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Diese Studie konzentriert sich auf die leistungsbasierte Optimierung passiver und aktiver Brandschutzsysteme für Betontunnelliner, um ihre Widerstandsfähigkeit gegen Fahrzeugbrände zu erhöhen. Die Methode zielt darauf ab, die Lebenszykluskosten zu minimieren und gleichzeitig die Schutzeffizienz zu maximieren, wobei Faktoren wie Tunnelgeometrie, Verkehrsaufkommen und Umleitungsstrecken berücksichtigt werden. Die Forschung setzt fortschrittliche Simulationstechniken ein, einschließlich der Finite-Elemente-Thermoanalyse und Monte-Carlo-Simulationen, um brandbedingte Schäden und Reparaturanforderungen zu bewerten. Zu den wichtigsten Ergebnissen gehört die Identifizierung optimaler Brandschutzlösungen, die Investitionen und wirtschaftliche Verluste ausgleichen. Sensibilitätsanalysen beleuchten die Auswirkungen von Tunneldimensionen, Verkehrslasten und Umleitungsstrecken auf diese Lösungen. Die Studie kommt zu dem Schluss, dass Tunnel mit kleineren Geometrien, höherem Verkehrsaufkommen, höherem Schwerlastanteil und längeren Umleitungsstrecken umfassendere Brandschutzmaßnahmen erfordern. Darüber hinaus unterstreicht die Forschung die Bedeutung einer sorgfältigen Auswahl von Zielvorgaben, um die Entscheidungsfindung einzuschränken, und die Ineffizienz der Längslüftung im Vergleich zu stationären Feuerlöschsystemen und Brandschutzbrettern bei der Verringerung thermischer Schäden.
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Abstract
Several technologies such as forced ventilation, active fire-fighting systems, and passive fire protection systems are used in current practice to mitigate fire-induced structural damage to concrete liners in roadway tunnels. This paper outlines a new decision-making approach to optimize these strategies by weighing their initial investment and life cycle costs against their relative impact for enhancing the tunnel’s resilience to fire (quantified in terms of post-fire downtime and cost as a function of the severity of fire-induced damage). First, a concrete liner damage assessment tool is developed to account for the presence or lack of active and passive fire protection, as well as realistic uncertainties in the thermal properties of the materials used. Second, both the cost of protection and the potential economic loss due to stochastic vehicle fire hazards are quantified for a single mitigation strategy or a combination of strategies. The economic losses include the direct repair cost and the functionality loss and detour distance due to the duration of tunnel closure, which is determined as a function of concrete liner damage severity and the corresponding repair procedures. Third, a genetic algorithm is used to perform a multi-objective optimization, resulting in a Pareto front as the reference for the decision-making process. The objectives and constraints of the algorithm can be readily modified based on practical engineering requirements. The proposed approach is then used to evaluate the sensitivity of fire protection selection to tunnel geometry, traffic volume and composition, and detour distance during closure.
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1 Introduction
Vehicle fire hazards within roadway tunnels can cause considerable structural damage to the liner, therefore necessitating lengthy closures for repair. Tunnels are often bottlenecks in roadway transportation networks, and the loss or reduction of their functionality is accompanied by significant economic losses that include the investment necessary for tunnel repair, the economic impact of traffic detours and associated increases in driving time, and associated fatigue or live load impact to alternate roadways (which may have been designed for a much lower equivalent single axle loads (ESAL)). For instance, the heavy goods vehicle (HGV) fire in the Channel Tunnel between the UK and France in 2008 caused a two-day closure and limited service during the subsequent four-month repair, resulting in an estimated revenue loss of £185 million ($260 million in 2008) [1]. The 2013 closure of the Königshainer Berge tunnel in Germany caused damage amounting to €6 million on alternative routes that were not designed for HGVs, in addition to tunnel repair cost of approximately €2.2 million ($2.7 million) [2].
Active fire protection systems for tunnels include (1) fire detection devices; (2) longitudinal ventilation systems to facilitate smoke control and removal of hot gases; and (3) fixed fire-fighting systems (FFFS) to reduce the fire’s thermal impact and provide fire suppression. Passive fire protection for the tunnel structure includes (1) a secondary layer of concrete or other cementitious material applied to the exposed interior surface of the tunnel; (2) protective shielding material (such as fire resistive boards) mounted to the tunnel walls and ceiling; and (3) the addition of polypropylene fibers to the concrete mix to enhance its resistance to fire-induced spalling [3]. Tunnels will often use a combination of these approaches to enhance life safety during a fire event and reduce fire-induced damage and downtime.
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Fire protection is designed as a function of tunnel geometry (e.g., cross-section shape and size, tunnel length, and the spacing of cross-passages for egress), the importance of the tunnel in the transportation network (e.g., if the tunnel closure largely increases the cost of traveling or creates increases in traffic), and the available funding. NFPA 502 [4] states that fire protection used in a tunnel should enhance life safety and ensure support firefighter accessibility, minimize economic impact, and minimize structural damage. However, there are currently no available methods for quantitative assessment of the cost–benefit for implementation of fire protection to reduce structural damage to the liner and increase post-fire resilience.
This study presents a streamlined methodology to optimize the tunnel fire protection systems based on dual objectives of minimizing the total protection cost and maximizing the protection efficiency. Per the flowchart in Fig. 1, the procedure includes three main steps: (1) concrete liner damage assessment, considering the presence of passive protection; (2) quantification of the cost associated with implementation of active and passive fire protection, as well as the corresponding economic loss in the service lifetime due to vehicle fire hazards; and (3) two-objective optimization of cost–benefit, powered by a genetic algorithm. The approach incorporates realistic uncertainties pertaining to the fire itself as well as the material properties of the concrete liner and the passive protection materials. Each of these three steps can be conducted separately, which allows for independent stochastic parameter generation via Monte Carlo Simulation. By obtaining more detailed data or specific information from a tunnel owner/operator (i.e., the tunnel traffic data, recommended repair procedure and typical repair time, tunnel detour distances for a particular roadway network, etc.), this methodology can be readily adjusted to provide more pertinent evaluations for a specific tunnel and provide quantitative decision-making metrics for fire protection design.
Fig. 1
Flowchart of tunnel fire protection optimization. Abbreviation: \({\alpha }_{LV}\) ventilation installation decision parameter; \({\alpha }_{FFFS}\) fixed firefighting system installation decision parameter; \({\alpha }_{PB}\) thickness of fire protection board; \({HRR}_{i}\) specific heat release rate for scenario i; CDSF confined discretized solid flame model; AADT Average annual daily traffic; MCS Monte Carlo simulation
Sections 2 through 4 of this paper provide a thorough description of each of the three steps in Fig. 1. The proposed methodology is then applied to a generic case study for demonstration in Sect. 5. Within that case study, the proposed approach is used to evaluate the sensitivity of optimal solutions for fire protection selection to variations in tunnel geometry, traffic volume and composition, and detour distance during closure.
2 Step 1: Damage Assessment for the Concrete Liner
During a tunnel fire event, the concrete liners will experience the following [5]: (1) permanent material strength reduction due to dehydration of the chemically bound water from the concrete, (2) explosive spalling due to rapid movement of moisture and elevated pore pressure in the heated face, and (3) and increased localized compression stress due to the restraint of thermal expansion. These effects can negatively impact the integrity of the structural cross-section and can expose the deeper concrete material and steel reinforcement to direct heat exposure, thus resulting in progressive damage to the section. Passive fire protection can be applied as a barrier to heat penetration to the interior face of the liner, thus mitigating its increase of temperature. The thickness of the protection product can be tailored to meet a performance objective based on the expected thermal demands in the tunnel.
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For this study, passive fire protection consists of fire protection boards composed primarily of calcium-silicate. These commercially available board systems (such as PROMATECT® -H, PROMATECT® -T, PROMATECT®-TFX, and AESTUVER®-T [6, 7]) are durable in the tunnel environment (e.g., fire protection boards are required to have a service life of at least 25 years [8]), have been tested to the latest industry fire standards [4, 9], and can be easily repaired or replaced if damaged (by impact or fire exposure). This study utilizes the publicly available data for these materials [10‐13] to demonstrate their mitigation performance under fire-induced thermal demands.
The performance of the concrete liner when subjected to thermal loading is simulated with an iterative finite element (FE) thermal analysis that accounts for the onset of fire-induced spalling during the fire event [14‐16]. The damage assessment tool for the concrete liner is developed by implementing the thermal FE model via Monte Carlo simulation (MCS) with latin hypercube sampling (LHS) [17], to consider a wide range of fire exposure intensities as well as realistic uncertainties associated with the thermal properties of both the concrete and protection materials. The input parameters with their corresponding stochastic distributions are summarized in Table 2—the development of these inputs for this study is presented in Sect. 2.2 to follow. To develop the tool to classify the thermal damage to the liner, a suite of 2000 thermal FE analyses are then performed with these stochastic inputs for the concrete liner when unprotected and when protected with calcium-silicate boards of 6-mm and 15-mm thicknesses.
2.1 Thermal Analysis Approach
As shown in Fig. 2, a one-dimensional FE strip model is developed in SAFIR 2019.a6 [18, 19] to represent a section cut of the concrete liner. The thickness of the concrete cover is defined at 50 mm (2 in.) [20], while the total thickness of the liner is taken as 300 mm (11.8 in.) [21]. The cover to the reinforcement was finely discretized into 20 fibers (each 2.5-mm thick) to capture the onset of a thermal gradient as well as spalling. The outermost face of the reinforcement is marked in Fig. 2; however, the steel reinforcement is not discretely included in the thermal model because it represents a very small portion of the liner’s thermal mass relative to the concrete. Instead, a 19-mm (0.75-in.) thick concrete fiber is placed behind the end of the cover layers to represent the zone in which the reinforcement would be located. The remaining concrete beyond that reinforcement zone is meshed at a slightly finer 12.7-mm (0.5-in.) thickness per fiber. When passive fire protection is included, those layers (marked as blue in Fig. 2) are attached to the outermost surface of the concrete cover and are evenly meshed into 10 fibers regardless of the total thickness of the protection. Preliminary mesh sensitivity analyses determined that these levels of discretization were sufficient for capturing a convergent thermal response under fire exposure [22]. The convective coefficients for the fire-exposed and unexposed surfaces were taken as 25 W/m2-K and 9 W/m2-K, respectively, per Eurocode 1, Part 1–2 [23], and the resultant emissivity of those surfaces was defined at 0.7 per Eurocode 2, Part 1–2 [24].
Fig. 2
Generic concrete tunnel liner geometry and fiber model
Fire-induced spalling of the concrete liner is simulated by iteratively analyzing the thermal FE model in SAFIR, using a procedure developed by Guo et al. in a previous study [22]. As shown in Fig. 3, the undamaged liner section is initially analyzed for the full duration of fire exposure to the inside (i.e. bottom) face. The results of that first analysis are then scanned from start-to-finish for a temperature increase threshold which would signify the first onset of spalling. Experimental tests by Carlton et al. [25] indicated that NWC panels with moisture content by mass greater than 2.4% would experience explosive spalling under intense one-sided heating (resembling a hydrocarbon fire exposure per ASTM E1529 [9]) when the concrete surface temperature reaches approximately 450 °C. The tests also showed that the spall event would result in a loss of concrete to a depth where the concrete temperature had reached 150 °C (i.e., at a location where the heated water vapor would be actively contributing to the absorption of thermal energy within the concrete media, causing a “moisture clog” in the concrete matrix and an associated spike in pore pressure [26‐28]).
Fig. 3
Flowchart of thermal analysis and damage classification for the concrete liner
Figure 3 shows that when the outermost concrete layer reaches 450 °C, the concrete cover to the 150 °C depth as well as any passive fire protection materials on the surface are considered to have spalled. The analysis results beyond that time for the initially undamaged section are discarded, and the analysis is restarted at the next time step with the spalled material removed and the fire exposure now applied to the fiber just beyond the spall depth. The restarted analysis is performed for the remaining duration of the fire and then scanned again for the onset of a subsequent spall event using the same temperature thresholds. This iterative process is performed using a custom shell algorithm in MATLAB [29] to iteratively run the SAFIR thermal FE model, scan the results, and restart the model with any spalled concrete layers removed. The potential for spalling ends either when the end of the fire exposure is reached or the reinforcement layer is exposed (which is consistent with design guidance in Eurocode 2, Part 1–2 [24]).
The fire exposure is expressed as a heat flux magnitude and exposure duration, and the input for each run of the thermal FE model is randomly selected (in whole number increments) between 50 kW/m2 (which is a lower bound for causing thermal damage to the protected concrete liner, as will be shown in the following sections), and an upper bound of 400 kW/m2 (which just exceeds the maximum heat flux measured in tunnel fire experiments [30]).
2.2 Thermal Material Properties
The thermal properties of the concrete liner are based on stochastic models proposed by Jovanović et al. [31], with mean and standard deviation for temperature-dependent thermal capacities and specific heat expressed in Eqs. (1) and (2), respectively. The concrete thermal conductivity and specific heat at a given temperature are assumed to follow a Gamma distribution.
The variation in specific heat with temperature also has an additional modification to account for the increase that occurs above 100°C due to the vaporization of free water from the concrete matrix. Per Eurocode 2, Part 1–2 [24], the specific heat departs from Eq. (2) at 100°C and increases instantaneously to a value that is a function of the concrete’s initial moisture content u. The curve then plateaus between 100°C and 115°C, followed by a linear decrease from 115°C to 200°C, at which point it rejoins the Eq. (2) curve. The moisture content is considered to follow Gaussian distribution according to the investigation of tunnel constructed in nineteenth century [22]. Details on this modification are described further in Zhu et. al. [22].
Calcium-silicate fire protection boards can demonstrate stable thermal properties throughout a wide range of temperatures. Table 1 lists the published thermal properties of PROMATECT®-T, PROMATECT®-H PROMATECT®-TFX and AESTUVER®-T (which were specially developed for tunnel fire protection) along with PROMATECT®-L and PROMATECT®-MST (which were developed for other high temperature applications such as industrial furnace protection) [10‐13]. These data, plotted as discrete points in Fig. 4, are used as the basis for temperature-dependent expressions when calculating thermal conductivity and specific heat for the calcium-silicate fire protection boards:
where \(\alpha \) is binarily chosen as either 0 or 1 to create upper- and lower-bound curves (plotted as gray lines in Fig. 4) that envelope the discrete reported data. The density of the protection board at ambient temperature, \({\rho }_{20}^{PB}\), is taken as 890 kg/m3, which represents the mean value of the data listed in Table 1. The variation of density with temperature is assumed to follow the same equation for normal-weight concrete (NWC) from Eurocode 2, Part 1–2 [24].
Table 1
Thermal material properties of the fire protection board
Brand
Material type
Thermal conductivity (W/m–K)
Moisture content
Thickness (mm)
Specific heat (kJ/kg-K)
Density (kg/m3)
Boards developed for tunnel applications
PROMATECT® -H (RWS/HCM)
Calcium-silicate based
0.175 at 20 °C
0.190 at 100 °C
0.210 at 200 °C
5–10%
6–25 mm
0.92 at 400 °C
870
PROMATECT® -T
Calcium-silicate based
0.212 at 20 °C
5%
15–40 mm
N/A
900
PROMATECT®-TFX
Matrix Engineered Mineral Board
0.200 at 20 °C
5%
20–40 mm
N/A
950
AESTUVER®-T
Calcium-silicate based
0.175 at 20 °C
N/A
N/A
N/A
690–980
Boards developed for other applications (e.g. industrial furnaces, dryers)
PROMATECT®-L
Calcium-silicate based
0.08 at 20 °C
0.10 at 200 °C
0.11 at 400 °C
N/A
20–50 mm
0.96 at 400 °C
N/A
PROMATECT®-MST
Calcium-silicate based
0.22 at 200 °C
0.20 at 400 °C
0.20 at 600 °C
0.22 at 800 °C
5%
12–60 mm
0.95 at 400 °C
N/A
Fig. 4
Thermal conductivity and specific heat of the passive fire protection materials
The residual state of the concrete liner that results from each thermal FE analysis is classified as one of three potential damage states: superficial, moderate, or heavy. The damage level is defined relative to the post-fire repair procedure that would be required for the level of residual damage to the concrete liner. The criteria for each damage level are indicated in Fig. 3 and described as follows:
Superficial The temperature of the concrete surface (behind the protection material) does not exceed 300°C. The concrete surface would therefore be expected not to be discolored or cracked after fire exposure. The residual strength is only minorly affected by the fire. No major structural or material repairs are required, and the tunnel can be reopened to its full functionality following the event following post-fire cleanup and cosmetic repair.
Moderate The concrete surface temperature exceeded 300°C, and the concrete cover layer suffers permanent strength loss and potential spalling. The temperature of the reinforcement layer does not exceed 600°C as the remaining concrete cover provides adequate thermal insulation during the fire through burnout. Repair consists of removal of the deteriorated concrete to a depth of 19 mm behind the reinforcement, which is presented in Fig. 2. Removed concrete is then replaced with new material to restore the original dimensions. This repair process requires closure of the tunnel to traffic.
Heavy The temperature of the concrete surface behind the protection exceeds 450°C, and the concrete cover has spalled to the depth of the reinforcement. Preliminary analyses of cases where the entire concrete cover spalled indicated that reinforcement temperature was very likely to exceed 600°C, which results in permanent yield strength reduction [32]. The compromised portions of reinforcement should therefore be removed and replaced. Concrete should again be removed to a depth of at least 19 mm behind the reinforcement. Unsound concrete will likely be present beyond this point as well, and further removal should be performed as shown in Fig. 3 pending additional post-fire inspection (to ensure that the global structural integrity is not compromised). This repair process again requires closure of the tunnel (Table 2).
Table 2
Stochastic input parameters for developing the concrete damage assessment tool
Figure 5a, c, e present the full set of thermal analysis results (with the damage level color coded) for 2000 randomized simulations for three cases: no protection, 6-mm protection board, and 15-mm protection board. As expected, when thicker fire protection is used, a higher heat flux magnitude and longer exposure time are needed to reach higher damage levels. The results show identifiable boundaries between damage levels, and these divisions can be mathematically defined to classify the damage thresholds without the need to repeat the FE thermal analyses. Due to the stochastic analysis approach, the damage conditions have some overlap at these boundaries. To more easily define a single division between the damage states, the results are replotted in ln-ln space in Fig. 5b, d, e. These results are then evaluated using a supervised machine learning approach called the support vector machine (SVM) classifier with linear kernel function [33]. This algorithm searches for an optimal hyperplane that separates the data into two classes. For the case of inseparable data, this algorithm imposes a penalty on the length of the margin for each observation located on the wrong side. To provide a conservative estimation of the damage state (i.e., rarely classify the heavy damage case as moderate damage), the misclassification cost matrix in this case amplifies the penalty of the margin-violation observations of the higher damage classes by three times. As shown in Fig. 5b, d, e, the SVM linear boundaries from SVM (plotted with black dashed lines) effectively delineate the concrete liner damage regions while allowing only a few observations to escape to the lower damage level. The linear binary classification is performed with the built-in MATLAB function and expressed with Eq. (5).
where \({w}_{1}\) and \({w}_{2}\) are weighting factors and \(b\) is the bias term. These linear boundaries in ln-ln space can then be expressed as nonlinear functions in linear space in Fig. 5a, c, e, the shape of which resemble Pressure–Impulse (P–I) curves that are commonly used to classify blast-induced damage to structural elements [34].
Fig. 5
Damage classification tool for concrete panels without fire protection boards in a linear and b ln-ln axes; with 6-mm protection boards in c linear and d ln-ln axes; and with 15-mm protection boards in e linear and f ln-ln axes
The stochastic analysis and damage boundary determination is repeated for five additional protection cases, resulting in six protection board thicknesses ranging from 6 to 20 mm as well as the unprotected case. Table 3 summarizes the associated parameters to per Eq. (5) for the resulting boundaries between damage levels for each case. Figure 7a presents the relative locations (i.e., blue and red lines represent the superficial-to-moderate and moderate-to-heavy damage thresholds, respectively) in the ln-ln space.
Table 3
Parameters for damage classification boundaries per Eq. (5)
Parameters
Passive fire protection plan
No protection
PB: 6 mm
PB: 9 mm
PB: 12 mm
PB: 15 mm
PB: 18 mm
PB: 20 mm
Damage classification
Superficial -moderate
w1
1.852
3.485
3.440
3.580
3.820
3.940
4.170
w2
3.953
5.323
4.450
4.730
4.830
5.020
5.320
b
− 18.39
− 34.61
− 31.74
− 35.51
− 38.56
− 41.46
− 44.87
Slope
− 0.469
− 0.655
− 0.773
− 0.757
− 0.791
− 0.785
− 0.784
Moderate- heavy
w1
3.755
4.167
3.830
3.640
3.640
4.150
4.120
w2
5.538
6.378
5.480
5.770
5.380
5.670
5.670
b
− 35.19
− 45.98
− 41.83
− 44.67
− 44.11
− 49.20
− 50.18
Slope
− 0.678
− 0.653
− 0.699
− 0.631
− 0.677
− 0.732
− 0.727
Depth of further removal
\({d}_{removal}\)
\((95\%)\)
p1
0.020
0.026
0.032
0.038
0.042
0.048
0.050
p2
0.020
0.027
0.034
0.038
0.046
0.055
0.052
p0
− 0.139
− 0.211
− 0.280
− 0.335
− 0.408
− 0.492
− 0.491
*Value is not provided since the heavy damage case is not observed in the simulated cases
For concrete liners with heavy damage, the concrete layers located 19 mm (0.75 in.) past the reinforcement layer may experience a temperature exceeding 300°C, which again indicates significant concrete material damage [32, 35, 36]. This portion of the concrete must also be removed and replaced during repair. The additional removal depth (e.g., the distance past the initial 19 mm removal beyond the reinforcement) for the heavy damage classification can be determined as a function of the heat flux and exposure time for each protection thickness. As illustrated in Fig. 6 for the case of the 6-mm protection board, the additional concrete removal is plotted as a third axis from the ln-ln plot of heat flux magnitude and exposure duration. A regression plane is introduced to determine the additional required removal depth in the ln-ln space as a function of heat flux magnitude and exposure duration. The heavy damage cases are clustered in five vertical planes which correspond to the fiber discretization. The blue surface represents the mean regression of the data set. The green surface is then plotted parallel to the blue but with 95% of the data points below. That 95% confidence prediction of additional concrete removal can be expressed with Eq. (6):
where the non-dimensional coefficients \({p}_{1}\), \({p}_{2}\) and the bias term \({p}_{0}\) varies with the selected passive fire protection thickness and are listed in Table 3. Figure 7b presents the surfaces for further removal depth when the concrete liner is protected with fire protection boards of varying thicknesses.
Fig. 6
Further concrete removal depth of heavy damaged class and surface regression for prediction (protection board of 6 mm thickness)
The damage classification maps and the equations to determine additional concrete removal for the heavy damage cases provides the damage assessment tools for the generic concrete liner with and without passive fire protection boards. The damage state and resulting amount of concrete removal as any location on the tunnel liner can be easily estimated for the heat flux magnitude and exposure duration imparted by an enclosed fire. This enables the estimation of repair time and the associated length of tunnel closure. This approach can be readily extended or modified to consider other passive protection schemes, concrete liner reinforcement strategies, material uncertainties, etc.
3 Step 2: Quantification of Cost and Losses
3.1 Cost of Fire Protection
The implementation of a tunnel fire protection plan is typically accompanied by a significant financial investment. This includes not only the initial construction cost but also the operation and maintenance costs over the tunnel service lifetime.
Table 4 lists the publicly available cost information on the use of protection boards, FFFS, and longitudinal ventilation fire protection plans. During the tunnel service lifetime, the fire protection systems may need to be replaced by the end of their specific use lifetime (i.e., assumed here to be 20 years for the ventilation system), which requires re-investment. By summing the total cost for each individual fire protection method, the total present investment for a specific protection plan can be calculated.
Table 4
Investment data for tunnel fire protection methods
*Interpolated from data for 6 mm, 9 mm, 12 mm, and 20 mm board
The available data for initial cost of FFFS and their annual maintenance costs varies greatly, as noted in Table 4. For illustration, this study utilizes data presented at the STUVA Conference 2011 in Berlin [37]. For the tunnel fire protection board, the insulationshop.co website [38] provides the price of the Promat® fire protection board with a panel size of 2440 mm × 1220 mm and thickness ranging from 6 to 20 mm. The tunnel ventilation design depends on the dimensions of the tunnel (i.e., tunnel height, tunnel cross-section area, and shape; tunnel length; the number of lanes and tunnel gradient, etc.), design fire hazard intensity and location, and the power of the selected jet fans. Mosen Ltd. [39] provides an spreadsheet for making preliminary tunnel ventilation calculations and estimating the number of jet fans that should be installed for specific tunnel conditions. The net cost can then be calculated also via the same spreadsheet considering the annual installed power cost, installation cost, procurement cost, and yearly maintenance cost (with the write-off period of 20 years) as summarized in Table 4.
3.2 Economic Loss Due to Fire Hazard
The economic loss due to tunnel fire hazards consists of the repair cost and any traffic-related losses, as shown previously in Fig. 1. Both of these losses correlate to the tunnel damage state, which could be quantified with the calculated thermal impact and damage assessment tool developed in Sect. 2. It is assumed that the tunnel is entirely closed for the repair procedure, which results in traffic detours and slowdowns during the repair period duration. Partial tunnel operation during repair could also be considered in future study but is neglected here for simplification. The tunnel traffic volume and composition are used for both the expected damage assessment and the traffic-related loss calculation.
3.2.1 Tunnel Fire Impact Calculation via the CDSF MODEl
The confined discretized solid flame (CDSF) model calculates the thermal impact from an enclosed fire on a tunnel liner and can consider both natural and longitudinally ventilated conditions [21, 42]. With the computational efficiency of CDSF, the fire hazard’s intensity and associated uncertainties (i.e., fire impact attenuation by FFFS) can be fully considered in the stochastic analyses. The CDSF is a semi-empirical model developed and calibrated based on both experimental data and numerical simulation. The CDSF model can accommodate tunnels of various shapes with rounded ceilings (i.e., horseshoe and circular), dimensions (i.e., 2-lane and 3-lane), and ventilation conditions (i.e., natural and longitudinal ventilation).
The CDSF model calculates the thermal impact on the tunnel liner from an enclosed fire as a combination of radiative and convective heat flux under either naturally or longitudinally ventilated conditions. The radiative portion is calculated by modeling of the confined flames as a solid 3D object that radiates emissive power to the surrounding environment. As illustrated in Fig. 8, the configuration of this radiative emitter is related to the fire’s peak heat release rate (\({\dot{Q}}_{max}\), in MW), the tunnel geometry, and the presence of a ventilation system operating at critical velocity. The surface of the solid flame model is discretized into i rectilinear elements, each of which emits radiation normal to its surface. The radiative heat flux on the tunnel liner (which is meshed into j panel elements) is calculated as a summation of the contribution from all the discretized flame surfaces that can ‘see’ the target liner element:
where \({E}_{i}\) (kW/m2) and \({A}_{i}\) (m2) are the emissive power and the area of the ith element on the fire surface, respectively; \({r}_{i\to j}\) is the distance or “radius” from the center of fire surface element i to target j; \({\theta }_{i}\) represents the absolute angle between the radius vector and the fire element’s normal vector; and \({\theta }_{j}\) is the absolute angle between the radius vector and the target surface normal vector. The geometric parameters are illustrated in Fig. 8a.
Fig. 8
CDSF model configurations for evaluating the thermal impact of enclosed fire on a circular tunnel liner
The radiative emissive power, considered to be a fraction of the fire’s HRR, is assigned to the surfaces of the 3D solid element flame model. If unconfined, the entire height of the flame object would have the same emissive power on all surfaces. Since the tunnel will confine the flame height for large fires, the emissive power at the top of the flame model (colored in dark orange in Fig. 8 for both natural and longitudinal ventilated cases) is therefore amplified proportional to the loss in surface area for the flame object with reduced (i.e. confined) flame height. A full description of the confined flame object is provided in previous work by Guo et al. and Zhu et al. [21, 42].
The convective heat flux is associated with gas temperature and smoke distribution. For the naturally ventilated case, the convective region is considered to be located at the upper portion of the tunnel as shown in Fig. 8a, which matches observations of smoke layering during fire tests [21]. The convective heat flux decays longitudinally from the fire location in both directions. Also, the convective heat flux decays vertically at each liner lateral cross-section to represent smoke dissipation. For the longitudinal ventilation case, the convective region is considered downstream of the fire footprint and lengthens to the entire tunnel height, as presented in Fig. 8b. A full description of the convection field calculations in the CDSF model is also provided in the aforementioned previous work [21, 42].
The total peak heat flux \({\dot{q}}_{total,j}^{\prime\prime}\) to each tunnel liner element is calculated as the summation of the radiative heat flux and convective heat flux. For the naturally ventilated case, both the radiative and convective thermal effects are calculated individually for each liner element along the tunnel length. For the longitudinally ventilated case, the radiative heat flux is first calculated with the 3D flame model and used as the base to apply the convective heat flux in the downstream region, as shown in Fig. 8. Figure 9a illustrates the total heat flux distribution for the 3-lane tunnel with natural ventilation and a vehicle fire hazard with \({\dot{Q}}_{max}\) = 100 MW.
Fig. 9
Contours of a heat flux, b damage state, c repair time, and d repair cost for the naturally ventilated 2-lane circular generic tunnel with 6-mm fire protection board subjected to 100 MW fire hazard
where Etot (kJ) is the total calorific energy;\(\chi \) is the combustion coefficient conservatively taken as 1.0; \({\alpha }_{g,q}\) (kW/s2), and \({\alpha }_{D,q}\)(s−1) are the quadratic growth and exponential decay rates, respectively, which are related to the vehicle type. Per Ingason [43], the value of \({\alpha }_{g,q}\) increases as the peak HRR arises while \({\alpha }_{D,q}\) is relatively constant. The HRR time history curve is then normalized by \({\dot{Q}}_{max}\) and multiplied by \({\dot{q}}_{total,j}^{\prime\prime}\) to obtain a similarly shaped heat flux time history for each liner element.
The level of thermal damage to each tunnel liner element can be classified using the classification maps from Sect. 2.3, which requires two inputs: the peak heat flux magnitude \({\dot{q}}_{total,j}^{\prime\prime}\), and exposure duration \({t}_{d,j}\). The value for \({t}_{d,j}\) is calculated by taking the area integration under the heat flux time history curve for a given liner element and then dividing by \({\dot{q}}_{total,j}^{\prime\prime}\). This approximation of total fire exposure duration is conservative because it converts a quadratic time history to a constant duration of exposure at peak heat flux. Figure 9b shows the distribution of the damage to the concrete liner of a generic 2-lane circular tunnel when protected with the 6-mm protection boards, resulting from the heat flux from the 100 MW vehicle fire hazard in Fig. 9a.
3.2.2 Tunnel Closure Time for Repair
The post-fire tunnel repair requirements are based on the damage assessment contours such as that shown in Fig. 9b. The overall tunnel closure time to repair all damaged panels can be determined by applying estimated rehabilitation rates (expressed in units of m2/h) that correspond to each damage level, as noted in Table 5. These values are illustrative for this study, since no detailed specifications for tunnel fire repair is currently available according to a MassDOT (2021) investigation [44]. This approach could be readily updated if more exact or pertinent information were available.
No repair action is associated with superficial damage for the illustration in this study. For concrete panels that experience moderate or heavy damage, the area of protection boards that must be removed and reinstalled is assumed to be 1.5 times that of the panel, as noted in the comments in Table 5. This recognizes that fire damage to the protection boards will extend beyond the edges of a panel that experiences a substantive level of concrete damage. The repair procedure for a protected concrete liner consists of protection board removal as well as removal and replacement of any damaged concrete and reinforcement. The installation rate for the protection boards is based on a FireMaster ® installation process for 24 m2 boards on a curved tunnel wall [45]. In the absence of additional information, the removal rate is assumed to be equal to the installation rate for simplification.
Concrete repair is required for moderate damage, while both concrete and rebar repair are needed for heavy damage. The rate of these repair steps in Table 5 is generally based on data obtained by the project team from an authorized contractor for PennDOT. The shotcrete application rate is taken from the standard practice for shotcrete by the US Army Corps of Engineers [46], and the hydro-demolition concrete removal is based on the Channel tunnel repair after its 2008 fire event [47].
Figure 9c illustrates the total repair time for each discretized tunnel liner element based on the damage level contour in Fig. 9b. As would be expected, the total repair time for panels with heavy damage is much longer, due to the long repair duration associated with rebar replacement. Also, for this damage level, the higher thermal impact magnitude increases the volume of damaged concrete behind the reinforcement thus adding to the repair time (see the magnified area in Fig. 9c).
For each specific tunnel fire hazard, the entire tunnel closure time \({T}_{hr}\) (h) for repair is calculated by summing the repair time for each discretized tunnel concrete liner element via Eq. (11):
where n1, n2, and n3 are the number of panels of (1) superficial, (2) moderate, and (3) heavy damage, respectively; \(t({dm}_{j})\) with j = 1, 2, 3 represents the repair time associated with each damage state. Recall that superficial damage is assumed to require no post-fire rehabilitation and is therefore omitted from consideration in Eq. (11), though this could be readily changed in future applications of this approach.
3.2.3 Repair Cost
The tunnel repair cost following a fire event directly relates to the damage state of the tunnel liner which is estimated via Sect. 2.3. Similar to the tunnel closure time, the repair cost is calculated as the summation of the cost for each discretized tunnel liner element (\(rc({dm}_{j})\), with j = 1,2,3) and the fixed cost \({rc}_{fix}\) associated with construction site equipment as expressed in Eq. (12):
where n1, n2, and n3 are the number of panels with superficial, moderate, and heavy damage. The value of \(rc({dm}_{j})\) varies with respect to the protection material while \({rc}_{fix}\) is related to the highest level of concrete liner damage across all impacted panels.
The assumption is made that the fire protection board in the region of the fire event is removed for concrete liner inspection and is then replaced. Based on an economic feasibility study for fire protection board in tunnels [8], the cost for concrete repair for moderate damage in this study includes cleaning, high-pressure blasting, chiseling, and shotcrete repair. Heavy damage includes an additional cost of reinforcement replacement and deeper concrete remediation. Table 6 summarizes the detailed area-dependent costs for each damage level which enables flexible calculation of the repair cost for different fire exposure scenarios. Figure 9d plots the repair cost associated with each discretized liner element according to the damage state contour in Fig. 9b for the generic tunnel.
Table 6
Representative repair costs for the tunnel concrete liner
Heavy damage: Cleaning, high-pressure blasting, chiseling, reinforcement replacement, shotcrete
$267/m2 + $81,200 fixed cost
*Same as the cost for protection board installation in Table 4
3.2.4 Traffic-Related Loss
Closing the tunnel for repair after a fire hazard will necessitate traffic detours. The associated economic loss includes the additional expenditure on time and fuel for motorists. Additionally, the unexpected traffic diversion may produce congestion elsewhere, which further increases the time loss. The total tunnel traffic-related economic loss can be representatively estimated via Eq. (13):
where \({TC}_{total}\), \({TC}_{DT}\), \({TC}_{DF}\), and \({TC}_{jam}\) (USD) are the total traffic-related cost, detour time loss cost, detour fuel loss cost, and time loss due to traffic jams. \(DL\) (km) represents the detour distance; \({V}_{i}\) (i = 1,2,…) is the traveling speed (km/h) for different types of vehicles; and \(T{L}_{i}\) (USD/h) is the hourly rate for each type of vehicle. It is conservatively assumed that each car is occupied by one working adult, and the statutory minimum wage is considered as the hourly rate for this person. \(F{L}_{i}\) (USD/km) accounts for the fuel combustion differential for each vehicle type; and \({AADT}_{i}\) represents the corresponding annual average daily traffic.\({T}_{jam}\) (h) is the time of traffic jam due to the unexpected traffic division; and \({T}_{day}\) (days) is the period of tunnel repair, which depends on the tunnel closure time \({T}_{hr}\) calculated in Sect. 3.2.2 according to the tunnel fire impact and selected protection method. The detailed parameters for traffic-related economic loss are listed in Table 7. To note, the tunnel traffic (including the traffic volume in AADT) and the detour distance will depend on the specific tunnel location and usage. For tunnels with bidirectional traffic, the detour length and tunnel traffic should be evaluated separately for each direction.
Table 7
Parameters for traffic-related economic loss assessment
Small vehicles (≤ 3.5 tons)
Large vehicles (> 3.5 tons)
Hourly rate (USD/h):\(T{L}_{i}\)
10.25
23.2
Detour fuel cost (USD /km):\(F{L}_{i}\)
0.35
0.75
Travel speed (km/h):\({V}_{i}\)
80
50
Traffic jam time (hr/day):\({T}_{jam}\)
0.5
Tunnel closure days:\({T}_{day}\)
Determined by tunnel damage and associated tunnel closure time \({T}_{hr}\)
3.2.5 Associated Uncertainties
The tunnel damage and corresponding tunnel closure time for a given fire intensity (expressed as peak HRR) are greatly influenced by the following fire-related parameters: total combustion energy and active fire protection systems such longitudinal ventilation and FFFS, if present. Since the longitudinal ventilation system is typically designed to operate at critical velocity during a fire event for smoke removal, the effect of the ventilation system on the thermal impact is considered to be deterministic with that condition. The activation delay time of the longitudinal ventilation can be assumed to be negligible per the report following the Burnley tunnel fire event of March 2007 [5].
The combustion energy, which determines the thermal impact duration, varies according to vehicle type (e.g., car, bus, HGV, and tanker truck) and the associated combustible materials. For a specific fire hazard in this study, the combustion energy is randomly selected between a “lower bound” (per Guo et al. [16]) and “upper bound” (per Ingason [43]), following a normal distribution between the two as shown in Fig. 10a (with standard deviation calculated as one-third of the difference between the two curves at any point along the x-axis). The expressions for “upper bound” \({E}_{tot}^{up}\) (GJ) and “lower bound” \({E}_{tot}^{low}\)(GJ) are considered as a function of the peak heat release rate \({\dot{Q}}_{max}\) (MW) respectively in Eq. (14):
Fig. 10
Uncertainties associated with tunnel fire intensity
An FFFS is assumed to mitigate the fire intensity while attenuating the radiative and convective heat transfer to the concrete liner. NFPA 502 [4] recommends an FFFS-induced reduction for peak HRR of different types of vehicle fire (i.e., car, bus, HGV, and tanker truck) based on experimental data. Figure 10b shows the lower and upper bound of the reduction factor as a function of the fire HRR along with the mean value. A normal distribution was assumed between the upper and lower bound reduction factor curves, with standard deviation again calculated as one-third of the difference between the two.
The transmission of thermal radiation from the fire to the tunnel surface is reduced by the droplets in the water spray. Experiments have shown that radiative attenuation ranges from 50 to 83% depending on the type of water spray system, water pressure, and droplet size [37, 40, 48]. To avoid the complexity of accounting for these factors, a conservative value of 40% is applied as the attenuation factor for both radiative and convective thermal impact if the FFFS is installed.
To summarize, the uncertainties in quantifying the tunnel structural damage, closure time, and the corresponding economic loss when subjected to a fire hazard of specific intensity (expressed with HRR) are considered via Monte Carlo Simulation with Latin hypercube sampling, which is highlighted in the green region of the flowchart in Fig. 1. A confidence interval of 95% is selected to provide a reasonably conservative estimation, consistent with a previous bridge fire fragility study by Zhu et al. [49].
3.2.6 Lifecycle Cost Assessment
The expected economic cost due to vehicle fire hazards in tunnels can be correlated to the fire intensity, which is calculated following the steps introduced in previous sections, with the probability of fire occurrence expressed as follows:
where \({HRR}_{j}\) is the heat release rate defining the fire intensity j; \({TC}_{total}({HRR}_{j})\) and \(RC{(HRR}_{j})\) are the traffic-related cost and repair cost for a fire hazard of \({HRR}_{j}\), respectively. During the service lifetime of the tunnel, multiple fire hazards may occur—a present value calculation is therefore needed to assess the effectiveness of fire protection methods against their lifecycle investment. The tunnel fire frequency within a given time frame, Ff, can be calculated via Eq. (16) by considering the fire rate Rf (per million vehicle-km), tunnel traffic volume AADT of all types of vehicles within a given time frame (in millions of vehicles), and tunnel length L (in km).
$${F}_{f}={R}_{f}\times AADT\times L\times 365$$
(16)
The return period is then taken as 1/\({F}_{f}\). It is assumed that the fire hazard occurs in the first year and then iteratively reoccurs with the interval of the return period during the tunnel lifetime. An interest rate of 1.75% is applied to the value of present investment to calculate the total economic cost in the tunnel lifetime.
4 Step 3: Multi-Objective Optimization
According to NFPA 502, tunnel fire protection systems should (1) support firefighter accessibility, (2) minimize economic impact, and (3) mitigate structural damage [4]. In practice, a cost–benefit analysis of various fire protection systems can help engineers develop an identify optimal (or at least preferable) solutions given the vulnerability of the particular tunnel to fire-induced damage.
4.1 Selected Objectives and Constraints
The objectives of the tunnel fire protection plan may depend on the objectives for a given project. In this study, two sets of objectives are applied for comparison. In the first set, minimizing the lifecycle investment calculated according to Sect. 3.1 and minimizing the lifecycle cost presented in Sect. 3.2 are used as objectives. In the second set, the first objective is the overall financial flow, including both investment and loss (which is also used in the economic feasibility study by Roland et al. [8]), and the second objective is the efficiency of the protection in reducing the economic loss compared to the unprotected case. The objective of mitigating the structural damage per NFPA 502 is omitted because the structural damage is largely represented by economic loss according to the procedure introduced in Sect. 3.2. The most straightforward constraint for the optimization is setting the budget for the fire protection plan, which is often a primary constraint of most engineering projects. Moreover, to realize the firefighter accessibility goal, the installation of tunnel longitudinal ventilation could be used as a constraint.
4.2 Pareto Front Identification via Genetic Algorithm
Genetic algorithms (GAs) have become a popular choice for optimization problems with conflicting objectives (in this study, for example, minimize the investment and minimize the economic loss). Specifically, GAs have been used where optimal decisions need to be made in the presence of trade-offs between objectives [50]. This algorithm repeatedly evaluates the objectives at several trial points of the admission domain with several iterations (called “generations”) and provides a Pareto front that contains the optimal solutions. These solutions can be used as a reference for the final decision-making process. The multi-objective genetic algorithm optimization is performed via the embedded function in Matlab. The convergence criteria primarily depend on the spread of the Pareto Front. It converges when the average change in the spread of the Pareto Front over a specified number of generations (i.e., default number of 100) is less than a tolerance (i.e., default value of 1e−4).
5 Example Application
5.1 Generic Tunnel Description
A previous survey of the shape and dimension of tunnel inventory in the United States [21] showed that a typical roadway tunnel has a road width ranging from 8 to 13 m (26.2–42.7 ft), accommodates 2 to 3 travel lanes, and has a curved ceiling. The recent development and deployment of tunnel boring machines have made circular tunnel cross-section more common and likely will become the dominant tunnel cross-section in the future. For illustrative purposes, this section focuses on the southbound bore of the Lehigh Tunnel in Pennsylvania, which is a typical 2-lane circular tunnel with a length of 1335.6 m (4382 ft). Based on the generic template in Fig. 11, the height of the Lehigh Tunnel is \({H}_{T}\) = 7.78 m while the corresponding tunnel radius is \({R}_{T}\) = 5.28 m. According to the National Tunnel Inventory database for 2022 [51], the AADT of the southbound Lehigh Tunnel is 13,099 vehicles/day, while the detour distance is merely 1 mile. The composition of the tunnel traffic is based on the information provided by PennDOT in 2019 for a 2-lane urban tunnel in Pennsylvania [16] and labeled as Traffic 2 in Table 8. The fire rate (\({R}_{f}\)) in Table 8 is taken according to data from a study published in 2016 by PIARC [52]. The resulting tunnel fire HRR probability density distribution and cumulative density distribution are derived via Monte Carlo Simulation with the equation correlating the combustion weight with HRR value [53] and presented with solid lines (black and gray, respectively) in Fig. 12. Four traffic cases are examined, with Traffic 2 representing the baseline case provided by PennDOT for a 2-lane tunnel. Traffic 1, Traffic 3, and Traffic 4 listed in Table 8 and presented in Fig. 12 with dashed lines represent variations on the baseline case, in which the HGV percentage is changed to 0%, 20%, or 50% of the total traffic volume. The remaining vehicle proportions for those 3 cases use the same distribution as Traffic 2.
The tunnel fire protection can consist of one or more active and passive systems. For active protection, design variables \({\alpha }_{LV}\) and \({\alpha }_{FFFS}\) indicate the presence of longitudinal ventilation and FFFS, respectively. These two parameters are binary with a value of either 1 or 0, representing installed and active or not present, respectively. For passive fire protection, the variable \({\alpha }_{PB}\) is an integer selected from 0, 1, 2, 3, 4, 5, and 6 to represent the following protection board thicknesses: no protection, 6 mm, 9 mm, 12 mm, 15 mm, 18 mm, and 20 mm. The associated damage assessment map for each thickness is developed per Sect. 2. The lifecycle cost and loss due to tunnel fire hazards for a specific protection plan are represented as variables C and L, while the L0 is the loss for a baseline case with no fire protection. Hence the protection efficiency can be expressed as \(\frac{{L}_{0}-L}{C}\) (lifecycle loss saved via the application of fire protection, divided by the lifecycle cost).
The formulation of the optimization problem is to find \({\alpha }_{LV}\), \({\alpha }_{FFFS}\), and \({\alpha }_{PB}\) such that either of the two objective sets are met: (1) minimize both C and L; or (2) minimize C + L and maximize \(\frac{{L}_{0}-L}{C}\). The optimization is subjected to the following constraints: \({\alpha }_{LV}=0\text{ or }1\); \({\alpha }_{FFFS}=0 or 1\); \({\alpha }_{PB}=\text{0,1},2,\dots ,5\); and \(\frac{{L}_{0}-L}{C}\ge 0\) (for objective set 2 only). It should be noted that these constraints can be modified according to the tunnel conditions, available resources, and performance targets. For example, \({\alpha }_{LV}\) would remain set to a value of 1 if the longitudinal ventilation system is required to control not only smoke and ensure life safety during a fire but for providing air flow during normal operation of the tunnel.
5.3 Results: Baseline Tunnel Dimensions and Traffic Load
Performing one optimization analysis takes 5 to 10 min computational time with a typical desktop computer. As presented in Fig. 13, the Pareto fronts containing the optimal solutions are obtained via a GA for objective set 1. The gray circles represent every individual protection solution, with red or blue colored markers indicating optimal solutions. For example, the red triangles in Fig. 13a represent optimal cases for a fully unconstrained optimization for \({\alpha }_{LV}\), \({\alpha }_{FFFS}\), and \({\alpha }_{PB}\). A total of 14 optimal solutions are identified, with the leftmost solution (marked as \({S}_{IL}^{1}\)) representing the case of no passive protection. As expected, the economic loss significantly decreases as the amount of fire protection increases, as illustrated in the solution marked as \({S}_{IL}^{2}\) (12-mm thick protection board) and \({S}_{IL}^{3}\) (FFFS installed + 20-mm thick protection board). Solutions with more installed fire protection (active and passive) beyond \({S}_{IL}^{3}\) provide little additional reduction in potential losses for the additional investment.
Fig. 13
Optimal solutions for objective set 1 for the 2-lane southbound Lehigh Tunnel
The blue squares in Fig. 13b represent optimal cases where longitudinal ventilation must be installed for smoke control (i.e. the optimization is constrained to have \({\alpha }_{LV}\) = 1). The optimal solutions are reduced to 10 cases, and these solutions are generally not as optimal versus those with no longitudinal ventilation to the significant investment needed to install the ventilation system. For the solution marked as \({S}_{IL}^{4}\), no FFFS or fire protection board is implemented; and for \({S}_{IL}^{5}\), a 6-mm thick fire protection board is installed while the FFFS is not. For the solution of \({S}_{IL}^{6}\), which is on the Pareto front of both cases, both the longitudinal ventilation and FFFS are installed, while the protection board is as thick as 20 mm. All the optimal solutions are listed in Table 9.
Table 9
Summary of optimal fire protection solutions for the 2-lane southbound Lehigh Tunnel
Objective set 1
Protection options
Unconstrained
Constrained
1
(\({S}_{IL}^{1}\))
2
(\({S}_{IL}^{2}\))
3
4
5
6
7
8
9
10
(\({S}_{IL}^{3}\))
11
12
13
14
1(\({S}_{IL}^{4}\))
2(\({S}_{IL}^{5}\))
3
4
5
6
7
8
9
10(\({S}_{IL}^{6})\)
Longitudinal ventilation
✓
✓
✓
✓
✓
FFFS installation
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
PB thickness (mm)
6
✓
✓
9
✓
✓
12
✓
✓
✓
15
✓
✓
✓
18
✓
✓
✓
✓
✓
20
✓
✓
✓
✓
✓
Objective set 2
Protection options
Unconstrained
Constrained
1 (\({S}_{FE}^{2}\))
2 (\({S}_{FE}^{1}\))
1
2
3 (\({S}_{FE}^{4}\))
4 (\({S}_{FE}^{3}\))
Longitudinal ventilation
✓
FFFS installation
✓
✓
PB thickness (mm)
6
✓
9
✓
12
✓
15
18
20
It should be noted that the four constrained optimal points with the highest lifecycle investment in Fig. 13b are also on the Pareto front for the plot of unconstrained optimal solutions in Fig. 13a. These results indicate that the longitudinal ventilation, which is required for the sake of life safety and smoke management, is not as efficient as the FFFS and protection board for mitigating the tunnel fire damage and associated economic loss. Specifically, the previous study by Zhu et al. [42] has shown that longitudinal ventilation does not mitigate the maximum heat flux intensity or the affected area of the tunnel liner; rather, the ventilation at critical velocity merely shifts the location of the heat flux contour to be downwind of the fire location. The amount of thermal damage inflicted to the tunnel liner by an enclosed severe vehicle fire can even be slightly worse with the presence of longitudinal ventilation, since the most severe areas of heat flux exposure will be more widespread over the tunnel due to the airflow. The increased airflow can also amplify the combustion reaction of the fire itself and increase its HRR intensity [54‐56]; however, that effect is neglected in the CDSF models used in this study for simplification, so that the HRR is the same for cases with and without longitudinal ventilation.
Figure 14 presents the Pareto fronts using objective set 2, with the optimal solutions listed in Table 9. The solution marked as \({S}_{FE}^{1}\) represents the FFFS installations, while \({S}_{FE}^{2}\) has no protection. Some solutions which involve multiple modes of fire protection (i.e., FFFS + 9-mm thick protection board) are circled with a blue dash oval and are not optimal due to their high financial flow or low protection efficiency for the Lehigh Tunnel case. Note that the constrained cases with longitudinal ventilation installed leads to negative protection efficiency; again, this is caused by the ventilation pushing hot gases downstream and creating a slightly larger area of thermal damage. For the constrained cases in Fig. 14b, \({S}_{FE}^{3}\) represents the FFFS installation and \({S}_{FE}^{4}\) includes the 12-mm thick protection board. Overall, objective set 2 is more efficient in narrowing down the number of choices for decision-making.
Fig. 14
Optimal solutions for objective set 2 for the 2-lane southbound Lehigh Tunnel
5.4 Sensitivity to Changes in Tunnel Cross-Section
The size of the tunnel generally influences the fire impact magnitude and the investment in fire protection. According to the NTI database [51], most US circular tunnels are 2-lane tunnels, whose dimensions vary slightly based on the shoulder and width of the lanes. However, the 3-lane tunnel driven by the tunnel boring machine is becoming more common. The Lehigh Tunnel is therefore reanalyzed with its cross-section enlarged to resemble a typical three-lane circular cross-section (\({H}_{T}\)= 9.93 m \({R}_{T}\)= 6.98 m per Fig. 11). Objective set 2 is applied for the optimizations performed in this sensitivity analysis.
The unconstrained optimal solutions for the 3-lane circular tunnel are consistent with that for the 2-lane tunnel as presented in Fig. 15a. For the constrained cases with longitudinal ventilation installed in Fig. 15b, the solution of only longitudinal ventilation installation is optimal. This solution is not optimal for the smaller cross-section of the 2-lane tunnel (and actually results in negative protection efficiency) because the fire hazard is more confined [42], and the longitudinal airflow spreads the intensity of that confined fire over a wider area downwind. For the larger 3-lane tunnel, the fire is less confined and imparts a slightly lower heat flux intensity to the tunnel liner. Also, the protection boards are slightly less efficient for the larger 3-lane tunnel cross-section because of the larger installation area (and thus larger cost).
Fig. 15
Optimal solutions for objective set 2 for the 3-lane southbound Lehigh Tunnel with larger cross-section
The length of the tunnel can vary widely depending on the location. The 2-lane Lehigh Tunnel (with an actual length of 1335 m) is therefore reanalyzed with tunnel lengths of 100 m, 200 m, 800 m, 1600 m, and 3200 m for sensitivity analysis. As presented in Fig. 16, the length of the tunnel does not strongly influence the optimal solutions mainly because the investment in fire protection is almost proportional to the tunnel length (i.e., the number of jet fans, the area of fire protection board, and the installation of FFFS). Meanwhile, the protection efficiency decreases since a longer tunnel (with more driving distance) will have a higher probability of having a fire hazard via Eq. (16) since the fire rate is a function of miles driven.
Fig. 16
Optimal solutions for objective set 2 for the 2-lane southbound Lehigh Tunnel with varying length
As explained in Sect. 3.2, the tunnel’s traffic profile will influence the fire frequency, the overall expected tunnel fire impact intensity, and the economic loss due to the tunnel closure. According to the NTI database for 2022 [51], the traffic volume in US tunnels varies widely from below 100 vehicles/per day (i.e., McCoy’s Ferry tunnel) to over 250,000 vehicles/per day (i.e., Yerba Buena Crossing Tunnel). The number of HGVs as a percentage of the total tunnel traffic fluctuates from near 0% for tertiary roadways (NJ 29 tunnel), to 50% (i.e., Green River Tunnel) for tunnels with mixed traffic types, to over 90% for tunnels dedicated almost entirely to bus and truck transportation (i.e., College Hill Bus Tunnel). The sensitivity of the optimal fire protection solutions for the 2-lane circular Lehigh Tunnel examined by imposing two variations in the tunnel’s traffic load: (1) the total AADT is both multiplied and divided by factors 5, 10, and 20; and (2) the percentage of HGVs in the traffic composition is reduced to 0% (Traffic 1 in Table 8) and increased to 20% and 50% (Traffic 3 and 4 in Table 8). Recall that the HRR probability density curves associated with the change in HGV percentage were plotted previously in Fig. 12.
Figure 17a and Table 10 show a clear increase in protection efficiency for the application of fire protection boards when the overall AADT increases. In Fig. 17b, the percentage of HGVs generally influences the expected consequences (i.e., damage and corresponding repair cost and time). The optimal solutions remain unchanged for the HGV percentage from 0 to 20%, while the protection efficiency increases.
Fig. 17
Optimal solutions for objective set 2 for the 2-lane southbound Lehigh Tunnel with varying traffic volume and composition
Summary of optimal solutions for objective set 2 for the 2-lane Lehigh Tunnel with varying traffic volume
AADT
Optimal solution number
Protection plans
Lifecycle flow (million USD)
Protection efficiency (%)
Ventilation
FFFS
Protection board thickness (mm)
655
1
Natural
No
None
0.15
0.00
2
Natural
Yes
None
5.05
2.24
2620
1
Natural
No
None
0.39
0.00
2
Natural
Yes
None
5.14
6.56
13,099
1
Natural
No
None
3.61
0.00
2
Natural
Yes
None
6.17
65.1
65,495
1
Natural
Yes
None
25.5
1225
2
Natural
Yes
6
20.4
568
3
Natural
Yes
12
19.7
481
4
Natural
Yes
15
19.7
448
130,990
1
Natural
No
12
71.5
2279
2
Natural
Yes
None
84.0
4762
3
Natural
Yes
6
41.4
2206
4
Natural
Yes
9
38.0
2002
5
Natural
Yes
12
29.6
1867
6
Natural
Yes
15
25.0
1738
7
Natural
Yes
18
22.7
1615
261,980
1
Natural
No
12
246
8271
2
Natural
Yes
None
313
17,179
3
Natural
Yes
6
123
8071
4
Natural
Yes
9
106
7323
5
Natural
Yes
12
68.3
6856
6
Natural
Yes
15
46.0
6380
7
Natural
Yes
18
32.0
5937
8
Natural
Yes
20
27.4
5667
5.7 Sensitivity to Changes in Detour Distance
The detour distance directly affects the traffic-related loss calculated via Eq. (15). For the Lehigh Tunnel, the 1.6 km (1 mile) detour distance is relatively modest, but other tunnels have much more significant detour in the event of their closure, such as 30 km (20 miles) for the Big Witch Tunnel, 320 km (199 miles) for Zion-Mount Carmel Tunnel, and 684 km (425 miles) for Thimble Shoal Tunnel [51]. For some tunnels (i.e., Portage Creek Tunnel) there is no alternative passage. For a tunnel with multiple bores, the detour distance is considered to be zero since one of the other bores could enable temporary bidirectional traffic if a single bore were damaged by fire. The 2-lane Lehigh Tunnel (with baseline geometry, AADT, and the Traffic 2 profile per Table 8) is therefore reanalyzed for a variety of detour distances to examine the impact of this parameter on the optimal fire protection solutions.
Figure 18 shows the Pareto fronts for the cases where the detour distance increases from zero to 1.6 km (i.e. the actual distance for the Lehigh Tunnel) to 30 km, 320 km, 700 km, and 10,000 km (with the last value essentially representing the case where a feasible detour does not exist). The summary of optimal solutions in Table 11 shows that more fire protection is needed as the detour distance increases. For example, the zero-protection solution is eliminated as the detour distance increases to 30 km. In particular, the application of fire protection boards enables greater efficiency for the detour distances of 320 to 700 km.
Fig. 18
Optimal solutions for objective set 2 for the 2-lane southbound Lehigh Tunnel with varying detour distance during closure
Summary of optimal solutions for objective set 2 for the 2-lane southbound Lehigh Tunnel with varying detour distance during closure
Detour distance (km)
Optimal solution number
Protection plans
Lifecycle flow (million USD)
Protection efficiency (%)
Ventilation
FFFS
Protection board thickness (mm)
Zero detour (two-bore tunnel)
1
Natural
No
None
2.89
0.00
2
Natural
Yes
None
5.96
51.8
1.6
1
Natural
No
None
3.61
0.00
2
Natural
Yes
None
6.17
65.1
30
1
Natural
Yes
None
10.3
316
320
1
Natural
No
12
47.0
1374
2
Natural
Yes
None
52.0
2873
3
Natural
Yes
6
30.0
1328
4
Natural
Yes
9
28.5
1205
5
Natural
Yes
12
24.1
1126
6
Natural
Yes
15
22.1
1046
7
Natural
Yes
18
21.3
973
700
1
Natural
No
6
136
3511
2
Natural
No
9
116
3168
3
Natural
No
12
88.5
2979
4
Natural
Yes
6
49.4
2878
5
Natural
Yes
9
44.6
2612
6
Natural
Yes
12
33.4
2436
7
Natural
Yes
15
27.0
2264
8
Natural
Yes
18
23.5
2105
9
Natural
Yes
20
22.7
2006
10,000
1
Natural
No
12
1109
42,241
2
Natural
Yes
None
1444
88,293
3
Natural
Yes
6
527
40,762
4
Natural
Yes
9
439
37,018
5
Natural
Yes
12
259
34,542
6
Natural
Yes
15
148
32,091
7
Natural
Yes
18
77.5
29,870
8
Natural
Yes
20
52.2
28,440
9
Longitudinal
Yes
15
47.6
17,967
10
Longitudinal
Yes
18
35.1
17,191
6 Conclusion
This study presents a decision-making methodology to optimize the application of passive and active fire mitigation systems for roadway tunnels by both minimizing the lifecycle financial cost (including the initial investment and economic losses due to fire) while maximizing the protection efficiency. The optimization accounts for the performance of the concrete tunnel liner when subjected to fire hazard, the post-fire repair procedures, the tunnel traffic volume and composition, and detour distance during tunnel closure. The conclusion of this study are as follows:
The MATLAB-SAFIR thermal analysis approach is efficient for modeling the behavior of the fire-exposed concrete liner both with and without passive fire protection (including the onset of thermal spalling). Uncertainties associated with the thermal parameters of both concrete liner and protection materials are incorporated with MCS-LHS, and damage maps are developed to efficiently characterize the damage state of the protected concrete liner for a given fire hazard intensity.
By coupling the computationally efficient confined discretized solid flame (CDSF) model [21, 42] with the damage maps developed in this study, the contour of fire-induced liner damage along with the repair time and associated cost can be assessed when subjected to specific tunnel fire hazard.
The lifecycle investment for a specific fire protection plan is calculated to account for installation and maintenance costs, while the lifecycle loss due to a tunnel fire hazard is assessed based on the tunnel traffic and the detour distance.
Multi-objective optimization via a genetic algorithm was used to obtain optimal solutions for the fire protection systems under consideration. This optimization method is flexible in choosing the objectives and constraints to meet specific engineering requirements. Carefully choosing the appropriate objective sets can help narrow down decision-making choices.
The results of analysis indicate that tunnels with smaller geometry, higher traffic volume, higher HGV percentage, and longer detour distance requires more protection. The longitudinal ventilation, though required for life safety concerns, is not necessarily efficient for mitigating thermal damage to the concrete tunnel liner compared to the FFFS and fire protection boards.
Acknowledgements
Financial support for this project has been provided by the U.S. Department of Transportation (Grant #69A3551747118) via the University Transportation Center for Underground Transportation Infrastructure (UTC-UTI) at the Colorado School of Mines (CSM). The findings presented in this paper are the authors’ and not the US DOT, UTC-UTI, CSM, or any state DOT.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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