Mass properties of the cross section

The total area A of the cross section is 0.584 m² and its centroid is 1.54 m from the deck extrados.

The principal moments of inertia about the centroidal axis are Jx=1.665 m⁴ about the horizontal axis x and Jy=3.711 m⁴ about the vertical axis y.

The strength moduli are then W’_x=-1.081 m³ and W”_x=0.737 m³, about the x-axis, and W’_y=-0.639 m³ and W”_y=0.639 m³, about the y-axis.

2.3 Material

Materials are chosen according to EN 1993-1-1 [6] and EN 1993-2 [7]

The structural steel is S355J2 grade.

3 LOAD ANALYSIS

3.1 Structural self-weight

Since the steel density is γ^{=78.5 kN/m3}, the nominal self-weight of the bridge is:

kN/m 84 . 45 m 584 . 0 kN/m

78.5 ^{3 2}

1

, = A= × =

g_{k b} γ ^{. (1)}

To take into account the weight of other structural parts (transverse beams, bracings and so on), the value of g_k,1b, is increased of about 6%, so that the self-weight g_k,1 results

kN/m 6 . 48

06 .

1 _,1

1

, = _{k b} ≅

k g

g . (2)

3.2 Self weight of non structural elements

Self weights of non structural elements to be considered are those due to the waterproofing, to the 60 mm thick asphalt surfacing and to the safety barriers. For global verifications it is a reasonable approximation to consider these loads distributed per unit surface, by “spreading out” the weight of safety barriers.

All told, the equivalent uniformly distributed loads corresponding to the self-weight of non structural elements g_k,1p is about 2.2 kN/m², so that the load per unit length of the bridge gk,2 is influence surface, in order to maximize or minimize the considered load effect.

For example, to determine the maximum sag moments in the spans and the hog moment at the supports, the relevant influence surfaces, illustrated in figures 3, 4 and 5, must be considered.

According to EN 1991-2, it is necessary first to determine the total width of the carriageway w and the number of conventional lanes. The width w depends on whether the walkways are isolated from vehicular traffic by fixed safety barriers or by kerbs of sufficient height (>100 mm) or not. In the present case, walkways are potentially interested by vehicle traffic, as they are separated from the physical lanes only by road signs. For this reason, the width w is represented by the inner distance between the safety barriers, and therefore

Figure 3. Influence line for max sag moment in span 1 [m]

-21

Figure 4. Influence line for max sag moment in span 2 [m]

12.3168

Influence Line for Hog Moment at Support 2

Figure 5. Influence line for hog moment at support 2 [m]

As the influence surfaces for bending in box girders are cylindrical, i.e. they have rectangular cross section, to maximize bending moments the entire carriageway should be loaded. The number of notional lanes n_l is then given by

3 3

and the remaining area is 1.50 m wide,

m

Obviously, to maximize the torque coexisting with the maximum bending moment, it is necessary to maximize the load eccentricity, so obtaining the notional lane arrangement illustrated in figure 6.

Clearly, when load conditions maximizing the torque are explored, load eccentricities should be maximized and different lane arrangement should be considered, like the one illustrated in figure 7, where only two notional lanes need to be loaded.

notional lane n. 1

Figure 6. Notional lane arrangement on the carriageway (bending moment calculation)

notional lane n. 1

notional lane n. 2

10.5

3.03.0

Figure 7. Notional lane arrangement on the carriageway (torque calculation)

As known, EN 1991-2 calls for four separate static load models, being the single axle load model n. 2 (LM2) devoted only to local verifications.

For global verifications of the bridge in question, only load model n. 1 (LM1) is relevant.

In fact, load model n. 3 corresponding to special vehicles (LM3) and load model n. 4, crowd loading (LM4), are not accounted for, as the bridge is not interested by special vehicle transit and it is located in an extra-urban area. In this regard, it must be recalled that load models LM4 and LM3 need to be considered only when expressly required.

On the i th notional lane, the main load model LM1 provides for a tandem system of axles weighing αQi Qik, accompanied by a uniformly distributed load αqi qik, being αQi and αqi

the adjustment factors. In the present work it has been assumed αQi=αqi =1.0 for each lane, while the values Qik and qik are summarized in table 1. Only one tandem system should be considered per lane, placed in the most unfavourable position.

Table 1. Characteristic values for load model n. 1 (LM1) Notional lane Q_k [kN] q_k [kN/m²]

Lane 1 300 9.0

Lane 2 200 2.5

Lane 3 100 2.5

Remaining area 0 2.5

As said, when seeking a determined effect on the bridge, the LM1 must obviously be arranged in the most unfavourable position and the tandem systems, when present, need to be considered in full, that is, with all their four wheels.

By way of example, possible arrangements of the static traffic loads are represented in figures 8 and 9, corresponding to notional lane numberings discussed below and illustrated in figures 6 and 7, respectively.

2000 Q =300 kN

q =9 kN/m²

q =2.5 kN/m² q =2.5 kN/m²

1k Q _1k=300 kN

1k

2k 3k q_rk =2.5 kN/m²

2000

Q _2k=200 kN Q_2k =200 kN

2000

Q_3k =100 kN Q_3k =100 kN

Figure 8. Load condition corresponding to notional lane numbering in figure 6

2000

Figure 9. Load condition corresponding to notional lane numbering in figure 7

3.4 Wind actions

Wind actions can be represented by vertical and horizontal equivalent static forces.

Vertical force is orthogonal to the roadway plane, while the horizontal forces can be represented by two components, parallel and orthogonal to the bridge’s longitudinal axis, respectively.

The equivalent pressure exerted by the wind can be calculated through the expression

( ) ( )

² reference altitude over the ground, ze, and given by

( )

e r

( ) ( )

e e

[

v

( )

e

]

e z c z c z I z

c = ² ⋅ ₀² ⋅1+7⋅ . (7)

Expression (7) depends on the roughness coefficient cr, on the orography factor c0 and on the turbulence intensity I_v. The orography factor, taking into account any significant local variations in the site’s orography, can usually be assumed equal to 1.0. I_v and c_r, instead, are defined by the following expressions

( ) ( )

where k_i is the turbulence factor, usually set to 1.0. The terrain factor k_r, the roughness length z0 and the minimum height zmin depend on the terrain category.

As said, the bridge in question is located in an extra-urban area which can be classified in terrain category II, i.e. an area with low vegetation, such as grass, and isolated obstacles (trees, buildings) with separations of at least 20 obstacle heights. For terrain category II it results z0=0.050 m, zmin=2.0 m, kr=0.19.

The reference height z_e represents the distance between the lowest ground level to the centre of the bridge deck structure, disregarding other parts (e.g. parapets) of the reference areas Recalling that the intrados of the structure is 20.0 m above ground level, it is ze=20.0+(3.80/2) m= 21.90 (figure 10).

Figure 10. Evaluation of the reference height z_e for wind actions

As z_e>z_min, it results

According to EN 1991-2, the y-axis is assumed parallel to the bridge axis, x-axis is assumed horizontal and perpendicular to the y-axis, while the z-axis lies in the vertical plane containing the y-axis.

The equivalent static force F_wk,x in the x direction is given by

( )

e f x ref x total deck’s height d_tot exposed to wind.

When the bridge is unloaded the exposed height is 4.4 m, as the presence of two open safety barriers determines an increase of 0.6 m in the exposed height.

When the bridge is loaded the exposed height increases by 2.0 m, so becoming 5.8 m.

For unloaded bridge, the coefficient cf,x is

709

If must be noted that the simplified approach proposed in EN1991-1-4, suggesting to assume c_f,x=1.3, is generally unsafe-sided.

Since the webs of the box girder are inclined by the angle α=10° with respect to the vertical (figure 11), the coefficient cf,x calculated in (15) can be reduced by the factor η1

(

^{1 0.005 ;0.7}

)

^0.95

Figure 11. Web inclination allowing a reduction of the coefficient c_f,x

For loaded bridge, the coefficient c_f,x is

90

and also for loaded bridge the simplification cf,x=1.3 proposed in EN1991-1-4 is unsafe-sided.

Considering the reduction factor η₁ calculated before, the combination value ψ0wF_wk,x for the equivalent wind force for loaded bridge results (figure 18)

( )

, 1 , ^{0.6 1.307 1.9 0.95 5.8 8.21kN/m} 0

,

0_wF_wkx =ψ _wq_p z_e c_{f x}η A_{ref x} = ⋅ ⋅ ⋅ ⋅ =

ψ ^{, (19)}

being ψ0w=0.6.

It is interesting to note that in the Italian National Annex the exposed height of lorries it has been set equal to 3.0 m, instead of 2.0 m. In this case, as dtot=6.8 m, cf,x=1.988 and

( )

, 1 , ^{0.6 1.307 1.988 0.95 6.8 10.07kN/m} 0

,

0_wF_wkx =ψ _wq_p z_e c_{f x}η A_{ref x} = ⋅ ⋅ ⋅ ⋅ =

ψ ^{. (20)}

Figure 12. Equivalent static force F_wk,x (unloaded bridge)

Figure 13. Equivalent static force ψψψψ^0wFwk,x (loaded bridge)

Concerning the vertical action, lacking more precise data from wind tunnel tests, a value of ±0.9 can be assumed for the force coefficient cf,z. Since the reference area Aref,z is the horizontal projection of the bridge deck, Aref,z=b=11.6 m²/m, the equivalent static force for unit length F_wk,z is then

( )

, , ^1.307kN/m2

(

^0.9

)

^{11.6m 13.64kN/m}

,_z = _{p e} ⋅ _{f z}⋅ _{ref z} = ⋅ ± ⋅ =±

wk q z c A

F . (21)

to be applied with eccentricity e=0.25 b=0.25⋅11.6 m=2.9 m with respect to the longitudinal axis of the bridge.

According to EN1991-2, as F_wk,z is much lower than the permanent load (65.52 kN/m), it could be disregarded.

Finally, when relevant, equivalent static forces in the longitudinal y-direction (the bridge’s longitudinal axis) should be considered, which can be set equal to 25% of the forces in the x-direction.

3.5 Thermal actions

As the structure under consideration is a continuous beam resting on four supports, thermal actions induce displacements and stresses.

In order to account for thermal variations, two different contributions must be distinguished. A uniform thermal variation along the cross section and a non-uniform temperature variation along the section’s height, corresponding to situations where the top and the bottom of the bridge are at different temperatures, due to differential heating or cooling effects.

The former contribution does not provoke any stresses as long as the bridge can slide horizontally in correspondence to its supports and it will cause only a shortening or elongation of the structure’s line of axis (i.e. it is relevant only for design of bearings and expansion joints).

A typical uniform temperature variation ∆TU can be derived from EN 1991-1-5.

Assuming that in the site under consideration the maximum and minimum air shade temperatures with an annual probability of being exceeded of 0.02 are Tmax=40 °C and Tmin =-10 °C, respectively, the uniform bridge temperature components for a steel bridge result Te,max=56.6 °C and T_e,min=-13.3 °C (figure 14).

Setting the initial bridge temperature T₀ to 20 °C, the characteristic values of the maximum expansion and contraction ranges, ∆TN,exp and ∆TN,con, result so

C

Figure 14. Evaluation of T_e,max and T_e,min

The second non uniform contribution is clearly very significant for the bridge under consideration. EN 1991-1-5 offers two possible procedures to deal with it, provided that the surfacing thickness is not less than 40 mm.

The first, more accurate one, calls for applying rather complex thermal variation laws along the cross section’s height (figure 15), while the second instead makes use of simpler linear variations. Consequently, while the first variation laws require employing dedicated software for the structural analysis, simplified linear variations enable even manual calculations, at least up to a certain degree. In case of steel deck structures, simplified linear variations correspond to a raise in temperature of 18 °C for top warmer than bottom,

∆TM,heat=+18 °C, and to an increase of 13 °C for bottom warmer than top, ∆TM,cool=+13 °C, (figure 16).

Figure 15. Accurate temperature distributions along the height of the cross section

Figure 16. Simplified temperature distributions along the height of the cross section

4 STRESS CALCULATION

In document Guidebook-2 Design of Bridges (Page 168-177)