NONLINEAR SEISMIC BEHAVIOUR OF CONCRETE …oa.upm.es/38559/1/Tesis_master_Pedram_Manouchehri.pdf · spanarchbridgeismadeofsteel,concreteorcombinationofthesetwoas"CFST"1, astheresultofusingthesehighstrengthmaterials,verylongspanscanbeachieved

DEPARTAMENTO DE MECÁNICA DE MEDIOS CONTINUOS YTEORÍA DE ESTRUCTURAS

Escuela Técnica Superior de Ingenieros de Caminos, Canales y PuertosUniversidad Politécnica de Madrid

NONLINEAR SEISMIC BEHAVIOUR OFCONCRETE ARCH BRIDGES

Pedram Manouchehri

In Partial Fulfilment of the Requirements for the Degree of:Máster Universitario en Ingeniería de las Estructuras,

Cimentaciones y Materiales(Master of Science)

Advisor: Prof. Dr. Miguel Ángel Astiz Suárez

September 2015

Nonlinear seismic behaviour of concrete arch bridges

Universidad Politécnica de MadridMadrid, September 2015

La composición de este texto se ha realizado íntegramente en LATEX.

Pedram ManouchehriCivil Engineer

Escuela Técnica Superior de Ingenieros de Caminos, Canales y PuertosDepartamento de Mecánica de Medios Continuos y Teoría de Estructuras(Department of Continuum Mechanics and Theory of Structures)

Universidad Politécnica de MadridProfesor Aranguren s/nMadrid 28040

Teléfono: (+34) 91 336 66 59Correo electronico: [email protected]

Dedicated to my ParentsFor their endless support

and encouragements

Acknowledgement

This master thesis has been carried out at the Department of Continuum Mechanicsand Theory of Structures, Universidad Politécnica de Madrid , in September 2015.A number of people deserve thanks for their support and help. It is therefore mygreatest pleasure to express my gratitude to them all in this acknowledgement.

First and foremost I offer my sincerest gratitude to my advisor, Prof. MiguelÁngel Astiz Suárez, who has supported me throughout my thesis with his patienceand knowledge. I attribute the level of my Master’s degree to his encouragement andeffort and without him this thesis, too, would not have been completed or written.One simply could not wish for a better or friendlier supervisor.

My sincere thanks also goes to my fellow lab mates in bridge department SergioBlancas Saiz and Ángel Fernández Gómez for their companionship and helping meget through, also made many helpful suggestions.Last but not least, I would like to thank my friend Shadi Samavi for her helps inthree dimensional models and graphical figures.

Contents

1 Introduction 1

2 Modelling and assumptions 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Arch bridge description . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Geometric aspects . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 Deck connections . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.1 Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Reinforcement steel . . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 Elastomer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Finite element type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Seismic analysis 173.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 The mathematical system of dynamics . . . . . . . . . . . . . . . . . 173.3 Elastic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.1 Direct Response History Analysis: DRHA . . . . . . . . . . . 183.3.2 Modal Response History Analysis: MRHA . . . . . . . . . . . 193.3.3 Modal Response Spectrum Analysis: MRSA . . . . . . . . . . 21

3.4 Inelastic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4.1 Nonlinear Response History Analysis: NL-RHA . . . . . . . . 223.4.2 Modal Pushover Analysis: MPA . . . . . . . . . . . . . . . . 233.4.3 Extended Modal Pushover Analysis: EMPA . . . . . . . . . . 253.4.4 Coupled Nonlinear Static Pushover Analysis: CNSP . . . . . 28

4 The ground motion and discussion of results 314.1 Ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.1 Result of Elastic procedures . . . . . . . . . . . . . . . . . . . 324.2.2 Results of Inelastic procedures . . . . . . . . . . . . . . . . . 32

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

A Description of pushover analyses 40A.1 Modal Pushover Analysis: MPA . . . . . . . . . . . . . . . . . . . . . 40A.2 Extended Modal Pushover Analysis: EMPA . . . . . . . . . . . . . . 42A.3 Coupled Nonlinear Pushover Analysis: CNSP . . . . . . . . . . . . . 44

B Ground motion definition 48

Contents vii

Bibliography 51

Chapter 1

Introduction

Arch bridge structural solution has been known for centuries, in fact the simplenature of arch that require low tension and shear strength was an advantage as thesimple materials like stone and brick were the only option back in ancient centuries.By the pass of time especially after industrial revolution, the new materials wereadopted in construction of arch bridges to reach longer spans. Nowadays one longspan arch bridge is made of steel, concrete or combination of these two as "CFST" 1,as the result of using these high strength materials, very long spans can be achieved.The current record for longest arch belongs to Chaotianmen bridge over Yangtzeriver in China with 552 meters span made of steel and the longest reinforced concretetype is Wanxian bridge which also cross the Yangtze river through a 420 metersspan (see figure 1.1). Today the designer is no longer limited by span length as long

Figure 1.1: Wanxian Bridge: The current longest RC arch bridge with 420m span.Photograph by [Glabb 2012]

1CFST stands for Concrete Filled Steel Tubular

2

as arch bridge is the most applicable solution among other approaches, i.e. cablestayed and suspended bridges are more reasonable if very long span is desired (Acomparison between largest spans of every type is presented in figure 1.3). Like

0 100 m

0 300 ft

Airbus A380

RMS Titanic

© 2015 CMG Lee

Russky BridgeAkashi Kaikyō Bridge Great Belt Fixed Link East Bridge

Golden Gate Bridge (USA, 1932-37)

Chaotianmen Bridge Millau ViaductSydney Harbour Bridge

Forth Bridge (Scotland, 1882-90)Tower Bridge (England, 1886-94)Rialto Bridge (Italy, 1588-91)

Figure 1.2: A comparison of famous long span bridges. Figure by [Cmglee 2015]

any super structure, the economical and architectural aspects in construction ofa bridge is extremely important, in other words, as a narrower bridge has betterappearance, it also require smaller volume of material which make the design moreeconomical. Design of such bridge, beside the high strength materials, requiresprecise structural analysis approaches capable of integrating the combination ofmaterial behaviour and complex geometry of structure and various types of loadswhich may be applied to bridge during its service life. Depend on the design strategy,analysis may only evaluates the linear elastic behaviour of structure or considerthe nonlinear properties as well. Although most of structures in the past weredesigned to act in their elastic range, the rapid increase in computational capacityallow us to consider different sources of nonlinearities in order to achieve a morerealistic evaluations where the dynamic behaviour of bridge is important especially inseismic zones where large movements may occur or structure experience P−∆ effectduring the earthquake. The above mentioned type of analysis is computationallyexpensive and very time consuming. In recent years, several methods were proposedin order to resolve this problem. Discussion of recent developments on these methodsand their application on long span concrete arch bridges is the main goal of thisresearch. Accordingly available long span concrete arch bridges have been studiedto gather the critical information about their geometrical aspects and propertiesof their materials. Based on concluded information, several concrete arch bridgeswere designed for further studies. The main span of these bridges range from 100to 400 meters. The Structural analysis methods implemented in in this study areas following:

Elastic Analysis:

Direct Response History Analysis (DRHA): This method solves the directequation of motion over time history of applied acceleration or imposed loadin linear elastic range.

Modal Response History Analysis (MRHA): Similar to DRHA, this methodis also based on time history, but the equation of motion is simplified to

3

Figure 1.3: The arch of Almonte railway bridge during construction. Personalphotograph by Pedram Manouchehri. 25 Mar 2015.

single degree of freedom system and calculates the response of each modeindependently. Performing this analysis require less time than DRHA.

Modal Response Spectrum Analysis (MRSA): As it is obvious from its name,this method calculates the peak response of structure for each mode and com-bine them using modal combination rules based on the introduced spectra ofground motion. This method is expected to be fastest among Elastic analysis.

Inelastic Analysis:

Nonlinear Response History Analysis (NL-RHA): The most accurate strat-egy to address significant nonlinearities in structural dynamics is undoubtedlythe nonlinear response history analysis which is similar to DRHA but extendedto inelastic range by updating the stiffness matrix for every iteration. Thisonerous task, clearly increase the computational cost especially for unsymmet-rical buildings that requires to be analyzed in a full 3D model for taking thetorsional effects in to consideration.

Modal Pushover Analysis (MPA): The Modal Pushover Analysis is basicallythe MRHA but extended to inelastic stage. After all, the MRHA cannot solvethe system of dynamics because the resisting force fs(u, u) is unknown forinelastic stage. The solution of MPA for this obstacle is using the previouslyrecorded fs to evaluate system of dynamics.

4

Extended Modal Pushover Analysis (EMPA): Expanded Modal pushover isa one of very recent proposed methods which evaluates response of structureunder multi-directional excitation using the modal pushover analysis strategy.In one specific mode,the original pushover neglect the contribution of the direc-tions different than characteristic one, this is reasonable in regular symmetricbuilding but a structure with complex shape like long span arch bridges maygo through strong modal coupling. This method intend to consider modalcoupling while it take same time of computation as MPA.

Coupled Nonlinear Static Pushover Analysis (CNSP): The EMPA includesthe contribution of non-characteristic direction to the formal MPA procedure.However the static pushovers in EMPA are performed individually for everymode, accordingly the resulted values from different modes can be combinedbut this is only valid in elastic phase; as soon as any element in structure startsyielding the neutral axis of that section is no longer fixed for both responseduring the earthquake, meaning the longitudinal deflection unavoidably affectthe transverse one or vice versa. To overcome this drawback, the CNSP sug-gests executing pushover analysis for governing modes of each direction at thesame time. This strategy is estimated to be more accurate than MPA andEMPA, moreover the calculation time is reduced because only one pushoveranalysis is required.

Regardless of the strategy, the accuracy of structural analysis is highly dependent onmodelling and numerical integration approaches used in evaluation of each method.Therefore the widely used Finite Element Method is implemented in process of allanalysis performed in this research.In order to address the study, chapter 2, starts with gathered information aboutconstructed long span arch bridges, this chapter continuous with geometrical andmaterial definition of new models. Chapter 3 provides the detailed informationabout structural analysis strategies; furthermore the step by step description ofprocedure of all methods is available in Appendix A. The document ends with thedescription of results and conclusion of chapter 4.

Chapter 2

Modelling and assumptions

Contents2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Arch bridge description . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Geometric aspects . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 Deck connections . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.1 Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Reinforcement steel . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 Elastomer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Finite element type . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Introduction

Study of arch bridges under seismic loads requires a very precise modelling bothgeometrically and for material definition, the stability and efficiency of this com-plex structure is based on combination of these two factors; this chapter is mostlydevoted to define the models based on constructed bridges or design codes. In sec-tion 2.2 the geometric shape of concrete arch bridges is studied in order to achievereliable dimensions to define required models for various analyses of chapter 4. Insection 2.3 the materials mechanical properties are developed based on Model Code2010 [fib 2010]. Finally the finite element types are discussed in last section of thischapter.

2.2 Arch bridge description

2.2.1 Geometric aspects

Arch bridges, regardless of size all are following a similar idea: passing the load ofthe whole structure and what is standing on top of it to both sides of span throughan arch. This design philosophy does not require materials with high tensionalresistance, however the interaction of deck, piers and arch itself is a very criticalpoint in design of efficient and narrow arch bridges. A schematic figure is presented

2.2. Arch bridge description 6

below (Figure 2.1) to illustrate the dominant geometry of arch bridges includingseveral models analysed in chapter 4.From the statistic point of view based on table 2.1 the span length to rise ratio

Figure 2.1: Schematic dimensions of Arch bridges

of long span concrete arcs is between 4 to 6 (see figure 2.2), this is also supportedby other authors [Manterola 2006]. The models analysed in chapter 4 are generatedaccordingly.

No Name Span Rise S/R1 R/S Year Country

1 Wanxian Bridge 420 84.0 5.00 0.20 1997 China2 Qiubei Nanpanjiang

Bridge416 99.0 4.20 0.24 2015 China

3 Krk Bridge 416 67.0 6.21 0.16 1980 Croatia4 Krk-A Bridge 390 52.0 7.50 0.13 2005 Croatia5 Almonte Railway

Bridge384 81.0 4.74 0.21 2013 Spain

6 Zhaohua Jialing RiverBridge

350 93.6 3.78 0.26 2012 China

7 Jiangjiehe Bridge 330 55.0 6.00 0.17 1993 China8 Tajo Railway Bridge 324 -2 - - 2013 Spain9 Hoover bypass (Mike

O’Callaghan-Pat Till-man Memorial Bridge)

323 84.4 3.83 0.26 2010 United States

10 Gladesville Bridge 305 40.7 7.49 0.13 1964 Australia11 Friendship Bridge 290 78.0 3.72 0.27 1965 Paraguay-Brazil12 Infante D. Henrique

Bridge280 25.0 11.20 0.09 2002 Portugal

13 Bloukrans Bridge 272 62.0 4.39 0.23 1984 South Africa14 Arr bida Bridge 270 52.0 5.19 0.19 1963 Portugal15 Froschgrundsee Viaduct 270 65.0 4.15 0.24 2010 Germany16 Grmpen Viaduct 270 70.0 3.86 0.26 2011 Germany17 Fujikawa Bridge 265 - - - 2005 Japan

Continued on next page

Table 2.1: Concrete Arch Bridges-continued


No Name Span Rise S/R1 R/S Year Country

18 Sand Bridge 264 42.0 6.29 0.16 1943 Sweden19 Contreras Railway

Viaduct261 40.3 6.48 0.15 2009 Spain

20 Takamatsu Bridge 260 - - - 2000 Japan21 Los Tilos Arch 255 45.0 5.67 0.18 2004 Spain22 Wild Gera Viaduct 252 - - - 2000 Germany23 Chateaubriand Bridge 250 - - - 1991 France

Table 2.1: Concrete Arch Bridges

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

300 350 4000

2

4

6

8

10

Span length(m)

Rise/Span

Figure 2.2: Span to rise ratio over span length.

The cross section of arch is defined as a box girder with variable dimensions alongthe span, meaning the cross section is smaller in the middle and larger on foundationlevel. Depending on other factors such as concrete grade and reinforcement ratio,the cross section of arch at midspan is selected as 1/2 or 1/3 of larger section(seefigure 2.3)

The cross section of decks are modified according to span length for every model,the only similarity is the width which is designed to carry two lanes for two directionon highway standards, the figure 2.4 illustrates the general shape of deck section.

The first estimation of required height of deck is calculated based on equation1R/S: Rise to Span ratio, S/R: Span to Rise ratio2(-) Author was unable to access any source of information about the missed data.


Figure 2.3: Cross sections of an arch with 400 meter span lenght with largest andsmallest dimensions(all in meters)

Figure 2.4: Deck section (dimensions are in meters)

below, however its displacement is controlled during static analysis.

hdeck =l

20 ∼ 25(2.1)

where l is the distance between two consecutive column.

2.2.2 Boundary condition

The key point on stability of an arch is the constant distant between its two ends,hence the arch is always forcing its sides to move away. Therefore there must befixities to prevent them from sliding. Defining such boundary condition is easier forthe mathematical model, but on construction site building such huge foundationsmay be not economically acceptable and other type of bridges such as cable stayedor suspension bridges may become more applicable. However, this problem vanisheswhen the bridge is to be constructed in a valley or V-shaped gaps with stiff bedrock.Based on design strategy used in the project, the arch supports may be free to rotate(pin supported-see figure 2.5a) or fixed as in figure 2.5b, the fixed supports aremostly used for concrete arch bridges because of its massive arcs and low flexibilityof concrete, the same type of boundary condition has been used for arch-groundconnection in this work.

2.3. Materials 9

(a) Pin support (b) Fix support

Figure 2.5: Type of Arch to ground supports

2.2.3 Deck connections

The arch and piers are connected in a rigid form, therefore they are consideredas a single mass, but deck is standing on top of piers and it can be free to moveon special and limited directions. In the mathematical models of this work, thedeck is fixed laterally but it is free to move along traffic direction up to a specificpoint (illustrated in figure 2.6). It has been achieved using elastomer bearings, thismaterial is to be explained in section 2.3.3.

Figure 2.6: Degrees of freedom of Deck

2.3 Materials

The material employed in different parts of bridges analysed in chapter 4 are de-scribed numerically through advanced constitutive models, the material class, espe-cially concrete are mainly selected based on available long span bridges such as Krkand Wanxian [Bangzhu 2008]; eventually some other modifications were necessaryto achieve a more numerically stable model in order to see nonlinear behaviour ofstructure under seismic actions. The specification of concrete, reinforcement steeland elastomer are presented below in order:

2.3. Materials 10

Figure 2.7: The 3D simulation of a 400m span arch bridge, designed based on dataof this chapter for further analysis proposes

Figure 2.8: The 3D simulation of a 400m span arch bridge, designed based on dataof this chapter for further analysis proposes

2.3. Materials 11

2.3.1 Concrete

Two normal weight (less than 2600 kg/m3) class of concrete is used in every model,a C60 for arch and piers and a C40 for deck to reduce the dead load exerted formdeck to columns. The materials stress-strain behaviour are formulated in equation2.2 for compression and algorithm 1 for tension based on CIB-FIB model code 2010[fib 2010] and a summary of the properties of C60 class concrete is presented intable 2.2

Eci [Gpa] 40.7 Modulus of elasticity of con-crete at age of 28 days

ρ[kg/m3] 2600 Densityν 0.2 Poisson’s ratio 1

fck [Mpa] 60 Characteristic compressivestrength

∆f 8Ec [Gpa] 38.9 Reduced Modulus of elasticityEcl [Gpa] 26.2 Secant modulus from the ori-

gin to the peak compressivestress

εcl [%�] 2.6 Strain at maximum compres-sive stress

εc,lim [%�] 3.3 Fracture strain

Table 2.2: Summary of the properties: C60 class concrete (data are based on[fib 2010] - for more details, see figure 2.9)

σc =fcm

(ηk − η2

)η(k − 2) + 1

(2.2)

where

η =εcεc1

k =Eci

Ecl

As the formulations are completed, the stress-strain values are available for everypoint in range; the figure 2.9 illustrates C60 concrete properties for its absolutevalues.

2.3.2 Reinforcement steel

As it is described in section 2.2.1 , the arch’s concrete cross section is responsibleto carry compressional stress. However, under some circumstances this state is notsteady i.e. as a result of any applied force or displacement to arch, its symmetry

1The exact value might be slightly different but the value 0.2 is recommended for all classes by[fib 2010]

2.3. Materials 12

Algorithm 1 Tension defenition [fib 2010]

1. Calculate fctm

if fck 6 50 Mpa then

fctm = 0.3 · (fck)2/3

else

fctm = 2.12 · ln(1 + 0.1 · (fck + ∆f))

2. Calculate σct based on one of following conditions:

for εx 6 0.9fctm/Eci do

σct[εx] = εx · Eci

for 0.9fctm/Eci < εx 6 εct do

σct[εx] = fctm ·(

1− 0.1(

εct−εxεct−0.9fctm/Eci

))for εct < εx 6 εct + wl do

σct[εx] = fctm

(1− 0.8(εx−εct)

wl

)for εct + wl < εx 6 wc + εct do

σct[εx] = fctm

(0.25 − 0.05(εx−εct)

wl

)

Εci Εcl

fcm

ϵcl ϵc,lim

0.0005 0.0010 0.0015 0.0020 0.0025 0.0030ϵ

1×107

2×107

3×107

4×107

5×107

6×107

7×107

σ (Pa)

Compression

Tension

Figure 2.9: Compression and tension behaviour of a C60 class concrete (Absolutevalues).

2.3. Materials 13

Es [Gpa] 210 Modulus of elasticityρ[kg/m3] 7850 Densityν 0.3 Poisson’s ratiof0.2 [Mpa] 420 steel ’yield’ strengths are gen-

erally quoted in terms of the0.2% proof strength

εy [%] 0.2 Yield strainεf [%] 22 Maximum elongationfyk [Mpa] 466 Ultimate strength

Table 2.3: Summary of the properties: B400 steel. [EN-1992 ]

and equilibrium for forces will be violated, accordingly there will be tension on someregion of arch; to resist this huge source of tension over the narrow section of arch,the longitudinal reinforcements are necessary. To fulfill this requirement the rein-forcement area is defined about 2% of total cross section area of arch (∼ 160kg/m3)which may vary depending on material class, geometry or load magnitude. Thisarea is divided into 10-12 rebars along the arch length (see figure 2.10).

Figure 2.10: Still of rebar placement ( dimensions are not realistic)

The deck and piers are also being reinforced using the same technique, althoughpiers are containing heavier reinforcement especially at the intersection with archto overcome the high amount shear force (up to 250kg/m3). The material used forall models is B400 grade steel which is defined in Table 2.3. for a precise materialmodel, the line equation has been developed based on fib model code, as well asconcrete (see equation 2.3, figure2.11).

σElastic = εx · Es εx 6 εy (2.3a)

σPlastic =εx · Es

100+ Es · εy εy < εx < εf (2.3b)

2.3. Materials 14

0.005 0.010 0.015 0.020ϵ

1×108

2×108

3×108

4×108

σ (Pa)

Figure 2.11: Simplified stress strain behaviour of steel B400 with average 1% hard-ening.

E1 [Mpa] 600 Modulus of elasticity (Normal)E2,3 [Mpa] Negligible Modulus of elasticity (Transversal)G1,2,3 [Mpa] 0.9 ∼ 1.2 Shear modulus of elasticityρ[kg/m3] 3000 Densityν1,2,3 0.5 Poisson’s ratio

Table 2.4: summary of the properties: Elastomer bearing. [Stanton 2008, Mag 2013]

2.3.3 Elastomer

The deck and piers are connected using an elastomer type bearing, although varioustype of connections are suitable for this part, the elastomer type is found to be themost simple and accurate one in finite element modelling point of view.Hence theconnection must be constrained in lateral direction (see figure 2.6) author decided toinclude this property into material definition of connection to avoid computationalexpenses of surface contact problem in finite element analysis 1. Therefore the shearmodulus of elasticity in desired direction is set as a large value to make material verystiff in that transversal direction. Accordingly the elastomer bearing is introducedto finite element code as an anisotropic material defined by equation 2.4.

2.4. Finite element type 15

ε11

ε22

ε33

γ12

γ13

γ23

=

1/E1 −ν12/E2 −ν31/E3 0 0 0

−ν12/E1 1/E2 −ν32/E3 0 0 0

−ν13/E1 −ν23/E2 1/E3 0 0 0

0 0 0 1/G12 0 0

0 0 0 0 1/G13 0

0 0 0 0 0 1/G23

σ11

σ22

σ33

σ12

σ13

σ23

(2.4)

2.4 Finite element type

The Abaqus is a finite element code used for numerical analysis of this document. Ingeneral, Abaqus provides advanced nonlinear and reinforcement concrete modellingfeatures compared to some other FE programs. Along several element type availablein Abaqus library, the beam 2element type "B31" (Timoshenko beam in space) meetsthe conditions of presented model. The beam cross section selected for arch andpiers is "Box-section" which by default has 16 integration points (5 points in eachwall) and "arbitrary-section" for deck with three point Simpson integration schemeof each section (5 section-total of 10 integration points). To define reinforcement,the command "*Rebar=Beam" must be used, this feature place the rebars parallelto beam element direction based on exact position of them. (For details on rebarpositions see section 2.3.2 and figure 2.10)

To check the accuracy of abovementioned finite element code, a cylindrical re-inforced specimen with dimensions of 150/300 mm is tested with four independentreinforcement area(Non-reinforced, 2.5%, 5%, 10%). The observations of this testcan be summarized as following: Under compression, concrete itself works fine withboth solid and beam element, however the program does not provide the rebars dataindividually, therefore the effect of reinforcement on specimen must be observed fromthe results for whole package. By reversing the elongation of specimen the way itgoes under tension, concrete reacts primarily. After concrete cracks up to a certainlevel,( it must be defined to Abaqus using "Smeared cracking" feature), the programwill partially release the strain energy to rebar until complete fracture of concrete.Eventually, the total force will be carried by reinforcements. This behaviour is il-lustrated in figure 2.12. This feature provided by Abaqus (smeared cracking) will

1When deck moves laterally there must be some element to hold it, if the surface of deck and piersbecome in contact, the surface-contact calculation must be performed, which is computationallyexpensive. Alternatively, Abaqus provides a wide library of connections to fulfill this requirement,however by using this type of constrains, the program will not be able to produce a unique massmatrix of all element which is a necessity for external calculations of MPA and etc. (explained onchapter 3)

2The element library in Abaqus contains several types of beam elements. A "beam" in thiscontext is an element in which assumptions are made so that the problem is reduced to onedimension mathematically: the primary solution variables are functions of position along the beamaxis only. For such assumptions to be reasonable, it is intuitively clear that a beam must be acontinuum in which we can define an axis such that the shortest distance from the axis to anypoint in the continuum is small compared to typical lengths along the axis. [Abaqus 2013]

2.4. Finite element type 16

●

●

● ● ● ● ● ● ● ● ● ●●■

■

■■■

■■■■■■

■■■■ ■ ■ ■

■■■■■■■■■■■■■■■■

■■■■■■ ■■ ■ ■ ■ ■■■

■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■

■■ ■

◆

◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆

◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆

◆◆◆◆◆◆◆ ◆ ◆ ◆

◆◆◆

◆◆◆

◆◆

◆

◆

◆◆

▲

▲

▲

▲

▲

▲

▲

▲

▲▲

▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025

0

5.0×106

1.0×107

1.5×107

2.0×107

u(m)

Reaction(N

)

● No-RC

■ RC 2.5%

◆ RC 5%

▲ RC 10%

Figure 2.12: The effect of reinforcement on a specimen(0.5m2 ∗ 1m) under tension.

allow us to perform a very realistic analysis on models with composite materialsunder cyclic loads such as several complex bridges of chapter 4 subjected to strongground motion.

Chapter 3

Seismic analysis

Contents3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 The mathematical system of dynamics . . . . . . . . . . . . . 173.3 Elastic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.1 Direct Response History Analysis: DRHA . . . . . . . . . . . 183.3.2 Modal Response History Analysis: MRHA . . . . . . . . . . . 193.3.3 Modal Response Spectrum Analysis: MRSA . . . . . . . . . . 21

3.4 Inelastic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4.1 Nonlinear Response History Analysis: NL-RHA . . . . . . . . 223.4.2 Modal Pushover Analysis: MPA . . . . . . . . . . . . . . . . 233.4.3 Extended Modal Pushover Analysis: EMPA . . . . . . . . . . 253.4.4 Coupled Nonlinear Static Pushover Analysis: CNSP . . . . . 28

3.1 Introduction

Reliability of results of a complex structure such as concrete arch bridges of previouschapter, depend upon its analysis procedure, among many mathematical methodsavailable, the designer must select the most suitable one considering the importance,environmental conditions and material behaviour expectations during the service lifeof structure as well as error estimation of method. The first section of this chapterdescribes the general system of dynamics, the chapter continues with discussion ofelastic procedures and its limitations. Finally a discussion of inelastic methods andapplications is presented at the end of this chapter.

3.2 The mathematical system of dynamics

The dynamic response of a N-degree of freedom structure subjected to ground mo-tion is expressed as following system of differential equation 3.1 :

mu + cu + fs(u, u) = −mιug (3.1)

Where m and c are respectively the mass and damping matrices, u(t) is the relativedisplacement , fs(u, u) is the stiffness component of the force vector in the structure

3.3. Elastic analysis 18

which defines the relation between force and displacement and at last, ι representthe influence matrix connecting the degree of freedom of the structure and imposedaccelerogram directions (ug(t)).

3.3 Elastic analysis

Assuming the structure remains elastic under imposed acceleration, makes the fs(Eq. 3.1) a linear relation of force displacement and can be expressed as fs = ku

where k is elastic stiffness matrix. This leads us to the traditional form of systemof dynamics for linear elastic systems:

mu + cu + ku = −mιug (3.2)

To evaluate the solution of abovementioned equation, three strategies are presentedbelow:

3.3.1 Direct Response History Analysis: DRHA

DRHA evaluates the system of equations by means of step by step numerical inte-gration scheme, therefore DRHA produce a time history of results. If the numericalintegration remains stable, this method provides accurate results, however the maindrawback of this procedure is undoubtedly the large computational cost which isdue to solving the system of equations for every time increment up to total dura-tion of accelerogram. Among many other numerical integration methods available,the broadly accepted integration algorithm proposed by Hilber, Hughes and Taylor(HHT, [Hilber 1977]) is employed in this work because of its reliability to remain nu-merically stable in higher order vibrations. This capability becomes more importantin nonlinear type of integration (NL-RHA), which will be discussed in next chapters.The Hilber-Hughes-Taylor operator is a generalization of the Newmark operator withcontrollable numerical damping, the damping being most valuable in the automatictime stepping scheme, because the slight high-frequency numerical noise inevitablyintroduced when the time step is changed is removed rapidly by a small amount ofnumerical damping. The operator replaces the actual equilibrium equation (Equa-tion 3.1) with a balance of d’Alembert forces at the end of the time step and aweighted average of the static forces at the beginning and end of the time step[Abaqus 2013].

mu|t+∆t + (1 + αa)(cu|t+∆t + ku|t+∆t)− αa(cu|t + ku|t) = F|t (3.3)

u|t+∆t = u|t + ∆t u|t +∆t2

2[(1− 2βa) u|t + 2βa u|t+∆t] (3.4)

u|t+∆t = u|t + ∆t[(1− γa)u|t + γau|t+∆t] (3.5)


βa =1

4(1− αa)2 γa =

1

2− αa −1

3≤ αa ≤ 0

It was experienced that three dimensional nonlinear analysis of previously describedmodels, require a higher level of numerical damping. Therefore the above mentionedparameters are selected as α = −0.41421, β = 0.5, γ = 0.91421. This configurationin Abaqus is also known as "Moderate dissipation".

3.3.2 Modal Response History Analysis: MRHA

As long as the structure remains in elastic range, equation 3.2 can be evaluated asa single degree of freedom system by decoupling equation 3.2 by means of modaltransformation for the modes with frequency within the range of interest. In orderto do so, it is required to expand the displacement vector u(t) in terms of modalcontributions:

u(t) =

N∑i=1

φiqi(t) (3.6)

where φi is the vector of normalized displacement for degrees of freedom associatedwith i mode obtained from the real eigenvalue (Eq. 3.7), N is the number of modesbeing considered and qi is the generalized modal coordinate.

kφi = ω2i mφi (3.7)

By replacing equation 3.6 into equation 3.2, and pre multiplying each term byφTn the general elastic equation can be rewritten as:

N∑i=1

φTnmφiqi +N∑i=1

φTncφiqi +N∑i=1

φTnkφiqi = −φTnmιug(t) (3.8)

Due to orthogonality relations (φTnmφi = 0, φTnkφi=0)[Chopra 2007], all terms ineach part vanishes except i=n:

Mn = φTnmφnqn Cn = φTncφnqn Kn = φTnkφnqn

Where Mn is generalized mass matrix, Cn is generalized damping matrix and Kn isgeneralized Stiffness matrix, therefore:

Mnqn(t) + Cnqn(t) +Knqn = −φTnmιug(t) (3.9)

On the other hand, expanding the spatial distribution of seismic excitation in baseof eigenvectors which represents the vibration modes:

s = mι =N∑i=1

Γimφi (3.10)


Same as previous part, by premultiplying φTn to both sides of equation 3.10 andconsidering the orthogonality property of mass matrix, the participation factor Γn

is obtained:

Γn =φTnmι

φTnmφn=φTnmι

Mn(3.11)

Dividing Eq. 3.9 by Mn results:

qn(t) + 2ξωnqn(t) + ω2qn(t) = −Γnug (3.12)

Considering:qn(t) = ΓnDn (3.13)

The Single Degree Of Freedom (SDOF) system with imposed accelerogram is readyto be solved using a numerical integration method:

Dn(t) + 2ξωnDn(t) + ω2Dn(t) = −ug(t) (3.14)

Obtaining the displacement of structure can be easily done by reversing thechange in variables:

un = φnqn(t) = φnΓnDn(t) (3.15)

The resultant of every SDOF system belongs to one mode and as the number ofincluded modes increase, the accuracy of results increase as well. Up to now, modaldisplacement of structure has been calculated and using this data, the forces instructural elements are to be determined by applying these nodal displacementsinto static analysis procedure (equivalent static force). The forces in structuralelement may be obtained as:

fn(t) = kun(t) (3.16)

Using Equations 3.10, 3.15 and 3.16 leads to:

fn(t) = Γnmφn︸︷︷︸sn

ω2nDn(t)︸︷︷︸An

= snAn (3.17)

Finally , the equivalent static forces fn(t) can be expressed as product of twofactors:

• The modal static response or structural response under static analysis (rstn )by applying spatial distribution of each mode (sn).

• The pseudo-acceleration response sn of n-th mode SDOF system subjected toground motion ug(t)

It is concluded as:rn(t) = rstn An(t) (3.18)


3.3.3 Modal Response Spectrum Analysis: MRSA

As long as the system of dynamics remains elastic, displacements are recovered afterevery cycle of motion; therefore it is not required to solve the SDOF system for everysingle time step as in MRHA. Simply the peak displacement is enough for calculatingthe maximum forces in elements and design of the sections accordingly. The peakdisplacement or acceleration can be obtained from earthquake response spectrumSdn = Sdn(Tn, ξn), San = Sdn(Tn, ξn) which are a function of natural vibrationperiod (Tn) and modal damping (ξn) then pesudo acceleration can be modified as:

An = ω2nDn(t) = ω2

nSdn = San (3.19)

Here, the previously developed expression 3.18 is modified due to new form ofpesudo-acceleration Eq. 3.19 for response spectrum analysis:

rno(t) = rstn ω2nSdn = rstn San (3.20)

Where rno is the peak response of nth mode. The presented procedure is computa-tionally less demanding in comparison to MRHA also the results of both MRHA andMRSA at peak displacement of every individual mode are identical, however aftercombination of modes the MRSA might slightly over estimate the forces, this is theonly source of error in MRSA and the reason is the peak displacement of modes arenot occurring at the same time although several Modal combination strategies areavailable to combine them all together.

Modal combination rules

As previously mentioned, every system of equations in MRSA calculates the responseof structure and accordingly element forces for one individual mode. There areseveral combination rules available in order to combine the results. Some of themare represented here:

ABSSUM or absolute sum is a modal combination rule that assume all modalpeaks occur at the same time and their algebraic sign is ignored. This ruleprovides an upper bond to peak value of total response.[Chopra 2007]

ro 6N∑

n=1

|rno| (3.21)

SRSS This is short term for square root of sum or squares and provides a very goodestimation as long as the natural frequencies of modes are well separated. Thepeak modal responses are squared then summed for all number of includedmodes. Finally the resulting value goes under square root to complete the

3.4. Inelastic analysis 22

estimation [Rosenblueth 1951]:

ro '

√√√√ N∑n=1

r2no (3.22)

CQC The complete quadratic combination rule has been developed by several au-thors, which are mostly similar in formulation (i.e expression 3.23) but in corre-lation factor, in this work, the formulation developed by [Der Kiureghian 1981]has been implemented:

ro '

√√√√√√√N∑

n=1

r2no +

N∑i=1

N∑n=1︸︷︷︸

i 6=n

ρinriorno (3.23)

ρin =8ξ2(1 + βin)β

3/2in

(1− β2in)2 + 4ξ2βin(1 + βin)2

β = ωi/ωn (3.24)

The advantage of this rule over SRSS is that CQC overcomes the limitationof SRSS in response estimation of modes with close frequencies. To use theCQC, a damping coefficient must be introduced this value is about 5% in thestructures studied in this work1.

3.4 Inelastic analysis

The elements of structure may be pushed above their linearly elastic strength whilethe structure is under extreme seismic action and deflect accordingly, which maycause other deflections in structure under secondary P − ∆ effect. On the otherhand traditional design strategies suggest designing the element sections by onlyconsidering the elastic strength of material, however current trends in earthquakeengineering requires more rigorous and realistic procedures. By the same token, thestrategies such as performance based design are including material inelasticity toachieve a more economic design. Despite the elastic analysis, the previous modebased procedures are not valid here because the system of dynamics in Eq. 3.1cannot be directly decomposed in set of real vibration modes as kinelastic 6= kelastic

respectively fs(u, u) = kinelasticu 6= kelasticu. Therefore appropriate methodologiesare necessary to predict these displacement demands. In following sections, therigorous NL-RHA and approximate methods will be discussed.

1Damping within range of 1 − 5% is found to have very little effect on accuracy of modalcombination.[Chopra 2007]


3.4.1 Nonlinear Response History Analysis: NL-RHA

The most accurate strategy to address significant nonlinearities in structural dy-namics is undoubtedly the nonlinear response history analysis which is similar toDRHA (Sec.3.3.1) but extended to inelastic range by updating the stiffness matrix(k) for each iteration. This onerous task, clearly increase the computational costespecially for unsymmetrical structures that require to be analyzed in a full 3Dmodel for taking the torsional effects in to consideration. Although approximatemethods are available to calculate nonlinear response of structure, they can only beused during the design process but the final design must be based on NL-RHA.

3.4.2 Modal Pushover Analysis: MPA

The Modal Pushover Analysis is basically the MRHA but extended to inelasticstage. After all, the MRHA cannot solve the system of dynamics (Eq.3.1) becausethe resisting force fs(u, u) is unknown for inelastic stage. The solution of MPA forthis obstacle is using the previously recorded fs to evaluate system of dynamics.The force-deformation relationship (Capacity curve) can be achieved by means of astatic pushover analysis when monotonically increasing forces are applied and thispushover analysis must be performed for all included modes and directions sepa-rately. The peak response results from using this method, proposed by Chopra andGoel [Chopra 2002] and improved by them [Chopra 2004a, Chopra 2004b]. Here abrief mathematical derivation of this method is available:

Repeating spatial distribution sn from Eq.3.10:

s = mι =N∑i=1

Γimφi (3.10 revisited)

And assuming that the n-th component of the expanded excitation vector inmodal coordinates only affect the n-th mode,(which is not generally true in nonlinearanalysis, but it is assumed to be acceptable in building structures [Chopra 2002]).In that case, the contribution of i-th mode to displacement vector when sn is appliedto the structure is null, being i 6= n, therefore modal decomposition in Eq.3.17 canbe represented as:

u(t) =

N∑i=1

φiqi(t) ' φnqn(t) (3.25)

Then, considering the orthogonality of m and c (classical damping assumedhere), and Fsn = φTnfs(u, u) result:

Mnqn(t) + Cnqn(t) + Fsn = −ΓnMnug(t) (3.26)

Dividing the equation by generalized mass Mn leads to:


qn(t) + 2ξωnqn(t) +Fsn

Mn= −Γnug (3.27)

By statically imposing the specific load pattern s∗n to push the structure todesired displacement, the capacity curve is attained.

s∗n = mφn (3.28)

Integrating the Eq.3.27 results the time history of displacement, however themaximum value is required here2, thus:

umaxrn = φrn max

t[qn(t)] (3.29)

From this point forward, the MPA loses its time history, therefore a modal combi-nation strategy is necessary to combine the results of all modes.

Capacity curve transformation and Integration of the differential equa-tion

Previously mentioned capacity curves are made by recording the base shear forcesof structure Vbn (sum of all boundaries) and displacement of the control point (urn)in the nth mode pushover analysis under s∗n (Eq.3.28) and gravity load.The trans-formation 3of these values is required in order to be applicable in Eq. 3.27, this canbe done using the equality below which is illustrated in Figure3.1.

Fsn

Mn=VbnLn

(3.30a)

qn =urnφrn

(3.30b)

For sake of simplicity, the non-linear part of graph is idealized using equal arearule, finally the Fsn/Mn − qn relationship is available to implement in system ofdynamics. To evaluate qn, the differential equation 3.27 must be integrated in timedomain, in this work, the central difference integration scheme is combined with non-linear cycling algorithm 2 to generate an explicit incremental integration methodcapable of evaluating the cycling displacement of SDOF structure.

3.4.3 Extended Modal Pushover Analysis: EMPA

Expanded Modal pushover is a proposed method in Camara Ph.D. thesis [Camara 2011]to evaluate response of structure under multi-directional excitation uT

g (t) = (uXg (t), uYg (t), uZg (t))

using the modal pushover analysis strategy. In one specific mode,the original pushoverneglects the contribution of the directions different than characteristic one. This is

2small differences have been introduced in Eq.3.29 in comparison with the originalMPA[Chopra 2002]

3Further mathematical derivation of this transformation is available on [Chopra 2002]


Figure 3.1: Transformation of capacity curve

Figure 3.2: The final stress is obtained by "returning" the trial stress to the yieldsurface through a scaling, hence the denomination return mapping.[Simo 1997]


Algorithm 2 Return-Mapping algorithm for one dimensional, Rate independentplasticity. Combined Isotropic/Kinematic hardening. [Simo 1997]

1. Database at x ∈ B : {εpn, αn, qn}

2. Given strain fieldat x ∈ B : εn+1 = εn + ∆εn

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

σtrialn+1 := E(εn+1 − εpn)

ξtrialn+1 := σtrialn+1 − qn

f trialn+1 :=∣∣ξtrialn+1

∣∣− [σY +Kαn]

if f trialn+1 ≤ 0 then

Elastic step: set (•)n+1 = (•)trialn+1 & Exit

else

Plastic step: proceed to step 4.

4. Return mapping:

∆γ :=f trialn+1

E + [K +H]> 0

σn+1 := σtrialn+1 −∆γE sign(ξtrialn+1 )

εpn+1 := εpn + ∆γ sign(ξtrialn+1 )

qn+1 := qn + ∆γH sign(ξtrialn+1 )

αn+1 := αn + ∆γ


reasonable in a regular symmetric building but a structure with complex shape maygo through strong modal coupling, meaning the modal displacement vector (φn) willhave non-zero components on non-characteristic directions although they are muchsmaller. Therefore the load distributions of modes are three-dimensional as it isrepresented in Figure3.3.

Figure 3.3: Left:Load distribution s∗n, Right:resulting control point displacement:urn (Adapted from Camara Ph.D. thesis [Camara 2011])

Subjecting the system of dynamics to triaxial excitation, converts the right sideof Eq.3.1 to :

Peffective = −mιX uXg (t)−mιY uYg (t)−mιZ uZg (t) (3.31)

Placing Eq.3.10 into Eq.3.31 leads to:

mu + cu + fs(u, u) = −sX uXg (t)− sZ uZg (t)− sZ uZg (t) (3.32)

The Eq.3.11 also must be redefined for triaxial form:

Γjn =

φTnmιj

Mn; j = {X,Y, Z} (3.33)

By neglecting the non-linear modal coupling and introducing qn as generalizedthree-dimensional coordinate, the Eq.3.25 is reformed for J number of modes:

u(t) =

J∑i=1

φiqi(t) ' φnqn(t) (3.34)

Considering Fsn = φTn fs(q, ˙q), substituting Eq. 3.34 into triaxial dynamic sys-


tem(Eq.3.32) and finally pre multiplying it by φTn leads to:

Mn ¨qn(t) + Cn ˙qn(t) + Fsn = −Mn (ΓXn u

Xg (t) + ΓY

n uYg (t) + ΓZ

n uZg (t))︸︷︷︸

u∗g,n(t)

(3.35)

u∗g,n(t) being the equivalent acceleration history; dividing by Mn results:

¨qn(t) + 2ξωn ˙qn(t) +Fsn

Mn= u∗g,n(t) (3.36)

At this point, the EMPA is able to evaluate response history of structure ina three-dimensional form without neglecting the components other than dominantdirection. However, the capacity curve of all directions is necessary to generateFsn/Mn − qn relationship. to achieve that, the projection of 3D pushover on everydimension is recorded, i.e. ((V X

bn − uXrn), (V Ybn − uYrn), (V Z

bn − uZrn)). As in MPA,the base shear-displacement records must be transformed to Fsn/Mn − qn usingpreviously defined equations of MPA procedure, here modified for 3D format:

F jsn

Mn=V jbn

Ljn

; j = {X,Y, Z} (3.37)

qjn =ujrn

φjrn; j = {X,Y, Z} (3.38)

The combination of resulted vectors is necessary using following expressions forits applicability in the system of dynamics:

F jsn

Mn=

√(FXsn

Mn

)2

+

(F Ysn

Mn

)2

+

(FZsn

Mn

)2

(3.39)

qn =√

(qXn )2 + (qYn )2 + (qZn )2 (3.40)

Integrating Eq. 3.36, provides the three dimensional generalized displacementof nodes of structure subjected to the equivalent accelerogram u∗g,n, (see figure 3.3),thus the target displacement of analysis is obtained as :

umaxrn = φrn max

t[qn(t)] (3.41)

Where:φrn =

√(φXrn)2 + (φYrn)2 + (φZrn)2 (3.42)

Here, the mathematical derivation of EMPA is completed. Integrating the sys-tem of equations and modal combinations are to be done as in MPA procedure.Additionally a step by step description of procedures is available in Appendix A ofthis document.


3.4.4 Coupled Nonlinear Static Pushover Analysis: CNSP

The EMPA includes the contribution of non-characteristic direction to the formalMPA procedure. However the static pushovers in EMPA are performed individuallyfor every mode. Accordingly the resulting values from different modes (directions)can be combined as explained previously. But this is only valid in elastic phase; assoon as any element in structure starts yielding the neutral axis of that section isno longer fixed for both response during the earthquake, meaning the longitudinaldeflection unavoidably affects the transverse one or vice versa. To overcome thisdrawback of EMPA, a new method proposed by Camara and Astiz [Camara 2012]suggests executing pushover analysis for governing modes of each direction at thesame time.

s∗nX = mφXn

s∗nY = mφYn(3.43)

Figure 3.4: Schematic three-dimensional coupled load distribution s∗C in CNSP. ¯VbCis the magnitude of the base shear in 3D coupled pushover.(Adapted from Camaraand Astis article.[Camara 2012])

The final coupled load pattern (s∗C) results from the algebraic weighted ad-dition of both modal excitation vectors in order to obtain a coupled response inlongitudinal and transverse directions during the non-linear static analysis. Thesecomponents are multiplied by factor Λ, which takes into account the difference inspectral accelerations associated with each governing mode [Huang 2007]:

s∗C = ΛXs∗nX + ΛY s∗nY (3.44)


ΛX =SaX

max(SaX , SaY )

ΛY =SaY

max(SaX , SaY )

(3.45)

Where SaX and SaY are the spectral accelerations associated with the transversaldirections. It is worth mentioning that both s∗nX and s∗nY are three dimensional bytheir very nature hence there are non-zero values in directions other than dominant.Therefore s∗C is resulting from by combination of two sets of three vectors; thisconcept is illustrated in figure 3.4.

Performing pushover under s∗C load pattern, provides the base shear-displacementbehaviour( ¯VbC− ¯urC) and its projection on every dimension must be simultaneouslyrecorded to be applicable in the system of dynamics(see figure 3.5).

Figure 3.5: Summary of CNSP philosophy. ¯VbC and ¯urC respectively mag-nitude of the base shear and control point displacement in the 3D coupledpushover.[Camara 2012]

The achieved FsnX,Y /Mn − qnX,Y relationship now can be placed in previouslyderived formulation of EMPA as the rest of procedure is similar. Furthermore a stepby step description of CNSP is available at Appendix A of this document.

Chapter 4

The ground motion and discussionof results

Contents4.1 Ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.1 Result of Elastic procedures . . . . . . . . . . . . . . . . . . . 324.2.2 Results of Inelastic procedures . . . . . . . . . . . . . . . . . 32

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Ground motion

All bridges in following analysis are subjected to same type of ground motion, theaccelerograms are generated using SismoArtif program [SeismoArtif-2.1.0 ], a sim-ulation of a far field inter-plate regime with 6.5 moment magnitude and linear siteeffect by selecting a generic rock (V30=630 m/s). The total of 8 synthetic accelero-grams are generated based on EC8(type1-A) [EN-1998 ] elastic spectrum with peakground acceleration of 0.5 and 5 % damping. An example of aforementioned ac-celerograms and spectra are presented in figure 4.1 and the complete data can befound in appendix B .

4.2 Discussion of results

4.2.1 Result of Elastic procedures

The elastic analyses are completed using procedures described in section 3.3 . Thedirect integration method takes longer, as it needs to regenerate the stiffness matrixand solve the equation of dynamics for every time step. Second is Modal ResponseHistory Analysis which uses accelerogram as base motion to calculate the resultant ofeach mode and combine them all using CQC (for further details on CQC see section3.3.3 ) The time consuming part of this procedure is the fact that the method needsto go through the time history for all included modes. Finally the fastest elasticprocedure is Modal Response Spectrum Analysis, similar to MRHA it is a modebased method but despite MRHA, it is using spectrum as it evaluates the applied

4.2. Discussion of results 32

2 4 6 8 10 Time(s)

-0.3

-0.2

-0.1

0.1

0.2

0.3

0.4

Acc(g)

(a) The Accelerogram1 2 3 4 T(s)

0.2

0.4

0.6

0.8

1.0

1.2

Acc(g)

(b) The Acceleration spectrum

Figure 4.1: An artificial ground acceleration generated for this work

loads to structure. In this method the time history is not available therefore onlythe extreme deflected shape of structure is resulted.

The following graphics (Figures 4.2 and 4.3) are presented to illustrate the dif-ference in results of elastic procedures in same model:

1.6×108 1.8×108 2.0×108 2.2×108 2.4×108 2.6×1080

20

40

60

80

Section Force1 (N)

height

(bottomtotoofHalfArc)

DRHA

MRSA

MRHA

Figure 4.2: The normal forces over arch section of a 400m span arch bridge in elasticanalysis

A very critical factor in mode base methods is the number of modes that are tobe included, this matter depends on participation factor which means how effectiveone mode can be on general behaviour of whole structure the decision on numberof included modes plays a very important role in mode base procedures and theirresults. Figures 4.4,4.5 and 4.6 are representing the result of MRSA of severalindependent modes.

The number of included modes is even more important in inelastic mode basemethods especially when governing modes are to be selected in CNSP.


1×107 2×107 3×107 4×1070

20

40

60

80

Section Force2 (N)

height


DRHA

MRSA

MRHA

Figure 4.3: The shear force along arch height of a 400m span arch bridge in elasticanalysis

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0 1×107 2×107 3×107 4×107 5×107 6×107 7×1070

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(m)

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▽ Mode 16

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▲ Mode 22

▼ Mode 23

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Figure 4.4: The effect of individual modes on structure- Normal section forces overhalf arch of a 400m span (SF1).


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Figure 4.5: The effect of individual modes on structure- Longitudinal section forcesover half arch of a 400m span (SF2).

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Figure 4.6: The effect of individual modes on structure- Transversal section forcesover half arch of a 400m span (SF3).


4.2.2 Results of Inelastic procedures

When structure is subjected to extreme seismic actions, it is expected to behaveabove its linear range, accordingly the nonlinear procedures must be applied. De-signing a structure to perform beyond its elastic range will reduce the dimensionsor materials required strength, but as it is economically efficient, it is computa-tionally expensive when the only rigorous available method is NLRHA which is atime history procedure, the provided results by this method is considered as exactvalues. However, the NLRHA is extremely time consuming and implementing it ina design procedure is very frustrating although the final calculations must be basedon this rigorous method. An alternative solution is using the mode based estima-tions, which are previously explained in section 3.4. Application of either MPA,EMPA or CNSP requires the integration of the nonlinear equivalent SDOF sys-tem (For Example, see expression 3.27) in order to obtain the target displacement,which has been performed by employing a code developed by author in Mathematica[Mathematica-10.0 ].the equation of dynamics is solved while the algorithm 2 pro-posed by Simo and Hughes [Simo 1997] is performed in each time step to consider thenonlinear behavior and dependency with the history of the response Fsn = (qn, qn).An example of nonlinear SDOF cyclic loading is plotted in figure 4.7.

-0.4 -0.2 0.2 0.4 0.6 0.8

-1.0

-0.5

0.5

1.0

Figure 4.7: An example of force-deformation relation generated during evaluationof a SDOF system in MPA procedure.


1×107 2×107 3×107 4×107 5×1070

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40

60

80

Section Force2 (N)

height


MPA

NL-RHA

CNSP

MRHA

Figure 4.8: a 400m span showing slight yieilding during seismic action.

1.1×108 1.2×108 1.3×108 1.4×108 1.5×108 1.6×1080

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40

60

80

Section Force1 (N)

height


NL-RHA

CNSP

MPA

Figure 4.9: a 300m span acting in its Inelastic range. (Normal Forces)

0 2.0×106 4.0×106 6.0×106 8.0×106 1.0×107 1.2×1070

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40

60

80

Section Force2 (N)

height


NL-RHA

CNSP

MPA

Figure 4.10: a 300m span acting in its Inelastic range. (Shear Forces)


0 1×106 2×106 3×106 4×106 5×1060

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40

60

80

Section Force3 (N)

height


NL-RHA

CNSP

MPA

Figure 4.11: a 300m span acting in its Inelastic range. (Shear Forces)

0 5.0×107 1.0×108 1.5×108 2.0×1080

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40

60

80

Section Moment1 (N.m)

height


NL-RHA

CNSP

MPA

Figure 4.12: a 300m span acting in its Inelastic range. (Section Moment-Normalaxis)

0 5.0×107 1.0×108 1.5×108 2.0×1080

20

40

60

80


height


NL-RHA

CNSP

MPA

Figure 4.13: a 300m span acting in its Inelastic range. (Section Moment-Longituidinal)


0 1×107 2×107 3×107 4×1070

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40

60

80


height


NL-RHA

CNSP

MPA

Figure 4.14: a 300m span acting in its Inelastic range. (Section Moment-Transversal)

0 10 20 30 400

5

10

15

Element No

Error

(%)

CNSP

MPA

Figure 4.15: Percentage error of first 40 elements of arch (2 only modes included inMPA to compare with the coupling concidration of CNSP

4.3. Conclusion 39

57.5

13.5

1.35

NL-RHA MPA & EMPA CNSP0

10

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30

40

50

60

CPU Time (Minutes)

Figure 4.16: Comparison of Inelastic analysis calculation time

4.3 Conclusion

The general conclusion about the seismic analysis of arch bridge resulted from thischapter are summarized as following:

• The forces obtained by modal dynamics are generally larger than spectrumanalysis. The reason is the inability of MRSA to combine peak response in theexact time of occurrence, although modal combination rules are implementedto overcome this drawback, the estimation is not accurate as modal dynamicprocedure which is resulted by exact integration. Therefore the modal dynamicmethod is assumed to provide exact results in elastic range analysis.

• The MRSA found to be the fastest elastic approach. 1 This advantage makesit very appealing for designer to use it in early design phase; however, it mayfall on unsafe side and should not be referenced for final analysis.

• Both SRSS and CQC rules are providing almost equal and reliable resultswhile combining extreme seismic response. Therefore SRSS is selected to beused in calculation of MRSA as it has a simpler formulation.

• When the bridge is expected to act in its inelastic range, the nonlinear responsehistory analysis is the only method which designer can rely on. However, exe-cution of numerical analysis on a complex model is extremely time consuming

1the maximum observed calculation time among all models was 11 seconds including dataexport.

4.3. Conclusion 40

and requires large computational power; this problem can be omitted duringdesign phase by using MPA, EMPA or CNSP to reduce the calculation timeto minutes. It is experienced that in long span arch bridge, the CNSP providethe results within range of less than 20% error (almost similar to MPA andEMPA) with 10% of time required for MPA or EMPA.

Appendix A

Description of pushover analyses

This appendix illustrates the step by step procedure of MPA,EMPA and CNSPrespectively, the following is adopted form Camara Ph.D thesis [Camara 2011] sinceit is applicable here as well. although a step by step procedure of MPA is primarilypublished by Chopra [Chopra 2007].The procedures are presented in a general formand could be applied to any structure.

A.1 Modal Pushover Analysis: MPA

1. Apply first the gravity loads considering geometric nonlinearities (P-∆ effects).

2. Perform modal analysis from the deformed configuration of the structure, com-puting frequencies fn, participation factors Γj

n (j = X,Y, Z) and mode shapesφn up to the maximum frequency of interests;fmax=25 Hz. After the modalanalysis, a study about the participation of each mode below fmax in the globalresponse in terms of forces or displacements should be conducted, identifyingthe governing longitudinal fnX and transverse modes fnY ,MPA should involveboth dominant modes.the limit frequency which marks the end of the rangewhere 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 selectedbecause MPA does not consider the three-directional contribution of modes,and subsequently the analysis continues in 2D (either longitudinally, trans-versely or vertically depending on the governing direction of the specific mode).The control point is selected for each vibration mode as the node with maxi-mum modal displacement in the dominant direction. A proposed method toobtain the characteristic direction of the mode in presented in section 6.4.4.1,summarized in the expression below.

ΓDRn = max

j(Γj

n);whitj = X,Y, Z → DRn (A.1)

4. For the n − th mode within the range studied by pushover analysis (fn ≤fgov),develop the base shear versus control point displacement curve in thedominant direction, Vbn-urn, by means of nonlinear static analysis (pushover)ofthe structure when the load distribution s∗n=mφn is incrementally applied(self-weight included), ignoring the components in the direction different thanthe dominating one. The base shear (Vbn)is the sum of the shear recorded in

A.1. Modal Pushover Analysis: MPA 42

each foundation (towers, intermediate piers and abutments) in the dominantdirection of the mode, and the control point displacement urn is consequentlythe 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 3.30a and 3.30b, repeated here for convenience:

Fsn

Mn=VbnLn

(A.2)

qn =urnφrn

(A.3)

6. Idealize the real Fsn/Mn-qn curve for the equivalent SDOF system associatedwith n− th mode. Here, a bi-linear curve considering a modified ’Equal Area’rule presented in figure 3.1 is proposed because of its simplicity, but morerealistic curves may be considered. The kinematic properties of the nonlinearspring behavior in the SDOF system need to be addressed

7. Compute the peak generalized displacement qn of the n-mode inelastic SDOFsystem. Here, it is proposed to integrate the SDOF differential equation ofthe corresponding mode in time domain (repeated below), by implementingthe algorithm 2 .

qn + 2ξnωnqn +Fsn

Mn= Γnug (A.4)

8. Obtain the peak control point displacement urn with expression:urn=qn φrn,where φrn is the modal displacement in the dominant direction DRn of n-mode at the selected control point.

9. Interpolate with urn + uG(being uG the control point displacement in thedominant direction due to gravity loads) from the database of pushover anal-ysis, to obtain the combined effects of lateral loads and gravity due to n-modecontribution (rn+G).

10. Obtain the contribution of the n-mode to the seismic response exclusively,by extracting the effect due to the self-weight: rn=rn+G-rG, where rG is thecontribution 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 ofeach 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 fgov and lower or equalthan fmax=25 Hz.

A.2. Extended Modal Pushover Analysis: EMPA 43

14. Combine the participation of first modes with the higher mode effect (betweenfgov and fmax), employing SRSS combination rule to obtain the dynamic re-

sponse of the structure: rd ≈√r2nl + r2

el. Frequencies higher than fmax=25Hz are ignored.

15. Calculate the total demand (r) by combining the self-weight effect (rG) withthe dynamic contribution exclusively due to the earthquake (rd). Since thesign of seismic forces is lost in pushover procedures 1, two hypotheses aremade, considering both positive and negative signs in the earthquake responsesto take into account that the seismic input, and the consequent structuralbehavior, have alternating sign reversals.

r ≈ max(rG ± rd) (A.5)

A.2 Extended Modal Pushover Analysis: EMPA

EMPA proposed procedure is summarized next, repeating for completeness somesteps introduced in MPA:

1. Apply first gravity loads considering geometric nonlinearities (P- ∆ effects).

2. Perform modal analysis from the deformed configuration of the structure com-puting frequencies fn, participation factors Γj

n(j=X,Y, Z) and mode shapesφn up to the maximum limiting frequency; fmax= 25 Hz. Next, a study aboutthe participation of each mode below fmax in the global response in terms offorces or displacements is conducted, identifying the governing longitudinal(fnX) and transverse modes (fnY ), EMPA covers both dominant modes. Thelimit frequency which marks the end of the range where pushover is to beconducted is established as (fgov=max(fnX ,fnY ).

3. For the nth mode,included in the range studied by pushover analysis (fn ≤fgov)develop the base shear-control point displacement curve, Vbn-urn 2, bymeans of three-dimensional nonlinear static analysis of the structure when theload distribution s∗n=mφn is incrementally applied (self-weight included), nowfully considering its three-dimensional characteristics. Three pushover curvesare obtained while the structure is being pushed beyond the linear range, eachone associated with the longitudinal, transverse and vertical directions; (V X

bn -uXrn),(V Y

bn-uYrn),(V Z

bn-uZrn)(see figure 3.3 The base shear V j

bn is the sum of theshear recorded in each connection of the model to the ground in j-directionduring the pushover analysis of the nth mode, whilst the control point displace-ment ujrn is consequently the displacement of the control point in j-direction;this point, is selected as the node with maximum modal displacement regard-less of the direction where it is recorded.

1Pushover is equivalent to spectrum analysis in the elastic range2the bar symbol means ’magnitude’ of the three directional displacement or shear components.

A.2. Extended Modal Pushover Analysis: EMPA 44

4. Transform each V jbn-u

jrn pushover curves to F j

sn/Mn-qjn coordinates using ex-

pressions 3.38 and 3.39, repeated below:

F jsn

Mn=V jbn

Ljn

;withj = X,Y, Z (A.6)

qjn =ujrn

φjrn;withj = X,Y, Z (A.7)

5. Idealize the real F jsn/Mn-q

jn curve for equivalent SDOF system associated with

n-mode in j-direction. Here, a bi-linear curve considering a modified ’EqualArea’ rule presented in figure 3.1 is proposed because of its simplicity (morerealistic curves could be considered.) The kinematic properties of the non-linear spring behaviour in the SDOF system need to be addressed, definingthe loading and unloading branches appropriate for the structural system andmaterial.

6. Obtain a modular Fsn/Mn-qn from the three-directional results using expres-sions 3.39 and 3.40, repeated next:

Fsn

Mn=√

(FXsn)2 + (F Y

sn)2 + (FZsn)2 (A.8)

qn =√

(qXn )2 + (qYn )2 + (qZn )2 (A.9)

7. Compute the peak generalized modular displacement qn of the n-mode inelas-tic SDOF system. Here, the integration of the SDOF differential equation 3.36of the corresponding mode in time domain (repeated below) is proposed.

¨qn + 2ξnωn ˙qn +Fsn

Mn= −u∗g,n(t) (A.10)

where u∗g,n(t)=ΓXn uXg (t)+ΓY

n uYg (t)+ΓZn uZg (t), and ujg the accelerogram 3D

components.

8. Obtain the peak modular control point displacement urn with:

umaxrn = φrn max

t[qn(t)] (A.11)

9. Interpolate with urn+uG (being uG the modular displacement of control point

due to gravity loads; uG=√

(uXG )2 + (uYG)2 + (uZG)2 from the database of thethree-dimensional pushover analysis, in order to obtain the combined effectsof lateral loads and gravity due to n-mode contribution rn+G.

10. Obtain the contribution of n-mode to the seismic response exclusively, byextracting the effect due to the self-weight: rn=rn+G-rG, where rG is the

A.3. Coupled Nonlinear Pushover Analysis: CNSP 45

contribution of gravity loads acting alone.

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

12. Obtain the total nonlinear seismic response (rel) by combining the contributionof each studied mode using an appropriate combination rule, here CQC ruleis selected for this purpose.

13. Compute the higher mode seismic response (rel) by means of spectrum analysis(MRSA), including modes with frequencies higher than fgov and lower or equalthan fmax=25 Hz.

14. Combine the participation of first mode with the higher mode effect by meansof SRSS combination rule to obtain the dynamic response of the structure:rd ≈

√r2nl + r2

el.

15. Calculate the total demand by combining the effect due to self-weight (rG)with the dynamic contribution due to the earthquake exclusively (rd). Sincethe sign of earthquake forces is lost in pushover procedure, two hypotheses aremade, 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) (A.12)

A.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 first the gravity loads considering geometric nonlinearities (P-∆ effects).

2. Perform modal analysis from the deformed configuration of the structure, com-puting frequencies fn, participation factors Γj

n(j=X,Y, Z) and mode shapesφn up to the limit maximum frequency fmax=25 Hz. Next, a study aboutthe participation of each mode below fmax in the global response in terms offorces or displacements is conducted, identifying the governing 3 longitudinal(fnX) and transverse (fnY ) frequencies.

3. Expand the excitation vector of the dominant longitudinal (nX) and trans-verse (nY ) modes, and compute the weight of dominant nk-mode in directionj(αj

nk) by adding its expanded nodal forces (sjnk,i in node i) along the whole

3CNSP, unlike MPA and EMPA, only conducts one pushover analysis using a combination ofloads from the transverse and longitudinal governing modes,instead of performing nonlinear staticanalysis for all the modes with frequencies below or equal to fgov=max(fnX ,fnY )


structure (k=X,Y and j=X,Y, Z). Alternatively, the ratio could be modifiedsubstituting the summations by the corresponding participation factors.

αjnk =

Nnode∑i=1

sjnk,i

Nnode∑i=1

sjnX,i +Nnode∑i=1

sjnY,i

(A.13)

4. Considering the two dominant longitudinal (nX) and transverse (nY ) modes,develop the base shear versus control point displacement curve, ¯VbC-¯urC , bymeans of 3D nonlinear static analysis of the structure (self weight included),incrementally increasing the coupled load pattern s∗C of the expression 3.44(considering the fully 3D characteristic of the excitation vector s∗n):

s∗C = ΛY s∗nY + ΛXs∗nX (A.14)

Λj=Saj/max(SaY ,SaX), where Saj is the spectral acceleration associated withthe governing mode in j-direction, k=X,Y .This displacement is measured simultaneously in two points during the cou-pled pushover, corresponding to the control points of the longitudinal andtransverse governing modes (selected according to EMPA described above).

5. The contributions of each dominant vibration mode to the longitudinal andtransverse projections of the coupled capacity pushover curve, are computedby multiplying the projection in j-direction (V j

bC-ujrC) by the weight factor

αjnX or αj

nY expressed above. Three pushover curves are obtained per bothtwo dominant modes; (V j

bnX -ujrnX) and (V jbnY -u

jrnY ) with j=X,Y, Z. The base

shear V jbnX and V j

bnY is respectively the sum of the shear recorded in eachfoundation of the model in the j-direction of each dominant mode, whist thecontrol point displacement ujrnX and ujrnY is, consequently, the displacementin the j-direction for each governing longitudinal and transverse mode at thecorresponding control point in a respective way.

6. Transform each V jbnX -ujrnX and V j

bnY -ujrnY pushover curves respectively into

F jsnX/Mn-q

jnX and F j

snY /Mn-qjnY coordinates by means of:

F jsnk

Mnk=V jbnk

Ljnk

;withj = X,Y, Z and k = X,Y (A.15)

qjnk =ujrnkφjrnk

;withj = X,Y, Z and k = X,Y (A.16)

7. Idealize the real F jsnX/MnX -qjnX and F j

snY /MnY -qjnY curves for the equivalent

SDOF systems associated with each governing mode in j-direction. Here,it is proposed to use a bi-linear curve considering a modified ’Equal Area’


rule presented in figure 3.1 (more realistic curves maybe considered). Thekinematic properties of the nonlinear spring behavior in the SDOF systemneed to be simulated, defining the loading and unloading branches appropriatefor the structural system and material.

8. Obtain modular FsnX/MnX - qnX and FsnY /MnY - qnY relations from the di-rectional results using expressions (k = X,Y ):

FsnX

Mnk=

√(FXsnk

Mnk)2 + (

F Ysnk

Mnk)2 + (

FZsnk

Mnk)2 (A.17)

qnk =√

(qXnk)2 + (qYnk)2 + (qZnk)2 (A.18)

9. Compute the peak generalized modular displacement qnX of the governing lon-gitudinal nX-mode inelastic SDOF system, and analogously for the dominanttransverse nY -mode the peak qnY . Here, integrating the differential equation3.36 of the corresponding governing mode in time domain (repeated below fornk-mode) is proposed, following the algorithm 2.

¨qnk + 2ξnkωnk ˙qnk +Fsnk

Mnk= −u∗g,nk(t) (A.19)

10. Obtain the peak modular control point displacement of the governing modes;urnX and urnY respectively, with expressions urnX= φrnX max

t[qnX(t)] and

urnY = φrnY maxt

[qnY (t)], where φrnX and φrnY are the modular modal dis-placements at control point in longitudinal and transverse dominating modesrespectively.

11. Interpolate with urnX+uG (being uG the modular displacement of control

point due to gravity loads: uG=√

(uXG )2 + (uYG)2 + (uZG)2 from the databaseof three-dimensional pushover analysis particularized for the longitudinal gov-erning mode (VbnX -urnX), to obtain the combine effects of lateral loads andgravity due to nX-mode contribution; rnX+G. Analogously, the contribu-tion of the governing transverse mode nY ,rnY +G, is obtained by interpolatingurnY +uG in the pushover curve (VbnY -urnY ).

12. Obtain the contribution of nX-mode and nY -mode to seismic response exclu-sively,by extracting the effect due to self-weight: rnX= rnX+G-rG and rnY =rnY +G-rG, where rG is the contribution of gravity loads alone.

13. Combine the three-dimensional contribution of governing nX-mode and nY -mode through the CQC combination rule, obtaining rnl.

14. Compute the contribution of vibration mods different than the governing onesand below fmax=25 Hz, assuming their response elastic (rel). This elastic


effect is obtained by means of spectrum analysis (MRSA). Vibration modesabove 25 Hz are neglected.

15. Combine through SRSS rule the contribution of governing modes with thecontribution of modes below 25 Hz and different than the governing ones, inorder to obtain the dynamic response of the structure: rd ≈

√r2nl + r2

el.

16. Calculate the total demand by combining the self-weight effect rG with thedynamic contribution due to the earthquake exclusively rd . Since the signof earthquake forces is lost in pushover procedure, two hypotheses are made(taking into account the alternating nature of the seismic input), consideringboth positive and negative signs in the earthquake response.

r ≈ max(rG ± rd) (A.20)

Appendix B

Ground motion definition

The section 4.1 includes a brief description of the ground motion implemented inthis analysis. Every bridge is subjected to two accelerogram at the same time intransversal and longitudinal directions in order to simulate the three directionalnature of earthquake. The time history approaches are using the accelerogram asthe source of motion directly, but in response spectrum analysis, the combinationof spectrums is achieved using SRSS (discussed in section 3.3.3). However, it isrequired to analyze every model with several number of accelerogram, therefore thetotal of 8 synthetic accelerogram are generated based on EC8 (type1-A) [EN-1998 ]Elastic spectrum with peak ground acceleration of 0.5 and 5 % damping. Thesesynthetic ground motions are plotted in following figures of this appendix.

50

1 2 3 4 5 6 7 8 TargetSpectra

01

23

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Period(s)

Acc(g)

Figure B.1: Plot of 8 artificial Spectrums

51

1 2 3 4 5 6 7 8

02

46

810

-0.6

-0.4

-0.2

0.0

0.2

0.4

Time(s)

Acc(g)

Figure B.2: Plot of 8 artificial accelerograms

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