Dynamic investigation of footbridges

Dynamic investigation of footbridges Daia Zwicky Dr. sc. techn.

This text summarizes the experience made with the application of a recent publication on dynamic investigations of footbridges to projects in Zumaia (Bask country, Spain) and Medellín (Colombia). It also explains some important aspects when assessing dynamic behavior of ‘lively’ structures and reports on the execution and performance of time-history analyses. Referred publication The text refers largely to a methodology proposed in a recent publication: o ‘Assessment of Pedestrian-Induced Vibratory Behaviour of Footbridges’, edited by the Association Française de Génie Civil AFGC, June 2006 This publication provides a rather comprehensive overview on the state-of-theart of pedestrian-induced dynamic behavior of footbridges. Modern footbridges may exhibit insufficient dynamic properties due to low stiffness combined with relatively high masses. The publication was mainly initiated due to the insufficient dynamic behavior of the Solferino Bridge in Paris and the Millennium Bridge in London. It provides information in English and French, and can be ordered from the secretariat of AFGC, 28 rue de Saints Pères, 75007 Paris. A copy of this publication is incorporated in the Pedelta library. Important aspects of dynamic behavior Structural modeling The essential parameters of the dynamic behavior of a structure are masses, stiffness and damping. When modeling a structure, one should bear in mind that all structural and nonstructural elements contribute to a higher or lower degree to the dynamic properties of the structure in terms of mass, stiffness and damping. Special attention should be paid to the modeling of: o non-structural elements, e.g. mesh safety railings, sealing, pavement etc. Æ increases in mass, eventually in damping o connections, e.g. of transverse girders, deck planks etc.: hinged, fixed, semi-rigid… Æ (reductions in) stiffness o composed systems, e.g. transverse deck with longitudinal girders: local mass and stiffness distribution Æ changes in local and global natural frequencies, possibly favorable effects for dynamic excitation (see below) o bearings: material, construction type Æ changes in global stiffness, damping (depending on material) o piers Æ increase in mass (MDOF systems), reductions stiffness (longitudinally in general) o foundations: not infinitely stiff, friction soil-structure Æ reduction in global stiffness (MDOF systems), increase in damping Note: MDOF = multiple degree of freedom \\servidor\usuarios\documentos\dzwicky\Mis documentos\Papers\Papers dz\Dyn. investigations Zumaia Medellin\Prep_Pres_Zumaia_Medellin.doc2007-03-20

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Dynamic behavior of footbridges – analysis and assessment of results

Natural frequencies and mode shapes are mainly influenced by the distribution of mass and stiffness and by structural geometry. Structural response in terms of accelerations is mainly influenced by damping and mass, displacement velocities and deformations are additionally influenced by the natural frequency. It is therefore crucial to consider mass and stiffness distributions correctly and consistently, e.g. to also consider the mass of pedestrians when determining accelerations for a load case involving pedestrian loads. In composed systems – e.g., relatively stiff longitudinal girders combined with a (hinged) light transverse deck – the natural frequencies of the two local systems influence each other in terms of the overall frequency of the system. Generally, the lower frequency of a partial system is further decreased by the influences of the other system(s). Example. Single-span girder with L = 26 m, bending stiffness 500 MNm2 and a total mass of 500 kg/m. Transverse deck of GFRP planks with l = 2.6 m, bending stiffness 30 kNm2/m, hinged connection (single span) and distributed mass of 80 kg/m2 (incl. pedestrian mass of approx. 60 kg/m2). The 1st natural bending frequency of each of the two systems can be calculated from

f=

π 2L

2

EI (L in m, EI in kNm2, m in t/m) Æ flong = 2.32 Hz, ftrans = 4.50 Hz. m

The lowest natural bending frequency of the overall system is determined from

1 1 1 = 2+ 2 2 f f1 f2

Æ fmin = 2.06 Hz, hence a frequency decrease of approx. 11% and leading to a natural frequency that is now clearly in the range with highest risk of resonance (see below).

Risk of resonance Uncomfortable vibrations in structures can occur only, if the structure is excited at or close to one of its natural frequencies. It is thus essential to analyze the natural modes and frequencies (a.k.a. eigenmodes and -frequencies) of the structure investigated. Considering also the higher harmonics of dynamic pedestrian loading, it can be concluded that o for an eigenmode with vertical movement (vertical modes), there is only risk of resonance if the lowest natural frequency is below 5 Hz. The risk of resonance is highest for natural frequencies of 1.7…2.1 Hz. o for an eigenmode with lateral movement (lateral modes), there is only risk of resonance if the lowest natural frequency is below 2.5 Hz. The risk is highest for natural frequencies of 0.5…1.1 Hz. Pedestrian loading Since the excitation of a structure is induced by moving people, the risk of resonance is related to the walking frequencies of pedestrians, i.e. their pace rate (in steps per second). It has been shown experimentally that the force on a structure exercised by moving pedestrians depends on many parameters – such as gait, crowd density, physiology, clothing characteristics, surface roughness etc. –, but is principally of a periodic type. It exhibits natural frequencies for different gaits as shown in Table 1: 2007-03-20 dz

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Gait

Ground contact

Walking speed [m/s]

Pace rates [Hz]

Walking

Constant

1.1…2.2

1.6…2.4

3.3…5.5

2.0…3.5

Running Intermittent

The pace rate at normal, unhindered walking has a mean value of approx. 2 Hz (1.8…1.9 Hz acc. to more recent experiments).

The standard deviation of the pace rate is approx. Table 1 Stepping frequencies for different gaits. 0.2 Hz (0.175…0.220 Hz, depending on reference). The average walking speed is approx. 1.5 m/s. Fig. 1 shows some densities of walking pedestrians, and Fig. 2 shows people standing in a predefined space of 4.0 m2. From these figures, it becomes evident why movement is restricted with already rather low pedestrian densities.

Fig. 1

Walking pedestrian densities [from fib Bulletin Nr. 32].

It should be noted that already with a pedestrian density of 0.3…0.6 Persons/m2 (or a uniform pedestrian mass of 20…40 kg/m2), the freedom of movement is occasionally hindered. For densities of 0.6…1.0 Persons/m2 (or a uniform pedestrian mass of 40…70 kg/m2), the freedom of movement is clearly restricted. For very dense crowds with d ≥ 1.0 Persons/m2, the pedestrians are no longer able to choose pace freely. This leads to considerably decreased dynamic pedestrian loading. The vertical force exerted by pedestrians has the same frequency as the pace rate. Since the center of gravity of the human body moves laterally and longitudinally when walking, horizontal forces are also exerted laterally and horizontally.

Fig. 2

Densities of stationary people with 2.75, 4.25 and 5.75 Persons/m2 (or uniform pedestrian masses of 200, 300 and 400 kg/m2, respectively).

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The frequency for lateral forces is half the pace rate (left foot – right foot); for longitudinal forces, it is the same as for vertical forces. Usually, longitudinal forces are not governing design situations. They may have significant influence, if the structure is supported on slender, flexible piers. The lateral forces, however, should not be neglected in design (they were the main reason together with ‘lock-in’ effects1 for the temporary closure of the Solferino Bridge and the Millennium Bridge). Structural damping The structural damping ratio ξ represents the viscosity of the structure, i.e. the structure’s ability to dissipate kinetic energy. Structural damping mainly depends on structural material and construction details; its influence on the dynamic behavior primarily depends on the movement of the structure (not the frequency). That is why damping at ultimate limit state (ULS) is significantly higher than at serviceability limit state (SLS). The structural damping ratio usually has to be determined experimentally, but some empirical values for design purposes exist. Structural damping has a significant beneficial influence on the dynamic response of a structure. It should therefore be estimated conservatively. In Fig. 3, the value Ω represents the ratio of excitation frequency ω to natural frequency ω0 of the system. Resonance occurs at Ω ≈ 1 (a bit higher in theory, since it is depending on ξ as well – but ξmax = ca. 3%, hence no significant error).

Dynamic amplification

5.0

0% 10%

4.0

20% 30%

3.0

40%

2.0 1.0

Ω

0.0 0

Fig. 3

1

2

3

Dynamic amplification – influence of structural damping ratio and excitation frequency.

The maximum dynamic amplification of the structure’s response is given by Amax (Ω) ≈ 1 / 2ξ. Hence, for a damping ratio ξ = 0.5% – which is a realistic value for steel or steelconcrete-composite structures at SLS – the dynamic response is 100x higher than the static response! It should be noted that for ξ t 0 B Amax (Ω) t ∞ Î resonance!

As can be seen in Fig. 3, the structural response is already significantly amplified if the excitation frequency is still below (or above) the natural frequency of the system. Usually, only the structural response at resonance is of interest. However, such considerations may be made when composed systems are investigated (see below). As the damping ratio has such a strong influence on the structural response close to or at resonance, it is fundamental to consider conservative estimations and to apply ranges of damping ratios for sensitivity analysis purposes. Table 2 shows values for structural damping ratios ξ for SLS and ULS investigations. 1

´Lock-in effect’ or ‘forced synchronization’ refers to the gradual alignment of initially randomly walking pedestrians in terms of pace rates and phase shifts to a common frequency and shift, getting in phase with the movements of a structure. For horizontal structure movement, the pedestrian tends to counteract the movement; this is more commonly known as ‘sailor’s walk’. You can experience such an alignment every morning in the narrow and crowded tunnels of the Metro…

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Structural damping ratio ξ

Construction material

SLS

ULS

Reinforced concrete

1.3%

5%

Pre-stressed concrete

1.0%

2%

Composite (steel-concrete)

0.6%

2% (welded)

Steel

0.4%

4% (bolted)

Timber

1.0% (traditional connections) 1.5% (mechanical connections)

GFRP

2…3%

… …

Table 2 Structural damping ratios. Design parameters from codes Since the primary index of potentially insufficient dynamic behavior of a structure is related to the phenomenon of resonance, code rules mainly focus on limits for critical frequencies of structures. From the relevant codes, it can be deduced that if the lowest natural frequency lies above 5 Hz for vertical movement and 2.5 Hz for lateral movement, there is no need for refined dynamic investigations. Some regulations additionally give values for comfort thresholds, even if comfort is a highly subjective issue depending on the pedestrians’ age/fitness and position (stationary/moving), period of exposure to vibrations and the level of threshold considered (perception, disturbance, hazardous). However, it is important to note that pedestrians are much more sensitive to lateral displacements than vertical displacements. Concerning comfort thresholds, reference is usually made to critical values of acceleration since this (and the corresponding displacement velocity) is what the pedestrian feels in the first place. In general, this also implies the definition of corresponding load cases and checks at SLS level. Many codes, however, only define the threshold values, but do not define the corresponding load case. Generally, the acceleration limits for vertical vibrations defined by code specifications lie in the range of 0.5…0.8 m/s2. For lateral vibrations, the limits are in the range of 0.2…0.4 m/s2. It should be noted that fulfillment of specified limits of deflections for specific load cases (e.g. w ≤ L/1200 for frequent load combinations) does not necessarily lead to a structure with reasonable dynamic properties (see examples below). Correlation of dynamic parameters The dynamic parameters of acceleration a, displacement velocity v and displacement δ are interrelated by the natural frequency f: o a = (2π·f)2·δ o a = (2π·f)·v Accordingly, the displacement velocities will significantly differ for modal cases with the same frequency but different accelerations. 2007-03-20 dz

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A vertical displacement velocity of 24 mm/s (or 0.3 m/s2 at 2 Hz) is said to be perceptible by a pedestrian. For a vertical displacement of 10…20 mm, the pedestrian is said to adjust his movement to the movement of the structure, leading to potential vertical ‘lock-in’; this corresponds to an acceleration of at least 1.6 m/s2 or a displacement velocity of 125 mm/s at a frequency of 2 Hz; hence, being already rather uncomfortable (see above). For lateral vibration, tests have shown that ‘lock-in’ effects might already occur at significantly lower acceleration levels than accepted by code previsions. A pedestrian is said to adjust his gait and pace rate from a lateral displacement of approx. 2 mm. At a pace rate of 2 Hz, this corresponds to an acceleration of 0.3 m/s2 and a displacement velocity of 25 mm/s. This, again, clearly shows the pedestrian’s higher sensitivity to lateral movement. Countermeasures for structures with insufficient dynamic behavior o Increase of natural frequency (‘high-tuning’): The easiest approach to avoid inappropriate dynamic behavior is to keep the structure’s natural frequency outside the range of risk of resonance. The frequency depends on the square of the span and the square root of stiffness to mass ratio (see above). Hence, the most evident solution to increase natural frequency is a decrease of span length or an increase in stiffness while keeping masses constant. However, span lengths usually are given from other circumstances, and stiffness increase often implies mass increase as well. Hence, the most efficient stiffness increase with almost negligible mass increase is an increase in static height, i.e. decrease of slenderness. o Increase structural damping: Structural damping does not or only hardly affect natural frequencies but significantly changes the dynamic response of a structure in terms of accelerations. It may therefore be attempted to increase structural damping by application of bolted instead of welded connections, modify independent structural parts into composite elements, apply mesh safety railing systems with high damping, etc. However, at design stage, the structural damping ratio can just be estimated; it can only be determined exactly for the completed structure from dynamic load tests. o Install dampers: tuned mass dampers, viscous dampers etc. If a structure is critical in terms of dynamic behavior, it may be reasonable to provide the construction space for later installation of dampers at design stage. Further literature o Referred publication: ‘Assessment of Pedestrian-Induced Vibratory Behaviour of Footbridges’, Association Française de Génie Civil AFGC, 2006. o fib Bulletin Nr. 32 ‘Guidelines for the design of footbridges’, International Federation for Structural Concrete (fib), 2005. o Bachmann, H. et al. ‘Vibration Problems in Structures – Practical guidelines’, Birkhäuser, 2nd edition, 1997. o Bachmann, H., Ammann, W. ‘Vibrations in Structures – induced by Man and Machines’, International Association for Bridge and Structural Engineering IABSE, Structural Engineering Documents SED 3e, 1987. o CEB bulletin Nr. 209 ‘Practical guidelines […]’, 1991.

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Summary of methodology proposed by AFGC Design procedure The verification of the dynamic behavior of a footbridge according to the AFGC publication consists in the following steps: 1. Choose bridge class in terms of traffic and comfort level (in consultation with ownership – at least be able to advise a bridge class to the client) 2. Calculate natural frequencies (vertical, horizontal lateral, horizontal longitudinal) for two cases of mass distribution o Bridge with dead loads only o Bridge deck additionally loaded with 1.0 pedestrian / m2 (mp = 70 kg/m2, see Table 4) over entire surface 3. Assess level of risk of resonance with Fig. 4 for vertical and longitudinal vibrations and with Fig. 5 for lateral vibrations, respectively. 4. Investigate dynamic load cases acc. Fig. 6 for the chosen bridge class if risk of resonance is in ranges 1 to 3. 5. Calculate maximum accelerations and assess corresponding comfort levels acc. to Fig. 9 and Fig. 10, respectively. Frequency [Hz] 0

1

1.7

2.1

2.6

5

Range 1 Range 2 Range 3 Range 4 Fig. 4

Risk of resonance ranges for vertical and longitudinal vibrations.

Frequency [Hz] 0

0.3

0.5

1.1

1.3

2.5

Range 1 Range 2 Range 3 Range 4 Fig. 5

Risk of resonance ranges for lateral vibrations.

The ranges of risk of resonance have the following definitions: • Range 1

maximum risk of resonance

• Range 2

medium risk of resonance

• Range 3

low risk of resonance (for standard load cases)

• Range 4

negligible risk of resonance

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Bridge classes Bridge classes can be estimated / chosen from the following descriptions: • Class I

urban footbridge with dense traffic, with high concentrations of pedestrian traffic or frequently used by dense crowds (e.g. near train/metro station, tourist routes, protest marches).

• Class II

urban footbridge connecting populated areas, subject to considerable pedestrian traffic and sometimes loaded over its entire surface (e.g. footbridge in front of school).

• Class III

bridge with normal use, may sometimes be crossed by large groups of pedestrians but is never loaded over its entire surface (e.g. footbridge in suburban area).

• Class IV

bridge used very little, connects only lightly populated areas, provides pedestrian continuity in areas severed by a motorway (e.g. footbridge in remote area).

Dynamic load cases The dynamic load cases to be investigated reflect the following circumstances: • Case 1

sparse and dense crowds

• Case 2

very dense crowd

• Case 3

crowd complement (influence of 2nd harmonic of crowd) Range of natural frequency (risk of resonance)

Traffic

Bridge class

1

2

3

Very dense

I

2

2

3

Dense

II

1

1

3

Sparse

III

1

--

--

Fig. 6

Dynamic load cases to be applied for dynamic verifications.

Generally, the dynamic load cases are defined with a harmonic load that has to be applied in the most unfavorable position of the bridge, i.e. corresponding to the modal shape of the investigated natural frequency fi: • Vertical loading

Fdyn,v = d · Gdyn,v · cos (2π·fv·t) · η · ψi [N/m2 = Pa]

• Longitudinal loading

Fdyn,l = d · Gdyn,l · cos (2π·fl·t) · η · ψi [N/m2 = Pa]

• Lateral loading

Fdyn,t = d · Gdyn,t · cos (2π·ft·t) · η · ψi [N/m2 = Pa]

Where

d=

density of pedestrians acc. to bridge class, see Table 4

Gdyn.i = dynamic component of single pedestrian mass, see Table 3 η=

factor for equivalent pedestrians with same frequency and footfall correlation, see Table 4

ψi =

reduction factor to account for decreasing risk of resonance outside range 1 depending on investigated load case, see Fig. 7 and Fig. 8.

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Dynamic component of single pedestrian

Load case 1

2

3

Vertical

280

140

35

Longitudinal

280

140

35

70

35

7

Gdyn,i [N]

Lateral

Table 3 Dynamic components of pedestrian mass.

Bridge class

Crowd density d [ped./m2]

I

Note: the dynamic component does not include the static influence of pedestrian mass. This influence has to be analyzed separately. However, with the usually considered uniformly distributed static load (approx. 3…4 kPa), the influence of ‘effective’ pedestrian mass is covered.

Factor η for equivalent pedestrians for load case 1

2

3

1.0

--

1.85 · √(1 / N)

1.85 · √(1 / N)

II

0.8

10.8 · √(ξ / N)

--

10.8 · √(ξ / N)

III

0.5

10.8 · √(ξ / N)

--

--

Table 4 Crowd densities and factor for equivalent pedestrians. Where

ξ=

structural damping ratio acc. to Table 2

N=

total number of pedestrians on bridge deck = Adeck · d

η=

factor for consideration of number of equivalent pedestrians, i.e. uniformly distributed pedestrians with common pace rate and footfall correlation producing the same dynamic response as randomly distributed pedestrians with random pace rates (within limits) and random footfall as determined empirically.

ψ 1, 2

ψ 1, 2

Vertical / longit.

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

f [Hz]

0.0 0.5

Fig. 7

1.0

1.5

2.0

2.5

3.0

Lateral

f [Hz]

0.0 0.0

0.5

1.0

1.5

Reduction factor ψ1,2 for dynamic load cases 1 and 2.

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ψ3

ψ3

Vertical / longit.

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

f [Hz]

0.0 2.0

Fig. 8

3.0

4.0

5.0

Lateral

f [Hz]

0.0

6.0

1.0

1.5

2.0

2.5

3.0

Reduction factor ψ3 for dynamic load case 3.

Comfort levels and acceleration limits The comfort levels are defined with acceleration ranges, which have to be determined from the corresponding dynamic load case. Acceleration [m/s2] Level 1

0

0.5

1.0

2.5

Level 2 Level 3 Level 4 Fig. 9

Comfort levels for vertical accelerations.

Acceleration [m/s2] Level 1

0

0.10

0.15

0.30

0.80

Level 2 Level 3 Level 4 Fig. 10 Comfort levels for horizontal accelerations. Note: to avoid forced synchronization (‘lock-in’), the lateral accelerations are limited to 0.10 m/s2. The comfort levels have the following definitions: • Level 1

maximum comfort – acceleration practically imperceptible

• Level 2

medium comfort – acceleration merely perceptible

• Level 3

minimum comfort – acceleration felt but not unbearable

• Level 4

unacceptable accelerations

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The acceleration for pure bending mode shapes can be calculated from the maximum dynamic force Fmax (in kN per unit length or unit area) from • amax = (2ξ)-1 · (4·Fmax / (π·m))

with

o ξ = structural damping ratio acc. to Table 2 o m = mass per unit length or unit area Note: Fmax is calculated for t = 0. Note: mass m per unit length / area should also consider mass of pedestrians and number of pedestrians / m2 acc. to Table 4. It is assumed in general, that an average pedestrian has a mass of 70 kg.

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Application to footbridges of Zumaia and Medellín Footbridge in Zumaia

¾ Structural modeling The steel members of the longitudinal girders and the transverse girders are introduced with (vertical) offsets in order to allow modeling of the GFRP sheets with their proper height. The sheets’ lengths, however, are modeled from post to post axis; hence, the stiffness of the sheets tends to be modeled too soft. The GFRP deck planks are modeled as independent beams tending to be modeled too soft as well (transverse coupling of sheets Æ increased plate stiffness). Special attention is paid to the modeling of the connections of GFRP transverse girders with the bottom chords of the steel Vierendeel trusses. The following alternatives for the joint model are investigated: o Clamped for the profile’s strong axis, hinged for the other axes o Elastically clamped for the profile’s strong axis, hinged for the other axes o Pinned, horizontally eccentric: due to geometry of bottom chord o Pinned centric connection o Clamped for all axes It can be shown that the type of joint model has no influence on the order of natural modes and only little influence on the natural frequencies in the decisive range of f = 0…5.0 Hz; the further investigations are therefore carried out with clamped joints. The global stiffness of the structural model tends to be underestimated, thereby underestimating the natural frequencies on the safe side. ¾ Bridge class and comfort level For the Zumaia footbridge, it is decided to achieve bridge class I or II and considering acceleration ranges of medium to maximum comfort (see Fig. 9 and Fig. 10), with o Medium comfort level for bridge class I: a ≤ 1.0 m/s2, d = 1.0 Ped./m2 o Maximum comfort level for bridge class II: a ≤ 0.5 m/s2, d = 0.8 Ped./m2 2007-03-20 dz

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¾ Eigenmodes and -frequencies – risk of resonance Depending on the bridge class investigated, different pedestrian masses have to be considered as additional masses in the determination of eigenmodes and eigenfrequencies. The natural modes and frequencies are found to be (bridge class I / II with corresponding pedestrian masses): o 1st mode: (pure) vertical bending, with f1 = 2.51 / 2.56 Hz. o 2nd mode: torsion with slight lateral bending, with f2 = 3.11 / 3.15 Hz. o 3rd mode: lateral bending with torsion, with f3 = 3.57 / 3.63 Hz. o Further modes: fi ≥ 6.1 Hz

Fig. 11 Eigenmodes of Zumaia footbridge. The risk of resonance can thus be judged medium for mode 1, low for mode 2 (vertical movement) and negligible for mode 3 (lateral movement), see Fig. 4 and Fig. 5. ¾ Investigation of dynamic load cases and influence of structural damping Structural damping is assumed for GFRP according to Table 2 with ξ = 2…3% (practically whole deck in GFRP). The maximum accelerations are determined by simple calculations according to the equations introduced above. The accelerations result to be (for bridge classes I / II): o Mode 1:

amax = 0.75…1.15 / 0.3…0.4 m/s2

o Mode 2:

amax = 0.70…1.00 / 0.7…0.9 m/s2

It is necessary to realize that the accelerations for mode 2 are calculated as if for a bending mode. This is assumed to be on the safe side. The corresponding mode shape could also be judged as a mode with lateral movement and hence negligible risk of resonance; that is why it is judged appropriate to allow a medium comfort level. Mode 3 is not investigated in more detail since it exhibits negligible risk of resonance. The maximum deflection of the structure for frequent variable loads amounts to L/1000, a value indicating a rather high stiffness. However, compared to the results of the dynamic investigation the bridge provides “only” a medium comfort level. ¾ General assessment of dynamic behavior of Zumaia footbridge The modeling of sections and above all static height of the longitudinal girder underwent several iterations in order to reach a satisfactory dynamic behavior. The bridge is expected to provide sufficient dynamic behavior, i.e. practically unfelt to merely perceptible accelerations for rather high crowd densities. The accelerations highly depend on the structural damping. The damping of the final structure will therefore be evaluated in a dynamic load test. 2007-03-20 dz

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Footbridge in Medellín

¾ Structural modeling The structural system is very similar to that of the Zumaia footbridge. The same parameters are considered in terms of structural modeling of GFRP sheets, offsets of steel members and deck planks. However, the joints of the transverse girders in stainless steel are modeled as pinned (except girders named ‘end box’) and additional bending hinges (in weak axis) are introduced to model the effective length and joints of the deck planks.

Here as well, the global stiffness is likely to be underestimated and hence leading to safe estimations of the natural frequencies. ¾ Bridge class and comfort level For the Medellín footbridge, the following dynamic target values are chosen: o Medium comfort level for bridge class II: avert ≤ 1.0 m/s2, alat ≤ 0.1 m/s2 2007-03-20 dz

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¾ Eigenmodes and -frequencies – risk of resonance The natural modes and frequencies are found to be: o 1st mode: horizontal lateral bending, with f1 = 2.47 Hz. o 2nd mode: vertical bending, with f2 = 3.28 Hz. o 3rd mode: torsion, with f3 = 4.72 Hz. o Further modes: fi ≥ 6.0 Hz

Fig. 12 First three natural modes of Medellín footbridge. The risk of resonance can be judged low for all modes, see Fig. 4 and Fig. 5; nevertheless, investigation of dynamic load case 3 is required, see Fig. 6. ¾ Investigation of dynamic load cases and influence of structural damping Due to the higher percentage of steel members, the structural damping is assumed to vary ξ = 0.4…0.6…2.0…3%. This way, more or less the whole range of possible structural damping ratios is investigated. For the lateral bending mode 1 with f1 = 2.47 Hz with the characteristic values are determined from (for ξ = 0.4…0.6…2.0…3%) o amax = (2ξ)-1 · (4·Fmax / (π·m)) = 0.021…0.017…0.010…0.008 m/s2 o vmax = a / (2π·f) = 1.4…1.1…0.6…0.5 mm/s o δmax = a / (2π·f)2 = 0.09…0.07…0.04…0.03 mm The accelerations clearly indicate a maximum comfort level, see Fig. 10, and are as well undoubtedly below the critical value for potential ‘lock-in’ (forced synchronization). The maximum lateral deflection is also clearly below the critical value of approx. 2 mm for beginning pedestrian adjustment due to lateral displacement. Additionally, the bridge deck is clearly modeled with a too low horizontal stiffness (transverse coupling of deck planks) and therefore, the risk of ‘lock-in’ is overestimated. For vertical bending mode 2 with f2 = 3.28 Hz, the natural frequency can be confirmed with an accuracy of 1.2% by simple hand calculations which consider an average bending stiffness of the longitudinal girders consisting of the steel sections of top and bottom chord and the GFRP sheets. The dynamic characteristics are determined by the equations introduced above and result in (for ξ = 0.4…0.6…2.0…3%): o amax = 2.59…2.11…1.16…0.95 m/s2 o vmax = 125.8…102.7…56.2…45.9 mm/s o δmax = 6.1…5.0…2.7…2.2 mm It can be seen, that the limit value of acceleration can only be satisfied for a high damping of approx. 3%. Having the investigation of the more complex torsion mode 3 in mind, it is decided to perform a more refined time-history analysis. 2007-03-20 dz

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¾ Time-history analysis of modes 2 and 3 In order to perform a time-history analysis (THA), the time-dependent part of the pedestrian loading has to be modeled numerically and integrated over a certain period. The most important parameters to be paid attention to are: • active mass, including pedestrian mass • time-function of dynamic load, especially time-step increments • integration time limit, in order to cover transient effects ROBOT can automatically convert defined load cases into dynamic masses. For the definition of the time-step increments, the involved period time (= inverse of frequency) gives a good estimation when divided by 30 corresponding to 30 points to represent one complete period of the time-function. As a value for the maximum integration time, two to three times the time that a pedestrian needs to cross the bridge seems reasonable (also refer to walking speeds in Table 1). The applied procedure for the time-history is as follows: o introduce a unity load case q = 1.0 kPa on the deck planks o convert to mass with factor fm = 56 / (1.0·103/9.81) = 0.54936 o apply factor to unity load for static fq = d· Gdyn,v·10.8·√(ξ/N)·ψ = 65.4·10-3·√ξ

part

of

dynamic

load:

o multiply static part with time-function cos (2π·f2·t), defined in time-steps of 0.01 s Note: Gdyn,v = 70 N, see Table 3; N = ADeck·0.8 Ped./m2 = 3·25.5·0.8 = 61.2; ψ = ψ3 (f2 = 3.28 Hz) = 0.846; ξ = 0.4…0.6…2.0…3%. The results for vertical bending mode 2 present them as follows: o amax = 1.59…1.34…0.74…0.60 m/s2 o vmax = 77.5…65.3…36.0…29.4 mm/s o δmax = 3.8…3.2…1.8…1.4 mm

Fig. 13 Displacement velocity at mid-point of deck surface for ξ = 3.0%. From Fig. 13 as an example, it can be seen that the structure ‘needs’ approx. 5…7 s to adapt to the dynamic loading. This is what is referred to as transient part of the loading. The higher the damping, the sooner this transient part can be neglected. The comparison of hand calculations and THA show that the results of the latter are approx. 40% lower, hence requiring more explanations. The main reasons for this difference are the accounting in THA for: • local stiffness distribution (transverse stiffness, joints in deck planks and of transverse girders to bottom chords). 2007-03-20 dz

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• local mass distribution (main part of total mass concentrated in longitudinal girders). The consequence is that the local structure directly exposed to the load – i.e. the deck – is not excited at its natural frequency but considerably below, also see Fig. 3. Therefore, the local dynamic amplification is noticeably lower than for the global system leading to a decreased global dynamic amplification. Such effects might be generally considered with composed systems, e.g. a system with a longitudinal system and more or less independent structure for the transverse deck. However, the investigation has to consider the natural frequency of the global system. For the torsion mode 3 with f3 = 4.72 Hz, similar investigations are performed. The procedure of calculation is the same as for mode 2, except that ψ3 = 0.347. The resulting dynamic characteristics are: o amax = 0.32…0.26…0.15…0.12 m/s2 o vmax = 10.7…8.9…4.9…4.0 mm/s o δmax = 0.4…0.3…0.2…0.1 mm ¾ General assessment of dynamic behavior of Medellín footbridge

1.80

a max [m/s2]

Influence of damping ratio ξ Bending mode f = 3,28 Hz

Torsion mode f = 4,72 Hz

1.60

Potencial (Bending mode f = 3,28 Hz)

1.40

Potencial (Torsion mode f = 4,72 Hz)

1.20 y = 1.0343x -0.484 R2 = 0.9995

1.00 0.80 0.60

y = 0.2034x -0.4927 R2 = 0.9999

0.40 0.20

ξ [%]

0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Fig. 14 Assessment of dynamic behavior of Medellín footbridge. From Fig. 14 and Fig. 9 it is evident that for a structural damping ratio ξ of approx. 1%, a medium comfort level for the vertical bending mode can be achieved. This structural damping ratio is assessed as realistic or even on the safe side due to the high number of bolted connections (transverse girders, GFRP sheets), the too low stiffness considered for the deck structure and the neglect of the handrail structure. For the torsion mode, a maximum comfort level can be attested. It is therefore concluded that the Medellín footbridge will exhibit a satisfactory dynamic behavior; nevertheless, with rather large crowds on the bridge the vibrations will be clearly perceptible but should not lead to disturbance of the gait of single pedestrians. The maximum deflection of the structure for frequent variable loads amounts to L/1500, a value indicating a very high stiffness. However, compared to the results of the dynamic investigation the bridge may provide ‘only’ a medium comfort level.

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Some remarks on working with ROBOT in the Zumaia / Medellín bridges Modeling • Pay attention on offsets when introducing new bar members; if required, you have to activate the separately defined offset. • When you use tapered cross-sections (i.e. variable geometry along a bar), you should be aware that ROBOT calculates intermediate sections when you introduce other bars dividing these tapered bars. Unfortunately, ROBOT does not display the properties of these interpolated sections. You have to rely on it that the program does it right… • When new bar members are introduced that divide other bar members, the latter are changed into independent bars. This might lead to a significant number of bars. If you want to select these bars (e.g. to change something on their properties), ROBOT offers the possibility to select bars by the name of the cross-section. • You all experienced it: ROBOT offers great possibilities but is a bit unstable sometimes… So: save, save, save! Especially, when: o You completed a time-consuming modeling task, e.g. the introduction of plates and shell elements (‘panels’). o You intend to start a more extensive calculation. I always had a “crash” when starting THA without prior saving… o You intend to see the preview of printout or want to start its compilation. There is also the option to show only simplified views of the graphs – saves you a lot of waiting time… o You finished all the calculations for a structure, save it under a new name. The file size will considerably decrease, because ROBOT deletes some (obviously crazily big) log files. • If you want to use orthotropic shells, e.g. the GFRP sheets in the Zumaia and Medellín footbridges, you should be aware that the option ‘material orthotropy’ offered by ROBOT does not produce correct values for the stiffness matrices! The stiffness matrices of a ‘genuine’ orthotropic material with E-moduli Ex, Ey and Poisson ratios νx and νy (= Ey / Ex · νx!) can be calculated from: Membrane stiffness matrix D

Ex ⋅t 1 −ν x ⋅ν y

ν y ⋅ Ex ⋅t 1 −ν x ⋅ν y

ν y ⋅ Ex ⋅t 1 −ν x ⋅ν y

Ey ⋅ t 1 −ν x ⋅ν y

0

0

Flexural stiffness matrix K

0

Ex ⋅t 3 12 ⋅ (1 −ν x ⋅ν y )

ν y ⋅ Ex ⋅t 3 12 ⋅ (1 −ν x ⋅ν y )

0

ν y ⋅ Ex ⋅t 3 12 ⋅ (1 −ν x ⋅ν y )

Ey ⋅ t 3

12 ⋅ (1 −ν x ⋅ν y )

G ⋅t

0

0

0

0

G ⋅t 3 12

Shear stiffness matrix H

G ⋅t ∗

0

0

G ⋅t ∗

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Note: t* considers shear effective height, e.g. 5/6·t for rectangular cross section.

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Calculations and design • The very extensive and time-consuming time-history analysis should only be performed when there is a real perspective for better results than by a hand calculation. o The resulting data amount is huge (e.g. result file of Medellín bridge: 1.3…2.1 GB, depending on integration period…). This huge data makes ROBOT extremely slow and unstable. o I had several “crashes” during the filtering of data for results. In order to be able to handle the data, I split the calculations for the different structural damping ratios in separate files. o Because of the huge amount of date to be due, calculate the file on your local hard disk in order to be independent of the (occasionally) busy network. A file in ‘Mis documentos’ is not saved local but on the server. • Do not rely on the results of an elastic-plastic analysis of a structure with ROBOT without simplified hand calculations! I tried to do such an analysis but I could not trust the results… it was simply wrong. • Elastically clamped joints require a certain solver algorithm that may not be applied to bar members with offsets. These two features can therefore not be combined. • If you have to apply IAP rules for load combinations, i.e. load and resistance factor design (LRFD), be aware that these combinations are not predefined in the code regulations under ‘Tools Æ Job preferences Æ Actions’. You have to define it yourself and should (in this order!): o Select a LRFD-compatible code (e.g. EC) as a basic template. o Change the load factors acc. to IAP o For the load combinations choose formulas (4) and (19) for ULS, formulas (1) and (21) for rare/characteristic combinations in SLS, and formulas (1) and (20) for frequent SLS load combinations. o Change the name of the code (‘Norma’) o Close all ‘Job preferences’ windows o Open the ‘Job preferences’ again, select the created file, and confirm all further questions of the programme with ‘Si’. • GFRP sheets o

Of course, ROBOT does not check buckling of GFRP sheets (unless told to do so…); for corresponding formulas to calculate the buckling shear and normal stresses refer to ‘Composites for Construction’ by L.C. Bank (saved under G:\Publicaciones Técnicas\GFRP\NormasManuales).

o

The choice of sheet thickness (2·8 mm) for the Zumaia footbridge was initially done for the assumption that the GFRP sheets carry the whole shear, i.e. independent of their effective stiffness, and considering the shear buckling formula by Bank. The sheets showed significant reserves for SLS proofs.

o

To generate a significant structural damping, the connections of the sheets to the steel structure are designed with blind rivets (POP® rivets). The design is conducted for the differential normal forces in the adjoining chords, assuming equal distribution by the rivets.

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