N. Yanagisawa/Y. Imagawa/O. Ohyama/M. P. Rutner/A. Kurita · Fire safety of bridges – methodology supporting design and forensic evaluation
Fig. 6. Positive bending moment and shearing force interaction M-Q and temperature distribution
(
)
MpC,Q θ+ = σ y κ yu,θ A ud1 + κ yC,θ A Cd 2 + κ yw,θ σ y,Q A wd 3 (4) u
C
w
where: d1 " hc – x/2 tu/2 (mm) d2 " hc – x/2 hs – tC/2 (mm) d3 " hc – x/2 hw/2 tu (mm) Fig. 7 shows how the positive bending moment capacity depends on the temperature. Fig. 7 reveals that 81 % of the total positive bending moment capacity at normal temperature remains at 500 °C, which decreases to 26 % at 700 °C. Interpolation shows that the positive bending moment capacity equals the bending moment due to dead load at a temperature of 720 °C. 4) Negative bending moment failure at intermediate support of side girder Accounting for negative bending moment and shearing force interaction in the composite girder, the distance of the plastic neutral axis from the upper flange x is obtained from Eq. (5):
x=
(
NθC − Nθu + Nθw − σ ry A ru + A rC 2κ yw,θ σ y,Q t w
)
(5)
w
where: Nθu = κ yu,θ σ y A u u
θ
N w = κ yw ,θw σ y,Qh w t w NθC = κ yC,θ σ y A C C
Xry Aru ArC hw tw
yield strength of reinforcing bar sectional area of upper reinforcing bar sectional area of lower reinforcing bar depth of steel web plate thickness of steel web plate
The full bending plastic moment MpC,Q θ− taking into account the shearing force can be expressed as follows:
(
)
MpC,Q θ− = σ ry A ruu1 + A rCu 2 + Nθuu 3 + NθC C1
{
(
)
(6)
}
+κ yw,θ σ y,Q t w xu 4 + h w − x t wC 2 w
where: u1 " hc – c tu x (mm) u2 " c tu x (mm) u3 " tu / 2 x (mm) u4 " x / 2 (mm) C1 " hw – x tC / 2 (mm) C2 " (hw – x) / 2 Fig. 9 plots the negative bending moment capacity over the temperature and provides the critical temperature initiating the failure mode at 800 °C.
5 Time to failure
Fig. 7. Positive bending moment capacity depending on temperature
6
Steel Construction 10 (2017), No. 1
Four failure modes have been identified for the overpass and the remaining task is to find out which of the failure modes is most critical and most likely caused the failure of the 9-Mile Overpass. The approach we took to solve this was to calculate the duration of fire exposure required to