Practical Method for Estimating Vehicular Impact Force on Reinforced Concrete Parapets for Bridge Infrastructure Design and Management

Abstract

The AASHTO Manual for Assessing Safety Hardware (MASH) replaced the NCHRP Report 350 in 2009, becoming the new standard for evaluating safety hardware devices, including concrete bridge parapets; all new permanent installations of bridge rails on the National Highway System must be compliant with the 2016 MASH requirements after 31 December 2019, as agreed by the FHWA and AASHTO. However, due to the complexity of vehicular impact events, there are several different methods for estimating vehicular impact force on the parapets. They can be grouped into three main categories: theoretical, numerical and measurement methods. This paper presents a practical method based on analytical concepts for providing impact force estimates that can help bridge owners to evaluate the structural capacity of bridge parapets at a fraction of the cost of full-scale crash tests and finite element numerical simulations. This approach was developed based on fundamental dynamic principles and refined dynamic analysis of vehicle rigid-body motions during multi-phased impact events. Principles of impulse and momentum were first applied to determine both linear and angular velocities of a vehicle immediately after the initial impact; then coupled differential equations of motion were derived and solved to describe the vehicle’s plane-motion during the subsequent stage, which includes both translational and rotational movements. The proposed method was shown to be capable of providing reasonably accurate force estimates with significantly less demand for time and effort compared to other complex methods. These estimates can help infrastructure owners to make informed and sustainable decisions for bridge projects, which include selecting the most efficient bridge design alternatives, in a cost-effective and timely manner. Recommendations for future studies were also discussed.

1. Introduction

Bridge parapets are traffic barriers constructed on edges of bridge decks to prevent vehicular and pedestrian traffic from falling off the bridge, especially during collisions. The reinforced concrete parapet system shown in Figure 1 is a popular choice of bridge railing due to its low maintenance need and simple configuration [1]. All railings have important safety roles in the National Highway System (NHS) and must be properly designed. Currently the most widely used and accepted approach for reinforced concrete parapet design is the yield-line method that is published in the AASHTO Load and Resistance Factor Design (LRFD) Bridge Design Specifications, Section 13—Railings [2].

The AASHTO specification presents a set of equations for computing the impact resistance R_w (kN) based on the design information of concrete parapets, including geometry and reinforcing bar arrangement. AASHTO also includes design values of lateral force F_t that represents the vehicular impact force exerted onto the parapet for different types of vehicles and test levels. The design of a reinforced concrete parapet is deemed structurally sufficient for certain impact conditions when R_w (kN) is equal to or greater than the design lateral force F_t. Therefore, design values of impact forces are fundamental elements of any parapet design.

It should be noted that the current Section 13 of AASHTO Specifications [2] has not received a major revision since the LRFD specifications were adopted. The development of content in this section was heavily based on NCHRP Report 350: Recommended Procedures for the Safety Performance Evaluation of Highway Features [3]. In 2009, the Manual for Assessing Safety Hardware (MASH) replaced NCHRP Report 350, becoming the new standard for evaluating safety hardware devices, including concrete bridge parapets. MASH, which was last updated in 2016, incorporated significant changes and additions to the procedures for the safety performance of roadside safety hardware, including new test vehicles that better reflect the changing character of the vehicle fleet that is currently using the highway network [4]. Studies have shown that crash tests performed under MASH have greater severity compared to those done following the NCHRP 350 requirements [5]. In 2019, an agreement was made between AASHTO and the Federal Highway Administration (FHWA) to implement MASH and require all new permanent installations and full replacements of bridge rails on the National Highway System (NHS) in contracts let after 31 December 2019 be compliant with MASH requirements [6]. However, there are numerous concrete parapets that had been already constructed on existing bridges prior to the adaptation of the MASH standards, such as the example shown in Figure 2. Infrastructure owners regularly face the need to make important decisions regarding treatment of these existing structures in bridge rehabilitation projects. Replacing all existing parapets with new MASH-compliant structures is impractical and potentially wasteful. On the other hand, their ability to satisfy MASH impact requirements is largely unknown. Therefore, there is a need to develop effective approaches for evaluating the adequacy of existing or non-standard parapet designs, which can assist bridge managers in choosing practical alternatives for bridge rehabilitation projects, including replacement, retrofitting or minor preservation. As previously mentioned, estimating design impact force is the most critical component of this process.

Due to the complexity of vehicular impacts, there have been several methods that were developed to estimate impact loads, but no single most accurate method for estimating impact forces has been identified by the safety community [7]. As summarized in NCHRP Research Report 1109, the most-used methods for impact force estimation can be grouped into three categories: numerical, theoretical and measurement methods [7]. Numerical methods such as the Finite Element Method (FEM) have been the most popular tool recently thanks to advancement of their technology and the ability to incorporate a great amount of design information into complex analysis models. However, at the same time these methods are highly dependent on the accuracy and algorithms of commercial FEM programs, as well as the accuracy of inputs from the modelers. Alternatively, impact forces can be estimated through measurement methods that include the instrumented barrier test and crash-test inertia estimates. In fact, current impact load values published in the latest version of AASHTO were determined based on extrapolation of instrumented wall test results produced by Noel et al. [8] and Beason [9]. These wall tests were performed in accordance with NCHRP 230 [10], and then obtained impact forces were scaled to satisfy NCHRP Report 350 test conditions [7]. However, these tests are tremendously expensive, so it is cost-prohibitive to perform this method for a wide variety of vehicle impact conditions. Furthermore, force values have to be determined indirectly through acceleration measurements, and thus results are subjected to noise and can also be influenced by the locations of accelerometers or rate transducers on test vehicles.

Besides those two methods, impact forces can also be estimated by theoretical methods using established engineering theories to determine the behavior of impacting vehicles. Even though these analyses are generally performed using simplified impact conditions, they are cost-effective and relatively simple so they can be applied to many different crash scenarios and vehicle configurations with significantly lower demand for time and funding. Certainly, it should still be acknowledged that oversimplification affects the accuracy of the analysis’ outcomes. The first rigorous theoretical estimate was proposed by Olson et al. [11]. In this method, a closed-form solution was derived and presented to determine the average deceleration of a vehicle’s center of mass during impact, which can be used to compute the average lateral impact force based on Newton’s second law of motion. This method was later modified by Hirsch [12] through idealizing the stiffness of both barrier and vehicle as a linear spring and proposing a modification factor to be multiplied by the average lateral force value. These theoretical methods were built upon multiple assumptions, including neglect of rotational accelerations and constant vehicle deceleration throughout impact time. Subsequently, Ritter et al. [13] proposed another theoretical method that attempted to apply impulse and momentum theory. This process assumed a simplified triangular force pulse and used Olson’s method to determine the elapsed time between the first impact and the point at which the vehicle is completely oriented parallel to the wall. Therefore, it also adopted several assumptions of the other methods. In addition, this approach assumed the impact between the vehicle and the concrete parapet was perfectly elastic without any energy loss due to vehicle permanent deformation.

Due to the complex nature of the vehicular impacts on bridge parapets and inherent limitations of the above-mentioned approach, there is a need to develop a method for estimating impact load that is simple and convenient to use but at the same time can incorporate essential physics principles related to impact events. The goal of this study is to develop an improved yet cost-effective theoretical process for producing reasonably accurate estimates of vehicle impact force with fewer resources in comparison to expensive and time-consuming crash tests or refined FEM simulations. The proposed mathematical procedure is also capable of analyzing a wide range of vehicle designs and crash test conditions, so its versatility would be useful for analyzing impact scenarios involving vehicle or parapet configurations that have not been crash-tested for bridge management purposes.

This paper presents an improved analytical method for impact force estimation that was developed based on a more comprehensive application of general dynamic principles in different stages of vehicular impact. The proposed process first took into account both linear and angular impulses and momenta to study the vehicle’s motion during the initial impact when the vehicle’s front bumper hits the concrete parapet. Then, dynamic vector equations of motion were derived to track both translational and rotational movement of the vehicle body between the first collision and the second “tail-slap” strike, which occurs when the entire side of the body hits the wall [7]. The proposed process also incorporated effects of frictional force between the vehicle components and concrete surface and plastic deformation for additional accuracy improvement. A dynamic finite element analysis (FEA) was performed with certain idealized conditions to verify kinematic solutions of this analytical procedure. Lastly, the principle of impulse and momentum was applied again to estimate peak impact force using kinematic results from the plane motion analysis.

2. Methodology

2.1. Review of Theoretical Methods

Since this study is about estimating impact loads using analytical tools, previously performed theoretical approaches are reviewed and summarized in this section. Limitations of these approaches are also identified and discussed.

2.1.1. Olson Method

One of the earliest rational analytical approaches for estimating vehicle impact force was presented in NCHRP Report 86—Tentative Service Requirements for Bridge Rail Systems [11]. In this method, a simple mathematical model was developed to produce design equations based on observation of high-speed film records of crash tests and the basic principles of dynamics.

A closed-form solution was derived for determining the average deceleration of a vehicle’s center of gravity (C.G.) during the time between the first contact at the vehicle’s front corner and the time the vehicle becomes parallel to the barrier wall. In this simple model, at first the travel distance of the truck’s C.G. was simplified to be ${∆s}_{lat}$, shown in Figure 3, which is the lateral distance between two C.G. positions. Then, the elapsed time $∆t$ between the first hit and when the truck becomes parallel to the barrier was computed using ${∆s}_{lat}$ and half of the lateral component of the initial velocity $V_I\mathit{sin}\theta$, because this model assumed the vehicle’s lateral velocity at parallel position is zero. Based on those assumptions, the average lateral acceleration of C.G., $G_{lat,avg}$, was computed based on the simplified average velocity of ${{\displaystyle \frac{1}{2}}V}_I\mathit{sin}\theta$, gravitational acceleration $g$ and elapse time $∆t$:

$$G_{lat,avg}={\displaystyle \frac{{\frac{1}{2}V}_I\mathit{sin}\theta }{g∆t}}={\displaystyle \frac{V_I^2{\mathit{sin}}^2\left(\theta \right)}{2g\left\{AL\mathit{sin}\left(\theta \right)-B\left[1-\mathit{cos}\left(\theta \right)\right]+D\right\}}}$$

(1)

where: $G_{lat,avg}$ is the average vehicle deceleration, $V_I$ is initial velocity at vehicle center-of-gravity, $\theta$ is impact angle, $g$ is gravitational acceleration, $AL, B$ are vehicle dimensions and $D$ is barrier lateral deflection.

Based on this average deceleration, the average lateral impact force $F_{lat, avg}$ was computed using second Newton’s law:

$$F_{lat, avg}=G_{lat,avg}W$$

(2)

with $W=$ weight of vehicle.

As previously mentioned, this model was built upon numerous assumptions, which included neglecting the vehicle’s rotational velocity and acceleration. The effects of the first impulsive initial collision at the front bumper and friction force between concrete surface and the vehicle body were also ignored.

2.1.2. Hirsch Method

Since the Olson method produced the average value of acceleration and consequentially average impact force, Hirsch [12] introduced modifications to this method that can estimate the peak lateral impact force, $F_{lat,peak}$, which is more critical for concrete parapet design. In order to relate the average and maximum lateral impact forces, Hirsch idealized the stiffness of both barrier and vehicle as a linear spring and proposed the modification factor be multiplied by the average lateral force value.

$$F_{lat,peak}={\displaystyle \frac{\pi }{2}}{F}_{lat,avg}$$

(3)

Even though this is an improvement to Olson’s model from a safety standpoint, basic analytical limitations remain unchanged.

2.1.3. Ritter Method (Impulse–Momentum Method)

Ritter et al. [13] proposed a theoretical method that was the first attempt to incorporate vehicle impulse and momentum into the force estimate. This approach assumed a perfectly elastic impact condition, and therefore, the lateral component of separation velocity after front-bumper impact was equal to the lateral approaching velocity, just in opposite directions, as depicted in Figure 4 [13]. Based on this assumption, the lateral impulse during the entire collision event, $J_{lat}$, was estimated to be:

$$J_{lat}=2m{V}_I\mathit{sin}\theta$$

(4)

where $m$ is vehicle mass.

A triangular force pulse was then assumed as shown in Figure 5 [13], and peak lateral force was determined based on $J_{lat}$ as given below:

$$F_{lat,peak}={\displaystyle \frac{J_{lat}}{\frac{1}{2}\left(2∆t\right)}}={\displaystyle \frac{2m_v{V}_I\mathit{sin}\theta }{∆t}}$$

(5)

where $∆t$ is elapsed time between the first impact and the point at which the vehicle is parallel to the wall, and it is the same $∆t$ computed by Olson’s method [7]. It should be noted that this method still relied on Olson’s method to determine the elapsed time $∆t$, and, therefore, it also adopted the simplification of Olson’s method. In addition, this approach assumed the impact between the vehicle and the concrete parapet is perfectly elastic without any consideration for vehicle plastic deformation, and it did not analyze the second whole-body impact either.

2.2. Improved Theoretical Method Using Principles of Dynamics

This paper presents an improved analytical method for estimating vehicle impact force that takes into account rotational vehicle behavior in dynamic plane motion, one of the key factors that were neglected by other theoretical approaches. The proposed method also includes an analytical process that incorporates the frictional force effect between the vehicle body and concrete parapet surface, which is another omitted factor in the existing methods presented above. The impacting vehicle and bridge parapet are still modeled as rigid bodies; however, the effect of vehicle deformation/flexibility (crushing) at the front bumper during the first impact has been accounted for by bringing the coefficient of restitution into the analysis. The use of this coefficient allows the mathematical model to facilitate the partial loss of the original kinetic energy due to permanent deformation of the vehicle assembly and to improve the accuracy of the rigid body analysis.

The proposed analysis presented in this paper was divided into two stages. The first stage analyzed the initial impact when the vehicle first came into contact with the barrier surface at one of its front corners. This event lasted only a very short period, and impulse–momentum principles were employed to determine both the linear and the angular velocities of the vehicle right after the collision. The second stage analyzed the plane motion of the vehicle when it slid along the bridge parapet while concurrently rotating until the whole body hit the parapet wall. Dynamic equations of motions were derived to analytically describe the plane motion of the impacting vehicle and determine vehicle velocities at any point in time, including in the moment the vehicle body came into full contact with the wall. The solutions of these coupled dynamic equations were used to estimate the magnitude of the peak impact force.

2.2.1. Stage 1—Impulse and Momentum Analysis

Impulse and momentum principles were first implemented to determine the linear and angular velocities of the truck’s center of mass immediately after the first collision at the front bumper. These velocities for a specific point in time were subsequently used as initial conditions for the next stage of analysis.

In this study, the impacting vehicle is modeled as a thin rectangular plate PEHK having mass $m$ and dimensions $B, L$ as shown in Figure 6. Point P is the point of contact, point G is the center of mass and point C is the projection of G on the wall. Immediately before hitting the parapet wall, the vehicle body was assumed to move with an initial angular velocity ${\omega }_1=0$ and an initial linear velocity at its center of mass ${\overline{v}}_1={\overline{v}}_0$, where the single bar accent denotes vector form. As shown in Figure 6, vector ${\overline{v}}_0$ is in a direction forming angle $\theta$ with the vertical plane representing the parapet face, and has a constant magnitude of $v_0$. Vectors $\overline{i}$, $\overline{j}$, $\overline{k}$ are unit vectors in the $xyz$ coordinate system, with $\overline{k}$ being a unit vector pointing out of the page.

Let us assume the forces between the impacting vehicle and the parapet were impulsive in nature and were of very short duration. Also let $F_{imp}$ be the total linear impulse at moment of impact, which acts on both the wall and the rectangular plate at point of contact P, and $\beta$ be the angle between the $F_{imp}$ vector and parapet face.

• Coefficient of friction

It should be noted that angle $\beta$ is related to the friction effect between the truck body and the concrete parapet surface and can be determined based on the value of the kinetic coefficient of friction (${\mu }_k)$ between the concrete surface and the metal body of the vehicle front parts. Angle $\beta$ is defined by the following relationship (Equation (6)):

$$\beta =\mathit{arctan}\left({\displaystyle \frac{F_{imp-y}}{F_{imp-x}}}\right)$$

(6)

where $F_{imp-x}$ and $F_{imp-y}$ are components of force impulse $F_{imp}$ that are parallel and normal to the parapet wall surface, respectively, as shown in Figure 7. Since $F_{imp-x}={{\mu }_{k}F}_{imp-y}$ based on definition of ${\mu }_k$, angle $\beta$ representing frictional force effect can also be determined by:

$$\beta =\mathit{arctan}\left({\displaystyle \frac{1}{{\mu }_k}}\right)$$

(7)

• Linear Impulse and Momentum

The principle of linear impulse and momentum [14] provides this relation:

$${\overline{F}}_{imp}=m{{\overline{v}}_1}^{\prime }-m{\overline{v}}_1$$

(8)

where prime refers to velocities after collision. Force impulse and velocity vectors in Equation (8) can be further resolved into components along unit vectors $\overline{i}$ and $\overline{j}$:

$${\overline{F}}_{imp}=F_{imp}\mathit{cos}\beta \left(-\overline{i}\right)+F_{imp}\mathit{sin}\beta \left(\overline{j}\right)$$

(9)

$${{\overline{v}}_1}^{\prime }=v_{Gx}^{\prime }\overline{i}-v_{Gy}^{\prime }\overline{j}$$

(10)

$${\overline{v}}_1={\overline{v}}_0=v_0\mathit{cos}\theta \overline{i}-v_0\mathit{sin}\theta \overline{j}$$

(11)

Therefore, Equation (8) can be rearranged to be:

$$m{{\overline{v}}_1}^{\prime }={\overline{F}}_{imp}+m{\overline{v}}_1=\left(-F_{imp}\mathit{cos}\beta +m{v}_0\mathit{cos}\theta \right)\overline{i}+\left(F_{imp}\mathit{sin}\beta -m{v}_0\mathit{sin}\theta \right)\overline{j}$$

or:

$$v_{Gx}^{\prime }\overline{i}-v_{Gy}^{\prime }\overline{j}=\left(-{\displaystyle \frac{F_{imp}}{m}}\mathit{cos}\beta +v_0\mathit{cos}\theta \right)\overline{i}+\left({\displaystyle \frac{F_{imp}}{m}}\mathit{sin}\beta -v_0\mathit{sin}\theta \right)\overline{j}$$

(12)

• Equating $\overline{i}$ components:

$$v_{Gx}^{\prime }=-{\displaystyle \frac{F_{imp}}{m}}\mathit{cos}\beta +v_0\mathit{cos}\theta$$

(13)

• Equating $\overline{j}$ components:

$$-v_{Gy}^{\prime }={\displaystyle \frac{F_{imp}}{m}}\mathit{sin}\beta -v_0\mathit{sin}\theta$$

(14)

In addition, it was assumed that after the first impact the vehicle entered a plane motion with point P moving along the parapet wall and the whole vehicle body concurrently rotating counterclockwise about point P toward the wall. The relation of relative velocities after the collision between corner P and center of mass G [14] provides this kinematic equation:

$${\overline{v}}_G^{\prime }={\overline{v}}_P^{\prime }+{\overline{v}}_{{\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P$}\right.}}={\overline{v}}_P^{\prime }+ {\overline{\omega }}_1^{\prime }\times {\overline{r}}_{PG}$$

(15)

with ${\overline{v}}_{{\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P$}\right.}}=$ relative velocity of G with respect to point P, ${\overline{v}}_P^{\prime }=$ velocity of point P immediately after collision $=v_{Px}^{\prime }\overline{i}+v_{Py}^{\prime }\overline{j}$, ${\overline{\omega }}_1^{\prime }=$ post-impact angular velocity, and $r_{PG}=\sqrt{{\displaystyle \frac{L^2}{4}}+{\displaystyle \frac{B^2}{4}}}={\displaystyle \frac{\sqrt{L^2+B^2}}{2}}$.

Resolve vectors in Equation (15) into unit vector components:

$$v_{Gx}^{\prime }\overline{i}-v_{Gy}^{\prime }\overline{j}=\left(v_{Px}^{\prime }\overline{i}+v_{Py}^{\prime }\overline{j}\right)+{\omega }_1^{\prime }\overline{k}\times \left(-r_{PG}\mathit{sin}\phi \overline{i}+r_{PG}\mathit{cos}\phi \overline{j}\right)$$

(16)

Rearranging Equation (16) provides:

$$v_{Gx}^{\prime }\overline{i}-v_{Gy}^{\prime }\overline{j}=\left(v_{Px}^{\prime }-{\omega }_1^{\prime }{r}_{PG}\mathit{sin}\phi \right)\overline{i}+\left(-{\omega }_1^{\prime }{r}_{PG}\mathit{cos}\phi +v_{Py}^{\prime }\right)\overline{j}$$

(17)

With $\phi =$ angle between the concrete parapet wall and vector ${\overrightarrow{r}}_{PG}=\theta +\mathit{arctan}{\displaystyle \frac{B}{L}}$. Equating $\overline{i}$ and $\overline{j}$ components of Equation (19) yields these simultaneous equations:

• Equating $\overline{i}$ components:

$$v_{Gx}^{\prime }=v_{Px}^{\prime }-{\omega }_1^{\prime }{r}_{PG}sin\phi$$

(18)

• Equating $\overline{j}$ components:

$$-v_{Gy}^{\prime }=-{\omega }_1^{\prime }{r}_{PG}cos\phi +v_{Py}^{\prime }$$

(19)

From Equations (13), (14), (18) and (19) the following simultaneous equations can be derived:

$$\left\{\begin{array}{cc}-{\displaystyle \frac{F_{imp}}{m}}\mathit{cos}\beta +v_0\mathit{cos}\theta =v_{Px}^{\prime }-{\omega }_1^{\prime }{r}_{PG}sin\phi & \hfill \left(20\right)\\ \left({\displaystyle \frac{F_{imp}}{m}}\mathit{sin}\beta -v_0\mathit{sin}\theta \right)=-{\omega }_1^{\prime }{r}_{PG}\mathit{cos}\phi +v_{Py}^{\prime }& \hfill \left(21\right)\end{array}\right.$$

Experimental crash tests have shown that after the vehicle’s front bumper corner came in contact with the concrete parapet, this corner of the vehicle body started going through a deformation period, and some parts were permanently deformed. As a result, a portion of the original kinetic energy was lost through the plastic deformation of the vehicle components. In order to incorporate the effect of vehicle crushing realistically into this impact analysis, the velocity of point P in y-direction $v_{Py}^{\prime }$ was included in the formulation, even though point P was considered to move along the x-direction without separation from the parapet wall. In other words, $v_{Py}^{\prime }$ was generated as an analytical tool for including effects of vehicle flexibility and material plasticity in this simplified rigid body analysis. This parameter can be determined using the coefficient of restitution as described in the section below.

• Coefficient of Restitution

According to Beer et al. [14], the coefficient of restitution $e$ can be defined as the ratio of magnitudes of impulses corresponding to the period of restitution and to the period of deformation, and can be expressed as:

$$e={\displaystyle \frac{u-v_{Py}^{\prime }}{v_{Py0}-u}}$$

(22)

With $u=$ velocity of point P at the end of the deformation period, $v_{Py}^{\prime }=$ velocity of P in the positive y-direction as previously described and $v_{Py0}=$ velocity of P in y-direction before impact. It should be noted that before the first collision at the front truck corner, point P travels at the same constant velocity as the whole truck body, thus: $v_{Py0}=v_0\mathit{sin}\theta$.

Velocity at end of deformation period $u$ was taken as zero, because the wall was assumed to remain stationary and point P cannot travel beyond the wall surface. Based on Equation (22), velocity $v_{Py}^{\prime }$ can be determined based on $u=0$ and $v_{Py0}$:

$$v_{Py}^{\prime }=-e{v}_{Py0}=-e{v}_0\mathit{sin}\theta$$

(23)

Values of $e$ vary between 0 and 1, with $e=0$ for inelastic impacts and $e=1$ for perfectly elastic impacts [14]. $v_{Py}^{\prime }$ was substituted from Equation (23) into Equation (21) and then rearranged to obtain the following expression for $F_{imp}$:

$$F_{imp}=\left({\displaystyle \frac{m}{\mathit{sin}\beta }}\right)\left[\left(1-e\right)v_0\mathit{sin}\theta -{\omega }_1^{\prime }{r}_{PG}\mathit{cos}\phi \right]$$

(24)

• Angular Impulse and Momentum

Next, by applying the principle of angular impulse and momentum for plane motion of a rigid body, as shown in Figure 7, using center of mass G as the fixed reference point [14] and assuming there were no out-of-plane rotations, i.e., angular momenta about $\overline{i}$ and $\overline{j}$ were zero before and after impact, the following relationship can be obtained:

$$\left[\left(r_1\times m{\overline{v}}_1\right)+I_{Gk}{\overline{\omega }}_1\right]+\left({\overline{r}}_{GF}\times {\overline{F}}_{imp}\right)=\left(r_2\times m{{\overline{v}}_1}^{\prime }\right)+I_{Gk}{\overline{\omega }}_1^{\prime }$$

(25)

with $I_{Gk}$= moment of inertia about $\overline{k}$, which is one of the principal axes of inertia:

$$I_{Gk}={\displaystyle \frac{1}{12}}m\left(B^2+L^2\right)$$

(26)

and $r_1$, $r_2$ are moment arms of $m{\overline{v}}_1$, $m{{\overline{v}}_1}^{\prime }$ about point G, respectively. Thus, $r_1=r_2=0$. The system of impulse must be equivalent to the system of post-impact momenta because initial angular velocity ${\overline{\omega }}_1=0$. Equation (25) becomes the following:

$${\overline{r}}_{GF}\times {\overline{F}}_{imp}=I_{Gk}{\overline{\omega }}_1^{\prime }$$

(27)

Considering $r_{GF}= r_{PG}\mathit{sin}\left[\beta -\phi \right]$ = moment arm for impulse $F_{imp}$, Equation (27) becomes:

$$r_{PG}\mathit{sin}\left(\beta -\phi \right)F_{imp}\overline{k}=I_{Gk}{\omega }_1^{\prime }\overline{k}$$

(28)

From Equations (26)–(28) we can get the following relation:

$$r_{PG}\mathit{sin}\left[\beta -\phi \right]F_{imp}={\displaystyle \frac{1}{12}}m\left(B^2+L^2\right){\omega }_1^{\prime }$$

(29)

Therefore, the angular velocity of vehicle body right after the first collision can be obtained from Equation (29):

$${\omega }_1^{\prime }={\displaystyle \frac{6\mathit{sin}\left(\beta -\phi \right)F_{imp}}{m\sqrt{B^2+L^2}}}$$

(30)

Substitute $F_{imp}$ from Equation (24) to (30) to obtain this expression of ${\omega }_1^{\prime }$:

$${\omega }_1^{\prime }={\displaystyle \frac{{2(1-e)v}_0\mathit{sin}\theta }{\sqrt{B^2+L^2}\left[\frac{\mathit{sin}\beta }{3\mathit{sin}\left(\beta -\phi \right)}+\mathit{cos}\phi \right]}}$$

(31)

Rearranging Equation (30) we have:

$$F_{imp}={\displaystyle \frac{{\omega }_1^{\prime }m\sqrt{B^2+L^2}}{6\mathit{sin}\left(\beta -\phi \right)}}$$

(32)

By substituting $F_{imp}$ from Equation (32) into Equation (20) the linear velocity of point P immediately after initial collision can be obtained as:

$$v_{Px}^{\prime }=v_0\mathit{cos}\theta -{\displaystyle \frac{F_{imp}}{m}}\mathit{cos}\beta +{\omega }_1^{\prime }{r}_{PG}\mathit{sin}\phi$$

(33)

Angular velocity ${\omega }_1^{\prime }$ and linear velocity $v_{Px}^{\prime }$ values, which were computed in this stage for a particular point in time, were used as initial conditions for the next stage of analysis.

2.2.2. Stage 2—Plane Motion Analysis

After the initial impact at the front bumper of a vehicle occurs, it has been observed through crash tests and previous studies that the vehicle’s front corner starts to slide along the parapet while its whole body rotates toward the bridge parapet until it eventually hits the parapet wall in the “tail-slap” strike and the truck becomes parallel to the parapet wall. This section presents the methodology used to analytically track the plane motion behavior of the vehicle between the initial and second impacts.

Let us assume the vehicle can be modeled as a rigid rectangular plate PEHK, again with mass $m$ concentrated at the center of mass G and dimensions $B, L$, as shown in Figure 8. Like the analysis model in Stage 1, point P was considered the point of contact. The main interest of this analysis stage is to study the effects of rotational behavior and inter-surface friction; therefore, the analysis model was idealized as shown in Figure 8. Gravitational force, which is acting in the global Z-direction, was not included in this 2-dimensional model. Point P was constrained to slide with speed $v_P$ only in the direction aligned with the parapet face, while the vehicle body plate was allowed to concurrently rotate about point P with angular speed $\omega$. Thus, there were only 2 independent generalized speeds, $v_P$ and $\omega$ [15]. Positive directions of these speeds are shown on Figure 8. In addition, $\theta$ was defined as the angle between side PE of the plate and parapet surface, and $\gamma =\mathit{arctan}{\displaystyle \frac{B}{L}}$.

In this stage, the vehicle body enters the plane motion after hitting the parapet at the front corner, so initial conditions were imported from previous impulse and momentum analysis as follows:

• Initial angular velocity: ${\omega }_0={\omega }_1^{\prime }$ as determined by Equation (31).
• Initial linear velocity of point P: $v_{P0}=v_{Px}^{\prime }$ as determined by Equation (33).

In order to describe both the translational and rotational movement of the vehicle in this stage, dynamical equations of motion were derived using d’Alembert’s principle and the virtual work principle. Choosing corner P as the reference point, Greenwood [15] provides the Lagrangian form of the D’Alembert equation of motion for a system of $N$ particles:

$${\displaystyle \sum _{i=1}^N}\left[m_i\left({\dot{\overline{v}}}_i+{\ddot{\overline{\rho }}}_{ci}\right)·{\overline{\gamma }}_{ij}+\left({\overline{I}}_i·{\dot{\overline{\omega }}}_i+{\overline{\omega }}_i\times {\overline{I}}_i·{\overline{\omega }}_i+m_i{\overline{\rho }}_{ci}\times {\dot{\overline{v}}}_i\right)·{\overline{\beta }}_{ij}\right]=Q_j$$

(34)

where $m$ is particle mass; ${\overline{\rho }}_c$ is the position vector of the center of mass relative to reference point P; ${\overline{\gamma }}_{ij}$, ${\overline{\beta }}_{ij}$ are linear and angular velocity coefficients corresponding to generalized speeds ($v_P,\omega$), respectively; ${\overline{I}}_i$ is moment of inertia about reference point; $j=$ 1 to total number of constraints; and $Q_j$ are generalized applied forces, which are force effects due to both applied forces and moments. Equation (34) is the vector form of equations of motion, and dot accent is notation for time derivatives.

Since there is only one rigid body in this study, $N=1$ and $i=1$ and can be omitted in subsequent calculations. There are 2 constraints associated with 2 generalized speeds ($v_P,\omega$) in that order; thus, $j=$1 for $v_P$-terms and 2 for $\omega$-terms. To find the other parameters of Equation (34), we chose a local $xyz$ coordinate system with the origin placed at point P, and the $x$ body axis was aligned with the PE side of the vehicle plate. The orientation of the global coordinate system $XYZ$ is displayed in Figure 8, with the X-axis parallel to the parapet face.

As mentioned above, point P’s movement was restricted to only sliding along the parapet face in this analysis, so velocity of point P was defined as:

$${\overline{v}}_P=v_P{\overline{i}}_0$$

(35)

with ${\overline{i}}_0=$ unit vector in the fixed positive direction of ${\overline{v}}_P$. Considering again that $\overline{i}, \overline{j}, \overline{k}$ are unit vectors in local $xyz$ coordinate system, we can determine the following:

$${\overline{i}}_0=\mathit{cos}\theta \overline{i}+\mathit{sin}\theta \overline{j}$$

(36)

Linear velocity coefficients in Equation (34) were defined by Greenwood [15]:

$${\overline{\gamma }}_{11}={\displaystyle \frac{\partial {\overline{v}}_p}{\partial v_p}}={\overline{i}}_0$$

(37)

$${\overline{\gamma }}_{12}={\displaystyle \frac{\partial {\overline{v}}_p}{\partial \omega }}=0$$

(38)

For rotation terms in Equation (34), firstly angular velocity was identified as:

$$\overline{\omega }=\omega \overline{k}$$

(39)

which leads to corresponding angular velocity coefficients [15]:

$${\overline{\beta }}_{11}={\displaystyle \frac{\partial \overline{\omega }}{\partial v_p}}=0$$

(40)

$${\overline{\beta }}_{12}={\displaystyle \frac{\partial \overline{\omega }}{\partial \omega }}=\overline{k}$$

(41)

Next, position vector of center of mass relative to reference point P was determined based on body axes:

$${\overline{\rho }}_c= {\overline{r}}_{PG}=-{\displaystyle \frac{L}{2}}\overline{i}+{\displaystyle \frac{B}{2}}\overline{j}$$

(42)

It is noted from the kinematics of rigid bodies [15]:

$${\ddot{\overline{\rho }}}_c=\dot{\overline{\omega }}\times {\overline{\rho }}_c+\overline{\omega }\times \left(\overline{\omega }\times {\overline{\rho }}_c\right)=\dot{\omega }\overline{k}\times {\overline{\rho }}_c-{\omega }^2{\overline{\rho }}_c$$

(43)

Thus, Equations (42) and (43) give the following:

$${\ddot{\overline{\rho }}}_c=\left(-{\displaystyle \frac{B}{2}}\dot{\omega }+{\displaystyle \frac{L}{2}}{\omega }^2\right)\overline{i}+\left(-{\displaystyle \frac{L}{2}}\dot{\omega }-{\displaystyle \frac{B}{2}}{\omega }^2\right)\overline{j}$$

(44)

Moment of inertia about point P was then determined using Equation (26) and the parallel-axis theorem:

$$I=I_{Gk}+m{\left(\sqrt{{\displaystyle \frac{L^2}{4}}+{\displaystyle \frac{B^2}{4}}}\right)}^2={\displaystyle \frac{m}{12}}\left(L^2+B^2\right)+{\displaystyle \frac{m}{4}}\left(L^2+B^2\right)={\displaystyle \frac{m}{3}}\left(L^2+B^2\right)$$

(45)

$$\therefore \overline{I}·\dot{\overline{\omega }}={\displaystyle \frac{m}{3}}\left(L^2+B^2\right)\dot{\omega }\overline{k}$$

(46)

Based on boundary conditions and constraints we could also obtain the remaining rotational terms [15]:

$$\overline{\omega }\times \overline{I}·\overline{\omega }=\left(I_{yy}-I_{xx}\right){\omega }_x{\omega }_y\overline{k}=0$$

(47)

because there were no out-of-plane rotations in this analysis, and ${\omega }_x$ = angular velocity about x-axis = 0, ${\omega }_y$= angular velocity about y-axis = 0, $I_{yy}$ and $I_{xx}$ are moments of inertia about y and x axes respectively.

$${\overline{\rho }}_c\times {\dot{\overline{v}}}_P=\left(-{\displaystyle \frac{L}{2}}\overline{i}+{\displaystyle \frac{B}{2}}\overline{j}\right)\times \left({\dot{v}}_P{\overline{i}}_0\right)={\displaystyle \frac{1}{2}}{\dot{v}}_P\left(-L\mathit{sin}\theta -\mathit{Bcos}\theta \right)\overline{k}$$

(48)

based on Equations (35), (36) and (42).

The next step was to determine the generalized forces $Q_j$ due to applied forces and moments. There are two generalized force values $Q_1$ and $Q_2$ associated with two generalized speeds $(v_P$, $\omega$) and they can be determined using linear and angular velocity coefficients [15]:

$$\left\{\begin{array}{cc}{Q}_1=\overline{F}·{\overline{\gamma }}_{11}+\overline{M}·{\overline{\beta }}_{11}& \hfill \left(49\right)\\ Q_2=\overline{F}·{\overline{\gamma }}_{12}+\overline{M}·{\overline{\beta }}_{12}& \hfill \left(50\right)\end{array}\right.$$

where $\overline{F}$ and $\overline{M}$ are applied force and moment at point P, respectively. In order to determine $Q_1$ and $Q_2$ it is first observed that there are 2 external forces $R_f$ and $R_n$ acting on the vehicle body at point P, as shown in Figure 8. $R_n$ is reaction force normal to parapet wall and $R_f$ is friction force acting along parapet surface in the opposite direction of the motion. Hence, applied force $\overline{F}$ and moment $\overline{M}$ at point P are:

$$\overline{F}=R_f{\overline{i}}_0+R_n{\overline{j}}_0$$

(51)

$$\overline{M}=0$$

(52)

with ${\overline{j}}_0=$ unit vector in global $XYZ$ coordinate system. Based on Equations (49)–(52) and previously determined velocity coefficients, generalized applied forces were obtained:

• For $v_P$-equation:

  $$Q_1=\left(R_f{\overline{i}}_0+R_n{\overline{j}}_0\right)·{\overline{i}}_0+0=R_f$$

  (53)

• For $\omega$-equation:

  $$Q_2=0+0·\overline{k}=0$$

  (54)

Dynamic equations of motions could then be derived through the next 3 steps: Step 1—$v_P$ equation, Step 2—$\omega$ equation, and Step 3—constraint and kinematic equations.

• Step 1—The $v_P$ equation:

Consider ${\mu }_k$ to be the dynamic coefficient of friction between the truck frontal parts and the concrete parapet surface as described earlier in the paper.

From Equations (34) and (40):

$$m\left({\dot{\overline{v}}}_P+{\ddot{\overline{\rho }}}_c\right)·{\overline{\gamma }}_{11}+0=Q_1$$

(55)

$$\therefore m\left[\left({\dot{v}}_P\right){\overline{i}}_0+\left(-{\displaystyle \frac{B}{2}}\dot{\omega }+{\displaystyle \frac{L}{2}}{\omega }^2\right)\overline{i}+\left(-{\displaystyle \frac{L}{2}}\dot{\omega }-{\displaystyle \frac{B}{2}}{\omega }^2\right)\overline{j}\right]·{\overline{i}}_0=R_f$$

(56)

$$\therefore m{\dot{v}}_P+m{\omega }^2\left( {\displaystyle \frac{L}{2}}\mathit{cos}\theta -{\displaystyle \frac{B}{2}}\mathit{sin}\theta \right)-m\dot{\omega }\left({\displaystyle \frac{L}{2}}\mathit{sin}\theta +{\displaystyle \frac{B}{2}}\mathit{cos}\theta \right)={\mu }_k{R}_n$$

(57)

• Step 2—The $\omega$ equation:

From Equation (34) the following equation was extracted for rotation:

$$0+\left(\overline{I}·\dot{\overline{\omega }}+\overline{\omega }\times \overline{I}·\overline{\omega }+m{\overline{\rho }}_c\times {\dot{\overline{v}}}_P\right)·{\overline{\beta }}_{12}=Q_2$$

(58)

Equations (41), (46), (47) and (48) provide this equation:

$$\left[{\displaystyle \frac{m}{3}}\left(L^2+B^2\right)\dot{\omega }\overline{k}+0+{\displaystyle \frac{1}{2}}{m\dot{v}}_P\left(-L\mathit{sin}\theta -\mathit{Bcos}\theta \right)\overline{k}\right]·\overline{k}=0$$

(59)

$$\therefore {\displaystyle \frac{m}{3}}\left(L^2+B^2\right)\dot{\omega }+{\displaystyle \frac{1}{2}}{m\dot{v}}_P\left(-L\mathit{sin}\theta -\mathit{Bcos}\theta \right)=0$$

(60)

• Step 3—Constraint and Kinematic equations:

Plane motion analysis provides the kinematic equations for linear velocities at center of mass $G\left(X,Y\right):$

• In ${\overline{i}}_0$ direction: Equation (20) provides

$$\dot{X}=v_{GX}=v_P-\omega r_{PG}\mathit{sin}\phi$$

(61)

• In ${\overline{j}}_0$ direction: Equation (21) provides

  $$\dot{Y}={-v}_{GY}=-\omega r_{PG}\mathit{cos}\phi$$

  (62)

  since plastic deformation was not considered between the first and second impacts, so velocity of point P in Y-direction $v_{Py}^{\prime }=0$ for this stage.

where $(X,Y)$ are global coordinates of center of mass G, and:

$$\phi =\theta +\gamma$$

(63)

Lastly, based on direction of rotation angle $\theta$ shown in Figure 8 we can derive the last kinematic relationships:

$$\omega =-\dot{\theta }$$

(64)

$$v_P=\dot{s}$$

(65)

with $s=$ translational displacement of point P in the x-direction. Equations (57), (60), (64) and (65) are first-order differential equations that describe the complete dynamic motion of the vehicle rigid-body system [15]. Solving these 4 differential equations yields the following set of 4 first-derivative equations that can track the vehicle’s movement within the applicable domain and provide pertinent information about its motion at any point in time:

$${\dot{v}}_P={\displaystyle \frac{2{\omega }^2\left(L^2+B^2\right)\left[B\mathit{sin}\theta -\mathit{Lcos}\theta -2{\mu }_k{r}_{PG}\mathit{sin}\left(\theta +\gamma \right)\right]}{4B^2+4L^2-3\left(L\mathit{sin}\theta +\mathit{Bcos}\theta \right)\left[L\mathit{sin}\theta +\mathit{Bcos}\theta -2{\mu }_k{r}_{PG}\mathit{cos}\left(\theta +\gamma \right)\right]}}$$

(66)

$$\dot{\omega }={\displaystyle \frac{3{\omega }^2\left(L\mathit{sin}\theta +\mathit{Bcos}\theta \right)[B\mathit{sin}\theta -\mathit{Lcos}\theta -2{\mu }_k{r}_{PG}\mathit{sin}\left(\theta +\gamma \right)]}{4B^2+4L^2-3\left(L\mathit{sin}\theta +\mathit{Bcos}\theta \right)\left[L\mathit{sin}\theta +\mathit{Bcos}\theta -2{\mu }_k{r}_{PG}\mathit{cos}\left(\theta +\gamma \right)\right]}}$$

(67)

$$\dot{\theta }=-\omega$$

(68)

$$\dot{s}=v_P$$

(69)

Solutions of the above 4 governing equations can be obtained by using commonly available analytical computer programs such as MATLAB^® or Mathematica^®. Based on these solutions, velocities at center of mass G can also be computed:

$$\dot{X}=v_P-\omega r_{PG}\mathit{sin}\left(\theta +\gamma \right)$$

(70)

$$\dot{Y}=-\omega r_{PG}\mathit{cos}\left(\theta +\gamma \right)$$

(71)

It should be noted that the $v_P$_equation, Equation (57), and $\omega$_equation, Equation (60), satisfy dynamic equilibrium in $v_P$ and $\omega$ motion directions, or ${\overline{i}}_0$ and ${\overline{k}}_0$ in $XYZ$ coordinate system, respectively. ${\overline{k}}_0$ is the unit vector pointing out of the page. Equilibrium in the remaining global direction, ${\overline{j}}_0$, was not needed to be considered in the derivation of the above dynamic equations of motion, because as discussed there are only 2 generalized speeds ($v_P$, $\omega$) in this study.

2.2.3. Lateral Impact Force

Currently, the structural design of bridge parapets is performed using the highest impact force at extreme event limit state. Therefore, presented below is the estimation of the maximum impact force for design based on the second impact, which has been found to produce the greatest impact load by numerous previous studies [7,16,17,18,19,20]. This lateral impact force can be determined by applying the principle of linear impulse and momentum again. Lateral force impulse $F_{imp2}$ can be computed based on linear momenta in the direction normal to the wall:

$$F_{imp2}={mv}_{G1}^{\prime }-m{v}_{G1}=0-m{v}_{GY}$$

(72)

with $v_{G1}$ and $v_{G1}^{\prime }$ being lateral velocities of the vehicle at center of mass before and after the impact, respectively; $v_{G1}=v_{GY}=\dot{Y}=$ velocity at G in the Y-direction right before the second strike occurs, while $v_{G1}^{\prime }=0$ because lateral movement of the vehicle is assumed to be completely stopped by the presumably fixed wall. Number “2” is used in the subscript of $F_{imp2}$ to indicate this is the force impulse from the second impact and to distinguish it from $F_{imp}$ that is related to the first impact.

The peak lateral impact force $F_{lat-peak}$ can then be estimated based on impact duration $∆t$ assuming a triangular pulse pattern as shown in Figure 9. This pattern was chosen based on review of available force–time history data from crash tests and FEA simulations. Magnitude of $F_{imp2}$ is equal to the area under the force–time curve, and elapsed time $∆t$ is the duration when the vehicle’s velocity changes from $v_{G1}$ to $v_{G1}^{\prime }$. $F_{lat-peak}$ can be computed using the triangular area formula:

$$F_{lat-peak}={\displaystyle \frac{F_{imp2}}{\frac{1}{2}∆t}}={\displaystyle \frac{2F_{imp2}}{∆t}}$$

(73)

3. Results and Discussions

3.1. Impact Analysis for MASH TL-4 Conditions

For demonstration purposes, the improved analytical method was implemented to estimate impact force produced by the MASH TL-4 crash test conditions for Single-Unit Truck, which is commonly used in bridge design because it normally meets the preferred minimum MASH requirement for parapets on interstate highway bridges [21]. Standard MASH TL-4 impact requirements for Single-Unit Truck are listed in

Table 1. MASH TL-4 (SUT) Impact Requirements.

Analysis Inputs	SUT Impact Values
Impact Speed, $V_I$ (kph)	90
Impact Angle, $\theta$ (degree)	15
Vehicle Weight, $W$(kg)	9979
Vehicle Width, $B$ (mm)	2438
Vehicle Length, $L$ (mm)	10,000

.

3.1.1. Analytical Approach Presented in This Study

Following the process described in Section 2—Methodology, the set of first-order differential Equations (66) through (69) have to be solved first to determine generalized speeds and other pertinent information of the plane motion analysis. However, those differential equations are coupled and finding closed-form solutions for them can be challenging. In this study, the Mathematica^® version 14.1 computation program using the Wolfram^® version 14.1 programming language was employed to simultaneously solve coupled differential equations, produce solutions through “interpolating functions” [22] and display results in the form of graphic plots.

Inputs for the TL-4 Single-Unit Truck (SUT) impact analysis are displayed in Table 1, and output charts produced by Mathematica^® 14.1 are shown in Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. From these charts, vehicle linear velocity in the direction perpendicular to the wall, $v_{Gy}$, and other kinematic variables at the moment immediately prior to the collision can be determined.

At first, the analysis inputs in Table 1 were used to determine the linear and angular velocities immediately after first impact, ${\omega }_1$ and $v_1^{\prime }$, using impulse–momentum principles and relations developed in Stage 1 of the impact analysis that is presented in Section 2.2.1. For this study, the kinetic coefficient of friction was taken as ${\mu }_k=0.47$ for steel on concrete under high normal stress [23]. This value was found to be nearly constant for stress levels between 1 and 468.84 MPa (68,000 psi.). The coefficient of restitution was assigned the value of $e=0.22$ using motor vehicle structural restitution estimates based on data from vehicle-to-fixed-barrier crash tests performed by the National Highway Traffic Safety Administration (NHTSA) [24]. The value of $e$ for this study was selected using an impact velocity equal to the y-component of the initial velocity of the truck, $v_0\mathit{sin}\theta \approx 23.3$ kph. Then, these values were computed and used as initial conditions for Stage 2 analysis: ${\omega }_0={\omega }_1^{\prime }=0.707 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ and $v_{P0}=v_{Px}=\mathrm{25,053} \mathrm{m}\mathrm{m}/\mathrm{s}$ to solve Equation (66) through Equation (69) in Mathematica^® 14.1.

Another needed initial condition variable was dynamic rotation angle ${\theta }_0=0.26 \mathrm{r}\mathrm{a}\mathrm{d}$. After solving the coupled differential equations, the computer program generated results by producing plots of interpolating functions, and some key plots are displayed in Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. The scalar outputs are only valid within the time domain from the onset of plane motion action until the moment the whole vehicle body hits the wall and finishes rotating; therefore, the time when the tail-slap strike occurs needs to be identified first. This could be done by examining the output chart for angle $\theta$ (Figure 10). It can be observed from Figure 8 that when the vehicle stops rotating about corner P and becomes parallel to the wall, impact angle $\theta$ is reduced to zero. Based on Figure 10, the time correlated to $\theta =0$ is $t=0.38 \mathrm{s}.$ That is the time the tail-slap strike occurs, theoretically, and can be used to determine other parameters from the Mathematica^® 14.1 generated charts.

Specifically, from Figure 11 the vehicle linear velocity at center of mass in the direction normal to the wall is $v_{GY}=-3274 \mathrm{m}\mathrm{m}/\mathrm{s}$ at time $t=0.38 \mathrm{s}$, with the negative sign denoting that the vehicle’s center is moving in the opposite direction with the positive Y-direction shown in Figure 8. This velocity is the critical variable for determining impact force using Equation (72), therefore, extra steps were taken to verify this solution and are discussed in the next section.

3.1.2. FEA Verification Using ABAQUS

A dynamic explicit Finite Element Analysis (FEA) was performed in ABAQUS version 2025 to verify the analytical results of rigid body analysis for TL-4 truck impact, especially the velocity at center of mass, $v_{GY}$, right before the second impact occurs. Similar to the above analytical models, the impacting vehicle was represented by a thin steel plate having the same dimensions and boundary conditions of the vehicle rigid body in the analytical model. Those conditions ensure that the lower right corner is restricted to moving along the x-direction only, which is aligned with the parapet’s long side, and there are only two initial speeds: angular velocity about lower right corner of $\omega ={\omega }_1^{\prime }=0.707 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$, and translational velocity of this node in positive x-direction $V_1=\mathrm{25,053} \mathrm{m}\mathrm{m}/\mathrm{s}$. These speeds were set as Predefined Fields at Initial Step, and their magnitudes were obtained from Stage 1 of analytical analysis shown above in Section 2.2.1. Since the main interest was to obtain kinematic plane-motion information related to the vehicle body only, the analysis model was simplified and impact analysis was not included. In addition, the parapet wall was also modeled as a large concrete plate representing the rigid body in the analytical analysis and was fixed in all directions. Basic linear material information for concrete and steel was used in the program. Key details of the FEA models are summarized in

Table 2. Details of finite element models.

FEA Inputs	Vehicle	Parapet
Element Type	Shell	Shell
Element Thickness (mm)	53	53
Number of Elements	120	600
Number of Nodes	147	671
Material Properties: Density (kg/m3) Young’s Modulus (GPa) Poisson’s Ratio	Steel: 7800 200 0.3	Concrete: 2400 24.82 0.18
Interaction Type	General Contact
Interaction Property	Penalty Friction
Friction Coefficient	0.47
Boundary Conditions	Corner in contact with parapet is only free to slide along x-direction and rotate about z-axis	Fixed in all degrees of freedom
Predefined Field	Translational velocity: Vx = 25 m/s Angular velocity about contact point = 0.707 rad/s	None
Analysis Type	Dynamic, Explicit

as follows:

Representative outputs at the time-step right before impact, i.e., when the vehicle body almost came into contact with the parapet wall, are displayed in Figure 15, Figure 16 and Figure 17. Displayed FEA contour maps are for translational velocities at center of mass $v_{GX}$, $v_{GY}$ and rotational velocity $\omega$.

Outcomes of both FEA and analytical approaches are tabulated side by side in

Table 3. Comparison of FEA and analytical results.

Variables	FEA	Analytical
Time, $t$ (s)	0.37	0.383
$v_{GY}$ (mm/s)	3289	3274
$v_{GX}$ (mm/s)	23,305	23,270
$\omega$ (rad/s)	0.66	0.65
$s$ (mm)	8915	9420

for comparison. It was observed that key kinematics results, especially $v_{GY,}$ produced by these two independent analyses are very close. Therefore, it can be concluded that the multi-stage analytical process developed in this study was able to produce reliable results for rigid body motion analysis.

3.2. TL-4 Impact Force Estimate

For the MASH TL-4 case, in order to calculate the dynamic lateral impact force $F_{lat-peak}$ using Equation (73), $F_{imp2}$ and $∆t$ values are needed. While $F_{imp2}$ can be calculated using Equation (72) and $v_{GY}$ from the analytical process, impact duration $∆t$ needs to be approximated based on available time-history results of physical crash tests and refined FEA simulations. Applicable impact force–time history charts showing 50 ms average and SAE-filtered data of MASH TL-4 tests from various experiments are displayed in Figure 18 [16,17,18,19,20]. These experiments were conducted by various research organizations including the Midwest Roadside Safety Facility (MwRSF) at the University of Nebraska-Lincoln, NCHRP Program, and the Texas Transportation Institution (TTI) in cooperation with FHWA and the Texas Department of Transportation. The majority of these studies are documented in NCHRP Research Report 1109, which was recently completed in 2024. This NCHRP project has been the most recent comprehensive study about bridge railing design that was performed to support AASHTO’s efforts in revising Section 13—Railings of the LRFD Bridge Design Specifications [7]. Hence the results of these studies have been used to compute impact force and compared with analytical results as described below.

For this study, $∆t$ was obtained graphically by measuring the distance along the time-axis of the force–time history curves obtained from crash tests or refined FEA simulations between estimated beginning and end points of the second impact force impulse as shown in Figure 18a–f. First, the force–time curve was examined to identify the second sharp spike in force value that represents the “tail-slap” pulse. Then two vertical dashed lines were placed at data points with the lowest force values located just before the sharp increase and after the steep decline of this second peak of impact force, which are strong indicators of the beginning and end of a force impulse, respectively. Then $∆t$ value was taken as the distance between these two vertical lines measured along the time axis. It is recognized that the peak impact force determined by Equation (73) is considerably sensitive to the value of $∆t$, because $F_{lat-peak}$ is directly related to the reciprocal of $∆t$. Therefore, reliable estimates of $∆t$ are essential to determining peak lateral impact force. Based on the available records shown on Figure 18, most of the $∆t$ values estimated by this approach were found to fall within the 0.08 to 0.1 sec range, which is relatively narrow, indicating high consistency of the impact duration information extracted from experimental data in available literature.

Peak lateral impact force $F_{lat-peak}$ values for MASH TL-4 were then determined by the analytical method presented in this paper using test condition data pertaining to various full-scale crash tests and FEA simulations, and results were displayed in

Table 4. TL-4 peak lateral impact force—50 ms average.

Chart No.	Reference	$∆\mathit{t}$ (s)	${\mathit{F}}_{\mathit{l}\mathit{a}\mathit{t}-\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$ (kN)				$\mathbf{Adjustment} \mathbf{Factor}, {\mathit{\phi }}_{\mathit{a}}$ (§)
			Previous Studies		Analytical Method (This Study)
			Crash Test	FEA Simulation	$\mathit{e}=0.22$	$\mathit{e}=0$	$\mathit{e}=0.22$	$\mathit{e}=0$
(a)	MwRSF Research Report No. TRP-03-415-21 [16]	0.1	680		707	898.5	1.04	1.32
(b)	MwRSF Research Report No. TRP-03-403-21 [17]	0.1	595		680	872	1.14	1.46
(c)	NCHRP Project 22-20(02) [18]—Tall Vertical Wall	0.1	(*)	415	609	783	1.47	1.89
(d)	NCHRP Project 22-20(02) [18]—42-in. Barrier	0.08	(*)	351	765	978	2.17	2.78
(e)	TTI-FHWA/TX-12/9-1002-5 Report [19]	0.1		360	614	796	1.7	2.21
(f)	Cao et al.—ASCE J. Bridge Eng. [20]	0.18		375	351	440	0.94	1.17

(*) Crash test was performed for barriers mounted on MSE wall. (^§) Adjustment factor ${\phi }_a$ equal to analytical value divided by crash test or FEA simulation value.

together with impact loads determined by respective experiments using 50 ms moving average data. Two theoretical $F_{lat-peak}$ values were calculated for each referenced case using two values of restitution coefficient, $e=0.22$ and $e=0$. It is worth noting that as mentioned earlier when $e=0$ the impact is perfectly plastic and the original kinetic energy is completely lost due to vehicle deformation without any rebound, which is not realistic based on crash test observation. Therefore, $F_{lat-peak}$ values computed based on $e=0$ are extremely conservative and were included for comparison purposes only.

Recognizing that there are still assumptions and simplifications associated with these theoretical force values, an adjustment factor ${\phi }_a$ was calculated by dividing the theoretical $F_{lat-peak}$ by the crash test or simulated peak force value for each test case and displayed in Table 4. These ${\phi }_a$ factors reflect how much the analytical peak impact force deviates from the value produced by crash tests or FEA simulation. It should be noted that MASH TL-4 criteria cover certain variations of test vehicle configurations, so they contain ranges of allowable test values for vehicle length, weight, etc. Thus, even though all aforementioned crash tests and simulations satisfied requirements for MASH TL-4, they were performed with slightly different testing parameters such as different wheelbase lengths, gross weight, initial impact velocity, etc., resulting in slightly varying test conditions.

In addition, it is noted that besides the 50 msec moving average data used for force estimates, Cao et al. [20] also presents data processed by another filter method. Figure 18f displays force estimates based on both 50 msec average data and force–time histories processed via SAE-filter [25] with a 60-Hz cutoff frequence. It is observed that the SAE method filters extreme spikes of the force–time curve while preserving the dynamic nature of the raw signal in a reasonable manner, and the time history filtered by the 50 msec was excessively smoothed and appears to have different dynamical features than the raw data [20]. Based on information provided by Figure 18f, the 50 msec filter was found to considerably underestimate the actual demand from the MASH truck impact [20]. Specifically, the peak SAE-filter force is greater than the peak 50 msec value by an approximate factor of 1.33.

Therefore, theoretical impact forces were also compared to the experimental values determined by the SAE filter method that can better represent the peak loads as shown in

Table 5. TL-4 Peak lateral impact force—SAE filtered data.

Chart No.	Reference	$∆\mathit{t}$ (s)	${\mathit{F}}_{\mathit{l}\mathit{a}\mathit{t}-\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$ (kN)				$\mathbf{Adjustment} \mathbf{Factor}, {\mathit{\phi }}_{\mathit{a}}$ (§)
			SAE-Filtered Estimates		Analytical Method (This Study)
			Crash Test	FEA Simulation	$\mathit{e}=0.22$	$\mathit{e}=0$	$\mathit{e}=0.22$	$\mathit{e}=0$
(a)	MwRSF Research Report No. TRP-03-415-21 [16]	0.1	904 (‡)		707	898.5	0.78	0.99
(b)	MwRSF Research Report No. TRP-03-403-21 [17]	0.1	791 (‡)		680	872	0.86	1.1
(c)	NCHRP Project 22-20(02) [18]—Tall Vertical Wall	0.1		552 (‡)	609	783	1.1	1.42
(d)	NCHRP Project 22-20(02) [18]—42-in. Barrier	0.08		467 (‡)	765	978	1.64	2.09
(e)	TTI-FHWA/TX-12/9-1002-5 report [19]	0.1		479 (‡)	614	796	1.28	1.66
(f)	Cao et al.—ASCE J. Bridge Eng. [20]	0.16		500	396	494	0.79	0.99

(^§) Adjustment factor ${\phi }_a$ equals to analytical value divided by crash test or FEA simulation value. (^‡) Projected by this study based on ratio of SAE-filtered and 50 ms average data in Cao et al. [20].

. Where the SAE-filtered information is unavailable, this study projected an equivalent SAE-filter force estimate based on the provided 50 msec value, by multiplying the 50 msec peak load by the conversion factor of 1.33 mentioned earlier. It is recognized that this conversion factor was used as a theoretical tool and it is not intended to replace the SAE-filter, therefore, those projected values in Table 5 were included for illustrative purposes only.

It should be noted that the peak lateral force based on SAE-filtered data is also sensitive to $∆t$ values, but since the force–time history curve determined by this method appears to follow raw data more closely, $∆t$ obtained from SAE-filter curve is expected to be more accurate.

In comparison with the simplified impulse and momentum analysis that was previously done by Ritter et al. [13] and other analytical methods, the theoretical procedure developed for calculating force impulse presented in this paper was found to be more accurate thanks to the inclusion of more realistic impact conditions such as friction force and material plasticity. The lateral impact force values estimated for different crash-test and simulation scenarios were also found to be relatively close to the results obtained by expensive instrumentation and advanced FEA simulations. For the case of MASH TL-4, adjustment factor ${\phi }_a$ that is defined as the ratio between theoretical and experimental peak impact force values $F_{lat-peak}$ varies from 0.94 to 2.17 for various testing conditions, using 50 ms moving average data and the more realistic restitution coefficient of 0.22. When no restitution effect is considered, which is extremely conservative, ${\phi }_a$ ranges from 1.32 to 2.78 for those test conditions. The range of ${\phi }_a$ also reflects the somewhat variability of experiment and FEA simulation results presented in the available literature. When the SAE-filter method is factored in, the range of ${\phi }_a$ could be tightened to about 0.78 to 1.64. This data processing method, which is an alternative to 50 ms filter and was commonly used for crashworthiness studies, is known to better preserve the dynamic nature of raw impact signals [20]. In general, due to simplifications of the analytical process, the scatter of results is observed to be wide in some cases, which would be further studied in future research. However, considering how much less resources the analytical method required to produce those results compared to dynamic FEA or full-scale physical crash tests, the accuracy level of the theoretical procedure proposed by this study is considered reasonable. It is expected that after more associated assumptions and simplifications are addressed in future studies, this analytical tool could be further enhanced.

The relatively small deviations from experimental values for the particular case of MASH TL-4 illustrated that the proposed analytical method, which is cost-effective and convenient to use, can be reasonably employed to obtain impact force estimates for certain preliminary engineering analyses or management decisions. Even though the developed process needs to be supplemented with experimental data for certain variables including ${\mu }_k$ and impact duration $∆t$, these pieces of information appear to be readily available and generally consistent between experiments. For instance, $∆t$ period of 0.1 s can be a reasonable estimate for second impact duration in most cases based on several testing records. It should be noted that the approach presented in this paper for obtaining impact duration $∆t$, which is impulsive in nature, is still much more logical and accurate than the over-simplified procedure adopted by other theoretical methods.

Once set up with commonly used computer programs, the same procedure can be applied to analyze a wide range of crash test configurations and test vehicle designs with minimal modifications, and its application is not limited to just analyzing current MASH requirements. This tool will be particularly useful in the case where there is a need to estimate impact loads for a variation of standard MASH conditions, including different vehicle wheelbases or total weight, or other test levels as well. That can be done without performing complex FEA simulations or crash tests. Besides, the vehicle fleet continues to change over the years, and it is a matter of time until safety standards such as MASH need to be updated again with new test vehicle configurations. For example, Battery Electric Vehicle (BEV) has emerged as a fast-growing segment of commonly available, high-sales volume vehicles, and NCHRP Project 22-61 has been initiated to investigate their impact on roadside hardware based on concerns that BEVs have significant differences compared to conventional vehicles [26]. Whenever new vehicle designs need to be analyzed, this simple theoretical method would be a cost-effective tool to gain early understanding or to cross-check and support results of future simulations or instrumentation work with reasonable accuracy.

In addition, this analytical tool for load estimate can be coupled with other analytical methods for estimating impact resistance of parapet designs, such as the plate-theory based approach presented by Chuong and Malla [27], to complete a full theoretical procedure for evaluating impact resistance capacity of parapets on existing highway bridges. This comprehensive analytical process will help bridge owners to effectively determine if there is a need to replace or retrofit certain non-conventional or grandfathered bridge parapet designs without relying on user-influenced FEM or spending great amounts of resources to have crash tests performed. The main intention of developing this analysis is to provide infrastructure owners with a versatile and practical tool to evaluate concrete parapets on their bridges for management purposes using a minimum amount of time and funding, and the results of this analysis demonstrated that the proposed theoretical method is sufficiently accurate and reliable for this purpose.

4. Conclusions

This paper presents a new and improved theoretical method for estimating lateral impact force that was developed based on fundamental dynamic principles to study a vehicle’s behavior during complex impact events, including its translation and rotation movements after the initial collision at the front corner. The proposed process comprises a series of analytical steps that include the derivation of a set of first-order differential equations based on dynamic equations of motion to describe the vehicle’s movement over time up until the tail-slap strike. Friction and plastic deformation effects were also incorporated into the formulation of analysis equations.

The analytical procedure developed in this paper has been used to estimate the impact force produced by the MASH TL-4 Single-Unit Truck. Firstly, kinematic solutions were obtained by the computer program Mathematica^® and verified by an equivalent dynamic FEA performed in ABAQUS. Results from these two independent methods were found to be in good agreement. Next, the principle of impulse and momentum was applied again to estimate the peak lateral impact force using vehicle velocity at center of mass immediately before the second impact occurred and the duration $∆t$ of the force impulse that was graphically determined from relevant crash test and FEA simulation records. Then, impact loads estimated by the proposed theoretical process in this study were compared with those in the experimental data. The deviation of analytical impact load from other estimates was presented in the form of adjustment factors, which were computed as the ratio of impact loads being compared.

In the case of MASH TL-4 impacts, the adjustment factors reflecting differences between theoretical and 50 ms average experimental or simulated results were determined to be between 0.94 and 2.17 based on the use of the frictional coefficient ${\mu }_k=$ 0.47 and the restitution coefficient $e=$ 0.22, which are experiment-based estimates. It is also noticed that the theoretical estimates produced by this study tend to be closer to impact loads determined through crash tests than values from FEA simulations. Considering the relatively narrow ranges of adjustment factors computed for the presented scenarios, it can be concluded that the proposed analytical process produces reasonably accurate theoretical results without the high demand for resources typically needed by other complex methods. In comparison with other theoretical methods that had been developed for estimating vehicle impact force for parapet design and were summarized in NCHRP Report 1109 [7], this study has proposed a more rigorous yet practical physics-based method that can be used in the evaluation of a wide variety of impact configurations.

For infrastructure owners, this cost-effective and practical method is particularly helpful. When preliminary engineering decisions need to be made based on a quick assessment of a bridge parapets’ adequacy for meeting MASH or any potentially new standard requirements, the proposed method can be conveniently implemented to provide high-level structural analysis results without the need to spend extensive effort on costly and complex FEA simulations or physical crash tests.

Potential implementations also include estimating impact forces produced by different crash test configurations and variations of MASH test conditions. When a need to evaluate new test vehicle designs, such as Battery Electric Vehicle, arises, this analytical tool can provide early analysis, or cross-check and support future experimental results with reasonable accuracy. This load estimate process can also be combined with other analytical methods for estimating impact resistance to create a complete theoretical procedure to independently evaluate non-conventional or grandfathered parapet designs. In conclusion, the proposed analytical method of determining impact loads was found to be a sufficiently accurate and practical tool that infrastructure owners can use to efficiently evaluate bridge parapets for project management purposes, without spending a great amount of effort to have expensive and time-consuming FEA simulations or physical crash tests performed.

This improved analytical method can be further enhanced in the future by eliminating assumptions of the rigid body for the vehicle and modeling the vehicle with multiple connected parts (i.e., articulate model of the truck assembly), incorporating flexibility of the concrete barrier and rotation of the truck about the axis parallel to the parapet wall (i.e., tipping over the parapet). While this study incorporated the effect of vehicle deformation to a certain extent by using a restitution coefficient, modeling more realistic effects of the impact process and flexibility due to complex assembly of the vehicle body systems is beyond the scope of the current research. Therefore, those features may also be considered for further improvement of the analytical model in the future. In addition, this paper focuses on estimating the highest impact force produced by the second impact; however, the initial collision at the front bumper could be important for certain purposes outside the structural design of bridge parapets, and thus, it is recommended to be examined in future studies. With the future research enhancements mentioned earlier in this paragraph, the method developed in this study should help in reducing the observed differences between the theoretical impact force estimates and the experimental/refined FEA simulation data.