Title. Author(s)ALAMIRI, M. ASSAD; GOTO, YOSHIAKI. Issue Date Doc URL. Type. Note. File Information

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1 Title ULTIATE STATE O THIN-WALLED HOLLOW CIRCULAR STEEL DIRECTIONAL HORIZONTAL SEISIC ORCES AND TRI-DIRECT Athor(s)ALAIRI. ASSAD; GOTO OSHIAKI Isse Date Doc URL Type proceedings Note The Thirteenth East Asia-acific Conference on Strc Sapporo Japan. ile Information easec pdf Instrctions for se Hokkaido University Collection of Scholarly and Aca

2 ULTIATE STATE O THIN-WALLED HOLLOW CIRCULAR STEEL COLUNS SUBJECTED TO BI-DIRECTIONAL HORIZONTAL SEISIC ORCES AND TRI-DIRECTIONAL SEISIC OENTS. Assad. ALAIRI 1* oshiaki. GOTO 1 Department of Civil Engineering Nagoya Institte of Technology Japan Department of Civil Engineering Nagoya Institte of Technology Japan ABSTRACT Torsional moment acts at the top of colmns of bridge piers when elevated crved girder bridges or elevated girder bridges spported by inverted L-shaped bridge piers are sbjected to seismic accelerations. Therefore to evalate the safety of the bridge piers in the above types of bridges it is essential to consider the effect of the torsional moment on piers. Herein focsing on the effect of the torsional moment versatile interaction srface epressed in terms of bi-directional horiontal seismic force and tri-directional seismic moment components is derived by the so-called pshover analysis to epress the ltimate state of thin-walled hollow circlar steel colmns. The accracy and validity of the derived ltimate interaction srface is eamined by carrying ot nonlinear dynamic response analysis on inverted L-shaped bridge pier models nder variosly factored bi-directional horiontal seismic acceleration components. As a reslt it is observed that the derived ltimate interaction srface well predict the ltimate state of the pier models althogh it is a little conservative. Keywords: Steel piers Torsional moment Limit State Seismic design Nonlinear Analysis. 1. INTRODUCTION The elevated girder bridge piers are normally sbjected to 3D components of seismic accelerations. However or recent research (Goto and Ebisawa 010) revealed that the copling of two horiontal seismic acceleration components has a significant impact on the ltimate behaviors of elevated girder bridge piers. Therefore to ensre their safety it is essential to eamine the ltimate behavior of bridge piers nder bi-directional seismic accelerations and specify their ltimate state. or this prpose athors (Obata and Goto 007) first developed an accrate 3D loading system as well as a rigoros finite element nonlinear shell analysis (Goto et al. 006). By the se of these tools the ltimate behavior of thin-walled steel colmns sbjected to bi-directional horiontal seismic accelerations is etensively stdied (Goto et al. 006; Goto el at. 009) both analytically and eperimentally. Based on these stdies an interaction srface epressed in terms of bi-directional horiontal force components and bi-aial bending moment components acting at the top of thin-walled steel colmns was derived to epress the ltimate state of bridge piers (Goto el at. 009; Goto and Ebisawa 010). However seismic accelerations indce torsional moment at the top of the colmns of elevated crved girder bridge piers or the colmns of inverted L-shaped bridge piers as shown in igre 1. Therefore as a versatile interaction ation to predict the ltimate state of colmns sed for bridge piers it is important to consider the effect of the torsional moment component in addition to that of bi-directional horiontal force components and bi-aial bending moment components. Gao et al. (000) presented a formla to predict the ltimate states of inverted * Corresponding athor: assad@koo.ace.nitech.ac.jp resenter: assad@koo.ace.nitech.ac.jp 1

3 L-shaped piers nder ot-of-plane horiontal loading where torsional moment acts at the top of the piers. However this formla is only applicable to a specific case of the inverted L-shaped piers where an ot-of-plane horiontal seismic force and a dead load act at a single point on the cross beam. Herein in order to derive the aforementioned versatile interaction ation of thin-walled circlar steel colmns sed for bridge piers the eisting ltimate interaction srface (Goto and Ebisawa 010) for the colmns nder bi-directional horiontal force components and bi-aial bending moment components is modified to take into accont the effect of torsional moment component. The thin-walled circlar steel colmns considered herein have the radis-to-thickness parameter in the range of 0.04 R = ( R/ t)( σ / E) 3(1 ν ) 0.1. t y. ULTIATE STATE UNDER BI-DIRECTIONAL HORIZONTAL SEISIC ACCELERATIONS.1. Ultimate State of Thin-walled Steel Colmns The conventional Japanese highway bridge design code (Japan Road Association 01a) stiplates that the thin-walled steel colmn reaches its ltimate state at the limit point of the horiontal force vs. horiontal displacement crve. This criterion is reasonable both from theoretical and engineering viewpoints becase the limit point is theoretically a transition point from stable to nstable state. rthermore the thin-walled colmns sed as bridge piers normally reach this point de to plastification together with the local bckling (Goto et al. 006; Goto el at. 009). This implies that the limit point is an initiation point of serios damage that may be considered as an ltimate state of thin-walled steel colmns from engineering viewpoint. Under the seismic loads even if a colmn temporarily falls into instability state it can regain its stability after the major seismic acceleration is over. However once colmn deforms beyond the limit point the damage of colmn is likely to be evident. Herein the initial transition point from stability state to instability state that is identified by the mlti-dimensional elastic-plastic stability criterion (Hill 1958; Goto et al. 009) is defined as the ltimate state of thin-walled steel colmns nder bi-directional horiontal force components bi-aial bending moment components and torsional moment component. As is well known the static stability criterion cannot identify the occrrence of some types of dynamic instability phenomena. In these cases however if sway displacement increases large enogh to inflict damages on colmns these colmns will reach an instability state that can be identified by the static stability criterion... Stability Criterion of Thin-walled Steel Colmns nder Bi-directional Horiontal orce Bi-aial Bending oment and Torsional oment The stability criterion of mlti-dimensional elastic-plastic static theory (Hill 1958) generally classifies the ilibrim state of colmns as stable critical and nstable according to whether the nd order of work W given by ation (1) is positive ero or negative respectively. In or previos research (Goto and Ebisawa 010) the stability criterion was shown for the case and bi-aial bending moment where bi-directional horiontal force components ( y) components ( y) act at the top of colmns. In the present case the effect of torsional moment has to be added. As a reslt the nd order of work sed for the stability criterion is epressed as W = ( + y y + θ + y θy + θ)/ (1)

4 denote arbitrary small incremental components of bi-directional horiontal displacement components and rotational components arond 3 orthogonal coordinate aes( ) and denote the where ( y) and ( θ θy θ ) yfrom an ilibrim state. ( y) ( y) reslting increments of bi-directional horiontal force components bi-aial bending moment components and torsional moment component acting at the top of a colmn. Theoretically the stability of ilibrim state has to be eamined by ation (1) for all the possible increments of displacement and rotation components. ractically however it is difficlt to cover all the possible increments. Herein as an acceptable alternative W is eamined following the response history of a pier (Goto et al. 009). It mst be noted that the ero-crossing point of W from positive to negative so evalated yields only a sfficient condition for instability initiation point. However considering that the safety check in the conventional seismic design is made on the basis of displacements or forces calclated following the response history of strctres the proposed evalation method for W may be jstified at least within the framework of the conventional design method. 3. ULTIATE INTERACTION SURACE UNDER BI-DIRECTIONAL HORIZONTAL ORCES BI-AIAL BENDING OENTS AND TORSIONAL OENT 3.1. Definition of Ultimate Interaction Srface As an alternative method to identify the ltimate state of thin-walled steel colmns it will be convenient in practical seismic design to epress the ltimate state in terms of response force components or response displacement components acting at the top of the colmns. This is becase these physical qantities are easily obtained by seismic response analysis. According to or previos research (Goto et al. 009) the ltimate state of colmns epressed by displacement components is strongly inflenced by response history whereas the ltimate state epressed by force components is less inflenced. Therefore athors proposed to se the mechanical qantities sch as seismic force and moment components at the top of colmns to epress the ltimate state of colmns of bridge piers sbjected to bi-directional horiontal seismic accelerations and derived a versatile mlti-dimensional interaction srface defined in terms of these force and bending moment components by employing pshover analysis (Goto and Ebisawa 010). This mlti-dimensional srface is referred to as an ltimate interaction srface. rom nmerical analysis together with shaking table test it was confirmed that colmns ehibit instability behavior after the response force components and moment components nearly toches the ltimate interaction srface (Goto et al. 009; Goto and Ebisawa 010). Considering the accracy and validity of the abovementioned ltimate interaction srface the ltimate interaction srface of circlar thin-walled steel colmns with the effect of torsional moment is herein derived by modifying the eisting ltimate interaction srface that considers the effect of bi-directional horiontal force components and bi-aial bending moment components (Goto and Ebisawa 010). 3.. Ultimate Interaction Srface nder Bi-directional Horiontal orces and Bi-aial Bending oments The eisting ltimate interaction srface of thin-walled circlar colmns nder bi-directional horiontal force and bi-aial bending moment components (Goto and Ebisawa 010) is epressed as + y h y h + = 1 y () 3

5 where( y ) and( y ) are the force and bending moment components at the top of a colmn when the colmn reaches the ltimate state. h is the ivalent height. In the nmerical analysis thin-walled circlar colmns are observed to reach their ltimate state when the bending moment at the height of ( h h ) becomes al to the bending moment capacity of thin-walled cross section nder constant compressive force. and y are the maimm horiontal forces calclated by the nidirectional pshover analyses in the and y directions. If the colmns are fabricated within an allowable geometrical tolerance specified in Japanese design code (Japan Road Association 01b) and y can be considered to be approimately coincident with the maimm horiontal force for ideal piers withot geometrical imperfection. Therefore y is herein assmed. Empirically obtained formlas to calclate h and are shown elsewhere (Goto and Ebisawa 010) Ultimate Interaction Srface nder Bi-directional Horiontal orces Bi-aial Bending oments and Torsional oment Considering the circlar cross section of a colmn the shape of the interaction srface is rotational symmetry in terms of the ais that denotes the magnitde of the torsional moment. Therefore the ltimate interaction srface of a thin-walled circlar colmn nder bi-directional horiontal forces bi-aial bending moments and torsional moment shown in igre 1 (b) can be epressed as α y y + h h + + = 1 and are the bi-directional force bi-aial bending moment and torsional moment components acting at the top of a colmn when the colmn reaches the ltimate state. denotes the maimm torsional moment obtained by monotonically increasing twisting rotation θ arond ais at the top of a colmn nder constant compressive force. In order to stabilie the nmerical analysis to calclate the small geometrical initial imperfection with RR= 1/ 500 shown in igre 3 (c) or (d) is considered. RR=1/500 is specified by Japanese design code as an allowable tolerance (Japan Road Association 01b). α and are the crve-fitting constants. These constants are determined so that the interaction srface best fits the ltimate points obtained by pshover analyses that are carried ot into varios directions as shown in igre. These limit points are identified by the stability criterion ation (1). where ( y ) ( y ) or the ease of epression total ivalent horiontal force components ( y ) = ( + y ) and y = ( y h ) are hereinafter sed. h (3) defined as 4. VERIICATION O ULTIATE INTERACTION SURACE UNDER SEISIC ACCELALATIONS 4.1. Test odels The validity of the ltimate interaction srface given by ation (3) is eamined by compting the ltimate seismic behavior of thin-walled circlar bridge piers nmerically. An inverted L-shaped pier shown in igre 3 is adopted as a bridge pier model so that the seismic torsional moment acts at the top of the colmns in addition to the bi-directional seismic force and bi-aial 4

6 bending moment components. In this model elastic diaphragms with the thickness of 1mm are assmed to be installed to the internal hollow space of the circlar colmn with an al interval of d = 3R. In the pier model the colmn is discretied by for-node shell elements (S4R) while the cross beam with a concentrated mass of eccentric distance e is epressed by rigid beam element (C). Two types of the inverted L-shaped pier models sed in the analysis are smmaried in Table 1. In these models two kinds of eccentricity ratios are adopted. However the radis-to-thickness ratio parameter R t slenderness ratio parameter λ diaphragm interval ratio d/( R ) and aial force ratio y are the same. ost of these qantities are determined referring to the model sed by Gao et al (000). or the ease of obtaining the convergence of soltion two kinds of cross sectional geometrical imperfection modes with magnitde of RR= 1/ 500 illstrated in igre 3(c) and (d) are considered for the respective colmn models. The crve-fitting constants of the ltimate interaction srface epressed by ation (3) are determined for the above colmn model as α = and = To calclate the ltimate seismic behavior of the pier models the geometrically and materially nonlinear dynamic shell analysis with the three-srface cyclic metal plasticity constittive model is carried ot by the general prpose finite element software ABAQUS (007). As inpt bi-directional horiontal seismic acceleration waves variosly factored NS and EW components of JRT observed dring the 1995 Kobe earthqake are simltaneosly applied to the directions of and y aes respectively defined in terms of the inverted L-shaped pier model shown in igre 1. The factored bi-directional waves sed for dynamic response analysis are created by mltiplying the magnification factor C ranging from 0.1 to 1.0. Table 1: Geometrical roperties of ier odels ( a) e Z y y ( b) Z igre 1: Seismic inertial forces acting at the top of inverted L-shaped circlar pier Z Limit point obtained by shover Analysis Loading ath Interaction srface y igre : Loading path to derive the ltimate interaction srface ier model -e1 -e eh R = 1000( mm) ; t = 34( mm) ; h= 850( mm) ; hs = 00( mm) ; ass = 1000( ton) ; RR=± 1/500 ; d = 3R ; = 0.15 ; R = ( R t)( σ E) 3(1 υ ) = ; y λ = σ A = 0.3 y cr Shell element (S4R) ( a) Rigid beam element ass e h s t d d h ( b) () c ( d) R R R t R R igre 3: Analytical model of pier (a) Inverted L-shaped circlar pier (b) Ideal circlar cross section (c and d) Circlar cross section with initial imperfection y 5

7 4.. Validity of Ultimate Interaction Srface nder Bi-directional Seismic Accelerations and moment components( y ) at the top of colmns how colmns of the inverted L-shaped piers reach instability state. or this prpose trajectories of the response components of irst it is investigated in terms of the seismic response inertia force components ( y ) ( / / / ) y are epressed in comparison with the ltimate interaction srface. As an eample the trajectories are illstrated in igre 4 for the colmn of pier model -e nder JRT 0.1 and JRT 1.0. In this figre in order to show the 3D trajectories the projections of the trajectories on + y / / and / / y 1/( α) 1/( α) ( 1 ( / ) ) ( 1 ( / ) ) ( ) 1/( α ) planes are shown in igre 4. In igre 4 the notations sch as = 1 ( / ) ( ) 1/( α ) and y y 1 ( / ) = are sed to simplify the epression. As can be seen from igre 4 when JRT 0.1 wave with small acceleration magnitde is applied the trajectory is located inside the ltimate interaction srface and the colmn is stable. However when JRT 1.0 wave with larger acceleration magnitde is applied the otermost trajectory comes very close to the ltimate interaction srface and proceeds along the interaction srface instead of penetrating the srface straightly. When the trajectory almost toches the ltimate interaction srface the pier becomes nstable afterwards. This behavior is almost the same as that of a colmn sbjected to bi-directional horiontal seismic force components ( y ) (Goto et al. 009). Therefore the validity of the ltimate interaction srface can be confirmed for the present case in addition to the bidirectional where the 3D seismic moment components ( y ) horiontal seismic force components ( y ) acts at the top of the colmn. Ultimate interaction srface y Ultimate interaction srface y y y (a) ier -e nder JRT ath before the first instability point ath after the first instability point aimm response point of fσ/ fσ before first instability aimm response point of fσ/ fσ withot instability irst instability point (b) ier -e nder JRT igre 4: Trajectories of response force components and ltimate interaction srface Second the accracy of the interaction srface is eamined for two types of inverted L-shaped pier models -e1 and -e with the different eccentricity ratios of eh= 0.1 and 0.. These pier models are assmed to have two types of initial geometrical imperfection modes illstrated in - 6

8 igre 3(c) and (d) with magnitde of RR= 1/ 500. The ltimate behaviors of these piers are calclated by applying bi-directional horiontal components of variosly factored JRT waves. agnification factors considered herein are C = and 1.0. The accracy of the ltimate interaction srface (ation (3)) is eamined for the cases when instability occrs by comparing the maimm response points with the ltimate interaction srface. or this prpose the qantity fσ / fσ defined by ation (4) is sed. f y y Σ = Σ f (4) where and are the qantities on the point of the ltimate interaction srface where y the etended response vector ( / / / ) y intersects. f Σ and f Σ are schematically shown in igre 5. or the cases when instability occrs the maimm response points of fσ / fσ before the occrrence of the first instability are marked by filled circles in the space defined in terms of ( / y / / ) coordinates. On the other hand when instability does not occr dring the whole response history the maimm response points of fσ / fσ are similarly marked by empty circles. The 3D distribtion of the aforementioned maimm response points are epressed by the projections of these points on y / / + y plane are shown in igre 6. rom plane and ( ) igre 6 it is observed that all the filled circles are located very close to the interaction srface and the empty circles are inside the interaction srface. Therefore the interaction srface epressed by ation (3) well approimates in a conservative manner the ltimate states of colmns nder the and moment components copling of the seismic response inertia force components ( y) ( y ) at their top. However as can be seen from igre 6 the response torsional moment ratio is less than 0.. In this range the effect of the torsional moment on the ltimate behavior of colmns is too small to verify the accracy of the interaction srface in wider range. Therefore it will be necessary to se inverted L-shaped pier models with larger eccentricity ratios that eceed 0. in order to eamine the accracy of the interaction srface etensively althogh the piers with the eccentricity ratio eh 0.4 are not common in practice. 5. SUAR AND CONCLUSION In the view of the importance to ensre the safety of thin-walled hollow circlar steel colmns nder torsional moment in addition to the bi-directional horiontal seismic force and bi-aial bending moment components a versatile interaction srface is derived by the so-called pshover analysis to specify the ltimate state of thin-walled bridge piers. This ltimate interaction srface is herein epressed in terms of the torsional moment component together with the bi-directional horiontal force components and the bi-aial bending moment components acting at the top of the colmn. The validity and accracy of this ltimate interaction srface is eamined by carrying ot nonlinear dynamic response analysis on inverted L-shaped bridge pier models nder variosly factored bi-directional horiontal seismic acceleration components. As a reslt it is demonstrated that the derived ltimate interaction srface well predicts the ltimate state of the pier models althogh it is a little conservative. However the response torsional moment ratio / that 7

9 acts on the colmn of the inverted L-shaped pier models is a little too small to verify the accracy of the interaction srface in a wider range. Therefore in order to eamine the accracy of the interaction srface more etensively it will be necessary to se inverted L-shaped pier models with etremely larger eccentricity ratios althogh sch piers are not common in practice. igre 5: Schematic eplanation for f Σ and f Σ 0.5 Ultimate interaction srface 1.0 y y Instability occrs Instability does not occr REERENCES igre 6: aimm response points and ltimate interaction srface on y + y planes and ( ) ABAQUS/ Standard 6.7 User s anal (007). Hibbit Karlsson and Sorensen. Inc. Hill R. (1958). A General Theory of Uniqeness and Stability in Elastic lastic Solids. Jornal of the echanics and hysics of Solids 6 pp Gao Shengbin Usami Tstom and Ge Hanbin (000). Eccentrically Loaded Steel Colmns nder Cyclic Ot-of-lane Loading. Jornal of Strctral Engineering. 16(8) pp Goto oshiaki and Ebisawa Takemasa (010). Ultimate State of Thin-Walled Stiffened Sqare Steel Bridge iers Sbjected to Bi-directional Horiontal Seismic orces and Bi-directional Directional Seismic oments. roceeding Ninth acific Strctral Steel Conference Beijing pp Goto oshiaki Jiang Knsheng and Obata akoto (006). Stability and Dctility of Thin-Walled Circlar Steel Colmns nder Cyclic Bidirectional Loading. Jornal of Strctral Engineering. 13(10) pp Goto oshiaki raki asayki and Obata akoto (009). Ultimate State of Thin-Walled Circlar Steel Colmns nder Bidirectional Seismic Accelerations. Jornal of Strctral Engineering. 135(1) pp Japan Road Association (01a). Specifications for Highway Bridges. art V. Seismic Design Tokyo Japan. Japan Road Association (01b). Specifications for Highway Bridges. arts I and II. Common and Steel Bridges Tokyo Japan. (In Japanese) Obata akoto and Goto oshiaki (007). Development of ltidirectional Strctral sedo dynamic test. Jornal of Strctral Engineering. 133(5) pp