The Influence of Longitudinal Oil Tank Bodies on the Vehicle’s Rollover Mechanics

Abstract

In order to determine the influence of longitudinal oil tank bodies with different cross-sectional shapes on the vehicle's center of gravity position and the lateral overturning moment generated during cornering and braking, this paper conducts oscillation simulation calculations and analysis on longitudinal oil tanks with three different cross-sectional shapes. The optimal cross-section of the longitudinal tank is ultimately determined, providing a basis for the stability calculation of oil tankers.

1 Introduction

When oil tankers are in motion, the overall center of gravity of the vehicle changes in response to the sloshing of the liquid within the tank. Different tank shapes, as well as variations in the size and shape of the free liquid surface within the tank, result in varying degrees of liquid motion. Consequently, different cross-sectional tank shapes can impact the safety and stability of the vehicle differently during cornering and emergency braking processes [1].

2 Vehicle Rollover State Conditions

During the vehicle's turning and braking process, the critical indicators that influence the risk of vehicle rollover include lateral acceleration $a_{{}}$, roll angle $\theta$, and two-wheel load transfer rate LTR. Among these indicators, the LTR exhibits the highest level of reliability in determining whether a vehicle will roll over or not [2].

Denoting the bearing force of the inner wheels as $F_{L}$ and the bearing force of the outer wheels as $F_{R}$, we then derive the following equation [3]:

$$ LTR = \frac{{F_{R} - F_{L} }}{{F_{R} + F_{L} }} $$

(1)

• (1) When LTR = 0, there is no transfer of load between the two wheels. According to the equilibrium condition $G_{{\text{y}}} = G\cos (\theta ) = F_{L} + F_{R}$, it can be determined that $F_{L} = F_{R} = 1/2G{\text{cos}}(\theta )$, which means the forces on both wheels are uniformly distributed and stable throughout the vehicle's turning process. Consequently, the occurrence of a rollover accident is improbable under these conditions [4].

• (2) When 0 < LTR < 1, load transfer occurs with $F_{R} \ge F_{L} \ge 0$. During turning and braking at this stage, the inner wheel experiences less force than the outer wheel. If the vehicle is unloaded, it is in a semi-unstable state and is less likely to roll over. However, if the vehicle is partially loaded with liquid, the transient impact force $F_{1}$ generated by the liquid's movement may create a lateral torque. As this lateral impact force increases, LTR also increases. As LTR approaches 1, the vehicle becomes increasingly close to an unstable state, which can lead to the initiation of a rollover [5].

• (3) When LTR = 1, the inner wheel is in a suspended state with no force applied to it $F_{L}$ = 0, while the outer wheel bears the entire load as $F_{R} = G\cos (\theta )$. At this stage, the vehicle is already in an unstable state, but the actual occurrence of a rollover is determined by the impact force $F_{1}$ generated by the movement of the liquid inside the tank. If a rollover occurs, it will happen with the longitudinal axis of the outer wheel $m - m^{\prime}$ serving as the rollover axis.

As mentioned above, the occurrence of vehicle rollover involves the vehicle initially being in a state between semi-unstable and unstable, the load transfer rate LTR greater than zero (LTR > 0), as well as the lateral torque generated by the lateral forces produced by the liquid's movement at that moment [6].

3 Vehicle Rollover Mechanics Calculation

Based on the previously discussed conditions for rollover, it is evident that when LTR = 1, the vehicle is in an unstable state, with the inner wheel's load-bearing capacity as $F_{L} = 0$. The occurrence of rollover is then determined by the lateral impact forces generated by the movement of the liquid. The critical point at which a refueling truck experiences a rollover is when the clockwise moment $M_{{\text{z}}}$ equals the counter-clockwise moment $M_{{ - {\text{z}}}}$. To prevent a rollover, the condition of $M_{{\text{z}}} < M_{{ - {\text{z}}}}$ should be satisfied. According to the equilibrium conditions, it can be concluded as:

$$ \left\{ {\begin{array}{*{20}l} {(G_{0y} - F_{0y} ) \cdot \frac{L}{2} + (G_{1y} - F_{1y} ) \cdot L_{1} = M_{{\text{ - z}}} } \hfill \\ {(G_{0x} - F_{0x} ) \cdot H + (G_{1x} - F_{1x} ) \cdot H_{1} = M_{{\text{z}}} } \hfill \\ {M_{{\text{z}}} < M_{{ - {\text{z}}}} } \hfill \\ \end{array} } \right. $$

(2)

The rollover moment can be calculated as:

$$ M_{{F_{1} }} = F_{1y} L_{1} - F_{1x} H_{1} < (G_{0y} - F_{0y} ) \cdot \frac{L}{2} - (G_{0x} - F_{0x} ) \cdot H - G_{1x} H_{1} + G_{1y} L_{1} $$

(3)

where:

$$ \left\{ {\begin{array}{*{20}c} {G_{0y} = G_{0} {\text{cos}}(\theta )} \\ {G_{0x} = G_{0} {\text{sin}}(\theta )} \\ \end{array} } \right.\left\{ {\begin{array}{*{20}c} {G_{1y} = G_{1} {\text{cos}}(\theta )} \\ {G_{1x} = G_{1} {\text{sin}}(\theta )} \\ \end{array} } \right.\left\{ {\begin{array}{*{20}c} {F_{0y} = F_{0} {\text{cos}}(\theta )} \\ {F_{0x} = F_{0} {\text{sin}}(\theta )} \\ \end{array} } \right.\left\{ {\begin{array}{*{20}c} {F_{1y} = F_{1} {\text{cos}}(\theta )} \\ {F_{1x} = F_{1} {\text{sin}}(\theta )} \\ \end{array} } \right. $$

4 FLUENT Numerical Computation

The fluid computational model is imported into the ANSYS FLUENT computational module after being processed. Within the FLUENT computational environment, a User-Defined Function (UDF) program is invoked to input the initial conditions for simulating the emergency turning and braking process. This includes the initialization of the free liquid surface. The simulation calculations are then performed for two sets of modular refueling truck storage units with three different cross-sectional shapes, operating at load rates of 40, 55, 70, 85, and 95%.

Under the same filling rate condition, the liquid surface heights for three tank shapes are concluded in the Table 1:

Table 1 The height of the liquid level of the three tanks at the same filling rate

5 Simulation Results for Longitudinal Tank at Different Filling Rates

5.1 Simulation of Liquid Sloshing in a 40% Filling Rate

Under a 40% filling rate condition, the variation in impact forces exerted by the liquid on the tank walls for the three tank shapes, identified as Type ${\text{a}}\prime$, Type b′, and Type c′, is illustrated in Fig. 1. The trends in the alteration of impact forces in the lateral, longitudinal, and vertical directions exhibit a general similarity. The vertical direction experiences the most significant impact force, particularly at the tank bottom. The presence of transverse and longitudinal baffles inside the tank influences the impact forces differently, with the lateral impact force affecting the tank sidewall and the longitudinal impact force influencing the tank's front head to varying degrees. The maximum values of impact forces experienced in each direction are summarized in Table 2. When the maximum lateral impact force is generated, Fig. 2 depicts the fluid state, wherein the intensity of liquid sloshing is notably strong, and significant fluctuations in impact force are observed. Despite the limited lateral force, its effect on the stability of the tank truck is relatively minor [7].

Fig. 1

Tank body force change curve under 40% filling rate within 0–3 s

Table 2 Maximum impact value and tank bottom pressure value of 40% filling rate

Fig. 2

Flow pattern and pressure distribution at the maximum pressure of 40% filling rate

During the process of emergency turning and braking, the liquid exerts significant stress impact in the outward direction on the outer bottom of the tank as it turns and creates a negative pressure zone at the rear inside of the tank. The distribution of pressure on the tank's bottom is illustrated in Fig. 2. When examining the distribution of the maximum pressure area, the tank with a Type a′ tank has the largest maximum pressure area, followed by the Type b′ tank, with the Type C tank showing the lowest maximum pressure on the tank bottom. Looking at the maximum pressure values on the tank bottom, the tank with a Type A tank has the largest, followed by the Type B tank, and the Type C tank has the smallest maximum pressure on the tank bottom. Specific values for the maximum pressure on the tank bottom can be found in Table 2.

5.2 Sloshing Conditions at a 55% Filling Rate

Under a 55% filling rate condition, the changes in impact forces exerted by the liquid on the tank walls over time for Type a′, Type b′, and Type c′, are shown in Fig. 3. As depicted, the trends in the variation of impact forces in the lateral, longitudinal, and vertical directions are generally consistent. The impact force has the most significant effect in the vertical direction on the tank's bottom. Due to the presence of transverse and longitudinal baffles inside the tank, the impact of lateral force on the tank sidewall and the impact of longitudinal force on the tank's front head varied in their degree of influence. The maximum values of impact forces experienced in each direction are summarized in Table 2. When the maximum lateral impact force is generated, a significant accumulation of liquid at the front section of the tank can be observed (Fig. 4). Throughout this process, the intensity of liquid sloshing is notably strong, with significant fluctuations in impact force, while the lateral force is of moderate magnitude with insignificant effect on the stability of the tank [8].

Fig. 3

Tank body force change curve under 55% filling rate within 0–3 s

Fig. 4

Flow pattern and pressure distribution diagram at a maximum pressure of 55% filling rate

During emergency turning and braking, the liquid normally exerts substantial stress impact in the outward direction on the outer bottom of the tank as it turns, creating a zone of negative pressure at the rear inside of the tank. The distribution of pressure on the tank's bottom is depicted in Fig. 4. Based on the distribution of the maximum pressure area, the tank with a Type b′ tank has the largest maximum pressure area, followed by the Type a′ tank, with the Type c′ tank demonstrating the smallest pressure area. In terms of the maximum pressure values on the tank bottom, the Type c′ tank experiences the highest maximum pressure, followed by the Type b′ and the Type a′ tank, which has the lowest maximum pressure on the tank bottom. The maximum pressures on the tank bottom are summarized in Table 3.

Table 3 Maximum impact value and tank bottom pressure value of 55% filling rate

5.3 Sloshing Conditions at a 70% Filling Rate

Under a 70% filling rate condition, the changes in impact forces exerted by the liquid on the tank walls over time for the three tank shapes, designated as Type a′, Type b′, and Type c′, are depicted in Fig. 5. The graphs illustrate the general consistency in the trends of variation in impact forces in the lateral, longitudinal, and vertical directions. The most significant impact force occurs in the vertical direction on the tank's bottom. Due to the presence of transverse and longitudinal baffles inside the tank, the impact of lateral force on the tank sidewall and the impact of longitudinal force on the tank's front head are varied. The maximum values of impact forces experienced in each direction are summarized in Table 4. When the maximum lateral impact force is generated, a significant accumulation of liquid can be observed at the front section of the tank. Throughout this process, the intensity of liquid sloshing is notably strong, with significant fluctuations in impact force. The lateral force is relatively large, which has a significant impact on the stability of the tank truck [9].

Fig. 5

Tank body force change curve under 70% filling rate within 0–3 s

Table 4 Maximum impact value and tank bottom pressure value of 70% filling rate

During emergency turning and braking, the liquid normally exerts substantial stress impact in the outward direction on the outer bottom of the tank as it turns, creating a negative pressure zone at the rear inside of the tank. The distribution of pressure on the tank's bottom is depicted in Fig. 6. Based on the distribution of the maximum pressure area, the tank with a Type b′ cross-section had the largest maximum pressure area, followed by the Type a′ tank, with the Type c′ tank having the smallest pressure area. In terms of the maximum pressure values on the tank bottom, the Type c′ tank experienced the highest maximum pressure, followed by the Type a′, and the lowest maximum pressure on the tank bottom is observed in the Type b′ tank. The maximum pressures on the tank bottom are summarized in Table 4.

Fig. 6

Flow pattern and pressure distribution diagram at maximum pressure of 70% filling rate

Fig. 7

Tank body force change curve under 85% filling rate within 0–3 s

5.4 Sloshing Conditions at an 85% Filling Rate

Under an 85% filling rate condition, the changes in impact forces exerted by the liquid on the tank walls over time for Type a′, Type b′, and Type c′ tanks are depicted in Fig. 5. The graphs illustrate the general consistency in the trends of variation in impact forces in the lateral, longitudinal, and vertical directions. The most significant impact force occurs in the vertical direction on the tank's bottom. Due to the presence of transverse and longitudinal baffles inside the tank, the impact of lateral force on the tank sidewall and the impact of longitudinal force on the tank's front head are varied. The maximum values of impact forces experienced in each direction are summarized in Table 5. When the maximum lateral impact force is generated, a significant accumulation of liquid can be observed at the front section of the tank. Throughout this process, the intensity of liquid sloshing is notably strong, with significant fluctuations in impact force. The lateral force is relatively large, which significantly impacts the tank truck's stability [10].

Table 5 Maximum impact value and tank bottom pressure value of 85% filling rate

During emergency turning and braking, the liquid normally exerts substantial stress impact in the outward direction on the outer bottom of the tank as it turns, creating a negative pressure zone at the rear inside of the tank. The distribution of pressure on the tank's bottom is depicted in Fig. 8. Based on the maximum pressure area distribution, the tank with a Type b′ cross-section has the largest maximum pressure area, followed by the Type c′ tank, with the Type a′ tank having the smallest pressure area. In terms of the maximum pressure values on the tank bottom, the Type a′ tank experiences the highest maximum pressure, followed by the Type c′, and the lowest maximum pressure on the tank bottom is observed in the Type b′ tank. The maximum pressures on the tank bottom are summarized in Table 5.

Fig. 8

Flow pattern and pressure distribution diagram maximum pressure of 85% filling rate

5.5 Sloshing Conditions at a 95% Filling Rate

Under a 95% filling rate condition, the changes in impact forces exerted by the liquid on the tank walls over time for the Type A, Type B, and Type C tanks are presented in Fig. 9. The graphs illustrate that the trends in the variation of impact forces in the lateral, longitudinal, and vertical directions are entirely consistent. The most significant impact force occurs in the vertical direction on the tank's bottom. Due to the presence of transverse and longitudinal baffles inside the tank, the impact of lateral force on the tank sidewall and the impact of longitudinal force on the tank's front head are varied. The maximum values of impact forces experienced in each direction are summarized in Table 6. The fluid state at the time that the maximum lateral impact force is generated is illustrated in Fig. 10, in which a substantial accumulation of liquid at the front section of the tank can be observed. Throughout this process, the intensity of liquid sloshing is notably strong, and there are significant fluctuations in impact force. The lateral force is exceptionally significant, and it substantially impacts the tank truck's stability.

Fig. 9

Tank body force change curve under 95% filling rate within 0–3 s

Table 6 Maximum impact value and tank bottom pressure value of 95% filling rate

Fig. 10

Flow pattern and pressure distribution diagram at maximum pressure of 95% filling rate

During emergency turning and braking, the liquid normally exerts substantial stress impact in the outward direction on the outer bottom of the tank as it turns, creating a negative pressure zone at the rear inside of the tank. The distribution of pressure on the tank's bottom is depicted in Fig. 10. Based on the distribution of the maximum pressure area, the tank with a Type c′ cross-section had the largest maximum pressure area, followed by the Type a′ tank, with the Type b′ tank having the smallest pressure area. In terms of the maximum pressure values on the tank bottom, the Type a′ tank experienced the highest maximum pressure, followed by the Type c′, and the lowest maximum pressure on the tank bottom is observed in the Type b′ tank. The maximum pressures on the tank bottom are summarized in Table 6.

6 Comparative Analysis of Longitudinal Tank Sloshing Conditions at Different Filling Rates

6.1 Comparative Analysis of Lateral Impacts for Three Tank Shapes at Different Filling Rates

As illustrated in Fig. 11, the magnitude of lateral impact forces on the tank truck varies with changes in tank cross-section and filling rates. At a 40% filling rate, the lateral impact forces generated by the tanks with Type a′, Type b′, and Type c′ cross-sections are as follows: 22,115 N, 23,842 N, and 25,798 N, respectively. At a 55% filling rate, the lateral impact forces are as follows: 41,342 N, 41,918 N, and 44,163 N, respectively. At a 70% filling rate, the lateral impact forces are as follows: 66,500 N, 79,740 N, and 94,477 N, respectively. At an 85% filling rate, the lateral impact forces are as follows: 97,298 N, 108,000 N, and 108,294 N, respectively. Finally, at a 95% filling rate, the lateral impact forces are as follows: 152,831 N, 161,803 N, and 184,284 N, respectively. The maximum lateral impact force experienced by the tanks is positively correlated with the filling rate; a higher filling rate results in greater lateral impact forces, while a lower filling rate leads to smaller lateral impact forces. Specifically, when the filling rate is less than 85%, the Type c′ tank exhibits the highest lateral impact force, while when the filling rate exceeds 85%, the Type b′ tank experiences the highest lateral impact force.

Fig. 11

Comparison of the lateral impact force changes of three tanks under different filling rates

The flow pattern distribution diagrams reveal that the lateral force fluctuations resulting from tank sloshing exhibit similar patterns under different filling rate conditions. In the case of the modular tank truck, starting from the initiation of emergency braking during a turn and extending to approximately 2.00 s, the lateral impact force initially increases, then gradually decreases until it disappears. When the maximum lateral impact force occurs, the motion of the liquid inside the tank becomes highly turbulent. The liquid surges from the lower part to the upper part and from the rear to the front in the direction of the turn's outer side. A substantial amount of liquid accumulates on the outer side of the front half in the direction of the turn. Subsequently, the liquid undergoes a rolling motion, briefly lifting off and descending, releasing potential energy. With multiple iterations of rolling motion, the lateral impact force gradually subsides. As the tanker truck decelerates due to braking, the sloshing of the oil gradually ceases. Therefore, the maximum lateral impact force is closely related to the intensity of sloshing and the filling rate. The timing of the occurrence of the maximum lateral impact force is roughly consistent, as indicated in Table 7.

Table 7 Maximum lateral force appearance time at different filling rates

6.2 Comparison of Lateral Impact Forces Over Time for Different Cross-Sectional Tank Shapes at Equal Filling Rates

A comparative analysis of three distinct tank shapes, denoted as Type a′, Type b′, and Type c′, reveals variations in lateral impact forces at different filling rate conditions (40, 55, 70, 85, and 95%) over 0–3 s. As depicted in Fig. 12, the Type a′ tank units generally exhibit slightly lower impact forces than the Type b′ tank units. In comparison, the Type c′ tank units experience slightly higher forces than Type b′. From a lateral force perspective, the modular refueling truck comprised of Type a′ tanks exhibits the least detrimental influence on the tank structure during the turning and braking, followed by Type b′ tank, and Type c′ tank has the most pronounced impact. Examining the variational tendency, it can be inferred that the timing of the maximum lateral impact force is correlated with the filling rate. When operating under lower filling rate conditions, the maximum lateral impact force tends to manifest at a later point in time, resulting in an extended average duration of lateral impact forces. Conversely, in situations characterized by higher filling rates, the maximum lateral impact force can be reached earlier, leading to a diminished average duration of lateral impact forces.

Fig. 12

Comparison of the lateral impact force of three tanks with time at the same filling rate

6.3 The Variations in the Maximum Lateral Rollover Moments for Different Tank Cross-Sectional

The variations in the maximum lateral rollover moments for different tank cross-sectional shapes at various filling rates can be calculated using the torque formulas. When the fluid motion continues until around 0.65 s under 40% filling rate conditions, the lateral rollover moments for the three tank shapes reach approximately $1.86 \times 10^{4}$ N/m, $2.17 \times 10^{4}$ N/m, and $2.35 \times 10^{4}$ N/m, respectively. In the case of a 55% filling rate, the lateral rollover moments at approximately 0.65 s are $3.60 \times 10^{4}$ N/m, $4.02 \times 10^{4}$ N/m, and $4.24 \times 10^{4}$ N/m for the respective tank shapes. At a 70% filling rate, the lateral rollover moments that occur at around 0.42 s are $6.06 \times 10^{4}$ N/m, $7.97 \times 10^{4}$ N/m, and $9.46 \times 10^{4}$ N/m for the different tank shapes. For an 85% filling rate, the lateral rollover moments at approximately 0.37 s are $9.26 \times 10^{4}$ N/m, $11.2 \times 10^{4}$ N/m, and $11.2 \times 10^{4}$ N/m, corresponding to the various tank shapes. Lastly, at a 95% filling rate, the lateral rollover moments occur around 0.21 s and are $14.93 \times 10^{4}$ N/m, $21.35 \times 10^{4}$ N/m, and $19.54 \times 10^{4}$ N/m for the different tank shapes.

From the curves in Fig. 13 that depict the variation in maximum lateral rollover moments of tanks with Type a′, Type b′ and Type c′ cross-sections at different filling rates, it is evident that as the filling rate increases, the maximum lateral forces for all three tank shapes continuously rise. When the filling rate is below approximately 55%, the maximum lateral rollover moments of the three tank shapes are nearly equal. In the filling rate range of 55% to 85%, tanks with the Type b′ cross-section exhibit larger maximum lateral rollover moments than those with the Type c′ cross-section. For filling rates exceeding 85%, tanks with the Type c′ cross-section surpass those with the Type b′ cross-section in terms of maximum lateral rollover moments. Notably, for filling rates above 55%, tanks with the Type a′ cross-section consistently demonstrate smaller maximum lateral rollover moments compared to the Type b′ and Type c′ tanks, with the disparity becoming more pronounced as the filling rate increases.

Fig. 13

Variation of the maximum lateral moment of the three tanks with the filling rate

7 Conclusion

This study introduces the simulations of oil tank trucks composed of tanks with three different cross-sectional shapes, investigating the sloshing behaviors of the internal oil in the tanks under different filling rates (40%, 55%, 70%, 85%, and 95%) during emergency braking while making sharp turns. The research focuses on factors that directly impact the tanks, including the impact forces, tank structural stress, and the lateral rollover moments during the turning and braking process. The following conclusions are drawn:

• (1) During sharp turns with braking, the oil in the tank generates impact forces in the lateral, longitudinal, and vertical directions. Impact forces are most significant in the vertical direction, primarily affecting the tank's bottom. The presence of transverse and longitudinal baffles within the tank effectively reduces the impact forces on the outer side of the tank's walls and the frontal section of the tank. The sloshing of the liquid exerts a significant stress impact on the outer side of the tank's front bottom, leading to noticeable stress concentration and the creation of a negative pressure zone behind the inner side of the tank.

• (2) For tanks with a longitudinal orientation, as the filling rate increases, the intensity of liquid sloshing weakens gradually, while the magnitude of the impact force steadily increases. This leads to an enhanced impact on the stability of the tanker truck. At a filling rate of approximately 85%, the liquid sloshing in the tanker truck becomes intense, accompanied by a substantial lateral force.

• (3) Comparing the rollover moments of tanker trucks with longitudinal tanks of different cross-sectional shapes (Type a′, Type b′ and Type c′), it is found that the tanker truck with the Type a′ cross-section experiences smaller maximum rollover moments than Type b′ and Type c′. This difference becomes more pronounced as the filling rate increases.