Dynamic analysis of pedestrian bridges with FEM and CFD

Transcription

1 Dynamic analysis of pedestrian bridges wit FEM and CFD Krzysztof Zoltowski Piotr Zoltowski Gdansk University of Tecnology KBP Zoltowski (Consulting Engineers), Gdansk Summary Te paper describes a numerical study of dynamic response of a footbridge under pedestrian load and flow field created by big truck passing underneat. Standard FEM formulation is used to compute te response of te structure but te main problem discussed is ow to define loads. Pedestrian load function is described in te first part of te paper. Lock in effect (interaction between pedestrians and te deck) for vertical vibrations is explained and te proposition of simple nonlinear dynamic model is presented. 1. Introduction Modern footbridges are more and more sensitive to variable load created by moving pedestrians. Tere are several reasons for tat. Quickly expanding road infrastructure, force us to build footbridges just to reconnect one piece of land separated now by te new road or railway. Tese bridges are delicate because te main live load acting on tem comes from pedestrians. Very often tere is no place for a pier and long spans are only feasible solutions. It appens often on te igway or in te city centers because of a visual aspects or collisions wit underground facilities. Additionally new ligt and igly stressed materials are used for constructing. Finally ligtweigt structures are coupled wit modern arcitecture and often armed wit cable stays and suspension systems. In consequence mass and stiffness of structures decrease so muc tat te structures are sensitive to dynamic excitation. Te dynamic problems of footbridges ave been subject of recent studies. Main reason of new engineering callenges related to footbridges is teir low excitation energy. Tis makes tem vulnerable to vibrations. Periodic load from pedestrians and wind can accelerate a bridge to te level wic can be dangerous for structure itself or at least to large to be tolerated. Recent studies describe accelerations criteria admissible for user comfort, but often lack descriptions of loads. Terefore definition of loading is key aspect for proper implementation of tese criteria.. Dynamic response under pedestrian action. Tis part focuses on te definition of uman induced dynamic load in te relation to body weigt, pacing rate, density and interaction between pedestrian and vibrating deck (lock in effect).te response of a pedestrian bridge structure can be calculated wit a commercial computer FEM programs if te dynamic load is defined. Bacman [1] as defined a pedestrian load function for Hz frequency. Kerr and Bisop [] presented experimental results and conclusions for pace rates between 1.6 and. Hz.

2 .1. Load function from single pedestrian. A special experiment as been done to develop more universal load functions. Te strategy was to examine several umans walking wit different pace frequency on te experimental set-up and to develop a load function. Te stiff steel double arm lever was constructed. First arm was connected to te digitally driven dynamic ydraulic jack. On te second arm a fitness treadmill wit electric steering was placed. (Fig.1). Force cells were used as connectors between treadmill and a steel lever. Dynamic load induced by walking person can be measured and stored in data base. Experimental set-up allows additionally to measure loads on te vibrating deck. Te interaction between walking person and vibrating structure can be also observed. Fig.1. Fitness treadmill on te vibrating lever A test was executed to find out a formula for uman ground reaction forces during walking wit different pace rate. 3 subjects were tested on te run way wit pace frequency 1.3 Hz to.6 Hz. Basing on te test results a simple formula for uman walking load was proposed (Eq. 1)..7 A.6.5 test upper bound lower bound pace freq. [Hz] Fig. First armonic amplitude of load function (Eq. 1) For eac single test, amplitude A as been calculated separately on te assumption given in Eq.. It means tat function F (t) as te same root mean square level RMS as original signal q(t). Tis condition in effect creates equivalent (different from origin) load wic gives te same response of bridge structure as origin. On fig. calculated first armonic amplitude A for F(t) is sown. Basing on te presented result a simple formula for amplitude A was developed (Eq.3). F (t) = BW{1+ A[sin( ω t ) +,5 sin( ω t + π ) +,5 sin( 3ω t + π )]} (1)

3 Z s = P s Z s = T o F ( t )dt T P s = t t1 [ q( t ) q t t 1 av ] dt () ω (3) A =,4 +,6BW,84 π BW body weigt [kn] q(t) uman induced load - experimental result ω - pacing rate- result of Fourier analysis of test signal.. Vertical lock in effect A walking person adapts to and syncronizes is/er motion in frequency and pase wit vibrating deck. Tis penomena depends from te uman personal features and vibration caracteristic of te deck ( Bacmann [1]). To develop a nature of tis penomena a laboratory test was executed. A nature of te uman beavior during walking on te vibrating fitness treadmill was te purpose of te test. On te plot (Fig.3) istory of vibrations of test platform (Fig.1) is presented. Amplitude of displacements is growing from zero to te constant value of 1 mm. Te standard test was divided into four pases. Preliminary pase. Te tested object was allowed in tis pase to walk freely to te moment wen e got used to walk on te fitness treadmill. Second pase (~1 s - first measured pase) it is te marc on te stabile platform. Tird pase (~1 s) it is te marc on te platform. Amplitude of vibrations is growing linearly to determined value. Fourt pase (~s) it is te marc on vibrating platform wit constant amplitude. Since a dynamic load is te practical consequence of walk, variation of dynamic reaction under fitness treadmill was tested. History of dynamic pedestrian load is presented on Fig.4. Fig. 3. History of deck vibrations Fig.4. History of pedestrian load during test (time ~15s to ~5s is pase 3) History of te load (Fig.4) is sowing te reliable regularity. Pedestrian in te pase 3 (increasing of te vibration amplitude) is loading te platform more strongly tan before (no vibration) and after (constant amplitude). Participants of te test determined te pase 3 as uncomfortable. Te analysis of te movie sows, tat pedestrian in te pase 4 is adapting te step to vibrations in order to reduce forces in legs and in consequence load action on te platform. Making te assumption tat te dynamic effect of walking as te origin in te vertical movement of te body masses it is possible to define te simple ypotesis of adapting.

4 Vertical lock in effect during walk on te vibrating deck as an origin in natural adapting mecanism wic reduces forces in uman body. Tis effect can be reaced if te canter of uman mass ( stomac ) gets as small acceleration as possible. Te lock in effect is a syncronization of te frequency of steps and te frequency of te deck coupled wit te pase sift. Analysis of test results and movies are pointing, tat te moment of foot contact wit te deck during walk occurs wen te deck is moving down. Te lock in effect can be simulated by a simple kinematics formula in wic pase sift is determined by minimum of RMS (root mean square level) from dynamic force acting on uman legs. Assumptions: pedestrian is represented as concentrated mass and one leg (te lengt of te leg is variable in time independently from deck), fig.5. ( t ) = H sin( ω t + ϕ ) deck is vibrating undependably form pedestrian y ( t ) = U sin( ω t ) b b Fig.5. Pedestrian and vibrating deck simple model. M (t) y (t) ω ω b - uman mass - distance from uman mass to te deck (lengt of te leg) - vertical displacement of te center of uman mass - paceing rate of te pedestrian - frequency of te deck y ( t ) = ( t ) + d( t ) y( t ) = U sin( ω bt ) + H sin( ωt + ϕ ) (4) Force in uman leg: P ( t ) = M y ( t ) = M ω bu sin( ωbt ) + ω H sin( ωt + ϕ ) (5) RMS (root mean square level) from dynamic force acting on uman legs (Z s ) T y ( t, ϕ ) dt o Z s( ϕ ) = M (6) T dz Force in uman leg is minimal if : s ( ϕ ) = ϕ = π dϕ

5 .5. Single degree of freedom Te simplest model of SDOF system can simulate te penomena of lock in effect (fig.6). ' Fig.6. A simple model wit a single degree of freedom yb M b + Cyb + Kyb + ( + yb )M = (7) ' yb M b + Cyb + Kyb + M + yb M = (8) ' M = F F (t) is defined in (1) yb M b + Cyb + Kyb + F + yb M = (9) ' yb M b + Cyb + Kyb + L = L = F + yb M (1) Equation (1) specifies a SDOF system wit nonlinear external load L (t). If te load and bridge frequency are equal and te load function is sifted by π in pase (to function of displacement of SDOF system- lock in effect) we can easily find out tat L (t) is decreasing in time to: F (t) = A[,5 sin( ω t + π ) +,5 sin( 3ω t + π )] in case if Asin( ω t ) = yb M Tis situation sows tat a lock in effect can accelerate structure to te finite value. If we ignore damping in (1) and -nd and 3-rd armonic in (1) te final acceleration of te deck wit first natural frequency of Hz (average pedestrian mass of 75 kg) is 6 m/s and amplitude of deflection is,38 m... Verification Te response of two footbridges as been measured under controlled pedestrian load to verify te proposed load function. First bridge is a steel continuous beam wit ortotropic deck over Nogat river in Malbork and second is cable stayed steel bridge over Woloska street in Warsaw (fig.7) Fig. 7. Bridge over Nogat river in Malbork and over Woloska street in Warsaw. Bridge in Malbork as a total lengt 168,3 m divided to 7 spans (16,5 m, 4, m, 39, m, 4,, 16,5 m). First natural frequency is Hz. Bridge was loaded by 3 pedestrian walking wit syncronized pace rate 1.6, 1.8, and Hz. Footbridge over Woloska street is a pure steel structure. Main span is 6,9 m long. First natural frequency is 1.78 Hz. Bridge was loaded by 8 pedestrians walking wit syncronized pace rate Hz. Numerical models for bot bridges were created wit

6 SOFiSTiK FEM system. An interactive procedure of intelligent moving load (equation no. 1) was implemented to ASE module. Results of te test and simulation are presented on fig.8 and 9. Simulation wit oter ( Hz) pace rate gives good compatibility too. 3 [m/s] Y i [m/s] Z i t1i () ti () Fig.8. Malbork - max. acceleration of te main span. Test (left) and simulation (rigt ) [m/s] Y j [m/s] Z i t1j () 7 1 ti () Fig.9. Woloska - max. acceleration of te main span. Test (left) and simulation (rigt) 7 3. Footbridge excitation from air impact of trucks passing underneat 3.1. Problem background Te dynamic action of truck s compression wave on ligtweigt footbridge as been noticed first (to our best knowledge) by Firt [3]. Te problem was mentioned as a potential barrier on furter ligtening footbridges troug introduction of ligtweigt materials. Following study of literature as sown tat tere was no interest in tis action before. Tis kind of loading as been omitted as not significant for traditional eavy bridges mainly because it acts in opposite direction to dead load and also because tere were no problems wit tis kind of action in e past. Similar researc, owever not directly related to analyzed problem was performed by Sin Park [4] and Fujii Ogawa [5]. Teir work was related to fast railway and problems of trains entering te tunnel or passing eac oter in te tunnel. Bot teams used CFD for unsteady flow simulations. Bot teams also used moving mes tecnology and interfaces to simulate flow around objects moving against eac oter. Obtained results were verified by scaled-down testing. Concluding tese works, forces on tunnel walls and train body are related to: - tunnel geometry - train body geometry - train speed - clearance between train body and tunnel walls Te procedure used for calculation of loading force on te footbridge span due to air impact from truck passing underneat resembles te approac of Sin Park [4] and Fujii Ogawa [5]. Te following variables were cosen for te analysis: - footbridge clearance 4.7m wic is minimum allowed for newly designed expressways in Poland, - truck eigt 4m wic is maximum allowed by transportation administration, - veicle speed 9km/ maximum allowed speed for trucks in Poland, Te FEM model of footbridge over Wołoska street in Warsaw (Fig. 7, rigt poto) was selected for evaluation as te very good model verified by in situ testing was available.

7 3.. Solution procedure Simulations were performed using FLUENT 6. CFD code, based on finite volume metod. Flow was treated as incompressible wic is justified by small Mac numbers around.1. Terefore only momentum equations and continuity equations were solved. In order to account for turbulence a two equation k ε turbulence model was used. Tis model is based on turbulence kinetic energy k and its dissipation rate ε. Te k equation is derived from exact solution, owever ε equation is derived from averaged Reynolds stress tensor. Terefore molecular viscosity effects are omitted and fully turbulent flow is assumed. Airflow around a veicle moving at a constant speed can be assumed to be steady in ideal conditions. Wen te veicle wit its steady pressure disturbance zone moves in vicinity of anoter object, te pressure disturbance zone must deform and steady flow assumption is no longer valid. Steady object and moving body introduce new pressure disturbances, wic can be interpreted also by means of forces acting on te bodies. Te problem of a veicle moving under a footbridge is not a typical task for wind tunnel testing. Tis task requires a setup in wic fluid (air) is not moving and te veicle (truck) is moving. Typical wind tunnels allow simulations of moving air around not moving objects. Tey don t allow simulations of object moving relative to eac oter. Te simplest way to setup testing for truck and footbridge is to use real footbridge and real truck. CFD offers a solution to suc simulation. Interfaces and sliding mes tecnique were used to connect stationary zones wit moving zone. Sliding mes tecnique allows coupling of zones even if te nodes on two sides of te interfaces do not coincide. Te equation system is modified dynamically wit nodes canging teir positions. Computational domain was divided to tree areas: - stationary upper part wit footbridge cross-section - sky - moving mes zone wit truck sape - lane - stationary lower part wit road surface road Domain division sceme and used boundary conditions are presented on Fig. 1. symmetry periodic edges outflow grid interface sky inlet 5m/s moving mes 5m/s lane wall floor grid interface Fig. 1. Boundary conditions and CFD calculations setup Truck geometry was simplified to very basic sapes. Te truck wit trailer and 4 ft. cargo container (Fig. 11) was used as a representative of common trucks found on Polis roads. 3m 3m.5m road surface 1m.5m m.5m Fig. 11. Truck geometry

8 .78m 4.5 m Fig. 1. Footbridge cross-section Te full 3D solution of te truck passage simulation was not possible due to limited computational power. Fine sape of footbridge cross-section required fine finite volume grid, wic exceeded reasonable size. Terefore a D simulation was performed wit exact sape of footbridge crosssection (Fig 1). Anoter 3D simulation was made wit footbridge cross-section simplified to a bounding box. Tis 3D simulation along wit a D simulation of a bounding box footbridge crosssection allowed to estimate of correlation coefficient between loading on te footbridge in a D and 3D case. Wile tis assumption is quite roug, it allowed estimation of forces acting on footbridge on te basis of D simulation results. Te forces on te deck obtained from D simulations are presented on Fig. 13. Flow velocities in te moment wen truck approaces te footbridge are sown on Fig vertical force lateral force Force[N] 1-1 -,5 1 1,5,5 3 Time Fig.13. Results of basic D calculations (v=9km/) wit applied correction formula Fig. 14 Flow velocities wen truck approaces te footbridge

9 Te correction of D simulation results into pseudo-3d is done troug a formula (11) were: p D () t t - D model load function [ N / m] p 3D ()- approximate load function [ N ] b - truck widt, b =. 5[ m] () t p ( t) * b c p3 = (11) D D * c - spatial distribution correction coefficient. Value of c equal.55 as been found as appropriate for considered geometries Parametric study Basing on te presented procedure, parametric calculations were performed. D models were used wit key parameters influencing load functions canging wit every calculation. All analyses were referenced to basic setup mentioned above. Parameters cosen for te analysis are structure clearance and veicle speed. Te results of te analyses are not multiplied by c scaling coefficient, because it applies only to te geometry used in 3D test. Canging structure clearance significantly canges spatial outflow and ence requires additional 3D calculations. Tese were very time consuming and D results are presented only for comparison purpose. Additional study was performed to account for te effect of a truck convoy. Tis was done to evaluate te risk of periodic excitation. Raising te clearance of te structure seems to be an effective way of reducing forces acting on footbridge deck. Vertical loading functions for various structural clearances are sown on figure ,7m - 5m/s 5,m - 5m/s 4,7m - 5m/s 15 force [N] 1 5,5 1 1,5, time Fig. 15. Vertical load function wit different structural clearances Veicle speed also as an effect on magnitude of loading. A truck traveling wit excess speed of 16km/ (35m/s) causes twice as ig peak load as te truck traveling at allowed speed of 9km/ (Fig.16).

10 ,7m - 5m/s 4,7m - 3m/s 4,7m - 35m/s force [N] 1-1,5 1 1,5, time Fig. 16. Vertical load function wit different veicle speeds A case wit several trucks traveling one after anoter was also evaluated. Tree convoy setups were studied wit trucks spaced wit 1m,.5m and 35m. In every case only te first truck caused impact same as single truck. Te following trucks were traveling in aerodynamic wake and forces caused by tem on te footbridge deck were significantly smaller. Loading function on te deck from a passage of four trucks spaced at.5m is presented on Fig force [N] x time Fig. 17. Vertical load function from 4 truck wit.5m spaces between trucks, v=9km/ 3.4. Structural dynamics analysis Te examination of dynamic response of footbridge over Woloska street was performed wit use of FEM model of structure (Fig. 18) and SOFiSTiK DYNA module using direct integration of equations of motion. Beam elements were used to represent main structural members. Te steel ortotropic deck was modeled wit equivalent beam elements. Te FEM model as been verified by field testing.

11 Fig. 18 Footbridge FEM model for transient dynamic simulations Te road under footbridge as two carriageways terefore it was critical to define rigt point to apply load from air impact. Te most unfavorable case was wen truck was traveling on te left lane of te carriageway furter from pylon. Point on te main span coincident wit te left lane was also te igest point of te first natural vibration bending mode of te footbridge. Te CFD calculated loading components (Fy, Fz) were applied at tis point..3 [ m/s ] Fig.19. Point of maximum accelerations in deck. Response to load defined on Fig.14. [ mm ] [ mm ] Fig.. History of max. vertical displacement in stay nr 1 and nr 4. Response to load defined on Fig.4 Time step of.5s was used. Total simulation time was set at 6s to allow free damped vibrations after truck as passed. Te accelerations of te deck are presented on Fig 19. Maximum vertical accelerations of.3m/s were recorded during dynamic simulations. Tese values are acceptable in terms of pedestrian comfort, owever tey are noticeable by standing people. Anoter interesting result was obtained for stay cables. Fig. sows displacements of te stay cables nr 1 and nr 4. Tese results point tat te problem of steel fatigue in te stays due to repeating excitation migt be wort cecking. Wile te evaluated footbridge is not te ligtest one (steel ortotropic deck), observable amplitudes of vibrations were recorded in te simulation. Tis could ave been muc worse for a FRP-decked footbridge. Terefore air impact from trucks passing underneat could be a problem for a ligtweigt footbridge over a motorway.

12 4. Conclusions Design procedures for modern footbridges are subject of continuous researc. Muc effort is spent on dynamics. Te complexity of te problems is mainly related to couplings between varying loading and transient structure response. Terefore proper understanding of loading is necessary for a successful design. Wile researced procedures take time to spread into design practice, a flexible and powerful software tools reduce tis time significantly. Incorporation of intelligent loading to SOFiSTiK dynamics solvers and coupling tem wit CFD allows modeling multi-pysics problems. Sample applications of tese features ave been presented in tis paper. Tis unique interactive load structure procedure allows us to estimate comfort properties of te structure te next step after te function, structural strengt and serviceability. References 1. Bacmann H., Vibration problems in structures, Basel, Switzerland, Birkäuser Kerr S.C., Bisop N.,W.,M.; Human induced loading on flexible staircases. Engineering Structures., 3/1 3. Firt I., New Materials for Modern Footbridges, Footbridge Conference 4. C-H. Sin, W-G. Park, Numerical study of flow caracteristics of te ig-speed train entering into tunnel. Mecanics Researc Communications 5. K. Fujii, T. Ogawa, Aerodynamics of ig speed trains passing by eac oter. Computers&Fluids Vol. 4 no.8, 1995