Calculating a Legally Enforceable Yellow Change Interval For Turning Lanes in California by Jay Beeber, Executive Director, Safer Streets L.A., Member ITE and J. J. Bahen, Jr., P.E., Life Member National Motorist Association Introduction When set correctly, the yellow interval time should not create situations where motorists are forced to unintentionally run the red light. That is, motorists should not encounter a Type I Dilemma Zone, an area of roadway, within which, if a motorist is present when the yellow signal illuminates, he will neither be able to stop comfortably and safely nor legally enter the intersection before the onset of the red phase. The theory behind calculating the minimum yellow interval is that to eliminate the Type I Dilemma Zone, the traffic engineer must set the yellow interval to at least the time it takes for a vehicle to traverse the critical stopping distance. In order to calculate this time, one must first, a) calculate the critical stopping distance and then, b) calculate how long it will take for a vehicle to cross that distance. For vehicles traveling at a constant velocity, the two equations can be combined into one equation (the ITE Kinematic Formula - See Appendix A) since the velocity used to calculate the critical stopping distance (the initial velocity) is the same as the velocity used to calculate the travel time across that distance. However, where a vehicle does not, or cannot, maintain a constant velocity (such as in a turning lane) the two equations cannot be combined (the ITE Kinematic Formula cannot be used) and the two calculations must be made separately using different velocities. The following is a step by step explanation of how to calculate a legally enforceable yellow change interval for turning lanes in the State of California. The following assumptions are made throughout: 1. Per CVC 22351(a), regardless of travel lane, speeds not in excess of the posted or prima facie speed limit are lawful. Therefore, this protocol assumes that a driver's initial approach speed will be at least the posted or prima facie speed limit. 2. In order to simplify the calculation and generate a chart of minimum yellow change intervals for turning lanes in California, we will assume that vehicles slow down from their initial approach speed and cross the stop bar at 20 mph (30 fps). This represents a curve radius of approximately 95 ft. Where the curve radius in known to be substantially different than 95 ft, practitioners are encouraged to perform the full calculation detailed below to determine the speed at which vehicles cross the stop bar. 3. The sample calculations used throughout this paper are based on a roadway posted at 35 mph.
Calculating the Critical Stopping Distance The Critical Stopping Distance = Perception/Reaction Distance + Braking Distance Critical Stopping Distance Equation Adding the distance the vehicle travels during the perception/reaction time to the vehicle's braking distance provides the critical stopping distance. This is the absolute minimum length of roadway a motorist requires in order to bring his vehicle to a safe and complete stop after the onset of the yellow signal and can be calculated using the following equation: d c =v i t pr + v i 2 2a Where: d c = the critical stopping distance v i = the initial approach velocity measured at the critical distance (for turning lanes, assume the posted or prima facie speed limit) t pr = the driver's perception/reaction time (assume 1.0 sec per ITE) a = the deceleration rate of the vehicle (assume 10 fps 2 per ITE) Note that if a driver is closer to the intersection than the critical distance when the yellow light illuminates, based on the laws of motion, the driver is forced to keep going. If he chooses to stop, his vehicle will travel farther than the distance remaining between his position and the limit line and his vehicle will come to a stop beyond the limit line, within the intersection or possibly beyond the intersection. DRIVERS CLOSER TO THE INTERSECTION THAN THE CRITICAL STOPPING DISTANCE WHEN THE YELLOW LIGHT ILLUMINATES MUST KEEP GOING. Sample d c =(51.3) x(1)+ (51.3)2 =183.1 ft 2x10 Note that under the assumptions above, if a driver is closer than 183.1 ft to the intersection when the yellow light illuminates, based on the laws of motion, the driver is forced to keep going. If he chooses to stop, his vehicle will travel farther than the distance remaining between his position and the limit line and his vehicle will come to a stop beyond the limit line, within the intersection or possibly beyond the intersection.
Calculating the Time to Traverse the Critical Stopping Distance Since motorists who are in the section of roadway closer than the critical stopping distance when the yellow light illuminates must keep going, the yellow signal must remain lit long enough to give this driver at least enough time to cover the distance to the intersection before the light turns red. Yellow times set shorter than this amount will create a dilemma zone for drivers. I. Driver Continues at Initial Velocity (not in a turning lane) For the driver who continues at his initial velocity, the minimum time needed to traverse the critical stopping distance can be calculated using the following formula: t y = d c v i Where: t y = minimum time needed to traverse the critical stopping distance = minimum yellow time d c = the critical stopping distance v i = the initial approach velocity at the critical distance t y = 183.1 =3.53 seconds 51.3 For the driver who continues at his initial velocity, the minimum time needed to traverse the critical stopping distance is 3.53 seconds. II. Driver Slows Down on Approach to Negotiate the Turn If a driver wishes to decelerate at the assumed rate of 10 fps 2 to a lower velocity in order to negotiate the turn, there exists a critical deceleration point that can be calculated based on the driver's initial velocity and the final velocity he wishes to achieve. This is the closest point to the limit line at which a driver can begin his deceleration and achieve his desired turning velocity. The following discussion assumes that a turning driver does not begin to decelerate until he must do so. In other words, that he does not begin to decelerate until he reaches the critical deceleration point. Base on this assumption, there exists a deceleration zone within the critical stopping distance between the critical deceleration point and the limit line. Under this model, the critical stopping distance then consists of two zones, a non-deceleration zone (ndz) where the driver maintains his initial velocity and a deceleration zone (dz) within which the driver decelerates to a velocity at which he can negotiate the turning movement.
We can express this as: d c =d ndz +d dz The total time to traverse the critical stopping distance (the minimum yellow interval) would therefore be the sum of the time to traverse the non-deceleration zone (the distance from the initial point of the critical stopping distance to the critical deceleration point) plus the time to traverse the deceleration zone. We can express this as: t tot =t ndz +t dz Calculating the Turning Velocity Although little research has been conducted on the speed at which turning vehicles enter an intersection to negotiate a turn, one method of determining that speed is to use the curve design speed (sometimes called the maximum safe speed or advisory speed ) which has been published by the Institute of Transportation Engineers for this calculation. On roadways with no banking, the equation to determine the curve design speed reduces to: v cds = 15 x R x f
Where: v cds = Curve design speed (mph) R = Curve Radius (ft) = 30 ft in this case f = Side friction factor; for speeds of 20 mph or less (as in this case), f = 0.28 Where the point of curvature is at the limit line, the curve design speed (turning velocity) is assumed to be the speed at which the vehicle crosses the limit line. v cds = 15 x 30 x 0.28=11.2mph=16.5 fps As stated in the introduction, we will assume that vehicles cross the stop bar at 20 mph (30 fps). Calculating the Critical Deceleration Point and Length of the Deceleration Zone Using the equations of motion, if we know the vehicle's initial velocity, the final velocity (speed at which the vehicle crosses the limit line), and the deceleration rate, we can calculate both the time it takes to decelerate from the initial velocity to the final velocity and the distance traveled during that time. Step 1 - Calculate the time to decelerate from the initial velocity to the final velocity. This is the time to traverse the deceleration zone and is given by: t dz = v i v f a Where: t dz = time to traverse the deceleration zone v i = the initial approach velocity measured at the critical distance (for turning lanes, assume the posted or prima facie speed limit) v f = final velocity at the limit line (assume 30 fps) a = deceleration rate (assume 10 fps 2 per ITE) Sample t dz = (51.3) (30) =2.13 seconds 10 It therefore takes a driver 2.13 seconds to traverse the deceleration zone t dz.
Step 2 - Calculate the distance traveled while decelerating from the initial velocity to the final velocity. This is simply the average velocity v av = v i+v f 2 multiplied by the time calculated in Step 1: d dz =t dz x v av This gives us the length of the deceleration zone d dz. v av = 51.3+30 =40.7 fps 2 d dz =(2.13) x(40.7)=87 ft The length of the deceleration zone d dz is therefore 87 ft. Step 3 - Subtracting this distance from the total critical stopping distance gives the length of the nondeceleration zone d ndz : d ndz =d c d dz d ndz =(183.1) (87)=96.1 ft The length of the deceleration zone d ndz (where the driver continues at his initial speed) is 96.1 ft. Step 4 - Since the driver's vehicle remains at its initial velocity while traversing the non-deceleration zone, the time to traverse the non-deceleration zone is simply the length of the non-deceleration zone divided by the initial velocity: t ndz = d ndz v i t ndz = 96.1 =1.87 seconds 51.3 The vehicles travels for 1.87 seconds before the driver begins to slow down.
Step 5 - Adding the time to traverse the non-deceleration zone to the time to traverse the deceleration zone as calculated in Step 1 gives us the total time to traverse the critical distance which again is the minimum yellow interval time needed to eliminate the dilemma zone. t y =t tot =t ndz +t dz t y =(2.13)+(1.87)=4.0 seconds For a roadway posted at 35 mph, the minimum yellow change interval to eliminate the dilemma zone would be 4.0 seconds. This is a full 1.0 second longer than the 3.0 second minimum yellow change interval time for dedicated turning lanes currently permitted by the CA MUTCD and 0.4 seconds longer than the 3.6 second yellow change interval time for through lanes. The following table sets out the minimum yellow change interval for posted speed limits in California: Table 4D-102xx (CA) POSTED SPEED or UNPOSTED PRIMA FACIE SPEED Minimum Yellow Interval MPH Seconds 25 3.0 30 3.4 35 4.0 40 4.6 45 5.3 50 5.9 >50 6.0 Yellow change interval times set shorter than those in the above table will create a dilemma zone and force some drivers to run the red signal. This puts at risk all roadway users including pedestrians, bicyclists and other motorists.