Dynamic Behaviour of Timber Footbridges

Transcription

1 Contança RIGUEIRO MSc PhD Student EST-IPCB Dynamic Behaviour of Timber Footbridge Graduated in Civil Engineering in Univ. of Coimbra (1992). MSc, Univ. of Coimbra (1997). João NEGRÃO PhD Aitant Profeor Dept. Civil Engineering Univ. Coimbra, PORTUGAL Graduated in Civil Engineering in Univ. of Coimbra (1981). MSc, Univ. of Oporto (1987). PhD, Univ. of Coimbra (1997). Aitant Profeor at the Dept. Civil Engin., Univ. of Coimbra. Carlo REBELO PhD Aitant Profeor Dept. Civil Engineering Univ. Coimbra, PORTUGAL Graduated in Civil Engineering in Univ. of Coimbra (1980). MSc, Technical Univerity of Libon (1987). PhD, Univ. of Karlruhe, Germany (1992). Aitant Profeor at the Dept. Civil Engin., Univ. of Coimbra Summary In ENV1995-2:1997 (Eurocode 5-Part 2), a implified method i propoed for checking the erviceability limit tate (SLS) of vibration in timber bridge ubjected to the action of tream of pedetrian. The procedure etimate the value of peak acceleration in the deck under uch condition. Thi paper undertake a comparion between the value upplied by that method and thoe obtained through a Finite-Element baed numerical analyi. A number of load cae i analyed, in which variable value of the number of pedetrian and of it walking frequency i conidered. The probabilitic model ued to generate the dynamic loading correponding to thee crowd are alo decribed. 1. Introduction Timber and timber-baed material, namely glued-laminated timber, have been uccefully ued in pedetrian or low-traffic hort to medium pan bridge. However, the low denity and modulu of elaticity of thee material may reult in exceive vibration caued by dynamic loading, particularly in the cae of ynchronized action of pedetrian. In ENV1995-2:1997 [1], a implified method i propoed for checking thi erviceability limit tate in current timber bridge, uch a ingle-pan and 2- or 3-pan continuou bridge, under the action of pedetrian crowd or vehicle. In the verion to come of that tandard, a ignificant part of thee proviion are likely to be removed, leading to further information lack. The vibration SLS criteria tated in EC5-2 conit of limiting the maximum value of the

2 acceleration in the deck, o that fatigue rik in tructural element i minimized and an appropriate level of comfort may be enured. Thi tudy compare the value provided by the implified method with thoe calculated from a detailed tep-by-tep dynamic analyi. Reference to precription of other tandard on thi topic are made whenever relevant. Prior to proceeding with the tep-by-tep analyi, the generation of dynamic loading correponding to the croing of only one or a tream of pedetrian had to be undertaken. The procedure ued for thi purpoe i decribed in ection Pedetrian Load Model The characterization of the action caued by a pedetrian involve conidering different motion pattern uch a walking, running or rhythmic jumping. Motion type typical of thoe activitie likely to take place in a footbridge, uch a walking and running, are of particular interet in the context of thi tudy. Rhythmic jumping may become relevant in ituation of vandalim, though it i not a uual ue condition. For the analyi of the dynamic behaviour of footbridge excited by tream of pedetrian, load function were conidered which repreent the variation with time of the force caued by the pedetrian on the deck. Thee function depend on a number of factor whoe characterization i important, uch a the tep frequency, the peed of the pedetrian motion and the tep length [2]. 2.1 Individual load The load function may uually be defined in term of a Fourier erie, N p () i in 2 i i 1 F t G G if t (1) where G repreent the weight of the reference pedetrian, i G the amplitude of the ith harmonic, i the correponding phae angle with repect to the firt harmonic, i the ordinal number of the term within the erie and N the number of harmonic conidered, uually equal to 3. The Fourier coefficient i may have the value lited in Table 1, according to the type of pedetrian motion under conideration. In the range 2,0-2,5, different motion type are poible, but running or rhythmic jumping lead to more evere dynamic coefficient than walking. The load function for the former type of motion i different from that of the latter, becaue it how a maximum value correponding to the contact of the foot with the deck, followed by ome time with no contact at all. Such function may, therefore, be modelled a a periodic equence of triangular or emi-inuoidal impule with a duration t c equal to the contact period between the foot and the deck. Table 1 Fourier coefficient for the firt three harmonic Type of motion Walking Running / Rythmic jumping Model in the time domain Triangular model Half-ine model Frequency Range 1 f 1, 5 0,43 0, 38 f 0,1 0,1 1,5 f 2, 5 0,43 f 0, 380,15 0, 125 2,0 f 4,0 co 4 i co i f 0,1 i ftc i f 2 c t i ftc if t 2 c Bachmann and Ammann [2] propoe the phae angle to be conidered with the value 1=0, 2= 3= /2. However, thi iue will be irrelevant in mot ituation, except when the repone i not dominated by reonance in one of the load harmonic, which will not be uually the cae.

3 Foot-Deck contact time t c [ec] t c = 1.8f Frequency f [] Figure 1: Foot-to-deck contact period t c The relation between the foot-to-deck contact period, t c, and the tep period, T p =1/f, may be obtained from the relation hown in Figure Spatial uperpoition of individual load In order to compute the overall effect of more than one pedetrian moving on the deck, a computer programme wa developed which overlap the effect of individual pedetrian. According to Matumoto et al (1978) [3], the ditribution of tep frequencie for pedetrian walking on a footbridge deck may be cloely repreented through a Gauian ditribution with a mean value of 2,0 and a tandard deviation of 0,18. A there i no available data concerning the variation of running frequencie, the ame ditribution function and the ame mean to tandard deviation ratio wa conidered for thi type of motion. The recent tandard till under approval, pren1991-2:2001 (EC 1) [4], recommend the conideration of two dynamic type of load, beide the action of a ingle pedetrian: one reulting from a group of about 10 pedetrian and the other from a continuou crowd of pedetrian moving on the deck. For the overlap of the individual load correponding to ingle pedetrian, the previouly referred probability ditribution law for the tep frequency wa ued. It wa alo aumed that the time delay between pedetrian entering on the bridge follow an exponential rule, in uch a way that the maximum pedetrian denity on the deck i 1,5peron/ m 2. Conidering that there i a trend for people to walk on the right hand ide of a footbridge, an aymmetric ditribution with the mean value at one quarter of the deck width wa conidered for the variable that define the trajectory along the deck of the bridge for each pedetrian. 3. Vibration limit tate in pedetrian bridge 3.1 Maximum value according to EC5 and EC1 Both EC1 and EC5 propoe expreion for the approximate evaluation of the maximum vertical acceleration reulting from the motion of a pedetrian or a tream of pedetrian on the deck of a footbridge. In the cae of EC5, the numerical procedure for a tream of pedetrian i that of equation (2), in which M m l b repreent the total ma of the deck, l i the pan of the bridge, b i it width and m i the ma per unit area. In the ame expreion, i the vicou damping ratio, n the number of tep required to cro the bridge, K a account for the upport condition (ee Table 2) and kvert, f account for the fundamental frequency (ee Figure 2-a). a 165k K max, vert vert, f a 2n e M For the determination of the effect caued by a pedetrian tream paing on the deck, the following expreion applie: (2)

4 a 165 0,027lbk K max, vert vert, f a 2n e M The EC1 propoe an expreion imilar to thee, with only light modification in the value of the coefficient to be ued. The expreion (4) i therefore uitable to evaluate the maximum value of the acceleration in the deck, caued by a crowd of pedetrian. In it, the coefficient K v i obtained from the graphic of Figure 2-b and the remaining parameter are a previouly defined. a max, vert 210K v 2n e M Table 2 Factor K a for different upport condition 1 Span ratio L 1 /L Continuou, 2 pan Continuou, 3 pan 1,0 0,7 0,6 0,8-0,8 0,6-0,9 (3) (4) k vert,f Kv f vert f v [] (a) (b) Figure 2 Relation between coefficient by the EC5(a) and the EC1(b) K vert, f and the fundamental frequency f vert, a precribed 4. Example 4.1 Structural model The decribed methodology wa ued in the dynamic analyi of three type of footbridge correponding to imply upported, two- and three-pan deck and to a cable-tayed deck. The total length of the deck wa kept contant and equal to 30 metre for the different typologie. For multipan deck, the pan length wa contant for all pan. Although the cable-tayed typology i not covered by the pecification included in pren 1995 or in the pren it wa aimed, with thi numerical tudy, to tet their applicability in thi cae. An aymmetric footbridge, with a main pan of 20,0m and a ide pan of 10,0m wa conidered. The typical cro ection i repreented in Figure 6 for the ingle-pan imply upported cae. The dimenion of the cro-ection for the tiffening girder are given in the table included in Figure 3 for the topologie conidered in thi tudy. In order to get comparable reult between the different tructure, the deign wa performed under imilar erviceability condition. Main and tranvere beam are made of trength grade GL24h glued laminated timber according to EN1194 (2002) [5]. Service cla 2 wa conidered, in term of the EC5-Part 1. Although irrelevant for thi tudy, it hould be emphaied that, in a real ituation, the tructural element mut be mounted on open air making the conideration of ervice cla 3 1 K a =1 for imply upported deck.

5 Glulam girder 135 x 1500 Tranvere beam 100 x 300 // compulory. Concerning the load, the elf-weight and a uniformly ditributed live load of 3kN/m 2 with hort duration wa conidered. To generate the dynamic load a 0.05 time tep wa conidered. Since the time integration uing the Newmark method wa made for a 0.01 time tep, a linear interpolation within the load time tep wa performed in order to make the integration poible. The damping ratio related to critical damping wa conidered to be 1%, a recommended by the EC5-2. Structural ytem Simply upported Continuou, 2 pan Continuou, 3 pan Cable-tayed (main pan: 6,0-7,5-7,5) bxh (mm, girder) 135x x x x400 Figure 3 Cro ection of the deck for all upport condition 4.2 Reult and dicuion At thi tage the comparion i made between the maximum acceleration value obtained from the load model preented before and thoe preconized by the Eurocode EC5 and EC1 a well a thoe obtained from other code, namely the Britih Standard 5400 [6], the Canadian ONT [7] and pecialized bibliography (Pernica, [8]). Following the rule commended by recent pre-tandard, prec5 and prec1, the load model wa applied in three different loading ituation concerning pedetrian activity, namely one ingle pedetrian, one group of 10 pedetrian and finally a contant tream of pedetrian croing the bridge with a pre-etablihed denity over the deck. Thi denity wa conidered to be 0.5 pedetrian/m 2 leading to about 45 pedetrian imultaneouly croing the 30 meter pan bridge. Concerning the data given in Table 3, it hould be treed that ditinction wa made between the ENV1995-2:1997 [1] and the new project pren [9], ince the claue concerning continuou beam are withdrawn in the lat one. Beide that, the given value refer to maximum acceleration ituation, o that the reonance of one harmonic of the load, the firt or the econd, with the firt eigenfrequency i the main loading ituation. However, pedetrian bridge are motly ued by walking people at pace rate of about 2, o that thi ituation i alo conidered in the load model for the group of people, in order to have reference value for the mot probable maximum acceleration in normal uage. Reult are included in the firt column of the Load Model reult. The reult from different ource ummarized in Table 3 how ome catter, which i not comprehenible in ome cae. A a matter of fact, for the implet cae, the one-pan deck loaded with one pedetrian croing the bridge with a pace rate coincident with the eigenfrequency, the pren give a ubtantially lower maximum acceleration (1.2m/ 2 ) than the one computed (2.3 m/ 2 ). However, the value obtained from thi code for the effect of a continuou tream of pedetrian acting in reonance with the tructure (12.1 m/ 2 ) compare well with that one obtained from the load model (12.3 m/ 2 ). The draft of the pren only conider the cae of a group of pedetrian (about 10). The value obtained from thi code compare relatively well with thoe computed for walking people croing the bridge at about 2 pace/ec. However they are ubtantially lower than thoe computed for the ituation of reonance, a it can be een when comparing the correpondent column with the lat column in the Table 3. The methodology given in Pernica [8] i limited to pace rate up to 2.5, o that, in thi cae it can be applied only in the three-pan deck ituation.

6 Table 3 Maximum vertical deck acceleration obtained from the different ource (m/ 2 ) Type of tructure Eigenfrequency Load harmonic N. of pedetrian ENV pren BS 5400 Pernica Draft pren Load Model 2 pace rate Reonance Onepan deck Twopan deck Threepan deck Cable tayed bridge t Stream nd Stream nd Stream t Stream Concluion Some dicrepancie were detected when comparing the reult obtained from the code, pecially the pren and the draft of the pren and the reult from the Load Model preented in the firt part of the paper. Thee dicrepancie hould be avoided, at leat for the implet ituation when only one pedetrian croe a imply upported bridge. For group of 10 pedetrian the pren compare well with the computed value if the dynamic load i due to walking people croing the bridge deck at about 2 pace/ec. In reonant condition at higher frequencie the maximum acceleration predicted by the load model i much higher than the one predicted by the referred code. 6. Reference [1] ENV :1997 Eurocode 5: Deign of timber tructure Part 2: Bridge [2] Bachmann, H. e Ammann, W., 1987 Vibration in Structure Induced by Man and Machine, Journal of IABSE, Structural Engineering International, vol 5, nº [3] Matumoto, Y., Nihioka, T. Shiojiri, H. Matuzaki, K., 1978, Dynamic Deign of Footbridge, IABSE Proceeding, P-17/78, pp [4] pren (Draft tage 34) General action Traffic load on bridge; Annex X (informative) Dynamic model of pedetrian load. (Augut 2001) [5] EN1194: Timber tructure Glued laminated timber Strength clae and determination of characteritic value. [6] Britih Standard Intitution, 1978, BS5400, Part 2, Appendix C, Vibration Serviceability Requirement for Foot and Cycle Track Bridge. [7] Ontario Highway Bridge Deign Code. 1983, Ontario Minitry of Tranportation, Toronto. [8] Pernica, G. Dynamic load factor for Pedetrian movement and rhythmic exercie, Canadian Acoutic, pp 18, 2, 3-18,1990. [9] pren 1995 (Draft tage 34) Eurocode 5: Deign of timber tructure Part 2: Bridge