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1.0 Introduction
1.1 Description of occurrence
1.1.1
On 06 July 2013, at approximately 0115 Eastern Daylight Time (EDT), Montreal, Maine & Atlantic (MMA) freight train MMA-02 carrying petroleum crude oil derailed in the town of Lac-Mégantic, Quebec. Numerous tank cars ruptured, lost their content and a fire ensued.
1.2 Engineering services requested
1.2.1
A request was received from the Transportation Safety Board of Canada (TSB) Eastern Regional Operations - Rail/Pipeline office to estimate the derailment speed of individual tank cars involved in the occurrence.
1.3 Background
1.3.1
The train consisted of 5 head-end locomotives, 1 VB car (special purpose caboose), 1 box car, and 72 Class 111 tank cars carrying flammable liquids (petroleum crude oil, UN 1267, Class 3). Sixty-three (63) tank cars and the box car derailed. The 9 tail-end tank cars had not derailed and were subsequently pulled back to Nantes as part of the emergency response. The last tank car to derail (NATX 310470, consist no. 65) was basically undamaged and it was re-railed and pulled back from the pileup during the wreckage clearing operations.
1.3.2
The investigation revealed that when the emergency brake signal was ON, the speed of the train was 65 miles per hour (mph).^{Footnote 1}
1.3.3
The train stopped after tank car NATX 310470 (63^{rd} tank car) derailed.
1.3.4
The train was on a downhill grade (G) of about 1.2% before the point of derailment.^{Footnote 2}
1.3.5
The length of a representative tank car (between the coupler pulling faces) was 60 feet^{Footnote 3} (18.30 meters). The gross weight of the derailed tank cars was 128 tons (116 metric tons) on average.
2.0 Analysis
2.1
Figure 1 is a diagram schematically illustrating the derailment of the freight train. It is presumed that at the moment when the ${k}^{\mathrm{th}}$ tank car derails (Figure 1a), it has the same speed as that of the remaining train. Thus, by estimating the speed of the remaining train, the derailment speed of the ${k}^{\mathrm{th}}$ tank car can be obtained.
2.2
The estimation of the speed of the remaining train is carried out based on the kinetic energy equilibrium method. There are several assumptions for the model:
The remaining train is a rigid body.
The point of derailment for all the tank cars does not change. This implies that when the ${k}^{\mathrm{th}}$ car derails, the remaining train would still remain on the track and push it forward until the ${(}^{k}\mathrm{th}$ car arrives at the same spot and begins to derail, as shown in Figure 1a through 1c.
When the ${k}^{\mathrm{th}}$ car derails, its sudden deceleration would produce an impeding force ($F$) against the travelling of the remaining train due to the interaction between the derailed tank car and the remaining train. Essentially the impeding force is a reaction force to the force applied by the remaining train on the derailed tank car to overcome its friction with the ground in order to push it forward. A detailed analysis of the impeding force against the remaining train is presented in Appendix A. Based on this analysis, a constant impeding force value can be used for a first order estimation of the speed of the remaining train using the energy equilibrium method.
The first derailed car was a box car (CIBX 172032), which had a length of 69 feet and a gross weight of 105 tons. For the purpose of simplifying the calculation, it will be treated as having the same physical parameters as the derailed tank cars (see paragraph 1.3.5).
2.3
The train has $N$ tank cars ($N=73$ in this case) and each tank car has a mass $m$ (in this case, $m=116$ metric tons=116,000 kg).
2.4
Presuming the ${k}^{\mathrm{th}}$ car is derailed, at the beginning of its derailment (as shown in Figure 1a), the parameters of the remaining train are:
speed, ${V}_{k}$
mass of the remaining train, $(N-k)\cdot m$
impeding force from the derailed tank car against the remaining train, $F$
(here, G is the downhill grade of the track and g is the gravity constant 9.8 m/s^{2})^{Footnote 4}
kinetic energy of the remaining train
${E}_{k}=\frac{1}{2}(N-k)\xb7m\xb7{V}_{k}^{2}$
(2)
2.5
At the end of the ${k}^{\mathrm{th}}$ car’s derailment (Figure 1c), the moment when the body of ${k}^{\mathrm{th}}$ car is completely pushed outside the track by the remaining train and the ${\mathrm{(k-1)}}^{\mathrm{th}}$ car arrives at the derailment point, but still remains on the track, the parameters of the remaining train are:
speed, ${V}_{k}^{\prime}$
mass of the remaining train remains the same, $\mathrm{(N-k)\xb7m}$
impeding force $\mathrm{(F)}$ from the derailed car against the remaining train still remains a constant
The kinetic energy equilibrium for the remaining train during the process of the ${k}^{\mathrm{th}}$ tank car’s derailment (over the distance $S$ which is the length of the car) is:
${E}_{k}^{\prime}={E}_{k}-{Q}_{d}+{Q}_{g}$
(4)
Here:
${Q}_{d}$ is the loss of kinetic energy of the train due to overcoming the impeding force $F$ over the distance $S$:
${Q}_{d}=F\cdot S$
(5)
${Q}_{g}$ is the kinetic energy gained by the train due to the gravity force ${f}_{\mathrm{gr}}^{}$ pulling on the train over the distance $S$: