Simplified Model and Design
Equations for the Collision between Over-high Truck and Bridge Superstructures Yan-sheng Zhang, Tsinghua University, China Xiao Lu, Tsinghua University, China Xin-zheng Lu, Tsinghua University, China Shui-tao He, Tsinghua University, China
Abstract Recently,
the accidents of collisions between over-high trucks and bridge superstructures happen
frequently, which seriously threaten the safety of bridges and city traffic
system. Based on the simulation of collision between over-high truck and bridge
superstructures with nonlinear finite element model, this paper discusses the
main factors that influencing the collision load, and proposes corresponding
simplified calculation model and design equations. The results of the simplified
model and the design equations are compared with the finite element model and
the comparison shows that both the proposed simplified model and design
equations are conservative and accurate enough for engineering applications. Keywords: Over-high truck,
Bridge substructures, collision, Simplified model, Design equations |
1.
Introduction
Recently, the accidents
of collision between over-high truck and bridge superstructure happen
frequently which seriously thread the safety of the bridge and the normal operation
of urban traffic^{ [1-5]}.
According to a report declared by the Department of Transportation of Beijing ^{[1]}, there were
about 50% bridge superstructures had been impacted by over-high trucks in
Beijing City, and the number of bridges damaged due to this reason was more
than 20% of all damaged bridges. Currently, China does not have systematic
research on this problem, and also lacks of design or protection
countermeasures. In order to reduce the
loss caused by the collision between over-high truck and bridge superstructure
and to suggest proper design and prevention for bridge superstructure, the
accurate collision load and its influencing factors must be known. Due to the
difficulties of experiments on the collision between vehicle and bridge, this paper
uses nonlinear finite element (FE) model to study the collision process. And
after a number of parametric discussions, simplified calculation model and
design equations are proposed for the design of collision between over-high
truck and bridge superstructures. |
2.
Collision load and its influencing factors based on
nonlinear FE model
Due to the difficulties
of experimental research on the collision between over-high truck and bridge
superstructure, numerical simulations based on nonlinear FE models are adopted
to study the collision process, which are implemented on the general-purpose FE
software, MSC.MARC. 2.1 FE models
The vehicle FE model will
greatly influence the simulation results. This work firstly uses the
double-axle truck, provided by American National Crash Analysis Center (NCAC)
to study the process of collision, and three more typical Chinese vehicles^{
[6]}, which
are Dongfeng 145 container truck, Dongfeng 3208 tipper truck and the Dongfeng EQ140 cement tank truck, are
modeled to discuss the influence of different truck types. The FE models of the
trucks are shown in Fig.1(a)-(d).
The carriage of the
truck, which is a thin-wall structure, is the mainly impacted zone. So the
constitutive model of Cowper-Symonds model is adopted for the carriage of the
truck which can consider the yielding, hardening and rate effect of steel
material. And the parameters of Cowper-Symonds model are proposed by Liu et al.^{[7]} via steel thin-wall beam impact tests.
The friction coefficients between the wheel and the pavement, or between the
carriage and the bridge, are proposed by Ref [8, 9]. According to the actual
types of bridge superstructure in the collision accidents, three typical urban
bridges, including a simply supported prestressed concrete (PC) T girder
bridge, a simply supported steel box-concrete slab composite bridge and the
Chegongzhuang Bridge, which is a three span PC box girder bridge, were modeled
and analyzed^{[10-13]}.
And the details of the bridge parameters can be found in Ref [6]. In Ref [6], the results of the
proposed FE model are compared with the tests which shows a good agreement and
verifies the rationality of the propose FE model.
2.2 Collision load and its influencing factors
Based on the proposed FE
model, the collision between different trucks, different bridges, and different
speeds are simulated and some of the results are shown in Fig. 3 and 4. More
details can be found in Ref[6].
Generally, because the stiffness, strength and the self-weight of the bridges
are much larger than those of the trucks, the types of the bridge have little
effect on the results. In contrast, the time-history of the collision load is
mainly affected by the vehicle parameters. The collision load of the tipper
truck and the tank truck are much larger than that of the container truck when
they have the similar velocity and self-weigh.
3.
Simplified model
Though proposed FE model
can accurately calculate the collision load, it is too complicated to derive a design equation. So a
simplified model is needed based on the FE model. According to results of
FE models, the following simplifies have little effect on the results^{[6]}: (1) ignoring
the friction forces between carriage and bridge; (2) ignoring the friction forces
between wheel and pavement; (3) ignoring the gravity of the vehicle; (4)
simplifying the bridge to be a rigid wall. Based on the above assumptions,
the simplified model of the collision between over-high truck and bridge
superstructure is established (Fig.5). The results of FE analysis show that the
displacement response of the over-high truck is the combination of the
translations in horizontal and vertical directions and rotation around the rear
axle. So the mass of the truck can be concentrated to the rear axle with corresponding
rotation inertia and rigid arms (Fig.5(a)). The kinetic coordinate system has
three degrees of freedom (x, y, ¦È). And the origin of the coordinate
is located at the initial position of the rear axle (Fig.5(b)). H and L are the vertical and the
horizontal distance between collision region and rear axle. J, m
and V are the rotational inertia, the mass and the initial velocity
of the vehicle, respectively. Because of the horizontal
collision forces F_{x }and the vertical collision forces F_{y}^{ }(Fig.5(a)), there is obvious plastic deformation of the
carriage in the impacted region. Hence, perfect elastic-plastic springs are adopted
to model it (Fig.5(b)), where,
k_{x} and k_{y} are the initial compressive
stiffness of the horizontal and vertical springs (but if in tension, they equal
to 0), F_{px} and F_{py} are the yield forces of the
horizontal and vertical springs. In the same way, the support forces F_{w} (Fig.5(a)) of the pavement to the
truck can also be simulated by a vertical spring and the compressive stiffness
of the wheel is k_{w} (if in tension, it equals to 0).
Fig.5
Simplified model and kinetic coordinate-system According to Fig.5, the
following equations (Eq.1) can be established.
where, x-Hsinq , y + Lsinq are total deformations of the horizontal
and vertical springs, and dp_{x }, dp_{y }are the accumulated
plastic deformation of the horizontal and vertical springs, respectively. The parameters of the
simplified model can be divided into two categories: the first category of
parameters can be easily determined by the types of truck and its loading
condition, including the mass m, the rotational inertia J, the compressive
stiffness of wheel k_{w}, the lengths of rigid arms H and L and the initial velocity V. And they are shown in the Table.1. In contrast, some
other parameters are relatively difficult to determine, such as the compressive
stiffness of impacted region k_{x}£¬k_{y}
and the
yielding forces F_{px}£¬F_{py}. Hence, static compression
numerical experiments are implemented to determine the value of k_{x}£¬k_{y}£¬F_{px} and F_{py}. The details of the static compression numerical experiments
and curves are shown in Ref [6].
Though the collision between the bridge and the truck is a dynamic process,
which has some differences to the static compression, these differences were
not so big. And the experiments and calculations of static compression are much
easier than the dynamic ones. Therefore, the static compression numerical
experiments are adopted to determine the relationship of forces and deformation
of the carriage. And Table 2 shows the values of k_{x}£¬k_{y}£¬F_{px}
and F_{py} of different trucks. Table.1 The value of basic
parameters about different vehicles
Table.2 the values of J
£¬k_{x}£¬k_{y}£¬F_{px}£¬F_{py} in the simplified model
Taking the collision
between composite beam and tank truck with the initial velocity of 60km/h for
example, the histories of collision forces are shown in Fig.6. And the collision
loads of tipper truck and tank truck, which are the control load cases for the
bridge damage, are compared in Table 3, in which the effective collision load F_{m} equals to the average collision load
of 0.1s around the peak load^{[14]}. It can be found that the results of simplified model
agree well with the FE results and they are conservative.
Fig.6
Comparison of collision forces for the collision between tank truck and
composite beam (V=60km/h) Table.3
Comparison of collision load obtained by different methods
The time-history of collision
forces obtained from the simplified model is applied to the bridge superstructure,
and the displacement responses are calculated and compared with the results of
FE model. Fig.7 shows the comparisons in which the maximum error of
displacement response is only 16.30% and the simplified model is conservative. 4.
Design equations of impact load
Although the simplified
model makes a great progress in simplifying the solution of the collision
between over-high truck and bridge superstructure, differential equations (Eq(1))
are still needed to be solved, which is too difficult for engineering design.
Therefore, the following section will propose the design equations which are
suitable and convenient for engineering design.
The collision loads
between the over-high truck and the bridge superstructure in both directions
can be simplified to half-sine time-history curves, just as shown in Fig. 8 and
Eq. 2, where i=1,2 represents the horizontal and vertical directions,
respectively. So if the total impulse I_{i}_{ }and the maximal collision load F_{max,i}, which are the
area under the half-sine time-history curve and the peak point of the half-sine
time-history curve respectively, are obtained, then the whole time-history
curve can be determined.
By referring to the related work on
the collision between ship and bridge piers which is sufficiently studied^{[15-18]}, The
following equations are proposed to calculate I_{i }and F_{max,i} in Eq. (2).
Eq. (3) is based on the momentum
conservation theorem approach, and Eq. (4) is based on
energy conservation theorem approach, in which a_{x}£¬a_{y} , b_{x}£¬b_{y} are dimensionless factors. After a number of computations,
it is found that a_{x}£¬a_{y} , b_{x}£¬b_{y} are controlled by the following three dimensionless factors:
the resistance of rotation J/[m(L^{2}+H^{2})], the ratio of stiffness k_{y}/k_{x} and the ratio of arm of force L/H. By changing the value of J/[m(L^{2}+H^{2})]¡¢k_{y}/k_{x}¡¢L/H in the actual range of engineering application, the corresponding values of a_{x},a_{y },b_{x} and b_{y} can be derived from the simplified model without considering the plasticity of carriage. Typical values of a_{x},a_{y },b_{x} and b_{y} are listed in the Table.4 and other value of these parameters can be obtained by linear interpolation. Taking into account of
the plasticity of carriage, a magnification factor of 1.35 should multiply to
the value of a_{y} in Table 4, while the other parameters a_{x}£¬b_{x}
and b_{y} can still using the value in Table 4. In order to verify the
accuracy of the design equations, the results calculated by design equations are
compared with the results of simplified model and FE model. Typical comparison is
shown in Fig. 8, which is the collision time-history between composite beam and
tank truck with an initial velocity of 60km/h. Table 3 shows the comparison of
the impact impulse I and the impact forces F_{m} for tipper truck and tank truck
with different speeds. Due to the complicated plastic behavior in the carriage,
the maximal error of impact forces F_{m} is larger than that of impact impulse I. But generally, most
errors are smaller than 15% and the design equations are conservative. |
Table.4 The value of
parameters in Eq (3~4) without considering the plasticity of carriage
5.
Conclusion
This paper simulates the
collision process between the over-high truck and the bridge superstructure.
Parameters that influencing the collision process are discussed and simplified
calculation model and design equations are proposed to calculate the collision
load. By comparing to the FE results, the simplified model and the design
equation are accurate enough and conservative. AcknowledgementThe authors are grateful for the financial support
received from National Science Foundation of China (No: 50808106)). References£º [1]
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