Novel Crashworthy Device for Pier Protection from Barge Impact

Barge impact is a potential hazard for bridge piers located in navigation waterways. Protective structures of different types, for example, dolphin structures, artificial islands, and guiding structures, have been widely used in bridge designs against barge impact. However, such structures often imply high cost and suffer from difficulties in installation as well as maintenance challenges. ,is paper aims to devise and investigate a new type of crashworthy device which is comprised of vertically supported impact cap connected to the bridge pier using a series of steel beams in a frame-type arrangement. ,is sacrificial steel structure is designed to form plastic hinges for energy dissipation whilst limiting the force transmitted to the protected pier. ,e dynamic analysis of the proposed crashworthy device subjected to barge impact is conducted using a simplified impact model previously developed by the authors. ,e parametric studies in this paper show that the proposed device has a large energy dissipation capacity and that the magnitude of impact force transmitted to the bridge pier can be dramatically reduced. In addition, an optimization model is proposed in this paper to achieve the cost-optimized design of the crashworthy device for a given impact scenario with constraints as per the prescribed design requirements.


Introduction
Bridge piers located in navigation waterways are often threatened by vessel impact due to the increment of vessel transportation volume.It was pointed out by Manen and Frandsen [1] and Larsen [2] previously that at least one major vessel-bridge collision accident of serious consequences occurred each year on average in the past.Barge collisions upon bridge structures were also frequently reported.Such collisions can often lead to catastrophic consequences including human casualties and economic losses; thus substantial investigations regarding the quantification of barge impact loading and dynamic structural responses have been conducted in recent years [3][4][5][6][7][8].
Different protection measures are being employed to protect bridge piers from vessel impact loading or reduce the damage of bridge piers during impact.As one of these protection measures, independent protective structures such as dolphin structures are frequently used in bridge designs.Such structures were, for example, adopted and installed for long-span bridges such as the Rosario-Victoria Bridge in Argentina [9], the Rhine Bridge in Kehl, Germany [10], and the American Sunshine Bridge [11].e advantage of such independent protective structures is that they can absorb high impact energy and protect bridge piers from direct contact with the vessels.However, such independent protective structures require high cost and suffer from durability problems and challenges regarding installation and maintenance.In addition, reconstruction of such independent protective structures after being damaged by vessel impact is often expensive, if at all possible.Other protective structures, for example, artificial islands [12] or guiding structures [13], are also frequently used.However, these structures suffer from problems as those mentioned above.
e problems related to the above mentioned protective structures have led to the investigation of bridge protections from impact by strengthening the bridge piers themselves, for example, with carbon fibre-reinforced polymers (CFRPs) [6,14].Such strengthening techniques can improve the pier resistance; that is, the pier undergoes less damage during impact.However, such technique cannot reduce the maximum impact force; therefore, it is effective for pier protection but not for barge protection [6].In addition, the studies by Sha and Hao [6] indicate that the effectiveness of the CFRP strengthening technique is very limited regarding the maximum pier displacement.
is paper aims to devise a novel crashworthy device which is comprised of a supported or oated cap connected to the pier using steel beams arranged in a frame-type manner.During a high-energy barge impact, many plastic hinges form in the proposed device, enabling it to absorb large amounts of impact energy through plastic deformations.Such crashworthy device is easy to install, maintain, and restore after an impact event.rough the choice of con guration, plastic moments, and postyield hardening, the maximum force transmitted to the main pier can be designed to not exceed an allowable force that is acceptable by the main pier.
To investigate the e ectiveness of the proposed crashworthy device for barge impact, the simpli ed impact model previously developed by the authors is employed in this paper to conduct dynamic analysis of the device subjected to barge impact.e simpli ed impact model transforms the highly nonlinear full barge impact model (FBIM) into a coupled multi-degree-of-freedom model (CMM).e accuracy and e ciency of CMM were thoroughly assessed for di erent impact scenarios [8]. is paper employs such simpli ed impact model to investigate the energy dissipation capacity of the proposed device and the magnitude of impact force transmitted to the bridge pier by the steel beams during impact for di erent structural con gurations.
e parametric studies in this paper indicate that the proposed device has a large energy dissipation capacity for barge impact and can signi cantly reduce the maximum impact force transmitted to the bridge pier during impact.To achieve cost-optimized design of such device for a given impact scenario, a mathematical optimization model is proposed in this

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Advances in Civil Engineering paper with constraints as per the prescribed design requirements.Examples presented in this paper show that the optimum con guration of the device can be obtained for di erent impact scenarios using the proposed optimization model.

Configuration of the Device
During a barge impact, a portion of the impact energy is transformed into the residual kinetic energy of the barge while the rest of the impact energy is dissipated through the plastic deformations of the barge and the impacted structure.In order to protect both the bridge pier and the barge, it is necessary to devise a crashworthy device which is easy to install, maintain and restore and can absorb large portions of the impact energy through plastic deformations.In this way, the energy absorbed by the barge and the pier during impact would be low, and consequently, both the barge and the pier can remain in the linear range, that is, elastic (undamaged), through the limitation of the force transmitted.e con guration of the proposed crashworthy device is depicted in Figure 1 for a sample bridge pylon foundation to be protected.e cap structure is designed to collect the local impact forces and is vertically supported for its self-weight by a structure that does not provide a signi cant lateral restraint, for example, a exible pile system or a oating structure.
e cap is connected to the pylon foundation using a series of steel beams of I cross section arranged in multiple frames.
e individual steel beam legs are connected bending sti , for example, through welds, such that a force applied to the cap generates bending moments in the steel structure.Several planes of beam units can be installed to provide the energy dissipation capacity and the elastic and plastic deformation behavior desired.In Figure 1, three planes are shown, which can be braced against each other to avoid out-of-plane stability failure.
For simpli cation purposes, several assumptions are adopted herein: (1) the lateral resistance of the cap's supporting structure is ignored, (2) the pylon foundation is assumed to be rigid, and (3) the cap is assumed to be rigid and is modeled using a lumped mass.Based on these assumptions, the structure shown in Figure 1 can be simpli ed into the cap steel beam structure, as shown in Figure 2. e cap can move freely in the horizontal direction whilst its vertical movement is constrained.Advances in Civil Engineering

Overview of CMM
e CMM previously developed by the authors simpli es the complex nite-element barge model into a nonlinear massspring model (MSM) and models the pier column using discrete masses and bre beam elements [8], as shown in Figure 3, where m b is the lumped barge mass and v b is the impact velocity.
As per previous studies [3,4,7], the force-deformation curve of the barge bow during impact (curve 1) generally includes a linear increase of impact force until the force peak is followed by an abrupt decrease when the barge bow yields, as shown in Figure 4, where u b is the barge bow deformation and F is the impact force.en the impact force roughly reaches a plateau until the unloading stage.
e shape of curve 1 can be regarded as the superposition of two curves-a triangular curve (curve 2) and a bilinear curve (curve 3), as shown in Figure 4. Two nonlinear springs which act in parallel are thus introduced to represent the barge bow resistance.e force-deformation curves of the two nonlinear springs are taken to be bilinear and triangular, respectively, as shown in Figure 5, where u 1 and u 2 are the yielding deformations of two springs, respectively; F sy is the yielding force of the rst spring; F sp is the peak force of the second spring; and x is the spring deformation.By coupling MSM with the column at the impact position, the CMM is developed to predict the dynamic barge impact process e ciently, as shown in Figure 3.
e MSM parameters are determined by an optimization model which minimizes the integration error of impact force time histories determined by CMM and FBIM, respectively.e quality of CMM regarding the prediction of impact force time history and dynamic pier responses was assessed in large detail for di erent impact scenarios in [8] by using FBIM as the benchmark model.e validated CMM is thus used for the studies herein.

Simplified Impact Model Based on CMM
In this section, the simpli ed impact model is developed based on CMM for dynamic analysis of the proposed crashworthy device subjected to barge impact.As shown in Figure 6, the steel beams are modeled using discrete masses and bre beam elements.e MSM is coupled with the cap which is expected to contact with the barge when impact occurs.It is assumed in this study that the beam elements undergo no shear deformations or torsional deformations.e stress-strain curve of the beam steel is bilinear in this study.
e MATLAB code was written with the bre method for solving the numerical model illustrated in Figure 6. e code was previously veri ed by detailed nite-element simulation results from LS-DYNA [8].Geometric nonlinearity of beam elements is analyzed using the corotational approach for problems of large displacements and small strains.e basic idea is to decompose the motion of the element into rigid body part and pure deformational part.A local coordinate system, which moves and rotates with the element's overall rigid body motion, is de ned, and the deformational part is measured under this local coordinate system [15]., curve 1; , curve 2; , curve 3.

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Advances in Civil Engineering width.e cap mass (m c ) is taken to be 100.0ton.e total length of the beam units in one plane (l csb ) is taken to be 15.0 m. e number of planes of beam units (N pl ) is taken to be one in this study.e information of the prespeci ed parameters regarding the device is tabulated in Table 1.ree parameters, that is, beam cross-section dimension, yielding strength of beam steel (f bs y ), and number of beam units in one plane (N bu ), are considered for the parametric studies herein.For comparison purposes, the following baseline simulation is conducted: beam cross-section dimension as follows: h bi 0.75 m, w bi 0.50 m, t fi 0.05 m, and t wi 0.03 m; beam steel yielding strength of 350.0 MPa; and beam unit number of two in one plane.e parameter MC bi , which means the ratio of the studied beam cross-section dimensional parameters, that is, h bi , w bi , t fi , and t wi , to the respective beam cross-section dimensional parameters used for the baseline simulation, is denoted herein.

Beam Cross-Section Dimension.
ree beam crosssection dimensions corresponding to MC bi of 0.8, 1.0, and 1.2, respectively, are considered herein.e energy absorbed by the device (W csb diss ) during impact corresponding to each beam cross-section dimension is shown in Figure 7, where the total impact energy (W total ) and the ratio (r csb diss ) of energy   , 2); , W total ; , the ratio of energy absorbed by the device after impact to the total impact energy.
Advances in Civil Engineering absorbed by the device after impact to the total impact energy are also presented.Figure 7 shows that a large portion of the impact energy is absorbed by the device and that the increase of beam cross-section dimension reduces the energy absorbed by the device due to the decrease of structure deformation caused by the increase of structure sti ness, as shown in Figure 8, where the maximum bending moment diagram of the structure during impact and the structure de ection after impact corresponding to each beam crosssection dimension are presented.
e moment-curvature relationship of each beam cross section and the moment-rotation relationship of a single steel beam corresponding to each beam cross-section dimension are shown in Figure 9, where c is the curvature of beam cross section and θ b is the relative rotation angle of two boundary sections of a single steel beam.Figures 8 and 9 show that the plastic hinges which form during impact are located at the upper part of the structure, that is, the horizontal beams at the top and the upper part of the vertical beams, where the maximum bending moment exceeds the corresponding yielding moment of beam cross section.e formation of plastic hinges enables the device to absorb a large portion of energy during impact, as shown in Figure 7.
e lower part of the structure, that is, the horizontal beams at the bottom and the lower part of the vertical beams, undergoes only elastic deformations, as Figure 8 shows., MC bi 0.8; , MC bi 1.0; , MC bi 1.2.
6 Advances in Civil Engineering e time histories of impact force on the pier with the device corresponding to each beam cross-section dimension and without the device, respectively, together with the reduction ratio (r f ) of maximum impact force when the device is used, are shown in Figure 10, which shows that the maximum impact force can be signi cantly reduced when the device is used.e increase of beam cross section would increase the magnitude of impact force due to the increase of cross-section bres.

Yielding Strength of Beam Steel.
ree steel yielding strengths of 250.0 MPa, 350.0 MPa, and 450.0 MPa, respectively, are considered herein.e energy absorbed by the device (W csb diss ) during impact corresponding to each steel yielding strength is shown in Figure 11, where the total impact energy (W total ) and the ratio (r csb diss ) of energy absorbed by the device after impact to the total impact energy are also presented.Figure 11 shows that the increase of steel yielding strength reduces the energy absorbed by the device during impact.is is because the structure resistance increases with the increase of steel yielding strength and consequently the structure undergoes smaller deformations, as shown in Figure 12.
e time histories of impact force on the bridge pier with the device corresponding to each steel yielding strength and without the device, respectively, together with the reduction ratio of maximum impact force when the device is used, are shown in Figure 13, which shows that the increase of steel yielding strength would increase the magnitude of impact force on the pier due to the increase of structure resistance.

Number of Beam Units.
e devices of one beam unit, two beam units, and three beam units, respectively, are considered herein.e energy absorbed by the device (W csb diss ) during impact corresponding to each beam unit number is shown in Figure 14, where the total impact energy (W total ) and the ratio (r csb diss ) of energy absorbed by the device after impact to the total impact energy are also presented.Figure 14 shows that the increase of beam unit number slightly reduces the energy absorbed by the device.is is because the structure becomes sti er when more beam units are used, as indicated in Figures 15 and 16 which show that the structure undergoes smaller de ections and smaller deformations during impact when more beam unit number is used.
e time histories of impact force on the pier with the device corresponding to each beam unit number and without the device, respectively, together with the reduction ratio (r f ) of maximum impact force when the device is used, are shown in Figure 17, which shows that the magnitude of impact force on the pier increases when beam unit number increases due to the increase of structure sti ness.

Cost-Optimized Design of the Device
e studies in the previous section have shown the great potentiality of the proposed crashworthy device for pier protection from barge impact due to its large energy dissipation capacity during impact.In order to achieve cost-optimized Figure 10: Impact force time histories on the bridge pier for the whole impact process (a), for the rst 0.10 s of impact process (b) corresponding to di erent beam cross-section dimensions, and the reduction ratio of maximum impact force versus MC bi (c).
Advances in Civil Engineering design of such device for a given impact scenario, a mathematical optimization model is proposed in this section with constraints as per the prescribed design requirements.

Mathematical Optimization Model.
For a given barge mass (m b ) and impact velocity (v b ), when the number of planes of beam units (N pl ), the yielding strength of beam steel (f bs y ), and the maximum allowable impact force F allow max on the bridge pier are speci ed, the device can be designed in such a way that the design requirements are satis ed and the required cost is minimized by using minimum amount of steel.e design of the device can thus be transformed into an optimization problem where the number of beam units in one plane (N bu ), the four dimensional parameters of the I cross section, that is, h bi , w bi , t fi , and t wi , and the length of each single steel beam l sb are optimized.For simpli cation purposes, the four dimensional parameters of the I cross section are assumed to satisfy the relationships as tabulated in Table 2. e optimization model and the corresponding constraints are described as follows: minimize: , W total ; , the ratio of energy absorbed by the device after impact to the total impact energy.

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Advances in Civil Engineering subject to: where m sb is the total mass of steel beams (ton), A I is the area of the I cross section (m 2 ), ρ bs is the mass density of beam steel (ton/m 3 ), N l bu and N u bu are the lower bound and upper bound of the number of beam units in one plane (-), respectively, h l bi and h u bi are the lower bound and upper bound of the depth of the I cross section (m), respectively, l l sb and l u sb are the lower bound and upper bound of beam length (m), respectively, F max is the maximum impact force on the bridge pier during impact (MN), D max cap is the maximum cap displacement during impact (m), and D allow max is the maximum allowable cap displacement (m).e value of D allow max is taken to be N bu l sb to avoid the contact of adjacent vertical beams during impact.Figure 14: Time histories of energy absorbed by the device (a) corresponding to di erent beam unit numbers and the ratio of energy absorbed by the device after impact to the total impact energy, respectively, versus beam unit number in one plane N bu (b)., W csb diss (N bu 1); , W csb diss (N bu 2); , W csb diss (N bu 3); , W total ; , the ratio of energy absorbed by the device after impact to the total impact energy.
Advances in Civil Engineering ree variables, that is, N bu , h bi , and l sb , are included in the optimization process.
e sequential quadratic programming (SQP) [16] is used for solving the proposed constrained optimization problem.

Application Example.
In this section, the optimum con gurations of the devices corresponding to several different impact scenarios are obtained using the proposed optimization model.e combinations of three barge , N bu 1; , N bu 2; , N bu 3.
x (m) x (m) Figure 16: Maximum bending moment diagrams of the structures during impact and de ections of the structures after impact corresponding to di erent beam unit numbers (unit: MNm)., original shape of the device; , deformed shape of the device.(a) N bu 1; (b) N bu 2; (c) N bu 3.

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Advances in Civil Engineering masses, that is, 181.4 ton (empty barge), 952.6 ton (half loaded barge), and 1723.7 ton (fully loaded barge), and three impact velocities, that is, 1.0 m/s, 3.0 m/s, and 5.0 m/s, are considered herein.Each impact scenario is labeled as IS ij , where i denotes the number index of barge mass varying from 1 to 3 and j denotes the number index of impact velocity varying from 1 to 3, as tabulated in Table 3. e cap surface which contacts with the barge is at and is 3.0 m in width.e cap mass (m c ) is taken to be 100.0ton.e number of planes of beam units (N pl ) is taken to be two.e yielding strength of beam steel (f bs y ) is taken to be 350.0MPa, and the maximum allowable impact force (F allow max ) on the bridge pier is taken to be 5.0 MN. e information of the prespeci ed parameters regarding the structure is tabulated in Table 4.
e optimum parameters generated by the proposed optimization model, the total masses of beam steel, and the con gurations of optimum devices corresponding to di erent impact scenarios are tabulated in Table 5. e total number of beam units (N total bu N bu × N pl ) and the total mass of beam steel (m sb ) used by the optimum device plotted against total barge impact energy (W total ) are shown in Figure 18, which shows that N total bu increases to four, corresponding to two beam units in one plane, when W total reaches around 5.0 MNm. is is because the maximum cap displacement (D max cap ) increases with the increase of W total , as shown in Figure 19.When W total reaches around 5.0 MNm, D max cap increases to such a level that two beam units in one plane are needed to increase the maximum allowable cap displacement (D allow max ) based on (7), enabling D max cap to be lower than D allow max to satisfy the design requirement.It is also shown in Figure 18 that m sb approximately shows linear dependency on W total , indicating that m sb is approximately directly proportional to barge mass while an increase of impact velocity could lead to a roughly quadratic increase of m sb .
e maximum cap displacements (D max cap ) and maximum impact forces (F max ) on the pier corresponding to di erent impact scenarios are tabulated in Table 6, which shows that for each impact scenario, D max cap is smaller than D allow max and F max is smaller than F allow max (5.0 MN); thus the design requirements can be satis ed for all impact scenarios.e maximum impact forces (F unprot max ) on the pier for di erent impact scenarios without using the optimum devices are also tabulated in Table 6 along with the reduction ratio (r f ) of maximum impact forces when the optimum devices are used.It is shown in Table 6 that the optimum device can signi cantly reduce the maximum impact force on the pier by more than 90.0% for di erent impact scenarios.Table 6 indicates that for a given impact velocity, F unprot max is not strongly in uenced by barge mass. is phenomenon has been explained in detail in [8].It is also indicated from Table 6 that D max cap is often close to or Figure 17: Impact force time histories on the bridge pier for the whole impact process (a), for the rst 0.10 s of impact process (b) corresponding to di erent beam unit numbers, and the reduction ratio of maximum impact force versus beam unit number N bu (c).
, without the device; , N bu 1; , N bu 2; , N bu 3.  20, which shows that the horizontal beams at the top and two vertical beams in the middle experience apparent plastic deformations after impact, enabling the devices to absorb high energy during impact, as shown in Figure 21.

Summary
is paper devised a novel crashworthy device for pier protection from barge impact and conducted parametric studies to investigate the effectiveness of the proposed device using the simplified impact model.A mathematical optimization model was developed with constraints as per the prescribed design requirements to achieve costoptimized design of the device for a given impact scenario.
e studies indicate that the proposed crashworthy device has a large energy dissipation capacity due to the   Advances in Civil Engineering formation of plastic hinges in the structure during impact.ese number and location of plastic hinges, and consequently the energy that can be absorbed, is determined by the design of the frame-like steel beam arrangement.
e studies show that the magnitude of the impact force transmitted to the main bridge pier can be dramatically reduced when the device is properly designed and installed and that the maximum force transmitted can be chosen as part of the device design.A mathematical optimization model proposed in this paper can be used for obtaining the optimum   ) and the total impact energy ( ) during impact corresponding to the impact scenarios (a) IS 31 , (b) IS 32 , and (c) IS 33 .14 Advances in Civil Engineering configuration of the device which satisfies the design requirements for a given impact scenario.e concept proposed here can be extended further to other configurations, for example, symmetrical or entwined arrangements which avoid vertical displacements or reduce the device's overall dimensions, respectively.e device concept presented and the analysis model adopted have the potential to rationalize ship impact protection and thus to provide cost-effective future protection solutions.

Figure 1 :
Figure1: e structure connecting the cap and the bridge pier using steel beams of I cross section for a sample bridge pylon foundation.

Figure 2 :
Figure 2: Con guration of (a) the cap steel beam structure and (b) I cross section of steel beams.N bu : number of beam units in one plane; l sb : length of each single steel beam; m c : cap mass.

Figure 6 :
Figure 6: Simpli ed impact model based on CMM for dynamic analysis of the proposed device subjected to barge impact.

Table 1 :N
Prespeci ed parameters for parametric studies of the device.pl number of planes of beam units 1 l csb total length of beam units in one plane 15.0 m ρ bs mass density of beam steel 8020.0 kg/m 3 E bs elastic modulus of beam steel 200.0 GPa E bs t tangent modulus of beam steel 1.5 GPa ε bs u failure strain of beam steel 0.25 Pier w p pier width 6

Figure 7 :
Figure7: Time histories of energy absorbed by the device (a) corresponding to di erent beam cross-section dimensions and the ratio of energy absorbed by the device after impact to the total impact energy, respectively, versus MC bi (b)., W csb diss (MC bi 0.8); , W csb

Figure 8 :Figure 9 :
Figure 8: Maximum bending moment diagrams of the structures during impact and de ections of the structures after impact corresponding to di erent beam cross-section dimensions (unit: MNm)., original shape of the device; , deformed shape of the device.(a) MC bi 0.8; (b) MC bi 1.0; (c) MC bi 1.2.

Figure 11 :
Figure 11: Time histories of energy absorbed by the device (a) corresponding to di erent yielding strengths of beam steel and the ratio of energy absorbed by the device after impact to the total impact energy, respectively, versus yielding strength of beam steel f bs y (b)., W csb diss (f bs y 250.0 MPa); , W csb diss

Figure 12 :
Figure 12: Maximum bending moment diagrams of the structures during impact and de ections of the structures after impact corresponding to di erent yielding strengths of beam steel (unit: MNm)., original shape of the device; , deformed shape of the device.(a) f bs y 250.0 MPa; (b) f bs y 350.0 MPa; (c) f bs y 450.0 MPa.

Figure 13 :
Figure 13: Impact force time histories on the bridge pier for the whole impact process (a), for the rst 0.10 s of impact process (b) corresponding to di erent yielding strengths of beam steel and the reduction ratio of maximum impact force versus yielding strength of beam steel f bs y (c)., without the device; , f bs y 250.0 MPa; , f bs y 350.0 MPa; , f bs y 450.0 MPa.

Figure 15 :
Figure 15: Time histories of cap displacement corresponding to di erent beam unit numbers (a) and maximum cap displacement D max cap

Figure 18 Figure 19 :
Figure18: e total number of beam units N total bu and the total mass of beam steel m sb used by the optimum device versus barge impact energy W total (N pl 2).

Figure 20 :
Figure20: Maximum bending moment diagrams of the optimum devices during impact and de ections of the structures after the impact corresponding to the impact scenarios (a) IS 31 , (b) IS 32 , and (c) IS 33 (unit: MNm)., original shape of the device; , deformed shape of the device.

Figure 21 :
Figure 21: Energy absorbed by the optimum device () and the total impact energy ( ) during impact corresponding to the impact scenarios (a) IS 31 , (b) IS 32 , and (c) IS 33 .

Table 2 :
Relationships of I cross-section dimensional parameters.

Table 3 :
Impact scenarios considered for structure optimization.

Table 5 :
Optimum parameters, total masses of beam steel, and configurations of optimum devices corresponding to different impact scenarios.

Table 4 :
Prespecified parameters for structure optimization.

Table 6 :
Maximum cap displacements and maximum impact forces on the pier using or without using optimum devices corresponding to di erent impact scenarios.