Impact factor method for design of bridge foundations under ship collisions

Abstract

Due to the complexity involved and limited study on the topic, the equivalent static method, adopted in the current codes for structural design of bridges under ship collisions, does not take into account the dynamic amplification effect correctly. In this article, impact factor method is proposed to estimate the response of bridge’s piers and foundations, as a better alternative of the equivalent static method. Through refined numerical simulations of ship-rigid wall collisions for nine typical ships under various impact velocities, 81 impact force time-histories are obtained. The period-dependent impact factor is defined, and empirical function of it is proposed and parameters in the empirical function are determined by the 81 sample impact force time-histories. Finally, both impact factor method and dynamic time-history method are used to estimate the responses of piers and foundations of two example bridges, and the precision of impact factor method is discussed.

Keywords

design of bridge foundations impact factor method period-dependent impact factor precision ship collision

Introduction

Vessel–bridge collision demand analysis aims at studying the response of bridge structure under vessel collision. Nowadays, the equivalent static load method, the finite element numerical simulation method, and the simplified dynamic analysis method are mostly used for demand analysis of vessel–bridge collision.

The equivalent static method is employed in most codes and specifications (Knott and Damgaard, 1990; Knott, 2010; Vrouwenvelder, 1998) for vessel collision design of bridges. However, the problem is that code formulas may provide impact force estimations that differ from one another (Wang et al., 2006). In addition, the dynamic characteristics of vessel–bridge collision are ignored.

For more than 10 years, the finite element model (FEM) has been widely used in numerical simulations of ship–bridge collisions (Consolazio and Cowan, 2003, 2005; Ocakli et al., 2004; Wang and Chen, 2007, 2009), and validation verifications of the FEM have been conducted by various academic organizations (Association for Structural Improvement of the Shipbuilding Industry (ASIS) 1993; International Ship and Offshore Structures Committee (ISSC), 2003). Although almost all dynamic characteristics can be considered, higher requirements to the researchers and heavy calculation are required for the finite element numerical simulation method, which is not practical for engineering applications.

Therefore, efforts have been paid to develop the simplified methods to estimate the impact response of bridge under vessel collisions. For example, impact response spectrum analysis method for barge-bridge collision has been proposed (Cowan, 2007; Fan and Yuan, 2012; Wang and Yu, 2014). However, the time duration of vessel–bridge collision is far shorter than the duration of seismic ground motions. Therefore, the concept of seismic response spectra and corresponding modal combination methods might not be suitable to apply to the impact response estimation for vessel–bridge collisions.

In this article, an impact factor method (IFM) for design of bridge piers and foundations under vessel collisions is developed. The period-dependent impact factor (PDIF) is developed based on numerical simulations of vessel–bridge collisions. The response of bridge piers and foundations may be estimated by static response multiplied by the impact factor. Finally, the precision of IFM is verified by two example bridges.

Impact force time-history of ship–bridge collision

In this article, nine typical ships are chosen with deadweight tonnage (DWT) ranging from 500 to 50,000, as listed in Table 1, and the configurations of the ships are illustrated in Figure 1.

Table 1.

Feature data of example ships.

DWT (ton)		50,000	30,000	10,000	12,000	5000	3000-1	3000-2	1000	500
Loaded displacement (ton)		62,000	43,028	18,542	16,700	6710	3962	6273	1210	797
Main dimensions	Length (m)	190	186.41	139.3	137	99.9	86.8	108	62	50
	Width (m)	32.26	26	20.8	22.4	16.8	16.2	17	10.6	8.2
	Depth (m)	17.2	15	11.2	11	7.8	4.7	8	3.4	3.72
	Drift (m)	12.2	11	7.8	7.8	6.1	2.8	4.2	2.6	3.1
	C_b	0.854	0.85	0.87	0.746	0.696	0.837	0.86	0.82	0.8
	Hs (m)	20	18	14.6	12	11	7	11.1	4.32	5.79
Bow shapes		Bulb bow	Bulb bow	Bulb bow	Bulb bow	Bulb bow	Bulb bow	Bulb bow	Clipper bow	Clipper bow

Figure 1.

Configurations of typical ships.

To obtain the impact force time-history of ship–bridge collision, the simplified computational model for ship-rigid wall collisions as shown in Figure 2 is used instead of the complex ship–bridge components collision cases. The FEMs of typical ships by software LS-DYNA are illustrated in Figure 3 for the ship-rigid wall head-on collisions. The reliability of the numerical simulation has been well stated in the literatures (Consolazio and Cowan, 2003, 2005; Liu and Gu, 2003; Wang et al., 2013; Wang and Gu, 2001; Yuan and Harik, 2009).

Figure 2.

Ship-rigid wall head-on collision model.

Figure 3.

Finite element models of ship bows.

The collision velocity is taken as 1.0, 1.5, 2.0, 2.5, 3.0, 3.5 4.0, 4.5, and 5.0 m/s. The numerical computations for the ship-rigid wall head-on collisions associated with nine ships and nine velocities are finished, and totally 81 samples of impact force time-histories are obtained. Example curves of the impact force time-history are illustrated in Figure 4.

Figure 4.

Example curves of the impact force time-history.

Period-dependent impact factor

Definition of PDIF

And the dynamic equation of motion of a single degree-of-freedom (SDOF) system with unity mass can be expressed as follows

\overset{\cdot\cdot}{δ} (t) + 2 ξ ω \overset{\cdot}{δ} (t) + ω^{2} δ (t) = \bar{p} (t) = p_{\max} \cdot p (t)

(1)

where ξ is the damping ratio; ω is the circle frequency of SDOF; δ(t) is the time varying displacement; and $\bar{p} (t)$ is the impact force time-history; p_max is the maximum value of $\bar{p} (t)$ and p(t) is equal to $\bar{p} (t) / p_{\max}$ .

The maximum dynamic response of SDOF system under the impact force time-history can be determined through the combined use of static response calculation and a dynamic magnification factor which is defined as PDIF

P D I F = \frac{δ_{m}^{d}}{δ_{m}^{s t}}

(2)

where $δ_{m}^{d}$ is the maximum dynamic displacement of SDOF system under the impact force time-history, $\bar{p} (t)$ , and $δ_{m}^{s t}$ is the static displacement response of SDOF under p_max, that is, $δ_{m}^{s t} = p_{\max} / ω^{2}$ , as shown in Figure 5.

Figure 5.

Responses of SDOF system.

The solution of equation (1) can be expressed by Duhamel integration as follows

δ (t) = \frac{1}{ω \sqrt{1 - ξ^{2}}} \int_{0}^{t} \bar{p} (τ) \exp [- ξ ω (t - τ)] \sin ω \sqrt{1 - ξ^{2}} (t - τ) d τ

(3)

Following the definition of PDIF (equation (2)) and the solution of $δ (t)$ (equation (3)), one can obtain

P D I F = \frac{1}{\sqrt{1 - ξ^{2}}} \cdot ω {| \int_{0}^{t} p (τ) \exp [- ξ ω (t - τ)] \sin ω \sqrt{1 - ξ^{2}} (t - τ) d τ |}_{\max}

(4)

where |.|_max means absolute maximum value.

Samples of PDIF

By solving equation (4) excited by impact force time-histories given in section “Impact force time-history of ship–bridge collision,” the 81 PDIFs are obtained, and two examples are shown in Figure 6 with λ = t_d/T, where t_d is the equivalent impact duration and T is the free vibration period of SDOF.

Figure 6.

Example curves of the result of PDIF.

It is reasonable to use t_d/T as the horizontal axis for short-time impact action (Chopra, 2012). To define t_d, it is assumed that the impact force time-history might be equivalent to a half-sine function, as shown in Figure 7. According to the equal-impulse principle, t_d is the equivalent duration of the sample impact force time-history and is defined as follows

\int_{0}^{t_{d}} p_{\max} \sin (π \frac{t}{t_{d}}) d t = I_{0}

(5)

t_{d} = \frac{π}{2} \frac{I_{0}}{p_{\max}}

(6)

where I₀ is the impulse corresponding to the sample impact force time-history.

Figure 7.

Equivalent duration definition, t_d.

The 81 samples of PDIF are shown in Figure 8 for damping ratio, ξ, of 0.05. One can observe from Figure 8 that the PDIF is related to the ship’s size, impact velocity, and λ. It is observed that the PDIF might be divided into three phases, the first phase increases significantly with λ, the second phase nearly maintains a constant in average meaning and the third phase approaches to 1.0 as λ increases.

Figure 8.

Samples of PDIF.

PDIF for designs

Empirical expression of PDIF

For bridge’s design under ship collisions, an empirical expression of PDIF with statistical meaning is necessary. Generally, contributions of very higher modes (say, natural frequencies higher than 6 Hz) to important structural responses are insignificant as shown in Figure 14 later, that is, rough values for the third phase might not lead to large errors to the structural responses. Therefore, it is proposed that the design value of PDIF (named as DPDIF) takes the following expression

PDIF = {\begin{matrix} k λ, 0 < λ ⩽ λ_{1} \\ k λ_{1}, λ > λ_{1} \end{matrix}

(7)

where k and λ₁ are the statistical coefficients. The graph of the function corresponding to equation (7) is shown in Figure 9.

Figure 9.

Shape of the DPDIF.

By least square fitting based on the 81 samples of PDIF, the sample values of k and λ₁ are obtained as listed in Tables 2 and 3. According to Tables 2 and 3, only small dispersion of k and λ₁ is observed, that is, the standard deviations of k and λ₁ are small. Therefore, it is proposed that the mean values of k and λ₁ may be used in designs, with k = 3.273 and λ₁ = 0.396. Two examples of the DPDIF and the corresponding sample PDIF are shown in Figure 10.

Table 2.

Sample values of λ_1.

V (m/s)	DWT (ton)
V (m/s)	500	1000	3000-1	3000-2	5000	10,000	12,000	30,000	50,000
1	0.414	0.356	0.414	0.424	0.405	0.365	0.406	0.405	0.395
1.5	0.353	0.386	0.395	0.374	0.466	0.369	0.376	0.413	0.364
2	0.407	0.394	0.457	0.422	0.506	0.378	0.427	0.387	0.355
2.5	0.413	0.414	0.504	0.396	0.402	0.373	0.445	0.329	0.358
3	0.473	0.428	0.406	0.394	0.404	0.381	0.405	0.358	0.364
3.5	0.415	0.417	0.425	0.362	0.387	0.366	0.408	0.343	0.384
4	0.396	0.427	0.484	0.404	0.378	0.357	0.368	0.366	0.402
4.5	0.384	0.393	0.423	0.354	0.387	0.348	0.347	0.365	0.417
5	0.417	0.416	0.405	0.354	0.398	0.382	0.382	0.384	0.363

Table 3.

Sample values of k.

V (m/s)	DWT (ton)
V (m/s)	500	1000	3000-1	3000-2	5000	10,000	12,000	30,000	50,000
1	3.214	3.353	3.404	3.397	3.419	3.357	3.294	3.119	3.338
1.5	3.332	3.446	3.409	3.213	3.339	3.423	3.359	3.269	3.377
2	3.191	3.413	3.116	3.291	3.279	3.427	3.257	3.179	3.401
2.5	3.135	3.397	2.745	3.344	3.345	3.419	3.131	3.395	3.339
3	2.968	3.323	3.081	3.369	3.331	3.419	3.144	3.307	3.289
3.5	3.179	3.120	3.217	3.348	3.384	3.263	3.263	3.302	3.177
4	3.066	3.199	3.087	3.336	3.406	3.306	3.292	3.267	3.095
4.5	3.044	3.236	3.239	3.321	3.241	3.389	3.319	3.221	3.111
5	3.156	3.309	3.307	3.366	3.273	3.321	3.280	3.256	3.296

Figure 10.

Examples for DPDIF samples: (a) 12,000 DWT and (b) 50,000 DWT.

Empirical expression of p_max and t_d

According to equations (2) and (7), to use empirical PDIF in designs, it is necessary to know the empirical formulations of p_max and t_d. For the 81 samples of impact force time-histories, the samples of p_max and t_d are listed in Tables 4 and 5, respectively.

Table 4.

Sample values of p_max (MN).

V (m/s)	DWT (ton)
V (m/s)	500	1000	3000-1	3000-2	5000	10,000	12,000	30,000	50,000
1	2.61	3.05	6.54	10.00	16.82	29.39	16.39	38.48	40.72
1.5	3.87	3.07	9.02	15.89	18.19	31.00	21.16	34.03	50.66
2	3.39	3.85	12.37	16.30	18.64	31.27	26.45	55.61	70.73
2.5	4.79	4.08	14.56	18.21	24.80	36.89	34.22	57.28	82.81
3	5.18	4.61	14.47	19.47	30.29	38.19	45.94	70.99	108.19
3.5	4.65	7.17	13.83	25.06	34.03	60.31	47.47	89.07	137.28
4	6.52	7.17	13.56	25.90	38.56	74.04	51.50	98.93	147.72
4.5	8.62	8.77	15.34	37.37	40.16	70.93	55.31	124.09	148.99
5	8.32	8.40	15.15	38.17	39.40	70.39	57.31	121.04	152.53

Table 5.

Sample values of t_d (s).

V (m/s)	DWT (ton)
V (m/s)	500	1000	3000-1	3000-2	5000	10,000	12,000	30,000	50,000
1	0.508	0.706	1.051	0.834	0.675	1.069	1.720	1.879	2.546
1.5	0.520	1.040	1.139	0.772	0.934	1.503	1.962	3.412	3.053
2	0.781	1.083	1.094	1.007	1.208	2.008	2.080	2.768	2.953
2.5	0.700	1.265	1.153	1.097	1.136	2.099	1.998	3.382	3.090
3	0.768	1.331	1.372	1.235	1.112	2.404	1.788	3.203	2.885
3.5	0.993	1.001	1.684	1.104	1.153	1.757	2.017	2.952	2.635
4	0.816	1.129	1.943	1.215	1.165	1.640	2.119	2.991	2.780
4.5	0.687	1.043	1.931	0.953	1.240	1.907	2.215	2.708	3.088
5	0.793	1.206	2.171	1.051	1.406	2.139	2.390	3.054	3.334

It is observed from Tables 4 and 5 that the p_max and t_d are related to the loaded displacement of ship and the impact velocity. The following functions are proved to be more accurate when compared with other empirical expressions

p_{\max} = α_{1} M^{β_{1}} V^{γ_{1}}

(8)

t_{d} = α_{2} M^{β_{2}} V^{γ_{2}}

(9)

where p_max is in MN, t_d is in second, M is the ship loaded displacement (ton), V is the impact velocity (m/s), and values of α₁, α₂, β₁, β₂, and γ₁, γ₂ are statistical factors.

By statistical analysis of the data of Tables 4 and 5, we obtain that α₁ = 0.0224, β₁ = 0.8726 and γ₁ = 0.6782, and α₂ = 0.0578, β₂ = 0.3393 and γ₂ = 0.1505, respectively. Indexes, ε₁ and ε₂, are introduced to evaluate the precision of equations (8) and (9), as defined follows,

ε_{1} = \frac{(p_{\max}^{e m p} - p_{\max}^{s a m})}{p_{\max}^{s a m}} and ε_{2} = \frac{(t_{d}^{e m p} - t_{d}^{s a m})}{t_{d}^{s a m}}

(10)

where $p_{max}^{e m p}$ and $t_{d}^{e m p}$ are the empirical values of p_max and t_d by equations (8) and (9); $p_{\max}^{s a m}$ and $t_{d}^{s a m}$ are the sample values of p_max and t_d in Tables 4 and 5. Figure 11 shows the distribution of ε₁ and ε₂. It shows that most values of ε₁ and ε₂ for p_max and t_d are in the range of ±20% and do not exceed a value of 80%. Actually, with existing reasons affecting the collision, such as bow type, bow stiffness, ship loaded displacements, and so on, the dispersion of the impact force is caused. With accumulating much more samples, the statistical significance of equations (8) and (9) could be improved.

Figure 11.

Relative error of empirical values of p_max and t_d: (a) ε₁ and (b) ε₂.

IFM for design of bridge piers and foundations

The dynamic equation with multi-degree of freedom for bridge under impact of ships may be expressed as follows

M \overset{\cdot\cdot}{u} + C \overset{\cdot}{u} + K u = s \cdot \bar{p} (t)

(11)

where M, C, and K are the mass matrix, damping matrix, and stiffness matrix of a bridge structure, respectively; u represents the displacement response vector of the bridge; s = {0,0, … ,0,1,0, … ,0}^T is the spatial distribution vector of the impact force time-history produced by ship collision and $\bar{p} (t)$ take its meanings as before.

Follow the general theory of dynamics of structures (Chopra, 2012), taking linear transform

u = Ф q

(12)

where $Ф = [\begin{matrix} φ_{1} & φ_{2} & \dots & φ_{n} \end{matrix}]$ represents the mode-shape matrix, which is normalized by mass matrix. The generalized components in vector q are called the normal coordinates of the bridge structure and may be solved through

{\overset{\cdot\cdot}{q}}_{j} + 2 ξ_{j} ω_{j} {\overset{\cdot}{q}}_{j} + ω_{j}^{2} q_{j} = Γ_{j} \bar{p} (t), j = 1, 2, \dots, N

(13)

where q_j is the generalized coordinates of mode j, ξ_j is the modal damping ratio of mode j, ω_j is the circle frequency of mode j, $Γ_{j} = ϕ_{j}^{T} s$ is the modal participation factor, and N is the number of dynamic degrees of freedom. Comparing equations (1) and (13), q_j = Γ_jδ_j, then equation (13) could be expressed as follows

{\overset{\cdot\cdot}{δ}}_{j} + 2 ξ_{j} ω_{j} {\overset{\cdot}{δ}}_{j} + ω_{j}^{2} δ_{j} = \bar{p} (t)

(14)

where δ_j is the time varying displacement of SDOF system under $\bar{p} (t)$ .

For any response, r(t) of bridge structure may be expressed as (Chopra, 2012) follows

r (t) = \sum_{j = 1}^{N} r_{j} (t) = \sum_{j = 1}^{N} r_{j}^{st} ω_{j}^{2} δ_{j} (t)

(15)

where r_j(t) is the response of mode j and $r_{j}^{s t}$ is the static response of mode j under the load of s_j, where s_j = Γ_j M ϕ_j is the component of s for mode j.

Assuming that the peak values, $r_{j}^{\max}$ , of r_j(t) for all modes have the same sign and occur at the same time, then the peak value r_max of the total response can be obtained by equation (15)

\begin{matrix} r_{\max} \approx \sum_{j = 1}^{N} r_{j}^{\max} = \sum_{j = 1}^{N} r_{j}^{s t} ω_{j}^{2} δ_{j m}^{d} = \sum_{j = 1}^{N} r_{j}^{s t} ω_{j}^{2} δ_{j m}^{s t} \cdot P D I F_{j} \\ = p_{\max} \sum_{j = 1}^{N} r_{j}^{s t} \cdot P D I F_{j} \end{matrix}

(16)

in which $δ_{j m}^{d}$ is the peak value of δ_j(t), $δ_{j m}^{s t} = p_{\max} / ω_{j}^{2}$ and PDIF_j is the dynamic magnification factor of mode j. Equation (16) is called the summation method (SUM).

Based on equation (16), it is further assumed that PDIFs for all modes are equal to the value of PDIF_max corresponding to the mode which contributes greatest to the total response, thus

\begin{matrix} r_{\max} \approx p_{\max} \sum_{j = 1}^{N} r_{j}^{s t} \cdot P D I F_{j} \approx p_{\max} \cdot P D I F_{\max} \sum_{j = 1}^{N} r_{j}^{s t} \\ = P D I F_{\max} \cdot r^{s t} \end{matrix}

(17)

where r^st is obtained by solving the following static equation

Ku = s \cdot p_{\max}

(18)

Equation (17) is called the IFM. To determine PDIF_max, mode contribution ratio, β_j, is introduced to evaluate the response contribution of mode j

β_{j} = \frac{r_{j}^{\max}}{r_{\max}}

(19)

where $r_{j}^{\max}$ and r_max take their meanings as before and the value of β_j could be either positive or negative.

However, it is not practical to use β_j to determine PDIF_max, because time-history analysis is necessary to get the modal response. Thus, another index, $β_{\max}^{Γ}$ , is introduced to determine PDIF_max, as defined below, $β_{\max}^{Γ}$ defined as follows is used to determine PDIF_max

β_{\max}^{Γ} = \frac{| Γ |_{\max}}{\sum_{j = 1}^{N} | Γ_{j} |}

(20)

where |Γ|_max = max{|Γ₁|, |Γ₂|, … ,|Γ_N|}.

The validity of the assumptions adopted by SUM and IFM will be stated in section “Examples and discussions” through two example bridges and the precision of SUM and IFM will be discussed.

Examples and discussions

Basic information and analysis models of the example bridges

To demonstrate the precision of SUM and IFM, both IFM and dynamic time-history method are employed to estimate the responses of two example bridges, one is a cable-stayed bridge and the other one is a continuous girder bridge. Such bridge types are quite common for bridges crossing navigation channels.

The cable-stayed bridge is of a length of 1430 m with five spans and two towers. Its span arrangement is 110 m + 240 m + 730 m + 240 m + 110 m. The group pile foundation of the pylon consists of 60 piles of 104 m long with variable diameters of 2.5 and 3.2 m. The group pile foundations of the auxiliary piers and the side piers are designed with 18 and 12 piles, respectively. The concrete grade of the foundation is C30.

The continuous girder bridge includes five spans each measuring 55 m. The group pile foundation of the pier consists of seven inclined piles with diameter of 1.6 m (Figure 12).

Figure 12.

Configurations of the example bridges. (a) The cable-stayed bridge: (i) overall elevation, (ii) tower and group pile foundation, and (iii) auxiliary pier and group pile foundation. (b) The continuous girder bridge: (i) overall elevation and (ii) pier and group pile foundation.

The bridge components are modeled with beam element. In order to accounting for the soil–pile interaction of the cable-stayed bridge, discrete springs are used to simulate the soil. The piles of the continuous girder bridge were fixed at the position of the riverbed. FEMs for static and dynamic analysis are illustrated in Figure 13.

Figure 13.

Finite element models of the example bridges: (a) the cable-stayed bridge and (b) the continuous girder bridge.

Impact loads and analysis cases

To demonstrate the precision of SUM and IFM, five kinds of response analysis of example bridges are performed as listed in Table 6.

Table 6.

Analysis cases.

Computing method for responses		Acting position of impact force			Direction of impact force
Symbol	Method	Bridge	Component	Acting point	Direction of impact force
TH	Sample time-history analysis	Cable-stayed bridge	Tower	Center of the pile cap	Transverse to the direction of bridges
SUM	Sum of the maximum modal responses		Tower
IFM-A	IFM, both p_max and PDIF come from samples		Auxiliary pier
IFM-B	IFM, p_max comes from samples, and PDIF from empirical formulation (equation (7))	Continuous girder bridge	P₂ pier	Center of the pile cap
IFM-C	IFM, both p_max and PDIF from empirical formulation (equations (7)–(9))	Continuous girder bridge	P₂ pier	Center of the pile cap

The sample impact force time histories come from ships of 50,000 DWT and 5000 DWT with impact velocity of 3 m/s for the cable-stayed bridge, respectively; 5000 DWT with 3 m/s, 1000 DWT with 4 m/s, and 500 DWT with 2 m/s for the continuous girder bridge, respectively.

Precision of IFM

By conducting calculations in Table 6, β_j of the case TH for the two example bridges is shown in Figure 14. f in Figure 14 is the modal natural frequency in Hertz. The symbol meanings of CPCT/5000, CPCAP/5000, and CPCP₂/1000 will be explained later in Tables 7 and 8.

Figure 14.

Modal response features. (a) Tower of the cable-stayed bridge: (i) β_j and modal frequency and (ii) β_j and time reaching $r_{j}^{\max}$ . (b) Auxiliary pier of the cable-stayed bridge: (i) β_j and modal frequency and (ii) β_j and time reaching $r_{j}^{\max}$ . (c) Continuous girder bridge: (i) β_j and modal frequency and (ii) β_j and time reaching $r_{j}^{\max}$ .

Table 7.

Precision of SUM and IFM impacted at center of pile cap of the cable-stayed bridge.

Force position/ship size	Resp	TH	SUM	IFM							Response ratio, η
				λ	A		B		C		SUM/TH	A/TH	B/TH	C/TH
				λ	r_st	PDIF _max	r_st	PDIF _max	r_st	PDIF _max	SUM/TH	A/TH	B/TH	C/TH
CPCT/50,000	D_p	12.02	12.50	5.93	10.59	1.27	10.59	1.30	10.19	1.30	1.04	1.12	1.14	1.10
	M_pier	7.10	3.09		0.49		0.49		0.47		0.44	0.09	0.09	0.09
	Q_pier	2.65	0.42		0.10		0.10		0.10		0.16	0.05	0.05	0.05
	M_pile	19.73	20.54		17.99		17.99		17.32		1.04	1.16	1.18	1.14
	Q_pile	1.91	1.98		1.80		1.80		1.74		1.04	1.20	1.23	1.18
CPCT/5000	D_p	3.80	4.04	1.56	2.96	1.56	2.96	1.30	2.27	1.30	1.06	1.21	1.01	0.77
	M_pier	3.99	1.56		0.14		0.14		0.10		0.39	0.05	0.04	0.03
	Q_pier	0.86	0.22		0.03		0.03		0.02		0.25	0.05	0.04	0.03
	M_pile	6.25	6.62		5.03		5.03		3.86		1.06	1.26	1.04	0.80
	Q_pile	0.60	0.64		0.50		0.50		0.39		1.06	1.30	1.08	0.83
CPCAP/50,000	D_p	9.40	9.51	14.69	8.88	1.24	8.88	1.30	8.55	1.30	1.01	1.17	1.22	1.18
	M_pier	14.70	3.30		6.74		6.74		6.49		0.22	0.57	0.59	0.57
	Q_pier	4.88	5.17		0.36		0.36		0.35		1.06	0.09	0.10	0.09
	M_pile	34.58	34.88		32.63		32.63		31.40		1.01	1.17	1.22	1.18
	Q_pile	6.32	6.37		5.96		5.96		5.74		1.01	1.17	1.22	1.18
CPCAP/5000	D_p	2.60	2.64	5.66	2.48	1.22	2.48	1.30	1.90	1.30	1.02	1.16	1.24	0.95
	M_pier	6.79	0.86		1.88		1.88		1.44		0.13	0.34	0.36	0.28
	Q_pier	1.40	1.44		0.10		0.10		0.08		1.02	0.09	0.09	0.07
	M_pile	9.54	9.63		9.12		9.12		6.99		1.01	1.16	1.24	0.95
	Q_pile	1.74	1.76		1.67		1.67		1.28		1.01	1.16	1.24	0.95

CPCT: center of pile cap of tower; CPCAP: center of pile cap of auxiliary pier; D_p (mm): displacement at the impact position; M_pier (MN m): moment at impacted pier (tower) bottom; Q_pier (MN): shear force at the impacted pier (tower) bottom; M_pile (MN m): moment at the pile top; Q_pile (MN): shear force at the pile top.

λ: corresponding to the mode with the greatest contribution to the total response.

Table 8.

Precision of SUM and IFM impacted at center of pile cap of continuous girder bridge.

Force position/ship size	Resp.	TH	SUM	IFM							Response ratio, η
				λ	A		B		C		SUM/TH	A/TH	B/TH	C/TH
				λ	r_st	PDIF _max	r_st	PDIF _max	r_st	PDIF _max	SUM/TH	A/TH	B/TH	C/TH
CPCP₂/5000	D_p	71.30	76.00	3.94	61.48	1.23	61.48	1.30	46.84	1.30	1.07	1.06	1.12	0.85
	M_pier	25.39	22.36		14.02		14.02		10.68		0.88	0.68	0.72	0.55
	Q_pier	26.22	57.63		1.20		1.20		0.91		2.20	0.06	0.06	0.05
	M_pile	20.79	21.78		18.67		18.67		14.23		1.05	1.11	1.16	0.89
	Q_pile	1.86	1.96		1.95		1.95		1.48		1.06	1.29	1.36	1.04
CPCP₂/1000	D_p	18.50	18.83	4.00	14.60	1.27	14.60	1.30	18.83	1.30	1.02	1.00	1.02	1.32
	M_pier	6.05	12.64		3.33		3.33		4.29		2.09	0.70	0.71	0.92
	Q_pier	6.83	14.78		0.28		0.28		0.37		2.16	0.05	0.05	0.07
	M_pile	5.25	5.34		4.43		4.43		5.72		1.02	1.07	1.10	1.41
	Q_pile	0.46	0.48		0.46		0.46		0.60		1.03	1.26	1.29	1.67
CPCP₂/500	D_p	8.20	8.38	2.76	6.88	1.22	6.88	1.30	7.76	1.30	1.02	1.02	1.09	1.23
	M_pier	3.15	1.42		1.57		1.57		1.77		0.45	0.61	0.65	0.73
	Q_pier	3.02	3.14		0.13		0.13		0.15		1.04	0.05	0.06	0.06
	M_pile	2.41	2.40		2.09		2.09		2.36		1.00	1.06	1.13	1.27
	Q_pile	0.22	0.22		0.22		0.22		0.25		1.00	1.23	1.31	1.48

CPCP₂: center of pile cap of P₂; λ: corresponding to the mode with the greatest contribution to the total response; D_p (mm): displacement at the impact position; M_pier (MN m): moment at impacted pier (tower) bottom; Q_pier (MN): shear force at the impacted pier (tower) bottom; M_pile (MN m): moment at the pile top; Q_pile (MN): shear force at the pile top.

Figure 14 shows the following valuable information:

Only a few modes contribute the most part to the total response, called “important modes” and the left modes contribute a small part to the total response;

β_j of the “important modes” have the same sign. Then, according to equation (19), $r_{j}^{\max}$ of the these modes also have the same sign;

The modal natural frequencies, f (in Hz), of the “important modes” are very closely spaced. In other words, these modes reach the peak response nearly at the same time and PDIFs of the “important modes” are nearly the same.

The mode with the greatest contribution to the total response determined by $β_{\max}^{Γ}$ is identical to that determined by equation (19). Therefore, it might be feasible to use $β_{\max}^{Γ}$ to determine PDIF_max for convenient use of IFM. For other responses of components below the impact position, the same conclusion could be drawn.

To evaluate the precision of SUM and IFM, define the following ratio

η = \frac{r_{SUM}}{r_{TH}} or η = \frac{r_{IFM}}{r_{TH}}

(21)

where r_SUM and r_IFM represent the response obtained by SUM and IFM, respectively, and r_TH represents the peak response by time-history analysis. The responses of the two example bridges are listed in Tables 7 and 8, in which the symbol meanings are explained in Table 6.

From Tables 7 and 8, some observations can be reached as follows:

Both SUM and IFM are suitable to calculate the responses of bridge components below the impact position, but are not suitable for responses above the impact position. Specifically, for responses below the impact position, the relative error does not exceed 7% for SUM and is mostly in the range of ±20% for IFM. However, for responses above the impact position, the precision of SUM and IFM is poor, which is highlighted in bold. For instance, η of M_pier of A/TH for CPCT/50,000 is 0.09 and for CPCP₂/5000 is 0.68;

Compared with IFM, SUM could get more accurate results for responses below the impact position. SUM assumes that the peak values of modal response r_j(t), j = 1, 2, … , N, have the same sign and occur at the same time. According to Figure 14, for “important modes,” the assumptions of SUM are quite reasonable and lead to minor error (not more than 7% for the two example bridges); IFM assumes that PDIFs for all modes equal to PDIF_max, which is a relatively stronger assumption than SUM. For the two example bridges, the errors of IFM are smaller than ±20% in general, and slightly larger than SUM;

By Comparing the IFM precision of responses below the impact position through the methods of A, B and C, it reveals that neither of these is constantly better than the other two. For instance, η of D_p for CPCT/50,000, CPCT/5000, and CPCP₂/5000 gets the best precision by methods C, B, and A, respectively. In fact, the errors of IFM come from two aspects, one is the assumptions of IFM, as stated in section “Impact factor method for design of bridge piers and foundations”; and the other is the statistical errors of equations (7)–(9) for DPDIF, p_max and t_d. Therefore, it does not mean that the precision of IFM using DPDIF, p_max, and t_d is always poorer than that using sample values;

Figure 15 shows that the modes with significant contributions to D_p are the same as that to M_pile and Q_pile, and even β_j of these modes are nearly equal for D_p, M_pile, and Q_pile, respectively. Therefore, for different responses of components below the impact position, values of PDIF_max could be taken equal when using sample values for PDIF, as shown in Tables 7 and 8;

According to Table 2, λ₁ < 0.5. From Figure 15, it shows that λ of all the modes with non-zero contributions to the total response are larger than λ₁. Therefore, DPDIF takes its value at the second part in Figure 9, that is, DPDIF takes a constant value in statistical sense. In this case, IFM is identical to SUM. Furthermore, larger λ might be expected for bridges with smaller spans, and therefore, IFM is suggested to estimate the responses of bridge components below the impact position for design under ship impact.

$β_{\max}^{Γ}$ could be used to determine PDIF_max. However, with using DPDIF, it becomes unnecessary to determine PDIF_max with $β_{\max}^{Γ}$ when λ > λ₁ for “important modes” because DPDIF takes a constant value in the second phase.

Figure 15.

λ for considered modes. (a) Tower of the cable-stayed bridge: (i) CPCT/50,000 and (ii) CPCT/5000. (b) Auxiliary pier of the cable-stayed bridge: (i) CPCAP/50,000 and (ii) CPCAP/5000. (c) P₂ of the continuous girder bridge: (i) CPCP₂/5000, (ii) CPCP₂/1000, and (iii) CPCP₂/500.

Conclusion

Based on the ship rigid wall head-on collision models of 9 typical ships, 81 impact force time-history curves are obtained by performing numerical simulations with LS-DYNA. Using the 81 samples of the impact force time-history, 81 PDIFs are developed considering the effect of impact force durations. Through statistical analysis, design DPDIF is developed based on the 81 samples of PDIF. Then, IFM is developed and its precision is discussed by two example bridges. The key points of the article are concluded as follows:

The PDIF might be divided into three phases, the first phase increases significantly with λ, the second phase nearly maintains a constant in average meaning and the third phase approaches to 1.0 as λ increasing. Actually, rough estimates of PDIF for the third phase lead to minor errors to the structural responses. Hence, for convenience of applications, a two-segment expression is used for DPDIF;

IFM and SUM are only suitable for response estimates of bridge components below the impact position;

For the two example bridges, IFM is identical with SUM if DPDIF is used, although SUM is of higher precision than IFM with using sample values for PDIF_j;

PDIF_max for different responses of bridge components below the impact position are identical, for instance, D_p, M_pile, and Q_pile for the two example bridges analyzed in this article;

$β_{\max}^{Γ}$ could be used to determine PDIF_max; however, with using DPDIF, it becomes unnecessary when λ > λ₁ for “important modes,” because DPDIF takes a constant value in the second phase.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was financially supported by the National Natural Science Foundation of China (Project Number: 51438010, 51278373) and the National Key Basic Research Program (Project Number: 2013CB036305).

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Impact factor method for design of bridge foundations under ship collisions

Abstract

Keywords

Introduction

Impact force time-history of ship–bridge collision

Period-dependent impact factor

Definition of PDIF

Samples of PDIF

PDIF for designs

Empirical expression of PDIF

Empirical expression of pmax and td

IFM for design of bridge piers and foundations

Examples and discussions

Basic information and analysis models of the example bridges

Impact loads and analysis cases

Precision of IFM

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References

Empirical expression of p_max and t_d