
Calculating Critical Speed  A MotorVehicle Crash Reconstruction Method Fraught with Error
John C. Glennon, D.Engr., P.E. August 2006 (copyright) Introduction The term critical speed as typically used in motorvehicle crash reconstruction refers to using the centripetal acceleration equation to calculate that speed at which a vehicle will allegedly lose control as a function of cornering radius, tirepavement friction, and crossslope. Two common analyses of critical speed are applied as follows
Basic Physics on Vehicle Cornering Sir Isaac Newton tells us that a body moving in a straight line will continue in a straight line unless acted on by an external force. A body moving on a circular path with constant speed will have a changing velocity (directional speed) due to the body's changing direction. This velocity change with time, called centripedal acceleration, has a radial direction toward the center of the circular movement and is given by the following equation: Since Newton's second law tells us that a force has to act on the body to produce an acceleration, the following is true: substituting Equation 1 into Equation 2, the following can be written: The lateral force on a vehicle moving in a circular motion on a pavement surface is produced by the frictional force between the tires and the roadway as follows: where f = the friction demand between the tire and roadway Substituting Equation 4 into Equation 3 yields Converting Equation 5 to allow for the velocity, S, in mph, and accounting for the roadway curve superelevation, e, yields the familiar form of the centripedal acceleration equation. The Critical Speed for a Horizontal Roadway Curve Critical speed used in this context is a term for the speed at which a vehicle will lose lateral control on a given roadway curve. The normally flawed procedure is to assume the highway curve radius as the critical path taken by the vehicle. On the surface, this procedure appears reasonable since the driver is apparently attempting to drive around the roadway curve. This theory fails, however, when it's used to compute the speed at loss of control on a dry surface with a reasonably good tirepavement friction value of, say 0.70. The procedure assumes that the driver exactly followed the roadway curve radius with a steadystate (not braking or speeding up) lateral acceleration (commonly called centrifugal force) of 0.7 g's. Studies consistently show, however, both that drivers' normally steer instantaneous path radii that are sharper than the roadway curve and also that drivers cannot tolerate steadystate lateral accelerations greater than about 0.30 g's. Therefore, a more likely scenario is that the driver, because of inattention or surprise, proceeded into the roadway curve without any initial steering and then needed to turn a sharper path than the roadway curve to stay on the roadway, thereby generating a temporal lateral acceleration greater than 0.7 g's at a speed much lower than the speed calculated by assuming the roadway curve radius is the vehicle path radius. Another likely scenario is where the driver entered the roadway curve too fast, generating .30–0.35 g's of lateral acceleration. Because the driver could not tolerate this level of lateral acceleration, he tried to compensate by flattening his path and/or braking, either of which could induce loss of control. As an illustration, consider the study by Glennon and Weaver^{1} that shows that drivers consistently steer sharper radii than the radius of the roadway curve. This study provides the following table: Table 1
Percentiles Of Vehicle Path Versus Roadway Curve Path If, for example, a ten–percentile driver (only 10% of all drivers use a sharper path) is considered as the criterion, the critical vehicle path on a 8degree roadway curve can be calculated as 9.92 degrees or a 577–foot path radius on this 716–foot roadway curve radius. Using the results of a study by Rice, et.al ^{2}, which shows that most drivers cannot handle steadystate lateral g levels greater than about 0.30, the critical speed with this lateral acceleration on an 8degree roadway curve with say 6% superelevation can be calculated as follows: This speed contrasts with the usual practice of using the roadway curve radius of 573 foot, with, say, a tire–pavement coefficient of 0.70, which would give the following calculations: As the reader can see, the theoretical critical speed calculation is far greater than the more realistic empirical calculation. The following table shows a series of similar calculations, demonstrating the dramatic difference between the theoretical critical speed and a more realistic critical speed: Table 2
Comparison Of Critical Speed Calculation Methods In summary, given that all drivers drive a path radius sharper than the roadway curve and given than most drivers cannot tolerate lateral accelerations much greater the 0.3 g's, a more realistic critical speed calculation should take these expected characteristics into account to arrive at an expected loss of control speed. The Myth of Calculating LossofControl Speeds from Yaw Mark Radii Most crash reconstruction books present a method for calculating the lossofcontrol speed for an outofcontrol vehicle that left yaw marks on the pavement. Yaw marks are tire marks characterized by diagonal striations left by a rotating tire sliding partially sideway on the roadway. Let's be entirely clear that this reconstruction method is presented in several wellused accident reconstruction texts. However, just because the method first showed up in one prominent text and subsequently has been copied by several "me to" texts doesn't make it right. This "critical speed" method must withstand the rigors of scientific inquiry, which it does not. This critical speed method is misused to estimate the initial speed of an outofcontrol vehicle. It is erroneously based on the assumption that the steadystate centripedal acceleration equation can be applied using the radius of the yaw marks as input to the equation. This equation can be stated as follow: In reality, this equation explains the dynamics of a steadystate point mass object and, therefore, should not be applied using tiremarks that are associated with a rotating outofcontrol vehicle. The only time yawmark radius can ever be reasonably used to calculate vehicle speed is when a vehicle is steered right at critical slip and deposits yawmarks with the vehicle (marginally) under control. The general procedure for this faulty method has the following steps:
Because the chordoffset method is intended to measure a circular curve, which for this application is applied to a spiral curve, because the measurement must be made downstream from the lossofcontrol point where the vehicle is now rotating around its center of mass, and because the radius calculated by the chordoffset method is very sensitive to small measurement errors, this method can produce substantial error in estimating lossofcontrol speed. The alternative to calculating speed from the centripedal acceleration equation by measuring the radius of yawmark is, of course, to treat the crash as any other singlevehicle crash where the reconstruction considers the dissipation of energy through skidding, rollover, collision with fixed objects, etc. References
About the Author
Dr. John C. Glennon is a traffic engineer with over 45 years experience. He has over 120 publications. He is the author of the book "Roadway Safety and Tort Liability" and is frequently called to testify both about roadway defects and as a crash reconstructionist. RESUME OF JOHN C. GLENNON Book , books,pavement edge drop expert, pavement edge drop off expert, guardrail expert, work zone safety expert, construction zone safety expert, roadway hydroplning expert, traffic engineering expert, traffic sign expert, traffic signal expert, pavement marking expert, highway safety expert 
