Follows the theoretical assessment of the forces arising during the impact of a car front end with the rear end of a truck equipped with a rigid underride guard.
The calculations presented here were based on the work of Prof. GEORGE RECHNITZER [1].Centered impact
Assumptions:
 both vehicles are traveling in the same straight trajectory;
 the impact is essentially plastic;
 no lateral displacement of the vehicles occurs after the impact;

after the impact both vehicles remain in contact as a single mass (m_{1} + m_{2}) at speed v_{3.}
From the conservation of momentum:
m_{1} v_{1} + m_{2} v_{2} = (m_{1} + m_{2}) v_{3} (1)v_{3} = (m_{1} v_{1} + m_{2} v_{2})/ (m_{1} + m_{2}) (2)
where:
m_{1} = mass of the truck (kg)
m_{2} = mass of the car (kg)v_{1} = velocity of the truck before impact (m/s)v_{2} = velocity of the car before impact (m/s)v_{3} = velocity of both vehicles after impact (m/s)
Before impact, the kinetic energy of the vehicles is:
E_{0} = 0.5(m_{1} v_{1}^{2} + m_{2} v_{2}^{2}) (3)
And after impact:
E_{1} = 0.5(m_{1} + m_{2})v_{3}^{2} (4)where:
E_{n} = kinetic energy (J)
The energy lost during impact is:
(5)
Taking the closing velocity v_{a} = (v_{1} + v_{2}), we obtain:
(6)
The work done by the average force arising during the collision in crushing the car is:
F.s = D E = E_{1} – E_{2} (7)where:
F = average force acting during the impact (N)
s = car crush (m)and
(8)
Substituting eq. (8) in eq. (6), we obtain the average force acting between the vehicles during the impact:
(9)
Eq. (9) shows that the average force acting during the impact is function of the truck and car masses, closing speed and car crush distance (considering a rigid guard).
In order to verify the influence of the truck mass in the force acting during the impact, let us take three different kinds of cars (“small”, “medium” and “large”). For each category we will consider two models, one produced in Brazil and one produced abroad. Crushing data for Brazilian models were obtained from a Brazilian industry which asked us not to divulge the names of its models. Crushing data for the foreign models were obtained at the NHTSA (National Highway Traffic Safety Administration) web site [2].
Table I presents the data employed to calculate the forces (crush distance for centered impact against rigid flat barrier).
TABLE I
Vehicle 
mass (kg) 
crush distance (m) 
impact velocity (m/s) 
Dahiatsu Charade 
1,015 
0.3861 
13.33
(48 km/h) 
Chevrolet Beretta 
1,442 
0.5105 
Buick Century 
1,749 
0.587 
Brazilian small car 
1,100 
0.511 
13.89
(50 km/h) 
Brazilian medium car 
1,350 
0.497 
Brazilian large car 
1,750 
0.816 
Table II shows the average force acting during impact calculated according eq. (9):
TABLE II
Truck mass (kg) 
3,500 
5,000 
10,000 
20,000 
40,000 
Dahiatsu Charade 
181 kN 
194 kN 
212 kN 
222 kN 
228 kN 
Chevrolet Beretta 
178 kN 
195 kN 
220 kN 
234 kN 
243 kN 
Buick Century 
177 kN 
196 kN 
225 kN 
244 kN 
254 kN 
Brazilian small car 
158 kN 
170 kN 
187 kN 
197 kN 
202 kN 
Brazilian medium car 
189 kN 
206 kN 
231 kN 
245 kN 
253 kN 
Brazilian large car 
138 kN 
153 kN 
176 kN 
190 kN 
198 kN 
Table II presents the average dynamic impact loads acting during the impact. Experimental results obtained by BEERMANN [3] show that the ratio of quasistatic crush loads to dynamic mean axial buckling loads for closedhat section members (similar to front structural members of cars) ranges from 1.30 to 1.56 (average value = 1.40), with no influence of the speed within 30 to 50 km/h. Dividing the values of Table II by 1.40 we obtain the corresponding quasistatic crush loads that can be used for design purposes. These quasistatic loads are presented in Table III.
TABLE III
Truck mass (kg) 
3,500 
5,000 
10,000 
20,000 
40,000 
Dahiatsu Charade 
129 kN 
139 kN 
151 kN 
159 kN 
163 kN 
Chevrolet Beretta 
127 kN 
139 kN 
157 kN 
167 kN 
174 kN 
Buick Century 
126 kN 
140 kN 
161 kN 
174 kN 
181 kN 
Brazilian small car 
113 kN 
121 kN 
134 kN 
141 kN 
144 kN 
Brazilian medium car 
135 kN 
147 kN 
165 kN 
175 kN 
181 kN 
Brazilian large car 
99 kN 
109 kN 
126 kN 
136 kN 
141 kN 
Average 
122 kN 
133 kN 
149 kN 
159 kN 
164 kN 
According to the data presented in Table III, an underride guard able to resist an impact at50 km/h of a hypothetical average car should be designed to resist the following quasistatic loads at the drop arm level (P_{2}):
TABLE IV
Truck mass 
< 5 ton. 
510 ton. 
1020 ton. 
2040 ton. 
Quasistatic load to be applied at the drop arm level (P_{2}) 
133 kN 
149 kN 
159 kN 
164 kN 
Offset impact
Unfortunately we were not able so far to get the crush data necessary to assess the force acting during an offset collision. So the assessment of these force will be based on the experimental results obtained by RECHNITZER et al. [4] e MARIOLANI et al. [5], who designed underride guards according to the quasistatic strength requirements proposed by BEERMANN [3], that is, 150 kN at the drop arm level (P_{2}) and 100 kN at the center of the main beam (P_{3}) and 300 mm from the outermost parts of the vehicle (P_{1}).
Both underride guard were successfully tested at 50 km/h, what allows one to suppose that the ratio of 1.5 between the load at the drop arm level and the load at the center of the beam and near its outermost part is satisfactory.
Based on this ratio (1.5) we suggest that underride guards should satisfy the following quasistatic strength requirements to be able to resist the impact of an AVERAGE car at 50 km/h:
Table V
Truck mass 
< 5 ton. 
510 ton. 
1020 ton. 
2040 ton. 
Strength near the outermost part of the truck (P_{1}) 
90 kN 
100 kN 
105 kN 
110 kN 
Strength at the drop arm level (P_{2}) 
135 kN 
150 kN 
160 kN 
165 kN 
Strength at the center of the main beam (P_{3}) 
90 kN 
100 kN 
105 kN 
110 kN 
Example of “half safety”
Comparing the guard strength suggested above with the test forces required by the new American and the Brazilian (= European) standards, we can easily conclude that the trafficauthorities do not know what a collision means…
References
 RECHNITZER, G. – “Design Principles for Underride Guards and Crash Test Results”. Notes for SAE Heavy Vehicle Underride Protection TOPTEC, April 1516 1997, Palm Springs, USA.
 NHTSA (National Highway Traffic Safety Administration) Vehicle Crash Test Data Base. URL: http://wwwnrd.nhtsa.dot.gov/database/nrd11/veh_db.html
 BEERMANN, H.J. – “Behaviour of Passenger Cars on Impact with Underride Guards”. Int. J. of Vehicle Design, vol. 5, nos. 1/2, pp. 86103, 1984.
 RECHNITZER, G.; SCOTT, G. & MURRAY, N.W. – “The Reduction of Injuries to Car Occupants in Rear End Impacts with Heavy Vehicles”. SAE Paper 933123. 37^{th} Stapp Car Crash Conference Proceedings, San Antonio, Texas, USA, November 810, 1993.
 MARIOLANI, J.R.L.; ARRUDA, A.C.F; SANTOS, P.S.P; MAZARIN, J.C. & STELLUTE, J.C. – “Design and Test of an Articulated Rear Guard Able to Prevent Car Underride”. SAE Paper 973106. VI International Mobility Technology Conference and Exhibit, São Paulo, Brasil, October 2729, 1997
