Dynamic Response of Orthotrpic Curved Bridge Deck

8/12/2019 Dynamic Response of Orthotrpic Curved Bridge Deck

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Compur trs & Struct ures Vol IR. No I. PP. 27-32. 1984 0045-7949184 3.00 t 00Punted in Great Britam Pergamon Prers Lfd

DYN MICRESPONSEOFORTHOTROPICCURVED BRIDGE DECKS DUE TO

MOVING LO DS

S. s. EYtCivil EngineeringDepartment, ndian Institute of Technology,Kharagpur-721302, ndiaand

N. BALASUBRAMANIAN~Military Engineering Services, India

Received 15 March 1982; received for publication 21 October 1982)Abstract-The dynamic response of horizontally curved bridge decks simply supported along the radial edgesunder the action of the moving vehicle is investigated. The bridge deck is idealised as a number of finite strips withorthotropic elastic properties. The stiffness and mass matrix of an individual element were derived using ahomogeneous differential equation of an orthotropic plate in polar co-ordinates. The vehicle is idealized as a sprungmass moving at a constant speed in a circular path parallel to the central line of the bridge. The unsprung mass ofthe vehicle is assumed to be always in contact with the bridge surface during its motion. Viscous damping is takeninto account for both bridge and vehicle. Dynamic deflections and moments are presented for the mid-point of thebridge deck and the values have been compared with the available analytical solution.

damping matrixNOTATION

bending rigidity/unit length in radial directionbending rigidity/unit length in angular directiontorsional rigidity/unit lengthbending rigidity due to coupling of the curvatures in theorthogonal directions due to Poissons ratioconcentrated dynamic forceacceleration due to gravitystiffness matrixmass matrixbending moment/unit length in the radial directionbending moment/unit length in the angular directiontwisting moment/unit lengthharmonic numberload vectorinner radiusouter radiusmean radiusKirchoffs edge reaction with outward drawn normal inthe direction of rKirchoffs edge reaction with outward drawn normal in thedirection of 0deflection of the middle surface of the platebridge deck angle subtended at the centrePoissons ratio in the radial directionPoissons ratio in the angular directionND,H/D,square or rectangular matrixdiagonal matrix

INTRODUCTIONWhen a vehicle traverses a bridge, the increase of bend-ing moments and deflections compared with thoseproduced under static loading, has generally been ac-counted for by the use of an impact factor dependentonly upon the span of the deck, applied to static designconditions[ 11. It has been observed that in some cases

+Assistant Professor.fAssistant Executive Engineer,

the recommended factor may considerably under esti-mate the effects that have been measured in practice[2-51. In addition, there is the possibility of reduced fatiguelife, which has been studied by Tung[6] and also thediscomfort and alarm experienced by pedestrian userswhen the level of vibration of bridge exceeds humantolerance level [7].The extensive use of curved slab bridges in the con-struction of highway system has drawn the attention ofseveral research worker in the response anlysis of suchstructure subjected to moving vehicle. The work done sofar is based on the idealization of the bridge deck as acurved beam, curved thin walled open section girder[l,91 or as a curved box girder using finite element [ lo] andfinite strips[ll]. A survey of recent work done in thisfield has been compiled by Huang[ 121.

In the present paper the vehicle model of Smith[l3] isused. The bridge deck is idealized using the concept offinite strip method in polar coordinate but the differencelies in the derivation of element properties. The methodpresented in the paper avoids the polynomial represen-tation and minimization procedure associated with finitestrip method. The element stiffness and mass matriceswere derived using the governing differential equation oforthotropic plate in polar coordinates. The study islimited to predicting dynamic deflections and moments ofthe bridge using high precision element rather than pre-dicting the effect of several variables such as bridge deckand vehicle parameters upon the dynamic response ofthe bridge structure.Modal analysis is adopted for determining the dynamicdeflection and moments since the method is particularlysuitable for linear structures with many degrees offreedom. The equation of motion in generalized coor-dinates are solved by the Runge-Kutta method usingGills variations.Structure idealizationThe bridge structure as shown in Fig. 1 constitutes ageneral type of deck slab encountered in the present day

27


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S. S. DEY and N. BALASUBRAMANIAN

Fig. 1. Simply supported composite curved slab-beambridges.highway system, i.e. curved slabs with radial and crossradial girders which is generally treated as an equivalentorthotropic plate. This is idealized as a number of con-centric parallel strips as shown in Fig. 2. The propertiesof each strip are regarded constant within the strip butcan vary from strip to strip. The dimension and coor-dinate system of a single element are indicated in Fig. 3.Vehicle idealizationThe vehicle is represented by a single degree freedomsystem comprising sprung and unsprung masses withviscous damping included in the suspension. The ideal-ized mechanical system is shown in Fig. 2. It is assumedthat the vehicle travels at a constant velocity along thecircular pathand the unsprung wheel is always in contactwith the road surface which is smooth. The centrifugal

Fig. 2. Idealization of curved bridge deck and vehicle system.

Fig. 3. Individual elements.

force on the vehicle is taken as counteracted by thesuper-elevation in the bridge deck.Derioation of element stifiness and mass matricesThe detlection w within the strip may be expressed as

uw(r, 0) = C W(r) sin@fl=, dwhich satisfies the simply supported boundary conditionsat the :.adial edges of the element. It ensures

w=o (2)

at 8=0and B=#.The requirement that eqn (1) should satisfy the homo-geneous differential equation of the plate in polar coor-dinatesDa%+2lJ a4wr a4 r 2 ar 2ae2

+I& 8~ j 2Dr a37 2f; a wZr4 ae4 r ar- r a r a eD, a*w+2( 0, +H) a% Raw- 7- i ? - - T- - - aB +7~=0 (4)r

w = x (Anrmlt B.rm2 t C, r m3+D,rq) sin NB (5)=1where A,, B,, C,, and Q, are four arbitrary constantscorresponding to harmonic number n and m, (s = 1,2,3,4) are the roots of the auxiliary equation resulting fromeqn (4). The roots for the nth harmonic are given by

m,=lt ~(l-i-a+2N~)~(~(l+a t 2N* f i ) * - c u( N2- ) ' ) 1 ' 2 ] 2 ( 6 )

where4 Ha=-- fi=,andN=n.0, , 4

The four constants A,, B,, C, and D, are determined


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Dynamic response of orthotropic curved bridge decks due to moving oadsfrom the boundary conditions at the curved edges of an In compact form

29

element as, at r = rivi = - v,

Mi =t Mpandat r-r*

v,= v,M,=-M,

where

aw 1 a--Do yq-yrg )IUsing conditions (7a) and (7b) in eqn (5) gives

sin Nt? - H,RI- G,R:*H,R;

whereG, = Dr{m,(m, - 2)*t 2/3(1- m,) - a(m, - N3))H, = DI{m,(m, - 1)t a(m, -N)}

The geometricedges are

and

where

Hence,

{d} = sin NB[U] {A} (161i.e. {A} = & [ U-1 {d}. (17)

Substituting (A) in eqn f 12) yields{FJ = [U u- II4 (18)

(7b) orvi = I&l 44 (19)

where [KC] is the stiffness matrix of the individual ele-ment.Element mass matrix

From eqn (5), the deflection w for the nth harmonic(8) can be written as,

w, = {R.JT{A} sin NO. (20)(9) Substituting eqn (17) in the above equation

w, = {RJT] U-lldI= IS,1 d). (21)

Rf = r ms-3)Rf* = rkms-3)

R: = -2)R; = -2)p = {(H t 2D,&/D,}N*. (11)

The subscript s takes the values 14. In compact form

G2RT-f&R;- GzRT*H2R;G,R$ GaRt- HJR; - H,R;- G,R: - G,R:H,R: H,RI; I

{F) = sin NOIL]{A}. (12)boundary conditions at the two curved

at r = ri wi = w, i = (b

at r=ro wo= w, cbo=d)

d=$.

(13)

(14)

The element mass matrix can be computed as follows:[M,]= hp/r,Ir{S.}T{S,}sin NOrdrdOi 0

= hp~[li-][lg{R,h.R~}Trdr] [Vl. (22)riThe stiffness and mass matrix of the whole structure isobtained by computing the properties of the individualstrips in succession and fitting them into global system ofcoordinates.Free vibration analysisThe equation of motion for an undamped structurecorresponding to nth harmonic can be written in theform,

WfnlbinI kfn11qnI{Ol (23)where {qn} is the vector of nodal line displacements androtations for the complete system, [M.] is the systemmass matrix and [ZLI is the system stiffness matrix. Thestandard eigen value formulation of the problem istherefore

[[&I - wt%KlIIYnI = {OI (24)where 6_* is the eigen value and {Y,} is the correspond-

(15)


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30 S. S. DEY nd N. B L SUBR M NI Ning eigenvector. An iterative technique was adopted forsolving eqn (24). The eigenvalue for any particular har-monic may be arranged in an ascending order of mag-nitude in a frequency matrix as follows:

[%Z _ zIn1= W2, . (25). . _ 2

wm n 1where m is the total degrees of freedom for the entirestructure. In addition the corresponding eigenvector maybe arranged in a modal matrix

[Y.l= [{YlJIY2HY3~~ WmIl (26)where each column represents the corresponding eigen-vector with reference to a particular mode.Forced vibration analysisThe equation of motion of forced vibration of thestructure may be expressed as

[Kl{ii.~+[Win}+ tL1Iq.J = iQ.1 (27)where [D] is the viscous damping matrix and {Q.} is thecolumn matrix of external forces. If the system is pro-portionally damped, these equations of motion can bedecoupled by introducing coordinate transformation be-tween the nodal line parameters {q_} and the generalizedcoordinates {p,,}as

14) = [ Ylbl. (28)Application of this transformation to eqn (27) gives

TLJItinI+Kl{dJ+ Kl{PJ=m (29)where the diagonal matrices are given by

IL1 = [YlT [Kl[Yl[ELI = [YlT [Dl[Yl (30)]CJ = 1YlT [&I [Yl.

The generalized force vector corresponding to general-ized coordinates is given by{QJ = 1YnlIQJ. (31)

In this analysis only one strip is loaded and the loadvector for this strip is represented by Vf.).The loadvector for the complete structure and the modal matrixmay be partitioned as follows:

Since the other strip is unloaded, it follows that {Qi} and{Q2} will be zero. [Z.] is the matrix of elements cor-responding to the loaded strip only.Using the relations (32) the generalized force vectorreduces to{Q] = Wll Vf. (33)

By applying the principle of virtual work, it is found that(f.)= FH (34)

where F is the concentrated dynamic force applied bythe vehicle and {s,} is equal to {S.} with local coor-dinates (r, @) substituted by (r,, vdr,) where r, is theradius of the concentric arc path on which the vehiclemoves, v is the velocity of vehicle and t is the time takenafter the vehicle enters the bridge to arrive at the pointunder study.Finally, eqn (29) reduces to the form asW+ r-LrrKl~dI+ ~W{PJ= ~LF[zITuI. (35)

The value of the dunamic force, F, applied by the vehiclemay be written down by equilibrium consideration.Assuming that the coordinate z represents the verticaldeflection of the sprung mass beneath its rest positionunder gravity, F may be expressed as

F=M,g-M.ItK,(z-w)tC,(i-i) (36)where the mass of the vehicle M,, is the sum of thesprung and unsprung parts, M, and M. respectively. Thedeflection of the bridge underneath the unsprang massmay be expressed by eqn (21) which is modified to thefollowing form by using eqns (28) and (32) as

w = ISNZl{P}. (37)Similarly the velocity and accleration ofthe bridge sur-face under the unsprung mass may be written as

k = ~S]ElU.] (38)and

i+ = {sItzl{ti~. (39)Substituting eqns (37)-(39) in eqn (36), the dynamic forcebecomes

F=M,g-M,{S,}[Z.l{B,}+F, (40)where the spring force F, is given by

F, = K(z - {S.HZl{p])+ CJ%%NZl~~l). (41)It may be noted that

C, =2&M,& (42)where A, is the proportion of critical damping in thevehicle suspension.

On substituting eqns (30) and (45) into eqn (39), theuncoupled equations may be written in the form,Ml {PI= PI (43)

where[Al = [II + MuEl[Ll -Vl{Sn} (44)

IW=(Fs +gM)~Ll~[ZITIS.}T-2A,~itl[Zl{lj}-~{(45)


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Dynamic response of orthotropic curved bridge decks due to moving oads 31Table 1.nd

Ff=_-2.MS w 1 strips First harmonicfrequency No. natural frequency, HzIt may be noted that in the above equation, the dampingmatrix [I,- [H] has been replaced by 2A,G,[Z] whereA, is the proportion of critical damping in the fundamen-tal mode of the bridge. This is based on the assumptionthat there is a constant damping in each mode. RungeKutta-Gill integration scheme was adopted to solve themodal equation of motion (43) instead of direct in-tegration technique. This algorithim is self starting, fastin convergence and well adopted to use with computer.From eqn (43), we get

{ii} = [A]-{II}. (47)In order to use this technique, the equations are writtenin the first order form by introducing the following sub-stitutions

where(48)

andaz=.Y,vz=&dz=r

Once this transformation is carried out, the solution ofthe equations now proceed using a standard algorithm.The deflections, velocities and acclerations of thebridge surface are calculated by using eqns (37)-(39)respectively.Dynamic bending moments in any strip for the nthharmonic can be calculated by using eqn (5)(49)

Numerical studiesThe example attempted deals with the analysis of aconcrete curved bridge deck for which analytical solu-tions are available in [14] is used as a check on the

1 3.90652 13.48723 46.32434 110.6765 209.663

accuracy of the present method. The dimensions andparameters of the bridge deck are as follows: innerradius (RI) = 12.0 m; outer radius (RO) = 18.0 m; 4 =45; t = 48 cm; pC = 0.2; EC= 2.5 x lokg/m; and unitweight of concrete p = 244.65kg sec/m*.The rigidities of the slab are calculated as follows:D, = De = Ec t32(1_ P,2) = 0.24 x 1Okg m*/mDfi = i (1 - p,)D, = 0.96 x 10 kg m*/mD, = p,D, = 0.48 x 10 kg m*/m.

In case of composite slab bridges, the aboveparameters can be determined either experimentallyor by using the formulae suggested by Heins andHails[lS].

Free uibration analysisThe natural frequencies have been calculated and thefirst five frequencies are listed in Table 1.The fundamental frequency of the bridge is found to be3.9065Hz. This is in close agreement with the resultspublished in [14] where the first natural frequency pre-dicted as 3.725 Hz.Dynamic responseFor dynamic response analysis the characteristicsvalues for a typical rigid axle vehicle was choosen:sprung mass (MS)= 21800kg; unsprung mass (MU)=5450 kg; total mass (My) = 27250kg; natural frequency =3.0 Hz; stiffness of suspension spring = 3939000N/m;and proportion of critical damping = 0.03.The figure given for damping is based on test on fullscale bridge with dynamically recorded wheel loads ofheavy vehicle conducted by Biggs et al. [16].

Table 2.

Vehiclepositionon spanRadial TangentialDeflection moment moment

No. of No. of No. of w,cm & kgcm & kgcmstrips harmonics modes Static Dynamic x lo- x 10-4Presentanalysis 0.5 7 1 16 0.85 1.05 2.88 I 96Results of [15] 0.5 10 - 1.586 2.53 1.43Dynamic incrementration for deflection

= DIR = Instantaneous dynamic value-static value with load in same position as in dyn, casestatic value with load at midspan


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32 S. S. DEY and N. BALASUBRAMANIAN1 5r ,DYNAMlC:;:I

I I0 02 0.L 06 - 0.8 10 vt

0r

0 02 01. 06 0% 1.0 vt@rf

Fig. 4. Dynamic response of curved bridge deck at the mid-span as the vehicle moves along the central line of thebridge deck.

The initial conditions required for response cal-culations are stated below: initial compression of vehiclesprings = 7.00 mm; proportion of critical damping inbridge = 0.01; and velocity of vehicle = 15 mlsec.Dynamic deflection and momentsDynamic deflection and moments at midspan have beencomputed. The values are shown in Table 2. The valueshave been compared with those published in [141and areseen to be in general agreement. The variation ofdeflection and moments at the midspan of the deck as thevehicle moves along the central line of the span areshown in Fig. 4.

CONCLUSIONSA high precision element with the displacement func-tion which satisfies the plate bending differential equationas well as boundary conditions is presented. The pro-cedure may be identified as an analytical finite stripmethod which leads to efficient solution of the bridgedeck with few elements. Convergence is achieved merelyby increasing the number of harmonics rather than byincreasing the number of elements. The results show thatthere is significant variation of response across thetransverse section of the bridge. The dynamic analysis

carried out with the proposed formulation gives reason-ably accurate results even with one harmonic term.

REFERENCES1. Standard Specification for Highway Bridges. The AmericanAssociation of State Highway Officials, Washington, D.C.(1969).2. L. T. Oehler, Vibration susceptibilities of various highway

bridge types. J. Struct. Dia., ASCE 83(ST4). -41 (1957).

P. F. Csagoly, T. I. Campbell and A. C. Agarwal, Bridgevibration study RR181. Ministry of Transportation andCommunications,Ontario (1972).R. Shepherd and R. J. Aves, Impact factors for simpleconcrete bridges.Pm. Inst. Ciu. Engrs 5.5,Part 2, 191-210(1973)..I. M. Biggs, H. S. Suer and .I. M. Louw, Vibration of simplespan highway bridges. J. Struct. Dia. ASCE 83(ST2), l-32(1957).6. C. C. Tung, Life expectancy of highway bridges to vehicleloads. Proc. ASCE. 95(EM6), 1417-1428 1969).7. D. R. Leonard, Human tolerance levels for bridge vibration,Road ResearchLaboratory,Crowthorne,England,Rep. No.34 (1%6).8. P. 0. Christian0 and C. G. Culver, Horizontally curvedbridges subjected to moving load. J. Slruct. Div., ASCE 95,1615-1643, 1969).9. S. Komatsu and H. Nakar, Fundamental study on forcedvibration of curved girder bridges. Trans. JSCE 12, Part I,37-42 (1970).IO. R. 0. Rabizadeh and S. Shore, Dynamic analysis of curvedbox girder bridges. 1. Struct. Div., ASCE 101, 1899-1912(1975).II. Y. K. Cheung and M. S. Cheung, Free vibration of curvedand straight beam-slab or box-Girder bridges. IABSE 32,Part II, 41-52 (1972).12. T. Huang, Vibration of bridges. Shock & Vib. Digest S(3).61-76 (1976).13. J. M. Smith, Finite strip analysis of the dynamic response ofbeam and slab highway bridges. Earthquake Engng S?ruct.Dyn. 1, 357-370 1973).14. V. X. Kunukkasseril and R. Ramakrishnan, Dynamic res-ponse of circular bridge decks. Earthquake Engng Struct.Dyn. 3(3), 217-232 (1975).15. C. P. Heins and R. L. Hails, Behaviour of stiffened curvedplate model. J. Struct. Div., ASCE 95, 2353-2370 1%9).16. J. M. B&s, H. S. Suer and J. M. Louw, Vibration ofsimple-span highway bridges. Trans., ASCE 124, 291-318(1959).

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Dynamic Response of Orthotrpic Curved Bridge Deck