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Analytical Modeling of HorizontallyCurved Steel Girder Highway Bridges forSeismic AnalysisEbrahim Amirihormozakia, Gokhan Pekcana & Ahmad Itaniaa Department of Civil and Environmental Engineering, University ofNevada at Reno, Reno, Nevada, USAPublished online: 13 Jan 2015.
To cite this article: Ebrahim Amirihormozaki, Gokhan Pekcan & Ahmad Itani (2015) AnalyticalModeling of Horizontally Curved Steel Girder Highway Bridges for Seismic Analysis, Journal ofEarthquake Engineering, 19:2, 220-248, DOI: 10.1080/13632469.2014.962667
To link to this article: http://dx.doi.org/10.1080/13632469.2014.962667
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Analytical Modeling of Horizontally Curved SteelGirder Highway Bridges for Seismic Analysis
EBRAHIM AMIRIHORMOZAKI, GOKHAN PEKCAN,and AHMAD ITANI
Department of Civil and Environmental Engineering, University of Nevada atReno, Reno, Nevada, USA
This article introduces a generic modeling approach that is suitable for static and dynamic analy-sis, and response assessment of highway bridges with varying levels of irregularities. The proposedapproach and modeling recommendations are based on grillage modeling rules that allows explicitrepresentation of various types of details and components. The validity and accuracy of the proposedapproach is demonstrated against three-dimensional finite element models as well as experimen-tally recorded response various benchmark bridges. While achieving remarkable accuracy, therequired analysis time was also reduced up to 80%, making the proposed approach suitable forcomputationally intensive studies.
The nonlinear response history analysis (nRHA) is by far the most comprehensive methodto establish realistically the seismic demand on highway bridges. As various nRHA toolsand software has become more widely available and efficient, a large number of researchstudies have examined performance of highway bridges subjected to various types of load-ing conditions [Seo, 2013; Wibowo, 2012; Caner, 2002; Chang, 2000; Watanabe, 1998;Dicleli, 1995]. Although the new generation of analysis tools facilitates complex nRHA, therequired computational effort is still very significant. Therefore, it is necessary to developmodeling techniques that require significantly less time to generate and analyze computa-tional models. However, this must be achieved without any loss of accuracy in the responsepredictions.
The simplest modeling approximation utilizes a single beam representing the super-structure with the lumped mass, stiffness, and section properties of the superstructure.In such models, the beam element is connected rigidly to the top of the bearings at supportsincluding bents and abutments. Memory et al. (1995) showed that the use of single beamwith high transverse stiffness leads to accurate results for bridges with straight geometry,simple supports and continuous superstructure. The single beam model provides approx-imations to the dynamic properties that are sufficiently accurate for preliminary seismicstudies of long multi-span bridges. However, the primary limitation of the simplified beamelement model is its inability to accurately model 3D boundary conditions such as those
Received 22 May 2013; Accepted 3 September 2014.Address correspondence to Ebrahim Amirihormozaki, Department of Civil and Environmental Engineering,
University of Nevada at Reno, Mail Stop 258, Reno, NV 89557, E-mail: [email protected] versions of one or more of the figures in the article can be found online at www.tandfonline.com/ueqe.
Curved Steel Girder Highway Bridges for Seismic Analysis 221
that arise from the connections of the plate girders to the bents. In addition, load path anddamage to components cannot be simulated accurately. Furthermore, single beam idealiza-tion is not appropriate for longitudinally asymmetric or skewed, continuous superstructureswith complex dynamic characteristics due to irregular geometry. In case of horizontallycurved bridges, a single beam idealization cannot capture the torsional characteristics andassociated response of the superstructure due to both static and dynamic loading. Recently,Aviram et al. [2008a] presented nonlinear modeling and analysis guidelines that utilize sin-gle linear-elastic beam-column element to model bridge superstructures. However, theseguidelines are limited to ordinary standard bridges.
Meng et al. [2002] proposed a dual-beam stick model for dynamic analysis of skewbridges. It consists of two beam-column elements connected by massless rigid bars. Theserigid bars are connected to the beam-column elements in such a way that only rotationsabout the longitudinal axis of the beams are allowed. The mass, section area, torsionalconstant, and moment of inertia about the horizontal axis of deck section are divided bytwo and assigned to each beam element. Spacing of the beam elements is based on havingthe same mass moment of inertia about the vertical axis of the bridge. However, simpli-fied modeling approaches proposed by Meng [2002] and Abdel-Mohti and Pekcan [2008]for skew bridges are not necessarily appropriate for modeling of horizontally curved steelgirder bridges. This is due to the fact that transverse response of the bridge deck introducesrelative deformations both in-plane and along the girder depth due to finite stiffness of slaband girders.
Another commonly used method is known as the Grillage Method in which the super-structure is modeled as a grillage [Meng et al. 2002; O’Brien and Damien, 1999; Memoryet al. 1995; Hambly, 1990]. This approach eliminates the shell elements representing theslab in the superstructure and uses beam elements to model the slab in both directions[Maleki, 2002; Hambly, 1990]. Coletti and Yadlosky [2005] indicated that the grillagemodels are not generally recommended for bridges with severe curvature since they areunable to capture the torsional response.
AASHTO Guidelines for Steel Girder Bridge Analysis [AASHTO, 2011] introducedfour methods to analyze steel girder bridges including 2D grid analysis methods, gener-alized grid analysis methods, plate and eccentric beam analysis methods, and 3D FEManalysis methods. In 2D grid analysis method, also known as grillage analysis, the structureis divided into plane grid elements with three degrees of freedom at each node. In gen-eralized grid analysis method, as a modification of a 2D grid analysis, more degrees offreedom are considered. Modeling of cross frames or diaphragms with consideration ofshear deformation in addition to flexural deformation and modeling of the warping stiff-ness of open cross section shapes are some typical enhancement of this method. Plateand eccentric beam analysis methods is a variant of a 2D grid/grillage analysis modelwhere the deck is modeled using plate or shell elements. The girders and cross frames aremodeled using beam elements which are located along the neutral axis of the individualgirders or cross frames. The category 3D FEM Analysis method is meant to encompassany analysis/design method that includes a computerized structural analysis model wherethe superstructure is modeled fully in three dimensions. Accordingly, girder flanges aremodeled using line/beam elements or plate/shell/solid type elements; girder webs usingplate/shell/solid type elements; cross frames or diaphragms using line/beam, truss, orplate/shell/solid type elements; and deck using plate/shell/solid elements.
NCHRP 620 [2008] introduced 3D spine beam and grillage analogy as two reliablemethods to model and analyze the superstructure of horizontally curved concrete box girderhighway bridges. In the spine beam model, a 3D frame element is divided into a series ofstraight segments located along the centerline of the superstructure of the bridge. In grillage
analogy method, the superstructure is modeled as a 3D grid of beam elements in whichthe superstructure is comprised of both longitudinal and transverse beams located at thevertical center of gravity of the superstructure. The modeling and analysis requirements forvarious type of load cases such as dead load, live load, prestressing, and response spectrumanalyses are also presented in this document.
Recently, Kappos et al. [2012] presented various methods that can be used for theseismic assessment and design of bridges, along with their range of applicability and rela-tive accuracy. These methods are presented as the most advanced inelastic analysis methodssuch as inelastic static (pushover) and response history analyses with the inclusion of allbridge components including columns, abutments, bearings, and shear keys.
In this article, a set of generic recommendations for the development of simplified 3Dbeam-stick models for primarily horizontally curved steel girder bridges is presented. Thevalidity and accuracy of the proposed approach are demonstrated through a series of static,modal, elastic and inelastic (nonlinear) response history analyses in comparison with 3DFE models. These comparisons demonstrated remarkable accuracy and reductions in therequired analysis time. Efficiency of the single beam model commonly used in engineeringpractice to evaluate the seismic demand on column and shear keys are also investigatedagainst both proposed simplified and 3D FE models. It is noted that nonlinear responsecharacteristics of various materials and critical components including columns, steel rein-forced elastomeric bearings, shear keys, soil, piles, and pounding are explicitly modeled.A comprehensive model of abutment configuration is presented as well. For the verificationof the proposed modeling approach and comparisons, OpenSees [McKenna and Feneves,2011] and SAP2000 [CSI, 2011] were used.
2. Superstructure Modeling
Bridge superstructure which consists of R/C deck, steel girders, and cross frames, is con-nected to the substructure through bearings, bents, and abutments. In general, due to largein-plane stiffness, superstructure is relatively more rigid than the substructure particu-larly in the transverse direction. Furthermore, the superstructure in conventional designis detailed such that it remains linear-elastic while most of the nonlinearity takes place inthe substructure when subjected to high intensity ground motions. However, realistic elasticproperties must be used to account for the effect of cracked concrete deck, e.g., a modifi-cation factor of 0.4 may be assigned to the elastic modulus of the deck slab [Carden et al.,2005]. While complex 3D FE models can be utilized to represent all components of super-structure explicitly, single beam models representing the entire superstructure with lumpedproperties is known to be more efficient. However, as previously noted, single beam mod-els cannot capture the superstructure torsion and dynamic response of horizontally curvedbridges. Properties of single beam models and a proposed simplified approach intended toaddress the deficiencies of single beam models are explained in the following sections.
2.1. Conventional Single Beam Model
For most ordinary highway bridges, single beam (spine) modeling approach is commonlyadopted despite its lack of accuracy. In this approach, a linear-elastic beam element isplaced along the centerline of the superstructure. Since all of the nonlinearities are expectedto take place in the substructure, linear-elastic geometric, and material properties areassigned to this beam element. The single beam representing the superstructure is con-nected to the top of bearings through a group of rigid links. In case of horizontally curvedbridges, the single beam is divided into multiple linear segments to better represent the
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Curved Steel Girder Highway Bridges for Seismic Analysis 223
.
. c.g
di
ri
c.g. of the element
FIGURE 1 Definition of di and ri to calculate the mass moment of inertia of thesuperstructure.
curved geometry of the superstructure. Tributary translational mass of each segment islumped at its ends. Rotational mass of the superstructure is also determined based on Eq. (1)and assigned to the segments’ ends equally:
Mr =∑n
i=1
mid2i
12+ mir
2i , (1)
where mi denotes the mass of each cross section segment (including the concrete slab,girders webs and flanges), di is the widest dimension of the segment, and ri stands for thedistance of each segment to the center of gravity of the superstructure (Fig. 1).
In Eq. (1), the first term(mid2
i /12)
is the rotational mass of each superstructure seg-ment about its individual axis and the second term mir2
i is its rotational mass about thecenter of gravity of the superstructure section. Assignment of superstructure rotational masssignificantly increases the accuracy in approximating the vibration modes and dynamicresponse of bridges in the transverse direction [Aviram et al., 2008a].
2.2. Proposed Enhanced Beam-Stick (BS) Model
The proposed beam-stick modeling approach introduces several improvements over singlebeam-stick model. These improvements allow accurate prediction and assessment of staticand dynamic response of horizontally curved highway bridges. Various aspects and detailsof the proposed modeling approach are discussed in what follows.
2.2.1. Geometric Definition and Properties. The superstructure is modeled using a grillagerepresenting the deck and the girders. The deck elements are connected rigidly to beamelements representing the girders. Discretization of superstructure cross section into lon-gitudinal beams is illustrated in Figs. 2 and 3. Each girder-beam has the properties of thecorresponding steel girder and is located along respective neutral axis. The centerline ofedge-beams are located at a distance equal to 0.3ts from the edges of the deck slab where ts
is the slab thickness. For solid slabs, the vertical shear flow is assumed to occur at 0.3ts fromthe edge of the slab [Hambly, 1990]. The two edge-beams represent the vertical shear flowof the slab with a spacing of 0.6ts and a height equal to the thickness of the slab as shownin Figs. 2 and 3. Main-beams are located at the level of slab mid-thickness and are locatedsymmetrically about the axis passing through the thickness of the web of the steel gird-ers. The main-beams are connected to girder-beams using rigid links (Fig. 4). The spacingbetween main-beams on interior girders should not be taken more than the girder spacing.This limitation of the width of the slab is based on the effective slab width for compositebeams as defined in AASHTO [2014], and work performed by Chen et al. [2007]. Theadditional-beams are used to account for the remaining parts of the deck slab, as shown in
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224 E. Amirihormozaki, G. Pekcan, and A. Itani
Girder-Beam Girder-Beam
Additional Beam
Main-Beam Main-Beam
Edge-Beam
ts
0.3 ts 0.3 ts
Edge-Beam
S S
FIGURE 2 Location of longitudinal beams.
S S
S0.6 ts
2 (L-0.6 ts)
LL
FIGURE 3 Tributary area of each longitudinal beam.
Transverse-Beam
Rigid Link Rigid Link Rigid Link
Transverse-Beam Transverse-Beam Transverse-Beam
FIGURE 4 Connection of beams.
Fig. 2. It is noted that all of edge-, main-, and additional-beams are located at the elevationof thickness of the slab and have the properties of the part of the slab that they represent.These longitudinal beams are connected by means of transverse-beams. The properties ofeach transverse-beam can be determined considering a rectangular cross section of the slabit represents.
The transverse beams are placed at the locations of cross-frames at a minimum, how-ever additional elements may be used to improve the accuracy of the nRHA results. Basedon grillage method rules, the spacing between two successive transverse-beams can be upto three times the distance between longitudinal beams; use of smaller spacing does notaffect significantly the accuracy of analysis results. The nominal stiffness of the deck slabsthat are represented by the individual transverse beams are assigned to these elements.However, since the deck mass are assigned to the longitudinal beams, the mass of thetransverse beams are taken as zero. It should be noted that the longitudinal and transversebeams are rigidly connected to each other without any releases. A 3D configuration of theproposed beam-stick model is shown in Fig. 5.
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Curved Steel Girder Highway Bridges for Seismic Analysis 225
FIGURE 5 3D Configuration of the proposed beam-stick model.
2.2.2. Property Modifications and Assignments. Certain adjustments and modifications tothe elastic properties of superstructure elements are necessary to ensure satisfactory accu-racy. First modification to the grillage beams (both longitudinal and transverse) results fromthe fact that there is no interaction between the axial force and bending in two perpendiculardirections in the grillage. In order to address this fact, it is enough to set the Poisson’s ratioof the grillage beams to zero [Hambly, 1990]. Additional modifications to the properties ofthe grillage beams can derived by considering a differential slab element with unit dimen-sions (Fig. 6). If u, v,, and w are the deformation of the cube in X, Y, and Z, respectively,the curvatures (k) in X, Y, and XY directions are defined by:
FIGURE 6 Internal forces in a cube of the slab.
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226 E. Amirihormozaki, G. Pekcan, and A. Itani
kxx = ∂2w
∂x2, kyy = ∂2w
∂y2, kxy = ∂2w
∂x∂y. (2)
Furthermore,
εij = −z∂2w
∂xi∂xj= −zkij, (3)
where εij denotes the strain oriented in jth direction on the face of the cube perpendicularto ith direction, and z is the distance of the fiber from the horizontal axis passing throughthe center of the face. Therefore, the axial and shear stresses are expressed as:
σxx = E
1 − v2
(εxx − vεyy
) = −zE
1 − v2
(kxx + νkyy
)(4)
σyy = E
1 − v2
(εyy − vεxx
) = −zE
1 − v2
(kyy + νkxx
)(5)
τxy = τyx = 2Gεxy = −2zGkxy, (6)
where E, G, and v are the elastic and shear modulus of elasticity and the Poisson’s ratio ofthe slab, σxx and σyy are the normal stress on the planes perpendicular to x and y directions,εxx and εyy denote normal strain on the planes perpendicular to x and y directions, τxy is theshear stress oriented in y direction on the plane perpendicular to x direction, and τyx is theshear stress oriented in x direction on the plane perpendicular to y direction, respectively.
By calculating the shear, moment and torsion due to these stresses and equating themto classic formulations for torsional constant and moment of inertia of a rectangle, it is pos-sible to find the necessary modification factors for the properties of the beams representinga definite width of the slab. Torsional constant of the beams can be obtained by integrationof the shear stress around the center of the rectangle:
txy =∫ ts
2
− ts2
−zτxydz = G
(ts3
6
)kxy. (7)
The torsion (txy) can be also expressed as:
txy = GJkxy, (8)
where J is the torsional constant of the cube face. Therefore:
J = ts3
6. (9)
This is half of the classical formula for torsional constant. As a result, a modification factorof 0.5 should be assigned to the beams representing the slab. In a similar manner, it ispossible to show that no modification factor is necessary for the moment of inertia of thebeams. This finding is valid for both X and Y direction.
In summary, the properties of the longitudinal beams at slab level and transverse beamsincluding shear area, cross-sectional area, and moment of inertia are the properties of
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Curved Steel Girder Highway Bridges for Seismic Analysis 227
the part of the slab they are representing except the torsional constant which should bedecreased by half. Furthermore, tributary mass of each stick is lumped at its ends.
3. Cross Frames
Cross frames are considered as part of the superstructure. However, since the forces inthe cross-frames over the support locations (abutments and bents) are significantly higherthan forces at intermediate cross-frames, only the support cross-frames are modeled explic-itly. However, the effect of intermediate cross-frames on the behavior of the superstructureshould be considered. To take into account the effect of intermediate cross-frames, a rota-tional constraint about the alignment of the superstructure has been assigned to all nodes atthe cross section of each intermediate cross-frame (Fig. 5).
Furthermore, cross-frames are recognized as critical components to transfer seismicforces from the superstructure to the bearings in steel girder bridges. In particular, crossframes over the supports may experience inelastic behavior during large earthquakes.Support cross frames may be designed to act as ductile components to dissipate energyand reduce the shear demand on columns through their inelastic deformations [Bahrami,2009]. “Uniaxial Pinching4 Material” in OpenSees [X] with appropriate floating pointvalues may be used to model hysteretic response of cross frames. Nonlinear axial force-deformation hysteretic properties of diagonal cross-frames are modeled using two TakedaNL-Link elements in SAP2000 [CSI, 2011; Carden et al., 2005]. The first element capturesthe compression envelope and has elasto-plastic properties of the tension part of the bracewith a yield force related to the point where the elastic post-buckling zone in tension turnsinto plastic zone in tension. This tension yield force is the residual force in the memberwhen it straightens after a buckling excursion due to plastic hinging in the member. Thesecond NL-Link captures the remaining tension properties on the member.
4. Reinforced Concrete Columns
Columns are part of the substructure where inelastic deformations and plastic hingingare expected to occur during earthquakes. Therefore, specific detailing requirements areestablished to provide sufficient confinement. Non seismically designed columns gener-ally do not meet the minimum requirements which may lead to undesired failure modes.Several models are proposed to model the shear strength of the reinforced concrete columns[Watanabe and Ichinose, 1992; Aschheim and Moehle, 1992; Priestley et al., 1994; Sezen,2004; Elwood, 2004; Ghannoum, 2007]. Elwood [2004] developed a uniaxial material tomodel shear failure of columns. This model captures the shear strength degradation afterfailure and takes into account subsequent degradation of the beam-column element repre-senting columns which is defined in series with uniaxial material. Plastic hinges may format both ends of the columns with plastic hinge length of:
where L is the length of column from the point of maximum moment to the point of momentcontra-flexure, fye is the expected yield strength of column longitudinal reinforcing steelbars, and dbl is the nominal diameter. Furthermore, an effective moment of inertia, Ieff
is determined and used following guidelines established by AASHTO [AASHTO, 2014].
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228 E. Amirihormozaki, G. Pekcan, and A. Itani
Accordingly, the effective (cracked) moment of inertia is calculated using [AASHTO, 2014,Sec. 5.6.2]:
EcIeff = My
�y, (11)
where My and �y correspond to the moment and curvature of the section at the first yieldingof the reinforcing steel in the column section. In addition, because of the great reduction ofthe torsional stiffness of the concrete members after first cracking, a modification factor of0.2 is applied to the gross torsional moment of the inertia of the column section [AASHTO,2014; Sec. 5.6.5]:
Jeff = 0.2Jg, (12)
where Jeff is the effective torsional (polar) moment of inertia of reinforced concrete section,and Jg is the gross torsional (polar) moment of inertia of reinforced concrete section.
The unconfined concrete compressive strain at the maximum compressive stress (εc0)and the ultimate unconfined compression strain (εsp) based on spalling are taken as0.002 and 0.005, respectively [AASHTO, 2014]. Mander’s constitutive model [Mander,1988] is implemented for confined concrete. Stress-strain constitutive models for uncon-fined and confined concrete are depicted in Fig. 7.
Two types of forced-based elements are provided in OpenSees to model reinforcedconcrete columns, named “beam-With-Hinge” and “nonlinearBeamColumn”. In the firstelement, plastic hinging is localized at the element’s ends while the middle part of the ele-ment is modeled as linear elastic. The second element called the “nonlinearBeamColumn”element is based on force formulation and spread plasticity along the element. A bettermass distribution along the column can be obtained using this element by dividing it intosegments. Moreover, there is no need to specify cracked section modifiers for each col-umn section. A fiber section representing the column section is defined and assigned tothis element. The fiber section is discretized into 10 radial and 12 tangential fibers whichtake into account the axial and flexural stiffness and PMM interaction for the column ele-ments. Several studies have recently suggested that the modeling accuracy is not necessarilyimproved by increasing the number of fiber elements beyond these values [Sadrossadat-Zadeh et al., 2007; Aviram et al., 2008a; Abdel-Mohti and Pekcan, 2008]. Cracked shear
FIGURE 7 Concrete stress-strain model.
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Curved Steel Girder Highway Bridges for Seismic Analysis 229
and torsional stiffness of the column section is specified for the fiber section. The con-stitutive relationships for cover and core concrete, and steel are defined for these discretefibers. “Concrete01” and “ReinforcingSteel” uniaxial materials in OpenSees can properlyrepresent concrete and steel reinforcements, respectively. Finally, the “geometric transfor-mations” between element local and global axes must be defined as user input for individualelements which can also include P-Delta effects.
5. Bearings
Bearings can be the most critical elements in the bridge load path connecting the superstruc-ture to the substructure. Bridge bearings are regularly subjected to vertical and horizontalloads due to gravity, traffic, wind, earthquake, or other sources (e.g., thermal, creep, andshrinkage). Bearings can be categorized into two generic types as steel bearings and mod-ern bearing types. Steel bearings, which have been historically used in non seismicallydesigned bridges, are selected for some of the analytical development presented in this arti-cle. It is noted that the majority of these bridges with steel bearings are still in service inparts of the U.S. Realistic force-deformation relationships have been identified based onearlier experimental studies [Mander et al., 1996]. Mander et al. [1996] introduced reli-able hysteretic models for sliding, fixed, and rocker bearings. The hysteretic response ofthese bearing types can be modeled using a combination of nonlinear elements available invarious analyses tools such as OpenSees.
Modern type bearings with clear advantage of being ductile and multi-rotational canbe categorized into two broad categories of Elastomeric bearings and High- load Multi-rotational bearings. Plain elastomeric bearing pads (PEP), steel reinforced elastomericbearings, and lead-rubber bearings (LRB) are the most commonly used elastomeric bear-ing types. Pot bearings, Disc bearings, and Spherical bearings are known as High-loadMulti-rotational bearings which are typically designed to allow rotation in the superstruc-ture. These bearings may also allow relative translation in one or more directions whenelastomeric bearing pads and/or polytetrofluoroethylene (PTFE) stainless steel sliders areincorporated in their design. Reinforced elastomeric bearings are the most common type ofbearings used in seismically designed bridges. These bearings are not vulcanized to the topand bottom plates but they are bolted instead to provide nominal lateral resistance and sub-sequently allowing them to slide on the concrete base. Based on AASHTO [2012], shearmodulus of the elastomer may be in the range of 0.62 (0.09) and 1.31 MPa (0.19 ksi).In this article, a shear modulus equal to 0.97 MPa (0.14 ksi) is chosen as an average value.Bearings are fixed in compression and free in all rotational degrees of freedom. Behaviorof bearings is considered to be elasto-perfectly plastic with an initial stiffness:
Ke = GA
hrt, (13)
where G is shear stiffness of the rubber, A denotes the area of rubber pad, and hrt is theheight of the rubber layers. The yield force is the minimum of the shear force correspond-ing 150% shear strain in the bearing and the friction strength of the pad-concrete surfaceobtained as:
Fy = µPm, (14)
where Pm is the maximum compressive load (kN, kip) and μ is the coefficient of friction[Schrage, 1981]:
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230 E. Amirihormozaki, G. Pekcan, and A. Itani
FIGURE 8 A representative hysteresis of an elastomeric bearing obtained in OpenSeesplatform.
µ = 0.05 + 0.4
σn, (15)
where σn is the normal stress on the elastomeric pad in MPa.Bolts connecting top plate to the concrete substructure in elastomeric bearings are
modeled as elastic elements in parallel to elastomeric bearings which are automaticallyremoved during response history analysis when they reach their maximum capacity.
In order to define an element representing elastomeric bearing in OpenSees platform,an “elastic perfectly-plastic” material in two perpendicular directions is assigned to a “two-node link” element. An elastic material with high stiffness is used to model axial behaviorand all of the rotational degrees of freedom are released. This element is in parallel withanother “two-node link” element representing bolts which are to be removed at the onsetof reaching their maximum shear capacity during the response history analysis. An illus-tration of hysteretic response of elastomeric bearings in shear direction is shown in Fig. 8.“ElastomericBearing” element in OpenSees library can also be used to model bearingsproperties in all directions. It is noted that appropriate force-deformation properties forother types of bearings must be assigned to the bearing elements in general.
6. Shear Keys
Shear keys are typically used to provide transverse restraint to the superstructure at theabutments and bents under service loads and earthquakes. Typically, for the design of high-way bridges in seismic zones two distinct conditions are assumed (1) shear keys remainintact and (2) shear keys fail completely. These two conditions are intended to establishconservative bounds on the deformations and forces in other structural elements such asbent columns, piles, etc. [Caltrans, 2006]. Shear keys that are designed to be sacrificialserve as fuses to limit the force transferred to the abutments and piles. The transverse forcefrom the superstructure to the substructure is transferred by the shear keys, however theyfail before the columns and piles reach their maximum capacity. Sacrificial shear keys maybe interior or exterior [Megally, 2002]; internal shear keys are placed along the width of the
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Curved Steel Girder Highway Bridges for Seismic Analysis 231
FIGURE 9 Hysteretic behavior of shear keys.
superstructure whereas external shear keys are located at the two edges of the superstruc-ture. Exterior shear keys are usually recommended for new construction because they areeasier to inspect and repair. Interior shear keys are assumed to be R/C blocks or a keeperbracket adjacent to the bearings.
In the present study, shear keys are assumed to be as R/C blocks for both bents andabutments. Transverse shear keys are located at the abutments only in transverse directionand at bents in both longitudinal and transverse directions. The maximum capacity of theshear keys are assumed to be equal to 30% of the dead load reaction for the shear keys atthe abutments [Caltrans, 2006]. However, the capacity of the shear keys at the bents are1.2 times the shear force due to plastic hinging at the bottom of column. The initial gapvaries from 25 to 75 mm (1 –3 in) and the maximum shear key deformation to the completefailure point is taken as 75 mm (3 in). An “Elastic Perfectly-Plastic Gap Uniaxial” materialis utilized to model the shear keys in OpenSees. A high initial stiffness and negative post-yield stiffness is assigned to shear keys which follow an elastic perfectly-plastic behavior,as shown in Fig. 9. The stiffness values were determined following the experimental resultsreported by Megally et al. [2002]. In addition, a “uniaxialMaterial MinMax” material isalso specified to mark the failure the shear keys (100 mm including the 25 mm initial gapin Fig. 9) and beyond which the shear key elements are removed during the response historyanalyses.
6.1. Soil-Abutment-Structure Interaction
Soil-abutment-structure interaction is due to soil resistance in passive, active, and trans-verse directions. It is assumed that the piles provide the primary resistance in the activeand transverse directions. The active and at-rest pressure due to embankment fill are neg-ligible relative to passive pressure. Therefore, abutment response in passive direction isdue to a combination of soil behind the back-wall and piles that provide support to theabutment. When the abutment wall pushes against the soil, both the back-wall soil andthe pile stiffness contribute until the capacity of soil is reached beyond which the resis-tance due to the soil remains constant. The gap between the deck and the back-wall canbe modeled using a compression-only gap element with a high stiffness that activates afterthe contact is established. As illustrated in Fig. 10, the elements considered in the longi-tudinal direction include bearings which ensure the force transfer to the abutment prior
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-m: Abutment mass distributed over a number of nodes.-PS: uniaxial element modeling combined soil (compression-only) and supporting piles-IG: two uniaxial elements in parallel that represent (1) Impact + Gap, and (2) Longitudinal Bearing
Explained in Figure 11
FIGURE 10 Abutment in longitudinal direction.
-PL: piles in transverse direction-BRT (for SD bridges): bolts in tension and shear, bearings and shear keys in transverse direction.-BRT (for NSD bridges): steel bearings in transverse direction.
FIGURE 11 Abutment configuration in transverse direction.
to closure of the gap. Piles and soil behind the back-wall are the two elements whichsupport the abutment in the longitudinal direction. Pounding takes place when the gap isclosed. Therefore, an element representing the impact is defined connecting the superstruc-ture and the abutment. After gap closure, forces are transferred directly to the abutmentand then to the piles and the soil behind the back-wall. A mass representing the mass ofthe concrete abutment has been assigned to nodes representing the abutment back-wall(Fig. 11). Aviram et al. [2008b] suggests that consideration of the participating abutmentmass has a critical effect on mode shapes and consequently the dynamic response of thebridge. Embankment critical length and its mass participation have been proposed by manyresearchers [Kotsoglou and Pantazopouloi, 2006; Zhang and Makris, 2002; Werner, 1994].Aviram et al. [2008b] assumed a nominal mass proportional to the superstructure dead
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Curved Steel Girder Highway Bridges for Seismic Analysis 233
load at the abutment, including contribution from structural concrete as well as the partic-ipating soil mass. In the transverse direction, as shown in Fig. 11, forces are transferredthrough the bearings to the abutment and then to the piles. In the present study, contribu-tion of wing-walls in transverse direction, the embankment transverse stiffness and its massmay be ignored due to small wing-wall length; however, these components can be readilyincorporated.
The force-deformation characteristic of the soil behind the back-wall can be modeledbased on the recommendations from Caltrans [2006]. Accordingly, the initial abutmentstiffness can be expressed as:
Kabut = ki × w ×(
habut
1.7m
)SI Units
Kabut = ki × w ×(
habut
5.5ft
)US Units,
(16)
where kiis the initial embankment fill stiffness, taken as 14.35 kN/mmm (20 kip/in
ft ), w is thewidth of the back-wall (m, ft), and habut is the height of the back-wall (m, ft). The ultimatecapacity of the abutment is due to only backfill soil and can be expressed as:
Pbw = Ae × (239kPa) ×(
habut
1.7m
)SI Units
Pbw = Ae × (5.0ksf ) ×(
habut
5.5ft
)US Units,
(17)
where Ae is the effective abutment area (mm2,ft2). The coefficient (239 kPa, 5.0 ksf ) isintended to account for the static shear strength of a typical embankment material. Effectiveabutment area is determined as:
Ae = hbw × wbw, (18)
where hbwand wbware the height and width of the back-wall, respectively.Passive soil behavior behind the back-wall can also be represented using the modified
hyperbolic force-displacement (HDF) equation proposed by Shamsabadi [2007], which isimplemented as “Hyperbolic Gap Material” in OpenSees. This material exactly follows theHyperbolic equation when all parameters are defined. The general equation for this materialin OpenSees is:
F (x) = x1
Kmax+ Rf
x
Fult
, (19)
where x is the deformation of soil in passive direction, Kmax is taken as the initial stiff-ness, Rf denotes the failure ratio (taken 0.7), and Fult is the ultimate (maximum) passiveresistance. A representative hysteresis obtained in OpenSees is illustrated in Fig. 12.
In addition to providing vertical support, piles contribute to transferring of the lateralforces in the abutment to the soil. Piles usually are designed to remain linear-elastic in theevent of an earthquake. However, their realistic modeling is necessary to accurately capturethe abutment response in both longitudinal (active and passive) and transverse directions.
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234 E. Amirihormozaki, G. Pekcan, and A. Itani
FIGURE 12 A representative hysteresis of soil behind the back-wall.
FIGURE 13 Hysteretic behavior of abutment in transverse direction.
To account for the group effect of the piles, the equivalent number of piles may be deter-mined based on spacing, diameter, and configuration. In this paper, initial stiffness of 7kN/mm (40 kips/in) and ultimate strength of 119 kN (27 kips) for each pile as recom-mended by Caltrans [1990] is utilized with elasto-perfectly plastic response characteristics.These values are for standard 45 and 70 ton, and 406 mm (16 in) Cast-In-Drilled-Hole(CIDH) piles. Choi [2002] introduced a tri-linear model for the piles. A representativehysteretic response of piles in transverse direction is presented in Fig. 13.
7. Pounding
Impact between the deck and abutments or between two decks occurs due to out-of-phasemotion of the two parts and pounding occurs at the interface. Pounding can cause crushingof the concrete at the interface, unseating, and also damage to columns, bearings, abut-ments, restrainers and shear keys. Researchers have used mainly two different methods tomodel the pounding, namely, the contact element approach [Maison and Kasai, 1990, 1992]and the stereomechanical approach [Goldsmith, 1960]. The stereomechanical method isless favorable due to required modification of the velocities at the colliding bodies at theinstant of the pounding. Muthukumar [2003] developed two simplified contact force-based
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Curved Steel Girder Highway Bridges for Seismic Analysis 235
FIGURE 14 A sample hysteresis of an impact element in OpenSees platform.
model for modeling the impact. One idealization is an inelastic truss element with a gap andthe other is an inelastic truss in parallel with a linear link element. Both models are basedon the Hertzdamp contact model. “Impact” material in OpenSees material library is utilizedto model pounding between deck and abutments in this study. This material agrees with thehysteretic response introduced by Muthukumar [2003] as inelastic truss contact element.Maximum penetration (δm) is taken 25.4 mm (1.0 in) [Muthukumar, 2003; Nielson, 2005;Padgett, 2007] with a coefficient of restitution of 0.7 which was used for the verificationanalyses in this article. However, it is not necessarily recommended as a unique value appli-cable to all types pounding surfaces. The gap defined in this material is taken as equal tothe gap between the deck and back-wall. A sample hysteretic response of impact elementis illustrated in Fig. 14.
8. Proposed Beam-Stick Model Verification
In order to verify the accuracy of the proposed modeling approach, two horizontally curvedsteel girder bridges are selected. Models were developed using both the proposed approachand the deck-shell modeling approach using SAP2000 (Fig. 15). The superstructure indeck-shell model generated by SAP2000 Bridge Modeler is a combination of shell ele-ments for slab and 3D frame element representing steel girders. Overall properties of thebenchmark bridges are summarized in Table 1.
In both proposed beam-stick and deck-shell models, column plastic hinges are modeledusing the P-M2-M3 hinge definition in SAP2000 with the fiber section assigned to thesection. A typical hysteretic curve and corresponding NL-Links were defined to representthe cross frames. All of the bearings at bent locations are considered pin, i.e. they canrotate but are fixed against transitional movement in all directions. Three cases of boundaryconditions with respect to bearings at abutments are modeled as shown in Table 2.
Gravity load, modal, and response history analyses are performed for both BM01 andBM02 to verify the proposed beam-stick model. The deflections of inner, interior and outergirder of the BM01 under dead load at the mid span of the middle span are shown inTable 3. It can be seen that the proposed model accurately predicts the dead load deflec-tions. Furthermore, the proposed model can accurately capture the girder shear forcesand moments as well as the bearing reactions due to static/gravity loading conditions[AmiriHormozaki et al., 2013].
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236 E. Amirihormozaki, G. Pekcan, and A. Itani
FIGURE 15 Simplified beam-stick models for (a) BM01 and (b) BM02.
Modal analyses are conducted to compare the vibration periods and mass participa-tion factors associated with the first 15 modes. As shown in Fig. 16, Table 4, and Table 5,excellent agreement was achieved for all abutment types. In order to verify the load pathand sequence of hinge formations, nonlinear pushover analyses were conducted using firstand second mode shapes. Pushover curves for the TFRR BM01 benchmark model areshown in Fig. 17 which compares lateral load vs. displacement at the top of bents forboth proposed BS and 3D FE models. Similar comparisons have been made for bridgeswith other boundary conditions. Last but not the least, nonlinear response history analy-ses of both BM01 and BM02 are performed by applying Northridge ground motion record(1994) with two components concurrently in the longitudinal (in chord direction) and trans-verse (direction perpendicular to chord) directions. Forces in the bearings, cross-frames,columns, displacement of deck at abutment, displacement at top of the bent, hysteresis ofplastic hinges at the top and bottom of columns are monitored. Critical damping ratio wasassumed as 5 percent. For the bridges considered in this study, very good agreements canbe seen in Figs. 18 and 19.
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Curved Steel Girder Highway Bridges for Seismic Analysis 237
TABLE 1 Properties of two benchmark bridges
First Benchmark Bridge(BM01)
Second Benchmark Bridge(BM02)
Class Continuous three span composite steel girder horizontallycurved bridge
Origin FHWA-fundedexperimentalinvestigation at theUniversity of Nevada,Reno (UNR)
Five-span bridge designed byCALTRANS
Number of Spans 3 5Span Length (m) 32, 46.3, 32 35.7, 46.3, 66.4, 65.2, 64.6Subtended Angle (degree) 104 80.5Bentcap Type Single-Column Dropped
Free Model Free in all transitional and rotational directionsTFRR Model Tangentially free, radially restrainedPinned Model Freely in all rotational directions but completely fixed in all transitional
directions
TABLE 3 Dead load deflection of three girders in BM01 (mm)
Abutment Type Model Type Inner Girder Interior Girder Outer Girder
TFRR BS Model 2.51 3.78 4.953D FE Model 2.26 3.51 4.78Difference (%) 11 8 4
Pinned BS Model 2.44 3.81 5.033D FE Model 2.26 3.56 4.90Difference (%) 8 7 3
As it was mentioned earlier, one of the primary objectives of the proposed model isto reduce the time required to run nRHA. Durations of analyses of BS model for differentmodels with nonlinearity in cross-frames are presented in Table 6. Accordingly, proposedBS model reduces analysis time by approximately 80%. Clearly, the preceding discus-sion and presentation suggest that the proposed simplified modeling approach is a suitablecandidate for computationally intensive studies [AmiriHormozaki et al., 2014].
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238 E. Amirihormozaki, G. Pekcan, and A. Itani
FIGURE 16 Comparison of natural periods of (a) BM01 with free abutment,(b) BM01 with TFRR abutment, (c) BM02 with TFRR abutment, and (d) BM02 withPinned abutment.
TABLE 4 Comparison of mass participation factor in six degrees of freedom (BM01 withTFRR Abutment)
FIGURE 17 Pushover curves with (a) first mode and (b) second mode as load patterns.
9. Experimental Verification of the Proposed Modeling Approach
It is noted that this study was part of a larger project that involved shake-table testingof a total of five distinctly different configurations of a curved bridge at the Universityof Nevada, Reno. Each test used the same superstructure but with different bearingsor columns or end cross-frames, namely: (1) conventional bearing and column details(conventional bridge); (2) seismic isolation at all supports (full isolation); (3) seismicisolation (lead-rubber bearings) and buckling restrained braces at abutments (hybrid protec-tive systems or hybrid isolation); (4) soil-structure interaction at the abutments (abutmentpounding); and (5) soil-structure interaction at the bents (footing rocking on foundation)[Monzon et al., 2013a,b,c]. The subject bridge was 2/5 scale model of a three-span, 44.2 mlong with 24.4 m radius at the centerline, and with 3.7 m wide concrete deck on three steelgirders. Experimentally recorded response data is being processed by the project partic-ipants, which will be used to present experimental and analytical response comparisonsin subsequent publications. However, some of the preliminary results are used to verifythe proposed grillage-based modeling approach in this article. For this purpose, the most
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240 E. Amirihormozaki, G. Pekcan, and A. Itani
FIGURE 18 (a) Radial displacement of the node at top of the column (Free BM01), (b)resultant moment at the plastic hinge at the base of the Column (TFRR BM01), (c) axialforce in the inner bearing at bent 4 (Free BM02), and (d) acceleration at the top of thecolumn (TFRR BM02).
FIGURE 19 Force-deformation of the two NL-Links of (a) proposed BS model and (b)3D FE model of BM01.
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Curved Steel Girder Highway Bridges for Seismic Analysis 241
TABLE 6 Duration of response history analysis of BS and 3D FE models
Response History Analysis Duration (min)
Analysis Case Proposed BS-Model 3D FE Model BS/FE (%)
challenging configuration that involved highly and unusually nonlinear response due topounding at the seat-type abutment locations was selected. A concrete backwall in serieswith a set of nonlinear springs were used to simulate the presence of the approach embank-ment. An initial seismic gap was included based on the design service loads. Completedetails of the tested-configuration can be found in Wieser [2014] and Wieser et al. [2014].The versatility of the proposed approach allowed detailed and accurate representation of thebridge-abutment interaction as depicted in Fig. 20 (with reference to Fig. 10). Accordingly,the girders were seated on teflon slider bearings, and a dash-pot element was used inparallel with the impact element to model the damping mechanism that was introducedexperimentally [Wieser, 2014].
A total of three historical earthquake records, namely, 1994 Northridge (Sylmar sta-tion, 360), 1940 Imperial Valley (El Centro), 1968 Tokachi-Oki (Hachinohe), was used inthe experimental program. The ground motions were scaled to achieve target site-specificdesign spectra and were applied incrementally to establish system and component capac-ities while monitoring the damage progression [Monzon 2013b,c]. Herein, some of theselected experimental responses due to Sylmar motion scaled to 75% and 100% of thedesign spectra are used for analytical verification. It is noted that the shear keys during the75% test failed, hence shear keys were not present during the subsequent 100% test. As canbe seen in Figs. 21–23, very good agreement between the experimental and analyticalresponse histories have been achieved.
10. Further Use of Model for Preliminary Design Purposes
AASHTO LRFD [2012, Article 4.7.4.3] establishes the minimum analysis requirementsfor multi-span bridges as summarized in Table 7. As it can be seen in this table, multimode
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242 E. Amirihormozaki, G. Pekcan, and A. Itani
FIGURE 21 North abutment shear key response (Sylmar at 75% intensity).
FIGURE 22 Deck displacement response (Sylmar at 100% intensity).
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Curved Steel Girder Highway Bridges for Seismic Analysis 243
FIGURE 23 South bent column hysteresis response (Sylmar at 100% intensity).
TABLE 7 Minimum analysis requirement for seismic effects of multi-span bridges
Other Bridges Essential Bridges Critical Bridges
Seismic Zone Regular Irregular Regular Irregular Regular Irregular
1 ∗ ∗ ∗ ∗ ∗ ∗2 SM/UL SM SM MM MM MM3 SM/UL MM MM MM MM TH4 SM/UL MM MM MM TH TH
∗: No seismic analysis required; UL: Uniform load elastic method; SM: Single mode elastic method;MM: Multi-mode elastic mode method; TH: Response history method
elastic method (MM) is the minimum required analysis method for irregular, essentialand critical bridges. Caltrans [2006] introduced three different analysis methods includingEquivalent Static Analysis (ESA) (same in concept with UL in AASHTO LRFD), ElasticDynamic Analysis (EDA) (as linear elastic multi-modal spectral analysis), and InelasticStatic Analysis (ISA) (also referred as “pushover” analysis). Multi-modal response spec-trum analysis is known as one of the most versatile methods to achieve demand on bridgesfor the purpose of preliminary design. Analytical models with varying complexities fromsingle-spine to full finite-element may be required depending on the overall geometry(straight, skewed, curved) and conditions at the support locations. However, in most ofthe cases, extreme boundary conditions are considered to establish upper-bound seismicdisplacement and force demands on the critical structural components.
Second benchmark bridge (BM02) has been modeled using single-beam, proposed-beam, and 3D FE models and the result of maximum column displacement, and maximumshear in transverse direction in the bearings have been compared. In case of single-beam
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244 E. Amirihormozaki, G. Pekcan, and A. Itani
model, to achieve acceptable accuracy in predicting the cap beam moments, hence deadload reactions, the cap beam was modeled with large stiffness (Fig. 24). It is noted that thebending results in compression force in the interior bearings and tensile force in the exteriorones. Comparison of maximum resultant displacement at the top of columns under responsespectrum analysis is shown in Fig. 24. As illustrated in this figure, the maximum resultantdisplacements from response spectrum analysis which is used to design columns are nearlythe same for all three cases. Therefore, response spectrum analysis of single-beam modelmay be suitable for the prediction of maximum column displacements. However, the single-beam model with abutments restrained in transverse direction cannot capture the realisticdistribution of maximum transverse bearing forces, primarily due to the fact that the capbeam has to be modeled to act as a rigid body. Clearly, the proposed beam model predictsthese forces accurately, as illustrated in Fig. 25.
FIGURE 24 Dead load reaction in bearings at bent assuming a rigid body bent cap.
FIGURE 25 Maximum resultant displacement at top of the four bents with free bearing atabutments and pinned bearings at bents.
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Curved Steel Girder Highway Bridges for Seismic Analysis 245
FIGURE 26 Maximum transverse shear in the bearings at bents for the bridge with pinnedbearings at bents and bearings free in longitudinal direction and restraint in transversedirection.
11. Summary and Conclusion
A very efficient and accurate simplified modeling approach for primarily horizontallycurved steel girder bridges is introduced in this article. The proposed model is suitable forall phases of preliminary design, static and dynamic response assessment. The modelingapproach can be readily adopted for highway bridges with other geometries and types, andallows direct and easy incorporation of other types of (linear as well as nonlinear) ele-ments and components. The accuracy and validity of the proposed models were verifiedagainst 3D FE counterparts. The analysis time that is required for nonlinear response his-tory analyses was reduced by 80%, making the modeling approach especially appropriatefor studies that require substantial number of analyses such as fragility curve development.Furthermore, recommendations for the selection and development of hysteretic propertiesof various critical bridge components such as shear keys, reinforced concrete columns, steelreinforced elastomeric bearings, pounding between the deck and back-wall, and soil behindthe back-wall are introduced and demonstrated using OpenSees. An abutment configurationincluding all bridge components that can properly capture the abutment response in trans-verse, active and passive direction is presented. The validity of single stick bridge modelingapproach commonly used by design engineers is evaluated and it was demonstrated that itcannot accurately capture the bearing force and deformations, although it may be suitablefor the prediction of column displacements from response spectrum analysis.
The proposed modeling approach was used to model the experimentally recordedresponse of a 2/5 scale model of a curved bridge which involved one of five highly non-linear configurations. The preliminary comparison of experimental vs. analytical responseverifies the efficiency and accuracy of the proposed approach.
Funding
This study was funded in part by the Federal Highway Administration (FHWA) underContract DTFH61-07-C-00031: Improving the Seismic Resilience of the Federal AidHighway System
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