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    ESTIMATING THE FORCES ACTING DURING A COLLISION
    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 (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)


      where:

        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:

                 (5)


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

                               (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

                                                  (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 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.


    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 at 50 km/h of a hypothetical average car should be designed to resist the following quasistatic loads at the drop arm level (P2):
    TABLE IV
    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 traffic authorities do not know what a collision means...

    References

    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: http://www-nrd.nhtsa.dot.gov/database/nrd-11/veh_db.html
    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.