Analytical Investigation of the Effects of Additional Load Mass on the Fundamental Frequency of Pedestrian Beam Bridges

Abstract

The aspect of resonant vibrations due to pedestrian movement is of great significance in engineering practice. Therefore, understanding the variations in the natural frequency of bridge structures under varying positions of additional mass is of particular interest. This paper presents a procedure for the straightforward determination of the natural frequencies of a beam pedestrian bridge for various positions of pedestrians or a service vehicle based on derived analytical solutions. The calculation takes into account the inertial effects of the additional load mass, modeled as either uniformly distributed or concentrated. The importance of additional load mass effects on the fundamental frequency of a beam pedestrian bridge and its dynamic response to a moving pedestrian load is demonstrated on a bridge example. The proposed solutions are also applicable to other girder system structures with uniform mass and stiffness along their span.

1. Introduction

Modern pedestrian bridges are evolving to be more innovative in their design, not only serving functional purposes but also acting as prominent urban landmarks. Advances in structural understanding and new materials have enabled longer spans and more slender structures [1,2]. In engineering practice, particularly concerning pedestrian bridges, resonant vibrations due to pedestrian traffic are highly significant. This is particularly understandable because footbridges are increasingly designed with lower frequencies and lighter weights, which can cause their fundamental vibration frequency to align with the walking frequency of pedestrians [3,4]. This alignment often leads to potential issues with vibration comfort, highlighting the importance of understanding how the natural frequencies of bridge structures change with different positions of additional masses.

As a result of specific incidents related to footbridge vibration issues over the past several decades such as the Millennium Bridge in London [5,6,7] and the Solferino Bridge in Paris [8,9], research has focused on enhancing design standards that consider pedestrian loads, traffic density, and comfort requirements [10]. Structures with natural frequencies outside the range of 1–3 Hz, which is typically induced by pedestrian loads [11], are not at risk of resonant vibrations according to many regulations and standards and, therefore, do not require further dynamic analysis.

According to the Eurocode [12], dynamic analysis is necessary for pedestrian bridges if their natural frequencies fall within the critical range of 0.5 to 5.0 Hz. Meanwhile, the AASHTO LRFD Guide Specifications for the Design of Pedestrian Bridges [13] indicate that pedestrian-induced vertical vibrations are negligible if a bridge’s fundamental vertical frequency exceeds 3.0 Hz, and transverse vibrations are negligible if the transverse frequency exceeds 1.3 Hz. If these conditions are not met, AASHTO recommends conducting a dynamic analysis according to European Standards, specifically following SETRA [14] for methods of analysis, loading cases, and criteria for acceleration acceptance. The serviceability assessment is generally based on a comparison between the pedestrian-induced acceleration and a suitably defined limit value [15].

In dynamic analysis, a significant aspect involves monitoring the shift in the fundamental frequency caused by additional masses, such as groups of pedestrians standing in specific sections of a bridge while others are in motion. It is essential to pinpoint the critical mass configuration capable of pushing a system into an undesirable frequency range, potentially inducing resonant vibrations. Such vibrations could compromise pedestrian comfort and raise concerns regarding a bridge’s stability. The first and simplest way to simulate people’s presence on a structure is by modeling human occupants as added masses [16,17], which can effectively explain reductions in natural frequencies. However, to realistically explain a dynamic response, it is preferable to use models that consider human–structure interaction when simulating the presence of people on a footbridge [18,19,20,21,22,23,24,25]. Analytical models for analyzing the vibrations of a beam under the action of a harmonic moving mass [26] and moving mass with constant [27] and variable speed [28], as well as for the evaluation of vertical vibrations of footbridges using response spectrum [29], have been proposed.

Regarding the current codes of practice [14,30], the influence of a static pedestrian mass, corresponding to up to 5% of a bridge mass, can be ignored since it results in a decrease in natural frequency of less than 2.5%. Several studies, including those by Ellis et al. [31] and Maraveas et al. [32], some relying on full-scale measurements, have demonstrated that passive humans (sitting and standing) significantly impact the dynamic properties of a structure. Typical findings indicate a notable reduction in vibration response and slight alterations in a structure’s natural frequency and damping [33,34].

Busca et al. [35] introduced a method to predict changes in a structure’s modal parameters due to passive individuals, requiring accurate knowledge of the empty structure’s modal characteristics, where each passive individual is modeled by their apparent mass and integrated into the structure without restrictions on the degrees of freedom. In their study, Yang et al. [36] investigated frequency variations in a vehicle–bridge interaction system subjected to a moving mass, which can be significant from the aspect of the movement of a service vehicle on a pedestrian bridge. They employed a three-dimensional model that incorporated the effects of inertial and centrifugal forces from the moving mass into the bridge’s motion equations. O’Sullivan et al. [37] suggested that the mass of pedestrians, along with any additional weight they carry in urban environments, could significantly contribute to the vibrations experienced by a bridge.

In addition to the effects of the additional mass on the dynamic characteristics of bridges, the influence of the stiffness of different structural parts on the dynamic properties of a timber bridge was experimentally analyzed [38]. Human–structure interactions were also analyzed in the case of aluminum footbridges [39]. Recently, studies have focused on the vibration control of pedestrian bridges using dynamic vibration absorbers [40,41,42].

This paper introduces a procedure that relies on analytical solutions to straightforwardly determine the natural frequencies of a beam pedestrian bridge with additional mass, which, to the best of the authors’ knowledge, has not been derived before. It applies to various positions of pedestrians or service vehicles on a bridge, where their mass is modeled either as uniformly distributed or concentrated. In one bridge example, the importance of the additional load mass effects on the fundamental frequency of a beam pedestrian bridge and its time–history response due to the dynamic moving pedestrian load is highlighted.

2. Problem Statement

Let us consider the transverse vibrations of a beam with constant bending stiffness EI and mass μ along span L, as shown in Figure 1. It is assumed that the effects of rotary inertia of the cross-section and shear deformation on bending are negligible in the vibration analysis. Furthermore, taking into account the assumption that the energy dissipation linearly depends on the mass and stiffness of the structure, the partial differential equation for the considered problem of the transverse vibrations of the beam can be defined as follows [43,44]:

$$\mu {\displaystyle \frac{{\partial }^2\nu \left(x,t\right)}{\partial t^2}}+2\beta \mu {\displaystyle \frac{\partial \nu \left(x,t\right)}{\partial t}}+EI\alpha {\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{{\partial }^4\nu \left(x,t\right)}{\partial x^4}}\right)+EI{\displaystyle \frac{{\partial }^4\nu \left(x,t\right)}{\partial x^4}}=p\left(x,t\right).$$

(1)

In Equation (1), v(x,t) represents the deflection at point x and time t. The load on the bridge, whether stationary or moving at a constant speed (c = const.), is described by the force p(x,t), which, without considering its inertial effect, is defined as a deterministic disturbance force. The second and third terms in Equation (1) represent damping forces, where the damping constant β expresses the proportionality of the damping force with respect to the mass μ and constant α expresses that with the stiffness EI of the system.

Using modal analysis and considering the orthogonality of eigenfunctions, Equation (1) can be reduced to a system of independent equations of the following form:

$${\ddot{\eta }}_r\left(t\right)+2{\xi }_r{\omega }_r{\dot{\eta }}_r\left(t\right)+{\omega }_r^2{\eta }_r\left(t\right)={\displaystyle \frac{1}{M_r}}{F}_r\left(t\right),$$

(2)

where η_r(t) denotes a principal coordinate of the r-th mode, ω_r is the natural circular frequency of the r-th mode of the system, and ξ_r represents the relative damping of the r-th mode.

In Equation (2) M_r stands for the generalized mass of the r-th mode of the system, which is defined as follows:

$$M_r=\mu {\int }_0^L{V}_r^2\left(x\right)dx,$$

(3)

and F_r is the generalized force of the r-th mode:

$$F_r\left(t\right)={\int }_0^{L}p\left(x,t\right)V_r\left(x\right)dx.$$

(4)

Applying the Laplace transform to Equation (2), the most general expression for the dynamic deflection of a beam, as a solution of Equation (1), based on which various specific cases can be analyzed, results in the following [43]:

$$\begin{array}{ll}v\left(x,t\right)={\displaystyle \sum _{r=1}^{\infty }}{\displaystyle \frac{V_r\left(x\right)}{M_r}}& {\int }_0^t\left[{\int }_0^{L}p\left(u,\tau \right)V_r\left(u\right)du\right]{\displaystyle \frac{\mathit{sin}{\omega }_{dr}\left(t-\tau \right)}{{\omega }_{dr}}}{e}^{-{\xi }_r{\omega }_r\left(t-\tau \right)}d\tau \\ & +{\displaystyle \sum _{r=1}^{\infty }}{\displaystyle \frac{V_r\left(x\right)}{{\int }_0^L{V}_r^2\left(x\right)dx}}[\mathit{cos}{\omega }_{dr}t{\int }_0^L{v}_{0r}\left(u\right)V_r\left(u\right)du\\ & +\mathit{sin}{\omega }_{dr}t{\int }_0^L{\displaystyle \frac{{\xi }_r{\omega }_r{v}_{0r}\left(u\right)+{\dot{v}}_{0r}\left(u\right)}{{\omega }_{dr}}}{V}_r\left(u\right)du]e^{-{\xi }_r{\omega }_{r}t},\end{array}$$

(5)

where V_r(x) denotes the eigenfunction of the r-th mode, $v_{0r}\left(x\right)=v_r\left(x,0\right)\mathrm{a}\mathrm{n}\mathrm{d}{\dot{v}}_{0r}\left(x\right)={\dot{v}}_r\left(x,0\right)$ are the initial conditions, τ is the time variable of force p, u is the position variable, and ω_dr is the circular frequency of free damped vibrations of the r-th mode of the system, which is defined as follows:

$${\omega }_{dr}\left(t\right)={\omega }_r\sqrt{1-{\xi }_r^2}.$$

(6)

The effects of additional load mass on the free transverse vibrations of an elastic oscillatory system (Figure 2) with constant bending stiffness EI and mass μ along span L are the main focus of this research.

The constant mass μ, per unit length of the beam, is increased by an additional uniformly distributed mass μ_q, per unit length λ, located at the position defined by the abscissa x₀. It is assumed that the contact between the beam and the additional masses is continuously maintained.

Such a configuration of additional masses to a beam structure is most commonly encountered in pedestrian bridges, where pedestrians either stand still or move at a speed represented by c, with x₀ = ct. The problems of undamped vibrations, starting from Equation (1), and neglecting the damping forces (second and third terms in Equation (1)), can be described by the following differential equation:

$$\mu {\displaystyle \frac{{\partial }^2\nu \left(x,t\right)}{\partial t^2}}+EI{\displaystyle \frac{{\partial }^4\nu \left(x,t\right)}{\partial x^4}}=p\left(x,t\right).$$

(7)

Considering the inertial effect of the additional mass µ_q as the force p(x,t), whose position is defined by the Heaviside function:

$$p\left(x,t\right)=-{\mu }_q{\displaystyle \frac{{\partial }^2\nu \left(x,t\right)}{\partial t^2}}\left[H\left(x-x_0+{\displaystyle \frac{\lambda }{2}}\right)-H\left(x-x_0-{\displaystyle \frac{\lambda }{2}}\right)\right],$$

(8)

one can obtain differential equation of undamped free transverse vibrations of a beam with an additional load mass [44]:

$$\mu {\displaystyle \frac{{\partial }^2\nu \left(x,t\right)}{\partial t^2}}+EI{\displaystyle \frac{{\partial }^4\nu \left(x,t\right)}{\partial x^4}}=-{\mu }_q{\displaystyle \frac{{\partial }^2\nu \left(x,t\right)}{\partial t^2}}\left[H\left(x-x_0+{\displaystyle \frac{\lambda }{2}}\right)-H\left(x-x_0-{\displaystyle \frac{\lambda }{2}}\right)\right],$$

(9)

with boundary conditions dependent on the type of support of the beam system, as well as initial conditions:

$$v\left(x,0\right)=v_0\left(x\right); \dot{v}\left(x,0\right)={\dot{v}}_0\left(x\right).$$

(10)

Employing modal analysis and introducing an approximation for the inertial effect of the additional masses as following:

$${\mu }_q{\displaystyle \frac{{\partial }^2\nu \left(x,t\right)}{\partial t^2}}\cong {\mu }_q{\ddot{\eta }}_r\left(t\right)V_r\left(x\right),$$

(11)

differential Equation (9) is reduced to a system of r independent equations, which can be symbolically represented in the following form:

$$\mu \ddot{{\eta }_r}\left(t\right)V_r\left(x\right)+EI{\eta }_r\left(t\right){\displaystyle \frac{d^4{V}_r\left(x\right)}{d{x}^4}}+{\mu }_q\ddot{{\eta }_r}\left(t\right)V_r\left(x\right)\left[H\left(x-x_0+{\displaystyle \frac{\lambda }{2}}\right)-H\left(x-x_0-{\displaystyle \frac{\lambda }{2}}\right)\right]=0,$$

(12)

$$\begin{array}{c}\ddot{{\eta }_r}\left(t\right)\left[\mu {\int }_0^L{V_r}^2\left(x\right)+{\mu }_q{\int }_0^L{V_r}^2\left(x\right)\left[H\left(x-x_0+{\displaystyle \frac{\lambda }{2}}\right)-H\left(x-x_0-{\displaystyle \frac{\lambda }{2}}\right)\right]\right]\\ +EI{\eta }_r\left(t\right){\int }_0^L{\displaystyle \frac{d^4{V}_r\left(x\right)}{d{x}^4}}{V}_r\left(x\right)dx=0,\end{array}$$

(13)

$${\ddot{\eta }}_r\left(t\right)+{{\overline{\omega }}_r}^2{\eta }_r\left(t\right)=0,$$

(14)

where η_r(t) denotes a principal coordinate of the r-th mode and ${\overline{\omega }}_r$ is the corresponding circular frequency of the r-th mode of the system with the additional mass:

$${{\overline{\omega }}_r}^2={\displaystyle \frac{EI{\int }_0^L\frac{d^4{V}_r\left(x\right)}{d{x}^4}{V}_r\left(x\right)dx}{\mu {\int }_0^L{V_r}^2\left(x\right)+{\mu }_q{\int }_0^L{V_r}^2\left(x\right)\left[H\left(x-x_0+\frac{\lambda }{2}\right)-H\left(x-x_0-\frac{\lambda }{2}\right)\right]dx}}$$

(15)

$${{\overline{\omega }}_r}^2={\displaystyle \frac{\frac{EI}{\mu }\frac{{\int }_0^L\frac{d^4{V}_r\left(x\right)}{d{x}^4}{V}_r\left(x\right)dx}{{\int }_0^L{V_r}^2\left(x\right)}}{1+\frac{{\mu }_q}{\mu {\int }_0^L{V_r}^2\left(x\right)}{\int }_0^L{V_r}^2\left(x\right)\left[H\left(x-x_0+\frac{\lambda }{2}\right)-H\left(x-x_0-\frac{\lambda }{2}\right)\right]dx}}$$

(16)

For ${\overline{\omega }}_r$ in Equation (16), Relation (17) applies as follows:

$${\overline{\omega }}_r={\displaystyle \frac{{\omega }_r}{\sqrt{1+G_r}}}$$

(17)

where:

$$G_r={\displaystyle \frac{1}{M_r}}{\mu }_q{\int }_{x_0-{\displaystyle \frac{\lambda }{2}}}^{x_0+{\displaystyle \frac{\lambda }{2}}}{V}_r^2\left(x\right)dx.$$

(18)

It should be noted that M_r in Relation (18) represents the generalized mass of the r-th mode of the system without the additional mass µ_q.

Since the contribution of higher modes is typically of lesser significance to the overall vibrations, it is evident that Relation (17) is acceptable for the fundamental mode (r = 1), and for several subsequent modes where deviations become more pronounced, particularly for higher modes when the ratio of additional mass to bridge mass is larger.

In the case of pedestrian bridges, it is practically important to understand the change in the frequency of the fundamental mode due to the pedestrian or service vehicle mass, which Relation (17) satisfies, specifically for r = 1.

3. Analytical Solutions for Beams with One, Two, and Three Equal Spans

The application of the presented theoretical procedure will be demonstrated on beams with span L, specifically for a beam with three equal spans (L = 3l), a beam with two equal spans (L = 2l), and a beam with a single span (L = l).

Analysis was conducted for three characteristic cases of additional mass configurations. The first case involves a uniformly distributed mass μ_q over length λ, the second case concerns the approach of a uniformly distributed mass μ_q on the beam, and the third case examines the impact of a concentrated mass m. The dynamic models of those three cases are further presented in Section 3.1, Section 3.2 and Section 3.3.

Sinusoidal vibrations were considered, with the following eigencharacteristics applying to all three beams:

• The eigenmode shape function for mode r, which satisfies boundary conditions:

$$V_r\left(x\right)=\mathit{sin}{\displaystyle \frac{r\pi x}{l}},$$

(19)

• Natural frequency of the r-th mode:

$${\omega }_r={\displaystyle \frac{r^2{\pi }^2}{l^2}}\sqrt{{\displaystyle \frac{EI}{\mu }}},$$

(20)

• Generalized mass of the r-th mode:

$$M_r=\mu {\int }_0^L{\mathit{sin}}^2{\displaystyle \frac{r\pi x}{l}}dx=\mu {\displaystyle \frac{L}{2}}, L=nl, n=1,2,3.$$

(21)

For a beam with a single span, eigencharacteristics (19)–(21) apply to all modes, while for beams with two and three equal spans, they pertain to the fundamental mode and higher modes only of sinusoidal nature. Since emphasis is placed on the fundamental mode, the analysis for higher modes of non-sinusoidal nature is omitted here.

3.1. Influence of Additional Mass over Length λ

The frequency ${\overline{\omega }}_r$ for this case is determined from Equation (17) in accordance with Relations (19)–(21) as follows:

$$G_r={\displaystyle \frac{{\mu }_q}{M_r}}{\int }_{x_0-{\displaystyle \frac{\lambda }{2}}}^{x_0+{\displaystyle \frac{\lambda }{2}}}{V_r}^2\left(x\right)dx={\displaystyle \frac{{\mu }_q}{M_r}}{\int }_{x_0-{\displaystyle \frac{\lambda }{2}}}^{x_0+{\displaystyle \frac{\lambda }{2}}}{sin}^2\left({\displaystyle \frac{r\pi x}{l}}\right)dx={\displaystyle \frac{{\mu }_q}{M_r}}·{\displaystyle \frac{l}{2}}·\left({\displaystyle \frac{\lambda }{2}}-{\displaystyle \frac{1}{r\pi }}sin{\displaystyle \frac{r\pi \lambda }{l}}cos{\displaystyle \frac{2r\pi x_0}{l}}\right),$$

(22)

$${\overline{\omega }}_r={\displaystyle \frac{{\omega }_r}{\sqrt{1+G_r}}}={\omega }_r{\left[1+{\displaystyle \frac{{\mu }_q}{M_r}}·{\displaystyle \frac{l}{2}}·\left({\displaystyle \frac{\lambda }{2}}-{\displaystyle \frac{1}{r\pi }}sin{\displaystyle \frac{r\pi \lambda }{l}}cos{\displaystyle \frac{2r\pi x_0}{l}}\right)\right]}^{-{\displaystyle \frac{1}{2}}},$$

(23)

$${\overline{\omega }}_r={\omega }_r{\left[1+{\displaystyle \frac{{\mu }_q}{M_r}}{\displaystyle \frac{l}{2}}\cdot {\Phi }_a\left(\lambda ,x_0\right)\right]}^{-{\displaystyle \frac{1}{2}}},$$

(24)

where:

$${\Phi }_a\left(\lambda ,x_0\right)=\left({\displaystyle \frac{\lambda }{l}}-{\displaystyle \frac{1}{r\pi }}\mathit{sin}{\displaystyle \frac{r\pi \lambda }{l}}\mathit{cos}{\displaystyle \frac{2r\pi x_0}{l}}\right).$$

(25)

Equation (24) is applicable to beams with one, two, or three equal spans. For such beams, the generalized mass M_r should be determined as specified in Equation (21). The specific expressions for the frequencies of systems with these span configurations are provided in

Table 1. Frequency due to the influence of additional mass μ_q over length λ.

Dynamic Model of the Bridge	Circular Frequency
	${\overline{\omega }}_r={\omega }_r{\left[1+{\displaystyle \frac{{\mu }_q}{\mu }}·{\Phi }_a\left(\lambda ,x_0\right)\right]}^{-{\displaystyle \frac{1}{2}}}$
	${\overline{\omega }}_r={\omega }_r{\left[1+{\displaystyle \frac{{\mu }_q}{2\mu }}·{\Phi }_a\left(\lambda ,x_0\right)\right]}^{-{\displaystyle \frac{1}{2}}}$
	${\overline{\omega }}_r={\omega }_r{\left[1+{\displaystyle \frac{{\mu }_q}{3\mu }}·{\Phi }_a\left(\lambda ,x_0\right)\right]}^{-{\displaystyle \frac{1}{2}}}$

.

Changes in fundamental frequency due to additional mass μ_q along length λ = 0.2l, obtained using the expressions in Table 1 and depicted through the ratio of circular frequencies, are presented in Figure 3 for bridges with one, two, and three equal spans. One may conclude that the additional mass in the mid-span has a larger effect on the natural frequency. Increasing the ratio of additional mass to bridge self-mass amplifies this effect.

3.2. Influence of the Arrival of a Uniformly Distributed Mass

The frequency ${\overline{\omega }}_r$ for this additional mass case can be obtained using Equations (24) and (25), where introducing the shift x₀ = λ and rearranging with a translation of x₀ by λ/2 can be represented as follows:

$$G_r={\displaystyle \frac{{\mu }_q}{M_r}}{\int }_0^{x_0}{V_r}^2\left(x\right)dx={\displaystyle \frac{{\mu }_q}{M_r}}{\int }_0^{x_0}{sin}^2\left({\displaystyle \frac{r\pi x}{l}}\right)dx={\displaystyle \frac{{\mu }_q}{M_r}}·{\displaystyle \frac{l}{2}}·\left({\displaystyle \frac{x_0}{l}}-{\displaystyle \frac{1}{2r\pi }}sin{\displaystyle \frac{2r\pi x_0}{l}}\right),$$

(26)

$${\overline{\omega }}_r={\displaystyle \frac{{\omega }_r}{\sqrt{1+G_r}}}={\omega }_r{\left[1+{\displaystyle \frac{{\mu }_q}{M_r}}{\displaystyle \frac{l}{2}}\cdot \left({\displaystyle \frac{x_0}{l}}-{\displaystyle \frac{1}{2r\pi }}sin{\displaystyle \frac{2r\pi x_0}{l}}\right)\right]}^{-{\displaystyle \frac{1}{2}}},$$

(27)

$${\overline{\omega }}_r={\omega }_r{\left[1+{\displaystyle \frac{{\mu }_q}{M_r}}{\displaystyle \frac{l}{2}}\cdot {\Phi }_b\left(x_0\right)\right]}^{-{\displaystyle \frac{1}{2}}},$$

(28)

where:

$${\Phi }_b\left(x_0\right)=\left({\displaystyle \frac{x_0}{l}}-{\displaystyle \frac{1}{2r\pi }}\mathit{sin}{\displaystyle \frac{2r\pi x_0}{l}}\right).$$

(29)

Relation (28) is relevant for beams with one, two, or three equal spans. For the analysis of these beams, the generalized mass M_r should be calculated according to Relation (21). Detailed expressions for the frequencies of such systems are listed in

Table 2. Frequency due to the influence of the arrival of uniformly distributed mass μ_q.

Dynamic Model of the Bridge	Circular Frequency
	${\overline{\omega }}_r={\omega }_r{\left[1+{\displaystyle \frac{{\mu }_q}{\mu }}\cdot {\Phi }_b\left(x_0\right)\right]}^{-{\displaystyle \frac{1}{2}}}$
	${\overline{\omega }}_r={\omega }_r{\left[1+{\displaystyle \frac{{\mu }_q}{2\mu }}\cdot {\Phi }_b\left(x_0\right)\right]}^{-{\displaystyle \frac{1}{2}}}$
	${\overline{\omega }}_r={\omega }_r{\left[1+{\displaystyle \frac{{\mu }_q}{3\mu }}\cdot {\Phi }_b\left(x_0\right)\right]}^{-{\displaystyle \frac{1}{2}}}$

.

Changes in fundamental frequency due to arrival of uniformly distributed mass μ_q defined based on Table 2 and compared with the circular frequencies are presented in Figure 4 for bridges with one, two, and three equal spans. It is evident that the largest effect on the natural frequency is obtained when additional mass is distributed over the entire span (x₀/l = 1 for a bridge with one span, x₀/l = 2 for a bridge with two equal spans, and x₀/l = 3 for a bridge with three equal spans). Similar to the previous case, increasing the ratio of additional mass to bridge self-mass amplifies this effect.

3.3. Influence of Concentrated Mass

The influence of concentrated mass on a pedestrian bridge structure may be more significant for loading from service or firefighting vehicles, whose movement in some cases may be permitted but practically negligible during pedestrian traffic.

The solution for this case can be obtained from Equations (24) and (25) if one considers λ to be infinitely small and the mass μ_qλ to have a finite value m. Then, as λ→0, $\sin {\displaystyle \frac{r\pi \lambda }{l}}$ becomes ${\displaystyle \frac{r\pi \lambda }{l}}$; thus, the general solution for ${\overline{\omega }}_r$ in this case results from the following:

$$G_r={\displaystyle \frac{{\mu }_q}{M_r}}{\int }_{x_0-{\displaystyle \frac{\lambda }{2}}}^{x_0+{\displaystyle \frac{\lambda }{2}}}{V_r}^2\left(x\right)dx={\displaystyle \frac{{\mu }_q}{M_r}}{\int }_{x_0-{\displaystyle \frac{\lambda }{2}}}^{x_0+{\displaystyle \frac{\lambda }{2}}}{sin}^2\left({\displaystyle \frac{r\pi x}{l}}\right)dx={\displaystyle \frac{{\mu }_q}{M_r}}·{\displaystyle \frac{l}{2}}·\left({\displaystyle \frac{\lambda }{2}}-{\displaystyle \frac{1}{r\pi }}·{\displaystyle \frac{r\pi \lambda }{l}}cos{\displaystyle \frac{2r\pi x_0}{l}}\right)={\displaystyle \frac{m}{M_r}}{sin}^2{\displaystyle \frac{r\pi x_0}{l}}$$

(30)

$${\overline{\omega }}_r={\displaystyle \frac{{\omega }_r}{\sqrt{1+G_r}}}={\omega }_r{\left[1+{\displaystyle \frac{m}{M_r}}{sin}^2{\displaystyle \frac{r\pi x_0}{l}}\right]}^{-{\displaystyle \frac{1}{2}}},$$

(31)

$${\overline{\omega }}_r={\omega }_r{\left[1+{\displaystyle \frac{m}{M_r}}\cdot {\Phi }_c\left(x_0\right)\right]}^{-{\displaystyle \frac{1}{2}}},$$

(32)

where:

$${\Phi }_c\left(x_0\right)={\mathit{sin}}^2{\displaystyle \frac{r\pi x_0}{l}}.$$

(33)

For beams with one, two, or three equal spans, the generalized mass M_r should be computed following Relation (21). The frequency expressions corresponding to these configurations are summarized in

Table 3. Frequency due to the influence of concentrated mass m.

Dynamic Model of the Bridge	Circular Frequency
	${\overline{\omega }}_r={\omega }_r{\left[1+{\displaystyle \frac{m}{0.5\mu l}}\cdot {\Phi }_c\left(x_0\right)\right]}^{-{\displaystyle \frac{1}{2}}}$
	${\overline{\omega }}_r={\omega }_r{\left[1+{\displaystyle \frac{m}{\mu l}}\cdot {\Phi }_c\left(x_0\right)\right]}^{-{\displaystyle \frac{1}{2}}}$
	${\overline{\omega }}_r={\omega }_r{\left[1+{\displaystyle \frac{m}{1.5\mu l}}\cdot {\Phi }_c\left(x_0\right)\right]}^{-{\displaystyle \frac{1}{2}}}$

.

For this analyzed case of additional concentrated mass, the changes in the fundamental frequencies of bridges with one, two, and three equal spans, depicted through the ratio of circular frequencies, are presented in Figure 5. Similar conclusions apply to this case as for the scenario with additional mass distributed over length λ.

4. Application Example

The importance of additional load mass effects on the fundamental frequency of a beam pedestrian bridge, as well as its time–history response and acceleration due to dynamic moving pedestrian loads, is illustrated through a specific example. A simply supported timber footbridge with a span length L = 25 m, bending stiffness of EI = 2.016 × 10⁶ kNm², mass per unit length μ = 400 kg/m, damping ratio β = 0.015, and an additional mass equivalent to that of a light service vehicle (m = 5000 kg) is considered. The numerical demonstration presented here is solely intended as an illustrative tool, utilizing assumed parameters. Importantly, these parameters correspond to a hypothetical pedestrian bridge scenario.

The bridge’s first-order vertical frequency was 5.64 Hz. According to Eurocode [12], it is unnecessary to verify whether the acceleration response under pedestrian loads meets criteria for walking comfort. Considering a pedestrian bridge with an additional mass equivalent to that of a light service vehicle, it is concluded from Figure 5a that placing the additional mass at mid-span would result in the greatest change in the natural frequency. Thus, according to Expression (32) and Table 3, the frequency of the system with the additional mass was determined to be $\overline{f_0}=$ 3.98 Hz. Given that the frequency of the bridge, with a service vehicle present, corresponds to the running frequency, there is a possibility that the bridge could resonate with a pedestrian jogging across it, thereby experiencing excessive vibrations in terms of the serviceability limit state.

The maximum vertical acceleration of the bridge with the additional mass was determined by assuming a resonant pedestrian jogging at a constant speed. This result was compared with the response of the structure without the additional mass under the same pedestrian movement conditions. Both obtained results were evaluated according to the comfort criteria proposed by relevant standards.

The pedestrian’s dynamic load [45,46] was modeled as a pulsating load P(t), moving across the bridge at a constant speed c:

$$P\left(t\right)=180 sin\left(2\pi \overline{f_0}t\right) \left[\mathrm{N}\right],$$

(34)

$$c=0.9·\overline{f_0}=3.59 \left[{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}\right].$$

(35)

The dynamic response was calculated using Equation (5), where the acceleration of the bridge was computed as the second derivative of deflection at x = L/2, and pedestrian load, P(x,t), is defined using the Dirac delta function as follows:

$$P\left(x,t\right)=P\left(t\right)\cdot \delta \left(x-ct\right).$$

(36)

Figure 6 shows the time histories of the dynamic response at the mid-span of the pedestrian bridge due to a resonant pedestrian load, considering the bridge with additional mass.

Figure 7 shows the time histories of the dynamic response at the mid-span of the pedestrian bridge due to the same pedestrian load without the additional mass. Therefore, the pedestrian load P(x,t), as given by Equation (36), was not in the resonant domain of the bridge.

Human perception of vibration is complex, with different individuals, and even the same person at different times, experiencing the same vibration source differently [41]. To address pedestrian discomfort from excessive vibrations on footbridges, various countries have developed comfort assessment methods based on acceptable vibration limits for the human body.

SETRA [14] categorizes vertical vibrations by maximum accelerations into three comfort levels: up to 3.535%g (practically imperceptible), between 3.535%g and 7.07%g (merely perceptible), and from 7.07%g to 17.675%g (perceptible but tolerable), where g denotes gravitational constant (approximately 9.81 m/s²). These levels reflect theoretical accelerations of a bridge deck under pedestrian load, with accelerations exceeding 25%g deemed unacceptable.

On the other hand, the European standard [12] defines the maximum acceptable vertical accelerations of a bridge as 0.7 m/s² to fulfil the serviceability limit state. It should be noted that this clausula is not mandatory, allowing each country to define different criteria in their national annexes. In the case of the national annex of the Republic of Serbia [47], the maximum acceptable vertical acceleration of a footbridge depends on factors such as bridge site usage, route redundancy, bridge structure height, and exposure aspects. For example, if a bridge serves as a primary route in a major urban center and has a height of less than 4 m, according to this national annex, the maximum acceptable vertical acceleration for the footbridge is 1.1 m/s².

Based on the presented analysis, it can be concluded that the hypothetical bridge structure met the serviceability limit state regarding vibrations when additional load mass was not considered. However, in the presence of additional load mass, this criterion was not fulfilled.

5. Conclusions

The derived analytical solutions for variations in the fundamental frequency of a pedestrian beam bridge due to additional mass, modelled as either uniformly distributed or concentrated mass, can be applied to a specific subset of pedestrian beam bridges where the bending stiffness and mass distribution along the span remain constant. It is acknowledged that the inherent dynamic characteristics of the oscillatory system (bridge without pedestrians) are known. It is essential to emphasize that additional masses always lead to a reduction in the fundamental frequency that depends on their size and position on the bridge.

The presented analytical solutions are highly suitable for both qualitative and quantitative analyses of the problem of fundamental frequency variation, which is always of particular interest in engineering practice. Since the fundamental frequency is a relevant parameter for the vibration analysis of pedestrian bridges, based on the derived analytical solutions or the presented diagrams, it is easy to determine when neglecting the inertial effects of added masses is acceptable.

The application example of a hypothetical pedestrian bridge scenario, with additional mass from a light service vehicle that is 50% of the bridge’s self-weight, showed that the vertical frequency of the bridge decreased by approximately 30% compared with the same bridge without the additional mass. Furthermore, for the same pedestrian excitation, the maximum vertical acceleration of the bridge increased by approximately 50 times due to the additional mass of a light service vehicle at the bridge’s mid-span. This example demonstrates the importance of understanding changes in bridge frequency due to additional mass, illustrating how pedestrian comfort when crossing the same footbridge without and with additional mass can vary from maximally acceptable to totally unacceptable levels. This conclusion points out that presence of an additional mass on the footbridge should not be neglected in the serviceability limit state regarding vibrations, emphasizing the necessity to develop existing standards in this area.

To fully validate the proposed analytical solution for determining the natural frequencies of a pedestrian bridge with additional mass, experimental testing and a comparison of the experimental and analytical results should be conducted. This will be the focus of the authors’ future work. Further research can also be directed toward developing an analytical solution for additional mass effects on footbridges with non-uniform bending stiffness and mass along the span, aspects commonly encountered in real engineering practice but not covered in the analysis presented in this paper.