JCSS Workshop on Reliability Based Code Calibration

Reliability-based Design of New Expansion Joints for the Little Belt Suspension Bridge

Poul Linneberg Pedersen COWI A/S,

Abstract This paper describes the practical experience with reliability-based design of new expansion joints for the Little Belt Suspension Bridge. The methodology is based on the concept that a bridge does not necessarily have to fulfil the specific requirements of a general code, as long as the overall level of safety defined by the code is satisfied. The purpose of this approach is to reduce the required displacement capacity of the new expansion joints compared with the existing ones and thereby to minimise the maintenance costs in the future without compromising the level of safety. The safety evaluations are based on probabilistic methods in which uncertainties can be taken into account consistently. In this present case, temperature load, bridge-specific traffic load and structural response (influence areas) are modelled as uncertain parameters. Further, proof-loading is used for calibration of the structural response from Finite Element calculations. Based on the above approach the capacity of the new expansion joints are reduced up to 60 % of the previous capacity.

Keywords: Temperature load, proof-loading, stochastic traffic load modelling, safety level and calibration of companion load factors and partial safety factors.

1 Introduction Design of new expansion joints for the Little Belt Suspension Bridge could have been done according to a general approach for safety evaluation of an existing bridge based on codes and regulations. In most cases, the codes are based on a partial safety factor format and include a large degree of generalisation in terms of safety and load specifications. Further, the codes must include general rules applicable for many types and various geometries of bridges. The fact that the codes generalise is efficient because the load and safety calculations become easy and extra costs due to the generalisation are typically marginal in the budget of a new bridge. However, it is not always suitable when dealing with operation management and rehabilitation of existing structures (including bridges such as the Little Belt Suspension Bridge).

The methodology, used in this project, is based on the concept that a bridge does not necessarily have to fulfil the specific requirements of a general code, as long as the overall level of safety defined by the code is satisfied. The purpose of this approach is to reduce the required capacity of the new expansion joints and thereby minimise the maintenance costs in the future without compromising the level of safety.

In the reliability-based approach uncertainties can be taken into account consistently. In this present case temperature load, bridge-specific traffic load and structural

1 JCSS Workshop on Reliability Based Code Calibration response (influence areas) are modelled as uncertain. Uncertainties, included in the structural response, are e.g. physical uncertainties in the use and identification of materials and the simplification in the structural model. Furthermore, the bridge- specific traffic load takes the heavy truck frequency into account so that the probability of a heavy truck meeting another heavy truck is lower for a bridge with a relatively low heavy truck frequency than for a bridge with a high heavy truck frequency. 2 The Little Belt Suspension Bridge The Little Belt Suspension Bridge is build during the period 1965-70 and it is located in Denmark. The Road Directorate is the owner and COWI A/S is the consultant presently consulting in operation management and rehabilitation. The bridge carries a motorway with 6 lanes (3 in each direction) across the Little Belt, connecting in the west with the island of in the east, see Fig. 1. The Little Belt Suspension Bridge has an overall length of 1700 m and an overall width of the bridge deck of 24.5 m. The Suspension Bridge has a length of 1080 Figure 1 Little Belt Suspension Bridge. The island of m, a main span of 600 m and two Funen is in the front of the picture. adjoining spans each with a length of 240 m. Elevation of the Little Belt Suspension Bridge is shown as Fig. 2.

Figure 2 Elevation of the Little Belt Suspension Bridge.

The bridge deck consists of a steel-box girder, which is seen from Fig. 3, where the existing expansion joints from 1970 are presented.

Figure 3 Expansion joint at one of the two adjoining towers (left) and at one of the two pylons (right).

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The new expansion joints will probably be based on the same principle as the existing ones using steel-manufactured coverplates, which slides on steel-trestles. 3 Characteristic temperature load In the original design basis from 1965 a maximum and minimum temperature of +/- 35 °C was used.

According to the Danish Code of Practice for Loads for the Design of Structures [1], maximum and minimum shade air temperatures, with a return period of R years, are:

1− K ln(−ln(1− 1 )) T max = T max 1,max R (1) R 50 − − 1 K1,max ln( ln(0.98))

max = o Where T50 36.0 C and K1,max= 0.026. The temperature model is described in detail in Sorensen [2]. Coefficient of variation of the Gumbel distribution is equal to 0.03.

1− K ln(−ln(1− 1 )) T min = T min 1,min R (2) R 50 − − 1 K1,min ln( ln(0.98))

min = − o Where T50 31.0 C and K1,min= 0.10. Coefficient of variation of the Gumbel distribution is equal to 0.14.

Based on shade air temperatures, the extreme temperatures in the bridge deck are calculated according to the Eurocode 1 - Part 2.5 [3]. According to the above mentioned, approximately +/-45 °C is estimated as maximum and minimum temperature.

Further, Eurocode 1 - Part 2.5 prescribes that maximum and minimum temperatures should be increased/decreased with 10 °C for expansion joint displacements. 4 Characteristic traffic load The capacity of the existing expansion joints are +468 mm/-499 mm for the main span and +/- 71 mm for the adjoining spans. In this paper rotations of the bridge deck around a horizontal axis are not discussed.

A stochastic traffic load model is used to establish new capacity requirements. For a detailed description the reader is referred to Madsen et al. [4] and Ditlevsen [5]. For calculation of the characteristic traffic load effect (98% percentile), the yearly extreme distribution has to be estimated. This is done for the following four traffic situations:

1. Congestion in one direction and no vehicles in the other direction 2. Free flowing traffic in one direction and no vehicles in the other direction 3. Congestion in one direction and free flowing traffic in the other direction 4. Congestion in both lanes

The mathematical formulation of the stochastic traffic load model is based on a Markov chain model for vehicle alternation in congested traffic, influence lines for

3 JCSS Workshop on Reliability Based Code Calibration each load effect and Poisson processes describes occurrence of congested traffic situations in time. A detailed description of the model is given in Madsen et al. [4].

4.1 Input In the following scheme input-parameters for the traffic load model at the main span are given. A discussion of selected parameters is below the scheme.

Variable Symbol Value Number of lanes (total) 6 Length of influence line L 540 m Length of car [4] l 6 m Weight of car [4] Pc 10 kN Mean weight of truck [4] µW 188 kN Standard deviation of truck weight [4] σW 151 kN Percentage of trucks in traffic f 0.20 Platoon length parameter [4] γ 0.93 Fraction of trucks in inner lane (1) in congestion [4] 0.60 Fraction of trucks in middle lane (2) in congestion [4] 0.30 Fraction of trucks in outer lane (3) in congestion [4] 0.10 Frequency of congestion in one direction υ 5.5 /year

Frequency of congestion in both direction υ12 0.0076 /year Mean duration of congestion [4] µD 1.8 hours Common speed for trucks and cars V 75 km/hour

Intensity of cars λc 3500/hour/direction Intensity of trucks λt 900 /hour/direction Fraction of trucks in inner lane (4) in free traffic [4] 0.00 Fraction of trucks in middle lane (5) in free traffic [4] 0.10 Fraction of trucks in outer lane (6) in free traffic [4] 0.90 Fraction of cars in inner lane (4) in free traffic [4] 0.10 Fraction of cars in middle lane (5) in free traffic [4] 0.30 Fraction of cars in outer lane (6) in free traffic [4] 0.60 Reference time in years corresponding to a 6 hours situation with 0.25 maximum traffic intensity Furthermore, influence areas and squared areas are computed and used as input in the traffic model.

Length and weight of car: According to Weigh-In-Motion measurements performed by Vägverket at E6 near Torp in the eastern part of Sweden from June 1993 to August 1994, the mean truck length is estimated as approximately 17 m with 2 m between the trucks, see Getachew et al. [6]. The traffic situation in the eastern part of Sweden is assumed similar to the situation at the Little Belt Suspension Bridge in Denmark.

The traffic load model is based on an assumption that the length of a truck is twice the length of a car. The length and weight of a car and fraction of trucks are therefore corrected to account for this fact (not in the above table).

Weight of a truck: Weight of a truck is assumed similar to the one used for design of the Link, see Madsen et al [4].

Percentage of trucks: Percentage of trucks should cover future traffic intensities. The Road Directorate measured a percentage of trucks of 15 in 1999 nearby the Little Belt Suspension

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Bridge. This value is projected as far as the year 2016 with an annual increase by 2 % in a similar fashion as traffic analysis of the Great Belt Link.

Platoon length parameter: This parameter is corresponding to the mean number of trucks in a platoon in freely flowing traffic. This value is based on traffic surveys of motorways in Denmark performed by Mohr [7].

Traffic intensity: Traffic intensities must also be projected to take into account future increase in traffic. Based on measurements from the Road Directorate the annual average increase is evaluated to 5 % and also projected until 2016. It is important to ensure that the projected intensity is lower than the actual capacity.

Frequency of congestion: The frequency of a full loading length L is according to Ditlevsen [5]:

υ = ε µ − (− L ) ( L)exp µ −L (3)

ε is the congestion intensity and equal to 3.848⋅10-3 /day/km/lane based on data from a 6 lane motorway (A9) from Munich in the northern direction and used on the Great Belt Link. µ is calculated from:

µ=E[Λ]/3+L (4)

E[Λ] is the mean queue length which according to Munich data and also used on the Great Belt Link is equal to 18.6 km.

The frequency of congestion in both directions is given by:

2 υ12=2υ µD (5)

µD is the mean duration of congestion. 4.2 Proof-loading Initially a calculation of displacement requirements where all loads were included was performed for the expansion joints at the main span. These calculations yield displacement requirements above the ones estimated in the original design basis from 1965. Based on the fact that the main part of the displacements arises from traffic load and the fact that the Finite Element Model never had been calibrated to real measurements, a proof-loading program was established.

The Road Directorate planned in corporation with COWI A/S the program for the proof-loading. The Danish Army kindly made seven heavy vehicles available for the program, each with a dead load of approximately 75 tons and a length of 18 and 20 meters respectively.

The heavy vehicles were parked in the slow lane on the eastside of the main span and on the east of the adjoining spans. Six different load arrangements were

5 JCSS Workshop on Reliability Based Code Calibration established: three on the main span and three on the adjoining span with a total of seven vehicles and four vehicles respectively. The displacement of the expansion joints without the seven heavy vehicles was measured twice, before and after the proof-loading program, because of normal traffic situation on the five remaining lanes on the bridge.

Measurement of displacements where carried out at eight places: at north and south in both ends of the main span and both ends at the eastern of the adjoining spans. The duration of each measurement was 10 minutes with a registration of displacements each minute. The duration and frequency were determined based on the fact that:

• Measurements should be done manually with a folding rule against a fixed measurement-arrangement • A wish to reduce the impact of other heavy vehicles passing the bridge • Minimising other dynamic influences from e.g. wind • A relatively short overall duration of the proof-loading program

Proof-loading took place on the 8th of October 2001 between 19-22 hours implying relatively low traffic intensity (400-800 vehicles/hour/direction). The temperature was relatively stable at 14-15 °C during the entire proof-loading program.

Based on the proof-loading program mean values and standard deviations of the displacements were calculated at each load step. Due to the measurement equipment the accuracy of each measurement was approximately 1 mm.

The main conclusions from the measurements are listed below. All conclusions are valid for displacements of both the main span and the eastern of the adjoining spans.

• Displacements are in general symmetric in the bridge length axis. • Differences in displacements at both ends of the main span and the eastern of the adjoining spans were registered. • Means of measurements during "unloaded" bridge at the beginning and the end of the program are rather different (between 0.4 and 11.7 mm). Due to a relatively low temperature difference during the program, the reason must be that the expansion joints have not moved back to their original position after proof-loading. • Standard deviations are not systematic with either location of the measuring point or the loading level.

Measurements are used for calibration of the load effect (influence area). To include the statistical uncertainty due to measurement errors, the area of influence lines and the squared area of influence lines are modelled as stochastic variables with normal distributions N(µ,σ) in the traffic load model. The coefficient of variation for the main span is determined from the load arrangement with maximum load on the bridge (seven vehicles on the main span). This is reasoned due to the fact that measurement errors must be less for a high loading level because of larger displacements. Based on this assumption, the coefficients of variation for the area of influence function are 0.10 and 0.12 for the squared area of the influence function.

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4.3 Results The characteristic load effects and coefficients of variation are calculated with the described stochastic traffic load model with influence areas calibrated by the proof- loading results. The statistical parameters are estimated by fitting a distribution function to the outcome of the load model. Characteristic expansion joint displacements are shown for the main span in Table 1.

Traffic load case V Displacement [mm] Congestion in one direction and no vehicles in the other direction 0.13 +/- 262 Free flowing traffic in one direction and no vehicles in the other direction 0.06 +/- 125 Congestion in one direction and free flowing traffic in the other direction 0.10 +/- 348 Congestion in both lanes 0.10 +/- 371 Table1 Characteristic expansion joint displacements for the main span toward/away from the pylon. V is the coefficient of variation.

4.4 Sensitivity analyses Sensitivity analyses have been performed for the two dominating traffic cases: Freely flowing traffic in one direction and congestion in the other direction (3), and congestion in both directions (4). The selected parameters are:

• Mean weight of truck • Fraction of trucks • Mean duration of congestion • Intensities of vehicles

From these analyses all parameters is of high importance. However, it is not the same degree of importance in the two dominating traffic cases. One should keep this in mind before a parameter is excluded from further investigations. The sensitivity analysis is presented in Fig. 4 in terms of the reliability elasticity coefficient ep=(dβ/dp)⋅p/β for the four parameters.

Mean w eigth of truck

Fraction of trucks Congestion in both Mean duration of directions (4) congestion Freely flow ing traffic in one direction and congestion in Intensity of trucks the other direction (3)

Intensity of cars

-5 -4 -3 -2 -1 0

ep

Figure 4 Reliability elasticity coefficients. 5 Other loads Other loads include breaking forces, interim crash barriers and wind.

Breaking load is according to [8] a 500 kN load, which is half of the one used in the original design from 1965.

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Interim crash barriers, which are prescribed during various road works, have a weight of 9 kN/m. In the most unfavourable case the load acts on half of the main span or at the adjoining span one at a time.

Wind load is in the original design from 1965 estimated as quasi static with a horizontal pressure of 150 kg/m2 when the bridge is loaded and 250 kg/m2 when the bridge is unloaded. The original description is judged accurate enough due to the fact that wind load response only account for less than 10 % of the total displacement and the aerodynamic response of the structure is fairly well known from wind tunnel tests, see Ostenfeld et al. [9], [10] and wind measurements Jensen [11]. Otherwise a stochastic response analysis could be relevant after comparing the first significant eigenfrequency with the autospectrum of turbulence. 6 Calibration of companion load factors and partial safety factors The required lifetime of the new expansion joints is 50 years and capacity requirements are calculated according to a Serviceability Limit State (SLS) because displacements above the capacity is unacceptable for the normal use of the bridge, see NKB [12].

6.1 Level of safety Level of safety is based on recommendations in different references as shown below:

• In ENV 1991-1 Part 1 [14] the following recommendations for the SLS are: - Target reliability index β>1.5 (2.9) for a irreversible limit state and a reference period of 50 years (1 year)

• In ISO 2394 [14] the following is given for the SLS: - Target reliability index β>1.5 (2.9) for a irreversible limit state and a reference period of 50 years (1 year) - Target reliability index β>0 (2.2) for a reversible limit state and a reference period of 50 years (1 year)

• In NKB [12] the annual probability of failure for the SLS should be chosen in the following interval: - 0.01-0.1 (corresponding to a target reliability index β in the interval 1.3 - 2.3)

Based on the above recommendations a target reliability index equal to 1.75 is chosen. It should be noticed that an appropriate target reliability index could have been evaluated within the framework of decision theory.

6.2 Companion load factors In what follows a description of the calculation of companion load factors and the results are presented.

Extreme value of the combined load effect from temperature and traffic during a reference period T:

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N = max N(t) = max(N temperature (t) + N traffic (t)) t (8) []0; T []0; T

Where Nt is a stochastic variable. The distribution function is not readily obtained causing Turkstra's rule to be applied as a simple approximation.

The companion load factors are calculated according to Eurocode 1 [13] or ISO 2394 [14] i.e. a method based on Turkstra's rule and the design value format. In this project the Ferry Borges-Castanheta load-model is used for combination of loads.

It is assumed that traffic and temperature loads can be described by square-wave processes as show in Fig. 5, in which the load is different from zero in short time intervals during the reference period T.

Q Q 1,max Assumptions made when establishing 1 this model are described in e.g. ISO 2394 [14]. Time τ 1 The distribution function for individual T loads given that they are larger than 0 Q 2 Q 2,max are Fq1() for Qt1() and Fq2 () for Qt2 (). Time

τ2 The distribution function of the maximum load within the time period T [,0 T ] becomes: Figure 5 Square-wave processes Q1(t) and Q2(t).

()−=ri T ==ri  10pqi = FqQ ()() FqQ ()  ,i 12, (9) ii()−+ri >  ()10ppFqiii () q pi is the probability that Qi in a specific interval τi is greater than zero and ri=T/τi.

The companion load factors can be estimated for the two cases:

If the distribution functions Fq1()and Fq2 ()are known the method is described in e.g. ISO 2394 [14].

If the distribution functions Fq1() and Fq2 () are not known approximations can be obtained from known yearly extreme distributions. Assume that the distribution function F 1year, max (q) for the yearly maximum load Qt() is known. It is assumed that Qi i µ Fqi () can be approximated by a normal distribution with expected value i and σ standard deviation i . In most cases this is a reasonable approximation. If ri is the µ σ number of repetitions for one year, then i and i can be determined from:

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r   q − µ  i (1− p ) + p Φ j i  = p (10)  i i  σ  j   i 

if two different values of p j are used. Corresponding to a given value of p j the value q is determined from the known yearly distribution function F 1year, max (q) : j Qi

F 1year, max (q ) = p (11) Qi j j

The time period T , to be used when determining the load companion factors can either be T = 1 year or T = expected lifetime of the structure considered. If the parameters ( β , r , r and distribution functions FqT () and FqT ()) corresponding to 1 2 Q1 Q2 either T equal to 1 year or the expected lifetime of the structure, then almost the same load companion factors will be determined.

In the following the companion load factors are calculated considering:

Leading traffic load - companion temperature load Leading temperature load - companion traffic load

Corresponding to the previous used notation Qt1() is the temperature load with τ ν τ ν max,pos = 3 hours, max,pos = 1/year, max,neg = 4 hours, max,neg = 0.5/year and Qt2 () is the traffic load with τ = 1.8 hours, ν = 5.5/year. This results in the companion load factors given in Table 2.

Main span Ψ 1 Ψ 2 max pos. temp. 0,95 0,77 / 0,89 / 0,82 / 0,82 max neg. temp. 0,69 0,72 / 0,86 / 0,78 / 0,78

Table 2 Companion load factors for the main span. For breaking load and wind a companion load factor of 0.6 is used. This is the same value as used in the design of the Oresund Link [15]. The companion load factor for the weight of interim crash barriers is chosen as 1.0.

6.3 Partial safety factors The partial safety factors are estimated according to the Design Value Method, please refer to e.g. ISO 2394 [14].

Based on the above method the partial safety factors listed in Table 3 are estimated. It should be noticed that the model described in the previous section approximates the distribution of the annually maximum load.

γtraffic γtemperature Max. positive temperature 0.96 0.87 / 0.93 / 0.89 / 0.89 Max. negative temperature 0.86 Table 3 Partial safety factors for traffic and temperature load for the four traffic situations for the main span.

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The partial safety factors for "other loads" as described in chapter 6 are equal to 1.0. This is in correspondence with [8] for the SLS. 7 Comparison of existing and new capacity requirements In table 4 the original design basis from 1965 is compared with the new capacity requirements described in this paper. Only the maximum load cases are listed.

Original design from 1965 New capacity requirements Main span (towards/away from pylon) +/- 780 mm +/- 640 mm Max pos. temp. and leading traffic Adjoining span at pylon (towards/away from pylon) + 380 mm / - 280 mm + 299 mm / - 233 mm Max pos. temp. and leading traffic/temp. Adjoining span at adjoining tower (towards/away from pylon) +/- 280 mm + 185 mm / - 114 mm Max pos. temp. and leading traffic

Table 4 Original design requirements and new capacity requirements including rotation of bridge deck around a horizontal axis. It is seen that the displacement requirements for the expansion joints have been reduced considerably leading to minimised maintenance costs in the future without compromising the level of safety.

Furthermore, the stochastic traffic load model is based on actual parameters such as weight and fraction of heavy trucks, congestion frequency and intensity of vehicles so that an unambiguous evaluation of capacity can be performed if traffic situation should change in the future. 8 References [1] Code of Practice for Loads for the Design of Structures DS 410, Dansk Standard, 1998 [2] J.D. Sorensen, Max and min temperatures in Denmark, Aalborg University, 1996 [3] Eurocode 1: Basis of design and actions on structures - Part 2-5 Actions on structures - Thermal actions, 1 issue 1991 [4] H. O. Madsen and O. Ditlevsen, Stochastic Traffic Modelling for the Eastern Bridge, 1990 [5] O. Ditlevsen, Pulse Process of Congested Traffic load Effects, Storebaelt, Review Board No. 3, Task Group on Reliability Analysis of the Bridge Across the Eastern Channel, 1990 [6] A. Getachew and R. Karoumi, Generating Site-Specific Vehicle Data Using Monte-Carlo, IABSE International Conference Malta, 2001 [7] G. Mohr "Traffic on Motorways - A descriptive study of 3 motorways in Denmark", Storebælt, Review Bord no. 3 Task Group on Reliability Analysis of the Bridge Across the Eastern Channel, 1990 [8] Beregnings- og belastningsregler for vejbroer, Vejdirektoratet, 1984 (In Danish) [9] C. Ostenfeld, E. G. Frandsen and G. Haas, Aerodynamic Investigations for the Superstructure, Motorway Bridge across Littlebaelt, Publication X, 1970

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[10] C. Ostenfeld, E. G. Frandsen and G. Haas, Model Tests for the Superstructure of the Suspension Bridge, Motorway Bridge across Littlebaelt, Publication XI, 1970 [11] M. Jensen, Wind Measurements, Motorway Bridge across Littlebaelt, Publication XIII, 1970 [12] Recommendation for Loading- and Safety Regulations for Structural Design, NKB-rapport no. 36, 1978 [13] Eurocode 1: Basis of design and actions on structures - Part 1: Basis of design, 2. Issue, 1991 [14] ISO/DIS 2394 General Principles on reliability for structures, Revision of the first edition (ISO 2394:1986), 27.11.96 [15] The Oresund Link Contract No. 3 - Bridge Design Requirements - Volume 1 Design, 1995

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