research article buckling instability behavior of steel

Research ArticleBuckling Instability Behavior of Steel Bridge under Fire Hazard

Ying Wang and Muyu Liu

Hubei Key Laboratory of Roadway Bridge & Structure Engineering, Wuhan University of Technology, Wuhan 430070, China

Correspondence should be addressed to Muyu Liu; [email protected]

Received 28 December 2015; Accepted 24 March 2016

Academic Editor: Veljko Milutinovic

Copyright © 2016 Y. Wang and M. Liu.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Failure of buckling instability will most likely occur before the displacement reaches the allowable value of the code when a tankerburns under the steel bridge. This research focuses on critical buckling stress of bridge under fire hazard and a thermal analysismodel of a steel bridge is established by FDS (Fire Dynamics Simulator). Thermal parameters of the steel are determined by thepolynomial fitting method. Temperature field and elastic modulus of the bridge changing with time are calculated by determiningthe heat release rate function of tanker. Critical buckling stress of the bridge web and bottom floor changing with time is calculatedaccording to steel floor buckling theory. Finite element software ANSYS is used to verify the result. Results show that when a tankeris burning for 17 minutes, critical buckling stress of steel web will be reduced to 𝜏crl,𝑙𝑤(𝑇) = 19.1MPa and 𝜎crl,𝑙𝑤(𝑇) = 38.8MPa, whichis less than the web stress (𝜏 = 19.6MPa, 𝜎 = 39.8MPa) caused by dead and live load. So steel web will be the first to show shearflexural bending buckling failure. Displacement in the midspan will reach 35.4mm at this time, which was less than the allowabledisplacement (50mm) set by standard. The best rescue time of the bridge under fire hazard is within 15 minutes.

1. Introduction

City viaduct and overpass were the key components of mod-ern transportation. Once fire hazard occurred, it would causegreat losses. On April 29, 2007, a tanker carrying about 32thousand and 600 liters of gasoline crashed into a landmarktower on the highway overpass and then overturned inMichigan [1]. The bridge was bending by the heat generatedby the fire that resulted in 250-meter-long bridge collapse.The elastic modulus of floor would decrease rapidly withthe increase of temperature when steel bridge was under firehazard, leading to serious deformation or collapse duringfire. It was likely that the failure of buckling instability of thesteel bridge would occur before the displacement reached theallowable value of the code under fire hazard.Therefore, it wasimportant to study the temperature variation and bucklingcharacteristics of the steel bridge under fire hazard.

In recent years, some scholars began to study the transienttemperature field and the mechanical properties of steelbridge under fire because of its tremendous harm. Garlocket al. [1] presented a detailed review of actual fire incidents,case studies related to fire hazards, and postfire assessment

and repair strategies in bridges. The study pointed out thatthe number of damaged bridges caused by fire is nearly3 times more than that caused by earthquake. Kodur andNaser [2] analyzed the mechanical properties of the steel-concrete composite beam bridge under high temperatureby the method of heating curve. The research showed thatfire resistance performance of steel-concrete composite beambridge was much better than the steel bridge. Transienttemperature field and the failure mode of the steel-concretecomposite bridges under fire were studied by means of finiteelement method and verified by experiments [3]. Mechanicalbehavior of simply supported beam bridge under fire wasanalyzed by ANSYS [4]. Relationship between displacementof midspan and time was presented under fire and finally thebridge’s destruction time was obtained.

In current fire resistance study of bridge,most researchersconsidered that the bridge would be damaged when dis-placement of the bridge under fire is exceeding the allowablevalue of the code [5, 6]. But for the steel bridge, criticalbuckling stress of bridge would be reduced greatly dueto rapid increase of temperature and decrease of elasticmodulus. Therefore, it was likely that buckling failure would

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 8024043, 11 pageshttp://dx.doi.org/10.1155/2016/8024043

2 Mathematical Problems in Engineering

Steel thermal parameters Fire heat release rate of tanker

Thermal analysis model of two-spancontinuous steel box girder bridge by FDS

Characteristics of the temperature of the steel bridge under tanker fire

Elastic modulus varied with time of steel bridge under high temperature

Finite element model of two-spancontinuous steel box girder bridge under fire by ANSYS

Calculation method of bottom floor and side web based on buckling theory

Buckling instability behavior of steel bridge

Failure time and rescue time of steel bridge under fire

Figure 1: Conceptual scheme of buckling instability behavior of steel bridge under fire hazard.

occur before the displacement of bridge reaches the allowablevalue of the code. In this study, the critical buckling stressvariation of the web and the bottom floor of steel bridgewas analyzed and failure time of steel bridge under hightemperature was obtained. Results showed that the failureof buckling instability occurred before the displacement inmidspan reached the allowable value of the code of the steelbridge under fire hazard. The best rescue time of bridgewas obtained and this can provide reference for bridge fireresistance design. Conceptual scheme of buckling instabilitybehavior of steel bridge under fire hazard is shown in Figure 1.

2. Project Summary

The bridge researched in this study is a 2 × 30m continuoussteel girder bridge located in Wuhan, Hubei, China, Gusaotree overpass of Tianhe District Second Highway Engineer-ing. Vertical clearance under the bridge is 4m and the heightof the girder is 1.68m, as shown in Figure 2(a). The positionin which tanker fire happened is shown in Figure 2(b).

Material of the bridge is steel Q370qD [7]. The yieldstrength (fy) of steel Q370qD is 370MPa and the elasticitymodulus (𝐸) of steel Q370qD is 210GPa at room temperature.The top of girder width is 9m and the height of girder at thedesign of line is 1.68m, both sides of the cantilever box girderare 1.75m long, cantilever root height is 0.5m, and end heightis 0.25. Number and dimensions of the different groups of ribsare shown in Table 1. Cross-sectional layout of the bridge isshown in Figure 3.

3. Steel Thermal Parameters andFire Heat Release Rate

3.1. Expressions for the SteelThermal Parameters. The thermalparameters used in this study include thermal conductivity(W/(m⋅K)), specific heat (J/(kg⋅K)), thermal expansion coef-ficient, and elastic modulus (MPa). Thermal parameters ofsteel vary with temperature observably, and the value of theresearch results is relatively discrete. Therefore, this studyfirstly summarizes the value of thermal parameters in [8–11].Then, according to the principle of least square method, ther-mal parameters are fitted by polynomial fitting to minimizethe value error.The polynomial fitting expressions of thermalparameters are obtained as follows.

Cubic polynomial fitting of thermal conductivity of steelis obtained as

𝜆 = 2.15 × 10−8

𝑇3

− 4.22 × 10−5

𝑇2

− 1.03 × 10−2

𝑇

+ 51.08.

(1)

Quadratic polynomial fitting of specific heat of steel isobtained as

𝑐 = 4.64 × 10−4

𝑇2

+ 11.3 × 10−2

𝑇 + 486.37. (2)

Linear polynomial fitting of thermal expansion coeffi-cient of steel is obtained as

𝛼 = 9.10 × 10−9

𝑇 + 1.08 × 10−5

. (3)

Mathematical Problems in Engineering 3

Table 1: Dimensions of steel box girder.

Position Thickness Type of stiffeners Thickness of stiffeners Number of stiffenersTop flange 20mm, 24mm T-rib Web of 12mm, flange of 10mm 22Bottom flange 20mm, 24mm T-rib Web of 12mm, flange of 10mm 14Side web 16mm Rib plate 12mm 9Middle web 16mm Rib plate 12mm 3Diaphragm 12mm — — —Note: the spacing of the diaphragm is 1.5m in the longitudinal direction of the bridge.

22.13022.060 22.20026

28

22

18

14

10

6

Ground Ground

120

(a)

1.68

30 30

Ground Tanker fire

4

Longitudinal direction

(b)

Figure 2: (a) Overpass of Tianhe District Second Highway Engineering in Wuhan city (m). (b) Position in which tanker fire happened (m).

2.75 2.75

4.5 4.5

9

5.5

Figure 3: Cross-sectional layout of steel box girder bridge (m).

Polynomial fitting of elastic modulus of steel is obtainedas

𝐸 (𝑇)

𝐸

= 3.55 × 10−18

𝑇6

− 2.07 × 10−14

𝑇5

+ 4.00

× 10−11

𝑇4

− 3.16 × 10−8

𝑇3

+ 9.02 × 10−6

𝑇2

− 0.0014𝑇 + 1.038,

(4)

where 𝑇 is temperature (∘C).Thermal parameters varied with time are shown in

Figure 4. 𝐸 is elastic modulus of steel at room temperatureand 𝐸(𝑇) is elastic modulus of steel at temperature 𝑇∘C. InFigure 4(d), the steel Q370qD loses almost all of its elasticmodulus between 1000∘C and 1200∘C. In the range between1000∘C and 1200∘C, elastic modulus of steel is close to 0.

So polynomial fitting curve of elastic modulus presents ahorizontal trend between 1000∘C and 1200∘C.

3.2. Heat Release Rate Function. Heat release rate growthmodels are widely studied in the field of tunnel and somescholars studied the heat release rate growth models basedon the fire experiment in tunnel in the past few years. Thereare many kinds of fire heat release rate growth models toanalyze fire hazard. Linear growth model, square growthmodel, and exponential growth model are usually used. Thegrowth model is composed of two parts: the maximum heatrelease rate (𝑄max) and the recession function [12].

The whole process of fire heat release rate is basicallyreflected through these heat release rate growth models.Linear model is simplest in the mathematical model of heatrelease rate and there are large deviations with the growth


0 200 400 600 800 1000 120010

15

20

25

30

35

40

45

50

55Th

erm

al co

nduc

tivity

(W/(

mK)

)

Temperature (∘C)

(a) Thermal conductivity

0 200 400 600 800 1000 1200400

600

800

1000

1200

1400

Spec

ific h

eat (

J/(kg

K))

Temperature (∘C)

(b) Specific heat

0 200 400 600 800 1000 12000.000010

0.000012

0.000014

0.000016

0.000018

0.000020

0.000022

Ther

mal

expa

nsio

n co

effici

ent

Temperature (∘C)

(c) Expansion coefficient

0 200 400 600 800 1000 1200

0.0

0.2

0.4

0.6

0.8

1.0

Temperature (∘C)

E(T

)/E

(d) Elastic modulus

Figure 4: Polynomial fitting curve of steel thermal parameters.

curve of the actual fire heat release rate. The mathematicalmodel of the heat release rate under different control condi-tions is given by exponential model, and the growth curveis also most consistent with the actual fire. In this study,the exponential growth model is used to model the fire heatrelease rate.

The heat releasemodel of the exponential heat release ratefire is given by

HRR = 𝑄max ⋅ 𝑛 ⋅ 𝑟 ⋅ (1 − 𝑒−𝑘⋅𝑡

)

𝑛−1

⋅ 𝑒−𝑘⋅𝑡

, (5)

where HRR is the heat release rate function (MW), 𝑄 is themaximum heat release rate (MW), 𝑛, 𝑟, and 𝑘 are parametersunder different fire scenarios, and 𝑡 is the fire time (s).

Seven kinds of heat release rate models under fire scenar-ios are given by foreign research [13], as shown in Table 2,and the corresponding fire heat release rate curve is shown inFigure 5.There is very little current research on bridges in fire,but characteristic of fuel combustion in the bridge is similar tothat in the tunnel.This research takes some tunnel fire curvesas a consequence of the absence of bridge fire curves.

From Figure 5, the heat release rate peak of scenario 7 istoo high, and its temperature growth rate is much faster thanother scenes. This fire scale is most dangerous. Therefore, thefire scene used in this study is a 200MW oil tanker burningnear steel bridge.

4. Results of Thermal Analysis Model

4.1. Thermal Analysis Model. The thermal analysis model oftwo-span continuous steel box girder bridge is establishedby FDS (Fire Dynamics Simulator). In this research, theemployed version of FDS is FDS 5.0. FDS was developedby NIST (National Institute of Standards and Technology)in America. FDS is a powerful fire simulator and FDSsimulates fire scenarios using computational fluid dynamics(CFD) optimized for low-speed, thermally driven flow. Thisapproach is very flexible and can be applied to fires rangingfrom stove-tops to oil storage tanks. FDS models can predictsmoke, temperature, carbon monoxide, and other substancesduring fires. The results of these simulations have beenused to ensure the safety of structures before construction.


Table 2: The value of fire heat release rate exponential growth model.

Firescenarios Fire type Total heat energy

𝐸tot (GJ)Maximum heat release rate

𝑄max (MW) 𝑛 𝑟 𝑘(∗10−3

)

1 Two cars 14.4 8 2.1 2.04 1.132 Bus 52.5 25 2.0 2.0 0.953 Truck + bus 75 75 5 2.44 2.444 Heavy truck (low exposure) 150 100 3.2 2.28 1.525 Heavy truck (high exposure) 225 150 3.2 2.28 1.526 Truck + bus + 6 cars 367 175 2.6 2.17 1.047 Tanker 1500 200 1.1 1.27 0.17

20406080

100120140160180200220

Hea

t rele

ase r

ate (

MW

)

Times (s)

Scenario 1Scenario 2Scenario 3Scenario 4

Scenario 5Scenario 6Scenario 7

1000 2000 3000 4000 5000 6000 7000 800000

Figure 5: Heat release curves under different fire scenes.

FDSmodel validation was proved bymany researchers. Alos-Moya et al. researched a bridge failure due to fire usingcomputational fluid dynamics and finite element modelsand pointed out that FDS models were able to simulate theobserved response of the bridge under fire hazard [14].

In the bridge model, the coordinate axis center is locatedin the center line of the base floor of bridge abutment, thelongitudinal direction of the bridge is 𝑥 direction, transversedirection of the bridge is 𝑌 direction, and 𝑧 direction isvertical, as shown in Figure 6.

The calculation area of 𝑋 direction is 80m, dividedinto 80 segments. The 𝑌 direction is 15m, divided into 45segments. The 𝑍 direction is 10m, divided into 50 segments.The whole computing model is divided into 180000mesh.

In this FDS model, two kinds of sensors are used.Thermocouples are set up to measure gas temperature andwall sensors are set up to measure bridge temperature. Theinitial temperature is set at 20∘C, and the wind speed isnot considered. The cross-sectional temperature measuringpoints arrangement of steel box girder is shown in Figure 7to measure transient temperature variation of bridge undertanker burning. Because the cross section is symmetrical

Y

Z

X

Figure 6: Coordinate axis of thermal analysis model.

along the centerline, test points are only arranged on +𝑌

direction.Wall sensors are set up in the girder to measure thetemperature of each part. The bottom floor has 10 test points,from D1 to D10, spacing 300mm. The roof floor has 8 testpoints, from D11 to D18, spacing 600mm.The outer web has8 test points, from F1 to F8, spacing 200mm. The inner webhas 8 test points, from F9 to F16, spacing 200mm.

Bottom floor in negative moment area is in the stateof compression and the side web is in the state of bendingshear stress. Negative moment area webs are most prone toinstability for that web critical buckling stress in negativemoment area is far less than the other position and thetemperature of the side web and the bottom floor rises fastestunder hazard. So the fire source is located in negativemomentarea.Thefire is set to 12m long, 2mwide 200MWtanker. Fireis 4m under the bridge because clearance height of bridge is4m. Fire center coordinates are (0, 0, −4m). The heat releaserate (MW) function is

HRR = 200 × 1.1 × 1.27 × (1 − 𝑒−0.00017⋅𝑡

)

1.1−1

× 𝑒−0.00017⋅𝑡

.

(6)

In this paper, special value for soot yield was consideredin the FDS model. In the FDS model, soot yield changeswith the heat release rate of oil tanker. Heat release rate ofoil tanker was defined as the source of fire, and soot yieldcould vary with heat release rate. Temperature of soot wouldtransfer towards the bridge through convection radiation.


2.75 2.75

D1D2D3D4D5D6D7D8D9D10F1F2F3F4F5F6

F12F13F14F15F7

F8

F11F10

F9

F16

4.5 4.59

D11D12D13D14D15D16D17D18

7 × 0.6

7×0.2

7×0.2

9 × 0.3

Figure 7: Temperature measuring point arrangement of steel box girder (m).

So temperature of steel would change with time due to thetanker fire and soot yield.

4.2. Results of Thermal Analysis. In this research, abutmentsof the bridge use concrete C50. Elasticity modulus of C50 is3.45 × 104MPa at room temperature. Thermal conductivityof C50 is 1.52W/(m⋅K) at room temperature. Thermal con-ductivity of deck of the bridge (steel Q370qD) is 51W/(m⋅K)at room temperature. Thermal conductivity of abutmentsis far less than thermal conductivity of deck of the bridge.That means abutments have poor thermal conductive ability.When fire happens, heating rate of abutments is very slow.Temperature of abutments is very low and it is far lessthan temperature of the deck of the bridge. Mechanicalproperties of abutments change little under fire. So in thispaper, calculation of abutments does not need to be doneunder fire.

Temperature of the steel bridge in the process of thecontinuous 60-minute combustion is increasing rapidly.Transient temperature of steel girder, the heat flux density,and transient temperature of surrounding air are calculated.

Wall sensors are set up in the girder to measure thetemperature of each part. The calculation results show thattemperature from D1 to D10 has little difference, so as F1 toF8, F9 to 16, and D11 to D18. So in this paper, the averagetemperature of bottom floor, the average temperature ofmiddle web, the average temperature of side web, and theaverage temperature of top floor were used to calculate thebuckling stresses of the girder.

By taking the average temperature of bottomfloor,middleweb, side web, and top floor, the average temperature of thecorresponding parts is obtained, as shown in Figure 8.

When fire hazard happens, source of fire would burn andrelease heat first. As a result, heat release rate would increase.Then temperature of the air around fire would increase basedon the heat. Air around fire then transfers heat to the bridgethrough convection and radiation. As a result, temperatureof bridge would increase. Temperature rising rate of bridge

lags behind fire heat release rate for that it takes some timefor the bridge temperature to rise during convection andradiation. In Figure 8, temperature maintains a horizontaltrend between 1200 s and 3600 s. In Figure 5, when timereaches 1200 s, the heat release rate drops to about 180MW,and when time reaches 3600 s, the heat release rate drops toabout 150MW; 150MW∼180MW has enough heat to let thebridge temperature maintain a horizontal trend. So Figure 8shows curves with no decay. If the heat release rate drops tosome value which is very low (e.g., under 100MW) over time,Figure 8(d) may decay.

From the graph, the temperature of the bottom floorand the side web under fire rose rapidly and reached thehighest temperature.The highest temperature of bottomfloorexceeded 1000∘C. The highest temperature of top floor andmiddle web was relatively low, and highest temperature ofmiddle web was close to 600∘C. Each part of the warmingtrend is consistent, and the highest temperature is about 20minutes.

The elastic modulus of steel at high temperature willchange greatly. According to formula (4), the elastic modulusof bottom floor, middle web, side web, and top floor underhigh temperature is obtained, as shown in Figure 9.

5. Buckling Stress Calculation Method forSteel under High Temperature

From Figure 8, temperature of side web is much higher thanthat of middle web. So side web is more dangerous thanmiddle web. Therefore, the critical bending shear bucklingstress expression of the side web and the critical bucklingstress of the stiffened bottom floor bearing axial pressureunder variation temperature are derived, respectively.

Elastic stability theory of steel floor is used in this section.The critical bending shear buckling stress expression of theside web and the critical buckling stress of the stiffened bot-tom floor bearing axial pressure under variation temperature


0 500 1000 1500 2000 2500 3000 3500 40000

200

400

600

800

1000

1200

Time (s)

Tem

pera

ture

(∘C)

(a) Bottom floor

0 500 1000 1500 2000 2500 3000 3500 40000

100

200

300

400

500

600

Time (s)

Tem

pera

ture

(∘C)

(b) Middle web

0 500 1000 1500 2000 2500 3000 3500 40000

200

400

600

800

1000

Time (s)

Tem

pera

ture

(∘C)

(c) Side floor

0 500 1000 1500 2000 2500 3000 3500 40000

50

100

150

200

250

Time (s)

Tem

pera

ture

(∘C)

(d) Top floor

Figure 8: Average temperature varied with time of steel bridge.

are derived, respectively [15–17]. The concrete derivationprocess is as follows.

5.1. Calculation Method of Bottom Floor Based on BucklingTheory. The area ratio 𝛿

𝑙

of the bottom floor is the ratio ofthe area of a rib 𝐴

𝑙

to the area of floor section 𝑏 × 𝑡:

𝛿𝑙

=

𝐴𝑙

𝑏𝑡

=

100 × 10 + 160 × 12

350 × 20

= 0.417. (7)

The stiffness ratio 𝛾𝑙

of the bottom floor is the ratio ofbending rigidity EI

𝑙

of connection position of stiffer and floorto product of𝐷 and floor width 𝑏:

𝛾𝑙

=

EI𝑙

𝐷𝑏

=

12𝐼𝑙

(1 − 𝜐2

)

𝑡3

𝑏

=

12 × 41992333 × (1 − 0.32

)

350 × 203

= 163,

(8)

where 𝜐 is Poisson’s ratio, 𝑡 is floor thickness, and 𝑏 is stiffenedfloor width.

According to [12], the stiffer is rigid when the aspect ratioof the stiffened floor meets condition 𝛼 ≤ 𝛼

0

:

𝛾∗

𝑙

=

1

𝑛

[4𝑛2

(1 + 𝑛𝛿𝑙

) 𝛼2

− (𝛼2

+ 1)

2

]

=

1

8

[4 × 82

(1 + 8 × 0.417) 0.5822

− (0.5822

+ 1)

2

]

= 46.78,

(9)

where 𝛼0

=4√1 + (𝑛 + 1)𝛾

𝑙

=4√1 + (8 + 1)163 = 6.19.

𝑎 = 1600, and 𝑎 is width of subplate divided bystiffeners.

𝑏 = 2750, and 𝑏 is length of subplate divided bystiffeners.

𝛼 = 𝑎/𝑏 = 1600/2750 = 0.582; 𝑁 is the number ofweb segmentations.


0 500 1000 1500 2000 2500 3000 3500 40000.0

0.2

0.4

0.6

0.8

1.0

Time (s)

E(T

)/E

(a) Bottom floor

0 500 1000 1500 2000 2500 3000 3500 40000.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

Time (s)

E(T

)/E

(b) Middle web

0 500 1000 1500 2000 2500 3000 3500 40000.0

0.2

0.4

0.6

0.8

1.0

Time (s)

Tem

pera

ture

(∘C)

(c) Side floor

0 500 1000 1500 2000 2500 3000 3500 4000

0.90

0.92

0.94

0.96

0.98

1.00

1.02

Time (s)

E(T

)/E

(d) Top floor

Figure 9: Elastic modulus varied with time of steel bridge.

So the bottom floor stiffener is rigid, and the criticalbuckling stress coefficient for bottom floor stiffener withoutconsidering the constraints of the web on the bottom is

𝑘 = 4 (𝑛 + 1)2

= 324. (10)

Therefore, the critical buckling stress of bottom stiffenedfloor under room temperature is

𝜎crl =𝜋2

𝐸 (𝑛 + 1)2

𝑡2

3 (1 − 𝜐2

) 𝑏2

= 3187MPa. (11)

From this, it is seen that the critical buckling stressof the bottom floor is a function related to the elasticmodulus 𝐸 of steel. When fire hazard happens, bottom flooris directly affected by the fire temperature, and the bottomfloor is heated. The elastic modulus of bottom stiffening rib

is decreased. According to the change function of elasticmodulus and temperature of steel of (4), it can be seen that

𝜎crl (𝑇) =𝜋2

𝐸 (𝑇) (𝑛 + 1)2

𝑡2

3 (1 − 𝜐2

) 𝑏2

. (12)

The bridge will be suddenly destroyed due to hightemperature buckling, when 𝜎crl(𝑇) is less than the actualstress of floor, causing serious consequences such as bridgecollapse.The critical buckling stress of bottom floor when fireis burning for 17 minutes is 𝜎crl(𝑇) = 51MPa according to (4)and (12).

5.2. CalculationMethod of SideWebBased on BucklingTheory.Derivation process of calculation method of web buckling isvery complicated.Derivation process contains a large numberof differentials and integrals. So we give the critical bucklingstress of web under bending load and under shear load basedon [15]. The detailed derivation process is in [15].


I

J

46

2 J

x

yz

62

1

3

5

7

3K

8

4

L

5

1I

X

Y

Z

K, L𝜃

Figure 10: Shell63 geometry.

Critical buckling stress of side web under bending load is

𝜎crl,𝑙𝑤 =𝜋2

𝐸𝑘𝑡𝑤

2

12 (1 − 𝜐2

) ℎ𝑤

2

. (13)

When 𝑎𝑤

> ℎ𝑤

, it can be seen as follows:

𝜓 =

𝜎2

𝜎1

= −0.5,

𝑘 = 7.81 − 6.29𝜓 + 9.78𝜓2

,

(14)

𝜎crl,𝑙𝑤 (𝑇) =𝜋2

𝐸 (𝑇) 𝑘𝑡𝑤

2

12 (1 − 𝜐2

) ℎ𝑤

2

, (15)

where𝜓 is bending gradient and 𝑎𝑤

, ℎ𝑤

, and 𝑡𝑤

are the width,height, and thickness of floor, respectively.

Critical buckling stress of side web under shear load is

𝜏crl,𝑙𝑤 =𝜋2

𝐸𝑘𝑡𝑤

2

12 (1 − 𝜐2

) 𝑎𝑤

2

,

𝑘 =

{{{

{{{

{

5.34 + 4.0 (

ℎ𝑤

𝑎𝑤

)

2

(𝑎𝑤

> ℎ𝑤

) ,

4.0 + 5.34 (

ℎ𝑤

𝑎𝑤

)

2

(𝑎𝑤

≤ ℎ𝑤

) ,

𝜏crl,𝑙𝑤 (𝑇) =𝜋2

𝐸 (𝑇) 𝑘𝑡𝑤

2

12 (1 − 𝜐2

) 𝑎𝑤

2

.

(16)

Web critical buckling stress under bending shear stressstation should meet the condition of

(

𝜎

𝜎crl,𝑙𝑤 (𝑇))

2

+ (

𝜏

𝜏crl,𝑙𝑤 (𝑇))

2

≤ 1. (17)

Web critical buckling stress theoretical value under roomtemperature is 𝜏 = 80.3MPa, 𝜎 = 167.1MPa from (15) and (16)and web critical buckling stress is 𝜏 = 18.5MPa, 𝜎 = 38.4MPawhen tanker fire is burning for 17 minutes.

6. Buckling Instability Finite Element Analysisof Steel Bridge under Fire

The fine steel shell model is established to get center spandeflection under high temperature, bending share ratio, andbending stress gradient because the bending share ratio andbending stress gradient of the bridge are unknown. Thevalidity of the theoretical formula is verified by establishingindependent web and stiffening floor model. The finiteelement software ANSYS [18] is used to establish the modelof two-span continuous steel bridge, and the element type ofsteel beam is shell63.There are 63254 units and 102458 nodesamong thewhole bridge.The version ofANSYS isANSYS 14.0in this research.

Shell63 has both bending and membrane capabilities.Both in-plane and normal loads are permitted. The elementhas six degrees of freedom at each node: translations inthe nodal 𝑥, 𝑦, and 𝑧 directions and rotations about thenodal 𝑥-, 𝑦-, and 𝑧-axes. Stress stiffening and large deflectioncapabilities are included. A consistent tangent stiffnessmatrixoption is available for use in large deflection (finite rotation)analyses.

Temperatures may be input as element body loads at the“corner” locations (1–8) shown in Figure 10: “shell63 geome-try.” The first corner temperature T1 defaults to TUNIF. If allother temperatures are unspecified, they default to T1. If onlyT1 and T2 are input, T1 is used for T1, T2, T3, and T4, whileT2 (as input) is used for T5, T6, T7, and T8. For any otherinput pattern, unspecified temperatures default to TUNIF, asshown in Figure 10.

Gravitational acceleration in this paper is 9.8m/s2. Thedensity of steel is 7850 kg/m3. Gravity loads of the bridge are3560 kN.

Thebucklingmode, web shear stress, andweb and bottomstress of the bridge under room temperature are calculatedunder the action of dead load and lane load.

From the calculation results, it is seen that the bendingstress gradient of web is −8.5MPa/(36.8MPa).

The side web model is established independently sur-rounded by simply supported sides because its heating rateis far greater than middle floor and the highest temperature


Table 3: Floor stress when tanker is burning for 17 minutes.

Floor position Actual stress Theoretical value ofcritical buckling stress

Finite element value ofcritical buckling stress

Side floor 𝜏 = 19.6MPa 𝜏 = 18.5MPa 𝜏cr1,lw(𝑇) = 19.1MPa𝜎 = 39.8MPa 𝜎 = 38.4MPa 𝜎cr1,lw(𝑇) = 38.8MPa

Bottom floor 𝜎 = 37.1MPa 𝜎cr1(𝑇) = 51.0MPa 𝜎cr1(𝑇) = 65.7MPa

is far higher than middle floor according to the results ofFigure 8. Bending shear ratio is 2.03 : 1 and bending gradientis −0.5. Critical buckling stress value from finite elementcalculation is 𝜏 = 95.9MPa, 𝜎 = 193MPa.

Critical buckling stress of side web is 𝜏 = 92.3MPa,𝜎 = 187.3MPa; 𝜏 = 83.5MPa, 𝜎 = 169.6MPa; 𝜏 = 44.3MPa,𝜎 = 89.9MPa; 𝜏 = 19.1MPa, 𝜎 = 38.8MPa, when the fireis burning for 5 minutes, 10 minutes, 15 minutes, and 17minutes, respectively, according to the elasticmodulus of webchanging with temperature under fire.

Stiffened bottom floor model is established indepen-dently surrounded by simply supported sides. Critical buck-ling stress of finite element calculation under uniform pres-sure and room temperature is 𝜎 = 3025MPa.

Critical buckling stress of bottom floor is 𝜎 = 2883MPa,𝜎 = 2240MPa, 𝜎 = 445MPa, and 𝜎 = 65.7MPa, when thefire is burning for 5 minutes, 10 minutes, 15 minutes, and17 minutes, respectively, according to the elastic modulus ofbottom floor changing with temperature under fire.

From finite element calculation, it is seen that criticalbuckling stress will be reduced to 𝜏crl,𝑙𝑤(𝑇) = 19.1MPa,𝜎crl,𝑙𝑤(𝑇) = 38.8MPa, when tanker is burning for 17 minutesand at this time critical buckling stress does not comply with(17).Therefore, the webwill be the first to show bending shearbuckling failure.

Critical buckling stress of bottom floor will be reduced to65.7MPa, when the fire is burning for 17minutes, higher thanactual stress 𝜎 = 37.1MPa caused by dead and live load underhigh temperature. So bottom floor will not be destroyed in 17minutes, as shown in Table 3.

Elastic modulus of each floor in the negative momentregion is reduced by (4) to calculate the growth of displace-ment in midspan under fire load. Displacement values of thebridge are 21.2mm, 25.7mm, 31.5mm, and 35.4mm, whenbridge is at room temperature, burning for 10 minutes, 15minutes, and 17 minutes, respectively.

From displacement analysis, it is seen that displacementinmidspan of steel bridge will increase, and the displacementwill increase from21.2mm to 35.4mm in 17minutes, less thanthe allowable value of the code [19] (L/600 = 50mm), whereL is the length of the span.

The calculation results show that buckling instability ofthe side web of the steel bridge will be the first to occur beforethe allowable value of the bridge is reached.

7. Conclusions

(1) Thermal analysis model of two-span continuous steelgirder bridge is established by fire dynamics softwareFDS. Thermal release rate function of tanker fire

is obtained to simulate tanker burning. Transienttemperature field and elastic modulus changing withtime in the negative moment region of the bridge arecalculated.

(2) Critical buckling stress of web and the bottom floorchanging with time is obtained by calculating thebuckling behavior of the bridge under high temper-ature of fire. Stress of side web is greater than thecritical buckling stress under tanker burning for 17minutes and the bending shear buckling failure willbe the first to occur. At this time, displacement inmidspan is stillmaintainedwithin the allowable rangeof specifications.

(3) Temperature of web and bottom floor rises very fastwhen tanker burns for 15 minutes to 17 minutes,which is a very dangerous moment, while temper-ature of web and bottom floor is maintained at arelatively low level during the first 15 minutes, and thebridge has a large safety reserve. So the best rescuetime of the steel bridge under tanker fire is within15 minutes. The bridge is likely to be destroyed if thetanker burns more than 15 minutes.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This paper was financially supported by the National NaturalScience Fund, “Stability of High Web in Long-Span Steel-Concrete Composite Bridge Based on the Elastic RotationalRestraint Boundary” (no. 51378405), and the Natural ScienceFund of Hubei Province (no. 2013CFA049).

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