3.7.1 General

Railway bridges have many specific aspects to be taken into account in design, some of them are out of the scope of this book. However, some important aspects related to actions, to be referred to here, are:

• load models for railway traffic
• dynamic load effects
• traction, braking, centrifugal forces and nosing forces
• action effects due to track–structure interaction

3.7.2 Load Models

Most European countries adopt the UIC Recommendations, considered also in EC1‐2 [5], where the following load models are specified:

• Load Model 71 and load model SW/0 for normal rail traffic on main railways,
• Load model SW/2 to represent heavy loads,
• Load model HSLM for high speed passenger trains, considered to be trains with speeds exceeding 200 km h⁻¹,
• Load model ‘Unloaded Train’, to represent the effect of an unloaded train.

It is important to note that the load models tend to induce the effects of railway traffic, but they do not attempt to represent by themselves the actual loads from trains. The UIC 71 train [5], here denoted as the load model 71, or simply LM 71, is the typical train load model for design of most railway bridges. The reader may refer to EC1‐2 [5] for the consideration of the remaining load models.

Load model 71 is represented in Figure 3.14. It consists of four axle loads of 250 kN, with the distribution shown in the figure and a UDL of 80 kN m⁻¹ under an unlimited length. These values are characteristic values that, according to EC1‐2, may be multiplied by a factor 0.75 ≤ α ≤ 1.46 on lines carrying rail traffic that is lighter or heavier than the normal rail traffic. This load model should be located at the most unfavourable position for the element under study, taking into consideration that the UDL, or even the concentrated loads, may not act at a certain loaded length if these are the most severe loading conditions for the element under consideration. An example is given in Figure 3.15 for a loading pattern case for the positive bending moment of a continuous beam bridge deck.

The loads of Load Model 71 should be multiplied by the appropriated dynamic load factor, Φ, with values between 1.0 and 2.0, which takes into account the dynamic amplification of load effects due to the overall vibration of the bridge or the local vibration modes of specific structural elements under consideration. This aspect of dynamic factors is considered in more detail in the next section.

For bridges carrying two tracks, Load Model 71 should be applied to one or two tracks in order to induce the most severe effects to the element under study. For bridges with three or more tracks, LM 71 should be applied to any one track, any two tracks or 0.75 times the load model to three or more tracks.

The loads of LM 71 shall be applied with a certain eccentricity to take into account possible lateral displacements of the vertical wheel loads on the tracks. In EC1‐2, this is considered by adopting a ratio of wheel loads on all axles up to 1.25/1.00 on any one track. This applies both to the concentrated load Q_vk as well as to the distributed load q_vk of LM 71 (Figure 3.16). For maximum eccentricity, one obtains e_max = 0.055 r = r/18. Hence, for a standard European gauge track of r= 1450 mm, one has an eccentricity of 81 mm, inducing a torsional moment on the deck under traffic loading.

3.10.1 Basic Concepts

To introduce the problem of dynamic load effects in roadway and railway bridges, a single degree of freedom system is adopted, with mass M, elastic stiffness K and viscous damping (i.e. damping proportional to the velocity) defined by the coefficient C. When a constant force P is suddenly applied, the basic equation for dynamic equilibrium [8, 9] is:

(3.13)

where and denote, respectively, the second and first derivatives of the displacement with respect to time, that is, the acceleration and the velocity. Assuming the system is at rest at t = 0, that is, initial conditions = 0, u = 0, the dynamic response for the undamped system is:

(3.14)

where is the circular frequency of the undamped system. The term P/K represents the static deflection u_sta and, hence, the dynamic amplification factor Φ, defined as the ratio between the maximum dynamic deflection u_dyn and the static one u_sta = P/K is given for the undamped system by:

(3.15)

Let the viscous damping be introduced through the damping ratio ξ, where C_cr = 2 M ω is the critical damping coefficient. It represents the smallest amount of damping for which the free vibration response changes from an oscillatory to exponential decay response. The system now vibrates with a response as shown in Figure 3.17, including a negative exponential term yielding a decaying vibration, defined by:

(3.16)

Here, is the damped vibration frequency of the system, differing very little from the undamped vibration frequency, since for most bridges ξ << 5%.

The dynamic coefficient Φ is therefore reduced with ξ and is still obtained from Eq. (3.15), yielding for the damped system at instant t = π/ω_d :

(3.17)

as shown in Figure 3.17. The internal elastic force R = K u is reduced as well in the same proportion. Now consider a moving load P = M g on a simply supported bridge, with velocity v as shown in Figure 3.18. The response of the bridge, modelled here as a simple supported beam with bending stiffness EI and mass per unit length constant μ, is defined by the vertical displacement w = w(x, t) obtained by solving the partial differential equation:

(3.18)

where δ(x − v t) represents the Dirac delta function [8, 10], that is, δ = ∞ for x = v t (at the point of application of the concentrate load) and δ = 0 elsewhere, such that . Besides, in this equation, ω_d represents the damped system circular frequency depending on ξ.

The undamped beam has natural sinusoidal vibration modes (Figure 3.18) and natural circular frequencies ω_n, obtained from Eq. (3.18) for ξ = 0, ω_d = ω and P = 0, by:

(3.19)

Equation (3.18) may be solved by analytical methods and classical solutions that are available in the literature [10]. The solutions involve the parameter:

(3.20)

where v_cr represents the critical speed of the moving load introducing an unbounded vertical deflection in the undamped system, that is, the resonance condition as it happens in a single degree of freedom (SDF) system when the frequency of the force tends to be equal to the natural frequency of the system. Taking into consideration the value of the first natural frequency of the beam f₁ = ω₁/(2π), one obtains v_cr = 2f₁L. Hence, the maximum displacement yields a dynamic amplification factor Φ dependent on the natural frequency of the bridge, as well as on the speed, v, of the mass running on the bridge. Of course, Φ is dependent on the bridge damping ratio and tends to increase with the natural frequency for v to approach v_cr. For velocities sufficiently far away from v_cr, the dynamic amplification factor for mid‐span deflection, is approximately given by Φ = 1/(1 − α) for an undamped bridge.

Figure 3.19 shows some numerical results obtained for a simply supported span under a moving load at different speeds with respect to the critical speed and for a damping coefficient of 2%. If the irregularities of the carriageway of a highway bridge, or of the rail of a railway bridge are considered, the dynamic amplification factor is dependent on these irregularities as well. Besides, if the mass of the vehicle is not negligible with respect to the bridge mass, the effect of bridge‐vehicle interaction may be relevant with due account for the elastic and damping suspension system of the vehicle. A model is shown in Figure 3.20 for this case. All these effects can only be considered through a dynamic numerical model that considers the structure‐vehicle interaction and the irregularities of the carriageway of the track as a random process. This subject is out of the scope of this book and the reader is referred to specific literature [10–13].

The load models defined for road bridges already include the dynamic coefficient in the value specified for the characteristic traffic loads. Hence, specific consideration of dynamic load effects is relevant for railway bridges only.

3.10.2 Dynamic Effects for Railway Bridges

For railway bridges, the dynamic behaviour of the bridge should be considered, either on the basis of a dynamic analysis or based on a simplified procedure in which a static analysis is adopted but with the traffic loads (Section 3.6) multiplied by a dynamic coefficient. For maximum design speeds not exceeding 200 km h⁻¹, the requirements to allow a static analysis are established in EC1‐2 [5], and are related to the natural frequency of the bridge f₁ to be between a lower limit governed by dynamic impact criteria, defined by (Figure 3.21):

(3.21)

and an upper limit, governed by dynamic increments due to track irregularities, defined as

(3.22)

The natural frequency of the bridge should take into account the mass due to permanent actions. In Eqs. (3.21) and (3.22), L_Φ(m) is the span length for simple supported bridges or an equivalent span for other bridge types. Some simplified expressions to evaluate the natural frequencies of typical bridge cases are given in [14].

A detailed table is given in EC1‐2 to define L_Φ. It is important to note that L_Φ is related to the vibration mode of the element under consideration. So, for example, if one takes the requirements to allow an overall static analysis of a continuous girder or slab bridge deck, or a continuous portal frame, over n spans, L_Φ is the average span length. However, if one considers the cross girders of a steel grid made of a ballastless open deck, L_Φ is twice the length of the cross girder. For continuous bridge decks (slabs, slab girders or box girder type) with span lengths L_i, i = 1, 2, 3, …, n, one may take for L_Φ = k L_m, where k = 1.2 for n = 2 spans, k = 1.3 for n = 3, k = 1.4 for n = 4 and k = 1.5 for n ≥5 , and L_m the average span. In any case, one should never take L_Φ to be smaller than the maximum span length L_i.

For cases where static analysis is allowed, the dynamic factors by which the loads of LM 71 should be multiplied are defined as Φ₂ or Φ₃, dependent, respectively, on whether this is the case of a carefully maintained track or a track under standard maintenance. The expressions as proposed by EC1‐2 and UIC specification are as follows:

(3.23)

with 1.00 ≤ Φ₂ ≤ 1.67 and 1.00 ≤ Φ₃ ≤ 2.0, where L_Φ is the ‘determinant’ length referred to earlier. These dynamic coefficients have been determined for simply supported girders; the concept of determinant length L_Φ allows generalizing the expressions for Φ₂ or Φ₃ to other bridge design typologies. In some way, L_Φ is the span of the equivalent simply supported beam, defined from the influence line of the deflections in the actual bridge deck.

3.11.1 Design Wind Velocities and Peak Velocities Pressures

The wind is specified at a certain point X = {x_i} acting on a structure by its instantaneous velocity v = v{x_i, t}, which may be considered to be composed by an average velocity, v_m, and a turbulent component, v_t:

(3.24)

The average value v_m referred is to a certain time interval, Γ, during which wind speed is measured. Generally, Γ = 10 minutes and after a certain number of measurements one has a statistic distribution of the wind speed measured during a certain number of intervals. The basic wind velocity, v_b, is a reference velocity for a certain probability of being exceeded (usually 2%, i.e. with a return period T = 50 years) defined in the codes for the bridge site as a function of wind direction and time of the year, at 10 m above the ground and for a certain type of terrain (open field). From v_b, the so called mean wind velocity refers to a time interval of 10 minutes, with a probability, p, of being exceeded (usually 2%). A different matter is the design wind speed, v, defined in the codes by its characteristic value, v_k, which is associated with the gust wind speed on the site and which may be evaluated for a certain return period related to the design situation under consideration. For example, in the case of erection phases, the design wind speeds are generally lower. For checking the structure during these phases, one adopts a shorter return period, T, say T = 10 years for a construction time no greater than 1 year. The associated characteristic wind speed for T = 10 years and T = 50 years, may be related by [15]:

(3.25)

For T = 10 years, as for the erection phases, v_k, 10 = 0.88 v_k, 50, that is, a 12% reduction. In some cases, and for very relevant bridges, one may increase the return period from 50 to more than 100 or 120 years, generally the reference lifetime of the structure. For T = 120 years, one obtains v_k, 120 = 1.06 v_k, 50.

For design, and if no aerodynamic problems are relevant, only the peak wind velocities pressures are needed. However, aerodynamic instabilities should be investigated on the basis of mean wind velocities.

The design mean wind velocity, at a certain level z above the ground, may be determined by:

(3.26)

where c₀(z) is the orography factor, taken as 1.0 unless otherwise specified and c_r(z) is the roughness factor defined, for example, in Part 1‐4 of EC1 (denoted by EC1‐1‐4, [15]) by:

(3.27)

where k_r is the terrain factor, which is a function of the roughness length, z₀. In EC1‐1‐4 there are five terrain categories, being categories 0 and I associated to a bridge site on the sea or in a coastal area. Category IV is for a bridge at an urban site. Intermediate categories II and III correspond, respectively, to open country with low vegetation and isolated obstacles, or areas with regular cover of vegetation or buildings (such as villages, suburban terrain and permanent forests).

For example, one has z₀ = 0.003 m for roughness length in terrain category 0, z₀ = 0.05 m for category II and 1.0 m for category IV. The expression in Eq. (3.27) holds for z > z_min; that is, respectively, 1.0 m for terrain category 0 and 10 m for category IV. Otherwise, one takes c_r(z) = c_r(z_min). Using c₀(z) = 1.0 and c_r(z), according to the expressions given in EC1‐1‐4 [15], yields the curves given in Figure 3.23 for terrain roughness categories between 0 and IV. The design wind speed may be defined by multiplying the mean wind velocity by a factor to take turbulence into consideration:

(3.28)

where g is the ‘peak factor’ (that may be taken as 3.5 as in EC1‐1‐4) and I_v(z) the turbulence intensity at level z defined as:

(3.29)

where σ_v is the standard deviation of the turbulence assumed constant with z and k_l is a turbulence coefficient (k_l = 1.0 in [15]) and c₀ is the orography factor already defined. The expression is only valid above a minimum level z_min (between 1.0 and 15 m, respectively, for terrain roughness of categories 0 and IV) above the ground and for z ≤ 200 m. Below z_min, one assumes a constant minimum turbulence evaluated from Eq. (3.29) for z = z_min. From these considerations, one may define the peak wind speed, including the turbulence effects, as:

(3.30)

From the design wind speed, v, the peak velocity pressure on an element of the structure is evaluated by:

(3.31)

where ρ is the unit mass of the air taken, under normal conditions, as ρ = 1.25 kg m⁻³. From these expressions, the peak pressure can be written in terms of the wind pressure, q_b, associated to the basic wind velocity, v_b, by:

(3.32)

where c_e(z) is the exposure factor given by:

(3.33)

Figure 3.24 presents the values for the exposure factor as given in EC1‐1‐4 [15], allowing a direct calculation of the peak velocity pressure from the basic wind velocity, v_b, defined in the code for the bridge site.

3.11.2 Wind as a Static Action on Bridge Decks and Piers

Let a bridge deck (Figure 3.25) be considered under the action of a uniform wind flow of speed v_α acting at a certain angle of attack, α. The deck is subjected per unit length, to the following forces as denoted in Figure 3.25:

(3.34)

where is the wind pressure associated to v_α, and C_D, C_L and C_M are nondimensional coefficients designated as shape aerodynamic coefficients dependent on the geometry of the cross section of the deck and on the angle of attach α of the wind flow. The dimension d is a characteristic dimension of the cross section, as its depth, to which the shape coefficients are referred to. However, the shape coefficients can also be referred to the deck width b or, for example,C_M may be defined by expressing the moment as where b is the width of the cross section and .

For current bridge design cases, the drag force induced in a bridge component by the wind action is the most relevant one, and it is evaluated by (Figure 3.26):

(3.35)

where F_D is the horizontal wind force and A_ref, x is the reference area of the element exposed to the wind action and projected on a plan normal to the wind direction. For the wind actions on the bridge deck it is usual to define the forces F_D per unit length, where the deck is simulated by a rectangular envelope of width b and depth d. The drag coefficient, C_D is given as a function of the ratio , and for a reference area 1 × d one may adopt the values given in Figure 3.27 taken from the EC1‐1‐4 [15] for open or box sections.

In most cases, is above 4 and C_D lies between 1.0 and 1.3. For box sections with inclined webs, C_D may be reduced at 0.5% per degree of web inclination with the vertical, with a maximum reduction of 30% in any case.

When the bridge is under traffic (Figure 3.27) the depth of the vehicles must be considered when evaluating the exposed area, A_{ref, x}. For road bridges, this is considered by taking an additional depth of 2.0 m above the deck, while for railway bridges this is done by taking an additional depth of 4.0 m from the top of the rails on top of the total length of the bridge. For bridges under traffic, the dynamic wind pressure is reduced by taking a reduced basic wind velocity ( is limited to 23 m s⁻¹ for road bridges and 25 m s⁻¹ for railway bridges) to take into consideration the reduced probability of having the bridge under the characteristic value of the live loads together with winds blowing at very high speeds. For specific design cases, the maximum peak wind velocity may be limited (e.g. to 100 km h⁻¹) as a safety condition for the vehicles, which means the bridge should be closed to traffic if higher peak wind velocities are forecast.

For lift forces, unless some wind tunnel tests have been performed for the deck section or a specific C_L coefficient is defined in EC1‐1‐4 [15]; from Figure 3.27 one may take C_L = +0,9 or C_L = −0,9, and the vertical action as

(3.36)

where A_{ref, z} = b L is the in‐plan area of the deck length L.

The moment due to wind actions on the deck may be evaluated by assuming the drag force, F_D, acting at 60% of the equivalent depth (d_eq), measured from below and the lift force, F_L, at 25% of the width (b) measured from windward (Figure 3.27).

For the piers, one may adopt the force (drag) coefficients given for rectangular or polygonal sections in EC1‐1‐4 [15] or, in a simplified approach, take the equivalent rectangular envelope shape as given in Figure 3.28. The force coefficients are in general between 1.0 and 1.8. In tall piers, these forces are evaluated with a variable wind pressure, q, with respect to the height above the ground.

3.11.3 Aerodynamic Response: Basic Concepts

Wind‐tunnel tests, done on section models such as the one shown in Figure 3.29, are most useful to interpret and determine the dynamic response of the deck under wind flow‐aerodynamic response. A section model for a wind‐tunnel test consists of a scale model of a certain length of the deck, L, which is determined by the available width of the tunnel in order to avoid any blockage effects under the test wind flow due to the proximity of the walls of the tunnel. This sectional model is then attached at its extremities to springs simulating the elastic stiffness of the remaining structure. These springs are modelled with scale stiffness in order to reproduce the scale of the main frequencies of the deck, which for these types of problem are, basically, the vertical and torsion frequencies, f_b and f_t, respectively. Figure 3.29 presents the variation of the shape coefficients C_D, C_L and C_M with the angle of attack, α. These values may be adopted for the evaluation of the equivalent static wind forces on the deck referred to in the previous section. For example, for α = 0, one has C_D = 1.17, C_L = −0.494, C_M = −0.006.

Under certain wind flow conditions, the deck (or the sectional model in the wind tunnel) may experience several types of aerodynamic excitation (Figure 3.30), which are characterized by a motion of limited or divergent amplitudes of oscillation. In the first case, limited amplitude response, which may induce unacceptable stresses or fatigue, different cases may occur:

• Vortex induced oscillations associated with the so‐called vortex shedding alternately from the upper and lower surface of the bridge deck (Figure 3.30). The frequency of the vortices, dependent on the magnitude of the wind velocity, v, being close to the bending frequency of the deck, induce the risk of resonance if structural damping is not enough to reduce the amplitude of the oscillations.
• Turbulence response, due to the turbulent nature of the forces induced by wind gusts, the deck may oscillate if sufficient energy is induced at certain frequency bands close to the natural frequencies of the structure. This phenomenon may usually be neglected if both bending and torsion natural frequencies, f_b and f_t, are greater than 1 Hz. Otherwise, the dynamic effects of turbulence response should be considered on the basis of a dynamic analysis.

The case of divergent amplitudes of oscillation gives rise to aerodynamic instability of the deck, namely by galloping or flutter, as referred in the sequel. For details on basic references on bridge aerodynamic stability, the reader is referred to Refs. [17–19].

3.11.3.1 Vortex Shedding

The wind flow around a bridge deck, a pier or even a stay cable, as shown in Figure 3.30, may induce vortex shedding. The pressure difference between the two sides of the element under wind flow, due to vortex shedding, produces transverse forces normal to the direction of the wind and so possible bending vibrations. These vortices are known as von Karman vortices after studies of this phenomenon by von Karman at the beginning of twentieth century. If the frequency of the vortices is close to one of the natural frequencies of the structure, only damping can limit large displacements induced by a resonance phenomenon.

In vortex shedding, the intensity and frequency of the vortices are dependent on the wind velocity v and on the geometry (shape) of the deck, namely its depth/width ratio . The critical wind speed for a bridge deck, to induce vortex shedding, is given by:

(3.37)

where f_o is the natural frequency of the fundamental mode, d is the depth of the cross section and S_t is the Strouhal number [15, 19], equal to 0.2 for a cylinder; for a rectangular section (that may be taken as bridge deck envelope) this only varies between 0.1 and 0.09, when 0.5 < < 10. If the critical wind speed, v_cr,VS, given by Eq. (3.37) is low, transverse oscillations may occur under normal wind conditions.

However, the induced transverse forces due to deck oscillations are small for low critical wind velocities and if the structural damping is enough to reduce the amplitude of the oscillations the consequences of the phenomena of vortex shedding are reduced. The relevance of vortex shedding may increase when the deck critical wind speeds are high, yielding high transverse forces in the structure under wind flow oscillations.

In practice, safety precautions against resonant critical conditions are satisfied if the critical wind velocity for vortex shedding is at least 1.25 times greater than the mean wind velocity as defined in Section 3.11.1.

3.11.3.2 Divergent Amplitudes: Aerodynamic Instability

The cases of divergent amplitudes, inducing oscillations that may lead to collapse by aerodynamic instability of the structure, are associated with the following basic phenomena:

• Instability in pure torsion – torsional divergence
• Instability in pure bending – galloping
• Instability by flutter in a single mode – torsional flutter
• Interactive instabilities, in torsion and bending – classic flutter

The first mode corresponds to an induced torsional rotation of the deck that cannot be ‘reduced’ by its elastic torsion resistance; is like a static instability mode and is not critical for a bridge deck because it is associated with very high wind velocities.

The second mode, galloping, is a self‐induced crosswind bending mode vibration occurring when the apparent variable incidence of the wind, due to the motion of the deck, induces vertical forces amplifying the bending movement. Some cases of practical examples of galloping are referred in Ref. [20] for decks and in particular for pylons of cable stayed bridges with a single plane of cables. The example of the wind tunnel testing undertaken for the Second Panamá Bridge (Figure 3.31) is presented. Galloping may be appraised on the basis of the lift coefficient and to its variation with the angle of attack, α, of the wind flow. A necessary condition for gallop may be taken from sectional wind‐tunnel tests. For that, observing at α = 0 the value of the drag coefficient and the variation with α of the lift coefficient, the necessary condition for gallop, the den Hartog criterion, reads:

(3.38)

As C_D > 0, gallop cannot occur if the derivative of the lift coefficient at α = 0 is positive. For example, this condition is satisfied by the results for C_L presented in Figures 3.26 and 3.29. For the wind‐tunnel studies on the second Panama Bridge mast [20], for wind angles of attack between 5 and 6°, = −5,34. However, further stability studies have shown the resulting damping remains positive for a 100‐year wind return period, assuring aerodynamic stability [20].

If wind‐tunnel tests are not available, the critical wind speed for galloping may be estimated for bridges with main spans up to 200 m by,

(3.39)

where f_b is the natural bending frequency, C_g is a galloping coefficient, m is the mass per unit length of the deck, δ_s is the logarithmic decrement of structural damping, ρ the density of the air = 1.25 kg m⁻³ and d is the depth of the cross section of the deck. Galloping coefficients C_g may be taken from specific literature [15, 21]. In Ref. [21], values equal to 1.0 or 2.0 are proposed, respectively, for bridges with side overhangs less than or greater than 0.7 d, where d is the depth of the bridge deck. For aerodynamic stability problems, the logarithmic decrement δ_s may be taken to be 3% for steel bridges, 4% for composite bridges and 5% for concrete bridges. It should be noted that the damping coefficient is ξ ≅ δ_s/(2π) [17], results in ξ = 0.5% for steel bridges and 0.8% for concrete bridges.

Flutter in a single mode, stall flutter, is similar to gallop. It is a self‐induced vibration in a single periodic torsion mode of the deck; it is induced by a periodic torsion oscillation originating from a periodic variation of the flow around the deck. The stall flutter critical wind speed may be evaluated from the torsion natural frequency f_t (Hz) on the basis of the following simplified formula [19, 21]:

(3.40)

where τ is a coefficient depending only on the type of the cross section, such that τ = 5 in most cases. But, for cases of slab or box girder sections with b < 4 d, then τ = 12.

Classical flutter is mainly an aerodynamic instability due to the interaction of the bending and torsion forced vibration modes of the deck under wind action. In slender bridge decks, namely with large values of the length/width ratio (L/b) as is often the case in suspension bridges, and with a low frequency ratio f_t/f_b between the torsion and bending natural frequencies, there is risk of interaction between the basic oscillation modes of the deck. The structure, for a certain critical wind speed, may experience large amplitude coupled oscillations and in these cases, structural damping may not be enough to avoid a collapse by aerodynamic instability by classical flutter.

The theoretical background of this phenomenon goes to the interaction of two oscillation modes in dynamic stability of structures, as shown in a schematic form in Figure 3.22 for a simple two degrees of freedom system [22]. The excitation load parameter λ is, for example, the dynamic wind pressure, inducing a forced oscillation in the dynamic equilibrium of the structure. For λ = 0, the system has two natural vibration frequencies, f₀₁ and f₀₂; under increasing load λ, these frequencies tend to coalesce, but above a certain load level dynamic equilibrium is no longer possible, inducing unlimited amplitude oscillations. The critical load for flutter instability is then associated to the load at which the two forced vibration frequencies tend to coalesce, as shown in Figure 3.32. Classical flutter of bridge decks may be investigated by a variety of approaches, namely simple analytical approaches [15, 17, 19, 21] – Selberg formula, Kloopel diagrams, CECM (Convention Européenne de la Construction Métallique) tables, Scanlan Derivatives, Numerical Approaches or Wind tunnel testing [23].

The simplest approach, for the critical wind speed of flutter instability of a bridge deck, is to adopt the basic case of an ideal thin plate aerofoil for which the Selberg formula [21] holds,

(3.41)

where m is the mass per unit length of the deck, r is the polar radius of gyration of the bridge cross section, that is, , with I_o = ρ(I_y + I_z) the polar mass moment of inertia of the cross section and A its area, ρ is the air unit mass = 1.25 kg m⁻³, b is the width of the cross section and f_b and f_t are the natural bending and torsion frequencies of the deck. The Selberg formula can also be adapted to evaluate the flutter critical wind speed of bridges with main spans less than 200 m through a correlation factor η [19] (Figure 3.33),

(3.42)

The Selberg formula gives good approximations for the critical wind speed, provided that the ratio f_b/f_t > 1.5 and δ_s is about 0.05. From Eq. (3.41) the influence of the following parameters is apparent:

• the ratio f_b/f_t, yielding a low critical wind speed when the frequencies are close to each other; in practice, it is convenient to have f_b/f_t > 2.5 as a starting point to warranty aerodynamic stability; the η values may vary between 0.2 for a slab girder type deck to 0.6 for a standard box girder deck; for a streamline box girder deck the values may approach 0.9. Some values are given in Figure 3.33 and more detailed values may be taken from [19];
• the parameter m r/ρ b³ = [m/(ρ b²)] (r/b), from which the influence of the ratios m/(ρ b²) and (r/b) are apparent; the former m/(ρ b²) is dependent on the cross section of the deck but it varies approximately inversely with the width b; the last (r/b) is about 0.3 for a practical range of bridge decks varying only between 0.27 for single box girders with large deck overhangs to 0.32 for decks supported by main girders or cables at the outer edges, as referred to in [21].

It is clear from the Selberg formula that aerodynamic stability is very much improved by the torsion stiffness of the deck, which increases the torsion frequency, f_t. This is the case for modern suspension bridges with ‘streamline’ box deck sections as shown in Figure 3.22b.

For all cases of aerodynamic phenomena (gallop, stall flutter or classical flutter), safety should be checked by assuring a safety factor for the critical wind speed of at least 1.5 with respect to the mean wind velocity measured at the deck level.

3.13.1 Basic Concepts

Temperature actions in bridges are due to temperature distributions within the structural elements, namely induced by solar radiation, as shown in Figure 3.35a. The heat of hydration of cement in concrete bridges is another relevant action, in particular up to three to four days after concreting the element. In general, one has to deal with a nonlinear distribution of temperature within the element (Figure 3.35b), which is dependent on several parameters, namely related to local climatic conditions, the orientation of the element, geometrical and material properties of the element as well as on specific characteristics of the surfacing of the bridge deck. The effects of thermal actions may be evaluated by splitting it into the following components:

• the uniform temperature variation, measured from a reference temperature, T₀, being the temperature with respect to which the action effect is considered; for example, the erection (or completion) temperature or more generally the temperature at the relevant stage of the structural element at its restraint.
• the differential temperature variation, which yields a thermal gradient within a bridge component, namely in a bridge deck due to solar radiation or due to the heat of hydration of cement in a concrete bridge. This is illustrated in Figure 3.36, where a nonlinear distribution of the temperature in a bridge deck results from a certain period of heating and cooling of the upper surface of the bridge deck under the solar radiation effect, while at the bottom surface the temperature is controlled by the shade temperature [24].

The uniform temperature variations are associated with seasonal changes, while the thermal gradients within the deck, or within any other bridge element such as a pier shaft, are associated with daily thermal variations.

The uniform and differential temperature actions in bridges are specified in codes for actions. These temperature distributions are dependent on a set of parameters, namely, solar radiation, air temperature, the degree of nebulosity and wind velocity, bridge location and its orientation, the shape of the cross section, thermal properties of the material and the type and colour of the surfacing of the bridge deck.

Formally, the temperature distribution, T, in a bridge deck is determined from the Fourier heat equation [25]:

(3.44)

Where:

• ΔT = ∂²T/∂x² + ∂²T/∂y² is the Laplacian of the temperature T (x,y,t) at each instant, t, if assumed only dependent on the cross section coordinates x and y; that is, the same for all cross sections,
• k is the thermal conductivity, representing the rate of heat flow per unit area and per unit temperature gradient, k (W m⁻¹ °C),
• Q is the amount of the heat generated inside the deck per unit volume and per unit time, Q (W m⁻³), as for example the heat of hydration of cement in a concrete bridge deck,
• ρ (kg m⁻³) is the material density (unit weight), and finally,
• c is the specific heat of the material, that is, the quantity of heat required to increase the temperature of the unit mass by one degree, c (J kg⁻¹ °C⁻¹).

Equation (3.44) is a partial differential equation to be solved taking into consideration the temperature and the heat flow conditions at the boundary of the deck surface, which may be expressed as

(3.45)

where is the derivative of temperature in the direction normal to the boundary defined by the unit vector n, that is,

(3.46)

where n_x and n_y are the direction cosines of n and grad T denotes the temperature gradient at the boundary. In Eq. (3.45), q denotes the amount of heat transfer per unit time and per unit area of the boundary, with the units W m⁻².

To solve Eq. (3.46), the quantity of heat q in Eq. (3.44) has to be defined as the sum of three basic quantities – solar radiation, q_s, which is dependent on the total heat from solar rays reaching the surface per unit area and per unit time, the amount of heat transfer by convection, due to the temperature difference between the bridge surface and the air, which is dependent on the wind speed, the type of material and on the configuration and orientation of the surface, and finally on the amount of radiation from the surface to the surrounding air. The reader is referred to Ref. [26] for a detailed discussion on thermal and meteorological properties involved in the use of Eq. (3.44). This equation may be solved, for example, on the basis of a finite element approach, to find the temperature at the bridge deck at each instant and at each point. Standard computer programs [27] may be adopted for specific bridge problems and for parametric studies to define the thermal field in typical bridge decks. Figure 3.36 shows a typical example for a bridge deck. The temperatures at several hours of the day were evaluated from a finite element program along the depth of the cross section and measured in a deck model. The basic geometrical and thermal characteristics adopted for the numerical model are referred to in the figure.

3.13.2 Thermal Effects

For design practice, the temperature distributions in bridge decks are usually defined from values defined in the codes. Only in very particular cases numerical models based on the approach referred in the previous section to evaluate the thermal field have to be developed. Let the effect of a nonlinear temperature distribution T(z, t) be assumed in the case of a statically determinate structure as shown in Figure 3.37 – a simply supported bridge deck with a constant cross section. One assumes the temperature distribution T at each time instant t, as being the same for all the cross sections and dependent only on the z coordinate of the sections. For the case under consideration, the nonlinear temperature distribution yields a self‐equilibrated stress field (eigenstresses) in the bridge deck, but no internal forces since the structure is statically determinate. To show this, one may consider the evaluation of these stresses, by using the principle of superposition of elastic effects, as shown in Figure 3.37. Firstly, one introduces a longitudinal restraint at the end sections by introducing applied stresses defined by:

(3.47)

where E is the Young’s modulus of the material and α is the coefficient of thermal expansion. The thermal strains ε = α T(z, t) are restrained by σ₀. At the end sections with area A of the cross section, the stress resultants are given by:

(3.48)

By applying a force N₁ = − N₀ and a moment M₁ = − M₀ at the end sections and superposing the two stress fields, σ₀ and σ₁ (due to N₁ and M₁), the stress field as indicated in Figure 3.38 is obtained by simple beam theory:

(3.49)

where I_y is the moment of inertia with respect to the neutral y‐axis of the cross‐section. By introducing the definitions of N₁ and M₁ and taking into consideration Eq. (3.48), one obtains the stress distribution

(3.50)

So, the resulting stresses may be defined as

(3.51)

where ∆T_U and ∆T_M are the average (uniform) and linear temperature components defined as:

(3.52)

and ∆T_E is a nonlinear temperature component defined as

(3.53)

In conclusion, the temperature distribution within a specific bridge element T(z, t), may be split into an uniform temperature component, ∆T_U, a linear varying temperature component, ∆T_M, and a non‐linear temperature difference component, ∆T_E, the last one giving rise to a self equilibrated stress field σ_E – eigenstresses. On a more general form, as represented in Figure 3.39, the linear varying temperature component, ∆T_M, is a temperature gradient along the vertical z‐axis, ∆T_Mz, but also a temperature gradient along the y‐axis, ∆T_My.

For statically determined structures, ∆T_U and ∆T_M do not induce any internal forces or stresses in the structure, but only strains and deflections. The uniform temperature only affects the design of the bearings and expansion joints through the imposed displacements.

Effectively, ∆T_U and ∆T_M induce an elongation and a constant curvature deflection in the beam, according to the Bernoulli hypothesis of plane sections in a remaining plane. Since the structure is statistically determined, these deformations, as shown in Figure 3.40, do not induce any internal forces. The bridge deck elongates with a total displacement u at the free end section and takes a constant curvature, 1/R_M, given by

(3.54)

where ε represents the strain at a distance z of the neutral axis and ∆T_M is the difference of temperature between the top and bottom surfaces of the bridge deck with depth h, as shown in Figure 3.40. Only the nonlinear temperature component gives rise to a nonlinear stress field σ_E, which has the same shape of ∆T_E distribution as results from Eq. (3.51).

Let the case of statistically undetermined structures be considered. If the same bridge deck is now assumed to be longitudinally built in at the end sections (Figure 3.41), which, however, are assumed to be free for the displacements in the plane of the cross section, the uniform and linear thermal components do not induce any deformations (the longitudinal displacements are totally restrained) but internal forces are generated that are equal to the stress resultants, N₀ and M₀, due to the stresses, σ_0, restricting the longitudinal displacement and curvature in the simply supported case:

(3.55)

Now let the case of a continuous bridge deck be considered, as shown in Figure 3.42. The uniform temperature component induces a total displacement at the expansion joint, u, which is still given by Eq. (3.54), now with 2.5L being the length of the bridge deck. The nonlinear temperature distribution, ∆T_E, induces the same stresses as in the simply supported case shown in Figure 3.37.

The linear temperature component, ∆T_M, induces a linear bending moment diagram in the bridge deck, as shown in Figure 3.43, due to the induced reactions R that may be easily evaluated (Figure 3.44) by the force method of linear structural analysis. Using the symmetry and the principle of virtual work for evaluating displacements for the deflections at the internally supported sections of the released structure, one assumes a simply supported beam and, as given by Eq. (3.54),

(3.56)

The displacements in the base system due to the forces are

(3.57)

From compatibility , one obtains

(3.58)

Figure 3.43 shows the final elastic bending moments due to the dead load p and the linear temperature variation ∆T_M. Then, for a positive ∆T_M, the bending moments increase at the span sections due to ∆T_M and are reduced over the supports.

3.13.3 Design Values

The uniform temperature component depends on the maximum and minimum temperature variation experienced by the bridge, which is dependent on the maximum and minimum shade air temperature, T_max and T_min, and on the initial bridge temperature, T₀. So, the maximum expansion and contraction ranges of uniform bridge temperatures are defined as:

(3.59)

where the values of T_e, max and T_e, min are the uniform bridge temperature components that are dependent on T_max and T_min, as well as on the type of structural material. Figure 3.45 shows the correlation between the characteristic values of the shade air temperature and the uniform bridge temperature component, for steel, composite and concrete bridge decks. Characteristic values of shade air temperatures shall be obtained at the site location with reference to national maps of isotherms usually given in each country national annex of Part 1‐5 of EC1 (denoted as EC1‐1‐5 [28]).

As an example, for the case of a bridge site where T_max and T_min are, respectively, 40 and −10°C, if one assumes T₀ = 10°C, one obtains for a steel bridge deck, using Eq. (3.59) and Figure 3.45, ∆T_U, exp = 55−10 = 45°C and ∆T_U, cont = 10 − (−13) = 23°C. For the same site conditions but for a concrete bridge deck, one obtains ∆T_U, exp = 42 − 10 = 32°C and ∆T_U, cont = 10 − (−2) = 12°C.

The vertical temperature component with nonlinear effect is defined as well in EC1‐1‐5 for steel, composite and concrete bridge decks on the basis of typical temperature diagrams along the depth of the deck. An example of these diagrams for a steel box girder deck is represented on Figure 3.46. It is important to note that these diagrams include the linear temperature gradient ∆T_M and the nonlinear temperature gradient ∆T_E, together with a small part of component ∆T_U, which is included in the uniform bridge component. So, from Eqs. (3.52) and (3.53), these diagrams may be used to evaluate ∆T_M and ∆T_E.

In practice, the most important component of the non‐uniform temperature variation is the linear component ∆T_M. Reference values of the linear temperature difference component for different types of bridge decks – steel, composite and concrete decks – are given in the Table 3.2, based on upper bound values of the linear temperatures for representative samples of bridge geometries. In this table, ∆T_M represents the characteristic value of temperature difference between the top and bottom surface of the deck.

These values are based on depth of surfacing of 50 mm for road and railway bridges. For other depths of surfacing, these values should be multiplied by the factor k_surf as proposed in EC1‐1‐5 [28]. For example, for the case of a concrete box girder roadway bridge with a surface thickness of 75 mm the linear temperature difference is 8.5°C (= 0.85 × 10°C) for the top warmer than the bottom, and 5°C (= 1.0 × 5°C) for the bottom warmer than the top.

It is sometimes necessary to consider horizontal temperature gradients, for example, taking into consideration possible action effects due to different temperatures between the webs of a box girder deck or between opposite outer faces of concrete piers. If no other information is available, a linear temperature difference of 5°C between the outer edges of the bridge independently of the width of the bridge, or between the outer faces of concrete piers, is recommended.

Differences in the uniform temperature components between different main structural elements may have to be considered, for example, between the stay cables and the deck of a cable‐stayed bridge with values in the order of ±15°C.

3.17.1 Basis of Design

The design of bridges in seismic zones, such as in southern European countries, requires earthquake actions to be taken into consideration at concept, analysis and details of a design. At the concept stage, one aims to achieve a structure with a reliable response under seismic actions. During analysis, by modelling the seismic actions by static equivalent forces (acceptable for simple and small bridges) or by a dynamic analysis based on response spectrums, power spectrums or by a time history analysis [8]. In the detail design stage, by considering structural elements and connections with sufficient resistance and ductility under seismic actions.

It is understood bridges generally behave well for earthquakes of moderate intensity, say up to a magnitude in the order of five to six. For earthquakes with a higher magnitude, the following types of damage have occurred in several cases:

• collapse of the foundations by soil liquefaction under seismic actions
• damage in abutments
• falling of superstructure elements
• structural damage in piers or, less frequently, in the deck

Damages in abutments are generally caused by dynamic overpressures of the adjacent soil or due to forces induced by the deck. Falling of deck elements is usually due to characteristics of bridge bearings not being compatible with the order of magnitude of the horizontal displacements induced by the earthquake. Finally, damages in bridge piers are usually due to lack of ductility of these structural elements.

For current bridges, it is sufficient to consider the seismic action in horizontal directions; the vertical component should be considered in a few structures. That is the case for structures with vertical vibration modes associated to natural frequencies less than 1 Hz. In these cases, if seismic action is considered by response spectrums, the ordinates of the response spectrum correspondent to the vertical component may be considered to be two‐thirds of the ordinates of horizontal components. The simplified methods of seismic analysis, based on static equivalent forces, are currently limited to bridges with vertical piers, bridges with a straight or almost straight in plan alignment or slightly skew bridges.

Besides, the span lengths should not be significantly different and should be approximately symmetrical with respect to a vertical plan perpendicular to the axis of the bridge. Static equivalent seismic analyses are not applicable to curved bridges, frame bridges with inclined piers, arch bridges, cable‐stayed bridges and suspension bridges.

In the static equivalent seismic force method, actions may be considered separately in two horizontal directions, one along the longitudinal axis and the other perpendicular to the bridge axis. The equivalent static forces are obtained by multiplying the permanent loads of the bridge by a seismic coefficient β, currently with values between 0.02 and 0.3. This coefficient depends on the seismicity of the bridge site, namely on the peak rock acceleration, the type of soils at the foundations, the natural frequencies of the bridge (in particular, the fundamental natural frequency) and the behaviour coefficient to be adopted.

For simple bridges, the longitudinal seismic action in the deck may be simulated by a horizontal force applied at the deck with a value βG, where G represents the permanent load of the deck. This may be done because the longitudinal mode of vibration of the bridge may be modelled by a single degree of freedom system, represented by the deck translation.

However, for the transverse direction, one must take into consideration the transverse mode of vibration and a discrete model is adopted with concentrated masses (m_i) localized at the most flexible parts of the structure.

The static equivalent force distribution, which is close from the inertia force distribution associated to the fundamental mode of vibration, may be obtained by evaluating the deformed configuration of the structure under the action of forces β G_i proportional to the permanent loads and acting along the direction under consideration.

The discrete model of the deck is as shown in Figure 3.59 and the effective acceleration of a mass m for a SDF system is a_ef = (2πf)² u, where u(t) is the time dependent displacement and f the fundamental frequency. The associated inertia force is F = m a_ef = (2πf)² m u.

For a MDOF system with masses m_i, the characteristic value of the static equivalent forces are

(3.81)

where d_i are the displacements due to the loads G_i acting simultaneously in the direction where the seismic action is being considered and g the acceleration of gravity. In loads G_i, one includes the permanent loads as well as the quasi‐permanent values of the variable loads correspondent to the masses m_i.

Longitudinal and transverse behaviour should be analysed, due to different influences of the support conditions of the superstructure. Another aspect of seismic actions refers to the hydrodynamic forces induced by water surrounding piers. This is a complex solid–fluid interaction problem, but the simplest model consists of considering a mass of a cylinder of water, which vibrates with the pier.

When response spectrums or power spectrums need to be considered for the seismic analysis of bridges, a modal analysis is considered. The reader is referred to the specific literature [8, 33, 34] for these methods of seismic analysis. Design codes such as the Part 2 of Eurocode 8 (denoted as EC8‐2 [35]) establish the basis for seismic analysis of bridges.

3.17.2 Response Spectrums for Bridge Seismic Analysis

The most adopted method for seismic analysis of bridges is the Response Spectrum Method. The basis of the method is outlined here. A SDF system may be considered, for example, for a bridge with a three span continuous deck supported by two flexible piers and with longitudinal sliding bearings at the abutments. The deck under seismic longitudinal actions is assumed to be rigid and hence the structure can be modelled by a SDOF system (Figure 3.60). Under a seismic action defined by the ground motion record in terms of accelerations, ‐acting accelerograms, and the relative displacements of the mass M with respect to the foundation level is u. Hence the dynamic equation of motion (3.13) for P = 0 but with the inertia term replaced by is written as

(3.82)

Taking into consideration, C = 2 Mω ξ and k = Mω² where ω = 2πf, (3.82) is rewritten as

(3.83)

This differential equation may be solved for a certain ground motion record to find the displacements u = u(t) of the mass M. The value of peak response (maximum displacement) of the SDOF defined by its frequency f and damping coefficient ξ defines the response spectrum of displacements S_d (f, ξ). Solving for different frequencies f, a diagram (S_d = S_d (f) or in terms of the period T = 1/f, (S_d = S_d (T)) can be plotted for the response spectrum for different damping ratios ξ. The response spectrums are defined in codes (EC8‐1, e.g. Figure 3.61) for different geotechnical conditions and zones usually associated to peak ground accelerations at the bridge site.

If the maximum displacement of the mass M is designated by S_d (f, ξ), the SDOF system is subjected to a maximum force

(3.84)

The term S_a (T, ξ) = (4 π²/T²) S_d is the acceleration spectrum of the mass; for T = 0, that is, a system with infinite frequency (totally rigid system) like a rigid mass supported directly on the rock, S_a is the peak ground acceleration of the mass M. Figure 3.61 shows different response spectra in terms of damping ratios.

For a Multi Degree of Freedom System (MDFS), a modal analysis [8, 35] has to be carried out and Eq. (3.83) is generalized for each normalized modal coordinate U_i = 1, with i = 1 to n, as

(3.85)

where is the participation factor [8, 35] of mode i in the response, for a specific direction of ground motion, depending on the mode shape and mass distribution. Denoting the modal displacement j in the mode i by φ_ij, one has

(3.86)

The effective masses in mode i are defined as

(3.87)

A multi degree spectrum analysis is carried out for the complete seismic analysis of the structure. By defining the spectral acceleration for each mode i as S_a (T_i, ξ_i), the maximum elastic forces in mode i may be calculated by using the participation factors. The simplest approach to obtaining a 2D bridge structure modelled as n lumped masses (Figure 3.59 or the bridge structure in Figure 3.60 under a horizontal seismic action transverse to the plane of the structure), the maximum base shear due to mode i is obtained from the response spectrum and generalized masses, as F_{max, i} = M_{eff, i}S_a (T, ξ). However, total response cannot be obtained by adding maximum force values at each mode, because these maximum values do not occur at the same time. The simplest and most adopted combination rule to obtain maximum values is the square root of the sum of the squares of the modal responses. The reader is referred to [8, 35] for developments on the subject.

A final remark is made concerning the use of response spectrums as defined. These are elastic response spectrums that should be modified by ductility/behaviour factors to take into consideration the nonlinear behaviours of the bridge structure during an earthquake. The nonlinear behaviours are associated to cracking or yielding reducing the stiffness and modifying the effective vibration frequencies of the bridge structure. The spectrum accelerations are then modified and this nonlinear effect is usually considered by reducing the elastic forces calculated on the basis of the elastic spectrums or reducing directly the spectrum by the behaviour factor. In bridge structures these behaviour factors q depend on piers and abutments ductility, and are in the order of 1.5–2.5, yielding a considerable reduction in the force values obtained from the elastic spectrums. Indicative values can be found in EC 8‐2 [35]. However, inaccessible parts of the bridge, like foundations, and bearing design in general, total elastic forces are usually required by the codes. The reader is referred to [34–37] for further developments.