# Impact Response Spectrum for Vessel-Bridge Collision Design

Authors: Tao Fu; Jianpu Hou; Kai Wang; Xiaoqian Ren; Zhixin Zhu; Lingxiao Meng
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

An equivalent static method is employed in calculating the impact response of vessel-bridge collisions, in which the dynamic characteristics of vessel-bridge collisions are ignored. An impact response spectrum for vessel-bridge design is proposed according to the principle of the earthquake response spectrum. The impact force time-histories are obtained through a numerical simulation method using finite element software. The relationship between the displacement amplification factor of the single-degree-of-freedom, period, and damping ratio under the excitation of impact force-time histories is obtained. The regularity of the displacement amplification is summarized. The characteristics of the vessel-bridge collision are given. The three-stage expression and specific application steps of the impact response spectrum for the design of vessel-bridge collision are provided.


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## Body

## 1. Introduction

Many high-input and high-tech bridges have been built for crossing inland rivers, estuaries, and seas. At the same time, the tonnage and the size of shipping have also been increasing. On busy route segments, the collision between vessels and bridges could occur frequently. According to statistics [1], the collapse of bridges being hit by vessels is a frequent occurrence. Vessels colliding with bridges not only cause significant economic loss and casualty but also brings negative political consequences and serious environmental disruption. The collision-related safety of channel bridges is manifested in many aspects, such as bridge damage, vessel damage, loss of life, waterway interruption, environmental pollution, economic loss, and social impact.At present, vessel-bridge collisions are receiving widespread attention [2]. The concerned research mostly focused on impact force estimation [3], finite element numerical simulation [4–6], improvement in the design of bridge anti-collision measures [7–9], and vessel-bridge collision risk assessment [10–12]. For the static method, the estimation formula used in the design of vessel-bridge collision to determine the impact force might give inaccurate quantitative results, especially for elastic-plastic problems associated with complex systems. Most of the estimation formulas are based on specific model tests or special formulas dedicated to a single bridge. Therefore, the impact force obtained by different estimation formulas is not universal. Moreover, there could be controversy in the design of vessel-bridge collision. The dynamic characteristics of vessel-bridge collision are also ignored by this method. The finite element analysis of vessel-bridge collisions which is mostly carried out for typical tonnages and speeds lacks universal adaptability due to the difficulty in the data processing. During the finite element simulation process, numerous complex factors need to be considered, such as the nonlinear constitutive relationship of the structure, the nonlinearity of boundary conditions, the contact type in collision processes, and block damping. As a higher level of numerical simulation is required for users, it is not widely used in engineering design.To deduce the dynamic response of vessel-bridge collisions, the static equivalent method is usually adopted in most specifications [13–15]. Wang and Fan proposed that vessel-bridge collision design should transit from a static to a dynamic design method [16]. Wang et al. established the simplified collision model and forced vibration model, which promoted the application of dynamic analysis methods and finite element analysis in bridge engineering design [17]. [20] established the barge impact spectrum model while considering the dynamic amplification effect under vessel impact [18–21]. Fan et al. proposed the WAS method to understand the maximum response of the structure [22]. Wang and Yu established the vessel impact spectrum model based on 81 impact force time-history samples [23]. Wang et al. proposed the SUM method to superpose modes. The SUM method was transformed into the IFM method, after further simplification. The precision evaluation was conducted to verify the feasibility of the two methods [24]. Consolazio et al. extended the application range of the impact spectrum method. The method was able to solve the nonlinear response based on the equivalent linearization technique while improving the computational efficiency [18]. A mass point collision method and forced vibration method for vessel collision impact calculation were stipulated followed by Specifications for Collision Design of Highway Bridges (JTG/T 3360-02-2020) [25]. Moreover, the model selection and the damping ratio of structural vibration mode for each method were also stipulated.Most vessel-bridge collision accidents only cause local damage to the bridge structure, while leaving the whole bridge structure in an elastic state. During the seismic analysis of the structure, the seismic acceleration response spectrum defines the relationship between maximum acceleration response and the natural vibration period of the single-degree-of-freedom(SDOF) system, which have different damping ratios and natural vibration periods under the action of ground motion time-history curve. Then, the universal adaptive response spectrum was obtained according to a large number of seismic acceleration response spectrum statistics. Referring to the seismic response spectrum and vessel impact force time-history curve, the maximum displacement of the single-degree-of-freedom system was calculated, which has different natural vibration periods under the action of vessel collision force time-history. Thus, the amplification displacement coefficient spectrum can be obtained by measuring the ratio between displacement and static displacement under a maximum impact force. According to the function fitting, the response impact spectrum of vessel-bridge collision with statistical significance can be deduced by establishing the rules of the amplified displacement coefficient spectrum.The dynamic analysis of the single-degree-of-freedom system was carried out by using the time-history curve of impact force. Similar to the definition of the earthquake response spectrum, a group of response spectrums could be obtained, while summarizing them statistically based on the dynamic peak response and static peak response ratio. The response impact spectrum for vessel-bridge collision design is the enveloped response spectrum curve which was fitted out.


## 2. Response Principle of Single-Degree-of-Freedom System under Vessel Collision Force-Time History

Finite element models of eight vessels with different tonnages (500DWT,1000DWT, 3000-1DWT, 3000-2DWT, 10000DWT, 12000DWT, 30000DWT, and 50000DWT) were built. The finite element model of a typical vessel bow is shown in Figure1. The vessels were forced to impact rigid walls at different speeds. A total of 72 impact force time-history curves were obtained and were applied to the single-degree-of-freedom system [26]. The single-degree-of-freedom system is shown in Figure 2. Several impact force time-history curves are plotted as shown in Figure 3. Under the impact force Pt, the dynamic equation of single-degree-of-freedom system is as follows:(1)mUt′′+cUt′+kUt=Pt=Pm⋅P¯t.Figure 1
Finite element models of typical vessel. (a) 1000DWT. (b) 10000DWT.
(a)(b)Figure 2
Single-degree-of-freedom system.Figure 3
The time-history curves of impact forces. (a) The time-history curve of impact force 1. (b) The time-history curve of impact force 2.
(a)(b)(1) can also be expressed as(2)Ut″+2ξωUt′+ω2Ut=Ptm=Pm⋅P¯tm=Pm⋅P¯tm,ξ=c2km,ω=km,where, m, c, and k denote the quality of the particle, system damping, and stiffness of single-degree-of-freedom system, respectively. ξ and ω denote the system damping ratio and circle frequency of single degree of freedom. Pt and U represent the impact force time-history and displacement, respectively. Pm is the maximum value of Pt, and P¯t is equal to Pt/Pm. The dynamic response of equation (2) can be calculated by the Duhamel integral:(3)Ut=1ωm1−ξ2∫0tPτexp−ξt−τsinω1−ξ2t−τdr.The displacement amplification coefficientβ is used to describe the maximum dynamic response of the SDOF system, which is defined as the ratio between the maximum dynamic displacement and the maximum static displacement, and the formula is as follows:(4)β=UmUst,where Ust represents the static displacement response of the system under the action of impact force time-history peak, that is, Ust is equal to Pm/mω2 and Um denotes the maximum displacement response of the system under impact force time-history.According to the definition ofβ (3 and 4), β can be further expressed as follows:(5)β=ω1−ξ2∫0tP¯τexp−ξωt−τsinω1−ξ2t−τdτ,where … denotes absolute maximum value and damping ratio is taken as 0.05.


## 3. Impact Spectrum Properties of Single-Degree-of-Freedom System

The 72 time-history curves of impact force acting on the single-degree-of-freedom system with a damping ratio of 5% were used to obtain the displacement amplification coefficientβ curves with different natural vibration periods. The natural vibration periods range from 0.01s to 10s. The curve consisted of a logarithmic coordinate system based on 10, whereas the transverse coordinate contained the natural vibration period T and the ordinate was the displacement amplification coefficient β. Figure 4 shows the impact spectrum of vessel-bridge collisions classified by tonnages.Figure 4
The impact spectra of bridge-vessel collisions. (a) 1000DWT. (b) 3000DWT. (c) 10000DWT. (d) 12000DWT. (e) 30000DWT. (f) 50000DWT.
(a)(b)(c)(d)(e)(f)The following conclusions were obtained through statistical analysis:(1)
Under the same collision velocity and tonnage of the vessel, the natural vibration period of the structure had a great influence on the displacement amplification coefficientβ. For a relatively long period of the natural vibration, all curves decayed regularly. On the contrary, the time-history spectrum characteristics of impact force had a great influence on the structure. With the increase in the natural vibration period, the impact force with a short duration was equivalent to an impulse and the influence of spectral characteristics on the structure decreased.(2)
The tonnage of the vessel had a great influence on the displacement amplification coefficient under the same impact velocity. Therefore, the influence of vessel tonnage should be considered in curve fitting.(3)
Under the same condition of vessel tonnage, the velocity of vessel collision had a great effect on the displacement amplification coefficient while affecting the shape of the curve minutely.As shown in Figure5, all β curves were plotted in the same coordinate system to obtain the statistical shock spectrum of each vessel tonnage under a 5% damping ratio. It can be seen that the displacement amplification coefficient was between 1 and 2 in most intervals of the natural vibration period. Therefore, the equivalent static method which is used to solve the dynamic equation comes with risks during the practical application. Moreover, the obtained results could be smaller. For a longer natural structural vibration period, the β curve decayed below 1, resulting in smaller outputs and conservative in practice.Figure 5
The impact spectra of bridge-vessel collisions at various tonnages.The time of force acting on the structure was changed from 0.8s to 5.0s. The shock spectrum fitting of vessel collision can be divided into three sections as smooth increased period0−T1, fluctuation period T1−T2, and the attenuation period T2−∞. With the increase in the period, shorter natural vibration periods and centralized displacement amplification coefficient were obtained, indicating the system to be minutely influenced by load duration. During the fluctuation period, the displacement amplification coefficient fluctuated, indicating the beginning of load duration time effect on the reaction system. The irregular change of load spectrum characteristics had a great influence on the structural system. During the attenuation period, the natural vibration increased gradually, whereas the displacement amplification coefficient decreased, indicating that the load duration was the main factor affecting the curve. Thus, the static and dynamic characteristics were comprehensively considered in the drawing process of the displacement amplification coefficient curves, which included the peak effect of collision force-time travel curve, the duration of load, and the irregularity of load spectrum characteristics. The resulting outputs were of practical significance after fitting.When the natural vibration period is less than 0.01, the value ofβ is missing. Based on the shock spectrum image, the value of β was close to 1 for the short natural vibration period of the structure. For T equal to 0.01s, the structure system was absolutely a rigid body when β is equal to 1. Based on the statistical analysis, for T equal to 0 s, β (beta) is equal to 0.


## 4. Vessel-Bridge Collision Impact Spectrum Expression

According to Figure6, the vessel-bridge collision impact spectrum could be divided into three sections based on the trend of the curve. The curve grows steadily when the period of vibration is in a certain range, then the curve begins to fluctuate frequently, and finally, the curve enters the attenuation.Figure 6
Feature point extraction diagrams. (a) Example of feature point 1. (b) Example of feature point 2.
(a)(b)For the damping ratio of 5%, Table1 presents the values of T1 with different tonnages and collision velocities obtained after comparing the displacement magnification coefficient curves at different collision velocities and tonnages. As the values of T1 were relatively concentrated, the mean of T1 is equal 0.07s as the periodic value of the first feature point was considered.Table 1
T1 at different tonnages and impact velocities.
DWT/V50010003000-13000-21000012000300005000010.050.080.080.090.060.070.050.091.50.070.070.090.080.090.090.050.0820.070.120.070.080.080.060.090.072.50.060.070.060.070.080.070.080.0630.050.050.090.050.070.080.060.083.50.080.080.10.070.090.10.090.0840.080.070.080.090.080.080.060.084.50.060.090.080.070.060.080.070.0750.060.050.060.070.080.070.060.09Table2 presents the values of β1 with different tonnages and collision velocities for the damping ratio of 5％. The scatter plot is shown in Figure 7, where the values of β1 are discrete. Multivariate analysis shows that β1 had no obvious relationship with vessel velocity and tonnage, and therefore the values are discrete. Thus, the value of β1 is 1.3398.Table3 presents the values of T2 with different tonnages and collision velocities for the damping ratio of 5％. The scatter plot is shown in Figure 8. Based on statistical analysis, T2 demonstrated a regular change with the increase in vessel velocity and tonnage. Better fitting results were obtained according to the statistics with different tonnage curves at different vessel velocities. The fitted curve formula is(6)T2=0.1709DWT0.2841V0.2850.Table 2
β1 at different tonnages and impact velocities.
DWT/V50010003000-13000-21000012000300005000011.40111.45091.4071.22531.2771.47671.35681.24121.51.70311.46171.46441.24311.34321.41071.45311.332821.91241.42031.16341.32561.41281.42481.15361.1912.51.74291.41781.14581.13751.09821.24151.3761.17731.36661.42461.29721.2351.60111.24081.47221.01263.51.16971.30081.17151.57711.32181.11691.38381.210141.22971.38211.37991.56331.5581.00171.56861.17194.51.40211.46521.1431.38061.62341.1451.13091.024951.20761.41831.12411.47281.27941.15391.8851.2649Table 3
T2 at different tonnages and impact velocities.
DWT/V500100030001000012000300005000010.91.211.371.732.882.984.021.50.831.391.592.233.224.874.6820.611.451.9923.445.074.732.51.361.431.472.963.475.364.4931.141.151.753.313.435.685.253.51.181.952.923.333.385.345.1741.781.962.552.982.925.15.174.51.521.872.723.132.735.064.9351.461.883.563.284.315.295.88Figure 7
Scatter plot ofβ1.Figure 8
Fitting plot ofT2.Table4 presents the values of β2 with different tonnages and collision velocities for the 5% damping ratio. The scatter plot is shown in Figure 9. Multivariate analysis indicated that β2 had no obvious relationship with vessel velocity and tonnage, and therefore the values were discrete. Thus, the value of β2 is 1.2340.Table 4
β2 at different tonnages and impact velocities.
DWT/V50010003000-13000-21000012000300005000010.82481.26651.65261.7191.30091.36811.06291.42781.51.08761.48951.59011.17951.47891.29951.16741.37221.09521.46281.27411.47271.43591.31931.20451.38932.51.01751.55861.10351.46161.44461.17251.28821.231930.99141.54611.10251.38711.48541.09781.27831.24363.51.22941.1091.08211.33721.1751.25291.01411.131141.06641.17421.17731.43521.15131.11741.12811.03294.50.98391.04651.1931.16241.161.05611.09821.093151.03631.15951.21361.31271.0391.0271.09911.2003Figure 9
Scatter plot ofβ2.Table2 presents the values of β1 with different tonnages and collision velocities for the damping ratio of 5％. The scatter plot is shown in Figure 7, where the values of β1 are discrete. Multivariate analysis shows that β1 had no obvious relationship with vessel velocity and tonnage, and therefore the values are discrete. Thus, the value of β1 is 1.3398.According to the above analysis, the values ofβ1 and β2 were discrete. The maximum values of ββmax were obtained in the first or second sections of the curve. Due to the large fluctuation of β in the second section of the curve, the values were unstable. To present a better envelope during the fitting, make βmax equal to β1 and equal to β2 and take βmax as a curve fluctuation segment. The displacement magnification coefficients of βmax for the 72 curves were extracted and are presented in Table 5. The scatter plot of βmax is shown in Figure 10.Table 5
βmax at different tonnages and impact velocities.
DWT/V50010003000-13000-21000012000300005000011.5621.47421.65261.7191.33071.52531.44061.54921.51.77181.48951.59011.57021.47891.57281.45311.470221.91241.46281.73291.65991.41281.76831.51591.38932.51.74291.55861.58081.57841.44411.5691.3761.391631.43651.54611.79791.86241.48541.47681.47221.52433.51.55911.55111.61611.57711.32661.47591.38381.589241.56621.55371.73721.56331.5581.45091.56861.55764.51.72041.46521.56671.52791.62341.50071.55941.588051.42881.41291.62611.50091.45911.48141.8851.7532Figure 10
Scatter plot ofβmax.The analysis indicated that the values ofβmax were concentrated, and hence the mean values βmaxa (equal to 1.5667) were taken as the values of fluctuation section β. The data after the second feature point T2,β2 were used to fit the attenuation curve. The curve approximately conformed to the exponential function shape after data processing by Origin. The fitting formula is(7)β=β2exp−η1T−T2.To obtain the values of the attenuation indexη1, the attenuation sections including the 72 impact force time-history curves were fitted by Origin. Figure 11 shows the scatter plot of η1. The curves of attenuation index η1 with respect to vessel velocity and tonnage were fitted well by the software. The fitting formula of η1 is(8)η1=1.3998DWT−0.19980.4304V−0.6211+0.5820.Figure 11
Fitting plot of attenuation indexη1.For the damping ratio of 5%, the expression of the vessel impact spectrum is(9)β=7.9529T+1,0≤T≤T11.5567,T1≤T≤T20.3227+β2exp−η1T−T2,T≥T2,where(10)T1=0.07,T2=0.1709DWT0.2841V0.2850,β2=1.2340,η1=1.3998DWT−0.19980.4304V−0.6211+0.5820.Appropriate mode combination methods can be used to solve more accurate structural overall reflection. The SRSS method can deduce better results for structures with sparse natural frequencies. The paper considered the CQC method as it is more suitable for the mode combination rule in response spectrum analysis under vessel-bridge collision [27]. Even though the WAS method can better estimate the vessel-bridge collision response, the ideal vessel model based on WAS is quite different from the actual one [20].According to the three expressions of vessel-bridge collision design shock spectrum, its application can be summarized as follows:(1)
According to the peak expression of impact force (Pm is equal to 0.07721DWT0.57V), the peak values of impact force were obtained by using vessel velocity and tonnage. Subsequently, the maximum static displacements were obtained.(2)
The eigenvalues of bridge structures, such as modes and periods, were solved.(3)
The displacement response of the nth-order vibration mode of the bridge structure under the action of the peak impact force was solved.(4)
The CQC method was used to carry out the vibration mode combination to finally obtain the vessel collision response of the bridge structure. It utilized the displacement response of the first n-order array.


## 5. Conclusion

Based on the earthquake response spectrum, this paper calculated the peak displacement of vessel-bridge collision response. The impact force time-history curves under different vessel tonnages and velocities were obtained by numerical simulation of vessel-bridge collision. As they were applied to the single-degree-of-freedom system, the displacement amplification coefficients were obtained by dividing the peak values of dynamic displacement and static displacement. Through statistical analysis and fitting of a large number of displacement magnification coefficients, the statistical law of displacement magnification coefficients was obtained. Moreover, the characteristics of the vessel-bridge collision shock spectrum were summarized. Finally, the three-segment curve expression and application steps of the impact spectrum for vessel-bridge collision design were presented.



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