Keywords

1 Introduction

In recent years, several serviceability losses of footbridges have captured the attention of the Civil Engineering community. Although the problem of excessive vertical vibrations has been known for a long time, the most recent failures came rather unexpected, because they were all caused by excessive lateral vibrations.

A first example is the T-bridge in Japan, a cable-stayed footbridge that exhibited excessive lateral vibrations due to a large number of people crossing it in 1989 (Fujino et al. 1993; Nakamura and Kawasaki 2006). The bridge was later provided with tuned liquid dampers (TLD) to control the lateral vibrations. The same problems were experienced by the Solferino bridge in Paris (France). In 1999, after being opened to the public, it had to be closed down because of excessive vibrations (Blekherman 2007). The failure, however, that received by far the most attention from the public occurred when the Millennium Bridge in London exhibited excessive lateral vibrations on its opening day in June 2000. It was closed shortly after, and provided with viscous dampers for lateral vibrations and tuned mass dampers for vertical vibrations, even though the latter had not caused problems (Dallard et al. 2001). More recently, similar cases were the object of several research works (Tubino et al. 2016, Lai et al. 2017).

All the aforementioned cases of excessive vibrations are a consequence of the closeness between footfall frequency and the natural frequencies of footbridges (Zivanovic 2005; Fujino and Siringoringo 2016). Table 1 shows the typical frequencies of human footfall (Bachmann 2002). It should be noted that the frequency values refer to the vertical and longitudinal direction, for which the action of the two feet of a pedestrian (left and right) takes place in the same direction. For the lateral direction, the frequencies are equal to half, due to the opposite action of the two feet. This aspect has received wide attention in recent years, especially after the closure of the Millennium Bridge. The research work carried out made it possible to highlight some peculiarities of the lateral action due to walking pedestrians, like, for example, the presence of a motion-induced load, that takes place at frequencies comprised between 0.6 and 1.2 Hz (Ingólfsson et al. 2011).

The amplification of the response to the action of walking pedestrians due to the dynamic behavior of footbridges may explain the significant amount of vibration observed, and thus the discomfort experienced by pedestrians. In addition, synchronization may take place in the presence of significant crowding: upon perception of a vibration in the lateral direction, pedestrians tend to synchronize their footfall frequency with that of the structure to maintain balance.

Table 1. Typical values of footfall frequency (Bachmann 2002)
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A first simple way for checking the serviceability of footbridges as regards vibrations is to determine the natural frequencies (through testing or analytical and numerical models), and to check whether they fall within critical ranges. Table 2 summarizes the critical ranges as reported in some international standards and guidelines (EN 1990 2004; SIA 2003; HIVOSS 2008). Footbridges with frequencies falling within those ranges can still be designed, provided that refined dynamic calculations are carried out, taking into account dynamic loading and the damping of the structure (whose role is crucial in resonance conditions). Based on the results obtained, serviceability checks must be performed by comparing the expected maximum acceleration values with limit values that are defined on the basis of the direction (vertical or lateral) and the frequency of the vibration.

This paper presents a case study concerning the dynamic behavior of a footbridge recently built in Milan. Following an episode of excessive lateral vibrations, shortly after being opened to the public, the structure was subjected to a strengthening intervention on the piers. Different methods for evaluating the effectiveness of the strengthening are compared, in order to draw general and design-oriented conclusions.

Table 2. Critical ranges for the natural frequencies of footbridges
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2 Description of the Structure

The structure investigated in this study is a steel footbridge built in 2010 to connect the newly built station of Subway 2 Assago Forum with the parking lot of a public hall hosting important sport events and concerts (Assago Forum).

The total length of the footbridge is equal to 180 m. It consists of 5 spans of length equal to 12, 48, 30, 36 and 30 m, with four intermediate piers. The longest span crosses the urban stretch of the heavy-duty A7 highway from Milan to Genoa. The footbridge is supported by corbels belonging to a reinforced concrete frame at one end, and by an H-shaped steel frame at the other (visible in Fig. 1a).

The deck consists of a 4.80 m wide corrugated sheet, with a lightweight concrete topping. The corrugated sheet rests on secondary HEA200 beams of length 6 m, which run parallel to the longitudinal axis of the footbridge. At the two sides of the main walkway, there are two 1.60 m wide sidewalks (not open to the public). The supporting structure of the deck consists of HEA300 beams, arranged in rectangular modules of size 6 × 7.3 m2 (in the longitudinal and transverse direction, respectively). Each module is provided with horizontal bracings, consisting of two coupled C-shaped UPN120 elements (UPN140 in the vicinity of the piers).

The transverse section of the footbridge consists of pointed arches (visible in Fig. 3a), at a spacing of 6 m (Fig. 2). The two curved beams of each arch, referred to as “cusps”, consist of HEA260 members (HEB260 close to the intermediate piers and to the extremities). All the arches are connected at the top by HEA500 longitudinal beams. Between one arch and the adjacent ones there are diagonal bracings, consisting of C-shaped UPN180 elements (UPN200 in the vicinity of the piers that delimit the longest span). Above the main walkway there are small transverse arches resting on the cusps, and consisting of HEB160 beams that support the roof, together with other elements, in connection with the diagonal bracings.

As it was mentioned previously, the structure rests on four piers (Fig. 3) characterized by a peculiar V shape. By indicating the longitudinal axis that runs parallel to the footbridge from the station (Y = 0) to the parking lot (Y = 180 m), the four piers are located at Y = 15, 69, 105 and 159 m, respectively. Following the cited excessive vibration, the piers were stiffened by means of diagonal braces (diameter = 273 mm, thickness = 15 mm) acting in longitudinal and transverse directions at piers V2 (Y = 69 m) and V4 (Y = 159 m, Fig. 1b), and only in the transverse direction in V3 (Y = 105 m). The choice of the strengthening strategy was dictated by the peculiar shape of the piers, which is rather unfavorable in the case of lateral loads.

Fig. 1.
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Views of the footbridge: (a) in direction SW, (b) after the insertion of diagonal braces

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Fig. 2.
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Transverse section of the footbridge

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2.1 Numerical Model

The dynamic behavior of the footbridge was investigated by means of numerical analyses developed with the commercial finite elements software MIDAS GEN (V.1.1). Two models were set, in order to make comparisons between the dynamic behavior of the structure before (Model 1) and after pier strengthening (Model 2).

All the structure members were modelled by beam elements, with the exception of the deck that was modelled by means of isotropic plate elements. The stiffening contribution due to the eccentricity between the middle surface of the deck and the centroid of the supporting secondary beams was taken into account.

The (welded) rigid moment connections were modelled as perfect constraints. For joints with fin or connecting plates, a hinge constraint was assumed (Fig. 3).

Fig. 3.
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3D view of the numerical model.

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The supports of the deck were modeled by means of rigid links. It is thus assumed that the effect of the supports deformation is of minor importance in terms of global behaviour. The same type of connection was used to model the connection between the feet of the inclined elements of the piers and the pad footings. Model 1 consists of 1857 beam elements, while Model 2 features 1862 beam elements. In addition, a total of 540 plate elements were used in both models.

Along with permanent loads, aload due to crowd (1 person/m2) equal to 0.70 kN/m2.

(larger than 5% of the deck mass) was considered in all the analyses, as per the indications of HIVOSS (2006). Linear elastic behavior was assumed with an elastic modulus of 210·106 kN/m2 for steel and 31.5·106 kN/m2 concrete.

3 Dynamic Behaviour of the Structure

3.1 Modal Analysis

The search for the natural frequencies and modes of the structure (respectively, eigenvalues and eigenvectors) was first performed. Figure 4 shows the most significant modes in the transverse direction, for Model 1 and 2, respectively. The choice of these modes was based on natural frequency f and participating mass Mx.

As regards Model 1, the behavior is governed by the first mode (Mode 1, lateral) with a natural frequency that is very close to the footfall one. The second relevant mode (Mode 4, lateral), has a frequency that is sufficiently high for the risk of synchronization to be averted. For Model 2, the first notable consequence of the insertion of the diagonal braces is the fact that in the transverse direction, the first relevant mode (Mode 2, lateral) is characterized by a modal deformation that involves only part of the structure (the longest span): this is a consequence of the stiffer behavior in the transverse direction, and more specifically of the increased lateral stiffness of the piers, and results in an increase in the associated natural frequency. In this case, the second transverse mode (Mode 3, lateral) presents values of natural frequency and mass that are close to those characterizing the first mode.

Fig. 4.
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First lateral modes of the footbridge: (left) Model 1 and (right) Model 2.

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Table 3 summarizes the main results of the modal analysis, where the values obtained considering the footbridge with/without full crowd loading are compared.

Table 3. Natural frequencies and participating masses with/without full crowd loading.
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3.2 Model Validation

After pier strengthening, the dynamic behavior was monitored in two phases, during two concerts at the nearby public hall. The vibrations were measured using three geophones (A, B and C). Sensor A was close to the first support after the stairs (Y = 180 m); sensor B was on the pier next to the motorway (V2, Y = 69 m); sensor C was at the middle of the span that crosses the highway (Y = 42 m). In the second monitoring phase, sensor A was moved next to pier V3 (Y = 105 m).

The main results of the monitoring are summarized in Table 4. On the whole, there is a satisfactory agreement between the dominant frequencies detected experimentally and the values obtained by means of Model 2 in the presence of dense crowds (1.70, 2.01 and 3.76 Hz, respectively), with differences that are lower than 5% in all cases. The small difference in the frequency of the vertical vibration could be easily explained by considering that the number of people on the footbridge during monitoring was smaller than that assumed in the model.

Table 4. Dominant frequencies [Hz] measured during the monitoring campaign
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Table 5. Maximum number of persons for avoiding lock-in
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3.3 Linear Dynamic Analysis Under Moving Pedestrians

The traditional serviceability assessment for a structure of this type is based on linear analysis under the effect of a moving load, representing a single pedestrian, to determine the corresponding values of peak acceleration amax. The structural behavior under crowd loading can then be estimated by considering different criteria: (1) assuming that a number n of equal pedestrians arrived on the footbridge, and that their action is uncorrelated; the maximum acceleration amax,n is equal to amax multiplied by the square root of n (Matsumoto et al. 1978); (2) additional observations performed on actual bridges in the presence of crowd loading (Fujino et al. 1993) revealed that the effective number of pedestrians in synchronization can be assumed as 0.2 n. For large bridges (likely to be crossed by more than 25 people) the second assumption leads to larger values of amax,n in comparison to the first one.

The time-forcing function due to a single pedestrian is modeled as a traveling load, with components in all three directions, a speed of 2 paces/s and a pace length of 1 m. The time histories of the components in the vertical, longitudinal and transverse direction were taken from the literature (Andriacchi et al. 1977, Zivanovic et al. 2005). The values of acceleration were evaluated by modal superposition, at the nodes corresponding to the maximum modal displacements (Modes 1, 3 and 4 for Model 1; Modes 2 and 3 for Model 2). A damping factor equal to 0.4% was assumed.

The results are shown in Figs. 5 and 6 for the two models. The effects of pier strengthening become evident by comparing the maximum values of acceleration, with a decrease from 5 (Fig. 5a, Y = 84 m) to 2 mm/s2 (Fig. 6a, Y = 42 m). The discrete Fourier transform (DFT Figs. 5c and 6c) highlights the role played by the first modes. On the contrary, the acceleration at the western extremity (Y = 180 m) exhibits a significant increase, from 8 (Fig. 5b) to 12 mm/s2 (Fig. 6b): the DFT (Figs. 5d and 6d) clearly shows how the response is governed by a higher frequency component (5 Hz, Model 2, Fig. 6d), with the ensuing increase of the acceleration.

3.4 Linear Dynamic Analysis Under Load Models

The values of peak acceleration due to crowd loading synchronized with the natural frequencies of the footbridge can be estimated by means of harmonic load models (HIVOSS 2006; SETRA 2004), tuned to the natural frequencies that are most likely to be excited by synchronized crowd walking.

The harmonic load p(t) that represents the equivalent pedestrian stream is defined as:

$$ p(t) = P\cos \left( {2\pi f_s t} \right)n^{\prime}\psi $$

where P is the force due to a single pedestrian (= 35 N for lateral vibrations); fs is the step frequency, which is assumed as being equal to the natural frequency under consideration; n’is the equivalent number of pedestrians on the loaded surface of the footbridge; ψis a reduction coefficient accounting for the probability of the footfall frequency to approach the critical range of natural frequencies under consideration.

Fig. 5.
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Model 1: acceleration at (a) Y = 84 m, (b) Y = 180 m and DFT at the same sections (c,d)

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Fig. 6.
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Model 2: acceleration at (a) Y = 84 m, (b) Y = 180 m and DFT at the same sections (c,d)

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The distribution of the load is concordant with the modal shape corresponding to the natural frequency under consideration. The reduction coefficient ψ as a function of the natural frequency is defined from both HIVOSS and SETRA guidelines: the results show how the load models are relevant only for Mode 1 of Model 1 (without diagonal braces), characterized by a natural frequency f1 = 1.08 Hz. The corresponding value of ψHIVOSS is 0.58 (HIVOSS 2008), and ψSETRA = 1.00 (SETRA 2006).

The acceleration at midspan of Model 1, it can be evaluated by means of the numerical model. The limit asymptotic value, assuming ψHIVOSS, is equal to 0.28 m/s2 and can be easily obtained considering the following simplified SDOF expressions:

\(a_{max} = \phi_{i,max} \frac{{F_{eq,i} }}{{2\upsilon \sqrt {1 - \upsilon^2 } }} \approx \phi_{i,max} \frac{{F_{eq,i} }}{2\upsilon }\); \(F_{eq,i} = bp_{max} \int_0^L {\left| {\phi_{i,x} } \right|dL}\).

where amax is the maximum value of acceleration, Feq,i is the equivalent modal force associated with the load model, ν is the damping ratio, Φi,x is the modal displacement in the lateral direction, and Φi,max is the maximum modal displacement. Clearly, the application of the load model leads to values of effective acceleration that violate the comfort limit over a broad range of frequencies (1–5 Hz).

As for Model 2 (with diagonal braces), considering that all the lateral modes are characterized by natural frequencies larger than 1.5 Hz, lock-in is not likely to occur, because the intensity of the distributed load is negligible (ψHIVOSS = ψSETRA = 0).

3.5 Check of Criterion for Lock-in

The effect of the strengthening was last evaluated by means of a simple criterion proposed by Fitzpatrick et al. (2001).

The criterion is based on the following assumptions:

  • the force exerted in the transverse direction by pedestrians in synchronization conditions is approximately proportional to the motion of the structure,

  • the corresponding modal force is proportional to the number of pedestrians,

  • in resonance conditions, this force is counteracted by the damping force.

The criterion was applied to evaluate the effectiveness of the strengthening in terms of maximum number of pedestrians (see Table 5). The two most relevant modes (Mode 1 for Model 1; Mode 2 for Model 2) were considered; the results show how the stiffening of the structure, with the ensuing increase of the frequency, as well as the notable variation of the modal shape (represented by the coefficient ξ) cause the maximum number of people NL in Model 2 to increase by almost 50% with respect to Model 1.

4 Conclusions

Based on the results obtained, the following conclusions can be drawn:

  • modal analysis shows how the insertion of diagonal braces results in a significant increase in the frequencies of the most significant lateral modes, thereby significantly reducing the risk of synchronization;

  • the linear dynamic analyses confirm that the stiffening leads to a reduction of the expected maximum acceleration, especially for the central portion of the footbridge; as regards the extremity resting on a portal frame, the stiffening causes the acceleration to increase, mainly because the local response is governed by modes with higher frequencies;

  • in the presence of dynamic crowd loading in synchronization with the vibrations of the bridge, the stiffened structure is definitely less likely to undergo synchronization phenomena;

  • considering the maximum number of people needed to trigger lock-in, evaluated by means of a simplified criterion, the insertion of diagonal braces brings in an increase of approximately 50%.