi =

H 18

aT ,pi =

B

bT ,bi =

B

H 23

aT ,vti =

H 20

hT ,vti =

tc

tc

y Sección D – D:

Sección C – C:

LP 4, 8

H 23

H 23

H 18

tc

z

x z

y

y

x (tráfico)

D

⎛ ⎜

⎞ ⎟ bT ,bi ⎟ ⎜ +1+ ⎟⎟ ⎝2 2 ⎠

D

H i = 0, 5H

C

bT ,pi =

C

H

α ≈ tan ⎜ ⎜B

H 20

α

z

En esta torre solo es posible el atirantamiento lateral

≈B N +2+2

y

x (tráfico)

25

≈

bT ,bi

≈

2

Hi + bT ,pi tan α

bT ,bi

≈

2

ZTgi + 1 + 1

B = 25

Atirantamiento lateral:

≈ ΔT

2

z

hT ,vti 2

bT ,bi

α

x (tráfico) y

1

bT ,A =

H 15

z x

y

aT ,A = z

H 18

aT ,A 2

aT ,A

y x (tráfico)

4

− tc

tc

bT ,A 4

Figure B.6. Parameterized dimensions of the `A'-shaped tower without lower diamond in terms of the main span LP , since H = LP /4.8. A-LCP models.

B.3. Dimensions of the towers

335

Figure B.7. Parameterized dimensions of the `A'-shaped tower with lower diamond in terms of the main span LP , since H = LP /4.8. AD-LCP models.

336

Appendix B. Dimensions and characteristics of the models

B.3.1 Thickness of the tower sections The thickness of the concrete sections (tc ) in the anchorage area of the towers is obtained so that the maximum allowable compression

∗ fcd

is not exceeded when the

self-weight and the live load are applied to the structure.

The same value of the

thickness is considered in the rest of the tower since the stresses are lower.

For

constructive reasons, the vertical pier of the lower diamond is the exception, where a constant value of 0.45 m is considered regardless of the main span length. The self-weight of the composite deck (qpp ) depends on its height and cablesystem arrangement; this value is obtained numerically in an exact way for each studied model, but rounds

6.5

-

8.5

2

kN/m

(in accordance with the standard val-

ues for composite girders [Manterola 2006]). The dead load is estimated as

3.7

2

kN/m , and the live load is simplied as

qscu = 4

2

kN/m

qcm =

[IAP 1998] across the

whole width and length of the deck. The maximum allowable uniform compression

∗ = 10 MPa, which is lower than the design value in fcd HA-40 concrete; fcd = 40/1.5 = 26.67 MPa. As it was veried, the maximum stress ∗ along the towers due to the self-weight and the live load is below the limit fcd = 10 in the concrete is considered

MPa with the proposed thickness of the sections. Below, the procedure to obtain the thickness of the tower sections is presented. In practice, the results for the required thickness are nearly independent of the main span and tower shape, ranging reasonably the nal employed values between

tc = 0.50

m for

LP = 200

m and

tc = 0.40

m for

LP = 600

m (and

tc = 0.45

m for

the lower diamond in all the cases).

Towers with two anchorage sides (`H'-shaped and `A'-shaped towers) The most loaded section in the anchorage area is just below the lower anchorages, resisting all the corresponding weight of the deck, which is expressed in [kN] as;

Corresponding deck length

2 anchorages sides

F =

z}|{ 1 2

z (qpp + qcm + qscu ) B

}| { LP + LS 2

(B.8)

2

The required area of the anchorage section in [m ] (Anec ) is obtained as:

The

  1 LP (qpp + qcm + qscu ) B + LS F 2 2 ∗ fcd = 10 MPa = = Anec Anec proposed section is rectangular, with dimensions a and b given in

(B.9) terms of

the main span by gures B.3, B.6 and B.7, particularized in the anchorage area. In the case of models with `A'-shaped towers, the dimensions of the section dened in the legs above the deck are considered, since the anchorage area present variable cross-sections. Figure B.8 presents the scheme of the section and the calculation of the required thickness. The thickness of the section is obtained by considering the negative sign between the two solutions of the quadratic expression illustrated in gure B.8, resulting:

B.4. Characteristics of the foundations

337

(cuadratic equation)

Figure B.8. Sketch of the concrete section of the tower just below the lower anchorage (the worst case), and calculation procedure to obtain the required thickness to prevent ∗ compressions exceeding fcd = 10 MPa.

tc =

2 (a + b) −

q 4 (a + b)2 − 16Anec

(B.10)

8

Towers with one anchorage member (`Y'-shaped towers) The procedure presented above is also valid in this case, assigning all the corresponding force of the deck to the single anchorage member, which is in [kN]:

Corresponding deck length

1 anchorage member

F =

z}|{ 1

z (qpp + qcm + qscu ) B

}| { LP + LS 2

(B.11)

In this case the required area and thickness are obtained by means of expression (B.10), using the force obtained in B.11 for the required area dimensions in the anchorage zone (a and

b)

Anec .

The cross-section

are taken from gures B.4 and B.5.

B.4 Characteristics of the foundations The foundations of the intermediate piers and abutments are assumed innitely rigid, however, the 3D exibility of the subsoil at the connections with the towers has been considered in terms of displacements by means of elastic springs, whereas all the rotations are completely prevented due to the large foundation stiness. The soil-structure interaction problem is neglected, as it has been discussed in chapter 3, and therefore, the stiness of the springs is independent of the excitation. In order to obtain the exibility of the towers at their connection with the ground along the three principal directions, it is necessary to know the type of surrounding soil at these points and the dimensions of the foundations. Two types of ground soils, compliant with Eurocode 8 [EC8 2004], are considered in this research; rocky soil (type A) and soft soil (type D). The Young's modulus (E ) and Poisson coecient

338

Appendix B. Dimensions and characteristics of the models

(ν ) associated with these soils are presented below, along with the corresponding shear modulus (G) [Manterola 2006]: Rocky soil (TA);

ETA = 4 × 106 νTA = 0.3

2

kN/m

 → GTA =

ETA = 1.54 × 106 2 (1 + νTA )

2

kN/m

(B.12)

Soft soil (TD);

ETD = 0.5 × 106 νTD = 0.35

2

kN/m

 → GTD =

ETD = 0.18 × 106 2 (1 + νTD )

2

kN/m

(B.13)

The dimensions and type of foundations clearly depend on the stiness of the subsoil; typically, soft soils require deep foundations with piles in the towers of cablestayed bridges, whilst shallow foundations are only employed if the surrounding subsoil is very competent.

However, in order to simplify the study, only shallow

foundations are considered here, and their dimensions are obtained from a specic study about the foundations of constructed cable-stayed bridges, neglecting the inuence of the soil resistance in these proportions due to the lack of information

Bc

found by the author. The sensitivity of the seismic response on the dimensions of the foundations is presented below, in order to address the possible errors introduced by neglecting the resistance of the subsoil in the denition of foundation proportions and the possibility of deep foundation.

Lc

The results of the study about the dimensions of the foundations in constructed cable-stayed bridges are presented in table B.9, besides the proposed dimensions for the present work. The parameters employed in this table are described graphically in gure B.9.

Bridge plan:

Tower

Tower deck axis ( ; traffic)

(Traffic)

Figure B.9. Schematic representation of the tower foundations and their characteristic dimensions.

Puente

Luz (Lp)

Lc

Bc

Lc / b pi

Bc / api

B.4. Characteristics of the foundations

339

Bridge

LP

Lf Y direction

Bf X direction

Lf /bpi Y direction

Bf /api X direction

Donghai Helgeland Wandre Barrios de Luna Sidney Lanier Tatara

420 425 168∗ 440 374 890

56 22 7 10 35.5 43

28 11 8 15 21.8 25

2.24 1.83 2.18 1.84 3.03 1.72

3.50 1.83 1.92 2.50 3.13 2.94

2.1

2.1

Proposed

Table B.9. General dimensions of the foundations in several constructed cable-stayed bridges, and ratios proposed for the present thesis. Wandre bridge has only one tower and it has been marked with symbol (∗ ) to distinguish it from the rest. Figure B.9 describes the parameters included. Measurements in meters. The exibility to the displacement along the three principal directions is estimated by assuming spread footings and homogeneous semi-space subsoil medium, leading to the following expressions

4

[Manterola 2006];

"   0.75 #  GLf Bf   Vertical stiness (Z); kz = 0.73 + 1.54   (1 − ν) Lf     " #   0.85 Bf GLf Lf > B f 2 + 2.5 Transverse stiness (Y); ky =   (2 − ν) Lf        Lf Bf  0.2   G 1− Longitudinal stiness (X); kx = ky −  (0.75 − ν) 2 Lf (B.14)

G and ν

being respectively the shear modulus and Poisson coecient associated

with the corresponding subsoil (expressions (B.12) and (B.12)), whereas

Bf

and

Lf

are the dimensions of the foundation. Linear springs along the three principal directions have been employed at the bottom section of the tower to represent the aforementioned exibility, as it is de-

5

picted in gure B.10, whereas all the rotations are prevented .

Sensitivity to the dimensions of the foundations The inuence exerted by the dimensions of the foundations on the seismic response of the studied cable-stayed bridges is assessed here. `H'-shaped towers on soft soil

4 In

expression B.14, Lf is the largest side of the foundation and Bf the smallest one. The dimensions are obtained in each model from the proposed ratios presented in table B.9. 5 The dynamic contribution to the response of the vibration modes associated with the rotations in the foundations of the towers is negligible, however modes involving their displacements were found important (section 4.3).

340

Appendix B. Dimensions and characteristics of the models Pier of the tower

ky

z

y

kz

kx

x

Figure B.10. Modelling of the tower foundation. LP = 200 m, according to table B.9 the proLf = Bf = 8.6 m. Undoubtedly, the subsoil σu (closely related to its stiness) aects

are considered; if the main span is

posed dimensions of the foundations are

6

ultimate bearing capacity

of the

7

the dimensions of the foundations, and their minimum side

N → Lf = B f = σu = Lf Bf Where

N

r

is obtained as:

N σu

(B.15)

is the maximum axial load transferred from the leg of the tower to the

subsoil; neglecting the accidental loads,

N

is obtained by aggregating the contribu-

tions of the self-weight and the live load, resulting

N = 58

MN in the studied case.

If the allowable stress associated with soft soil (type D) is

σu = 2

2

kg/cm , then

Lf = Bf ≈ 17 m (in this case the most reasonable solution would be a deep founda-

tion), which is approximately twice the value proposed in this thesis, according to table B.9. If, on the other hand, the subsoil is rocky (type A) with

then

Lf = Bf ≈ 3

σu = 60 kg/cm2 ,

m, however, the dimensions of the lower piers in this case have

sides of 3.2 m (see gure B.3), and the foundations should be larger. Therefore, in order to cover all the range of solutions in terms of possible subsoils, the sensitivity study should take values of

Lf

ranging from half the one proposed in table B.9 (4.3

m) to twice this reference (17.2 m). Figure B.11 presents the extreme seismic reaction of the deck against the towers, expressed in terms of the factor aecting the side of the tower foundations with respect to the original solution;

kf = A/Ao ,

where

foundation employed in this sensitivity study, and

A = Lf = Bf is side Ao the side considered

of the in the

body of the thesis (proposed in table B.9). Both models with 200 and 600 m main span have been considered; from gure B.11, it is clear that the inuence of the stiness in the springs representing the foundations is negligible in terms of the reaction of the deck. The extreme seismic axial load along the legs of the towers with several dimen-

6 The ultimate bearing capacity is the maximum pressure which can be supported by the subsoil

without shear failure. 7 In `H'-shaped towers the foundation is square-shaped.

B.5. Cable-system arrangement

341

Figure B.11. Inuence of the dimensions of the foundations on the extreme transverse seismic reaction of the deck against the towers. `H'-shaped towers, soft soil (type D), θZZ xed in deck-abutments connections. Response spectrum analysis (MRSA).

sions for their foundations is depicted in gure B.12, considering models with 200, 400 and 600 m main span.

Again, the reduced eect of these dimensions on the

seismic response is highlighted; it may be concluded that reducing the dimensions to half their original value may aect the seismic response in the largest models, however, the dierences are below 10 % and such small foundations are not realistic taking into account the dimensions of the lower piers. On the other hand, modications below 0.1 % of the dominant vibration modes have been observed by the variation of the foundations in the mentioned range, and consequently the spectral accelerations are nearly independent of the dimensions of tower foundations. In light of the results obtained in this study, the following conclusions may be drawn; (i) the inuence of the foundation proportions is reduced if the seismic response of the tower is pursued; the type of foundation soil is paramount due to its eect in the design spectra, but not through the modication of the foundation characteristics and, therefore, it is valid to ignore the type of surrounding subsoil in the parametrization of the foundations and their associated stiness; (ii) distinguishing between shallow or deep foundations depending on the subsoil bearing capacity is not important if the global seismic response of the towers is studied, hence considering always shallow solutions is appropriate for the general study conducted in the present doctoral thesis, despite probably, a real project of a cable-stayed bridge would employ deep foundations.

B.5 Cable-system arrangement The longitudinal separation between cable anchorages along the deck in the main span (∆D,P ) is constant and equal to 10 m, which is a reasonable value in cable-

342

Appendix B. Dimensions and characteristics of the models

Figure B.12. Inuence of the foundation dimensions on the extreme axial load along the tower height. `H'-shaped towers, soft soil (type D), θZZ xed in deck-abutments connections. Ao is the original side of the foundation. Response spectrum analysis (MRSA). stayed bridges with composite deck [Manterola 2006]. This distance in the lateral span (∆D,S ) is reduced since the same number of cables (NT ) needs to be distributed in a length (LS ), which is smaller than half the main span. The geometrical description of the cable-system assembly was shown in gure B.1. The number of cables per half main span and cable plane (NT ) is obtained in terms of the main span length (LP ), imposing that

∆D,P = 10:

LP − ∆D,P LP − 20 NT = 2 = ∆D,P 20

(B.16)

Consequently, the separation between cable anchorages along the deck in the side spans (∆D,S ) is:

∆D,S =

LS NT

(B.17)

B.6 Characteristics of each stay. Cable cross-section The determination of the required area in each cable comprises an initial trial phase and a subsequent iterative process that takes into account the redistribution of

8

stresses in the bridge .

In the initial trial stage, the area of the cross-section in

each cable is obtained so that the vertical component of the cable force balances the self-weight and the live load over the corresponding deck length, being their stress

8 In the case of cable-stayed structures the redistribution is important because of its marked hyperstatic response.

B.6. Characteristics of each stay. Cable cross-section

343

equal to 40 % of the ultimate stress allowed in the prestressing steel of the cables (σT,ult ). Figure B.13 shows the areas of the deck which correspond to each stay in lateral and central cable-system assemblies, along with the calculation scheme.

Lateral Cable Plane (LCP):

Central Cable Plane (CCP):

B B

B

Contributing area of the deck corresponding to i-stay:

Contributing area of the deck corresponding to i-stay:

B

B

Calculation scheme: Calculation scheme:

Figure B.13. Contributing surfaces of the deck associated with each stay and calculation scheme to determinate their area, both in LCP and CCP models. Measurements in meters. By adding forces in vertical direction (Z ) in the illustrated scheme, and imposing equilibrium, the area of each stay is obtained:

X

(F )Z = 0 → FT,i sin(βi ) = (qcp + qscu )Aco,i

Where, particularized for the cable deck length corresponding to it and

βi

i, FT,i

is its force,

Aco,i

(B.18)

is the surface of the

its angle with respect to the deck.

Taking into account the values dened in paragraph B.3.1 for the the self-weight and the live load:

qcp = f (LP ) = qpp + qcm

h

2

kN/m

qscu = constant = 4 kN/m2

i ) (B.19)

The calculation of the trial cross-section area of the stay (AT,i ) is completed as follows:

σT,i =

FT,i (qcp + qscu )Aco,i → σT,i AT,i sin(βi ) = (qcp + qscu )Aco,i → AT,i = AT,i σT,i sin(βi ) (B.20)

344

Appendix B. Dimensions and characteristics of the models 9

Since the stress of the stay is limited to 40 % of its maximum resistance :

σT,i = 0.4σT,ult → AT,i

 B  (qcp + qscu )Aco,i Aco,i = ∆D → = 2  A 0.4σT,ult sin(βi ) co,i = ∆D B

(LCP)

(B.21)

(CCP)

Where the distance between consecutive anchorages along the deck (∆D ) depends on the span where they are located, as it has been explained in the preceding section. The cables employed are made of steel Y-1770, hence their maximum resistance is [ACHE 2007]:

σT,ult = 1770

MPa

(B.22)

Once obtained this rst trial value for the cable sections (AT,i ), the response of the model under self-weight is calculated.

Due to force redistributions, the stays

which connect the tower with areas of the deck constrained in vertical direction (abutments, intermediate piers and surrounding areas) receive more load, whereas the rest of the cables are unloaded. Consequently, the area is increased in the most loaded cables and reduced in the others. This process is iteratively repeated until the vertical deformation along the deck is reduced to a minimum value, verifying that the stress in each stay after the application of the self-weight is admissible. It is also checked that the self-weight of the deck only introduces compressions in the tower, but not bending moments, or in other words, the resulting horizontal force of the cable planes in the side span is equal to the horizontal force of the cable planes in the corresponding half of the main span (prior to the earthquake).

Simplied procedure Despite the aforementioned iterative procedure is the standard way to proceed and conducts to accurate distributions of cross-sections and prestress along the cablesystem, a simplied methodology has been adopted in this thesis due to the great number of cable-stayed bridges studied and to the fact that mainly the seismic forces are of a concern, not the eects of the self-weight or the fatigue of the stays. In the proposed simplied procedure, the trial area of the cables (AT,i , expression (B.21)) is considered as their nal cross-section, except in the case of the cables directly anchored to the abutments or the intermediate piers, whose area is incremented in order to balance the larger horizontal force exerted in the tower by the cable planes corresponding to the main span, because their slope is lower (since

LP /2 > LS ).

In gure B.14 this unbalanced force and the corresponding cables

anchored to vertically xed points in the deck are schematically represented.

9 The actual limit is 45 %, but the remaining 5 % is left to take into account other features, such as small extra-loads.

B.6. Characteristics of each stay. Cable cross-section

345

Figure B.14. Dierences in the horizontal forces exerted in the tower by the cable planes

in both sides (uniform load along the deck), besides the increment in the forces introduced by the cables in abutments (Habt ) and intermediate piers to balance them (Hpint ). The global forces exerted by the cable planes in both sides of the tower depicted in gure B.14 are obtained as follows:

Hp =

NT X

AT i,p σi,p cos βi,p

(B.23a)

AT i,s σi,s cos βi,s

(B.23b)

i

Hs =

NT X i

Where subindex

p

and

s

respectively indicate main and side spans.

If the same preload is assumed in all the cables

σi,p = σi,s = σ

(which is the

ideal resulting solution), the unbalanced force (∆H ) is:

NT X

∆H = Hp − Hs = σ

i

AT i,p cos βi,p −

|

NT X

! AT i,p cos βi,p

(B.24)

i

{z

}

∆A

It is supposed that this unbalanced force is resisted by the cables anchored to the abutments and intermediate piers in the side spans, maintaining constant their stress (σ ) and hence increasing their cross-section in

∆A∗

(which is safer than the

increase of their stress), proportionally to the inclination of each cable:

 ∆Aabt = ∆A

cos βabt cos βabt + cos βpint

 ∆Apint = ∆A



cos βpint cos βabt + cos βpint

(B.25a)

 (B.25b)

346

Appendix B. Dimensions and characteristics of the models

Finally, the area of the anchoring cables in the abutments (Aabt ) and intermediate piers (Apint ) is:

AT,abt

and

AT,pint

Aabt = AT,abt + ∆Aabt

(B.26a)

Apint = AT,pint + ∆Apint

(B.26b)

being the initial trial area of the cables anchored to the abut-

ment and intermediate piers respectively, obtained with expression (B.21). Note that the simplied procedure is independent of the prestress (σ ) since it is assumed to be equal in all the cables, thus facilitating the parametrization of the area of each cable exclusively in terms of the main span, which is paramount in the proposed parametrization of the models.

Appendix C Validation of the nonlinear model

It is essential to ensure that the assumptions employed in the representation of both the sectional and material behaviour lead to realistic load - displacement cyclic response in the proposed cable-stayed bridges. The purpose of this appendix is to verify the results obtained in the nonlinear seismic analysis with the developed Finite Element Models (FEM), by comparing the numerical results obtained with Abaqus [Abaqus 2010] and the experimental results oered by several researchers from laboratory tests on columns with dierent characteristics. Here, an experimental and numerical support for the assumptions made in nonlinear FEM is provided. First, a detailed description of the nonlinear material constitutive models is presented in this appendix, which is followed by the discussion about the appropriate nite element length considering the localization phenomena.

To accomplish this

task, numerical analysis with dierent types of nite element discretizations and boundary conditions are compared with monotonic and cyclic experimental results published by other authors on cantilever beams. This appendix is nished with the conclusions of specic studies about the sensitivity of the seismic response to several factors.

C.1 Materials: constitutive properties and assumptions C.1.1 Concrete The conned model of concrete proposed by Mander et al [Mander 1988] has been adopted, which is included in Eurocode 8 [EC8 2005a] and the Spanish seismic code [NCSP 2007], among others. The type of concrete considered is HA-40. When the elastic limit in compression (εcy ) is exceeded, the plastic deformation increases and the stiness of the material is reduced (`softening'). If the load is released beyond the linear range, the elastic stiness is somewhat lower than the initial

1

elastic slope due to the damage of the material , however this eect is neglected in the considered numerical model since it is not available using `beam' elements in Abaqus [Abaqus 2010]. If the concrete is subjected to tension, its response is linear and elastic up to the maximum compression in the unconned model (fc,t ), beyond this level, cracking is developed so fast that it is very dicult to observe in detail the behaviour. In

1 There are proposals to include the low-cycle fatigue damage in reinforced concrete structures under seismic excitations [Perera 2000].

348

Appendix C. Nonlinear nite element model

the employed model, cracking is a damage phenomena; elastic stiness is lost when cracking arises but the model does not track with individual cracks. The permanent deformation associated with the crack is not considered, it is assumed that the concrete cracks are completely closed when it is compressed again. According to Eurocode 2 [EC2 2004], the Young's modulus of the concrete along its initial elastic branch is

Ec = 35

GPa for HA-40, which is close to the values sug-

gested be seismic codes [EC8 2005a] [NCSP 2007] and research works [Mander 1988]. For this concrete category, the mean value of unconned concrete cylinder compressive strength is

fcm = fck + 8 = 48

MPa, where

fck = 40

MPa is the characteristic

strength. In order to consider the initial cracking of the reinforced concrete elements, Priestley [Priestley 2003] recommended the reduction of the Young's modulus of the concrete to half the value indicated above. Other authors agree with Priestley [Bonvalot 2006] [Cofer 2002], but in this study the cracking eects are specically included in the model of the concrete and hence the original modulus

Ec = 35

GPa

is employed without being reduced (in section C.4 the conclusions of a specic study on the sensitivity to this modulus is presented).

Constitutive behaviour of the concrete under compression The connement of the concrete is an essential feature for anti-seismic designs, since increases the resistance and, specially the ductility, which in turn magnies the capacity to dissipate energy by hysteresis loops.

This connement is achieved by

providing a triaxial stress state in the concrete through strong transverse reinforcement, and enough longitudinal rebars to ensure the yield of the transverse hoops under large compressions. A considerable number of research works and regulations propose constitutive models for conned concrete.

Here, the model of Mander et al [Mander 1988] is

adopted due to its broad acceptance in the nonlinear study of concrete structures, both from the standpoint of seismic regulations ([EC8 2005a], [NCSP 2007]) and research ([Priestley 1996], [Bonvalot 2006]).

Below, the model of Mander is pre-

sented, adapted in some details to Eurocode 8 [EC8 2005a]. Figure C.1 includes the stress-deformation response of the conned concrete employing the proposed model under monotonic compression, compared with the unconned concrete. The behaviour of the conned concrete is governed by the expression:

σc =

fcm,c xr r − 1 + xr

(C.1)

σc being the stress of the concrete (in [MPa]) associated with the total deformation (εc ) and fcm,c the maximum stress of the conned concrete under compression. The parameters x and r are obtained as: x=

εc εc1,c

(C.2a)

C.1. Materials: constitutive properties and assumptions

349

Figure C.1. Stress-strain curve of the conned and unconned concrete under monotonic compression loads (Mander et al [Mander 1988]).

r=

Ec Ec − Esec

(C.2b)

εc1,c = 0.5 % is the deformation of the concrete associated with the point fcm,c = 64.28 MPa (these values are justied below), Ec = 35 GPa and Esec = 12.9 GPa are respectively the tangent and se-

Where

of maximum compression whereas

cant elastic stiness. Both the maximum compression stress and the corresponding deformation may be obtained with the following expressions:

fcm,c

fcm = 48

  r σe 2σe = fcm 2.254 1 + 7.94 − − 1.254 fcm fcm    fcm,c εc1,c = 0.002 1 + 5 −1 fcm

(C.3)

(C.4)

MPa being the maximum stress of the unconned model.

Despite the model of conned concrete expressed in eq. (C.1) is nonlinear even for reduced deformations (see gure C.1), it may be considered that the softening branch starts when the stress exceeds

fcm ,

below that threshold the response is

assumed elastic and, consequently, the deformation marking the elastic limit is

fcm /Ec = 0.14 %.

Finally,

σe

εcy ≈

is the lateral eective conning stress when the hoops

yield, which may be obtained with the expression:

σe = Ke ρw fym ; fym = fs,y = 552

For rectangular hoops

MPa being the probable yield stress of the hoops,

(C.5)

Ke

is the

connement eectiveness factor associated with the reinforcement (which is considered

Ke = 0.6

2

in this thesis ) and

ρw

is the volumetric ratio of transverse

reinforcement, dened as the ratio between the volume of transverse reinforcement and the volume of surrounding concrete per unit length. Following the indications

2 A specic study about the eect in the results varying K has been conducted and the e conclusions are presented in section C.4.

350

Appendix C. Nonlinear nite element model

of the Spanish seismic code [NCSP 2007] about the reinforcement in conned members, the characteristic design of the section was included in gure 3.13, obtaining a transverse reinforcement ratio approximately equal to

ρw = 0.8

% (which is hardly

inuenced by the dimensions of the cross-section and is thus assumed independent of the model proportions, and in turn, of the main span length). This ratio yields

σe = 2.65 MPa by employing expression (C.5), and nally the aforementioned values of the extreme stress and associated deformation in equations (C.3) and (C.4) are obtained. The ultimate deformation of the conned concrete (εcu,c ) is a key value in the design and is suggested by Priestley [Priestley 1996] in a conservative way through the following expression, which has been also adopted in Eurocode 8 [EC8 2005a] and NCSP [NCSP 2007]:

εcu,c = 0.004 + Where

1.4ρs fym εsu fcm,c

(C.6)

ρs = 2ρw applies for the type of hoops considered in this study (rectanεsu is the deformation of the steel in the point of extreme stress, which is

gular), and

assumed typically the ultimate strain of the material (see gure C.3). In this thesis

εsu = 0.114

is considered. Priestley stated that expression (C.6) is valid for the de-

sign stage, and if it is applied to reinforced concrete members with connement under exure and compression, the resulting values are, at least, 50 % conservative. This expression is applied in the present work and, considering the values included so far, the ultimate deformation of the conned concrete under compression is

εcu,c = 2.6

%. The concrete of the cover, outside the conned core between external and internal hoops, is unconned and therefore the corresponding constitutive behaviour presented in gure C.1 should be assigned to this area of the section. However, the employed discretization, despite follows the `ber section' approach, does not allow the denition of dierent types of concrete in the section. Another way to take into account the reduction of the mechanical properties in the cover concrete is to reduce the dimensions of the sections, but the result of numerical analysis conducted eliminating completely the cover, drew the same seismic responses.

Therefore, it

has been preferred to assume the conned concrete extended to the whole crosssection in order to facilitate the comparison between inelastic and elastic seismic analysis (which considers the unlimited elastic stiness of the concrete

Ec ).

This

simplication is supported since the cover is reduced compared with the core, and the demand of plastic deformation in the compressed concrete area of the section during the earthquake is moderate, as it has been observed. Hence, signicant differences are not expected if reduced properties are assigned to the cover concrete. The same approach was selected by other research works [Bonvalot 2006]. The conned behaviour of hollow sections, like the ones employed in the towers, is dicult to provide. The constitutive response of these sections is currently the objective of several works [Papanikolaou 2009b] [Papanikolaou 2009a]. Some questions about the applicability of current conned concrete models in hollow sections arise

C.1. Materials: constitutive properties and assumptions

351

in light of observed experimental tests on hollow sections, with signicant spalling under moderate loads and complex characteristic interactions between bending and shear in advanced loading stages[Takahashi 2000]. Nonetheless, the study of these eects is not the objective of the present work, which adopts a code-supported and broadly accepted conned concrete model presented above, developed also for hollow sections.

Constitutive behaviour of the concrete under tension The tensile behaviour of the concrete is dicult to predict as it has been mentioned earlier and, therefore, there are few research works which publish the stress-strain curve of the concrete subjected to tension. The eects associated with the concrete-rebar interface, like dowel eect or reinforcement slip, among others, are indirectly taken into account by means of the tension-stiening behaviour presented in gure C.2, trying to represent the load transfer to the longitudinal rebar through the cracks. The tension-stiening reduce numerical instabilities and is a key aspect in the seismic response as it has been veried in this work.

Typically, the tension-stiening curve resulting from tests

on plain concrete is enlarged in the numerical analysis to involve these eects in a simplied way, here, the original response obtained from the tests of Mazars on plain concrete is employed, since the same seismic response has been observed with enlarged curves (section C.4). According to [Priestley 1996], the cracking stress of a concrete member under bending is 7.5 % of the maximum stress in compression (unconned), and other authors suggest percentages between 7 and 10 % (leading in this case to 3.4 to 4.8 MPa), Eurocode 2 proposes 3.5 MPa for HA-40. However, a reduced value of the tension-stiening is considered in this thesis in light of the experimental results obtained by Mazars et al [Mazars 1989], which is a safe assumption close to the values recommended by seismic codes; the maximum strength of the concrete under

fc,t = 3 MPa, and the associated deformation marking the start εc,crack = 0.0086 %. The complete tension-stiening curve employed in

tension is assumed of cracking

the analysis, and presented in gure C.2, is a simplication of the laboratory tests results.

C.1.2 Reinforcement steel The category of the employed steel is B-500 SD. Figure C.3 presents the stressstrain curve of the longitudinal reinforcing steel considered in this thesis, which is in compliance with the indications of Eurocode 2 [EC2 2004], Eurocode 8 [EC8 2005a] and the Spanish seismic regulation [NCSP 2007]. The Young's modulus of the steel is

Es = 210000

εsy = 0.26 %, the deformation εsu = 11.4 % (as it was mentioned in the stress is fs,y = 552 MPa and the maximum

MPa, the elastic limit deformation is

associated with the maximum stress is preceding section), the expected yield stress is

fs,t = 1.2fs,y = 665

MPa. To complete the picture, the deformation which

352

Appendix C. Nonlinear nite element model

Figure C.2. Experimental behaviour of concrete under tension (Mazars

et al

[Mazars 1989]) and proposed idealization of the stress-strain curve (tension-stiening) considered in this thesis (red doted line).

marks the end of the horizontal plateau is

εsh = 1.40

%. The behaviour of the steel

is the same under tension or compression since buckling of the rebars is not directly considered.

Figure C.3. Stress-strain curve of B-500 SD steel considered in this work under monotonic test (compression or tension)

Several analyses about the sensitivity of the seismic results regarding the hardening type (isotropic, kinematic or combined) has been performed, and the conclusions are presented in section C.4. Linear kinematic hardening is employed in order to reduce the computational cost, whereas the Bauschinger's eect is taken into account.

C.2. Mesh sensitivity: localization in hollow sections

353

C.2 Mesh sensitivity: localization in hollow sections C.2.1 Motivation Considering linear analysis, the shorter the element length provide more accurate results, however this is not necessarily true in nonlinear analysis, where plastic strains localize into narrower and narrower bands as the elements get shorter, which may introduce large errors in the prediction of the response [Needleman 1988], specially in terms of displacement ductility and if negative stiness (due to `softening') is present

3

beyond the linear response

[Légeron 2005]. The reason behind this undesirable ef-

fect is that strain-softening plasticity (like the one associated with the concrete and reinforcement steel), when it is deprived of a characteristic length, could lead to ill-posed boundary value problems generating load-displacement responses with excessive softening.

Several authors have proposed solutions to this problem; by

including visco-plasticity [Needleman 1988]; nonlocal formulation by modifying the softening plasticity and damage model [Bazant 1984] [Lu 2009]; or dening adapting meshing controls (ALE), among others.

However, these approaches undoubt-

edly complicate the picture and may introduce additional uncertainties, therefore, proposing optimized element lengths (constant throughout the analysis) which, employing standard material models, describe properly the load-displacement response of the structure beyond the linear range, seems very appealing [Bazant 1979].

In

this direction, several authors suggested `Beam' elements with the same length as the equivalent plastic hinge [Légeron 1998] [Légeron 2005], proposed by dierent research works [Park 1975] [Priestley 1996] [Dowell 2002]. As it was previously mentioned, the seismic behaviour of hollow sections members is currently the subject of numerous studies, since they are often used on bridges piers and their response may be rather complex; they present a characteristic interaction of bending and shear in advanced loading stages, and spalling of the inner cover in initial stages, among others [Takahashi 2000], furthermore, the eectiveness of the conning eect in hollow sections is not obvious [Papanikolaou 2009b] [Papanikolaou 2009a]. On the other hand, most of the laboratory tests conducted on concrete structures available to date are referred to simple cantilever columns with 2D response [Takahashi 2000] [Orozco 2002], whereas real hyperstatic structures develop complex redistributions which may aect the plastic hinge length in dierent parts. The main objective of this section is to study the optimum nite-element discretization in concrete frames with hollow sections, which form the towers of cablesupported bridges under 3D seismic excitation (where the axial load is signicantly variable due to the earthquake), based on simple estimations of the plastic hinge length.

3 Negative stiness may be present in the pushover analysis of cable-stayed bridges due to the geometric nonlinearity, favoured by the large level of axial load in the towers (exerted by the weight of the deck and the cable-system).

354

Appendix C. Nonlinear nite element model

C.2.2 Plastic hinge length Sometimes, the length of the plastic hinge is considered equal to half the depth of the section, as it was proposed by [Park 1975] and established by some seismic codes [NCSP 2007] in a simplied way. A more realistic solution was suggested by Priestly [Priestley 1996] in order to obtain the equivalent plastic hinge length in concrete structures, presented in chapter 3 and repeated here for convenience:

lp = 0.08 Lc + 0.022 fs,ye φbl ≥ 0.044 fs,ye φbl Where

Lc

(C.7)

is the distance between the plastic hinge critical section and the point

of contra-exure (in meters), and

Stress in [MPa]

fs,ye = 1.1fs,y = 607

φbl = 0.032

m is the diameter of longitudinal rebars

MPa is the design yield stress of reinforcement steel.

Eurocode 8 (part 2) [EC8 2005a] suggests an expression which yields results very similar to eq. (C.7). The proposal of Priestley takes into account the size of the structure, the solicitation and the boundary conditions by means of the length expressions are available in the literature (e.g.

Lc .

More advanced

Dowell and Hines [Dowell 2002]),

but more parameters are involved and hence unavoidable uncertainties arise. The expression (C.7) is widely accepted since more than fteen years and it has been veried under experimental tests by dierent authors, which supports its use in the present thesis. However, a certain concern appears when expression (C.7) is applied in hollow sections like the ones used in the studied towers, since it was developed for solid sections. In this paragraph, numerical analysis are conducted and contrasted with experimental results on hollow sections reported by other authors, observing accurate plastic hinge lengths when Priestley's expression is employed under certain conditions. The value of

Lc

is straightforward if the rst vibration mode of a cantilever

beam is considered, but it is uncertain when dealing with complex three-dimensional structures subjected to multi-directional ground motions, since dierent deformation and hence values of

Lc

4

are associated with dierent vibration modes . On the other

hand, if the structure is subjected to strong seismic shaking, and the limits of the linear behaviour are exceeded, the vibration modes change and consequently their mode shapes, which may yield dierent values of

Lc

as the inelastic plastic

deformation is developed. Furthermore, the expressions available for the estimation of the plastic hinge length, like the one presented in (C.7), have been developed from the experimental results in cantilever

5

beams, and the considered towers are

hyperstatic structures where redistributions may aect the length of the plastic hinge.

4 This

is the case of cable-stayed bridges, where the response of the towers in the transverse direction is analogous to a cantilever structure, whilst an encastred-pinned beam response could be associated with the longitudinal direction [Walker 2009] (veried in section 2.2.1). 5 There are a number of research works proposing expressions for the plastic hinge length in concrete frames, but their implementation in the studied towers is uncertain.

C.2. Mesh sensitivity: localization in hollow sections

355

Since it is impossible to consider several element lengths at the same time to optimize the response in dierent directions, a xed value is considered (constant along the earthquake); in accordance with the observed deformation of the towers

Lc = 0.2Htot as a compromise value for both longitudinal and transverse response of the tower (Htot is the total height of the tower, from the foundation level) and Lc = 0.5B for transverse struts (where B = 25 m is the width of the deck). It is veried in the following paragraphs that

during the seismic analysis, it has been selected

this simplied discretization is valid for the tower models.

C.2.3 Solid-section cantilever beams. Monotonic test Despite we are interested in the cyclic response, monotonic loads up to the structure collapse are more commonly applied in the laboratory tests. The study of the mesh sensitivity starts with the results given by Prota et al [Prota 2007], who studied square solid section with 300 mm side and 2000 mm high. The interested reader may obtain more details about the laboratory test in the mentioned reference. Dierent FEM have been developed in the present research to simulate this experimental test; `beam' elements with linear interpolation (B31), `beam' elements with quadratic interpolation (B32) and `brick' elements with explicit denition of longitudinal and transverse rebars embedded in the concrete section (C3D20). The models with `beam' elements employ the `ber section' approach and include shear deformation (`Timoshenko beams'). FEM with `brick' elements include a `damage

6

model' of the concrete . Figure C.4 includes the load-displacement curve obtained experimentally by Prota, compared with B31 FEM results considering several element lengths. The importance of the element length in the ductility of the FEM is observed, obtaining smaller plastic hinge lengths and ductilities by reducing the size of the element. However, in agreement with Légeron et al [Légeron 2005], the sensitivity of the maximum load to the element length is hardly appreciable. Both experimental and numerical results considered the collapse when the same ultimate deformation (in the steel or the concrete) is achieved. In order to study the error in the prediction of the ductility demand by FEM, the following relative error index is proposed:

e= Where

uref ult = 80

F EM uref ult − uult

uref ult

mm [Prota 2007] and

· 100

uFultEM

(C.8)

are respectively the ultimate

displacement given by the laboratory test and FEM in this case (dierent values are employed in the following sections). The evolution of this relative error in terms of the nite element length in the FEM proposed is collected in gure C.5, besides the corresponding plastic hinge

6 The `damage model', unlike the `smeared concrete' employed in beams, is able to represent the cyclic damage by reducing the elastic properties, which is not meaningful in monotonic tests, but it could be appreciable in the studies conducted in subsequent paragraphs.

356

Appendix C. Nonlinear nite element model

Figure C.4. Load-displacement curve obtained in a solid cantilever column under monotonic test (Prota et al [Prota 2007]), compared with B31 FEM solution proposed here, considering dierent element lengths. lengths given by the expression (C.7) (lp,Priestley rule of half the section depth (lp,0.5D

= 150

= 254

mm) and according to the

mm). In light of the results, the special

sensitivity of the ductility to the element length in models with `beam' elements is clear (the evolution of the error is almost linear with the element length), which is reduced by considering the computationally expensive `brick' elements (although bad results may be given by `brick' models if the element length is too large to capture properly the exure of the beam). On the other hand, the results considering quadratic elements are virtually identical to the ones given using linear elements with

7,8.

half length

The prediction of the plastic hinge given by Priestley is in this case

too large, probably due to the second term of the expression (C.7), which aims to take into account the penetration of the rebars, being not clearly applicable to the conditions of the laboratory test [Prota 2007]. The ultimate deformation of `brick' FEM with dierent discretizations is shown in gure C.6, remarking the plastic hinge. It is again veried that these models yield results independent of the mesh, provided that the element length is small enough to represent the global bending.

If B32 elements are employed with the plastic

hinge length predicted by `brick' FEM (lp,F EM

≈ 200

mm), reduced errors are

observed in gure C.5. Therefore, `brick' FEM practically overcome the localization problem but, unfortunately, they are too computationally expensive to represent the whole tower. However, `brick' models may be employed to address the localization phenomena in special cases, when no experimental results are available, as it is

7 This

is a reasonable result, since elements with quadratic interpolation aggregate an auxiliary node between element ends. 8 This result has been observed by simply dividing by two the horizontal axis scale in the results obtained with B32 elements.

C.2. Mesh sensitivity: localization in hollow sections

Relative error; e [%]

150

357

Beam FEM. B31 Beam FEM. B32 Beam FEM. B32 Lelem/2 3D brick FEM. C3D20

100

50

0

-50

50

100

150

200

Element length;

250

300

[mm]

350

400

Figure C.5. Relative error in the ductility predicted by FEM with dierent element lengths

(the reference is the experimental result of Prota et al [Prota 2007]). Monotonic test. Solid section. `Lelem /2' means that the horizontal axis is simply divided by two when the results are represented.' conducted in the next paragraph.

PE, PE(ZZ) +5.504e-01 -1.000e-03 -3.833e-03 -6.667e-03 -9.500e-03 -1.233e-02 -1.517e-02 -1.800e-02 -2.083e-02 -2.367e-02 -2.650e-02 -2.933e-02 -3.217e-02 -3.500e-02

z x

Figure C.6. Ultimate deformed shape and plastic deformation contours in 3D `brick' FEM with dierent element sizes. The plastic hinge length predicted is included. lp,F EM ≈ 200 mm.

C.2.4 Hollow-section cantilever beams. Monotonic test There are published results about laboratory tests on cantilevers with hollow sections under cyclic loads [Takahashi 2000], unfortunately, the ultimate load is not clearly established. A 3D `brick' FEM, illustrated in gure C.7, is employed here to simulate such test under monotonic loads.

The ultimate displacement obtained with this

model (which is independent of the element length as it was veried in the previous

358 section) is

Appendix C. Nonlinear nite element model uref ult = 73

mm.

The inclination of the cracking pattern in advanced

loading stages is well predicted in the 3D `brick' FEM, an inherent characteristic of hollow-section structures [Takahashi 2000].

ZY

ZY

X

X

Figure C.7. 3D `brick' FEM employed to simulate the laboratory test of Takahashi and Iemura [Takahashi 2000].

The expression (C.8) is employed again in order to obtain the relative error introduced by linear `beam' FEM (B31) with dierent element lengths in gure C.8, along with the values of the plastic hinge predicted by Priestley's rule, mm, and the half depth of the section, lp,0.5D

= 160

lp,Priestley = 167

mm. An important conclusion

may be extracted from gure C.8; if the structure has concrete hollow sections, like the towers of the proposed cable-stayed bridges, the optimum element length using linear `beam' elements (B31) is equal to half the plastic hinge length predicted by the expression of Priestley (C.7) or the rule of the half depth, which is similar to the result obtained with solid sections.

Hollow section test

150

Beam FEM. B31

Relative error; e [%]

100

50

0

-50

40

60

80

100

120

140

160

180

200

Element length; Lelem [mm]

Figure C.8. Relative error in the ductility predicted by FEM with dierent element lengths

(the reference is the result of `brick' FEM simulating the experimental test of Takahashi and Iemura [Takahashi 2000]). Monotonic test. Hollow section.

C.2. Mesh sensitivity: localization in hollow sections

359

Moreover, motivated by the real bending moment distribution of the towers, the eect of the boundary conditions has been studied. The result of simple and double cantilever beams under monotonic test up to failure was compared. No published research works were found by the author with such double-cantilever conguration, and hence the result of the `brick' model is employed again as the reference value. Figure C.9 presents both FEM ultimate deformations, along with the plastic hinge lengths predicted.

Considering the expression of Priestley (C.7) in the cantilever

lp,Priestley = 167 mm (Lc = L = 1.2 m), whereas in the double-cantilever lp,Priestley = 119 mm (Lc = L/2 = 0.6 m). Due to the second term in expression (C.7), which does not depend on the distance Lc , the sensitivity to the boundary model

conditions is reduced. The plastic hinge lengths predicted by Priestley are close to the reference results drawn by the `brick' FEM, however the rule of the half depth is deemed inappropriate since it does not depend on the boundary conditions, so it will be discarded for mesh optimization in the towers of the proposed bridges.

PE, PE(ZZ) (Avg: 75%) +2.182e-01 -1.000e-03 -1.777e-03 -2.554e-03 -3.331e-03 -4.108e-03 -4.885e-03 -5.662e-03 -6.440e-03 -7.217e-03 -7.994e-03 -8.771e-03 -9.548e-03 -1.032e-02 -1.033e-02

Figure C.9. Ultimate deformed shape and plastic deformation in `brick' FEM. Simulation

of the laboratory test of Takahashi and Iemura [Takahashi 2000] with dierent boundary conditions. Nonetheless, the concentration of plastic deformation of dubious origin in the center of the double-cantilever beam in 3D `brick' FEM should be remarked, probably due to the inappropriate representation of the shear stress, leading to excessive distortion of the elements in the center of the column.

C.2.5 Full cable-stayed bridge FEM The study about the localization phenomenon ends with the seismic analysis of one proposed cable-stayed bridge model (`H'-shaped towers with

LP = 200 9

on soft soil) employing dierent discretizations in the towers

m founded

with linear `beam'

9 The nite element length in this analysis is considered constant along the height of the towers and the transverse struts.

360

Appendix C. Nonlinear nite element model

elements (B31). Both Pushover (NSP) and nonlinear response history analysis (NLRHA) have been performed. Pushover was conducted up to the collapse in models representing exclusively the towers (with proper boundary conditions) in transverse and longitudinal directions independently, including in gure C.10 the results corresponding to a load pattern proportional to the rst mode (transverse or longitudinal). The following conclusions deserve to be remarked; (i) the transverse response of the towers is more aected by the element length than the longitudinal behaviour, due to the increased complexity of the former, caused by redistributions in the connections of the struts and both legs of the towers; (ii) the mesh sensitivity is larger the more inelastic the response; (iii) the ductility not always increases with the element length, as it was the case in the cantilever column, probably due to force redistributions mainly in the connections with the struts; and (iv) by modifying the element length more than 150 %, the ultimate displacement variation is lower than 40 %, which suggests that the localization phenomenon is not crucial even in advanced loading stages of the towers.

16

10

14 8

12 10

6

8 4

6

= 0.8 m = 1.5 m 0

50

100

150

200

250

= 1m = 1.5 m

2

= 2m 0

= 0.8 m

4

= 1m

2

= 2m 0

300

0

200

400

600

800

1000

1200

1400

1600

Figure C.10. Total base shear versus control point displacement (deck-tower connection)

pushover curves with dierent element lengths in the towers; (left box) transverse response; (right box) longitudinal response. Model representing exclusively the tower. H-LCP with LP = 200 m. Foundation soil type D.

Nonlinear dynamics, with dierent element lengths in the discretization of the towers, revealed that the seismic response earthquakes (ag

= 0.5

10

of full cable-stayed bridges under strong

g) is hardly inuenced by localization phenomena, as it is

demonstrated in gure C.11. Again, the mesh sensitivity is somewhat stronger in the seismic forces associated with the transverse response of the bridge (transverse shear

VY

10 The

and axial load

N ).

seismic response was studied in terms of forces and displacements.

C.2. Mesh sensitivity: localization in hollow sections

361

22 17

1.0

0.8

= = = =

0.6

0.4

0.5 m 1m 1.5 m 2m

0.2

0.0

0

10

20

30

40

4

8

12

16

1.5

3.0

4.5

6.0

Figure C.11. Extreme seismic forces recorded along the height of the tower with dierent element lengths (N ; axial load, VY transverse shear, VX longitudinal shear). H-LCP full model with LP = 200 m and θZZ xed. Soil type D.

C.2.6 Conclusions and application rules on the discretization of the towers The following conclusions have been extracted about the the localization phenomenon in simple columns and cable-stayed bridges towers:

•

The length of the plastic hinge proposed by [Priestley 1996] (expression (C.7)) is also valid in concrete members with hollow sections and dierent boundary conditions.

•

The optimum element size is half the length of the plastic hinge if linear `beam' elements (B31) are employed. Quadratic elements (B32) require the denition of an auxiliary node between both element ends, which slightly complicates the model denition, whereas the results are not improved in comparison with linear elements employing half the total length of B32.

•

`Brick' models are adequate to obtain reference values of the ultimate column displacement if no experimental results are available from laboratory tests, however, representing the whole tower with these elements is too expensive from a computational point of view.

•

The mesh sensitivity is reduced in the extreme seismic forces of the towers because of two reasons; (i) the element length is more inuencing in the prediction of displacements than maximum forces [Légeron 2005]; (ii) the threedimensional response is composed of the contributions of a number of vibration modes, some of them remaining completely elastic and hence without mesh sensitivity; and (iii) the seismic demand of inelastic deformation in the concrete is not generally extremely important, considering the proposed seismic

362

Appendix C. Nonlinear nite element model action (with

ag = 0.5

g) and the `H'-shaped tower model studied in this ap-

pendix (bridges with lower diamond and reduced spans could present larger mesh sensitivity due to important inelastic demands, chapter 8). The pushover analysis is more inuenced by the element length close to the failure point.

In light of the results, linear `beam' elements (B31), with `ber section' approach and longitudinal rebars specically dened, are validated for the nonlinear study of the towers in cable-stayed bridges subjected to seismic excitation. Taking into account the observed seismic deformation of the towers in order to obtain the distance

Lc ,

the following discretization with linear beam elements (B31) is employed:

Lower piers and inclined legs In light of the observed seismic response of the tower legs, a reasonable value for the distance between the foundation and the point of zero bending moment is 20 % of the total height

11

of the tower

Htot = 1.5H = 1.5LP /4.8

(see appendix B),

therefore, the following discretization is employed in the inclined legs (between the anchorage area and the level of the lower strut) and lower piers (below the lower strut):

Lleg elem =

lp 1 1 = (0.08 (0.2 Htot ) + 0.022 fs,ye dbl ) = (0.005 Lp + 0.427) 2 2 2 12

This expression

1.71

yields element lengths in the tower legs ranging from

m for bridges with main spans of

200

to

600

(C.9)

0.71

to

m.

Anchorage area In the anchorage area of the tower, a constant element length is proposed:

Lelem = 2

m, in order to facilitate the modelling due to the requirements of the anchorage locations.

This is deemed acceptable due to the reduced seismic demand in the

anchorage area, compared with the one in the foundation level.

Transverse struts It has been observed that the transverse struts behave almost like a double cantilever beam during the seismic excitation, and hence

Lc

is approximately half the length

of each strut, which, in order to simplify the model, is considered in all the cases equal to the deck width (B

11 It

= 25

m).

has been adopted Lc ≈ 0.2Htot both in longitudinal and transverse directions, which facilitates the discretization. 12 It is considered d = 32 · 10−3 m since it is the diameter of the external longitudinal rebars, bl which is dierent from the inner reinforcement (being the diameter of these rebars 25 · 10−3 m).

C.3. Experimental verication employing cyclic tests on columns Lstrut elem =

lp 1 = (0.08 (0.5 B) + 0.022 fs,ye dbl ) 2 2 Lstrut elem = constant = 0.71

363

(C.10)

m

C.3 Experimental verication employing cyclic tests on cantilever columns The proposed material constitutive behaviour and FEM have been veried under monotonic tests in the preceding section, however, the seismic excitation subjects the concrete members to strong cyclic loads, which origins cumulative sources of damage. In this section, the validation of the employed nonlinear FEM under cyclic loading beyond the linear range is performed by comparing its results with dierent experimental load-displacement curves published by other authors on cantilever columns. The optimum nite element discretization suggested above has been employed in this section; linear `Beam' elements with length equal to half the corresponding plastic hinge obtained by expression (C.7). Three research works have been selected; Takahashi and Iemura [Takahashi 2000] focused their work on columns with dierent hollow sections; Sakai and Unjoh [Sakai 2006] analyzed several columns with solid circular sections; nally, Orozco and Ashford [Orozco 2002] published results about columns with solid circular sections and cyclic loads which aim to represent the seismic action. The details of each research are briey included here, but a thorough discussion may be found in the corresponding papers.

C.3.1 Takahashi and Iemura's tests Figure C.12 illustrates one typical hollow section studied in the experimental program conducted by Takahashi and Iemura [Takahashi 2000]. Dierent columns were tested under quasi-static loads, varying the transverse reinforcement ratio (ρw ), the compressive load, and the slenderness ratio between the height of the column and the side of the transverse cross-section (l/d). Table C.1 summarizes the proposed studies, which have been simulated here with linear `beam' (B31) FEM. It is noteworthy that, observing the reduced longitudinal reinforcement employed in these tests, the eectiveness of the connement is compromised; in the opinion of the author of this thesis, the three-axial stress state required in conned concrete is dicult to be fully provided with such reduced amount of longitudinal reinforcement. In order to reduce the amount of information provided in this appendix, only the validation of the FEM with H4-4 test is presented in gure C.13, but the rest of the columns described in table C.1 have been also simulated. A good correlation was generally obtained between FEM and experimental cyclic results in hollow sections. The following conclusions may be drawn from the validation with these tests; (i) complex shear-bending interactions arise in advanced

364

Appendix C. Nonlinear nite element model

Figure C.12. Cross section and elevation of one typical column tested by Takahashi and Iemura [Takahashi 2000]. Measurements in mm.

Quasi-static test FEA Results

Figure C.13. Comparison between the FEM model proposed here and the quasi-static experimental results obtained by Takahashi and Iemura [Takahashi 2000]. H4-4 test (description in table C.1). FEM with linear `Beam' elements B31. Quasi-static load.

C.3. Experimental verication employing cyclic tests on columns Unit Column H4-1 H4-2 H4-4 H2-2

l/d 4.0 4.0 4.0 2.0

365

Axial applied stress [MPa] Stirrup Ratio [%] 0.0 3.7 3.7 3.7

0.34 0.17 0.34 0.34

Table C.1. Summary of the of the experimental program conducted by Takahashi and Iemura [Takahashi 2000].

loading stages, characteristic of hollow sections [Takahashi 2000], leading to a reduction of the stiness which is not fully predicted in `beam'-type FEM; (ii) the lower the reinforcement ratio, or the slenderness of the column (if this factor is lower than four,

l/d < 4),

the lower ductility and worst correlation between experimental re-

sults and FEM predictions, but the reinforcement and tower sections employed in this thesis avoid such poor response; and (iii) the `pinching' eect, which cause the reduction of the unloading stiness cycle to cycle, is not captured by the considered FEM, leading to a slight overestimation of the dissipated hysteretic energy with the proposed numerical model.

C.3.2 Sakai and Unjoh's tests The tests performed by Sakai and Unjoh [Sakai 2006] in a column with solid circular section under quasi-static loads are schematically represented in gure C.14. The comparison with FEM results is included in gure C.15, along with the numerical results of Sakai and Unjoh [Sakai 2006]. Good correlations were again obtained.

Figure C.14. Cross-section and elevation of one typical column tested by Sakai and Unjoh [Sakai 2006].

366

Appendix C. Nonlinear nite element model

Quasi-static test FEA Results (Sakai, 2006) FEA Results (this thesis)

Figure C.15. Comparison between the FEM model proposed here and the experimental and numerical results obtained by Sakai and Unjoh [Sakai 2006]. Quasi-static load.

C.3.3 Orozco and Ashford's tests Orozco and Ashford [Orozco 2002] conducted several tests of circular solid columns under dynamic cyclic loads in order to address the eect of near-fault records in bridge piers. Figure C.16 includes the main characteristics of the laboratory tests, whereas the time-history amplitude of the imposed load is described in gure C.17. The correlation with FEM results

13

is shown in gure C.18, observing a good re-

sponse with the numerical model.

Figure C.16. Cross section and elevation of the column tested by Orozco and Ashford

[Orozco 2002].

13 Only one pulse of the load per each stage is considered in the proposed FEM, since the damage of the elastic stiness is not included and, thus, the response under identical cycles is the same.

C.3. Experimental verication employing cyclic tests on columns

367

Figure C.17. Time-history of the imposed dynamic loads at the top of the column tested. Orozco and Ashford [Orozco 2002].

comprobacion de Bonvalot - esayos de Orozco

Experimental test (Orozco, 2002) FEA Results

Figure C.18. Comparison between the FEM model proposed here and the experimental results obtained by [Orozco 2002]. Dynamic load simulating a seismic event.

368

Appendix C. Nonlinear nite element model

C.4 Model optimization Specic dynamic and static studies

14

about the sensitivity of the seismic response

to several factors have been performed in order to address the level of uncertainties involved in the analysis, and to x these parameters to optimize the accuracy of the solution while reducing the computational cost. The results obtained are too large to be included in this appendix, which just summarizes the main conclusions in tables C.2 and C.3.

14 The model with `H'-shaped towers, 200 m main span, θ ZZ xed and foundation on soft soil has been considered in all the optimization studies conducted in this paragraph.

C.4. Model optimization

Number of integration points

369

More than 5 Integration Points (IP) through the depth are recommended if nonlinear behaviour with sign reversals is expected [Abaqus 2010].

It has been observed that

the seismic response of the bridge is almost the same by including 7 or 21 IP in the `beam' elements representing the towers, therefore 7 IP are considered along the whole tower.

Concrete Young's modulus

Two values of the concrete Young's modulus in the towers were considered; the value suggested by the codes

Ec ≈ 35

GPa and half

this quantity. Despite the ductility and calculation agility are larger with reduced values of this modulus, the extreme response along the towers is nearly the same and the recommended value

Ec = 35

GPa is

adopted.

Connement eectiveness Ke

The

original

model

of

[Mander 1988] propose

Mander

Ke = 0.6

et

al

in hol-

low sections, but Eurocode 8 [EC8 2005a] suggests

Ke = 1.0

if all the reinforce-

ment requirements are met.

It has been

observed that the seismic response is nearly the same considering both values and, therefore

Ke = 0.6

has been adopted since it

falls on the safe side.

Concrete tension - stiening

Several tension-stiening curves have been considered in accordance with the proposals of dierent authors.

It has been observed

that the eects caused by dierences between these curves are appreciable only in the initial stages of the excitation, close to the rst cracking and yielding in the tower. The proposal of Mazars et al [Mazars 1989] has been considered (gure C.2).

Table C.2. Summary of the conclusions in the specic studies about FEM.

370

Appendix C. Nonlinear nite element model

Reinforcement hardening

It was observed that the seismic response is independent of the type of hardening considered in the steel of the longitudinal rebars (linear kinematic or combined kinematic/isotropic). In order to reduce the computational time, linear kinematic hardening is considered.

HHT force residual tolerance

Automatic step-time selection is employed in HHT integration scheme (used in nonlinear dynamics), which requires dening (prior to the analysis) a force residual tolerance (Rt+∆t/2 ) that helps the nite element package to decide if the step needs to be reduced or not, before the next step-time is started [Abaqus 2010].

Dierent values of

this residual force have been provided, and the seismic results are very similar. value

Rt+∆t/2 ≈ 0.1Pchar =

The

106 N has been

adopted (Pchar is the characteristic reaction of the model during the earthquake, which is obtained in the model with 200 m main span and applied in the rest of structures regardless of their dimensions, which falls on the safe side), ensuring an accurate result even if the response remains elastic.

Table C.3. Summary of the conclusions in the specic studies about FEM (continued).

Appendix D Study about the spatial variability of the seismic action

Large-span cable-stayed bridges are long and hyperstatic structures where the eects of nonsynchronous input motions may be important; the seismic waves aecting each foundation along the structure are uncorrelated due to the nite propagation velocity or local changes in the mechanical properties of the surrounding subsoil. Nonetheless, the great exibility of these bridges reduces the problems associated with the asynchronous movement of the towers and other supports.

Despite the

multicomponent nature of the seismic signals is neglected in the body of the thesis, this feature is characteristic of cable-stayed bridges and deserves a specic study in order to address its possible inuence. According to Eurocode 8 (part 2) [EC8 2005a], the spatial variability of the seismic action should be considered in the analysis if one of the following conditions is satised; (i) the subsoils surrounding the foundation of dierent supports are dierent; (ii) if the deck is continuous and its length is larger than the limit value, xed as 40 % of the distance

Lg

(the seismic action is assumed completely uncorrelated

beyond this length); this normative suggests

Lg = 300

Lg = 600

m for rocky soils (TA) and

m for soft soils (TD), consequently, the asynchronism of the imposed

accelerograms needs to be considered if the length of the deck is larger than 240 or 120 m considering rocky or soft soils respectively, which is clearly below the total length of the considered structures, between 360 m (if

LP = 600

LP = 200

m) and 1080 m (if

m). In the Spanish code [NCSP 2007], the spatial variability should be

addressed if the total length of the deck is larger than 600 m. This appendix includes the main results of a study about the eect of the spatial variability in one of the proposed cable-stayed bridges; the model with inverted `Y'shaped towers, 400 m main span, lateral cable-arrangement, xed transverse rotation of the deck at the connection with the abutments (θZZ xed) and foundation on rocky soil (TA). First, the theory behind the phenomenon is presented in a practical way, besides the analysis approaches currently available for the designer, followed by a brief discussion of the results.

The appendix is closed extracting the main

conclusions. The study conducted in this appendix is completely elastic and the reference seismic action is the one proposed in chapter 5, but including the multi-component eects in dierent ways.

372

Appendix D. Spatial variability

D.1 Background theory and analysis framework The loss of synchronism in the seismic excitation between the support

i

and

j

may

be mathematically represented by the following expression, referred in the literature as the `coherency function':

γij (ω) = γij,1 (ω) · γij,2 (ω) · γij,3 (ω) Where

γ1 , γ2 , γ3

(D.1)

are functions that depend on the frequency

ω

and respec-

tively represent the causes of the loss of correlation in the seismic action according to Eurocode 8 (part 2) [EC8 2005a]; (1) the loss of correlation due to reections/refractions in the propagation medium,

γ1 ;

(2) the niteness of the wave-train

propagation velocity (vs ) and its angle of incidence at the surface (θ ),

γ2 ; and (3) the

ltering and local amplications originated by dierent soil conditions between the two supports,

γ3 .

If

γij (ω) = 1,

the seismic excitation of the support

with respect to the one aecting the support

i

is identical

j.

The loss of correlation caused by reections/refractions may be assumed negligible when exible structures with fundamental periods longer than 1 s are studied [Priestley 1996], which is the case of all the proposed cable-stayed bridges (where

T1 = 2 − 6

s, see chapter 4), hence

γ1 ≈ 1.

On the other hand, local soil conditions

are beyond the scope of this general study and require the analysis of the specic project location, therefore this eect is neglected here and

γ3 = 1.

Consequently,

only the delay of the accelerograms caused by the niteness of the propagation velocity and their angle of incidence is considered,

γij = γij,2 .

The asynchronous excitation of the supports along the bridge has two eects associated; (i) one component of the structural behaviour is entirely inertial,

Ei ,

and may be obtained by imposing exactly the same excitation in every foundation (its analysis is performed through any of the methodologies presented in chapter 6); (ii) the second component is due to the pseudostatic forces appearing as a result of the dierential movement of the foundations, excitation is merely synchronous.

Eps ,

being this term null if the

There are a number of procedures available to

take into account the pseudostatic component of the earthquake, or directly the total combined eect

Etot ,

with dierent levels of complexity and realism, which

may be grouped in general terms as follows:

1.

Static procedures based on imposed displacements at foundations: These are the simplest methods and the most removed from reality. The pseudostatic forces are assumed completely uncorrelated from the inertial contribution, and thus, their combination is performed through the SRSS expression:

Etot =

q

2 Ei2 + Eps

(D.2)

These pseudostatic forces are obtained separately, by imposing statically displacements at the foundations, trying to represent the worst possible eect

D.2. Static procedures based on imposed displacements

373

due to the spatial variability; according to Eurocode 8 [EC8 2005a], these displacements depend on the maximum ground displacement of the soil during the design earthquake (dg ), and tacitly on the propagation velocity of the wave

vs ,

whereas considering NCSP [NCSP 2007] or the procedure proposed by Priestley, they depend directly on this velocity. These procedures are presented in section D.2. Despite the static analysis may be inelastic, these approaches are intended for elastic calculations since the inertial eects are typically obtained through response spectrum analysis (MRSA), and the SRSS combination of both contributions is rooted in the superposition principle. 2.

Procedures based on the modication of the acceleration spectra: Through the analysis of the system of dynamics (2.7), separating the degrees of freedom corresponding to the supports from the rest, and employing statistical concepts, the acceleration spectra representing the synchronous excitation may be modied to take into account the spatial variability. There are several research works on this approach and it is included in the Eurocode 8 [EC8 2005a], but is beyond the scope of this study; the interested reader is referred to the reference paper of the procedure, published by Der Kiureghian and Neuenhofer [Der Kiureghian 1992]. These methodologies yield the total response of the structure,

Etot ,

combining the inertial and pseudostatic re-

sponses, and are suitable only for elastic responses. 3.

Procedures based on the time-domain analysis by means of asynchronous accelerograms and random vibration: The most general and rigorous procedure is to generate accelerograms from the matrix of spectral density functions

G(ω) which depend on the coherency functions, resulting ac-

celeration histories with the desired degree of correlation and frequency content [EC8 2005a].

In cable-stayed bridges, ignoring the eect of local soil condi-

tions, only the loss of synchronism due to the physical distance between supports and the nite wave propagation velocity may be considered (γij

= γij,2 ,

as it was stated before), consequently, the accelerogram imposed at any support

j

is exactly the same as the one aecting the support

by means of a certain time-lag

δt ,

i,

but delayed

maintaining the frequency content and the

acceleration ordinates of the record. This is the procedure recommended by [Priestley 1996] to conduct response history analysis with asynchronous excitations, and will be applied in one practical case in section D.3. The total response (Etot ) is obtained directly, and this approach is valid to consider both the spatial variability and the nonlinear seismic response of the structure.

D.2 Static procedures based on imposed displacements D.2.1 Eurocode 8 (part 2) proposal Two sets of imposed displacements (A and B) are proposed in Eurocode 8 [EC8 2005a] in order to assess the pseudostatic eects, applied separately along the supports of

374

Appendix D. Spatial variability

the bridge in the studied direction of the response;

•

d√ r imposed displacements in the same direction, being √ dr,i = Li dg 2/Lg ≤ dg 2 the one corresponding to the support i, where Li is the distance between the reference foundation and this support (dr,1 = 0, in this case the reference is the left abutment), dg is the maximum ground dis1 placement (dg = 0.1 m for rocky soil type A) and Lg was presented previously at the beginning of the appendix (Lg = 600 considering rocky soil type A). Set A is composed of

Accordingly, considering the cable-stayed bridge elevation with a main span of 400 m, the imposed displacements along the foundations for the set A are:

Set

  Left         Left

→ dr,0 = 0 mm √ 0.1 2 160 = 38 mm tower → dr,1 = 600√ A  0.1 2   Right tower → dr,2 = 560 = 132 mm    600 √    Right abutment → d = 0.1 2 = 141 mm r,3 abutment (reference)

(D.3)

These displacements are imposed separately in longitudinal (X) and transverse (Y) directions. It is noteworthy to mention again that, in the proposed model, the intermediate piers located between the abutments and the towers in the side spans only constrain the deck in the vertical direction, and therefore, they are not included in the imposed displacements, since the spatial variability of the vertical seismic action is not addressed in this appendix (neither is considered in studied seismic codes).

•

The possibility of opposite direction movements between adjacent supports is covered through the set B, and hence the resulting displacements are imposed in opposite directions between them. The displacement is obtained as

√

di =

±0.5βr Lαv,i dg 2/Lg , where Lαv,i is the average distance from one support βr is a factor which takes into account the magnitude of

to the adjacent, and

the opposite displacements between adjacent foundations, being recommended

βr = 0.5

when the surrounding subsoil is the same along the bridge (which is

the case in this study). Thus, the following displacements are obtained in set B (applied in longitudinal and transverse directions separately), considering the two possibilities in the opposite movement of the supports (one to the right and the adjacent one to the left, or vice versa) through the sign at the beginning of the following expression:

1 According

to Eurocode 8, the maximum ground displacement related to rocky soils (TA), type 1 spectrum and ground acceleration ag = 0.5 g is: dg = 0.025ag STC TD = 0.025·5·1·0.4·2 = 0.1 m, being S the amplication factor due to the soil class (S = 1 for the reference rocky soil type A), whereas TC and TD represent corner periods marking dierent areas of the design spectrum.

D.2. Static procedures based on imposed displacements

Set

375

√ 0.1 2 160 = ±10 mm abutment (reference) → d0 = ±0.25 600 √   0.1 2 160 + 400 tower → d1 = ±0.25 = ±17 mm 600 2 B √    160 + 400 0.1 2   = ±17 mm Right tower → d2 = ±0.25    600 2   √    0.1 2   Right abutment → d3 = ±0.25 160 = ±10 mm 600     Left           Left

(D.4)

The responses due to both sets of displacements are not combined, instead, the pseudostatic eects are studied separately in transverse and longitudinal directions, being the maximum response between set A and B, the result of the pseudostatic component in each direction (Eps ), which is nally combined with the inertial component (Ei ) by means of SRSS rule, expression (D.2), obtaining this way the total eects (Etot ) in each horizontal direction. The combination of the resulting components in the three principal directions (the vertical one without pseudostatic eects) is carried out through the SRSS rule, according to the methodology detailed in section 6.3.3 for classical modal spectrum analysis (MRSA). Figure D.2 presents the pseudostatic forces in the right tower, the furthest one

2,3,

from the epicenter of the earthquake

when the bridge is statically subjected to

the Eurocode 8 compliant imposed displacements in the elastic range.

D.2.2 Spanish code proposal: NCSP Only one set of displacements is suggested by the Spanish seismic code for bridges, NCSP [NCSP 2007], which are inversely proportional to the wave-propagation velocity

vs ,

inuenced in turn by the subsoil class; larger imposed displacements are

obtained as the soil gets softer, like considering the Eurocode 8, and meeting the physical reality. However, the sign of the displacements is not specied by NCSP, in this work they are assumed in the positive sign of

X

or

Y

axis.

i, separated a distance Li from the di = Li vg /vs ≤ 2dg , where vg is the maximum hori-

The imposed displacement in the support reference one, is obtained as

zontal velocity of the ground in the supercial layers due to the design earthquake;

vg = 0.2STC ag = 0.2 · 1 · 0.4 · 5 = 0.4

m/s according to NCSP. Assuming that the

wave-propagation velocity in a typical rocky soil(TA) is

vs = 2000

m/s, which is

recommended by NCSP. The following imposed displacements are obtained:

2 The

excitation is clearly asymmetric if the spatial variability is considered, and both towers should be therefore distinguished in the study, nonetheless, similar results have been recorded. 3 All the results presented in this appendix are referred to the right tower.

376

Appendix D. Spatial variability   Left         Left

→ d0 = 0 mm 0.4 = 32 mm tower → d1 = 160 2000 0.4  Right tower → d2 = 560 = 112 mm    2000      Right abutment → d3 = 720 0.4 = 144 mm 2000 With

vs = 2000

m/s

abutment (reference)

4.

(D.5)

Again, these displacements are imposed separately in

longitudinal (X) and transverse (Y) directions. The pseudostatic forces obtained according to NCSP are included in gure D.2. By comparing expressions (D.3) and (D.5), the displacements imposed for rocky soil considering NCSP and Eurocode 8 are similar (compare expressions (D.3) and (D.5)), which is reected in the pseudostatic forces collected in gure D.2.

D.2.3 Priestley's proposal Priestley [Priestley 1996] proposed the incorporation of the incidence angle

θ,

be-

tween the propagation direction of the wavefront and the bridge axis (see gure D.1), in the denition of the imposed displacements. This procedure is based on the

max ) arises close to the instant when

fact that the maximum relative displacement (dm

its derivative with respect to time becomes greatest, which is never above the maximum velocity of the seismic signal equal (vg

= 0.4

vg ;

on the safe side, both velocities are assumed

m/s). In this case, the sign of the displacements is considered the

same for all the imposed movements (in

X

or

Y

axis).

Plan of the bridge

Bridge axis

Seismic waves propagation

Figure D.1. Scheme of the incidence angle θ of the seismic wavefront propagation with

respect to the axis of the bridge in plan, and projections of the peak ground velocity (vg ) in one support. 4 The

wave-propagation velocity may be obtained as the velocity of the S-waves; vs = G/ρ, considering granite, the shear modulus and density round G = 11 GPa and ρ = 2700 kg/m3 respectively, hence vs ≈ 2000 m/s. However, the denition of the foundation stiness (appendix B) assumes a lower value of the shear modulus, G = 1.5 GPa expression (B.12), which is also a reasonable hypothesis for rocky soils and falls on the safe side. p

D.2. Static procedures based on imposed displacements

377

Observing gure D.1, the ground velocity is decomposed in the longitudinal and transverse components in order to obtain the extreme relative displacements in these directions, resulting the expressions

dmi,x = vg cos θδt,i

and

dmi,y = vg sin θδt,i

in longitudinal and transverse directions respectively. Taking into account that the time-lag of the accelerogram between supports is

δt,i = Li cos θ/vs

(gure D.1), the following expressions are obtained for the extreme

relative displacements respectively in longitudinal and transverse directions;

vg Li cos2 θ vs

(D.6a)

vg Li sin θ cos θ vs

(D.6b)

dmi,x = dmi,y =

Maximizing expression (D.6a), the maximum relative displacement in longitudinal direction is obtained when

θ = 0◦

and the value is

dmax mi,x = vg Li /vs ,

cor-

responding to the extreme case where the epicenter of the earthquake is aligned with the prolongation of the bridge axis (in this case the transverse relative displacement is null,

dmi,y = 0).

On the other hand, the maximum relative displace-

ment in transverse direction occurs if

dmax mi,y

= 0.5vg Li /vs =

θ = 45◦

(minimizing eq.

(D.6b)), resulting

dmax mi,x /2.

By assuming the most unfavourable case with Priestley's method (x ˙g

= vg ),

the

imposed displacements in longitudinal direction are the same than those proposed by NCSP (section D.2.2), whereas the transverse ones are half the values obtained with this normative. Therefore, one conclusion is obtained prior to the results of the seismic analysis; the static procedures proposed by the normative (either Eurocode 8 or NCSP) fall on the safe side in comparison with Priestley's proposal, where the incidence angle of the seismic waves is included (as it may be observed in gure D.2). The displacements obtained by Priestley's procedure clearly depend on the considered horizontal direction, obtaining the following:

   Left          Left          

max → dmax m0,x = dm0,y = 0 mm  0.4   dmax = 32 mm m1,x = 160 2000 tower →   dmax = 32 = 16 mm m1,y 2  0.4 max   dm2,x = 560 = 112 mm 2000  Right tower →      dmax = 112 = 56 mm   m2,y   2     0.4  max    dm3,x = 720 = 144 mm   2000   Right abutment →      dmax = 144 = 72 mm  m3,y abutment (reference)

2

Figure D.2 depicts the pseudostatic forces in Priestley's static procedure.

(D.7)

378

Appendix D. Spatial variability

D.2.4 Results obtained with the static procedures The pseudostatic forces obtained by means of the mentioned static procedures are presented in gure D.2, particularized for the right tower of the studied cable-stayed bridge; in general terms, the static procedure proposed by Eurocode 8 yields the safest prediction of the pseudostatic eects (considering the envelope of both sets of imposed displacements), furthermore, the inuence of the spatial variability in the transverse exure of the towers is larger in the set of displacements with opposite directions (set B, with positive or negative sign), which is only considered in the Eurocode 8 proposal among all the static procedures studied here. The forces obtained following the method suggested by Priestley are typically below at least one of the code-compliant methodologies. However, the most interesting result in this gure is the large pseudostatic longitudinal response of the towers in comparison with the transverse one, specially in the lower part of the anchorage area, due to the longitudinal constraint exerted by the cable-system. The comparison between the pseudostatic, inertial and total forces (obtained by combining the pseudostatic and inertial components by means of SRSS rule, with expression (D.2)) is depicted in gure D.3 for the pseudostatic longitudinal bending moment obtained with the Eurocode 8 methodology, verifying that the eect of the spatial variability is negligible by considering the simplistic static procedures, even for the variables which are more inuenced by this phenomenon (like the longitudinal bending moment

MY Y ), being only appreciable the dierence between total and

inertial responses in the areas where the constraint of the cable-system is higher, i.e.

around the bottom part of the anchorage area and the anchors of the tower

corresponding to the intermediate piers (z

∗

≈ 0.75).

The reason for the low con-

tribution of the pseudostatic component in these simplied procedures is the large weight given to the greatest term in SRSS rule (being this component the inertial forces obtained through elastic response spectrum analysis).

D.3 Dynamic procedure based on delayed accelerograms Assuming that the loss of coherency is negligible, which is admissible studying exible structures like the considered cable-stayed bridge, and accepting that the subsoil is the same along the length of the structure, it is valid to say that the accelerogram imposed in the support

i

is the same than the one aecting the reference left

abutment, but delayed a certain value of time

δt,i = Li cos θ/vs

[Priestley 1996], i.e.

the wave-propagation velocity is assumed constant and the waves are not dispersive [Abdel-Ghaar 1991a]. The accelerograms imposed in the reference abutment are the ones obtained from the elastic design spectrum compliant with Eurocode 8 (part 1) [EC8 2004] prescriptions, detailed in chapter 5. The incidence angle (θ ) aects the pseudostatic displacements of the supports in the horizontal plane (longitudinal and transverse plane), as it was mentioned in section D.2.3. The maximum eect of the spatial variability is pursued along all the directions, and consequently the incidence angle is considered

θ = 0◦

and

θ = 45◦

D.3. Dynamic procedure based on delayed accelerograms

Spatial variability. Static procedure

Dimensionless tower height; z*= z / Htot

1

0.8

0.6

0.4

EC8 - set A EC8 - set B (+) EC8 - set B (-) NCSP Priestley

0.2

0

0

200

400

600

800

1000

1200

Pseudostatic axial load; N [kN]

1400

1600

1800

Spatial variability. Static procedure

1

Dimensionless tower height; z*= z / Htot

379

0.8

0.6

0.4

EC8 - set A EC8 - set B (+) EC8 - set B (-) NCSP Priestley

0.2

0

0

5000

10000

15000

20000

25000

30000

Pseudostatic longitudinal bending moment; Myy [kNm]

Spatial variability. Static procedure Dimensionless tower height; z*= z / Htot

1

EC8 - set A EC8 - set B (+) EC8 - set B (-) NCSP Priestley

0.8

0.6

0.4

0.2

0

0

5000

10000

15000

20000

25000

30000

Pseudostatic transverse bending moment; Mxx [kNm]

Figure D.2. Pseudostatic forces (Eps ) obtained along the right tower height employing dierent static procedures based on imposed displacements. Y-LCP model, LP = 400 m,

θZZ xed and rocky soil (TA).

380

Appendix D. Spatial variability Spatial variability. Static procedure

Dimensionless tower height; z*= z / Htot

1

Total force; Etot Inertial force; Ei Pseudostatic force; Eps

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

20000

40000

60000

80000

100000

120000

140000

160000

Longitudinal bending moment; Myy [kNm]

180000

200000

Figure D.3. Longitudinal bending moment (pseudostatic, inertial and total) obtained in the right tower with the static procedure proposed in Eurocode 8 [EC8 2005a] based on imposed displacements (envelope of sets A and B). Y-LCP model, LP = 400 m, θZZ xed and rocky soil (TA). Spectrum analysis (MRSA). 5

for the longitudinal and transverse directions respectively . Considering a typical wave-propagation velocity in rocky soil

vs = 2000 m/s, the time-lags of the imposed

accelerograms associated with each foundation with respect to the reference (left) abutment are the following:

  Left            Left          

→ δt0,x = δt0,y = 0 s  160 cos (0)   δt1,x = = 0.080 s 2000 tower →  160 cos (π/4) δ = 0.057 s t1,y = 2000  560 cos (0)   δt2,x = = 0.280 s 2000  Right tower →    560 cos (π/4)  δ   = 0.198 s t2,y =   2000      720 cos (0)     δt3,x = = 0.360 s   2000   Right abutment →    720 cos (π/4)  δ  = 0.255 t3,y = 2000 5 The

abutment (reference.)

(D.8)

s

reduced interaction between transverse and longitudinal responses allows dierent values of θ, and therefore dierent time-lags for the generation of the accelerograms in these directions in the same analysis. However, if signicant coupling arises, these is not valid and two analyses should be performed, one per each value of θ.

D.3. Dynamic procedure based on delayed accelerograms

381

The vertical accelerogram is not delayed since the eect of the asynchronism in the vertical ground motion is not studied here (neither in seismic codes [EC8 2005a] [NCSP 2007]).

However, the eect of the vertical accelerograms is reduced when

dealing with cable-stayed bridges, as it is demonstrated in chapter 7.

D.3.1 Results obtained with the dynamic procedure The longitudinal and transverse bending moments recorded in time domain at the foundation level of the right tower are presented in gure D.4, comparing the results obtained by imposing synchronous accelerograms or delayed records with a wavepropagation velocity of

vs = 2000

m/s (the time-lags are collected in expression

(D.8)). Once again, the larger inuence of the spatial variability in the longitudinal response of the towers is veried by comparing the upper and lower boxes in gure D.4, however in this case the eect of the asynchronous excitation is benecial for the seismic behaviour of the bridge (upper box in gure D.4). Considering the same structure, the strong inuence of the accelerogram delay (governed by the wave-propagation velocity

vs

or the incidence angle

θ,

see gure

D.1) is clear from gure D.5, where one of the most sensitive responses to the spatial variability is presented during the seismic excitation, imposing a 3D record delayed with several reasonable propagation velocities; the longitudinal bending moment recorded at the bottom section of the anchorage area is included, considering (synchronism), gle

θ = 0◦

vs = 2000

m/s or

vs = 1330

m/s

6,

vs = ∞

whereas the same incidence an-

is assumed in this case for both horizontal and transverse accelerograms,

regardless of the considered velocity.

vs = 1330

m/s is a critical velocity in the stud-

0.30 s, which almost = 0.31), with signicant

ied bridge since it causes delays of the signal between towers of coincides with the vibration period of the 59th mode (T59

contribution to the longitudinal tower response (and is 10 times shorter than the fundamental longitudinal period). Figure D.5 also compares the results given by the dynamic procedure with the ones obtained with the static methodologies proposed by Eurocode 8 (section D.2.1), which is obviously time-independent, in order to provide the reader with a clear reference. The delay of the seismic action may be damaging or benecial for the same cable-stayed bridge, depending on the relative movement of the towers, since the asynchronous movement of these members deforms the deck and could positively redistribute the forces. The eect is more clear in longitudinal direction, because the towers are strongly related to the girder through the cable system, which was observed by comparing the longitudinal and transverse bending moments in gures D.2 and D.4. An interesting idea arises from the redistribution capabilities of the deck in cable-stayed bridges; if the main span is very long (say much larger than 400 m), the exibility of the deck is so high that it does not constrain the dierential movement of the towers, and consequently the eect of the spatial variability is

6 Both v = 2000 m/s and v = 1330 m/s are realistic values of the wave-propagation velocity s s in granite (soil class type A [EC8 2004]), hence, the results of EC8 static procedure are valid to compare the dynamic results employing both velocities to delay accelerograms.

382

Appendix D. Spatial variability

150000 100000 50000 0 −50000 −100000 −150000

0

5

10

15

20

0

5

10

15

20

150000 100000 50000 0 −50000 −100000 −150000

Figure D.4. Time-history of the longitudinal (upper box) and transverse (lower box) bend-

ing moment at the foundation level in the right tower. Model subjected to a single 3D record, considering synchronous and asynchronous excitation with dierent incidence angles to maximize the longitudinal and transverse lags. Y-LCP model, LP = 400 m, θZZ xed and rocky soil (TA).

D.4. Conclusions

383

150000

150000

100000

100000

50000

50000

0

0

−50000

−50000

−100000

−100000

−150000

0

5

10

15

20

−150000

0

5

10

15

20

Figure D.5. Time-history of the longitudinal bending moment at the lowest section of the

anchorage area in the right tower. Single 3D record considering synchronous and asynchronous excitation with dierent propagation velocities. θ = 0◦ . The result of the static procedure proposed by Eurocode 8 [EC8 2005a] is also included. Y-LCP model, LP = 400 m, θZZ xed and rocky soil (TA). expected to be reduced, despite the time-lags (δt,i ) are longer due to the increment of the spans (Li ); on the other hand, considering main spans much shorter than 400 m, the redistribution of the deck and its interaction with the tower by means of the cable-system is maximized, however, the delays of the seismic signals along the supports of the bridge are lowered due to the reduction of the distance between foundations.

D.4 Conclusions In this study, the eect of the spatial variability of the seismic action on the response of cable-stayed bridges was studied to assess the level of errors which may be introduced by ignoring this phenomenon in the body of the thesis. One of the proposed models has been considered in this appendix, with the most critical main span length (LP

= 400

m) in order to maximize the eects of non-uniform ground

motions. The following conclusions are drawn from the presented results:

•

The longitudinal response of the towers is more inuenced (positively or negatively) by the asynchronism of the excitation than the transverse one, since the cable-system strongly coerces these members in the longitudinal plane, relating them with the deck.

•

The wave-propagation velocity and the incidence angle of the wavefront with respect to the bridge axis are both important variables in the structural response, because of their eect on the time-lags aecting the accelerograms in adjacent supports. Similar conclusions were suggested in a number of research

384

Appendix D. Spatial variability works [Walther 1988] [Tuladhar 1999] [Abdel-Ghaar 1991a] [Nazmy 1992] [Soyluk 2004].

Signicant reductions or increments of the recorded seismic

forces may be observed by comparing the solution including the spatial variability or neglecting this phenomenon; in light of the results obtained with a critical wave-propagation velocity (selected in light of important vibration modes with longitudinal movement of the towers in the studied model), ignoring the spatial variability could lead to under-predictions of the response up to 20 % on rocky soil (TA) considering the worst scenario (right box in gure D.5), whereas probably larger dierences may be observed on soft soils (TD) due to the maximized time-lags (the wave-propagation velocity is lower).

•

The main span of the bridge decisively aects its sensitivity to the spatial variability; very large cable-stayed bridges present exible decks incapable of coercing the movement of the towers, which accommodate, without introducing extra forces, the dierential displacements due to the long time-shifts caused by great distances between supports; on the other hand, very short cable-stayed bridges are more likely to be aected by the redistribution effects caused by dierential movements due to the important stiness, however the time-lags are reduced.

It is reasonable to say that cable-stayed bridges

between 300 and 500 m main span are expected to be more sensitive to the asynchronism of the earthquake, but more research is required to clearly discern the critical dimensions. Despite obvious, it is worth to mention that the softer the surrounding subsoil, the more inuence of the spatial variability is expected due to larger signal delays.

•

Procedures based on static displacements imposed at the foundations, and included in seismic codes, simplify notably the analysis of spatial variability eects, which on the other hand always aggregates forces with respect to the synchronous excitation (this may be not the case in rigorous dynamic procedures), however, these methodologies may under-predict the response; in the studied case, the maximum dierence between uniform and non-uniform ground motions is only 3 % by considering the static analysis strategies (gure D.3).

Among the studied static procedures, the Eurocode 8 (part 2)

[EC8 2005a] proposal is generally the most conservative one, and includes the possibility of dierential movements with opposite signs.

The method

proposed by Priestley [Priestley 1996] introduces the incidence angle, distinguishing the longitudinal and horizontal directions in the asynchronism, which reduces the pseudostatic forces in the transverse response in comparison with code-compliant procedures.

Appendix E Validation of synthetic accelerograms

The aim of this appendix is to analyze the level of unrealistic features present in the synthetic accelerograms employed in this thesis, and described in chapter 5, by comparing the characteristics of these signals with the theoretic natural ones, based on empirical models developed by several authors. The comparison is centered in the energy, duration and phase content, among other considerations.

E.1 Denition of earthquake scenarios As stated in chapter 5, no attempt has been made by the author to establish an earthquake scenario in order to maintain the generality of the study, hence the only information about the seismology of the area is the Uniform Hazard Spectrum (UHS) provided by the seismic design code [EC8 2004].

However, an earthquake

scenario associated with the code spectrum has to be dened if the adequacy of the synthetic accelerograms aims to be studied, comparing their features with empirical models developed from real records. Unfortunately, there are no realistic scenario completely associated with the code UHS; such design spectrum is an envelope of the earthquake hazard with very dierent scenarios; typically the long period spectral acceleration area is dominated by large events located at long distances and the short period zone is controlled by weaker earthquakes produced in the vicinity of the site. Figure E.1 represents this envelope.

Sa

UHS = envelope

Large distant event

Small local event

T

Figure E.1. Schematic illustration of the Uniform Hazard Spectrum (UHS) as an envelope of contributions from dierent seismic sources, redrawn from [Reiter 1990].

386

Appendix E. Validation of synthetic accelerograms

Here, a reverse procedure in comparison with the recommended one is adopted; the design spectrum (UHS) is rst dened, and the earthquake scenario compatible with this spectrum is addressed afterwards. To accomplish this task, the predicted spectrum associated with the natural scenario is matched to the UHS as far as possible, providing more importance to the long periods (from 2 to 6 seconds studying cable-stayed bridges with main spans of 200 to 600 meters) to cover the fundamental modes in the studied models. Once the natural spectrum is matched to the design one, the information about the scenario behind the former is extracted. The vertical accelerograms will be studied here considering the same scenario established for the associated horizontal records, despite it is known that the governing vertical scenario is generally dierent than the horizontal one [Bommer 2004]. Therefore, the results obtained for vertical accelerograms should be quoted. The target UHS have been dened in section 5.2 for two extremely dierent grounds; rocky (TA) and soft soil (TD).

E.1.1 Empirical spectrum The empirical model for horizontal ground motions proposed by Chiou and Youngs [Chiou 2008] from the Next Generation of Attenuation (NGA) project is employed (PEER-NGA database). The comparison between the UHS and the empirical spectra is not straightforward due to the unrealistic shape of the design action, as it has been discussed above. The Type 1 design spectrum in Eurocode 8 [EC8 2004] is derived from moment magnitudes just under

MW = 7, but it may be observed that considering medium to long

rupture distances, the empirical Peak Ground Acceleration (PGA) is well below the value adopted in the design spectrum (ag

= 0.5

g) for

MW < 7,

because such large

PGA corresponds to a very high return period event, and this `abnormal' scenario is dominated by a very large magnitude and short rupture distance earthquakes. Considering that the maximum magnitude compliant with Eurocode 8 [EC8 2004] design spectrum is

MW = 7,

rupture distances lower than 10 km are required in

order to obtain similar PGA with the empirical model. Natural earthquakes with such reduced epicenter distances might involve near-fault eects, but no attempt to consider these special characteristics has been made in the procedure to generate articial records. The input parameters for the empirical model of Chiou and Youngs [Chiou 2008], some of them described in gure E.2 are;

•

Moment magnitude;

•

Rupture distance (closest distance from rupture to site);

•

`Joyner  Boore' distance (distance to projection of rupture);

MW Rrup

[km]

Rjb

[km]

• `RX ' distance [km] (site coordinate, measured perpendicular to the fault strike

from the RX surface projection of the updip edge of the fault rupture, with the downdip direction being positive)

E.1. Denition of earthquake scenarios

387

•

Average shear wave velocity of the uppermost 30 meters;

•

Depth to the top of the rupture;

•

Fault dip;

•

Rake angle;

Vs,30

[m/s]

Ztor

[km]

•

Width of the rupture;

•

Flag for site on/o the hanging wall;

FHW

•

Sediment depth (thickness of the near-surface sediments represented by the

Dip

◦

[ ]

Rake

◦

[ ]

W

[km]

1.0 km/s, Vs,30 in 8 + 378.78 )] [m] Z1p0 = exp[28.5 − (3.82/8) ln(Vs,30 depth to a shear wave velocity of

• ε;

m/s);

number of standard deviations above the median motion.

Figure E.2. Distances in a vertical cross-section through a fault rupture plane. The study of empirical models for the prediction of response spectra is beyond the scope of this appendix and also the doctoral thesis. The most important parameters for our purpose are; Moment magnitude (xed in

MW = 7),

rupture distance

and the average shear wave velocity of the uppermost 30 meters;

Vs,30

Rrup

[m/s].

RX Rrup . The depth to the top of the rupture, consistently with the soil conditions, is Ztor = 0 for rocky soil (type A) and Ztor = 5 km for soft soil (type D). The fault dip and ◦ ◦ rake angle are 90 and 0 respectively. The width of the rupture in both cases is 15 1 km. The site is considered o the hanging wall (FHW = 0), and ε = 0. The rupture distance Rrup of the earthquake scenario is established in order to Realistic values are going to be assumed in the rest of parameters:

Rjb

and

distances are considered always equal in magnitude to the rupture distance

t the associated empirical spectrum to the UHS in the range of periods between 2 an 6 s. The moment magnitude (MW are xed;

Vs,30

= 7)

and the characteristics of the ground

= 760 m/s for rocky soil (type A) and

Vs,30

= 250 m/s for soft soil

(type D). There are specic models for the prediction of vertical ground motions given the characteristics of the seismic scenario, like the model of Campbell and Bozorgnia

1 i.e.

it is considered on the footwall; RX < 0 in gure E.2.

388

Appendix E. Validation of synthetic accelerograms

[Campbell 2003], however they are not considered in this appendix, maintaining the horizontal earthquake scenario for the vertical direction (both horizontal scenarios have been considered in order to validate the vertical records since Eurocode 8 does not distinguish the soil typology in this direction).

E.1.2 Specic horizontal earthquake scenarios Suitable earthquake scenarios for the target code spectrum in soft and rocky soil are dened below, being their characteristics summarised in table E.1. These scenarios are compliant with Eurocode 8 UHS in the range of fundamental periods, however, the range of periods between 2 and 3 seconds is not very well represented by scenarios for both soft and rocky soil (gures E.3 and E.4), which includes the fundamental modes of cable-stayed bridges with main spans from 200 to 300 meters (chapter 4).

Soft soil (type D) Rocky soil (type A) 7 2 2 2 250 5 90 0 15 0 0.332

Moment Magnitude

Rrup [km] Rjb [km] RX [km] Vs,30 [m/s] Ztor [km] Dip [◦ ] Rake [◦ ] Rupture width

W

[km]

FHW Sediment depth;

Z1p0

[km]

7 2 2 2 760 0 90 0 15 0 0.023

Table E.1. Seismological characteristics of the earthquake scenarios compliant with Eurocode 8 spectra.

Soft soil (type D) In this case; PGA

= ag · S = 0.5 · 1.35 = 0.67

g and

Vs,30

= 250 m/s. A rupture

distance of only 2 km is required to obtain acceptable spectral accelerations (Sa) in the high-order period range and

Vs,30

= 250 m/s and

P GA ≈ 0.6

MW = 7),

g with the empirical model (employing

in such case the predicted and design response

spectra are depicted in gure E.3. Figure E.3 suggests that the design spectrum of EC8 [EC8 2005a] is unreasonably high for soft soils, specially in the short period area.

Rocky soil (type A) In this soil; PGA

= ag · S = 0.5 · 1.0 = 0.5

g and

distance of only 2 km is required to obtain PGA

Vs,30 = 760 m/s. ≈ 0.6 g with the

Again, a rupture empirical model.

E.2. Seismological features Engineer s point of view

1.8

1.8

EC8 elast ic design spect r um Empir ical model. Chiou and Youngs (2008)

1.4 1.2 1.0 0.8 0.6 0.4

1.4 1.2 1.0 0.8 0.6 0.4

0.2

0.2

0.0

0.0 − 2 10

0

1

2

3

4

Per iod; T [s]

5

6

Seismologist s point of view

1.6

Spect r al acceler at ion; Sa [g]

1.6

Spect r al acceler at ion; Sa [g]

389

7

EC8 elast ic design spect r um Empir ical model. Chiou and Youngs (2008) 10− 1

Per iod; T [s]

100

101

Figure E.3. Eurocode 8 elastic design target spectrum and predicted spectrum (model of Chiou and Youngs [Chiou 2008]). Soft soil (type D). Horizontal spectrum.

Both the predicted and the design response spectrum are represented in gure E.4. Unlike the result on soft soil, the t between the design spectrum and the empirically predicted one is adequate in a broad band of periods considering rocky soil; it must be recognized that most of the recorded accelerograms in PEER - NGA database correspond to rocky soil conditions, and hence the empirical model of Chiou and Youngs may be more appropriate for these conditions. Furthermore, the Eurocode 8 reference spectrum is precisely dened for rocky soil (TA).

Engineer s point of view

1.4

EC8 elast ic design spect r um Empir ical model. Chiou and Youngs (2008)

1.2

Spect r al acceler at ion; Sa [g]

Spect r al acceler at ion; Sa [g]

1.4

1.0 0.8 0.6 0.4 0.2 0.0

0

1

2

3

4

Per iod; T [s]

5

6

7

Seismologist s point of view

1.2 1.0 0.8 0.6 0.4 0.2 0.0 − 2 10

EC8 elast ic design spect r um Empir ical model Chiou and Youngs (2008) 10− 1

Per iod; T [s]

100

101

Figure E.4. Eurocode 8 elastic design target spectrum and predicted spectrum (model of Chiou and Youngs [Chiou 2008]). Rocky soil (type A). Horizontal spectrum.

E.2 Seismological features The purpose of this section is to establish comparative features which are predictable by means of empirical expressions for real earthquakes based on the earthquake scenarios dened above. These empirical models are not explained here, the interested reader is directly referred to the relevant papers.

390

Appendix E. Validation of synthetic accelerograms

No distinctions are made by the authors of the predictive empirical models between horizontal and vertical components, hence the models are considered applicable to both types of records. In addition to the empirical predictions considered below, the qualitative aspect of the acceleration, velocity and displacement time histories have been contrasted with El Centro (1940,

MW = 7.1)

benchmark record.

Peak Ground Acceleration (PGA) As stated before, the earthquake scenario is established by matching the spectral acceleration of its empirical spectrum with the target code one in the long period range (2-6 s).

Thus the PGA (spectral acceleration for a period equal to 0 s) is

not specically taken into account.

However, PGA is an Intensity Measure (IM)

historically considered in seismic studies, and here its simulated value with articial

2

accelerograms

is compared with the predicted one adopting the empirical model of

Chiou and Youngs [Chiou 2008]. Several researchers have demonstrated that PGA is an inecient Intensity Measure, specially for exible structures like cable  stayed bridges due to the great inuence of the long period area of the spectrum, which is weakly correlated to PGA (this fact may be observed in gures E.3 and E.4).

Arias intensity Arias Intensity (Ia , expression (5.1)) is an Intensity Measure (IM) that has been proved to be more ecient than PGA for geotechnical considerations. Arias Intensity measures the strength of the accelerogram, and its prediction is established here through the empirical model developed by Foulser  Piggott and Staord [Foulser-Piggott 2010].

Duration of the strong pulse phase The empirical model of Bommer et al [Bommer 2009] for the prediction of the Signicant Relative Duration (DSR ) is employed in this appendix, which is valid for moment magnitudes between 4.8 and 7.9, and rupture distances up to 100 km from the source, therefore it is valid for the earthquake scenarios considered (see table E.1).

Husid plot Husid plots study the rate of introduced seismic energy, being represented by the ratio between the accumulated partial and total Arias Intensity versus the time

2 Since

synthetic accelerograms t the complete range of the target UHS, from 0.03 to 6.6 s (section 5.4.2), Sa (Tmin = 0.03) ≈ PGA, thus the simulated PGA with this articial accelerograms will be approximately the value of ag employed to dene this design spectrum; PGA ≈ 0.5 g for horizontal accelerograms in rocky soil (type A); and PGA ≈ 1.35 · 0.5 = 0.675 g for horizontal records in soft soil (type D).

E.3. Results

391

elapsed since the earthquake begins. It is clearly inuenced by the interval of strong shaking of the record; the longer this interval the more linear results the Husid plot. The model of Staord et al [Staord 2009] is employed in order to predict the Husid plot produced by natural earthquakes.

E.3 Results For the sake of brevity, only the comparison results are summarized here, collecting the relevant conclusions at the end of the appendix. Two types of plots are presented; (i) the seismological features averaged from the considered set of 12 synthetic accelerograms, including not only the absolute values, but also their ratio with the predicted ones based on natural records compliant with the specied seismic scenarios; and (ii) the Husid plot with colour bands representing the relative duration

DSR,5−75 ,

comparing the articial results with the empirical model of Staord et al

[Staord 2009]. In each case, the articial records obtained for Modal Response History Analysis (MRHA) and those obtained for Non-Linear Response History Analysis (NL-RHA) are distinguished, due to the dierences in their generation procedure (chapter 5). Since the vertical spectrum in Eurocode 8 is independent of the type of foundation soil, the synthetic accelerograms obtained by tting this spectrum are compared with the empirical predictions for natural earthquakes compatible with both horizontal scenarios; only the comparison with the scenario in rocky soil (TA) are presented for vertical records, the results are similar on soft soil. In one of the studied cases, the results of Simqke [Gasparini 1999] have been also included, which is a well known software for the calculation of synthetic records from a specic target spectrum.

MRHA accelerograms. Soil D

5

11.8 10.4

Simulated / Predicted

4

3

Predicted natural record Proposed in this thesis. Synthetic Simqke. Synthetic

2

1

0

10.3

8.3 0.6

0.7

PGA [g]

0.7

2.6

Ia [m/s]

5.0

DSR,5−75 [s]

11.6

13.8 14.1

DSR,5−95 [s]

(a) Earthquake features

(b) Husid plot

Figure E.5. Horizontal records for soft soil (type D) employed in modal dynamics (MRHA).

392

Appendix E. Validation of synthetic accelerograms NL-RHA accelerograms. Soil D

5

Husid plot. NL-RHA. Soil D

1.0

Predicted natural record Propoposed in this thesis. Synthetic

3

5.3

2

7.9

1

0.6

0.7

2.6

5.0

11.6

13.5

Synthetic Cumulat ive ener gy / Tot al ener gy

Simulated / Predicted

4

0.8

75 %

Predicted

0.6

Proposed in this thesis. Synthetic 0.4

Pr edict ed nat ur al r ecor d

0.2

5% 0

PGA [g]

Ia [m/s]

DSR,5−75 [s]

0.0

DSR,5−95 [s]

0

5

(a) Earthquake features

10

Time; t [s]

15

20

(b) Husid plot

Figure E.6. Horizontal records for soft soil (type D) employed in nonlinear dynamics (NLRHA).

MRHA accelerograms. Soil A

5

Husid plot. MRHA accelerograms. Soil A

1.0

Synthetic

Simulated / Predicted

4

3

2

7.6

4.4

1

0.5

0.5

2.9

13.6

4.7

9.4

Cumulat ive ener gy / Tot al ener gy

Predicted natural record Proposed in this thesis. Synthetic

Predicted

0.8

75 %

0.6

Proposed in this thesis. Synthetic 0.4

0.2

Pr edict ed nat ur al r ecor d

5% 0

PGA [g]

Ia [m/s]

DSR,5−75 [s]

0.0

DSR,5−95 [s]

0

5

(a) Earthquake features

10

Time; t [s]

15

20

(b) Husid plot

Figure E.7. Horizontal records for rocky soil (type A) and employed in modal dynamics (MRHA).

NL-RHA accelerograms. Soil A

5

Husid plot. NL-RHA Rocky soil (t ype A)

1.0

Predicted natural record Proposed in this thesis. Synthetic

3

2 7.3

1

0.5

0.5

2.9

2.8

4.7

13.8 9.4

Cumulat ive ener gy / Tot al ener gy

Simulated / Predicted

4

Synthetic 0.8

75 %

Predicted

0.6

Proposed. Synthetic 0.4

0.2

Pr edict ed nat ur al r ecor d

5% 0

PGA [g]

Ia [m/s]

DSR,5−75 [s]

DSR,5−95 [s]

(a) Earthquake features

0.0

0

5

10

Time; t [s]

15

20

(b) Husid plot

Figure E.8. Horizontal records for rocky soil (type A) and employed in nonlinear dynamics (NL-RHA).

E.3. Results

393

MRHA accelerograms. Soil A

5

Husid plot. MRHA vertical accelerograms

1.0

Predicted natural record Proposed in this thesis. Synthetic

3

2

1

0

0.5

0.5

PGA [g]

2.9

13.9

6.8

4.1 4.7

Ia [m/s]

9.4

DSR,5−75 [s]

Cumulat ive ener gy / Tot al ener gy

Simulated / Predicted

4

Pr edict ed. Rocky Soil 0.8

Pr edict ed. Soft Soil 0.6

0.4

0.2

0.0

DSR,5−95 [s]

Proposed in this thesis. Synthetic

0

5

(a) Earthquake features for rocky soil (type A)

10

Time; t [s]

15

20

(b) Husid plot

Figure E.9. Vertical records employed in modal dynamics (MRHA).

NL-RHA accelerograms. Soil A

5

Husid plot. NL-RHA vertical accelerograms

1.0

Predicted natural record Proposed in this thesis. Synthetic

3

2

0

0.5

0.5

PGA [g]

2.9

Ia [m/s]

14.1

6.9

3.9

1

4.7

DSR,5−75 [s]

9.4

DSR,5−95 [s]

(a) Earthquake features for rocky soil (type A)

Cumulat ive ener gy / Tot al ener gy

Simulated / Predicted

4

Pr edict ed. Rocky Soil 0.8

Pr edict ed. Soft Soil 0.6

Proposed in this thesis. Synthetic 0.4

0.2

0.0

0

5

10

Time; t [s]

15

20

(b) Husid plot

Figure E.10. Vertical records in nonlinear dynamics (NL-RHA).

Global aspect of the record Figure E.11 includes the history of ground acceleration, velocity and displacement in one horizontal synthetic record associated with rocky soil (type A) and obtained

3

considering constant damping (employed in modal dynamic analysis, MRHA). For comparison purposes, gure E.12 illustrates the data registered during El Centro record (1940,

MW = 7.1),

which is employed traditionally as a benchmark far-fault

natural signal. Compared with natural records, both the synthetic acceleration and velocity traces seem reasonable, perhaps the aspect of the synthetic accelerogram is more stationary, but one should take into account the unreasonable shape of the target

3 It was generated with the proposed algorithm based on the work of Levy and Wilkinson [Levy 1975].

Appendix E. Validation of synthetic accelerograms

Displacement; ug [m]

Velocity; u˙ g [m/s]

Acceleration; u¨g [m/s2 ]

394

6 4 2 0 -2 -4 -6

0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4

A proposed synthetic record. MRHA. Rocky soil (TA)

0

5

10

15

20

0

5

10

15

20

0

5

10

15

20

0.05 0.00 -0.05 -0.10 -0.15

Time; t [s]

Figure E.11. History of accelerations, velocities and displacements of one synthetic hori-

Displacement; ug [m]

Velocity; u˙ g [m/s]

Acceleration; u¨g [m/s2 ]

zontal record obtained for rocky soil (type A) and modal dynamic calculations (MRHA).

3 2 1 0 -1 -2 -3 -4

0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 0.15

El Centro earthquake (1940, MW = 7.1)

0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

30

35

40

0.10 0.05 0.00 -0.05 -0.10

Time; t [s]

Figure E.12. History of accelerations, velocities and displacements of El Centro earthquake; Imperial Valley 5/19/40, El Centro array #9, 180 (USGS station 117).

E.4. Conclusions

395

spectrum imposed to the articial record. Associated with the accelerograms obtained for soft soil conditions (TD), there is an appreciable baseline drift at the end of the displacement history; in principle this permanent movement is harmless since it leads to rigid body motions in the structure. The proposed algorithm may be easily corrected to overcome this unrealistic feature, but it has been veried elsewhere that the eect is not important for the extreme seismic response [Abaqus 2010], and no attempt has been made to reduce the baseline drift in the proposed synthetic records.

E.4 Conclusions Keeping in mind that the predicted natural spectrum, associated with a realistic seismic scenario, cannot be matched to the UHS provided by design codes in the whole range of interesting periods, and assuming the inherent errors in empirical models, the following conclusions arise from the comparison between articial records and empirical predictions based on natural earthquakes:

•

The modied proposal of Levy & Wilkinson [Levy 1975] algorithm to obtain the synthetic accelerograms employed in this thesis, yields records closer to the predicted natural ones in comparison with Simqke [Gasparini 1999] results, in terms of Arias Intensity, Signicant Durations, rate of introduction of the energy during the seismic excitation and global aspect of the accelerogram (which are less stationary). This is probably due to the possibility of reducing easily the strong pulse interval (tsp ) in the proposed algorithm, getting closer to the natural results.

•

The total amount of energy (Arias Intensity) introduced by natural predicted earthquakes is generally lower than the one associated with articial records, which is specially true in soft soil scenarios, where material nonlinearities are expected to be strong in exible structures like cable-stayed bridges.

•

The envelope modulating function of maximum accelerations

F (t),

which is

strongly related to signicant durations (veried here), has been obtained from the recommendations of Priestley [Priestley 1996] and Eurocode 8 [EC8 1994]. Nevertheless, simulated signicant durations are larger than the predicted values for coherent natural records, and therefore, the rate of introduction of the energy (Husid plot) is lower for synthetic accelerograms.

•

The special procedure employed to obtain synthetic accelerograms appropriate for nonlinear dynamics (with Rayleigh damping) leads to more realistic Arias Intensities in such records. Hence, articial accelerograms closer to the natural ones are obtained in nonlinear calculations, which could be sensitive to this feature (elastic modal dynamics is independent of the energy content, duration and other accelerogram characteristics, the most important feature in elastic analysis is indeed the t of the spectrum).

396 •

Appendix E. Validation of synthetic accelerograms The overestimation in the Arias Intensity using articial accelerograms is due to unrealistically large durations of the strong pulse interval, which is directly related to the signicant duration

DSR,5−75 .

An easy way to solve

this problem, obtaining more realistic synthetic accelerograms, is to modify the modulating function

F (t)

(presented in gure 5.3) by reducing the strong

shaking interval (tsp ), resulting Husid plots more consistent with the natural

4

previsions . However, this implies a deviation from the recommendations of Eurocode 8 [EC8 1994]. Such procedure is not further developed here, adopting the normative modulating function, which is supported by the results of previous research works [Jangid 2004] [Quek 1990].

•

The Peak Ground Acceleration (PGA) obtained in the synthetic records is very close to the expected value (with variations between g, where

S = 1

S = 1.35 for P GAV = 0.9 · P GAH = 0.45

for rocky soil and

vertical accelerograms

±10

%);

P GA = 0.5 · S

soft soil, whereas in the g. The predicted PGA in

natural records is somewhat dierent, because the imposed t of the empirical spectra to the design one is done mainly in the range of long periods, whilst no constraint is imposed in the area of high frequencies, where PGA is strongly correlated.

•

The vertical synthetic accelerograms are observed to follow the same trends than horizontal ones associated with rocky soil, due to the good agreement with Eurocode 8 spectra, presenting Arias Intensity close to the predictions.

•

The frequency of the strong acceleration pulses is larger for rocky soil, because the horizontal target spectrum associated with this ground gives more prominence to spectral accelerations in the high frequency range than the spectrum in soft soil (see gure 5.1). The spectrum shape also explains the large contain of strong pulses in vertical synthetic records.

4 There

are other more comprehensive proposed envelope modulating functions F (t), like the model of Staord et al (2009) [Staord 2009] (the Husid plot recommended by these authors has been considered in the present appendix). Nevertheless, the denition of earthquake scenarios is required to dene DSR,5−75 , reducing the generality of the study.

Appendix F Pushover analysis in seismic codes and guidelines

The objective of code-compliant pushover analysis is to reduce the structure to an inelastic SDOF system properly dened, and usually related to the fundamental vibration mode.

F.1 Eurocode 8 - Part 2: Bridges Following Eurocode 8 (Part 2) [EC8 2005a], pushover is performed separately in both longitudinal and transverse directions. The investigation of two pattern loads is recommended;

•

Triangular:

The displacement at the level of the deck is assumed constant

and piers are subjected to linearly increasing loads from the foundation to their top:

s∗i = mi ·

zi Htot

(F.1)

s∗i , mi and zi are respectively the load, mass node i, whilst Htot is the height of the pier.

Where with

•

Principal Mode:

and height associated

The load pattern is proportional to the principal mode

in each direction, which is considered the one with the highest corresponding participation factor

Γ: s∗i = mi · φi,p

Where

φi,p

is the modal displacement at node

(F.2)

i

in the principal mode.

The target inelastic displacement demand of the equivalent SDOF (um ) is considered equal to the elastic displacement of the centre of mass of the deformed deck

1

(control point) when linear multi-mode spectrum analysis is performed , tacitly

1 According

to Eurocode 8 (Part 2) [EC8 2005a], the target displacement is considered equal to the maximum displacement obtained between two seismic elastic combinations; (1) longitudinal earthquake spectrum combined with 30 % of the transverse spectrum and (2) transverse earthquake spectrum combined with 30 % of the longitudinal spectrum.

398

Appendix F. Pushover analysis in seismic codes and guidelines

considering that bridges are exible enough to apply the `equal deformation' rule (um

≈ uo ,

see gure F.1), which is reasonable for vibration periods in the velocity-

and displacement-sensitive regions of the spectrum, i.e

T > TC

s

2.

This rule may

be deemed appropriate in cable-stayed bridges due to their important exibility, nevertheless, it has been veried in the present work that several inuencing longitudinal or transverse modes could be located in the acceleration-sensitive region

3

of the spectrum, specially if the foundation soil is soft , and therefore the `equal deformation' rule should be questioned. Eurocode 8 (Part 1) [EC8 2004] is prepared for this situation (when relatively sti modes inuence the response) and provides a simplied expression to estimate the inelastic target displacement of such periods (where

um

  uo TC ≈ 1 + (qu − 1) ≥ uo ; qu T

if

T TC uo = Sd(T, ξ)

being the peak displacement obtained by considering the initial

elastic SDOF properties (see gure F.1) and response, whereas

qu

T

the vibration period of this elastic

is the ratio between the acceleration in the elastic equivalent

problem (Sa(T, ξ)) and the inelastic SDOF system (Fy /Mef f , where mass of the

TC

4 SDOF

and

Fy

its elastic limit); hence

Mef f is the qu = Sa(T, ξ)Mef f /Fy . Finally,

is the aforementioned corner frequency.

Figure F.1. Elastoplastic system and its corresponding linearization. 2 Following

Eurocode 8 [EC8 2005a], in rocky soil class (TA) TC = 0.4 s, whereas in soft soil class (TD) TC = 0.8 s. 3 Table 6.1 includes the governing frequencies of several cable-stayed bridges on soft soil, and there may be appreciated that models with 200 m main span could be dominated by periods lower than TC = 0.8 s (TD), i.e. frequencies higher than f = 1.25 Hz. 4M ef f is the eective mass of the studied mode, obtained with eq. (4.3).

F.2. ATC-40

399

Anyhow, Eurocode 8 (Part 2) [EC8 2005a] discourages pushover analysis if the mass of some piers has a signicant eect on its dynamic behaviour, which is clearly the case in cable-stayed bridges due to the large mass of the towers, recommending nonlinear dynamics (NL-RHA) in such cases.

F.2 ATC-40 ATC-40 [ATC 1996] was one of the rst guidelines proposing pushover as a methodology to study the nonlinear seismic behaviour of buildings. The load pattern proposed in ATC-40 is the `Principal Mode' described above. The inelastic target displacement is estimated by means of an iterative procedure called `capacity-spectrum' method, where a sequence of elastic SDOF are

5

analyzed by updating the period and equivalent damping ratio .

This procedure

does not always obtain convergence, and when it does converge, the solution could be misleading [Chopra 2007].

The pushover analysis leads to the capacity curve,

6

which is expressed in ADRS coordinates . The reduced spectra, taking into account hysteretic damping by means of simplied rules focused on building structures, is represented along with this plot. The reduction of the spectrum due to such equivalent viscous damping (ξ ) may be done by using the reduction factor proposed by seismic codes (for example

η =

p

10/(5 + ξ) ≥ 0.55

is proposed by Eurocode 8

[EC8 2004]), which is known to present lack of physical basis. Shortcomings of ATC-40 methodology in the estimation of the inelastic displacement demand appear to have been rectied in FEMA-440 report [fem 2005], but the benets in the equivalent linearization detour is unclear when there are available expressions for the inelastic deformation ratio, like the one proposed by Eurocode 8 and presented in equation (F.3) [Chopra 2007]. ATC-40 suggests the possibility of including higher mode eect for high-rise buildings with fundamental period larger than 1 s, which are observed to be the most sensitive to such modes, by repeating the `capacity-spectrum' method for important modes dierent than the rst one.

F.3 FEMA-356 FEMA-356 [fem 2000] is a seismic guideline focused on building structures.

At

least two load distribution patterns must be considered separately and the most demanding results are selected.

The rst load distribution is to be selected from

among the following:

•

Principal Mode

•

Equivalent Lateral Force (ELF), analogous to triangular distribution.

5 The `capacity spectrum' procedure has been described by the author elsewhere [Camara 2008]. 6 ADRS coordinates represent a plot of spectral displacements (horizontal axis) versus spectral

accelerations (vertical axis).

400 •

Appendix F. Pushover analysis in seismic codes and guidelines MRSA pattern:

s∗

is dened with the storey shears obtained from the elastic

spectrum analysis (MRSA) of the structure.

The second pattern is either the `Uniform' distribution (where the load is equal to the mass associated with each point,

s∗i = mi )

or an adaptive one which varies

as the structure yields. There are many research works available in literature about the applicability of these load patterns to building structures, several concluding that `Uniform' distribution is not appropriate since it grossly underestimates the drifts in upper stories, and overestimates them in lower stories [Chopra 2007]. In the present work `Principal Mode' and `Uniform' patterns were studied in several cable-stayed bridges and `Uniform' distribution is also found to yield wrong results (see section 6.4.4.2). The target displacement of the SDOF system is estimated by means of the `coecient method':

 T 2 = C1 C2 C3 Sa 2π | {z } 

um

(F.4)

Sd

Where the elastic spectral displacement of the considered vibration mode (Sd) is increased by three coecients; (again,

um

C1

represents the inelastic deformation ratio

is the maximum inelastic displacement and

uo

um /uo

is the maximum displace-

ment of the equivalent elastic SDOF, see gure F.1) for SDOF systems with stable hysteresis loops (i.e.

C2

without stiness degradation, pinching, etc.), the coecient

considers the increase in the deformation due to the damage eects which are

not covered in

C1 ,

whereas

C3

takes into account the increase in the deformation

because of the negative post-yielding stiness arising from P-∆ eects. The principal problem is that values provided for such coecients are not prepared to be used in bridges, and some numerical results are not supported by research [Chopra 2007]. Improvements in coecients

C1

and

C2

were established in FEMA-440 [fem 2005]

report. If higher modes are important in the response of the structure, pushover is supplemented by the linear dynamic analysis according to FEMA-356.

Appendix G Step-by-step description of advanced pushover analysis

The advanced pushover analysis (NSP) employed in this thesis, and presented in chapter 6, are detailed here in step-by-step form; Modal Pushover Analysis (MPA) was proposed by Chopra and Goel [Chopra 2002] and is widely accepted in building structures, several contributions have been devised to adapt the methodology to the three-directional seismic response of cable-stayed bridges; The Extended Modal Pushover Analysis (EMPA) is a generalization of MPA suggested to fully consider the three-axial ground motions; and the Coupled Nonlinear Static Pushover (CNSP) is a procedure developed here to take into account the nonlinear couplings between governing transverse and longitudinal modes. The procedures are presented in a general form and could be applied to any structure.

G.1 Modal Pushover Analysis: MPA 1. Apply rst the gravity loads considering geometric nonlinearities (P-∆ eects). 2. Perform modal analysis from the deformed conguration of the structure, com-

fn , participation factors Γjn (j = X, Y, Z ) and mode shapes φn up to the maximum frequency of interest; fmax = 25 Hz. After the modal analysis, a study about the participation of each mode below fmax in the global puting frequencies

response in terms of forces or displacements should be conducted (see section 6.4.4.1), identifying the governing longitudinal (fnX ) and transverse modes (fnY ), MPA should involve both dominant modes. The limit frequency which marks the end of the range where pushover is to be conducted is established as

fgov = max(fnX , fnY ).

3. For the modes below the limit

fgov ,

their dominant direction (DRn ) is se-

lected because MPA does not consider the three-directional contribution of modes, and subsequently the analysis continues in 2D (either longitudinally, transversely or vertically, depending on the governing direction of the specic mode). The control point is selected for each vibration mode as the node with maximum modal displacement in the dominant direction. A proposed method to obtain the characteristic direction of the mode is presented in section 6.4.4.1, summarized in the expression below.

402

Appendix G. Step-by-step description of advanced pushover
ΓDR = max Γjn ; n j

4. For the

nth

with

j = X, Y, Z → DRn

(G.1)

mode, within the range studied by pushover analysis (fn

≤ fgov ),

develop the base shear versus control point displacement curve in the dominant

Vbn − urn ,

direction,

by means of nonlinear static analysis (pushover) of the

structure when the load distribution

s∗n = mφn

is incrementally applied (self-

weight included), ignoring the components in the direction dierent than the dominating one.

The base shear (Vbn ) is the sum of the shear recorded in

each foundation (towers, intermediate piers and abutments) in the dominant direction of the mode, and the control point displacement

urn

is consequently

the displacement of this point along the dominant direction in the

nth

mode

(it position depends on the vibration mode). 5. Transform

Vbn − urn

pushover curve to

Fsn /Mn − qn

coordinates using expres-

sions (6.37a) and (6.37b), repeated here for convenience:

Fsn Vbn = Mn Ln qn =

6. Idealize the real with

nth mode.

Fsn /Mn −qn

(G.2a)

urn φrn

(G.2b)

curve for the equivalent SDOF system associated

Here, a bi-linear curve considering a modied `Equal Area' rule

presented in gure 6.11(a) (section 6.4.1) is proposed because of its simplicity, but more realistic curves may be considered. The kinematic properties of the nonlinear spring behaviour in the SDOF system need to be addressed (see appendix H), dening the loading and unloading branches appropriate for the structural system and material. 7. Compute the peak generalized displacement

1 system .

qn

of the

n-mode

inelastic SDOF

Here, it is proposed to integrate the SDOF dierential equation of

the corresponding mode in time domain (repeated below), by implementing the algorithm presented in appendix H.

q¨n + 2ξn ωn q˙n +

Fsn = −Γn u ¨g Mn

urn = qn φrn , where φrn is the modal displacement in the dominant direction DRn of n-mode

8. Obtain the peak control point displacement

urn

(G.3)

with expression:

at the selected control point.

1 Dierent options are available in the literature in order to obtain the target control point displacement in pushover analysis, some of them have been presented in appendix F.

G.2. Extended Modal Pushover Analysis: EMPA 9. Interpolate with

urn + uG

(being

uG

403

the control point displacement in the

dominant direction due to gravity loads) from the database of pushover analysis, to obtain the combined eects of lateral loads and gravity due to

n-mode

contribution (rn+G ). 10. Obtain the contribution of

n-mode

to the seismic response exclusively, by

extracting the eect due to the self-weight:

rn = rn+G − rG ,

where

rG

is the

contribution considering the self-weight acting alone.

11. Repeat steps 4 to 10 for all modes with frequencies below or equal

fgov .

12. Obtain the total nonlinear seismic response by combining the contribution of each mode with frequency below

fgov ,

using an appropriate combination rule,

here CQC rule is selected for this purpose:

rnl .

13. Compute the higher mode seismic response (rel ) by means of spectrum analysis (MRSA) including modes with frequencies higher than than

fmax = 25

fgov

and lower or equal

Hz.

14. Combine the participation of rst modes with the higher mode eect (between

fgov

and

fmax ),

employing SRSS combination rule to obtain the dynamic re-

sponse of the structure: Hz are ignored.

rd ≈

q 2 + r2 . rnl el

Frequencies higher than

fmax = 25

15. Calculate the total demand (r ) by combining the self-weight eect (rG ) with the dynamic contribution exclusively due to the earthquake (rd ).

Since the

2 sign of seismic forces is lost in pushover procedures , two hypotheses are made, considering both positive and negative signs in the earthquake response to take

into account that the seismic input, and the consequent structural behaviour, have alternating sign reversals.

r ≈ max (rG ± rd )

(G.4)

G.2 Extended Modal Pushover Analysis: EMPA EMPA proposed procedure is summarized next, repeating for completeness some steps introduced in MPA: 1. Apply rst the gravity loads considering geometric nonlinearities (P-∆ eects). 2. Perform modal analysis from the deformed conguration of the structure, com-

fn , participation factors Γjn (j = X, Y, Z ) and mode shapes φn up to the maximum limiting frequency; fmax = 25 Hz. Next, a study about the participation of each mode below fmax in the global response in puting frequencies

2 Pushover

is equivalent to spectrum analysis in the elastic range.

404

Appendix G. Step-by-step description of advanced pushover terms of forces or displacements is conducted (see section 6.4.4.1), identifying the governing longitudinal (fnX ) and transverse modes (fnY ), EMPA covers both dominant modes. The limit frequency which marks the end of the range where pushover is to be conducted is established as

fgov = max(fnX , fnY ).

nth mode, included in the range studied by pushover analysis (fn ≤ fgov ), develop the base shear-control point displacement curve, V¯bn − u ¯rn 3 , by

3. For the

means of three-dimensional nonlinear static analysis of the structure when the load distribution

s∗n = mφn

is incrementally applied (self-weight included),

now fully considering its three-dimensional characteristics.

Three pushover

curves are obtained while the structure is being pushed beyond the linear range, each one associated with the longitudinal, transverse and vertical di-

X − uX ), (V Y − uY ) ,(V Z − uZ ) (see gure 6.14). The base shear (Vbn rn rn rn bn bn j Vbn is the sum of the shear recorded in each connection of the model to the rections;

ground in

j -direction

during the pushover analysis of

control point displacement point in

j -direction;

ujrn

nth

mode, whilst the

is consequently the displacement of the control

this point, is selected as the node with maximum modal

displacement regardless of the direction where it is recorded. 4. Transform each

j − ujrn Vbn

pushover curves to

expressions (6.46) and (6.47), repeated below:

j Vj Fsn = bn ; Mn Ljn

qnj =

ujrn φjrn

;

with

with

j /Mn − qnj Fsn

j = X, Y, Z

coordinates using

(G.5)

j = X, Y, Z

(G.6)

j Fsn /Mn − qnj curve for the equivalent SDOF system associated with n-mode in j -direction. Here, a bi-linear curve considering a modied

5. Idealize the real

`Equal Area' rule presented in gure 6.11(a) is proposed because of its simplicity (more realistic curves could be considered). The kinematic properties of the nonlinear spring behaviour in the SDOF system need to be addressed (see appendix H), dening the loading and unloading branches appropriate for the structural system and material. 6. Obtain a modular

F¯sn /Mn − q¯n

from the three-directional results using ex-

pressions (6.48) and (6.49), repeated next:

F¯sn = Mn

s



q¯n = 3 The

X Fsn Mn

2

 +

Y Fsn Mn

2

 +

Z Fsn Mn

2 (G.7)

q

(qnX )2 + (qnY )2 + (qnZ )2

(G.8)

bar symbol means `magnitude' of the three directional displacement or shear components.

G.2. Extended Modal Pushover Analysis: EMPA

405 q¯n

7. Compute the peak generalized modular displacement

of the

n-mode

in-

elastic SDOF system. Here, the integration of the SDOF dierential equation (6.45) of the corresponding mode in time domain (repeated below) is proposed, following the scheme presented in appendix H.

q¨¯n + 2ξn ωn q¯˙n + Where

F¯sn = −¨ u∗g,n (t) Mn

Y ¨Y (t) + ΓZ u Z u ¨∗g,n (t) = ΓX ¨X nu g (t) + Γn u g n ¨g (t),

(G.9)

and

u ¨jg

the accelerogram

3D components. 8. Obtain the peak modular control point displacement

u ¯rn

with:

¯ u ¯max qn (t)] rn = φrn max [¯

(G.10)

t

u ¯rn + u ¯G (being u ¯G the modular displacement of control point q Y 2 Z 2 2 due to gravity loads; u ¯G = (uX G ) + (uG ) + (uG ) ) from the database of the

9. Interpolate with

three-dimensional pushover analysis, in order to obtain the combined eects of lateral loads and gravity due to 10. Obtain the contribution of

n-mode

n-mode

contribution

rn+G .

to the seismic response exclusively, by

extracting the eect due to the self-weight: contribution of gravity loads acting alone.

rn = rn+G − rG ,

where

11. Repeat steps 3 to 10 for all modes with frequencies below or equal

rG

is the

fgov .

12. Obtain the total nonlinear seismic response (rnl ) by combining the contribution of each studied mode using an appropriate combination rule, here CQC rule is selected for this purpose. 13. Compute the higher mode seismic response (rel ) by means of spectrum analysis (MRSA), including modes with frequencies higher than than

fmax = 25

fgov and lower or equal

Hz.

14. Combine the participation of rst modes with the higher mode eect by means of SRSS combination rule to obtain the dynamic response of the structure:

rd ≈

q 2 + r2 . rnl el

15. Calculate the total demand by combining the eect due to self-weight (rG ) with the dynamic contribution due to the earthquake exclusively (rd ). Since the sign of earthquake forces is lost in pushover procedures, two hypotheses are made, considering both positive and negative signs in the earthquake response, which takes into account the alternating nature of the seismic response.

r ≈ max (rG ± rd )

(G.11)

406

Appendix G. Step-by-step description of advanced pushover

G.3 Coupled Nonlinear Pushover Analysis: CNSP The steps involved in the proposed coupled pushover (CNSP) are detailed next, repeating for convenience several features introduced in MPA and EMPA: 1. Apply rst the gravity loads considering geometric nonlinearities (P-∆ eects). 2. Perform modal analysis from the deformed conguration of the structure, com-

fn , participation factors Γjn (j = X, Y, Z ) and mode shapes φn up to the limit maximum frequency fmax = 25 Hz. Next, a study about the participation of each mode below fmax in the global response in terms puting frequencies

of forces or displacements is conducted (see section 6.4.4.1), identifying the

4

governing

longitudinal (fnX ) and transverse (fnY ) frequencies.

3. Expand the excitation vector of the dominant longitudinal (nX ) and transverse (nY ) modes, and compute the weight of dominant

j

nk -mode

in direction

j j (αnk ) by adding its expanded nodal forces (snk,i in node i) along the whole

structure (k

= X, Y

and

j = X, Y, Z ).

Alternatively, the ratio could be mod-

ied substituting the summations by the corresponding participation factors.

NP node j = αnk

i=1 NP node i=1

sjnX,i

sjnk,i

+

NP node i=1

(G.12)

sjnY,i

4. Considering the two dominant longitudinal (nX ) and transverse (nY ) modes, develop the base shear versus control point displacement curve,

¯rC , V¯bC − u

by

means of 3D nonlinear static analysis of the structure (self-weight included), incrementally increasing the coupled load pattern

s∗C

of expression (6.52) (con-

sidering the fully 3D characteristics of the excitation vector

s∗C = ΛY s∗nY + ΛX s∗nX Λj = Saj /max(SaY , SaX ), with the governing mode in

where

Saj

s∗n ): (G.13)

is the spectral acceleration associated

j -direction5 , k = X, Y .

The displacement is measured simultaneously in two points during the coupled pushover, corresponding to the control points of the longitudinal and transverse governing modes (selected according to EMPA described above).

4 CNSP,

unlike MPA and EMPA, only conducts one pushover analysis using a combination of loads from the transverse and longitudinal governing modes, instead of performing nonlinear static analysis for all the modes with frequencies below or equal to fgov = max(fnX , fnY ). 5 The component in j -direction of the coupled excitation vector (s∗ j ) is obtained by considering C the corresponding Λ applied to this direction, i.e. by selecting the acceleration spectrum in direction j when expression (6.53) is applied (if j = X, Y the spectrum is horizontal, if j = Z is the vertical design one).

G.3. Coupled Nonlinear Pushover Analysis: CNSP

407

5. The contributions of each dominant vibration mode to the longitudinal and transverse projections of the coupled capacity pushover curve, are computed by multiplying the projection in

j αnX or

j j -direction (VbC − ujrC )

by the weight factor

j αnY expressed above. Three pushover curves are obtained per both j j j j two dominant modes; (VbnX − urnX ) and (VbnY − urnY ) with j = X, Y, Z j j (see gure 6.15). The base shear VbnX and VbnY is respectively the sum of

j -direction ujrnX and ujrnY

the shear recorded in each foundation of the model in the

of each

dominant mode, whilst the control point displacement

is, con-

j -direction

sequently, the displacement in the

for each governing longitudinal

and transverse mode at the corresponding control point in a respective way.

j j VbnX − ujrnX and VbnY − ujrnY pushover curves respectively j j j − qnX and FsnY /Mn − qnY coordinates by means of:

6. Transform each

j into FsnX /Mn

j Fsnk Vj ; = bnk Mnk Ljnk j qnk =

7. Idealize the real

ujrnk φjrnk

;

with

j = X, Y, Z

with

j j /MnX − qnX FsnX

j = X, Y, Z

and

and

and

k = X, Y

k = X, Y

j j /MnY − qnY FsnY

(G.14)

(G.15)

curves for the equiva-

lent SDOF systems associated with each governing mode in

j -direction.

Here,

it is proposed to use a bi-linear curve considering a modied `Equal Area' rule presented in gure 6.11(a) (more realistic curves may be considered).

The

kinematic properties of the nonlinear spring behaviour in the SDOF system need to be simulated (see appendix H), dening the loading and unloading branches appropriate for the structural system and material. 8. Obtain modular

F¯snX /MnX − q¯nX

directional results using expressions

F¯snk = Mnk

s



q¯nk =

X Fsnk Mnk

q

F¯snY /MnY − q¯nY (k = X, Y ):

and

2

X qnk

 +


2

Y Fsnk Mnk

Y + qnk

2


2

 +

Z + qnk

Z Fsnk Mnk


2

relations from the

2 (G.16)

(G.17)

q¯nX of the governing nX -mode inelastic SDOF system, and analogously for the domtransverse nY -mode the peak q ¯nY . Here, integrating the dierential

9. Compute the peak generalized modular displacements longitudinal inant

equation (6.45) of the corresponding governing mode in time domain (repeated below for

nk -mode)

is proposed, following the algorithm in appendix H.

q¨¯nk + 2ξnk ωnk q¯˙nk +

F¯snk = −¨ u∗g,nk (t) Mnk

(G.18)

408

Appendix G. Step-by-step description of advanced pushover

10. Obtain the peak modular control point displacements of the governing modes;

urnX

and

urnY

respectively, with expressions

u ¯rnY = φ¯rnY max [¯ qnY (t)],

where

t

φ¯rnX

and

u ¯rnX = φ¯rnX max [¯ qnX (t)]

φ¯rnY

t

and

are the modular modal dis-

placements at control point in longitudinal and transverse dominating modes respectively. 11. Interpolate with

u ¯rnX + u ¯G

point due to gravity loads:

¯G the modular displacement of control q u Y 2 Z 2 2 u ¯G = (uX G ) + (uG ) + (uG ) ) from the database (being

of three-dimensional pushover analysis particularized for the longitudinal gov-

¯bnX erning mode (V

−u ¯rnX ), to obtain the combined eects of lateral loads and gravity due to nX -mode contribution; rnX+G . Analogously, the contribution of the governing transverse mode nY , rnY +G , is obtained by interpolating u ¯rnY + u ¯G in the pushover curve (V¯bnY − u ¯rnY ). 12. Obtain the contribution of

nX -mode

and

nY -mode

clusively, by extracting the eect due to self-weight:

rnY = rnY +G − rG ,

where

rG

to seismic response ex-

rnX = rnX+G − rG

and

nX -mode

nY -

is the contribution of gravity loads alone.

13. Combine the three-dimensional contributions of governing mode through the CQC combination rule, obtaining

and

rnl .

14. Compute the contributions of vibration modes dierent than the governing ones and below

fmax = 25

Hz, assuming their response elastic (rel ).

This

elastic eect is obtained by means of spectrum analysis (MRSA). Vibration modes above 25 Hz are neglected. 15. Combine through SRSS rule the contribution of governing modes with the contribution of modes below 25 Hz and dierent than the governing ones, in order to obtain the dynamic response of the structure:

rd ≈

q 2 + r2 . rnl el

16. Calculate the total demand by combining the self-weight eect dynamic contribution due to the earthquake exclusively

rd .

rG

with the

Since the sign

of earthquake forces is lost in pushover procedures, two hypotheses are made (taking into account the alternating nature of the seismic input), considering both positive and negative signs in the earthquake response.

r ≈ max (rG ± rd )

(G.19)

Appendix H Nonlinear SDOF: description and integration

In this appendix, the algorithm proposed by Simo and Hughes [Simo 1998] to integrate a nonlinear SDOF system with combined isotropic/kinematic hardening is described, which has been implemented in Matlab [Matlab 2011]. This algorithm is involved in advanced pushover procedures detailed in chapter 6 and appendix G. We consider now the nonlinear SDOF represented by a unit-length element with elastic and frictional behaviour.

Figure H.1 presents the schematic constitutive

model of a nonlinear SDOF system with yielding stress

σY .

Two extreme cases of

the hardening rule may be considered; the isotropic hardening assumes the yield surface only able to expand itself, with its center xed along the time; the kinematic hardening, however, considers the movement of the yield surface in the direction of the plastic ow, maintaining constant its dimensions (Bauschinger eect). The cyclic behaviour of real metals is somewhere in the middle of such extreme idealized responses, which is taken into account in the combined isotropic/kinematic hardening (Simo and Hughes [Simo 1998]). The failure criterion describing the solution in such case is:

f (σ, q, α) := |σ − q| − (σY + Kp α) ≤ 0 q

(H.1)

being the back stress, its evolution is dened by Ziegler's rule as:

q˙ = H ε˙p ≡ γH sign(σ − q) Where

H

is the kinematic hardening modulus,

is the rate at which slip takes place and

α

ε˙p

is the plastic strain rate,

(H.2)

γ≥0

is a `nonnegative function' of the amount

of plastic ow (slip) called internal hardening variable, being its evolution dened in a simplied way by

α˙ = γ .

Figure H.1. Constitutive characterization of the nonlinear SDOF.

410

Appendix H. Nonlinear SDOF integration

To address the numerical implementation of the cyclic loading in the nonlinear SDOF, the auxiliary variable called relative stress; convenience in algorithm 1. In algorithm 1,

E

ζ := σ − q

is dened for

is the elasticity modulus of the element representing the elastic

stiness of the nonlinear SDOF system, and

Kp

is the plastic modulus (slope of the

hardening response in the space of plastic deformations

σ − εp ).

The elastoplastic

tangent modulus (slope of the hardening response in the space of total deformations

σ − ε)

is obtained as;

βp = EKp /(E + Kp ).

Further renements of the kinematic hardening may be performed, for example

by considering its nonlinear evolution. However, the number of parameters required increases with the complexity of the constitutive behaviour. Since the study conducted in this thesis is general and no laboratory tests have been performed, it is preferred to simplify the model to the maximum point, yet maintaining an acceptable level of accuracy. Therefore combined isotropic/kinematic hardening with rate independent plasticity (presented in algorithm 1) is implemented in the advanced pushover analysis conducted in this thesis (chapter 6).

411

Algorithm 1

Return-mapping algorithm for one-dimensional, rate-independent

plasticity. Combined isotropic/kinematic hardening. Simo and Hughes [Simo 1998]

1. Database at

x ∈ B : {εpn , αn , qn }

2. Given strain eld at

.

x ∈ B : εn+1 = εn + ∆εn

.

3. Compute elastic trial stress and test for plastic loading:

p trial := E (ε σn+1 n+1 − εn )

trial := σ trial − q ζn+1 n n+1

trial := ζ trial − [σ + K α ] fn+1 p n Y n+1 IF

trial ≤ 0 fn+1

THEN

Elastic step :

(·)n+1 = (·)trial n+1

& EXIT

ELSE

Plastic step : Proceed to step 4.

ENDIF

4. Return mapping:

∆γ :=

trial fn+1 >0 E + [Kp + H]

trial − ∆γE sign ζ trial σn+1 := σn+1 n+1 trial εpn+1 := εpn + ∆γ sign ζn+1




trial qn+1 := qn + ∆γH sign ζn+1

αn+1 := αn + ∆γ







Appendix I Analytical model for the transverse reaction of the deck

This appendix collects the complete integrated expressions behind the analytical model proposed to obtain the extreme transverse seismic reaction of the deck against the towers.

The discussion about the background theory has been presented in

section 7.6.3, following the procedure proposed by Lord Rayleigh, which is based on a completely linear and elastic response. The results are particularized to the case where the transverse rotation of the deck at the abutments is completely released (θZZ free). Only the response associated with the rst transverse mode is obtained here. Considering the structure and the loads represented in gure I.1 with

q = 1 1

[N/m] (uniform load), the bending moment distribution is expressed as follows :

Figure I.1. Scheme of the static loads imposed to obtain the trial function (ψ) in order to estimate the fundamental mode shape. θZZ free in deck-abutments connection. x (L − x) − Rq1 x; x ∈ [0, LS ] 2

(I.1a)

x (L − x) − Rq1 LS ; x ∈ [LS , L/2] 2

(I.1b)

MAB =

MBE =

1 Only

half the total length of the deck needs to be studied due to symmetry conditions.

414

Appendix I. Transverse reaction of the deck Rq1 is the reaction in the deck due to the springs representing the towers q = 1 N/m. This value is unknown at this point, an it will be obtained by

Where when

imposing the compatibility constraints at the connection of the springs. The curvature (χ(x)) is immediately obtained from eq. (I.1), taking into account that the study in this case is completely elastic:

χ = M/EI

(where

E

and

I

are the

elasticity modulus and transverse moment of inertia of the deck). The distribution of curvature is integrated once to obtain the distribution of rotations (θ(x)), which is in turn integrated to extract the distribution of transverse displacements due to the unit load:

ψq1 (x).

The unknown value of

Rq1

results by considering that

the reaction in the springs due to the displacement at deck-tower connections is expressed by Hooke's law:

Rq1 = Kt ψq1 (x = LS )

Rq1

(I.2a)


Kt −LS L3 + 2 LS 3 L − LS 4 =−   12 LS 2 Kt L − 16 LS 3 Kt 24 EI +1 24 EI

(I.2b)

However, we are interested in the study of a distributed load which origins a unit displacement in the center of the span (in order to obtain the trial function normalized), thus a constant scale factor by imposing

q = a0,q

ψ

is applied. This value is obtained

ψ(x = L/2) = 1: 24 EI

a0,q = 3 LS L2 Rq1

− 4 LS

3

(I.3)

5 L4 Rq1 − 16

With these conditions, the normalized trial mode shape is:

ψAB = a0,q


x 12 LS L Rq1 − 4 x2 Rq1 − 12 LS 2 Rq1 − L3 + 2 x2 L − x3 ; x ∈ [0, LS ] 24 EI

(I.4a)


12 LS x L Rq1 − 12 LS x2 Rq1 ψBE = a0,q + 24 EI
−4 LS 3 Rq1 − x L3 + 2 x3 L − x4 a0,q ; x ∈ [LS , L/2] 24 EI

(I.4b)

Which particularized at the connection between the deck and the towers:

ψB = a0,q

LS 12 LS L Rq1 − 16 LS 2 Rq1 − L3 + 2 LS 2 L − LS 3 24 EI


(I.5)

By means of the principle of virtual work applied to the deformed deck with transverse displacement distribution

ψ,

the generalized properties of the structure

associated with the fundamental mode are estimated.

415 The generalized stiness for the fundamental transverse mode of the deck is obtained with:

  k˜1 = 2 

ZLS

 EI

2 d2 ψAB (x) dx dx2

0 Where

Kt

 2 ZL/2  2 d ψBE (x) 2 + EI dx + Kt ψB  dx2

(I.6)

LS

is the transverse tower stiness.

Substituting expression (I.4) in (I.6) and integrating:

k˜1 =

a20,q 2 2 2 720 L4S Kt L2 Rq1 + 1440 L2S EI L Rq1 − 1920 L5S Kt L Rq1 1440 EI 2 2 2 − 1920 L3S EI Rq1 + 1280 L6S Kt Rq1 − 120 L3S Kt L4 Rq1

− 240 LS EI L3 Rq1 + 160 L4S Kt L3 Rq1 + 240 L5S Kt L2 Rq1

(I.7)

+ 480 L3S EI L Rq1 − 440 L6S Kt L Rq1 − 240 L4S EI Rq1 + 160 L7S Kt Rq1 + 5 L2S Kt L6 + 12 EI L5 − 20 L4S Kt L4
+10 L5S Kt L3 + 20 L6S Kt L2 − 20 L7S Kt L + 5 L8S Kt

Analogously, the generalized mass for the fundamental transverse mode of the deck results from:



ZLS

 m ˜1 = 2

 ZL/2 2 m [ψBE (x)]2 dx + Mt ψB m [ψAB (x)]2 dx + 

0

m

(I.8)

LS

being the distributed mass of the deck and

Mt

the tower mass.

Which is integrated considering expression (I.4):

m ˜1 =

a20,q 2 2 2 3024 L2S m L5 Rq1 − 10080 L4S m L3 Rq1 + 181440 L4S Mt L2 Rq1 362880 EI 2 2 2 2 − 483840 L5S Mt L Rq1 + 16128 L6S m L Rq1 + 322560 L6S Mt Rq1 2 − 9216 L7S m Rq1 − 612 LS m L7 Rq1 + 1008 L3S m L5 Rq1

− 30240 L3S Mt L4 Rq1 + 40320 L4S Mt L3 Rq1 − 504 L5S m L3 Rq1 + 60480 L5S Mt L2 Rq1 − 110880 L6S Mt L Rq1 + 144 L7S m L Rq1

+ 40320 L7S Mt Rq1 − 36 L8S m Rq1 + 31 m L9 + 1260 L2S Mt L6 − 5040 L4S Mt L4 + 2520 L5S Mt L3 + 5040 L6S Mt L2
−5040 L7S Mt L + 1260 L8S Mt Finally, the property obtained with:

˜1 L

(I.9)

for the fundamental transverse mode of the deck is

416

Appendix I. Transverse reaction of the deck  ˜1 = 2  L 

ZLS

 ZL/2  mψAB (x) dx + mψBE (x) dx + Mt ψB 

0

(I.10)

LS

Again, substituting expression (I.4) and integrating:

˜1 = L

a20,q 10 LS m L3 Rq1 + 120 L2S Mt L Rq1 − 20 L3S m L Rq1 120 EI − 160 L3S Mt Rq1 + 10 L4S m Rq1 − m L5
−10 LS Mt L3 + 20 L3S Mt L − 10 L4S Mt

(I.11)

Once the generalized properties associated with the fundamental mode are known, its frequency and participation factor are easily obtained:

ω12 ≈

k˜1 m ˜1

(I.12)

Γ1 ≈

˜1 L m ˜1

(I.13)

The extreme seismic displacement of the connection between the deck and the towers results from:

umax yB,1 ≈ Γ1 ψ1 (LS ) Sa (f1 )

Sa (f1 ) (2πf1 )2

(I.14)

being the spectral acceleration of the transverse design spectrum (pre-

sented in gure 5.1) particularized for the fundamental frequency

f1 = ω1 /(2π).

And the extreme seismic reaction of the deck against the towers (or vice versa) is nally derived from Hooke's law:

Fy,deck,1 = Kt umax yB,1

(I.15)

In order to obtain the proposed tri-linear evolution of the total extreme reaction dened in section 7.6.3 (considering the contribution of higher modes), the maximum reaction of the deck against the towers caused by the rst mode in the model with 300 m main span (F1,max ) is required. particularized for

LP = 300

This force results from expression: (I.15),

m:

F1,max = Fy,deck,1 (LP = 300)

(I.16)

Appendix J Tributary mass of the deck: the `tower model'

By analyzing the transverse seismic behaviour of the towers in full cable-stayed bridge models (e.g.

gure 7.2) two eects were observed to contribute to the re-

sponse; one part of the demand is due to the own tower vibrating like a cantilever frame structure, whereas the rest of the response is exerted by the transverse reaction of the deck.

Consequently, the transverse response of the tower could be

simulated with a 2D model representing exclusively this element, and introducing

1

the reaction of the deck by means of a point mass connected to one leg

(section

3.2.3), allowing only its transverse displacement. On the other hand, the self-weight of the deck and the cable-system aecting the tower is represented by concentrated loads in the anchorages. This simplied `tower model' has the following drawbacks; (i) neglects the vertical and transverse seismic loads transferred from the cables to the tower, nonetheless they are deemed small since the eect of the vertical earthquake is reduced (section 7.4) and the transverse seismic action of the deck is mainly transferred through the connection between the deck and the towers, not by means of the cable-system (section 7.6.1); (ii) the concentrated mass representing the deck, behaves passively against the earthquake, receiving all the seismic excitation through the connection with the deck, despite the abutments and intermediate piers also contribute to its excitation in the real situation. This `tower model' is a simplied way rooted on physical basis which allows the study of a large number of control strategies in the tower with a reasonable computational cost. It has been thoroughly veried that the results of the tower model agree with the ones associated with the complete bridge. Furthermore, the study conducted in chapter 9 is based on the comparison of the response obtained uniquely with this model (with and without seismic devices), hence the errors introduced by the simplication are relativized. The tributary mass of the deck associated with the transverse seismic response of the tower is addressed in this section. The objective now is to establish the value of the concentrated mass which should be dened in the tower model to simulate the transverse seismic eect of the deck in the full cable-stayed bridge. Moreover,

1 The

connection between the deck and the towers is modied in dierent ways throughout chapter 9, however, in the study of the tributary mass of the deck, the original connection considered in this thesis is represented; sti connection between the deck and one leg of the tower (section 3.2.3).

418

Appendix J. Tributary mass of the deck: tower model

the tributary mass of the deck in the towers is interesting in order to design the seismic devices along the deck [Priestley 1996]. From the static point of view, and taking into account the longitudinal elevation of the bridge presented in gure 3.1, the tributary mass of the deck is directly the mass corresponding to half the side and main spans

Ltrib,S = (LS + LP )/2.

2

(the tributary static length);

However, signicant dynamic eects were observed in the

transverse seismic reaction of the deck against the towers (section 7.6); typically, a certain percentage of the tributary static weight of the deck vibrates freely in transverse direction due to the its large exibility, being more remarked as the main span gets larger. A contribution factor

Υ is dened as the percentage of the static tributary mass

of the deck (or length, since its density and cross-section are constant) employed in the 2D model of the tower which is required to obtain the most similar solution in comparison with the corresponding full cable-stayed bridge model:

Υ= Where

Md,2D

Md,2D 100 Mtrib,S

(J.1)

is the point mass representing the seismic eect of the deck in the

Mtrib,S is the mass of the deck corresponding Ltrib,S . In the static reference case; Υ = 100 %.

2D model of the tower, and static tributary length

to the

Taking advantage of the simplied model described in section 7.6.2 (without cable-system but including the whole deck), the analysis of the contribution factor has been conducted by means of spectrum analysis (MRSA) for a large number of main spans, imposing that the error in the transverse shear and bending moment at the foundation level falls within

±5

%. Figure J.1 includes this result in one of

the considered models, observing the decreasing trend of the contribution factor as the main span is increased above a certain level, approximately

300

m; below this

threshold, the increase in the transverse reaction of the deck with the main span is stronger than the increment of the mass corresponding to the deck (Mtrib,S ), which leads to larger values of

Υ

(exceeding in some cases 100 %, which means that the

dynamic reaction of the deck is larger than the static one considering the weight of the structure applied in transverse direction), nonetheless, above

LP = 300

m the

transverse reaction is maintained more or less constant (section 7.6) whereas the mass of the deck grows linearly, hence the contribution factor is reduced. The lower values of the tributary mass for rocky soil were expected because the reaction of the

3

deck against the towers is also smaller in this case . It should be recognized that the analysis presented in gure J.1, despite shedding some light on the trends governing the tributary mass, is based on simplied models

2 Notice

that the transverse movement of the deck in the intermediate piers of the main span is not coerced, and thus the distribution of spans in transverse direction is LS + LP + LS , where LS and LP are respectively the side and main spans. 3 The dierences between soil classes in terms of Υ are more evident as the main span becomes larger due to the dierences in the long period area (which governs bridges with large spans) between the spectrum associated with dierent soils.

419

Figure J.1. Percentage of the tributary mass of the deck Υ obtained from the simplied model (without cable-system) in terms of the main span. YD-CCP model with θZZ free in the connection between the deck and the abutments. Elastic spectrum analysis (MRSA).

and spectrum analysis, thus the results may be not accurate enough to be employed in the design of seismic devices conducted in chapter 9. On the other hand, despite employing the proposed analytical model to predict the extreme reaction of the deck (FY,deck , section 7.6.2) in the estimation of its tributary mass for dynamic analysis (Υ) would be very appealing, the relationship between both variables is not clear due to the contribution of the tower itself to the transverse forces, which becomes more important as the main span gets larger. Consequently, the value of

Υ

is obtained in a rigorous way with a trial & er-

ror procedure, until the distribution of transverse shear (VY ) and total deformation (εtot ) along the height of the tower in the 2D analysis is close enough to the result of the full cable-stayed bridge subjected to the same transverse earthquake, employing modal dynamics (MRHA) repeated for the twelve records. This procedure has been repeated for the ve types of towers, the two cable-system assemblies and considering both xed or free transverse rotation of deck-abutment connections.

Three

main spans have been studied; 200, 400 and 600 m. Only the extreme deformation with several contribution factors in one of the studied models is included in gure J.2, plots obtained for other towers and responses are not presented for the sake of brevity. Tables J.1 and J.2 summarize the proposed values in several cases (with foundation on soft soil). Although the results included in these tables are not strictly valid for rocky soil, where the transverse reaction of the deck is signicantly lower, accurate results have been generally observed in that case by employing the contribution factors obtained for soft soil if the main span is below 400 m, in agreement

420

Appendix J. Tributary mass of the deck: tower model 4

with gure J.1 .

Figure J.2. Extreme averaged total deformation (εtot ) along the tower height in the reference 3D full model and the `tower model' with dierent contribution factors Υ. H-LCP model. LP = 200 m. Soft soil (TD). θZZ xed in deck-abutments connection. Elastic modal dynamics (MRHA) with transverse earthquakes. Sign `+' tension; `-' compression. LP [m] 200 400 600

H-LCP

Y-LCP

YD-LCP

A-LCP

AD-LCP

θZZ xed

θZZ xed

θZZ xed

θZZ xed

θZZ xed

77 26 13

50 25 10

39 15 5

46 26 10

36 13 5

Table J.1. Summary of the contribution factors Υ [%] in cable-stayed bridges with lateral cable-system, expressing the tributary mass of the deck aecting the seismic response of the towers. θZZ xed. Soil type D. Elastic modal dynamics (MRHA). See gure 3.2 for the description of model keywords. Several valuable conclusions may be extracted from tables J.1 and J.2, taking into account the larger eectiveness of the seismic devices connecting the deck and the tower in transverse direction as the tributary mass is increased (i.e. higher values of

Υ), •

which is veried in chapter 9:

The tributary mass of the deck in transverse direction has a decreasing trend with the main span, which starts suggesting that above

LP = 400

m is more

eective to focus the seismic control strategy on the tower itself, ignoring the

4 The results collected in table J.2 (rigorous procedure) and gure J.1 (simplied procedure) for the same model are also in close agreement, except in the case of large models (LP = 600 m), as it was expected due to high order mode contributions.

421 LP [m] 200 400 600

Y-CCP

YD-CCP

YD-CCP

θZZ free

θZZ free

θZZ xed

100 45 13

100 50 13

40 21 8

Table J.2. Summary of the contribution factors Υ [%] in cable-stayed bridges with central

cable-system, expressing the tributary mass of the deck aecting the seismic response of the towers. Soil type D. Elastic modal dynamics (MRHA). See gure 3.2 for the description of model keywords. eect of the deck because it is reduced in comparison with the response of the tower itself (observed throughout chapter 9).

•

The connection between the deck and the abutments plays a fundamental role in the tributary mass below

400

m main span, as it was expected from

the conclusions extracted in chapter 7; larger tributary mass (even equal to the corresponding static value), and therefore larger eectiveness of devices between the tower and the deck, are obtained if the transverse rotation of this connection is released (increments of

•

Υ

around 50 % are shown in table J.2).

The tributary mass in models with `H'-shaped towers and main span of

200

m is comparatively higher than values observed in homologue structures.

The preceding results are focused on purely elastic analysis, however, the inelastic demand is expected to reduce the eectiveness of seismic devices for two reasons; (i) the reaction of the deck is reduced as the inelastic demand becomes more signicant (section 8.4) and, consequently, the performance of the seismic devices between the deck and the towers is also lower; and (ii) the hysteresis loops recorded in the behaviour of several tower sections increases the damping, being well known the reduced eectiveness of damping increments (like the ones introduced by seismic devices) beyond certain levels [Priestley 1996] [Villaverde 2009].

List of Abbreviations A-LCP `A'-shaped towers with lateral cable plane arrangement AD-LCP `A'-shaped towers with lower diamond and lateral cable plane arrangement BRB

Buckling-Restrained Brace

CCP

Central Cable Plane

CNSP Coupled Nonlinear Static Pushover CQC

Complete Quadratic Combination

DRHA Direct Response History Analysis EMPA Extended Modal Pushover Analysis FEA

Finite Element Analysis

FEM

Finite Element Model

H-LCP `H'-shaped towers with lateral cable plane arrangement HHT

Hilber, Hughes and Taylor integration algorithm

IP

Integration Point

LBC

Link Bearing Connection

LCP

Lateral Cable Plane

LRB

Lead Rubber Bearing

MD

Yielding Metallic Dampers

MDOF Multi Degree Of Freedom MECS Multiple Element Cable-System MRD

Magnetorheological Dampers

MRHA Modal Response History Analysis MRSA Modal Response Spectrum Analysis NGA

Next Generation of Attenuation

NSP

Nonlinear Static Procedures

OECS One Element Cable-System

424 PGA

List of Abbreviations Peak Ground Acceleration

SDOF Single Degree Of Freedom SL

Shear Link

SRSS

Square Root Sum of the Squares

TA

Rocky soil: Eurocode 8 soil type A

TADAS Triangular Added Stiness and Damping TD

Soft soil: Eurocode 8 soil type D

TLD

Tuned Liquid Damper

TMD

Tuned Mass Damper

UHS

Uniform Hazard Spectrum

VD

Viscous uid Dampers

Y-CCP Inverted `Y'-shaped towers with central cable plane arrangement Y-LCP Inverted `Y'-shaped towers with lateral cable plane arrangement YD-CCP Inverted `Y'-shaped towers with lower diamond and central cable plane arrangement YD-LCP Inverted `Y'-shaped towers with lower diamond and lateral cable plane arrangement

List of Symbols Note : Several symbols have been repeated along the thesis to maintain their classical

notation, however, they do not interfere with each other and the distinction is clear. Symbol (*) means that the parameter could adopt any form of the response. Only the most relevant symbols are included here.

Symbol αa αd j αnk βa βb βi βp βp,j βr Γ(·) Γjn γa γij γi,n p γSL ∆ε/2 ∆εp /2 ∆D,p ∆D,s ∆T ∆t δc δt εc εc,crack εmax c,t εcu,c εcy εc1,c ε0f εsh εsu εsy

Description

SI unit

Factor controlling the numerical damping in HHT

dimensionless

Damper velocity exponent

dimensionless

Weight of dominant

nk -mode

in

j

direction

dimensionless

Parameter controlling the stability in HHT

dimensionless

Angle of the tower legs above the deck Angle between the

i-cable

and the deck

Angle of the tower legs below the lower strut Elastoplastic tangent modulus for

j th

damper

rad rad rad N/m

2

Factor to consider opposite disp. between supports

dimensionless

Gamma function

dimensionless

Participation factor for

n-mode

in

j -direction

dimensionless

Parameter controlling the stability in HHT

dimensionless

Coherency function for spatial variability eects

dimensionless

Correlation coecients for CQC rule

dimensionless

Ultimate plastic rotation angle of the shear link

rad

Maximum strain amplitude in the yielding damper

dimensionless

Plastic strain amplitude in the yielding damper

dimensionless

Distance between cable anchorages in deck main span

m

Distance between cable anchorages in deck side span

m

Distance between cable anchorages in the tower

m

Step-time Gap between the deck and the tower leg Time-delay of an asynchronous seismic excitation

s m s

Normal total deformation in the concrete

dimensionless

Deformation marking concrete cracking initiation

dimensionless

Maximum deformation of concrete under tension

dimensionless

Ultimate deformation of concrete under compression

dimensionless

Elastic limit deformation of concrete under compression

dimensionless

Deformation during max. concrete compression force

dimensionless

Fatigue ductility coecient

dimensionless

Deformation marking the end of the steel plateau

dimensionless

Ultimate deformation of reinforcement steel

dimensionless

Elastic limit deformation of reinforcement steel

dimensionless

426

List of Symbols

Symbol η η η¯ j ηmodal Θ θ θ θjj θZZ ι ιj Λ λ ¯ λ ¯ λ µ νc , νs ξ ρc , ρs ρij ρw ρ∗ σ σ σc σe σtot σT,ult σu σy,f , σy,w Υ ϕp φn φnX , φnY φ φbl φi,p φrn φ¯rn φY χ

Description

SI unit

Damping correction factor for design spectra

dimensionless

Performance index of the lower strut

dimensionless

Average performance index of the lower strut

dimensionless

Modal eective mass ratio (cumulative) in

j -direction

Operator relating the damper force and relative disp.

dimensionless *

Angle between the damper axis and the horizontal line

rad

Incidence angle between the wave-train and structure axis

rad

Nodal rotation about

j -axis

(global)

Transverse rotation of the deck in the abutment

rad rad

Inuence matrix of the structure

dimensionless

j -column

dimensionless

of the inuence matrix

Ratio between spectral accelerations of governing modes

dimensionless

Ratio between the studied procedure and modal dynamics

dimensionless

Ratio Ratio

λ ¯ λ

averaged for the whole tower

dimensionless

averaged for the ve main spans

dimensionless

Mean response

*

Poisson's ratio of concrete and steel

dimensionless

Fraction of critical damping

dimensionless

Density of concrete and steel

kg/m

3

Modal coupling ratio

dimensionless

Volumetric ratio of transverse reinforcement

dimensionless

3

Fictitious density in homogeneized sections

kg/m

Normal stress

N/m

Standard deviation of the response

2

*

2

Normal stress in the concrete

N/m

Lateral eective conning stress

N/m

Normal stress due to the self-weight and earthquake

N/m

Maximum strength of prestressing steel

N/m

Ultimate bearing capacity of the subsoil

2 N/m

Yielding stress of the ange and the web

N/m

Transverse contribution factor of the deck Phase angle for signal generations Natural

nth

mode shape

2

2

2

dimensionless rad dimensionless

Governing longitudinal and transverse mode shapes

dimensionless

Diameter of reinforcement rebars

m

Diameter of longitudinal reinforcement rebars

m

Modal displacement for

2

i-node

in the principal mode

dimensionless

Modal displacement for control point in

dimensionless

Modular modal control point

n-mode displacement in n-mode

dimensionless

Transverse modal displacement Beam curvature

dimensionless 1/m

List of Symbols

427

Symbol ψ ψB ψE ψij Ω Ω0 Ω1 ωn ω0

Description

SI unit

Trial mode shape for the deck

dimensionless

Trial mode shape for the deck at tower connection

dimensionless

Trial mode shape for the deck at the span center

dimensionless

Load combination factor

dimensionless

Dissipation factor

dimensionless

Dissipation factor in bridges without seismic devices

dimensionless

Dissipation factor in bridges with seismic devices

dimensionless

nth

rad/s

Angular frequency for

mode

Angular frequency of the sinusoidal motion

rad/s

Roman symbols ABRB Ac Aco,i Ahom Anec As As AT,i At,SL ag an aSL avg a0 , a1 a0,q B Ba Bf b0 bSL CB Ccp Cd Cn Cy C1 , C2 , C2 c c

2

Area of the metallic core in BRB cross section

m

Area of concrete in deck cross-section

2 m

Surface of the deck corresponding to

i-cable

m

2 2

Homogeneized area in deck cross-section

m

Area required in tower critical cross-section

2 m

Area of steel in deck cross-section

m

Shear area of the shear link cross-section

2 m

Area of

i-cable

2

2

cross-section

m

2

Total area of the shear link cross-section

m

Design ground acceleration on type A ground Pseudo-acceleration for SDOF representing

nth

m/s mode

Distance between vertical stieners in the shear link Design ground acceleration in vertical direction

2

m/s m m/s

Rayleigh's damping coecients Scale factor to obtain the normalized trial function

ψ

dimensionless m

Maximum width of the triangular plate in TADAS

m

Smallest side of the shallow foundation

m

Width of the section between stieners

m

Total width of the shear link

m

Parameter to design stieners spacing in shear links Vertical stiness of each cable plane Damping coecient (initial damper stiness)

nth

2

dimensionless

Width of the deck (distance between cable anchorages)

Generalized damping for the

2

mode

dimensionless N/m

α N(s/m) d Ns/m

Factor depending on the main to side span length

dimensionless

Factors modifying the elastic displacement (FEMA-356)

dimensionless

Damping matrix of the structure Fatigue ductility exponent

Ns/m dimensionless

428

List of Symbols

Symbol DRn DSR,5−75 DSR,5−95 d d0 dg dmax m dp dr dSL E Ec EH , EV Ei EK EM Ep Eps ES ESd ESe ESp Es Esec Etot EV Ev,0 EW e ef evp eSL F Fa F (t) Fd Fdj Fd,max Fel,MD

Description Dominant direction for

nth

SI unit

mode

dimensionless

Signicant relative earthquake duration

s

Signicant relative earthquake duration

s

Continuum damage parameter

dimensionless

Depth of the section between stieners

m

Maximum ground displacement during the earthquake

m

Maximum relative displacement between supports

m

Cable horizontal projection

m

Foundation displacement in asynchronous excitations

m

Depth of the shear link

m

2

Young's modulus

N/m

Tangent elastic Young's modulus of concrete

N/m

Seismic excitation in horizontal and vertical direction

m/s

2

Inertial contribution of the seismic response

*

Kinetic energy

J

Dissipated energy by seismic devices

2

J

Young's modulus of cable prestressing steel

2

N/m

Pseudostatic contribution of the seismic response

*

Strain energy

J

Strain energy dissipated by damage

J

Recoverable strain energy

J

Dissipative plastic strain energy

J

2

Young's modulus steel

N/m

Secant stiness of conned concrete

N/m

2

Total contribution of the seismic response

*

Dissipated energy by viscous damping

J

2

Ernst's tangent cable modulus

N/m

Seismic input energy

J

Relative error of FEM in ductility predictions

dimensionless

Eciency factor for control strategies

dimensionless

Eccentricity of the lower diamond foundation

m

Length of the shear link

m

Force vector

N

Force vector due to seismic devices

N

Modulating function: ground acceleration envelope Damper reaction force Force in the dashpot representing the

dimensionless N

j th

damper

N

Maximum allowable damper reaction force

N

Yielding force of the metallic damper

N

List of Symbols Symbol Ffuse Fkj Fp,0 Fsn F¯sn Fy FY,b FY,crack FY,d FY,deck Fy,deck,1 FY,p FY,strut Fy,wind FY,yield F1,max fn fS ∗ fcd fck fcm fcm,c fc,t fgov flim fm fmax fnX , fnY fs,t fs,y fs,ye fv fy fθ G g H HA Ha Hi Htot

429 Description

SI unit

Failure load of the fuse restrainer

N

Force in the spring representing the

j th

damper

Initial cable preload

N N

Force stiness component of the inelastic SDOF

N

Modular force stiness component of the inelastic SDOF

N

Elastic limit of the inelastic SDOF system

N

Transverse global tower force above the deck connection

N

Force in the dampers when cracking in the tower arises

N

Transverse global tower force at the deck connection

N

Extreme reaction in the towers due to the deck

N

Extreme deck reaction in the towers for mode 1

N

Transverse global tower force below the lower strut

N

Increment of transverse force below the lower strut

N

Extreme transverse force in the damper due to the wind

N

Force in the dampers when yielding in the tower arises

N

Extreme deck reaction for mode 1 and 300 m main span

N

Equivalent static force vector for

nth

mode

Stiness component of the force vector

N N

2

Maximum allowable compression prior to the earthquake

N/m

Characteristic unconned concrete compressive strength

N/m

Mean unconned concrete compressive strength

N/m

Maximum conned concrete compressive strength

N/m

Tensile strength of concrete

N/m

Frequency of the governing mode Frequency of the higher mode in modal analysis Damper magnication factor

2 2 2 2

Hz Hz dimensionless

Maximum frequency included in advanced pushover

Hz

Governing longitudinal and transverse frequencies

Hz

2

Maximum strength of reinforcement steel

N/m

Yield strength of reinforcement steel

N/m

2

Design yield strength of reinforcement steel

N/m

2

First frequency with vertical deck deformation

Hz

First frequency with transverse deck deformation

Hz

First frequency with torsional deck deformation

Hz

2

Shear modulus

N/m

Earth gravitational acceleration

m/s

Height of the tower above the deck

m

Length of the anchorage area in the tower

m

Height of the triangular plate in TADAS

m

Tower height from the foundation to the deck

m

Total tower height, from the foundation to the top

m

2

430

List of Symbols

Symbol h Id I , IH Ia Ijj Im,x J Kel,MD Ke Kfuse KN Kn Kp Ks Kt Kt,max Kt∗ k k˜ kA , kB kf kpile kx , ky , kz L ˜n L Lαv,i Lc Lelem strut Lleg elem , Lelem Lf Lg Li LMD Ln LP LS lp lp,Priestley lp,0.5D Ltor Ltrib,S

Description

SI unit

Depth of the deck

m

Localization matrix for seismic devices

dimensionless

Transverse moment of inertia of the deck Arias intensity

m

4

m/s

Moment of inertia of the section along

j -axis

Deck torsional mass moment of inertia

m

4

kg m

2

4

Torsion constant

m

Elastic stiness in TADAS yielding damper Connement eectiveness factor

dimensionless

Stiness of the fuse restrainer

N/m

Axial stiness of the shear link Generalized stiness for the

N/m

N/m

nth

mode

N/m

2

Plastic modulus

N/m

Elastic shear stiness of the shear link

N/m

Transverse stiness of the tower

N/m

Maximum transverse stiness of the tower Dimensionless transverse stiness of the tower

N/m dimensionless

Elastic stiness matrix of the structure

N/m

Generalized modal stiness

N/m

Slopes for the tri-linear deck reaction curve Scale factor of the foundation (sensitivity analysis)

N/m dimensionless

Transverse cantilever stiness of the lower legs

N/m

Foundation stiness in the corresponding direction

N/m

Total length of the deck Generalized

m

Ln

kg

Average distance from one support to the adjacent

m

Length between the hinge and the contra-exure

m

Finite element length

m

Finite element length in the leg and the strut

m

Largest side of the shallow foundation

m

Maximum length for correlated seismic actions

m

Distance between the

i-support

and left abutment

m

Total length of TADAS yielding damper

m

= φTn mι

kg

Main span length

m

Side span length

m

Plastic hinge length

m

Plastic hinge length proposed by Priestley

m

Plastic hinge length by the half section depth rule

m

Free torsional length of the deck

m

Tributary transverse static length of the deck

m

List of Symbols Symbol Md Md,2D j Mef f,n Mjj Mn Ms MT Mt Mtrib,S MW m m m ˜ mi N N Nelem Nel,SL Nf Nm Nnode Np NT Nval NVD Ny,BRB n nk nX , nY Pef f Pchar q qcm qn q¯n qpp qscu qu Rpm Rq1 Rrup Rt+∆t /2 Ry

431 Description

SI unit

Mass of the deck

kg

Point mass representing the deck in tower model

kg

Eective mass for

n-mode in j -direction j -axis for the nth mode

kg

Bending moment along Generalized mass

Nm kg

Bending capacity of the shear link (plastic moment)

Nm

Total mass of the model

kg

Transverse eective modal mass of the tower

kg

Tributary transverse static mass of the deck

kg

Moment magnitude of the earthquake

dimensionless

Mass matrix of the structure

kg

Mass of the deck per unit length

kg/m

Generalized modal mass Mass associated with

kg

i-node

kg

Number of degrees of freedom in the structure

dimensionless

Axial load

N

Number of elements in the tower

dimensionless

Yielding axial load (acting alone) of the shear link

N

Number of cycles to failure

dimensionless

Number of modes included in signal generations

dimensionless

Number of nodes in the tower (excluding struts)

dimensionless

Number of triangular plates in TADAS

dimensionless

Number of stays per half main span and cable plane

dimensionless

Number of analysis steps where

η¯ 6= 0

dimensionless

Number of viscous damper units

dimensionless

Failure axial load of BRB

N

Mode number

dimensionless

Mode number for governing mode in

k = X, Y

direction

Mode number for longit. and transv. governing modes

dimensionless dimensionless

Excitation vector

N

Characteristic model reaction during the earthquake

N

Distributed load along the deck

N/m

Dead load of the deck Generalized coordinate for the

N/m

nth

mode

Generalized modular coordinate for the

2

m

nth

mode

m

Self-weight of the deck

N/m

2

Live load over the deck

N/m

2

Ratio of elastic and inelastic accelerations

dimensionless

Palmgren-Miner fatigue ratio Static transverse deck reaction imposing

dimensionless

q=1

N/m

N

Rupture distance to site

m

Residual force in HHT algorithm

N

Material overstrength

dimensionless

432

List of Symbols

Symbol r r rd rel rG rn rnst rnl rn,o rn+G S Sa SaX , SaY Sd s sj s∗n s∗C s∗nX , s∗nY s∗i T T TC , TD Tf Tv Ty t ta tc tf , tw tsp ttotal Uj u, u˙ , u ¨ u ¨ g (t) un u ¨ u ¨jg u ¨∗g,n u ¨T uG

Description

SI unit

Deck cross-section radius of gyration

m

Total response of the structure

*

Contribution of the earthquake to the response

*

Contribution of higher modes to the response (elastic)

*

Contribution of the self-weight to the response

*

Dynamic structural response for Static structural response for

nth

nth

mode

mode

*

Contribution of the earthquake to the inelastic response

nth

Peak structural response for

mode

Eect of lateral loads and gravity for

nth

*

* *

mode

Amplication factor due to the soil class

* dimensionless

2

Spectral acceleration

m/s

Spectral acc. of governing long. and transv. modes

m/s

2

Spectral displacement

m

Spatial distribution of the seismic forces

kg

Seismic force spatial distribution in

nth

Load pattern in pushover for

j -direction

mode

kg kg

Coupled pushover load pattern

kg

Load pattern for governing long. and transv. modes

kg

Load associated with

i-node

in pushover

Torsional moment

kg Nm

Vibration period

s

EC8 corner vibration periods for design spectra

s

Fundamental vibration period

s

First period with vertical deck deformation

s

First period with transverse deck deformation

s

Time

s

Thickness of the triangular plate in TADAS

m

Tower cross-section thickness

m

Thickness of the ange and the web in shear links

m

Duration of the signal strong-pulse phase

s

Total duration of the signal above threshold

s

Nodal displacement along

j -axis

m

Relative displacement, velocity and acceleration vectors

2 m/s

Accelerogram vector Contribution of

nth

m, m/s, m/s

mode to the total displacement

Relative acceleration of the SDOF system Accelerogram component along Equivalent accelerogram for

j -axis

nth

mode

Total acceleration of the SDOF system Control point displacement due to self-weight

m m/s

2

m/s

2 2

m/s m/s m

2

2

List of Symbols

433

Symbol um uo urn u ¯rn ¯rC u ujrC umax rn u ¯max rn uFultEM uref ult uY,t umax yB,1 upZ Vbn V¯bn V¯bC

Description

SI unit

Peak displacement of the inelastic SDOF

m

Peak displacement of the equivalent elastic SDOF

m

n-mode pushover displacement (n-mode pushover)

Control point displacement in Modular control point

m m

Control point disp. magnitude in 3D coupled pushover

m

j -projection

m

of control point disp. in coupled pushover

Target control point displacement (n-mode pushover)

m

Modular target control point disp. (n-mode pushover)

m

Ultimate displacement obtained in FEM

m

Ultimate displacement obtained in laboratory tests

m

Transverse displacement of the tower at deck level

m

Max. transv. deck displacement at tower level (mode 1)

m

Max. relative plastic displacement between link ends

m

Base shear in

n-mode

pushover

Base shear magnitude in 3D pushover for

N

n-mode

Base shear magnitude in 3D coupled pushover

N N

j VbC

j -projection

Vcrack Vel,SL Vj Vs Vs,30 k k VX,nX , VY,nY k k , VY,tot VX,tot vg vs wp x, x˙ , x ¨ x xd x ¨g x ¨g,max (t) x ¨t x0 Yvp y y∗ z z∗ zg zi

Shear in the shear link when cracking arises in the tower

N

Yielding shear of the shear link (acting alone)

N

of base shear in 3D coupled pushover

Shear force along

j -axis

N

Link shear strength

N

Shear wave velocity of the uppermost 30 meters Shear in node

k

due to the governing modes

Total shear in node

N

k

m/s N N

Maximum ground velocity during the earthquake

m/s

Propagation velocity of the seismic wave-train

m/s

Weight of the cable per unit length Relative displacement, velocity and acceleration (SDOF)

N/m m, m/s, m/s

Distance along the deck from the left abutment

m

Relative displacement of the damper along its axis

m

Accelerogram record in

x

direction

2

m/s

Envelope function of absolute ground accelerations

m/s

2

Total acceleration of a SDOF system

m/s

2

Amplitude of the sinusoidal motion

m

Transverse width of the vertical pier

m

Transverse distance to a generic section of the strut Strut transverse dimensionless distance Vert. distance of a generic section from tower foundation Dimensionless tower height

2

m dimensionless m dimensionless

Height of the deck gravity center (from lower deck ber)

m

Height between the deck and lower diamond connection

m

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Seismic behaviour of cable-stayed bridges: design, analysis and seismic devices Abstract:

Cable-stayed bridges, key points in the transport networks, are one

of the most challenging structures for the civil engineering community, and their collapse should be prevented even under extremely strong earthquakes. In this work, the seismic response of a large number of linear and nonlinear cablestayed bridges models has been studied, verifying the inuence of the tower shape, cable-arrangement, main span or foundation soil (among other features) on their behaviour when severe ground motions are imposed (compliant with Eurocode 8 specications).

Rigorous Finite Element Models have been employed, thoroughly

validated by the comparison with experimental testing. Due to excessive inelastic demand identied, and following the current trend of cable-stayed bridge design in seismic zones, the incorporation of yielding metallic and viscous uid dampers has been explored. On the other hand, a considerable eort has been made on the discussion of available analysis strategies and the development of advanced pushover procedures, appropriate for seismic responses with signicant coupling between structural members. Furthermore, an analytical model has been included in order to predict the seismic reaction of the deck against the towers, and modied algorithms were suggested to generate synthetic accelerograms to be used in linear and nonlinear analysis with Rayleigh damping. Valuable conclusions for the designer about the seismic behaviour of cable-stayed bridges have been obtained, regarding the inuence of several project decisions on the response of the structure, proposing seismic devices to avoid undesirable damage in the main components of the bridge and also suggesting new pushover procedures specically adapted to three-axially excited cable-stayed bridges.

Keywords:

Cable-stayed bridges; design; pushover; seismic devices; nonlinear dy-

namics; energy balance.