Efficient dynamic analysis of cable-stayed bridges under vehicular movement using space and time adaptivity

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Finite Elements in Analysis and Design 40 (2004) 407 – 424 www.elsevier.com/locate/nel

Ecient dynamic analysis of cable-stayed bridges under vehicular movement using space and time adaptivity Animesh Das, Anjan Dutta∗ , Sudip Talukdar Department of Civil Engineering, Indian Institute of Technology, Guwahati 781039, India Received 13 June 2002; received in revised form 6 January 2003; accepted 9 February 2003

Abstract The e2ects of random road surface roughness on the impact e2ects on cable-stayed bridge due to moving vehicles are investigated. The random road surface roughness is described by a zero-mean stationary Gaussian random process. The bridge is modeled by nite element method as a planar structure. Each moving vehicle is idealized as a single degree-of-freedom lumped mass system, in which a mass is supported by a spring and a dashpot. The impact e2ects of random road surface roughness on a cable-stayed bridge vary a lot, depending on the location and the structural components. The entire computation has been carried out in an adaptive nite element environment. It has been clearly demonstrated that the dynamic amplication factor, a signicant parameter for bridge design can become highly erroneous if the errors due to both space and time discretization are not kept within control. ? 2003 Elsevier B.V. All rights reserved. Keywords: Cable-stayed; Surface roughness; Vehicular impact; Adaptivity; Amplication factor

1. Introduction A cable-stayed bridge is a nonlinear structural system in which the girder is supported elastically at points along its length by inclined cable-stays. The nonlinear behavior is a result of the nonlinear axial force–displacement relationship for the inclined cable-stays due to the sag caused by their own dead weight, the nonlinear axial–bending force interaction relationship for deck and pylon and the large displacement which can occur in the structure under normal design loads. Fleming [1] has carried out a nonlinear static analysis of cable-stayed structure incorporating all those nonlinearities. Karoumi [2] has provided di2erent modeling aspect of cables. A cable can be considered as a bar element with equivalent tangent modulus of elasticity to account for sag e2ect. A cable can be ∗

Corresponding author. Tel.: +91-361-269-0321; fax: +91-361-269-0762. E-mail address: [email protected] (A. Dutta).

0168-874X/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-874X(03)00070-2

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considered as a combination of large numbers of straight beam segment with very small moment of inertia adequately modeling the curved geometry of the cable. Isoparametric element, which include the element curvature are also tried. The last two modeling approaches are found to be superior to the bar element approach and have almost similar performances. Pavement surfaces of roads on bridges have some irregularities, however, carefully they are prepared. It may be attributed to the inaccuracies in casting concrete, assembling pre-cast segments and the subsequent overlaying of water proong membrane and bituminous surfacing. Newly constructed pavement may be poorly nished or may have design features, such as construction joint, thermal expansion joints, etc. Pavements that have been in-service for sometimes often develop localized distresses due to application of heavy wheel load and environmental changes. Moving vehicles exert Guctuating forces on the bridge deck caused by its vibration resulted from surface unevenness. The Guctuating nature of the dynamic wheel load with increase in number of load repetitions may cause pavement degradation and fatigue damage. Many studies have been reported for the analysis of bridge response due to vehicles passing over at constant and variable speed. The development and application of computers opened up new and practically unlimited possibilities for the exact solution for highly statically indeterminate systems. Some of studies dealing with the bridge–vehicle interaction are summarized in Refs. [3–14]. Wang et al. [6], have studied the dynamic behavior of multigirder bridges due to vehicles moving across rough bridge deck. Chatterjee et al. [7] have carried out the dynamic analysis of multispan continuous bridges under a moving vehicular load by considering the interaction between the vehicle and the bridge pavement and the randomness of the pavement surface irregularity. Chatterjee et al. [8] have adopted a continuum approach for determining the vibration of cable-stayed bridges under the passage of moving vehicle. The analysis has been performed in time domain by considering the e2ects of Gexibility of towers, interaction between the vehicle and the bridge pavement, coupling of vertical and torsional motion of the deck due to eccentrically placed vehicles and the randomness of the bridge surface irregularity. Chatterjee et al. [9] have further conducted a similar study as in Ref. [8] by considering a cable suspension bridge. Huang and Wang [10], Wang and Huang [11] have studied dynamic response of a cable-stayed bridge due to moving vehicles across rough bridge deck and the impact on such bridge. They have generated the road surface roughness from the power spectral density (PSD) according to International Organization for Standardization (ISO) specication. Yang and Fonder [12] have studied the dynamic response of cable-stayed bridges under moving loads by taking a linear approximate bridge model; the nonlinear contribution of the stay cables is added by means of an iterative procedure. Caetano et al. [13] have presented an experimental investigation involving the study of the dynamic interaction between the cables and the deck/towers system in cable-stayed bridges. Au et al. [14] have studied the e2ects of random road surface roughness and long-term deGection of prestressed concrete girder and cable-stayed bridges on impact due to moving vehicles. In most of the works, an average road surface prole based on several simulations was obtained and then analyzed for evaluation of impact e2ect. The studies of vehicle-induced vibration on cable-stayed bridge have identied vehicle characteristics, vehicle speed, roadway surface irregularities, the damping characteristics of bridge and vehicle, etc. to be signicant factors a2ecting the dynamic response of such bridges. However, statistical analysis for the impact on such bridges owing to random road surface roughness are still very limited and needs to be addresses further. Moreover, since the impact factors evaluated from such studies bear great signicance from the point of view design of cable-stayed bridge, the evaluation

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of the same should be carried out as accurately as possible. The error occurs in the analysis due to space and time discretization should be kept under control for such accurate evaluation of the bridge response due to vehicle induced vibration. Using nite elements, linear dynamic transient response is usually solved either by mode superposition or by direct integration schemes. Procedures describing adaptive time stepping are available in Refs. [15,16] and also spatial mesh adaptation [17]. Wiberg and Li [18] have proposed a post-processed type of a posteriori estimates in space and also in time when direct integration is used for dynamic response evaluation. It updates the spatial mesh and time step so that the discretization errors are controlled. Wilson and Joo [19] have arrived at the nal mesh using Ritz vector as basis of transformation. In their investigation, the authors have made use of modal participation and amplication factor and obtained error estimates based on Babuska’s criterion using amplied Ritz modes. Cook and Avrashi [20] have discussed the procedure for estimating the discretization error of the nite element modeling as applied to the calculation of natural frequency of vibrations. Meshes are obtained corresponding to each mode. Dutta and Ramakrishnan [21] proposed a measure for discretization, which is a logical extension of Zienkiewicz and Zhu [22] error criterion by involving time integration to consider the variation of response with time. Dutta [23] has adopted the adaptive strategy given by Dutta and Ramakrishnan [21] for arriving at an optimal mesh for a prescribed domain discretization error limit and integrated with an adaptive time stepping procedure proposed by Zienkiewicz and Xie [15] with some modication. In the present paper, the random road surface roughness is simulated by zero-mean stationary Gaussian random process. For each sample of random surface prole, the dynamic response of the bridge is rst analyzed by nite element method. Statistical characteristics of dynamic response are then obtained based on the results of twenty simulations. A three span cable-stayed bridge is chosen for the study. Since nite element method has been utilized for response evaluation, the discretization errors in space and time may lead to erroneous result. Error estimation and adaptivity in both space and time has been adopted according to the procedure outlined by Dutta [23].

2. Generation of random road surface roughness For evaluation of the response of vehicle traveling over uneven road, it is customary to assume road surface as homogeneous. This signies that its probability structure is independent of a shift of the parametric origin. It is further assumed that probability density fuction is Gaussian and the datum plane has been chosen so that mean elevation is zero. The stochastic process can be simulated by way of the following series [14]: h(x) =

N


k cos(2!k x + ’k );

(1)

k=1

where k is the amplitude of the cosine wave, !k , the frequency within the interval [!l ; !u ] in which the power spectral density function is dened, ’k , the random phase angle with uniform probability distribution in the interval [0; 2]. x is the global coordinate measured from the left end of the bridge and N , the total number of terms used to build up the road surface roughness. The parameters k

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A. Das et al. / Finite Elements in Analysis and Design 40 (2004) 407 – 424 Table 1 Road surface classication Road surface condition Very good Good Average Poor Very poor

R (m2 =(m=cycle)) R 6 0:24 × 10−6 0:24 × 10−6 ¡ R 6 1:0 × 10−6 1:0 × 10−6 ¡ R 6 4:0 × 10−6 4:0 × 10−6 ¡ R 6 16:0 × 10−6 R ¿ 16:0 × 10−6

and !k are computed, respectively, by k2 = 4Sr (!k )S!; 

1 !k = !l + k − 2

(2) 

S! = (!u − !l )=N

S!;

k = 1; 2; : : : ; N;

(3) (4)

in which Sr (!k ) is the power spectral density function (in m3 =cycle) and !l and !u are the lower and upper cut-o2 spatial frequency (in cycle/m), respectively. The power spectral density function Sr (!k ) can be expressed in terms of the spatial frequency !k , of the road roughness as [14]  − ! R k for !l ¡ !k ¡ !u ; (5) Sr (!k ) = 0 elsewhere; where the parameter R is a spectral roughness coecient in m2 /(m/cycle). The exponent  is taken as 1.94. The road surface condition may be classied into ve classes according to ISO specication in terms of coecient R as shown in Table 1 [14]. 3. Structural stiness The bridge is analyzed by nite element method. Sti2ness of the deck and pylons are evaluated by considering the element, as beam-column element. The nonlinear behavior of these members in the structure, due to the interaction of large bending and axial deformations has been considered by introducing the concept of stability function [1]. All cables have been considered as a combination of a straight beam segment, of very small moment of inertia adequately modeling the curved geometry of the cable [2]. The total structural sti2ness matrix can now be computed for any deformed state of the structure by transforming the local coordinate member sti2ness matrices, into global structure coordinate system and generating the overall sti2ness matrix of the structure following standard assembly procedure. Since the local member sti2ness matrices are not constants, but depend upon the deformation state of the member, the structure sti2ness matrix will change continuously as the structure deforms. It is

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therefore necessary to recompute the structure sti2ness matrices for each deformed position of the structure, which is considered during the analysis. Thus due to the large displacements, which occurs for a structure like cable-stayed bridge, it will be necessary to continuously modify the geometry of the structure as it deforms.

4. Governing equations of motion To identify the important parameters that govern the response, the bridge and the vehicle are both represented by highly idealized models. An idealized two-dimensional bridge model is considered for this study (Fig. 1). The damping e2ect of the bridge is taken into account by assuming Rayleigh damping. The damping ratio for the bridge is taken as 1%. In the present study, a simple vehicle model is used. It consists of a single mass supported by a spring and a dashpot (Fig. 2). The vehicle has a point contact with the bridge deck and maintains that contact as it moves along the deck. For a sprung mass system moving along the bridge deck, two equations of motion have to be solved simultaneously. The rst equation represents the dynamic equilibrium for the bridge. The second equation is for the dynamic equilibrium of the sprung mass system. The interaction force between the pavement and the sprung mass system depends upon the deck displacement. Hence the two equations are coupled and need to be solved simultaneously. The equations of motions for the bridge are Mb uUb + Cb u˙b + Kb ub = Fbv :

(6)

The equation of motion for the vehicle is mv uU v + Fvb = 0;

(7)

where ub =uv are the DOF associated with the bridge/vehicle, respectively, Mb =mv , Cb =cv , Kb =kv are the mass, damping and sti2ness for the bridge/vehicle, Fbv =Fvb are the movement dependent forces acting on the bridge/vehicle. The equilibrium imposes the equality of the interaction forces at the same contact point. Fbv = Fvb at any instant Fbv = cv u˙t + kv ut

(8)

in which ut is the relative displacement between the tire and the bridge at a location of vehicle ˙ i − u˙ bi , where corresponding to ith DOF of bridge = (uv − h(x)i − ubi ), u˙ t = u˙ v − h(x) ˙ i = dh(x)i = dh(x)i d x = dh(x)i v; h(x) dt d x dt dx in which v is the vehicle velocity. x is the position of vehicle measured from the left end of the bridge, uv is the vertical displacement of the tire, h(x)i the road surface roughness under the tire and ubi the bridge vertical displacement under the tire corresponding to ith DOF of bridge.

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Fig. 1. A typical modied fan-shaped cable stayed bridge geometry.

Fig. 2. Idealized interaction system with exaggerated surface irregularity.

The coupled sets of equations of motions are solved by using Newmark- with predictor–corrector algorithm [24]. 5. Dynamic magni%cation factor In the presentation of results, the dynamic amplication factor (DAF) for displacement Ddj (xi ) for the section at xi corresponding to the jth simulation of random road surface roughness can be dened as Ddj (xi ) = jdm (xi )=sm (xi )

(9)

in terms of the maximum dynamic displacement jdm (xi ) and the maximum static displacement sm (xi ) of the section at xi due to the moving vehicles calculated for the jth simulation of random road surface roughness. Similarly, the DAF for bending moment and cable tension are evaluated. It is noted that the design is often governed by the total dynamic response, and therefore, the DAFs are normally calculated at locations where the static response themselves are critical. In order

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to evaluate DAF with condence, adaptive analysis is introduced, where errors in both space and time discretization are kept within prescribed limit. 6. Adaptive analysis 6.1. Space discretization error: Finite element stresses are calculated as  = DBz:

(10)

The approximate solution  containing discretization errors di2ers from the accurate solution ∗ . A post-processed value of stress ∗ , which is continuous have been considered as more accurate solution than the nite element stresses . The di2erence is the error e Thus, e = ∗ − :

(11)

The complete procedure of discretization error measure is given in Ref. [21]. However for completeness, the algorithm is given below. Algorithm for domain discretization error: (1) Solve MzU + Kz = q(t). (1.1) Using Newmark- method calculate dynamicresponse  for t[0; T ]  ˙ U + z(t U + St) , z(t + St) = z(t) + z(t)St + St 2 12 −  z(t) ˙ + St) = z(t) ˙ + St{(1 − )z(t) U + z(t U + St)}. z(t Rearranging the rst equation for displacement at (t + St), we can get   1 1 1 ˙ − U U + St) = z(t) − 1 z(t), z(t (z(t + St) − z(t)) − St 2 St 2 where  = 14 and  = 12 (1.2) for e = 1, n elements { Retrieve ze(t) from z(t) for j = 1, number of Gaussian point ( j) {Calculate e(t) = DB( j) ze(t) where D and B are usual elasticity and strain displacement matrices, respectively. ∗( j) Calculate smooth stress e(t) ( j) e(t) = DB( j) ze(t) Energy norm for error “e” at element level

1=2 ( j) T −1 ( j) ee(t) = " (ee(t) ) De ee(t) d" Energy norm for FEM solution at element level

1=2 ( j) T −1 ( j) ) De e(t) d" u R e(t) = " (e(t) }

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for the whole structure e(t) =



e2e(t)

1=2

1=2 u R (t) = u R 2e(t) } } (2) Overall domain discretization error

T ee(t) dt eav = 0 T

T u(t) dt uav = 0 T u(t) = [u R 2(t) + e2(t) ]1=2 Domain discretization error eav × 100% #= uav (3) Discretization error at element level For i = 1, n elements ei(t) {Error at element level, $i = eR m 1=2

u R 2(t) + e2(t) where eR m = #R n and #R is the permissible domain discretization error

T $i dt Time averaged error at element level, $i
= 0 T If # 6 #R and If $i
¡ 1 for i = 1, n STOP hi Else New element size, hinew = old% ($i
) Where % = 1=p for no singularity and % = 1=' for element close to singularity (p is the order of dening polynomial and ' is the strength of singularity) Endif } Go to step 1 6.2. Time discretization error In order to control the time discretization error in the time marching scheme, some means of estimating the errors should be introduced so that the steps can be suitably adjusted. The adaptive time stepping is aimed at maintaining largest possible step size while keeping the accuracy within the prescribed limit. In this work, the error measure proposed by Zienkiewicz and Xie [15] is adopted with some modication. The total time domain “T ” is divided into a nite number (nT ) of sub-domains. A uniform value of ‘St’ is maintained in a sub-domain and the time discretization

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errors are calculated. Time integration of errors is carried out over individual sub-domain interval to arrive at a new value of step size for that sub-domain. The local error in time is estimated as   1 2 (zUn+1 − zUn ); (12) e = St  − 6 where zUn+1 is acceleration vector at (n + 1)th time step zUn is acceleration vector at nth time step and  = 14 . The L2 norm of the local error is #t = e = St 2 ( − 16 )zUn+1 − zUn :

(13)

Usually, it is inconvenient to specify an absolute error tolerance. To measure the relative error, it may be dened as #R =

e ; (u)max

(14)

where (u)max is the maximum value of the corresponding norm of the displacement solution recorded during the computation. Time integration of error over sub-domain leads to #Ri =

eavei ; (u)max

where

Ti eavei =

0

e dt Ti

(15)

for i = 1; nT

Ti is the time interval of sub-domain ‘i’. In this section, the error #Ri in a sub-domain ‘i’ is used to calculate new step size for the next iteration so that the error is roughly equal to the prescribed tolerance. With this tolerance given as #Rt , an upper limit (1 #Rt ) and lower limit 2 #Rt is also specied, where the parameter 1 and 2 are in the ranges of 0 6 1 6 1 and 2 ¿ 1. When the error #Ri exceeds the upper limit the step size needs to be reduced and the new step size in a sub-domain ‘i’ may be predicted as  1=3 #Rt Stnew = Stold : (16) #Ri Similarly, if the error #Ri is smaller than lower limit, then the step size may be increased using the above expression. Hence the overall strategy can be written as below. Starting with a coarse mesh and a value of “St” for every time sub-domain, overall domain discretization error and discretization errors at element level are calculated and time discretization errors are calculated for every time sub-domain. Mesh is then adaptively rened based on the space discretization error values at element level and new values of “St” are obtained for every time sub-domain. Iteration is carried out until the mesh is an optimal one and time discretization errors in all the sub-domains are within the prescribed limit.

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7. Numerical simulations The moving vehicle, which has been modeled as single DOF dynamic system for the present study consists of a sprung mass of 31; 800 kg supported by a spring of sti2ness 9:12 × 106 N=m and a dashpot of damping coecient of 8:6 × 104 N s=m. A typical modied fan shaped cable-stayed bridge with a main span of 364 m as shown in Fig. 1 is modeled as a two-dimensional structural system. It may be noted that the bridge deck is hinged supported on Tower 1 but roller supported on Tower 2. Support connections are shown in details in Fig. 3. The relevant properties of the bridge deck and towers are given in Table 2. Young’s modulus, E and mass density, * of the stay cables are equal to 165 GPa and 9440 kg=m3 , respectively. Cross-sectional areas for the stay cables are given in Table 3. The damping e2ect of the bridge is taken into account by assuming Rayleigh damping. The damping ratio is taken as 0.01. In this study, R is taken as 0:12 × 10−6 , 3:0 × 10−6 and 15:0 × 10−6 m3 =(m=cycle) according to International Organization for Standardization (ISO) specications for the classes of very good, average and poor roads, respectively (Table 1). Twenty proles of random road surface roughness are generated for each type of road using the following parameters: !l = 0:01 cycles=m and !u = 3:0 cycles=m. The cut-o2 spatial frequencies are chosen in view of the practical size of a tire. N is taken as 1500. A typical vertical highway surface prole of average road surface is shown in Fig. 4. Dynamic responses are evaluated for each of the simulations corresponding to a particular vehicle velocity. The cable-stayed bridge has been discretized in nite element method by 26 numbers of beamcolumn elements for the bridge deck, 11 beam-column elements for each of the pylons and 3 beam elements for each of the cables. A time step of 0:483 s is chosen for the analysis. Time histories for some of the critical components are shown in Figs. 5–8 for a vehicle velocity of 22 m=s.

Fig. 3. Connection details between bridge deck and lower part of bridge tower.

Table 2 Properties of deck and towers of a cable-stayed bridge Part of structure

A (m2 )

I (m4 )

E (Mpa)

* (kg=m3 )

Bridge deck Bridge tower—above deck Bridge tower—below deck

6.00 14.20 35.80

4.70 30.00 40.0

31,600 31,600 31,600

2550 2550 2550

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Table 3 Cross-sectional area (A) of stay cables of a cable-stayed bridge Cable no.

A (m2 )

6 and 19 7 and 18 5 and 20 8 and 17 4 and 21 9 and 16 3 and 22 2 and 23 10 and 15 11 and 14 12 and 13 1 and 24

0.011 0.013 0.014 0.016 0.017 0.019 0.020 0.023 0.023 0.026 0.029 0.060

Fig. 4. A typical vertical highway surface prole of average road.

In order to assess the validity of the results, error estimation in both space and time is carried out. The error for space discretization is obtained as 19.5% and the error for time discretization has been obtained as 9.18%. Adaptive analysis has now been carried out for target accuracy in space as 5.0% and error limit in time as 0.1%. The rened discretized bridge structure contains 130 beam-column elements for deck, 21 elements for each of the pylons and 10 beam elements for each of the cables. A rened time step of 0:0483 s is used for the analysis for the vehicle velocity of 22 m=s. Time histories are obtained again for the rened analysis (Figs. 5–8) and plotted over the time histories obtained earlier corresponding to the initial analysis with both coarse mesh and time step. DAF are calculated corresponding to both the initial and rened analysis. The DAFs corresponding to the initial analysis with coarse mesh and coarse time step are compared with the DAFs obtained for

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Fig. 5. Comparison of vertical deGections at middle of the central span of bridge deck.

Fig. 6. Comparison of horizontal deGections at top of the Tower 2.

rened analysis (Figs. 9–12) for various structural components for the range of velocities considered with average road surface. Figs. 9 and 10 shows the DAFs for vertical displacement at middle of the central span of the bridge deck and DAFs for the horizontal displacement at top of the Tower 2, respectively. Fig. 11 shows the DAFs for the bending moment at base of the Tower 1. Fig. 12

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419

Fig. 7. Comparison of bending moments at the base of Tower 1.

Fig. 8. Comparison of axial force of Cable 18.

shows the DAFs for axial force of the Cable 18. The comparison of time histories as shown in Figs. 5–8 and the comparison of DAF as shown in Figs. 9–12 clearly indicate that unless the error in space and time are controlled, the response and DAF, which plays a signicant role in bridge design could be highly erroneous. The percentage change in DAF is found to as high as 5.6%

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Fig. 9. Variation of DAFs for displacements at middle of the central span of bridge deck for initial and rened analysis.

Fig. 10. Variation of DAFs for horizontal displacement at top of the Tower 2 for initial and rened analysis.

for vertical displacement at middle of the central span of the bridge deck, 24.2% for horizontal displacement at top of the Tower 2, 12.5% for bending moment at base of the Tower 1 and 13.2% for axial force of Cable 1 when the error in space and time discretization for the initial analysis is 19.5% and 9.18%, respectively, and the rened analysis is carried out for a target accuracy of 5% in space and 0.1% in time. Detailed studies are carried out by varying velocities from 14 to 38 m/s and by considering three types of road surfaces viz. very good, average and poor. DAF values are calculated while the space and time discretization errors are kept within a prescribed limit of 5% and 0.1%, respectively. Accurate DAFs for the responses considered above are shown in Figs. 13–16. Figs. 13 and 14 shows accurate DAFs for vertical displacement at middle of the central span of the bridge deck and DAFs for the horizontal displacement at top of the Tower 2, respectively. DAF values for vertical displacement at the middle of the central span of the bridge deck for very good surface are expectedly the lowest and substantially di2erent than that for poor road surface. However,

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Fig. 11. Variation of DAFs for bending moment at the base of Tower 1 for initial and rened analysis.

Fig. 12. Variation of DAFs for axial force of Cable 18 initial and rened analysis.

not much variation in the di2erence of DAF for di2erent surface types with velocities are observed, whereas quite noticeable changes in the di2erence in DAF for di2erent surfaces with velocities are observed for horizontal displacement at top of the Tower 2. Fig. 15 shows the DAFs for the bending moment at base of the Tower 1 and Fig. 16 shows the DAFs for axial force of the Cable 18. The values of DAF obtained using the adaptive nite element analysis module are reliable enough to be considered as bridge design criteria. The coecients of variations are calculated for the DAF for the structural components considered for study. It may be noted that the coecients of variations are rather low. The coecients of variations for the DAF for vertical displacement at middle of the central span range from 4.1% to 4.6% and those for the horizontal displacement at top of the Tower 2 range from 4.9% to 6.2%. Similarly for bending moment at base of the Tower 1 and for axial force of Cable 18, the coecients of variation range from 6.2% to 12.6% and 0.7% to 4.1%, respectively.

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Fig. 13. Variation of accurate DAFs for displacements at middle of the central span of bridge deck.

Fig. 14. Variation of accurate DAFs for horizontal displacement at the top of the Tower 2.

It is thus observed that DAF values for di2erent structural components can vary quite signicantly with velocities and with surface types. The values of DAF are quite important from design point of view and hence reliable, accurate computation of the same is a necessity. Thus error estimation and adaptivity should be integrated within the analysis in order to arrive at a reliable and accurate value of response parameter. 8. Conclusion The e2ect of random road surface roughness on the impact of cable-stayed bridge have been investigated and found to be signicant for some structural components. The analysis has been

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Fig. 15. Variation of accurate DAFs for bending moment at the base of Tower 1.

Fig. 16. Variation of accurate DAFs for axial force of Cable 18.

carried out along with error estimation and adaptive renement in both space and time. Thus, the reliability of the result is ensured. The integrated module of adaptive nite element analysis has been observed to perform well and the error converges to the desired level of accuracy with renement in both space and time. References [1] J.F. Fleming, Nonlinear static analysis of cable-stayed bridge structures, Comput. Struct. 10 (1979) 621–635. [2] R. Karoumi, Some modeling aspects in the nonlinear nite element analysis of cable-supported bridges, Comput. Struct. 71 (1999) 397–412. [3] J. Hino, T. Yoshimura, K. Konishi, A nite element method prediction of the vibration of a bridge subjected to a moving vehicle load, J. Sound Vibration 96 (1) (1984) 45–53.

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