Superelevation Criteria for Sharp Horizontal Curves on Steep Grades (2014) / Chapter Skim
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From page 40... ...
40 S E C T I O N 4 This section presents the analytical and simulation modeling work performed to investigate superelevation criteria for sharp horizontal curves on steep grades. Section 4.1 presents the stepby-step analysis approach which integrates both field and simulation data and is based upon an increasingly detailed analysis using progressively more sophisticated simulation models.
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From page 41... ...
41 To ascertain whether the operating point ( fx, fy) lies inside the friction supply ellipse for a given combination of cornering and braking demand, the constraint of Equation 15 must be met.
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42 Combining Equations 16 and 17 obtains a usable definition of the lateral friction margin: 1 (18) margin y,max x x,max 2 yf f f f f= − − This definition of the lateral friction margin therefore depends on the tire's demanded side force, fy, the demanded braking, fx, and maximum dimensions of the friction ellipse in the braking and lateral directions, fx,max and fy,max.
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43 the simulation results for passenger vehicles with a brief discussion on the simulation results for trucks. It is not until the last few steps (i.e., beginning with Step 7)
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44 Additional information about the road surface is needed to capture the full tire force curves in combined longitudinal and lateral skidding, across a range of skidding values from normal driving to full skids. In particular, the skid numbers only provide the skidding values and therefore do not give a good indication of tire forces transitioning from maximum friction to skidding friction conditions.
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45 tires. Figure 29 illustrates the friction supply curves for the maximum friction measurements, providing both average values and two standard deviations below the average values, for both the full braking and full cornering conditions.
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46 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0 10 20 30 40 50 Peak normalized braking force, F x,max /F z Fr eq ue nc y Data Fit with Mean = 0.878 and StdDev = 0.091 2% friction value (Mean - 2 ) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0 10 20 30 40 50 60 70 Peak normalized cornering force, Fy,max/Fz Fr eq ue nc y Data Fit with Mean = 0.711 and StdDev = 0.068 2% friction value (Mean - 2 )
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47 Figure 28. Distribution of maximum friction for longitudinal (braking)
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48 Figure 29. Passenger vehicle tire measurements of maximum wet-tire friction in longitudinal (braking)
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49 4.2.3 Summary of Key Results from Step 1 The results shown in Figures 29 to 32 allow comparisons between road friction measurements and the maximum side friction, fmax, used in the current AASHTO design policy for horizontal curves. The friction supply curves for both the lateral (cornering)
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50 a horizontal curve with no superelevation. From Equation 7, the relationship between radius, speed, and friction will be approximately: = + 0.01 (20)
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51 These equations can be simplified by substituting Equation 23 into Equations 21 and 22, and then simplifying the result using the friction factors from Equations 13 and 14 to obtain: = − 100 (24)
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52 For different braking values, the lateral friction margins change because the braking forces utilize some of the reserve lateral friction available. Three decelerations levels (0, -3, and -11.2 ft/s2)
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53 that this might not apply to more realistic (i.e., complex) vehicle models.
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54 available in the lateral direction and thus decrease the cornering abilities of the vehicle. Speed distributions of vehicles collected in the field were also used to confirm that simulations were using a range of speeds similar to those measured from vehicles at the actual field sites.
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55 was determined that the maximum lateral friction demand is required when the vehicle brakes after reaching steady state (Case 1)
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56 This model associates the rollover event with the onset of wheel lift, characterized by the normal load on the inside wheels going to zero (Fzi = 0)
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57 design speeds of 25 mph or greater, it can be deduced that wheel lift will not occur for passenger vehicles or trucks driving at the design speed on a curve designed according to current AASHTO policy. Note, though, that at design speeds below the scope of this research (i.e., design speeds of 10 and 15 mph)
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From page 58... ...
58 Table 21 is 0.56, and half this value is 0.28, which for practical purposes is the same as a rollover threshold value of 0.30. This suggests that the lateral accelerations in curves, and hence the maximum side friction values used for design, should be limited to values less than 0.30.
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59 rear to the front of the vehicle may significantly change the relative amounts of vertical tire force on each axle. If there is a curve on a grade, the cornering forces required from each axle remain proportional to the mass above each axle, not the weight.
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60 AASHTO minimum curve radii, this is not the case. Equation 36 for the steady-state bicycle model can be rewritten as: + = − 100 (38)
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61 stand that braking is sometimes (and intentionally) not distributed equally between axles.
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62 Table 23 shows the decelerations at which each vehicle's proportioning valve would initiate a reduction in rear tire braking force, for a level grade situation and for a 9% downgrade. Thus, of the four decelerations levels (0, -3, -11.2, and -15 ft/s2)
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From page 63... ...
63 When the longitudinal friction factor exceeds the longitudinal friction supply, fx,max, the lateral friction supply is assumed to be zero. Using the above equations for the steady-state bicycle model, the friction demand and friction supply analysis is performed for each individual axle.
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64 of the demand is more appropriate. Thus, the lateral friction margins are formulated as: = − = − (61)
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65 decreases the lateral friction margin by approximately 0.001. For curve-entry deceleration (ax = -3 ft/s2)
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66 with yaw rate and lateral velocity as the motion variables. The input variables in this model -- the steering input, d, and velocity of the vehicle -- are assumed to be under the driver's control.
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67 the vehicle's longitudinal orientation (vehicle's x-axis) to the tire's direction of heading (i.e., tire's x′-axis)
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68 Substituting these expressions into the equations of motion given earlier, Equations 63 to 69, the lateral dynamics (cornering) equation becomes: i i i i ( )
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69 a feedback-driver model) , and it is seen that the transient bicycle model is noticeably more conservative in predicting lateral friction margins on entry to the curve because of these assumptions of fully developed superelevation on the tangent approach.
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70 no steering inputs other than those to maintain the vehicle within the lane, and no braking inputs other than those to prevent the vehicle from accelerating on a downgrade. When analyzing the simulation results, there were effects at high and low speeds that caused disagreement between the transient model and the other models.
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71 < − 100 100 (78) 2 p e V gR e tangent < +100 1 1 (79)
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72 input in this case is a 1 s transition from the tangent steering value to the curve-keeping steering value, but the resulting tire forces show a small peak at the end of the steering transition. This peak is due to the additional forces necessary to accelerate the vehicle in rotation versus the steady forces necessary for maintaining the vehicle's spin and tire forces within the curve.
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73 and higher superelevations, the transient bicycle model estimates lower friction margins than the other models. Comparing the steady-state bicycle model and the pointmass model, the point-mass model predicts a rear-axle friction margin that is 0.006 higher than predicted by the steady-state model.
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74 found that the minimum friction margins occur at the front axle for all these cases and is caused by the front tires requiring additional friction during the transition from the tangent to the curve steering levels. This transition becomes increasingly abrupt with increasing superelevation.
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75 reduction in lateral friction margins if transitioned too quickly. 4.8.2.2 Effect of Curve-Entry Deceleration Another set of simulations were conducted to represent a mild deceleration within a curve.
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76 the same levels of deceleration. As before, this level is particularly small and dwarfed by the change in friction margins versus speed.
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77 vehicle behavior on curves with steep grades. To investigate how tighter curve geometries might affect friction margin, the friction margins were evaluated for curves with radii that were 80% of the AASHTO minimum-radius curves.
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From page 78... ...
78 a sine wave, and so this steering waveform was used as an idealization of the driver's input. To determine the appropriate duration of the lane-change portion of the steering input, field data were used for guidance.
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30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph) M in im um F y M ar gi n Vehicle: E-Class Sedan, Grade = -9% R = Rmin e = 0% Intermediate superelevations (4% increments)
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80 Sensitivity analyses of the steering input assumptions revealed several important points. Lane-change-steering amplitudes strongly depend on speed.
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81 vehicle-to-vehicle differences amount to approximately 0.06 in margins. The full-size SUV had the worst margins among the two-axle vehicles simulated here (articulated vehicles are studied in later sections)
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82 Figure 67 for an E-class sedan, in Figure 68 for an E-class SUV, in Figure 69 for a full-size SUV, and in Figure 70 for a singleunit truck. As observed before, for passenger vehicles, each percentage decrease in grade appears to reduce friction margin by approximately 0.001.
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From page 83... ...
83 values of all roads, and if the maneuver combination is at the worst timing and location within the curve, then skidding may occur. Thus, the results do not provide absolute pass/fail criteria for a road design; instead they serve as indicators of trends and identify combinations of designs and operational conditions that might cause concern.
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84 semi-trailers typically have multiple axles spaced close together longitudinally at the back of the cab and at the back of the trailer unit, these axles were each lumped into single representative axles for the simulations. A diagram outlining the model structure for the low-order tractor semi-trailer dynamic equations is presented in Figure 71.
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85 Figure 68. Lateral friction margins from transient bicycle model for E-class SUV (G 0% to 9%, e 4% to 16%)
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86 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph) M in im um F y M ar gi n Full-Sized SUV, e = 4% Grade = 0% Intermediate grades Grade = -9% 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph)
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87 Figure 70. Lateral friction margins from transient bicycle model for single-unit truck (G 0% to 9%, e 4% to 16%)
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Symbol Meaning Mass of tractor Mass of loaded trailer Mass of tractor and trailer together Distance from hitch to trailer axle Distance from hitch to trailer CG Deceleration along -axis , , Braking force (front, rear, trailer axle) , , Cornering force (front, rear, trailer axle)
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From page 89... ...
89 the approach tangent rather than the curve itself. Again, this is consistent with observations made from passenger vehicles.
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90 Figure 76. Lateral friction margins from transient bicycle model for tractor semi-trailer (G 0% to 9%, e 4% to 16%)
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91 because tractor semi-trailers were seen to take longer to reach steady-state after the curve entry than other vehicles, due to the trailer dynamics and the vehicle's larger mass. Figure 78 shows a comparison of the lateral friction margins for the transient model, the steady-state model, and the point-mass model for the lane-change maneuver for the tractor semi-trailer for 8% superelevation.
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92 30 40 50 60 70 80 −0.1 0 0.1 0.2 0.3 0.4 Speed (mph) M in im um F y M ar gi n Tractor Trailer, e = 4% Rear Axle Limits Margin Grade = 0% Intermediate grades Grade = −9% 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph)
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From page 93... ...
93 small: a 0.02 increase in margin occurs across a 16% superelevation change, or about 0.001 increase in margin increase per 1% of superelevation added. This, as observed earlier, is almost negligible.
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94 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph) M in im um F y M ar gi n Tractor Trailer, e = 4% Rear Axle Limits Margin Grade = 0% Intermediate grades Grade = -9% 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph)
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95 departments of transportation (DOTs)
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96 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph) M in im um F y M ar gi n Tractor Trailer, e = 8% Tangent Approach Limits Margin Rear Axle Limits Margin Grade = 0% Intermediate grades Grade = -9% F to R F to R F t o R R to T R to T R to T Speed (mph)
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From page 97... ...
97 overall shape of the margin curves and trends are unchanged. The biggest effect of the brake variations is to change the axle with the minimum force in the speeds from 45 to 60 mph to be the trailer axle instead of the rear axle of the tractor.
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98 not change as significantly as do passenger vehicles for combined lane changes and braking. This is primarily due to the longer length, slower response, and tires that are less sensitive to changes in loading conditions.
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99 these lateral deviation estimates will be less accurate due to the assumptions mentioned previously as well as due to approximations used in the derivation of the lateral deviation distance. To calculate the lateral deviation distance, some basic assumptions must be made about the vehicle within the skid.
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100 Figure 87. Lateral friction margins from steady-state bicycle ( left plots)
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101 Figure 88. Lateral friction margins from steady-state bicycle ( left plots)
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102 Figure 89. Lateral friction margins from steady-state bicycle (left plots)
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103 Figure 90. Lateral friction margins from steady-state bicycle ( left plots)
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104 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph) M in im um F y M ar gi n Constant Speed, ax = 0 ft/s2 SSD Decel, ax = -11.2 ft/s2 Tractor Trailer, e = 0% Emergency Decel, ax = -15 ft/s2 Grade = 0% Intermediate grades Grade = -9% 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph)
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105 Figure 93. Lateral friction margins while maintaining the same lane (left plots)
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From page 106... ...
106 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph) M in im um F y M ar gi n Constant Speed, ax = 0 ft/s2 Curve Entry Decel, ax = -3 ft/s2 Tractor Trailer, e = 0% Emergency Decel, ax = -15 ft/s2 Grade = 0% Intermediate grades Grade = -9% 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph)
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From page 107... ...
107 Figure 99. Lateral deviation distances (ft)
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108 the same lane on the approach tangent and through the curve)
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109 2.8125e-007 2.8125e-007 2.8125e-007 0.4 0.4 0.8 0.8 1.21 1.612.012.41 Speed (mph)
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110 Without ABS, the transient bicycle model predicts that this vehicle may not be able to maintain its position in the lane on a curve while undergoing stopping sight distance or emergency braking decelerations.
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111 CarSim also requests a desired speed profile for each simulation. Speed profiles for vehicles were generated using MATLAB for input into CarSim.
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From page 112... ...
112 grade of -6%, and superelevation of 4% and assuming a con stant speed throughout the maneuver and the intended tra jectory of the vehicle is within the same lane on the approach tangent and through the curve. The resulting lateral friction margins match fairly well between these two models for this mild, steady maneuver.
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From page 113... ...
113 model are shown in Figure 106 for a -6% grade and a 4% superelevation, assuming constant speed. The rearmargin minimums of the transient model agree with the frontmargin minimums from the multibody model.
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114 one axle exhibits slip. This can result in low margins for one axle of a closely spaced pair.
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115 Figure 108. Lateral friction margins from multibody model for all five axles of a tractor semi-trailer (V 25 mph, G 0%, e 0%, R 500 ft)
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Figure 109. Lateral friction margins from transient bicycle and multibody models for tractor semi-trailer (G 6%, e 4%)
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117 normal driving, Figure 111 compares the inside versus outside tire lateral friction margins for a full-size SUV. The overall differences between the inside and outside tire margins are only a maximum of about 0.06.
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From page 118... ...
118 grade with 0% superelevation) as determined by the simulations in Section 4.8 were further evaluated.
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From page 119... ...
119 low margins but predicts that there will be no skidding across the range of speeds. In contrast, the transient bicycle model predicts that the rear axle could skid or be close to skidding for nearly all speeds.
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120 Figure 117. Lateral friction margins from transient bicycle and multibody models for full-size SUV (G 9%, e 0% and 8%)
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From page 121... ...
121 Figure 119. Lateral friction margins from transient bicycle and multibody models for tractor semi-trailer (G 9%, e 8%)
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122 0 5 10 15 20 25 -5 0 5 D ec el er at io n (ft/ s2 ) Transient Bicycle Model Multibody Model 0 5 10 15 20 25 20 40 60 Sp ee d (m ph )
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From page 123... ...
123 friction margins are shown for the inside and outside tires in Figure 122 for the 0% superelevation case and in Figure 123 for the 8% superelevation case. As anticipated, the figures indicate that the weight transfer effects on lateral friction margins are higher for this maneuver than for a steady-curve traversal.
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124 zero or negative, the vehicle only skids for a short period of time (i.e., less than 1 s)
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125 For the sake of completeness in the analysis of trucks, this same maneuver on the same geometry of Figure 124 was simulated with the standard STAA Double twin-trailer truck (i.e., tractor semi-trailer/full-trailer)
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126 e = 0%, G = -9%, STAADouble Figure 125. Lateral friction margins from multibody model for tractor semi-trailer/full-trailer truck (Double)
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Figure 126. Margin trajectories for tractor semi-trailer/ full-trailer truck (Double)
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128 0 2 4 6 8 10 12 14 16 18 20 -20 -10 0 10 D ec el er at io n (ft/ s2 ) Transient Bicycle Model Multibody Model 0 2 4 6 8 10 12 14 16 18 20 0 20 40 60 Sp ee d (m ph )
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From page 129... ...
Figure 129. Lateral friction margins from transient bicycle and multibody models for tractor semi-trailer (G 6%, e 4%)
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130 a lane change (i.e., an even more aggressive maneuver)
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From page 131... ...
Figure 133. Lateral friction margin trajectories from transient bicycle and multibody models for E-class sedan [V 55 mph, G 9%, e 0% ( left plots)
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From page 132... ...
0 2 4 6 8 10 12 14 16 18 20 −0.5 0 0.5 1 Fr on t T ire M ar gi n Transient Bicycle Model Multibody Model Outside Tire Multibody Model Inside Tire 0 2 4 6 8 10 12 14 16 18 20 −0.5 0 0.5 1 Time (s) R ea r T ire M ar gi n Transient Bicycle Model Multibody Model Outside Tire Multibody Model Inside Tire 0 2 4 6 8 10 12 14 16 18 20 −0.5 0 0.5 1 Fr on t T ire M ar gi n Transient Bicycle Model Multibody Model Outside Tire Multibody Model Inside Tire 0 2 4 6 8 10 12 14 16 18 20 −0.5 0 0.5 1 Time (s)
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From page 133... ...
133 0 5 10 15 20 25 30 −0.5 0 0.5 1 Fr on t T ire M ar gi n Transient Bicycle Model Multibody Model Outside Tire Multibody Model Inside Tire 0 5 10 15 20 25 30 −0.5 0 0.5 1 Time (s) R ea r T ire M ar gi n Transient Bicycle Model Multibody Model Outside Tire Multibody Model Inside Tire 0 5 10 15 20 25 −0.5 0 0.5 1 Fr on t T ire M ar gi n Transient Bicycle Model Multibody Model Outside Tire Multibody Model Inside Tire 0 5 10 15 20 25 −0.5 0 0.5 1 Time (s)
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From page 134... ...
134 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 Fr on t T ire M ar gi n Transient Bicycle Model Multibody Outside Tire Multibody Inside Tire 0 5 10 15 20 25 -0.5 0 0.5 1 R ea r T ire M ar gi n Transient Bicycle Model Multibody Outside Tire Axle 2 Multibody Inside Tire Axle 2 Multibody Outside Tire Axle 3 Multibody Inside Tire Axle 3 0 5 10 15 20 25 -0.5 0 0.5 1 Time (s) Tr ai le r T ire M ar gi n Transient Bicycle Model Multibody Outside Tire Axle 4 Multibody Inside Tire Axle 4 Multibody Outside Tire Axle 5 Multibody Inside Tire Axle 5 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 Fr on t T ire M ar gi n Transient Bicycle Model Multibody Outside Tire Multibody Inside Tire 0 5 10 15 20 25 -0.5 0 0.5 1 R ea r T ire M ar gi n Transient Bicycle Model Multibody Outside Tire Axle 2 Multibody Inside Tire Axle 2 Multibody Outside Tire Axle 3 Multibody Inside Tire Axle 3 0 5 10 15 20 25 -0.5 0 0.5 1 Time (s)
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From page 135... ...
135 compared to the transient bicycle model, primarily because of the driver behavior used by the multibody simulation software. Most significantly, the multibody model shows that the ABS may be activated for some of these maneuvers even when the simulations are conducted on high-friction roads.
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From page 136... ...
136 change at 70 mph, predicted for the multibody simulation in situations where there are negative lateral friction margins. Note that the ABS was activating during the simulation, even for high-friction roads, which again indicates that the friction margins are extremely low.
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From page 137... ...
137 or for the most aggressive maneuvers. Additionally, using a high-friction road to generate friction demand values could inadvertently allow the vehicles in the multibody simulation to maintain path tracking artificially well.
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138 lane change and/or lane follow, but not actually decelerate at the intended amount. Further investigations confirmed that the deceleration rates (as measured at the vehicle's sprungmass center of gravity)
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From page 139... ...
139 what a human driver will do during a traversal of a curve, but it is also acknowledged that these inputs may no longer be the worst case because a worst-case driver response cannot be defined. When the deceleration values exceed curve-entry deceleration values, oscillations in the braking input caused by the simulation trying to maintain the desired deceleration become significant and can lead to negative margin predictions.
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From page 140... ...
140 relative severity of a maneuver with respect to wheel lift, is utilized. The metric is defined for each axle as: = − + (89)
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141 Figure 148. Rollover margins of individual axles for single-unit truck and tractor semi-trailer (G 9%, e 0% and 8%)
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142 vehicle will likely rollover. This is evidenced by the fact that the multibody simulation itself did not predict a rollover event, which the software is fully capable of simulating.
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143 Figure 150. Rollover margins of individual axles for single-unit truck and tractor semi-trailer (G 9%, e 0% and 8%)
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144 4.12.1 Analysis Approach On upgrades, the direction of the grade requires traction forces instead of braking forces to be applied on vehicles. While this generally makes braking efforts easier, for situations without braking it means that more of the friction margin may be used.
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145 Figure 152. Rollover margin time trajectories for individual axles for single-unit truck (V 50 to 65 mph, G 9%, e 0%)
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146 which can be rearranged by multiplying by V to obtain: i i i= − − ρ0 100 1 2 (94) 3P V mg G C A Vcrawl d The wheel-horsepower, P, is related to the rated engine horsepower, Peng; the static power load on the engine, Pstatic; and the rolling power coefficient, Croll, by the following: i= − − (95)
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147 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph) M in im um F y M ar gi n Constant Speed, ax = 0 ft/s2 Curve Entry Decel, ax = -3 ft/s2 E-Class Sedan, e = 0% Emergency Decel, ax = -15 ft/s2 Grade = 0% Intermediate grades Grade = 9% 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 Speed (mph)
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148 Figure 156 shows the same situation with lane changes on the upgrade. The simulations were conducted assuming the vehicle was initially traveling at the crawl speed on the grade.
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149 the emergency deceleration cases. Like the situation observed for passenger vehicles, there is a reduction of margin for the constant-speed cases due to the traction required to maintain speeds.
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150 speeds is low and hits a minimum of zero for speeds from 70 to 80 mph. As defined in Section 4.11, a rollover margin of zero implies that a wheel has lifted on the axle, which implies a zero normal force on that tire.
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151 Figure 158. Lateral friction margins for inside and outside tires from multibody models for tractor semi-trailer (G 9%, e 16%)
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152 30 40 50 60 70 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Ro llo ve r M ar gi n Speed (mph) e = 16%, G = 9%, TractorTrailer Axle 1 Axle 2 Axle 3 Axle 4 Axle 5 Figure 159.
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From page 153... ...
153 Figure 161. Trajectory of simulation inputs for transient bicycle and multibody models for tractor semi-trailer (V 70 mph, G 9%, e 16%)
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154 The predicted rollover margins for a superelevation of 8% are positive for all speeds, although a minimum of 0.1 for the 80 mph traversal is still rather low. It appears that the 12% superelevation will still result in wheel lift at a 70 mph design speed.
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From page 155... ...
155 a general rule, these results suggest that on upgrades of 4% and greater, the maximum superelevation should be limited to 9% for curves with design speeds of 55 mph and higher. Alternatively, if it can be verified that the available sight distance is such that deceleration at -11.2 ft/s2 is unlikely to be required on upgrades of 4% and greater, minimum-radius curves could be designed using a maximum superelevation up to 12% on these steep upgrades.
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From page 156... ...
156 rollover for a variety of vehicles types when traversing sharp horizontal curves on steep grades. The point-mass model was the simplest model considered, while the transient bicycle and multibody models are more complex and simulate vehicles using multiple axles and multiple tires, respectively.
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