BENDING INDUCED VERTICAL OSCILLATIONS DURING SEISMIC RESPONSE OF RC BRIDGE PIERS

Transcription

1 BENDING INDUCED VERTICAL OSCILLATIONS DURING SEISMIC RESPONSE OF RC BRIDGE PIERS Giulio RANZO 1, Marco PETRANGELI And Paolo E PINTO 3 SUMMARY The aer resents a numerical investigation on the behaviour of reinforced concrete bridge iers subjected to horizontal seismic inut. Scoe of the investigations is to quantify the henomenon of bending-induced axial vibrations. The results of a set of analyses conducted on single column bent systems indicate that flexural cracking roduces, in fact, significant axial vibrations. Quantification of the effects related to this henomenon can be determinant for the seismic assessment of existing bridges as well as for the design of new bridges. Likewise, erformance and design forces of bearings and other anti-seismic devices can be estimated with more accuracy, based on the exected level of combined vertical and horizontal acceleration resonse on decks. Shear resisting mechanisms should also be sensitive to these vibrations and shear failure anticiated when a reduction in the axial contribution to the section shear caacity occurs. A tentative equation for the rediction of this flexural-induced vertical acceleration comonent is roosed based on simlified section kinematics and elastic imact analysis. INTRODUCTION The field evidence [Ono et al.,1996] of the damaging effects due to axial vibrations in vertical members of RC structures during ast earthquakes, has attracted the attention of various authors. While the effects of vertical ground motion comonents on buildings and bridges [Paazoglou et al.,1996][elnashai et al.,1997] have been studied, no attention has been dedicated so far to the bending induced vertical accelerations in RC members subjected to inelastic oscillations. This tye of vertical oscillations is indeendent of the vertical ground motion inut and can induce vertical accelerations that are comarable to the horizontal ones [Petrangeli et al.,1997]. Scoe of the resent work is to quantify these bending induced vertical oscillations due to the rocking mechanism of r.c. bridge iers, with articular reference to systems in which the deck is made of multile girders suorted by large ca beams. In this kind of structures, very frequent in Euroean and Jaanese highway networks, these vibrations may have a significant effect on the general structural erformance. The frequency content and magnitude of the vertical motion associated with this effect is analyzed for different structures, with different natural eriods. Consequences on the erformance of bearings is also investigated. A simlified model, based on the cracked section kinematics, is develoed to redict the magnitude of bending-induced axial accelerations. THE ANALYSED STRUCTURES : GEOMETRY AND DIMENSIONING Three different structures, meant to be reresentative of tyical restressed concrete viaducts in seismic regions, have been analysed. The three structures have the same 3m san suerstructure and different ier heights: 6, 1 and 18m resectively. Each analysed structure is suosed to be art of a viaduct made of a sequence of equal 1 1 PhD Student, University of Rome La Saienza, Rome, Italy granzo@dsg.uniroma1.i Assistant Professor, University of Chieti G.D Annunzio, Pescara, Italy 3 Professor of Earthquake Engineering, University of Rome La Saienza, Rome, Italy

2 sans, simly suorted on iers of similar heights. The analysis of the seismic resonse of these structures in the transverse direction is then carried out on a D schematisation, taking into consideration one ier only with two half sans each side. The suerstructure, with a total latform width of 15.7m, is made of a.5m reinforced concrete slab connecting four restressed concrete girders as shown in Fig.1. This deck configuration requires a 11.5m wide ca beam, in order to seat four bearings with a centre to centre distance of 3.5m. The weight of one san has been assumed equal to 6kN, therefore a vertical load of 15kN acts on each bearing suort. An additional weight of 6 kn has been considered to account for the ca beam. Dimensioning of the ier cross section has been carried out so as to obtain a normalised axial load ' P f c Ag ' =.1 under self weight alone, as tyical for this kind of structures ( f c is the unconfined concrete comression strength and A g is the area of the gross section). The flexural caacity of the ier cross section, reflecting the actual situation of most existing viaducts, has been 15.7 dimensioned according to allowable stress criteria. For each structure, the design moment is comuted based on a constant resonse sectrum of.1g. The required flexural caacities are therefore roortional 1.8 to the ier height since the total mass is 1.7 roughly the same for the three cases; the base bending moments are comuted on a cantilever scheme, neglecting the influence of.5 deck torsional inertia and ca beam flexibility. The same hollow cross-section has been 1. adoted in the three cases with different amount of longitudinal reinforcing steel ρ l. section - 1m Pier 1.5 Table summarises the main design characteristics. The first yield moment M y (bending moment at first yield of longitudinal rebars) and the nominal moment M n (defined here as the bending moment at 5 times yield curvature) are indicated to conventionally define the mechanical roerties. Shear dimensioning of the three structures is omitted since the investigations are focused on axial-flexural couling, however it is assumed that adequate shear reinforcement is rovided to ensure a flexural dominant resonse when large inelastic dislacements occur. Tab. 1 Results of modal analyses Pier Height [m] ρ l [%] M y [knm] mm Pier height [m] mm Figure 1 Geometry of the analysed structures Tab. Pier roerties Mode number Mass % X dir. Mass % Y dir. Before analysing the nonlinear behaviour of these structures it is interesting to see the results of the modal analysis. Natural frequencies and articiating masses in x and y direction and modal shaes are indicated in table and fig.. When these structures are modelled with realistic flexibility for the ca beam and both vertical and rotational masses are included to account for the vertical loads of the suerstructure acting on the bearing suorts, higher T [sec.]

3 modes significantly influence the global behaviour. Esecially in the case of the short ier, a significant ercentage of horizontal modal mass is found in the second mode, which is of the double bending tye. Concrete cracking will therefore take lace in the to and bottom sections, ossibly increasing the hammering effect at bending reversal. In the 1 and 18 metre iers instead, Mode 1 T=.583s Mode T=.1s Figure Modal shaes Mode 3 T=.7s the deck rotational inertia is less significant when comared to the ier flexibility. The ier deforms mainly in simle bending with concrete cracking located at ier base only. THE NUMERICAL MODELS FOR NON-LINEAR TIME-HISTORY ANALYSES The three structures have been modelled by using a flexibility-based fibre beam element develoed by the authors [Petrangeli et al.,1999]. Each ier has been modelled using two fibre beam elements with three, four and five integration Gauss oints (monitoring sections) for the 6, 1 and 18 metres iers resectively. The number of integration oints has been selected in order to attain the same numerical recision in integrating the longitudinal strain field in the three structures, while maintaining the same tributary length to each integration oint. The ier ca has been modelled using linear elastic elements with equivalent mechanical roerties. Constitutive models for concrete and reinforcing steel adot state-of-the-art uniaxial stress-strain relationshis based on the work of Mander [Mander et al.,1988] and [Menegotto et al.,1977] resectively (see also fig. 5 and 6). In the concrete model, a crack-bridging branch has been introduced, roviding a smooth transition between the tensile and the comression branches. This feature was required in order to avoid an overestimation of the imulsive comonent of vertical acceleration at crack closure as a result of the abrut transition between the zero stiffness, zero stress cracked state and the reloading branches to comression. Mechanical roerties of the steel have been assumed as follows: yield strength = 4 MPa, ultimate strength = 57 MPa, Young s modulus = MPa, ultimate strain =.1. Mechanical roerties of the concrete are: unconfined strength = 35 MPa, confined strength = 4 MPa, strain at ultimate stress =.35, Young s modulus = 3 MPa, tensile strength =.5 MPa, fracture energy =.1 kn/m. The deck horizontal mass (6t) has been laced in one node only (as indicated in fig.3) to avoid axial (horizontal) vibrations in the ier ca beam; vertical masses have been laced instead at each beam suort (15t each) and at ier to (6t). The masses of the suerstructure are rigidly connected to 1/4 Deck Mass the ca beam. Total Deck Mass The first mode natural frequencies comuted with modal analysis have been used to quantify the viscous comonent of the structural daming. A viscous daming, in addition to the hysteretic one, has been considered in fact by means of a mass Elastic Ca Beam Ca Beam Mass roortional daming factor C, where, for elastic systems, C = ξωm with m the mass, ξ the ercentage of critical daming and ω H Nonlinear fibre elements the circular frequency. A value of 3% of Vertical Mass critical daming has been assumed in our Horizontal Mass case to be reresentative of all viscous Fixed Base daming comonents acting within the elastic structural resonse. Figure 3 Numerical model RESULTS OF NON-LINEAR ANALYSES A set of non-linear time-history analyses using the general urose F.E. code FIBER has been erformed using an accelerogram comatible with the EC8 resonse sectrum with PGA=.35g as horizontal ground motion 3

4 inut. The vertical comonent has been urosely ignored in a first stage, while it has been included in a second set of analyses to evaluate the couling effect on the ier axial resonse. Under the imosed horizontal ground motion, large inelastic deformations occur in the three structures. In all cases a lastic hinge forms at ier base, where longitudinal reinforcing bars reach (for the 6m ier) a imum strain of aroximately.%. Maximum base shears are 5kN, 15kN and 98kN for the 6m, 1m and 18m ier resectively. Moment curvature cycles at ier base are lotted in fig.4. In the same grahs the corresonding axial force time history is also reorted. The largest ductilities are found for the 6m ier with a curvature ductility µ φ = 8 9. For this ier, the inflection oint is located at.54h (with H full height of the ier), while for Moment (knm) Moment (knm) Moment (knm) base section Pier 1 (6m) Curvature (1/ m) base section Pier (1m) Curvature (1/ m) base section Ax. f orce (kn ) -6-8 time (sec.) Ax. f orce (kn ) -6-8 time ( sec.) Ax. f orce (kn ) -6 Pier 3 (18m) -8 time (sec.) Curvature (1/ m) the 1m and 18m is.66h and.69h resectively. Note that the imum axial force fluctuations are found for the 6m ier (+58% of additional comression and -35% in axial force reduction). None of the iers exerienced steel yielding in the to section below the ier ca beam. Concrete and steel stress-strain histories for the base section of the 6m ier are lotted in fig. 5 and 6. In the 6m ier a global dislacement ductility of about 5. is reached, comared to.5 and. in the 1m and 18m ier resectively. This remarkable difference in global damage is due to the inaroriate strength rovided by the allowable stress design criterion and to the flat design sectrum adoted (.1g). However, this result reflects the actual situation on existing viaducts where squat iers tend to have light longitudinal reinforcement ratios. Maximum dislacement drifts are in a range of.75% to 1% of the ier height. Maximum vertical dislacements at external bearing locations are insensitive to the ier height and are always around.5m. The imum resonse of the three structures is summarised in fig.7 (the 5% daming elastic resonse sectrum of the accelerogram used in the analyses is also reorted). Maximum accelerations at bearing locations are indicated for each structure as a function of their fundamental elastic flexural eriod. It can be seen that in the roosed examles, the deck horizontal imum acceleration does not vary significantly with ier height while the vertical acceleration does, due to varying axial/flexural eriod ratio as well as ca beam width/ier height ratio. As anticiated before, vertical acceleration Stress (kpa) extreme concrete fiber Pier 1 (6m) Stress (kpa) extreme steel fiber Pier 1 (6m) -3.E-3 -.E-3-1.E-3.E+ 1.E-3.E-3 3.E-3 Strain Figure 5 - Concrete -5.E-3.E+ 5.E-3 1.E- 1.5E-.E-.5E- Strain Figure 6- Steel 4

5 resonse is articularly high for the squat ier, where a eak value of.9g is found at external bearing location. Generally, bending-induced vertical accelerations decrease with increasing ier height as also confirmed by other analyses. Vertical acceleration of the outer bearings includes in fact also a geometric comonent due to the rotational acceleration of the ier ca beam itself. This comonent obviously decreases with decreasing ca beam width/ier height ratio. The distribution of the vertical acceleration along the ca beam from ier to to the external bearing location can be easily derived from the grahs of fig.7. The lots of fig.8 show the acceleration resonse sectra (with 5% daming) of the horizontal and vertical deck motion recorded at each bearing location and at ier to. The ground motion sectrum is lotted for comarison. These sectra rovide an idea of the frequency content and magnitude of the structural resonse. Their values for T= corresond to the imum accelerations lotted in fig.7 which are the ones to be used to evaluate the imum vertical and horizontal force transmitted between deck and iers. In fig.7, the eak below. sec. at the outer bearing location is clearly due to the selective amlification of the first vertical vibration mode of the ier (mode 3 in tab). It can be seen that the resonse sectra of the vertical accelerations tend to be more concentrated in a narrow band of frequencies as the ier flexural eriod increases, while those of horizontal Rigid Deck Model accelerations tend instead to 1 1. have a constant level of Deck Horizontal acc. (A) Pier to vert. acc. (D) resonse (i.e. frequency Internal bear. vert. acc. (B) Lateral bear. vert. acc. ( C) 1 1. indeendent) around.75g. Acceleration (m/ s) Resonse sectrum of the imosed ground motion 6m Pier 1m Pier 18m Pier M ain Flexural Period (sec.) Figure 7 Maximum resonse D A B C It can be noted from the results discussed above that on external bearings vertical acceleration resonse is equal or greater (u to a factor of for the 6m ier) than the horizontal one. With the ratio between vertical and horizontal force on bearings (R v /H) falling to such low values, unseating henomena are likely to occur. The most significant examles of these low values occurred during the analysis are reorted in table 3 for both internal and external bearing in the 6m ier. Note that the vertical reaction under self weight alone is equal to 15kN. Resonse at 8. sec. shows instead a case where the imum vertical reaction of the bearings is nearly twice the static one. During the analysis the external bearing exeriences a minimum ratio between vertical reaction R v and shear force H equal to.65, while the internal bearing has a minimum value of.6. Tab. 3 - Forces on bearings during earthquake resonse External Bearing Internal Bearing Time (sec.) Shear R v (kn) Shear R v (kn) If the effect of the vertical ground motion comonent is now introduced, a quantification of the relative imortance of the two different sources of axial vibrations can be made. The analyses carried out above were reeated again with inclusion of a vertical motion with eak ground acceleration equal to /3 of horizontal eak acceleration. This ground motion comonent is still comatible with the EC8 resonse sectrum and has the same duration and starting time ste of that of horizontal motion. In these analyses the two sources of axial vibrations are therefore taken into account and their effects aear combined. The values are in all cases larger than the corresonding ones in fig.7, where the ground vertical acceleration is absent, but the order of magnitude does not change. In other words, from the sot cases examined it would seem that the redominant contribution to the vertical resonse accelerations comes from the rocking mechanism, not from the vertical acceleration inut. SIMPLE MECHANICAL MODEL FOR AXIAL VIBRATIONS Acceleration (g) An attemt to establish a closed form aroximate relationshi between these axial vibrations and the ier 5

6 flexural resonse will be resented herein. The main assumtion is that cracked sections can be treated as rigid bodies during their motion induced by flexural resonse. In this idealisation the sections rotate about a oint that coincides with the osition of the neutral axis. The imact of the sections during bending reversal will be assumed elastic with initial conditions (i.e. values of velocity and acceleration) found from the rigid body motion assumtion. Assuming that lane sections remain lane the following relation holds between the curvature χ and axial elongation ε for a RC section: 1 ε = χ k d (1) where k is a scalar arameter ( k 1) defining the neutral axis deth (1-k)d. Let us now assume the flexural resonse of a generic section be 6m Pier described by a simle sinusoidal.75 Deck horiz. motion (A) Internal bear. vert. motion (B) function as follows:.5 Lateral bear. vert. motion (C).5 = π Pier to vert. motion (D) D A B C t χ( t ) χ sin (). Imosed ground motion T f with T f being the redominant flexural 1.5 eriod of the ier. The section axial 1. deformation, according to (1), can.75 therefore be written: ε ( t) = χ πt 1 sin k d Tf (3) where the absolute value of the sinusoidal function is taken since the axial elongation is always ositive. In order to obtain simle exressions for the velocity and the acceleration of the axial strain (3) we assume that the osition of neutral axis is fixed (i.e. k is constant), even though with increasing curvature the neutral axis tends to shift outwards (i.e. k increases). Axial dislacement, axial velocity and axial acceleration as a function of time have been qualitatively lotted in fig.9. The velocity is discontinuous for t = nτ f / (with n=1,...). In these characteristic oints, the section is subjected to a vertical imulse which reverses the dislacement direction, changing sign to the axial velocity. These oints corresond to the crack closure and the sudden shift of the neutral axis from one side of the section to the other (as deicted in fig.9). These imulses are the main cause of the vertical oscillations observed in the analyses. Outside these oints, the section is still subjected to a vertical acceleration as the result of the flexural resonse. This comonent of the acceleration, smaller than the imulsive one at bending reversal, is found as the second derivative of (3). Its imum value is : Acceleration (g) Acceleration (g) Acceleration (g) Deck horiz. motion (A) Period (sec) Lateral bear. vert. motion (C) Pier to vert. motion (D) Imosed ground motion 1m Pier Deck horiz. motion (A) Period (sec) Lateral bear. vert. motion (C) Pier to vert. motion (D) Internal bear. vert. motion (B) Imosed ground motion Internal bear. vert. motion (B) D A B C 18m Pier Period (sec) D A B C Figure 8 Accelration Resonse sectra on the deck 6

7 d ε [ a ] T f 4π 1 = = χ k d t n dt T T f f t n This comonent of the vertical acceleration is negligible when comared to that due to the imulse at bending reversal (see eq.(7)). In order to obtain an estimate for the magnitude of this imulse, the assumtion of rigid body motion must be relaxed and the imact at crack closure treated as an elastic rebound. We assume therefore that this elastic imact takes lace in a finite time interval t. With these hyotheses, the δt = T a T f (4) imulse amlitude can be comuted as: Dislacement, Velocity, Acceleration m v = t f dt= t m a dt (5) Simlified section kinematics T f T f 3 T f T f Figure 9 Section kinematics Imulse Dislacement Velocity Acceleration where m is the mass and f the inertia force. The velocity variation v that takes lace during the time interval t can be set according to Fig.9 by comuting the right and left limit of the first derivative of (3) for t nτ f /. The time interval t is tentatively set equal to one half the fundamental axial eriod of the ier-deck system ( t =T a /), so that t corresonds to the comressive semi-cycle of the ier elastic rebound. Therefore, the velocity variation within the secified time interval is : d v = ε π 1 = χ k d dt T t n Tf f If we assume that the imulse has a sinusoidal shae, we obtain from (5): π v π [ a ] k d t n T 4π 1 f = = χ (7) = t Ta T f where the π/ factor is found by integrating the sinusoidal imulse over the time interval t in (5). A simle relation between the ier imum horizontal acceleration resonse and the curvature imum acceleration can be easily obtained by assuming the flexural deformations taking lace in a localised lastic hinge at the column base. In this case we obtain that the imum acceleration at ier to is : a T l H d χ 4π h = β( f ) = c = lc Hχ (8) dt Tf where l c is the lastic hinge length, H the ier height and β(t f ) is the ordinate of the acceleration resonse sectrum for the ier redominant flexural eriod. The simlification associated with this kinematic mechanism is valid for single column bents in general and also for members that tend to oscillate in double bending, because imum curvature at column base is, in most cases, one order of magnitude greater than that at column to, where yielding of steel seldom occurs. Similarly, if the largest axial deformations take lace within the lastic hinge region, as it is for the lastic rotations, the imum value of the imulsive (a v,i ) comonent of the ier vertical acceleration can be written as: d Tf d avi, = lc = ( Tf ) k dt (9) ε 1 β Ta H f t= n T (6) 7

8 A comarison between the values found with eq.(9) and the results of the non-linear analyses for the rigid deck model are resented in the following table 3. Horizontal and vertical accelerations found from time-history analyses are indicated with β h and β v resectively. The flexural eriod of the iers T f has been calculated by using the secant flexural stiffness at imum resonse (average stiffness). The axial eriod T a instead has been comuted using the cracked elastic stiffness and the neutral axis deth evaluated at imum resonse. In this case study, both secant stiffness and neutral axis deth, were available from the results of the non-linear analyses; for design urose instead, they should be calculated by using the assumed structural ductility or the imum exected dislacement, if a dislacement based design aroach is being used. Tab. 4 - Numerical analysis versus Eq.(9) rediction Pier Height 6 β h 4.1 β v 3.96 T a [sec].1 T f [sec].69 (k -1/).7 a vi 3.19 β v /a vi The results from table 4 seem to indicate the soundness of the assumtions used to derive equation (9) and the caability of it to rovide a correct estimate of the magnitude of these axial vibrations. From the cases resented above it seems that little or no amlification of the axial motion is found between ier base and ier to. However, it is imortant to note that the axial inut found with (9) might have been overestimated due to the assumtion of section rigid body motion. Results obtained with eq.(9) are strongly affected by the value assumed for β h (T f ). In our case this value was given by the results of non linear time history analyses ( β h ), whereas in design it must be found from a design sectrum based on the imum exected ductility (i.e. behaviour factor). CONCLUSIONS Although exerimental results are needed to confirm the redictions of the numerical study resented herein, there is no doubt that a significant contribution to the vertical acceleration in RC iers subjected to seismic excitation is due to the rocking mechanism. This contribution is neglected in ordinary design, based on linear modal analysis and resonse sectra. The intensity of this vertical acceleration may in effect be greater than the structural resonse to the vertical comonent of the seismic inut motion. The effect of this additional motion in the vertical direction can be articularly severe on deck bearings, which may exerience the imum horizontal shear forces associated with very low vertical reactions roviding an additional exlanation for the widesread henomenon of bearing failure and deck unseating observed during ast earthquakes. Based on the reliminary investigations resented herein, it seems that equation (9) rovides a reasonable estimate of the vertical accelerations in bridge iers subjected to horizontal seismic inut motion alone. This equation could be used as a starting oint towards the definition of a design formula for the quantification of this additional vertical comonent to be used in the dimensioning of bridges in seismic areas. REFERENCES Elnashai, A.S., Paazoglou, A.J. (1997), Procedure and Sectra for Analysis of RC Structures Subjected to Strong Vertical Earthquake Loads, Journal of Earthquake Engineering Vol.1 n.1, , Imerial College Press. Mander, J.B., Priestley, M.J.N., and Park, R. (1988). "Theoretical Stress-Strain Model for Confined Concrete." Journal Struct. Eng., ASCE, 114(8), Menegotto, M. and Pinto, P.E. (1977). "Slender RC Comressed Members in Biaxial Bending." J.Struct. Engrg, ASCE, 13(3), Ono, K., Kasai, H., Sasagawa, M. (1996), U-down Vibration Effects on Bridge Piers, Secial is sue of Soils and Foundations, Jaanese Geothecnical Society, Paazoglou, A.J., Elnashai, A.S., (1996), Analytical and Field Evidence of the Damaging Effect of Vertical Earthquake Ground Motion, Earthquake Engineering and Structural Dynamics, Vol.5., Petrangeli, M. and Pinto, P.E. (1997). "Finite Element Modelling of the Hanshin Viaduct Failure in Kobe."Proc. of the Second Jaan-Italy Worksho on Seismic Design of Bridges, Rome,Italy. Petrangeli, M,. Pinto, P.E. and Ciami, V. (1999). "A Fibre Beam Element for cyclic bending and shear. Part I and II", J. Mech. Engrg., ASCE, 8