Estimating Forces During Collision

Estimating Forces
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


  • 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 (m1 + m2) at speed v3.

    From the conservation of momentum:
    m1 v1 + m2 v2 = (m1 + m2) v3                                 (1)v3 = (m1 v1 + m2 v2)/ (m1 + m2)                             (2)

    m1 = mass of the truck (kg)

    m2 = mass of the car (kg)v1 = velocity of the truck before impact (m/s)v2 = velocity of the car before impact (m/s)v3 = velocity of both vehicles after impact (m/s)

    Before impact, the kinetic energy of the vehicles is:

    E0 = 0.5(m1 v12 + m2 v22)                                         (3)

    And after impact:

    E1 = 0.5(m1 + m2)v32                                                 (4)where:

    En = kinetic energy (J)

    The energy lost during impact is:


    Taking the closing velocity va = (v1 + v2), we obtain:

    Image2                       (6)

    The work done by the average force arising during the collision in crushing the car is:

    F.s = D E = E1 – E2                                                   (7)where:

    F = average force acting during the impact (N)
    s = car crush (m)and


    Substituting eq. (8) in eq. (6), we obtain the average force acting between the vehicles during the impact:


    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).



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):



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 closed-hat 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.



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 (P2):



Truck mass < 5 ton. 5-10 ton. 10-20 ton. 20-40 ton.
Quasistatic load to be applied at the drop arm level (P2) 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 (P2) and 100 kN at the center of the main beam (P3) and 300 mm from the outermost parts of the vehicle (P1).

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. 5-10 ton. 10-20 ton. 20-40 ton.
Strength near the outermost part of the truck (P1) 90 kN 100 kN 105 kN 110 kN
Strength at the drop arm level (P2) 135 kN 150 kN 160 kN 165 kN
Strength at the center of the main beam (P3) 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…


  1. RECHNITZER, G. – “Design Principles for Underride Guards and Crash Test Results”. Notes for SAE Heavy Vehicle Underride Protection TOPTEC, April 15-16 1997, Palm Springs, USA.
  2. NHTSA (National Highway Traffic Safety Administration) Vehicle Crash Test Data Base. URL:
  3. BEERMANN, H.J. – “Behaviour of Passenger Cars on Impact with Underride Guards”. Int. J. of Vehicle Design, vol. 5, nos. 1/2, pp. 86-103, 1984.
  4. RECHNITZER, G.; SCOTT, G. & MURRAY, N.W. – “The Reduction of Injuries to Car Occupants in Rear End Impacts with Heavy Vehicles”. SAE Paper 933123. 37th Stapp Car Crash Conference Proceedings, San Antonio, Texas, USA, November 8-10, 1993.
  5. 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 27-29, 1997