Estimación de longitud crítica en pendientes ascendentes en caminos bidireccionales Estimation of critical length in ascending grades of two-lane rural roads

The critical length of grade on two-lane rural roads is used as criterion to assess the need of ascending lanes for heavy vehicles. To estimate it, a speed profile is needed, which depends on the road’s grade and length, and on the physical, dynamic and mechanical characteristics of heavy vehicles. The Chilean standard uses a speed profile for a design truck that have a weight-to-power ratio of 120 kg/kW and an entrance speed of 88 km/h. This unique model prevents from studying cases in which the entrance speed can be different or in which the design vehicle is significantly different to the design speed of standards due to local conditions and consequently, does not allow estimating realistic critical lengths of grades. In this work, speed profiles models based on acceleration and force equilibrium were simulated, considering weight-to-power ratios between 120 and 263 kg/kW, entrance speeds between 80 and 100 km/h. Several values of critical lengths of grades were estimated, which offers engineers more options when designing ascending slopes.


Introduction
Speed profiles describe how the operating speed of a heavy-vehicle varies as it drives up a grade.In ascending grades, design instructions use them to assess the need of projecting ascending grades under service level and security criteria, or to define the maximum slope and length of an ascending grade in order to omit an ascending lane.The main variables which explain the speed reduction are the slope and length of the grade, the vehicle weight-to-power ratio (WPR) and the speed at which the vehicle enters the grade (Fitch, 1994).For the same ascending grade geometry, the performance of similar heavy vehicles, related to WPR, may be completely different depending on the speed entrance.Likewise, considering the wide range WPR for vehicles, the performance may also be different, so in practice, each heavy vehicle will have a unique speed profile.
Standards use speed profiles considering a design vehicle, with a typical power, with a unique entrance speed to the grade, for uniform slopes up to 10% and lengths of grades that vary between 0 and 6 km (Arellano et al., 2014).In Chile, design instructions define the maximum longitudinal grades based on a standard speed profile adopted from the US' standards.This speed profile considers a design vehicle whose WPR is 120 kg/kW, a unique entrance speed of 88 km/h, an inclination range between 1% and 8% and a length of grade between 0 and 3 km (MOP, 2014).
The Chilean standard that uses only one speed profile does not allow considering diverse design conditions, as for example the effect of elements for speed reduction that limit the entrance speed or, on the contrary, wide courses that encourage drivers to use higher operation speeds.Likewise, the WPR proposed in the current Chilean Roadways Handbook does not necessarily represent the characteristics of the heavy weight vehicles fleet.Arellano et al. (2014) indirectly determined that it varies between 95 and 609 kg/kW, with predominance between 100 and 160 kg/kW.In consequence, relying on speed profiles, models and more entrance speed options of design vehicles which are more similar to the current fleet, offers a greater variety of design conditions that, in the end, allows improving the geometric designs.
Therefore, this study proposes new truck speed profiles for ascending grades, using different WPR and entrance speed combinations.Based on these profiles new critical length charts are proposed to define the critical longitudinal of grades for two-lane roads.
Speed theoretical models and road geometric design standards were analyzed that included speed profiles for the design of ascending grades.A speed theoretical model was simulated based on Bester (2000), and Rakha and Yu (2004) models.From these simulated models, nine speed profiles were obtained for the combinations of three WPR (120, 150 and 263 kg/kW) and three grade entrance speeds (80, 90 and 100 km/h).Using these profiles, nine new charts with the critical length of grade were prepared for speed acceptable reductions (ΔV) between 10 and 30 km/h and for each WPR and entrance speeds.Finally, the results obtained were compared with the critical length values recommended by the Chilean Roadways Handbook and the American design standard.In ascending slopes, speed decreases from an initial value (point B) to a balance speed (point D) (also known as crawl speed), which remains more or less constant until the end of the uniform slope (F).This deceleration is due to the increase of the resistive forces in the face of the forces provided by the engine (Rakha and Yu, 2004) reaching to a null value (point C).The balance speed corresponds to the maximum speed that a truck may attain travelling in a longitudinal slope.The magnitude and distance at which this speed is reached depend on (Archilla and Fernández de Cieza, 1996): the length of the ascending section, the slope of the ascending grade, the vehicle WPR, the height above sea level, the vehicle initial speed, the presence of horizontal curves and the lateral restrictions.

Speed models in ascending grades
Models that explain this behavior are classified in two categories (Arellano et al., 2014): those based on the dynamic performance and those based on the cinematic performance.Dynamic models describe the balance between the tractive force (F T ), the aerodynamic force (F A ), the rolling resistance (F R ) and the resistance associated to the weight (F G ).When the tractive force is greater than the sum of resistances, the remaining force (Ma) allows the movement (section AC or BD in Figure 1).When this remaining force is null (Ma=0), the vehicle travels at the balance speed (section CE and DF in Figure 1).In this category of models, those from Lee and Lee (2000), and Rakha et al. (2001) are the most important.
Equation 1 shows the Rakha et al. (2001) model based on the force balance.Equation 2shows the discrete solution of the differential equation: that allows obtaining the speed profile in function of the distance based on Equation 1. P (N) corresponds to the heavy vehicle effective power, a (m/s 2 ) is the acceleration, V (km/h) is the speed, F MAX (eN) is the maximum power, m (kg) is the mass, g (m/s 2 ) is the gravity acceleration, i (%) is the grade slope, t i (s) is a time instant in which the vehicle is in the position x i , a corresponds to the acceleration, Δt (s) is the integration time step and α i are the model's coefficients.The detailed expression can be read in Rakha and Yu (2004).

Speed profile in descending slope
Cinematic models create the speed profile based on the following variables: displacement, speed and acceleration.The acceleration was modelled using lineal specifications such as that of Bester (2000) (Equation 3) or non-linear such as that of Lan and Menendez (2003).
In Chile, Arellano et al. ( 2014) evaluated Bester's acceleration model ( 2000) (Equation 3) based on speed and position data obtained through a GPS device, for a WPR range between 100 and 160 kg/kW and an entrance speed of 93 km/h, to obtain the speed profiles.They determined that the entrance speed used in the Chilean Roadways Handbook (MOP, 2014) corresponds to percentile 82 of the speed distribution observed in the field and that the balance speed resulted 1.15 times greater than that predicted by the model described in MOP (2014).

Critical length in ascending grades
The critical length of an ascending grade is the maximum length over which a heavy vehicle travels without reducing its speed, in a certain amount, in respect to the slope entrance speed.The decrease in the operation speed of heavy vehicles increases the risk of accidents in two-lane roads, because light vehicles are compelled to surpass them.Glennon (1970) demonstrated that, independently of the media speed value of the road, the more a vehicle deviates from this media speed, the more there is a probability of suffering an accident.Figure 2 graphically shows how the accident rate varies when a heavy vehicle reduces the speed on a grade.In case it is not possible to limit the grade to its critical length, there is the option of designing an exclusive ascending lane for heavy vehicle.To estimate the critical length, all possible combinations are traced for: the grade (i in %), the slope length (L, in m), the distance from the beginning of the slope, where the vehicle reduces its speed ΔV km/h.To obtain this chart, it is necessary to have profile speeds such as those described in section 2 of this paper.

Calculation of the new values for critical speed in grades
The calculation of the new critical length values was developed in three stages.In the first one, the analysis conditions were defined; in the second one, speed profiles were simulated and in third one, critical length values were calculated.

Conditions for calculation
For this purpose, combinations of the entrance speed to the grade, WPR and the geometry of the grade profile were considered.The entrance speed to the grade was chosen considering project speeds associated to each design category defined by MOP (2014).The speeds used were 80, 90 and 10 km/h, which correspond to the Primary and Collector road categories, both for two-lane roads.This speed range also considers the types of plain, undulating and mountainous ground.Speeds associated to the upper categories were not considered, assuming that, in general, they correspond to roads with separated lanes and less inclined slopes, which due to their design, do not induce the same phenomena as two-lane roads do.Instead, lower categories lead to consider entrance speeds lower enough so the difference with respect to the balance speed is lower than the critical values; thus, so the accident rate is not as important as in the study cases.In relation to the WPR, three values were considered: 120, 150 and 263 kg/kW.The first corresponds to the one used at the present by MOP (2014); the second, is based on the results obtained by the Arellano et al. study (2014) and the third corresponds to heavy vehicles with a total gross weight of 90 tons, which are high-tonnage vehicles according to Díaz (2012).The latter was analyzed assuming that they may be used in Chile to reduce transportation costs.Geometry was specified considering the standards established by MOP (2014) for the profile geometric design: a slope range between 1and 8%, and a slope maximum length of 3 km.Rakha et al. model (2001) was simulated for each entrance speed, WPR and geometry combination using Equations 1 and 2. Afterwards, the Bester's model (2000) was used to estimate α and β values.Both models adjusted when the root mean square error (RMSE) value was under 0.5 km/h.Then, the values of α = 0.948 ± 0.17 and β = 0.043 ± 0.01 were obtained.The variability of these coefficients is due to the combinations of the entrance speed values, WPR and geometry used.The α values depend on the entrance speed, the WPR and on the geometry, in consequence, they are unique for each value combination of these variables.Considering the lower variability of β, a value of 0.043 was assumed.The α values used correspond to those calculated for each combination of geometry, WPR and entrance speed.With these parameters, 9 speed profiles were reconstructed based on distance using the Bester's model (2000) calibration procedure described in Arellano et al. (2014).Figure 4 shows the simulated profiles.

Estimation of the critical length values
Based on the speed profiles in Figure 4, the new critical length values were calculated for speed variations beginning from an entrance speed of 10 and 30 km/h each 5 km/h.Results are graphically summarized in Figure 5. Table 1 shows the critical length values for speed differences of 15 km/h and 25 km/h, and different entrance speeds and WPRs.Table 2 shows that for speed reductions of 15 km/h and 25 km/h, the proposed model exhibits lower critical lengths than those established in the AASHTO standards.For ΔV=15 km/h, differences vary from 42% for grades of to 16 % for grades of 8%.For ΔV=25 km/h, differences vary between 38% for grades of 3%, to 11% for grades of 8%.That implies that when using the AASHTO model in Chile, the critical length is being over-estimated and in consequence, the need of ascending lanes is being under-estimated.
In Table 3, it can be observed that for ΔV=40 km/h and grades between 3% and 5%, the of this paper does not the grade lengths, so it not to make comparisons.For ΔV=40 km/h and grades between 6 and 8%, critical lengths were higher than those given by the Roadways Handbook.The differences vary from 140% for a grade of 6%, 57% for a grade of 7% and 36% for a grade of 8%.For ΔV=24 km/h, the proposed values are higher than those given by the Roadway Handbook.The differences vary from 21% for a grade of 8% to 6% for a grade of 4%.

Conclusions
The objective of this work was to propose new values for critical length of grades, based on simulated new speed profiles and considering a wider variety of grade entrance speeds and weight-to-power ratios.
The results obtained show that the critical length of grades depends on the speed profile, which at the same time depends on the geometry, the and grade entrance speed of the vehicle.As the last two variables depend on the heavy-vehicle fleet and the vehicle operation, it is not recommendable to establish only one group of values for a design vehicle, but rather a group of values that accounts for the variability of speed entrance, WPR and vehicle fleet.
The work analyzed the case of high-tonnage configurations (WPR=263 kg/kW).This configuration which has being adopted in some Latin American countries in order to generate savings in transportation costs.The work demonstrates that for these configurations, the need of ascending lanes is substantially greater (between 40 and 60% greater) than for conventional heavy vehicles; therefore its eventual use will necessarily require an investment on ascending lanes for safety reasons.
Compared with the Chilean Roadways Handbook, and using the same design vehicle, the critical length values proposed in this paper are higher.On average, for ΔV=24km/h and ΔV = 40km/h, the proposed model gives values 35 and 77% higher.It is recommendable to use the values proposed in this study, since they consider more options for the entrance speed, they use a design vehicle similar to those travelling on Chilean roadways and consider a more rigorous speed reduction criterion that results in a reduction of accident risks.
Although Arellano et al. calibrated a model in Chile, in order to increase the model's generality it is necessary to characterize heavy vehicles with respect to their weight-topower ratio.Therefore, it is necessary to calibrate the Bester's model in plain ground to estimate the movement in relation to the weight-to-power ratio in motion and not in a static way or motionless.
Another aspect that should be considered to improve the estimation of critical length of grades is the validation of Glennon's model, since it allows establishing, in a more accurate way, which are the tolerable values for accident risk and from this assesment to establish the speed decreasing values that will be used to estimate the critical length.Specially, in this work the values used were those recommended worldwide by the geometric design standards (ΔV=15 km/h and ΔV = 25km/h) and those used in the Roadways Handbook mainly for comparative purposes (ΔV=24km/h and ΔV = 40km/h).

Figure 1
Figure 1 outlines two types of speed profiles depending on the sign of the uniform slope.
Figure3ashows the design graph used by the US design recommendations and Figure3bshows that of the Chilean standards calculated from the speed profile model described by the Transportation Research Board (TRB)(1994).

Figura 5 .
Figura 5. Gráficos para el cálculo de longitud crítica en base a perfiles de velocidad simulados Figure 5. Graphs for the estimation of the critical length based on simulated speed profiles

Table 2 .
Valores de longitud critica en pendiente de AASHTO (2011) y este estudio Critical length values of grades fromAASHTO (2011)and this study Tabla 3. Valores de longitud critica en pendiente de MOP (2014) y este estudio Table3.Critical Length values of grades from MOP (2014) and this study