Human induced vibrations in footbridges. Design guidelines.

RFS2-CT-2007-00033 Human induced Vibrations of Steel Structures Design of Footbridges Guideline Footbridge_Guidelines_EN03 - 08. September 2008 Design of Footbridges Guideline Table of Contents 1 Introduction ....................................................................................... 4 2 Definitions ......................................................................................... 4 3 Design procedure................................................................................ 6 4 Design steps ...................................................................................... 8 4.1 Step 1: Evaluation of natural frequencies........................................... 9 4.2 Step 2: Check of critical range of natural frequencies .......................... 9 4.3 Step 3: Assessment of Design Situation............................................. 9 4.3.1 Step 3a: Assessment of traffic classes.........................................11 4.3.2 Step 3b: Assessment of comfort classes......................................12 4.4 Step 4: Assessment of structural damping ........................................13 4.4.1 Damping model .......................................................................13 4.4.2 Damping ratios for service loads ................................................13 4.4.3 Damping ratios for large vibrations.............................................14 4.5 Step 5: Evaluation of acceleration....................................................14 4.5.1 Harmonic load models ..............................................................15 4.5.2 Response Spectra Method for pedestrian streams .........................19 4.6 Step 6: Check of criteria for lateral lock-in ........................................21 4.7 Step 7: Check of comfort level ........................................................21 5 Evaluation of dynamic properties of footbridges......................................22 5.1 Introduction .................................................................................22 5.2 Response measurements................................................................22 5.2.1 Measurements of ambient response for identification of critical natural frequencies .............................................................................23 5.2.2 Raw measurement of damping ratios associated with critical natural frequencies .............................................................................23 5.2.3 Measurement of the response induced by one pedestrian...............23 5.2.4 Measurement of the response induced by a group of pedestrians ....24 5.2.5 Measurement of the response induced by a continuous flow of pedestrians .............................................................................24 5.3 Identification tests.........................................................................24 5.3.1 Forced vibration tests ...............................................................24 5.3.2 Ambient vibration tests.............................................................25 5.3.3 Free vibration tests ..................................................................25 5.4 Instrumentation ............................................................................25 5.4.1 Response devices.....................................................................25 5.4.2 Identification devices................................................................25 1 Design of Footbridges Guideline 6 Control of vibration response...............................................................26 6.1 Introduction .................................................................................26 6.2 Modification of mass ......................................................................27 6.3 Modification of frequency................................................................27 6.4 Modification of structural damping ...................................................27 6.4.1 Introduction ............................................................................27 6.4.2 Simple measures .....................................................................27 6.4.3 Additional damping devices .......................................................28 7 References........................................................................................31 2 Design of Footbridges Guideline Table of frequently used symbols alimit Acceleration limit according to a comfort class [m/s²] amax maximum acceleration calculated for a defined design situation [m/s²] B width [m] d density of pedestrians on a surface [P/m²] f, finatural frequency for considered mode [Hz] fsstep frequency of a pedestrian [Hz] P force amplitude due to a single pedestrian [N] ( πft2cosP× ) harmonic load due to a single pedestrian [N] L length [m] m* modal mass [kg] M mass [kg] n number of the pedestrians on the loaded surface S (n = S × density) [P] n’ equivalent number of pedestrians on a loaded surface S [P/m²] p(t) distributed surface load [kN/m²] S area of the loaded surface [m²] δ logarithmic decrement for damping [-] μ mass distribution per unit length [kg/m] Φ(x) mode shape [-] ψ reduction coefficient account for the probability of a footfall frequency in the range the natural frequency for the considered mode [-] ξ structural damping ratio [-] 3 Design of Footbridges Guideline 1 Introduction Vibrations are an issue of increasing importance in current footbridge design practice. More sophisticated bridges (such as cable supported or stress ribbon footbridges) with increasing spans and more effective construction materials result in lightweight structures and a high ratio of live load to dead load. As a result of this trend, many footbridges have become more susceptible to vibrations when subjected to dynamic loads. The most common dynamic loads on footbridges, other than wind loading, are the pedestrian induced footfall forces due to walking or jogging. The fundamentals of these guidelines and the accompanying background document were prepared by: • Christiane Butz and Christoph Heinemeyer from RWTH Aachen University, Germany, • Andreas Keil, Mike Schlaich, Arndt Goldack and Stefan Trometer from Schlaich Bergermann und Partner, Germany, • Mladen Lukić, Bruno Chabrolin, Arnaud Lemaire and Pierre-Olivier Martin, from Centre Technique Industriel de la Construction Métallique, France, • Álvaro Cunha and Elsa Caetano from Faculdade de Engenharia do Universidade do Porto, Portugal. These guidelines are based on the results obtained within the research project RFS-CR-03019 “Advanced Load Models for Synchronous Pedestrian Excitation and Optimised Design Guidelines for Steel Footbridges (SYNPEX)” financially supported by the Research Fund for Coal and Steel (RFCS). Extra detail on the information given in these guidelines can be found in the accompanying Background Document [1] which also includes further references. 2 Definitions The definitions given here relate to the application of this guideline. Acceleration A quantity that specifies the rate of change of velocity with time (denoted as dv / dt or d2x / dt2), usually along a specified axis. Normally expressed in terms of g or gravitational units. Amplification The process of increasing the magnitude of a variable quantity, without altering any other property. Damper Device mounted in structures to reduce the amplitudes of vibration through a method of dissipation of energy. Damping Damping is any effect, either inherent to a system or specifically added for the purpose, that tends to reduce the amplitude of vibration of an oscillatory system. Damping is the energy dissipation of a vibrating 4 Design of Footbridges Guideline system with time or distance. For structures, the total damping consists of • Material and structural damping • Damping by furniture and finishing • Spread of energy throughout the whole structure Dynamic action Action that causes significant accelerations of the structure or structural members Modal mass = generalised mass A multiple degree of freedom system can be reduced to a combination of several single degree of freedom (SDOF) systems with coincident natural frequencies: m* k* π2 1 f= where f is the natural frequency, expressed in Hz k* is the modal stiffness m* is the modal mass. Thus the modal mass can be interpreted to be the mass activated in a specific mode shape. Mode of vibration A characteristic pattern assumed by a vibrating system in which the motion of every particle is a simple harmonic with the same frequency. Two or more modes may coexist in a multiple degree of freedom system. Natural frequency = The natural frequency is the frequency of free vibration of a system. For a multiple degree of freedom system, the natural frequencies are the frequencies of the modes of vibration. Each structure has as many natural frequencies and associated modes of vibration as degrees of freedom. They are commonly sorted by the amount of energy that is activated by the oscillation; the first natural frequency is that on the lowest energy level and is the most likely to be activated. eigenfrequency The equation for the natural frequency of a single degree of freedom (SDOF) system is: M K π2 1 f= where K is the stiffness M is the mass. The derivation of natural frequencies is described in chapter 4.1. The frequency f is the reciprocal of the oscillation time T (f = 1 / T). 5 Design of Footbridges Guideline Resonance A system is at resonance when any change in the frequency of a forced vibration, however small, causes a decrease in the response of the system. When damping is small, the resonant frequency is approximately equal to the natural frequency of the system (the frequency of free vibrations). Response spectrum A response spectrum is a plot of the peak or steadystate response (displacement, velocity or acceleration) of a series of linear single degree of freedom oscillators of varying natural frequency that are forced into motion by the vibration. The resulting plot can then be used to pick off the response of any linear system, given its natural frequency of oscillation. The response spectrum contains precise information about the distribution of vibration energy for various frequencies. Spectrum Description of any time dependant signal as a series of single-frequency components, each with an amplitude and, if appropriate, phase. 3 Design procedure An increasing number of vibration problems for footbridges encountered in the last few years show that footbridges should no longer be designed only for static loads, but also for the dynamic behaviour. The design should take into account the vibration performance of the footbridge due to walking pedestrians. It is important to note that there are currently no code regulations available. Although from designers’ point of view this lack of regulation allows a large amount of freedom and therefore a large variety of innovative bridge structures, it is nevertheless of vital importance that the bridge will meet comfort requirements which are required by the client or owner. The question “Will the footbridge meet the comfort criteria when vibrating?” plays an important role in the design process, as dampers are not only additional bridge furniture, but may need to be included in the design. The general principles of a proposed design methodology are given in Figure 3-1. 6 Design of Footbridges Guideline A client together with the consultant … … defines several design situations by combinations of possible traffic classes and required comfort levels: A design office… … and (for each “design situation ” chosen by the client)… Yes Improvement of the dynamic behaviour: Modification of structure Increase of natural frequencies Installation of dampers … checks: Is the footbridge comfortable? Comparison of the accelerations for each design situation with the corresponding comfort criteria … undertakes the structural analysis by - modelling the footbridge and - calculating its natural frequencies … calculates the accelerations and … “design situation: traffic class + comfort class” (1) “design situation: traffic class + comfort class” (2) … “design situation: traffic class + comfort class” (n) OK HiVoSS Guidelines No Figure 3-1: Methodology for the design The flowchart in Figure 3-2 shows how to check the dynamic behaviour of the footbridge in the design phase and how this guideline can be employed. The various steps mentioned in the flowchart will be discussed in section 4. Safety problems due to overstressing or fatigue may also occur and should also be considered in the design of footbridges - this guideline only treats reversible serviceability, as defined by the Eurocodes. Design rules for overstressing and fatigue are given elsewhere. It should be noted that all the usual verifications in Serviceability Limit State (SLS) and Ultimate Limit State (ULS) must be carried out according to the standards in use. 7 Design of Footbridges Guideline ξ Average Construction type Minimum ξ Prestressed concrete 0,50% 1,0% Composite steel-concrete 0,30% 0,60% Steel 0,20% 0,40% Timber 1,0% 1,5% Stress-ribbon 0,70% 1,0% 4.4.3 Damping ratios for large vibrations Intentional loads can produce large levels of oscillation in light footbridges, which lead to higher damping ratios, as listed in Table 4-6. Table 4-6: Damping ratios according to construction material for large vibrations Construction type Damping ratio ξ Reinforced concrete 5,0% Prestressed concrete 2,0% Steel, welded joints 2,0% Steel, bolted joints 4,0% Reinforced elastomers 7,0% 4.5 Step 5: Determination of maximum acceleration When one or several design situations (cf. section 4.3) are defined and the values for damping are determined (cf. section 4.4), the next step is to calculate the maximum acceleration amax for each design situation. There are various methods for calculating the acceleration of the bridge; this design guideline recommends using one of the methods shown in Figure 4-1, which will be discussed in the following chapters. 14 Design of Footbridges Guideline Harmonic load model for TC1 to TC5 Acceleration amax Finite Element Method SDOF Method Res p onse S p ectra Method A pp lication of load models Figure 4-1: Various methods for calculating the acceleration Note: It is important to check whether the acceleration calculated with the assumed damping parameters for large or small vibrations (cf. section 4.4) corresponds to the acceleration on the built structure (cf. section 5). Experience has shown that it is very difficult to predict the structural damping of the finished footbridge. Therefore, damping always has a broad scatter and consequently acceleration also has a broad scatter. 4.5.1 Harmonic load models 4.5.1.1 Equivalent number of pedestrians for streams Once a numerical model of the footbridge has been developed, the design situations and corresponding load models chosen and the damping ratios specified, the footbridge response can be calculated. Harmonic load models are required to calculate the acceleration when using either Finite Element methods or Single Degree of Freedom (SDOF) methods (cf. section 4.5.1.3). For the modelling of a pedestrian stream consisting of n “random” pedestrians, the idealised stream consisting of n’ perfectly synchronised pedestrians should be determined (cf. Figure 4-2). The latter would be synchronised only among themselves (without taking into account the influence of the vibrating structure on their footfall frequency). The two streams are supposed to cause the same effect on a structure, but the equivalent one can be modelled as a deterministic load. Q j Φ i (x) } n ⇔ Q j n’ / n Φ i (x) } n’ Figure 4-2: Equivalence of streams For the evaluation of the response with respect to group or pedestrian stream loading, the application of a distributed harmonic load along the bridge deck (simulating an equivalent number of pedestrians at fixed locations) meets almost all requirements for practical design of footbridges. Care is needed in the choice of the range of frequencies for which this kind of calculation makes sense. The problem of the influence of the structure on the 15 Design of Footbridges Guideline behaviour of the pedestrians is not taken into account and this can aggravate the response. 4.5.1.2 Application of Load models In the recommended design procedure, harmonic load models are provided for each traffic class TC1 to TC5 (cf. Table 4-3). There are two different load models to calculate the response of the footbridge due to pedestrian streams depending on their density: • Load model for TC1 to TC3 (density d < 1,0 P/m²) • Load model for TC4 and TC5 (density d ≥ 1,0 P/m²) Both load models share a uniformly distributed harmonic load p(t) [N/m²] that represents the equivalent pedestrian stream for further calculations: () ψn t fπP t p s×××= ')2cos( Eq. 4-1 where ( ) t fπPs 2cos× is the harmonic load due to a single pedestrian, P is the component of the force due to a single pedestrian with a walking step frequency fs, fs is the step frequency, which is assumed equal to the footbridge natural frequency under consideration, n’ is the equivalent number of pedestrians on the loaded surface S, S is the area of the loaded surface, ψ is the reduction coefficient taking into account the probability that the footfall frequency approaches the critical range of natural frequencies under consideration. The amplitude of the single pedestrian load P, equivalent number of pedestrians n’ (95th percentile) and reduction coefficient ψ are defined in Table 4-7, considering the excitation in the first harmonic or second harmonic of the pedestrian load (see Section 4.2). 16 Design of Footbridges Guideline Table 4-7: Parameters for load model of TC1 to TC5 P [N] Vertical Longitudinal Lateral 280 140 35 Reduction coefficient ψ Vertical and longitudinal Lateral 1. Harmonic 0 0,25 1 0 1,25 2,1 3,4 4,2 4,6 Frequency 2,31,7 0 1 00,70,5 1,2 1,7 2,11,0 2,4 2. Harmonic Frequency 2,5 Equivalent number n’ of pedestrians on the loaded surface S for load model of: TC1 to TC3 (density d < 1,0 P/m²): n’=S nξ×8,10 [1/m²] TC4 and TC5 (density d ≥ 1,0 P/m²): n’=S n85,1 [1/m²] where ξ is the structural damping ratio and n is the number of the pedestrians on the loaded surface S (n = S × d). The load model for pedestrian groups (TC1) takes into account a free movement of the pedestrians. Consequently, the synchronization among the group members is equal to a low density stream. In the case of dense streams (TC4 and TC5) walking gets obstructed, the forward movement of the stream gets slower and the synchronisation increases. Beyond the upper limit value of 1,5 P/m² walking of pedestrians is impossible, so that dynamic effects significantly reduce. When a stream becomes dense, the correlation between pedestrians increases, but the dynamic load tends to decrease. In Figure 4-3 a harmonic load p(t) is applied to the structure for a particular mode shape. 17 Design of Footbridges Guideline p ( t ) [N/mm²] Figure 4-3: Application of a harmonic load according to mode shape Φ(x) The harmonic load models above describe the loads induced by streams of pedestrians when walking along the footbridge. Some footbridges may be further affected by the action of joggers which is further described in [1]. 4.5.1.3 SDOF method Generally, the dynamic behaviour of a structure can be evaluated by a modal analysis, where an arbitrary oscillation of the structure is described by a linear combination of several different harmonic oscillations in the natural frequencies of the structure. Therefore, the structure can be transformed into several different equivalent spring mass oscillators, each with a single degree of freedom. Each equivalent single degree of freedom (SDOF) system (cf. Figure 4-4) has one natural frequency and one mass that is equal to each natural frequency of the structure and the accompanying modal mass. Figure 4-4: Equivalent SDOF oscillator for one natural frequency / vibration mode of the structure The basic idea is to use a single equivalent SDOF system for each natural frequency of the footbridge in the critical range of natural frequencies and to calculate the associated maximum acceleration for a dynamic loading. The maximum acceleration amax at resonance for the SDOF system is calculated by: ξm p δ π m p a2 1 * * = * * = max Eq. 4-2 where p* is the generalised load m* is the generalised (modal) mass ξ is the structural damping ratio and δ is the logarithmic decrement of damping. 18 Design of Footbridges Guideline 4.5.2 Response Spectra Method for pedestrian streams At the design stage it is not necessary to apply a time domain analysis in every case. The aim of a spectral design method is to find a simple way to describe the stochastic loading and system response that provide design values with a specific confidence level. It is assumed that: • the mean step frequency, fs,m, of the pedestrian stream coincides with the considered natural frequency of the bridge, fi, • the mass of the bridge is uniformly distributed, • the mode shapes are sinusoidal, • no modal coupling exists, • the structural behaviour is linear-elastic. The system response – “maximum peak acceleration” – was chosen as the design value. In the design check, this acceleration is compared with the tolerable acceleration according to the comfort class to be proofed. For different pedestrian densities, the characteristic acceleration, which is the 95th percentile of the maximum acceleration, can be determined according to the formulas and tables given below. This maximum acceleration is defined by the product of a peak factor ka,d and a standard deviation of acceleration σa: a a,d,d σka = max Eq. 4-3 Note: The peak factor ka,d serves to transform the standard deviation of the response σa to the characteristic value amax,d. In serviceability states, the characteristic value is the 95th percentile, ka,95%. Both factors are derived from Monte Carlo simulations based on numerical time step simulations of various pedestrian streams on various bridges geometries. The result is an empirical equation for the determination of the variance of the acceleration response: 2 2 1 2 * 2 i F k am σC ξkσ= Eq. 4-4 where 32 2 11 afafak ii ++= 32 2 12 bfbfbk ii ++= a1, a2, a3, b1, b2, b3 are constants fi is the considered natural frequency that coincides with the mean step frequency of the pedestrian stream ξ is the structural damping ratio C is the constant describing the maximum of the load spectrum 19 Design of Footbridges Guideline nkσFF = 2 is the variance of the loading (pedestrian induced forces) kF is a constant n = d×L×B number of pedestrians on the bridge, with d: pedestrian density, L: bridge length, B: bridge width m*i is the modal mass of the considered mode i The constants a1 to a3, b1 to b3, C, kF and ka,95% can be found in Table 4-8 for vertical accelerations and in Table 4-9 for lateral accelerations. Table 4-8: Constants for vertical accelerations d [P/m2] kFC a1a2a3b1b2b3ka,95% ≤ 0,5 1,20×10-2 2,95 -0,07 0,60 0,075 0,003 -0,040 -1,000 3,92 1,0 7,00×10-3 3,70 -0,07 0,56 0,084 0,004 -0,045 -1,000 3,80 1,5 3,34×10-3 5,10 -0,08 0,50 0,085 0,005 -0,060 -1,005 3,74 Table 4-9: Constants for lateral accelerations d [P/m2] kFC a1a2a3b1b2b3ka,95% ≤ 0,5 6,8 -0,08 0,50 0,085 0,005 -0,06 -1,005 3,77 1,0 7,9 -0,08 0,44 0,096 0,007 -0,071 -1,000 3,73 1,5 2,85×10-4 12,6 -0,07 0,31 0,120 0,009 -0,094 -1,020 3,63 Alternatively, for a simplified estimation of the required modal mass for a given pedestrian traffic to ensure a given comfort limit alimit, an expression is derived that is valid for fs,m = fi: () limit 4 k 3 2 k 1 * a ξ k 1.65ξ k n m+ ≥ i where m*i modal mass for the considered mode i n number of pedestrians on the bridge ξ structural damping coefficient k1 to k4 constants (cf. Table 4-10 for vertical bending and torsion modes and Table 4-11 for lateral bending modes) Table 4-10: Constants for required vertical modal mass d [P/m2] k1k2k3k4 ≤ 0,5 0,7603 0,050 1,0 0,5700 0,040 1,5 0,4000 0,468 0,035 0,675 20 Design of Footbridges Guideline Table 4-11: Constants for the required lateral modal mass d [P/m2] k1k2k3k4 ≤ 0,5 1,0 1,5 0,1205 0,45 0,012 0,6405 The design method was elaborated with beam bridge models. If the structural behaviour of a bridge differs significantly from that of beam bridge, limits of application of the spectral method may be reached. 4.6 Step 6: Check of criteria for lateral lock-in The triggering number of pedestrians for lateral lock-in, that is the number of pedestrians NL that could lead to a vanishing of the overall damping producing a sudden amplified response, can be defined as: k fm*ξπ8 NL= Eq. 4-5 where ξ is the structural damping ratio m* is the modal mass f is the natural frequency k is a constant (300 Ns/m approximately over the range 0,5-1,0 Hz). Another approach is to define the trigger acceleration amplitude when the lock-in phenomenon begins: 2 m/s15,0to1,0= inlock a Eq. 4-6 Recent experiments have shown the adequacy of both formulae to describe the triggering for lock-in. Note: Pedestrian streams synchronising with vertical vibrations have not been observed on footbridges. 4.7 Step 7: Check of comfort level According to the design verification methodology specified in Figure 3-2, the response calculated for the specified design situations and the corresponding load models has to be compared with the specified comfort limits given in Table 4.4. The non-compliance with those limits implies the need of measures that improve the dynamic behaviour of the footbridge. These measures include: • modification of the mass • modification of frequency • modification of structural damping • addition of damping 21 Design of Footbridges Guideline For an already constructed bridge, the simplest approach is based on the increase of structural damping, which can be achieved either by implementation of additional control devices, or by actuation on non-structural finishings, like the hand-rails and surfacing (cf. chapter 6). 5 Evaluation of dynamic properties of footbridges 5.1 Introduction The experimental characterisation of the dynamic behaviour of a footbridge may be an important component of the project and can be performed based on two different levels of complexity: • Level 1- Identification of structural parameters, with the purpose of calibrating numerical models and eventually tuning control devices. Natural frequencies, vibration modes and damping coefficients are the parameters of interest; • Level 2- Measurement of the bridge dynamic response under human excitation for assessment of comfort criteria and/or correlation with the simulated response. The adoption of one of the above mentioned strategies depends on the characteristics of the structure and on the aims of the study. Level 2 tests can be characterised as standard tests that should be developed at the end of construction of any potentially lively footbridge, providing important information for design and verification purposes. Based on the results of these tests, the bridge owner may decide whether to implement control measures or not. It should be noted that the use of experimental tests to check the comfort class of a specific footbridge requires the performance of measurements for all vibration phenomena and design situations considered in the development of design load models and involves the obtainment of characteristic values of the response. Level 1 tests are required when it is clear that the dynamic behaviour of the footbridge is beyond acceptability limits and control measures are necessary. The appropriate design of control devices requires an accurate knowledge of structural parameters, namely natural frequencies and vibration modes. The current chapter presents general guidelines for testing and data analysis of footbridges. 5.2 Response measurements The performance of Level 2 tests should consider the following items: 1. Identification of critical natural frequencies; 2. Identification of damping ratios; 3. Measurement of response induced by one pedestrian; 4. Measurement of the response induced by a small group of pedestrians; 5. Measurement of the response induced by a continuous flow of pedestrians. 22 Design of Footbridges Guideline The verification of acceptability limits of vibration for a particular pedestrian bridge should be based on the results of these tests, under consideration of the particular use of the bridge. 5.2.1 Measurements of ambient response for identification of critical natural frequencies Tests should preferably be conducted on the bridge closed to pedestrian traffic. Assuming that a preliminary dynamic analysis of the bridge has been conducted, providing an estimation of natural frequencies and vibration modes, the instrumented sections should correspond to the sections of maximum estimated modal response for the estimated critical frequencies. 5.2.2 Raw measurement of damping ratios associated with critical natural frequencies Raw estimates of the damping ratios associated with critical natural frequencies can be obtained from a simple free vibration test in which a pedestrian jumps / bends knees / bounces on a fixed location at a particular frequency, trying to induce resonant response of the bridge for the corresponding vibration mode. After a few cycles of excitation, the pedestrian action is suddenly interrupted and the free vibration response is recorded. This process should be repeated a number of times, in order to provide average estimates of damping coefficient as a function of amplitude of oscillation. 5.2.3 Measurement of the response induced by one pedestrian The tests described above provide an update of the expected critical natural frequencies. The response of the footbridge is now measured at the relevant sections (the maximum modal displacement section for each critical frequency), considering the motion of a single pedestrian over the bridge. Several types of motion should be explored, as a function of the frequencies of interest: • walking, for critical natural frequencies below 2,5 Hz; • walking or running, for critical natural frequencies between 2 Hz and 3 Hz; • running, for natural frequencies above 3 Hz. Given the random characteristics of excitation, a number of tests should be performed for each combination of frequency and motion., typically about 5. A metronome should be used to ensure the correct walking rate is obtained. The maximum acceleration and dynamic displacement (which can also be derived from acceleration) of the bridge should be recorded for each collected series, and the peak response induced by one pedestrian can be taken as the maximum of the peak responses from the various tests. The weight of the pedestrian should be noted. Whenever the bridge has a non-symmetric incline, the response should be recorded with the pedestrian travelling down the slope. 23 Design of Footbridges Guideline designed to split the critical frequency into two new frequencies (one below and another above the initial one), and the relative movement between structure and TMD allows for energy dissipation. Since the structural mass is much higher than the TMDs’, the movement of the TMD usually comprises large displacements when compared to the structure motion. Figure 6-2 shows two examples of TMDs, one vertical and one horizontal, installed in footbridges. Figure 6-2: Examples of installed TMDs 6.4.3.3 Pendulum dampers Pendulum dampers (cf. Figure 6-3) are a specific type of tuned mass dampers, which are used for suppressing horizontal vibrations. The main difference with a TMD is that no springs are used, except in cases when the frequencies to suppress are higher than 1 Hz. The mass is hung by truss elements, which reduces friction forces when compared to a normal horizontal support. Figure 6-3: Example of pendulum systems 6.4.3.4 Tuned liquid column dampers A Tuned Liquid Column Damper (TLCD) consists of U-shaped tube (cf. Figure 6-4), filled with a fluid (usually water), which properties are tuned in such a way that the forces at the base of the device, resultant from the movement of the liquid, counteract the horizontal movement of its support. This principle is therefore identical to that of a TMD. However, there are several advantages over other types of damping devices, such as easy tuning of frequency and damping, simple accommodation, simple construction and almost zero maintenance costs. 30 Design of Footbridges Guideline R Y B H H Y f Ah Ah Ab Figure 6-4: TLCD scheme The optimum damping of the TLCD should be the same as the analogue TMD. The TLCD has intrinsic damping due to fluid turbulence and, by inserting a control valve or an orifice plate in the horizontal tube, the TLCD damping can be further enhanced. However, there is no specific literature with information concerning the quantification of TLCD damping, so it must always be obtained from tests on the TLCD prototypes. 6.4.3.5 Tuned liquid dampers Tuned Liquid Dampers (TLDs) are passive control devices that consist of rigid tanks filled with liquid (cf. Figure 6-5) to suppress horizontal vibration of structures. Advantages like low cost, almost zero trigger level, easy adjustment of natural frequency and easy installation on existing structures have promoted an increasing interest in these devices. However, motion of the fluid can be highly nonlinear, since breaking of waves can occur for high vibration amplitudes. x L s h0 η a a Figure 6-5: TLD scheme 7 References [1] HiVoSS (Human induced Vibrations of Steel Structures): Design of Footbridges – Background document, September 2008. 31