Simplified calculation methods for all-vertical-piled wharf in offshore deep water

All-vertical-piled wharf is a kind of high-piled wharf, but it is extremely different from the traditional ones in some aspects, such as the structural property, bearing characteristics, failure mechanism, and static or dynamic calculation methods. In this paper, the finite element method (FEM) and theoretical analysis method are combined to analyze the structural property, bearing behavior and failure mode of the all-vertical-piled wharf in offshore deep water, and to establish simplified calculation methods determining the horizontal static ultimate bearing capacity and the dynamic response for the all-vertical-piled wharf. Firstly, the bearing capability and failure mechanism for all-vertical-piled wharf are studied by use of FEM, and the failure criterion is put forward for all-vertical-piled wharf based on the ‘plastic hinge’. According to the failure criterion and P–Y curve method, the simplified calculation method of the horizontal static ultimate bearing capacity for all-vertical-piled wharf is proposed, and it is verified that the simplified method is reasonable by comparison with the FEM. Secondly, the displacement dynamic magnification factor for the all-vertical-piled wharf under wave cyclic loads and ship impact loads is calculated by the FEM and the theory formula based on the single degree of freedom (SDOF) system. The results obtained by the two methods are in good agreement with each other, and the simplified calculation method of the displacement dynamic magnification factor for all-vertical-piled wharf under dynamic loads is proposed. Then the simplified calculation method determining the dynamic response for the all-vertical-piled wharf is proposed in combination with P–Y curve method. That is, the dynamic response of the structure can be obtained through the static calculation results of P–Y curve method multiplied by the displacement dynamic magnification factor. The feasibility of the simplified dynamic response method is verified by comparison with the FEM under different conditions.


Introduction
With the large-scale development of the transport ship, the port and wharf are gradually built in offshore deep water. The all-vertical-piled wharf, a new type of high-piled wharf, is suitable for the soft clay foundation in offshore deep water. The structure type of the all-vertical-piled wharf is different from that of the traditional high-piled one which involves the sloping piles or overlapping piles, so the structural property, bearing behavior and failure mode for the allvertical-piled wharf are very different from those of the tra-ditional ones. As the longer pile body, the larger structural flexibility and the basic period which is close to that of the wave load and impact load for the all-vertical-piled wharf, it is essential to research some problems, such as the bearing behavior, failure mode, simplified static calculation method and simplified dynamic response calculation method under the lateral loading for the all-vertical-piled wharf in this paper.
In the existing specification, the methods calculating the bearing capacity of the pile under the lateral loading in-clude m method, P-Y curve method and NL method (Brandenberg et al., 2013). The P-Y curve method, a composite foundation reaction method, is used to calculate the bearing capacity of the elastic pile in American Petroleum Institute (API, 2000) specification and the port engineering pile foundation specification (The Ministry of Transport of the People's Republic of China, 2012). However, whether this simplified static calculation method can be used to calculate the bearing capacity of the all-vertical-piled wharf is unclear. In addition, there is no simplified dynamic response calculation method for the all-vertical-piled wharf, but only the static calculation is not sufficient to engineering design, because the wharf is subjected to dynamic loading, such as the wave load and impact load (Zhao et al., 2007;Zhang et al., 2013). Li (2000) researched the natural oscillation and the dynamic response under wave loads for the open pile-supported beam-slab wharf. He thought that it was necessary to conduct the dynamic analysis for the design of the open pile-supported beam-slab wharf. Miura (2002) proposed the nonlinear static and dynamic design theory for the pile based on a large of project cases of failed piles. Hu et al. (2005) studied the nonlinear dynamic behavior of the pile under the lateral load, and analyzed the effect of the axial force on the stiffness of the structure in detail. Mostafa and EI Naggar (2004) simulated the pile-soil interaction based on the dynamic P-Y curve and the relationship of the frictional resistance and the axial displacement of the pile under the vertical load. And then the dynamic response of the fixed offshore platform under the wave loading was calculated by FEM. Luan et al. (2005) developed a simplified analytic method to determine the vertical simple harmonic vibration response on the basis of the modified Winkler foundation beam model and soil dynamics. However, the above researches were about the dynamic behavior and calculation of the single pile, and the researches on the dynamic response characteristics, the dynamic design calculation principle, and the simplified calculation method were very rare (Shafieezadeh et al., 2013;Chen et al., 2015).
In this paper, the finite element method and theoretical analysis method are combined to analyze the structural property, bearing behavior and failure mode of the all-vertical-piled wharf in offshore deep water, and to establish the simplified calculation methods determining the ultimate bearing capacity and the dynamic response for the all-vertical-piled wharf.

Finite element analysis method
The actual three-dimensional geometric characteristics, the elasto-plastic behavior of soils and the nonlinearity of contact surfaces and materials can be considered for the FEM. The method, which is unlimited by the structure geometry and boundary conditions, can not only accurately simulate the total loading process of the pile and the pile-soil interaction, but also consider many factors which affect the performance of the pile. And the method has a complete theory. Therefore, the FEM is used to research on the bearing capacity and deformation for all-vertical-piled wharf in offshore deep water in this paper.

Project profile
An actual wharf project in offshore deep water is analyzed in the paper. This port is for iron mine, and its port handling capacity is 9.0×10 7 T. The design ship type is 400000 DWT vessels. The major import wharf in the port is selected as the research object, and the length and width of the major import berth are 510 m and 35 m, respectively. The wharf is an all-vertical-piled and beam-plate structure. The center distance from the first pile to the last pile is 35 m. All piles are steel pipe piles with Q345 (Q345 means that the yield strength of the steel is 345 MPa), and the thickness and the diameter of which are 22 mm and 1800 mm, respectively. Parameters of the steel pipe pile and concrete material are shown in Table 1. The free height and the embedded depth of the pile foundation are 32.2 m and 45 m, respectively. The Morison formula used to calculate the wave load of the pile in 'Harbor Hydrological Specifications' (The Ministry of Transport of the People's Republic of China, 1998) is adopted to calculate the design wave load of the actual wharf project in offshore deep water. When the pile is subjected to the peak wave load in the design high water level and the total lateral wave load is the maximum, it is taken as the most unfavorable situation. The design ship impact load is 1600 kN which is the impact load when the ship docks. Detail hydrological conditions are listed in Table 2. Properties of soil layers are shown in Table 3.

Three-dimensional elasto-plastic finite element model
The cross-section of the above all-vertical-piled wharf is shown in Fig. 1. A row of lateral frames are taken as the researched object (Wang and He, 2014). The three-dimensional (3D) elasto-plastic finite element model with the pile-soil interaction is established by the ABAQUS software, as shown in Fig. 2. In order to reduce the effect of the boundary condition on the analysis results, the length of foundation soils for the front and back of lateral frames is 30 times the pile diameter in the lateral direction, and is 2 times the embedded depth of piles in the vertical direction. Frames and foundation soils on the left and right sides are sym-  metry constraints, and lateral constrains on the front and back sides. The bottom of foundation soils is the full constrain. Three-dimensional eight nodes reduced integral solid element C3D8R is used to simulate the frame and the foundation soil. The stress-strain relationship of soft clays is described by the perfect elasto-plastic model based on Mohr-Coulomb yield criterion. The pile-soil interaction is simulated by setting the master-slave interface. Because the elastic modulus of the pile is much larger than that of the soil, the surface of the pile body is the master surface, and the surface of the soil is the slave surface . The coulomb friction model and the hard contact are used for the interface in the tangential direction and the normal direction, respectively (Wang et al., 2016).
In previous analyses, the conclusion is that since the stiffness of the steel pipe pile is much larger than that of the soil, and the failure of the structure is caused by the insufficient bearing capacity of foundation soils (Qiu and Wang, 2015). The pile is often simulated by use of the linear elastic model. However, the bearing behavior and failure mode of the all-vertical-piled wharf are still unclear. So, the steel pipe pile is simulated by using the linear elastic model and the elasto-plastic model to analyze the bearing behavior and failure mode. Analysis results will be compared in this paper.

Loading coefficient and instability criteria
In order to illustrate the relationship between the applied load and the design load in the finite element numerical calculation, the loading coefficient is defined as (Wang et al., 2006): where P D is the design load, and P is the applied load.
There are displacement-controlled and load-controlled ways when the ultimate bearing capacity is calculated by the finite element method. When the foundation soil reaches the overall shear failure, the bearing capacity can be effectively calculated by employing the displacement-controlled way. However, when the instability failure mode is unclear, the instability failure mechanism can be obtained by employing the load-controlled way. Therefore, the displacement of the wharf in the soil surface is determined through gradually applying the load by the load-controlled way. Combined with the loading coefficient, the curve of the loading coefficientdisplacement is obtained. When the slope of the curve approaches zero, the load P corresponding to the slope is the ultimate bearing capacity P u of the wharf and the loading coefficient α is the safety factor.

Simplified calculation method of the static ultimate bearing capacity
The bearing behavior and failure mode of the all-vertical-piled wharf in the offshore deep water mentioned above are analyzed by the FEM. In order to express the calculated results clearly and conveniently, two key points are defined as follows: A is the node at the top of the pile, and B is the node of the pile in the mud surface.

Bearing behaviors and failure modes
Curves of the loading coefficient against the displacement obtained by the finite element method under wave loads are shown in Fig. 3. The figure shows that, when the pile is simulated by the perfect elasto-plastic model, the curves for the loading coefficient against the displacement appear the apparent asymptote with the gradual increase of the load, which indicates that the structure has already   failed. It can be seen that the safety factor K is about 16.17 on the basis of the instability criteria. However, the curves for the loading coefficient against the displacement never appear the apparent asymptote when the pile is simulated by using the elastic constitutive model, which indicates that the structure has not reached the limit bearing state according to the instability criteria. According to the stress field of the structure under the limit state shown in Fig. 4a, it can be found that the maximum stress of the pile is approximately 633 MPa, which is much larger than the yield stress of the steel pipe pile. So, it indicates that the structure has already reached failure. Thus, it can be seen that the pile for the allvertical-piled wharf in offshore deep water should be simulated by the perfect elasto-plastic model, which is different from the traditional analytical method. Otherwise, the calculated results are wrong.
When piles are simulated by using the elasto-plastic constitutive model under the limit state, some conclusions can be obtained in Fig. 5 that the displacement of the pile which is closer to the mud surface is smaller, and the displacement of the pile which is buried in the deep soil and fixed is very small, but the displacement of the pile body is relatively large because the displacement at the top of the pile is large. The maximum stress of the pile body is about 385 MPa which reaches the yield stress according to the stress field distribution of the structure shown in Fig. 4b, so the plastic deformation occurs. Then, it makes the displacement increases rapidly and the structure fails eventually. It is concluded that the plastic failure of the pile body is the controlled factor for the instability of the structure under the lateral loading.
The plastic deformation only occurs at the shallow soil near the soil surface under the limit state, as shown in Fig. 6.
The plastic deformation does not occur for the soil at the tip of the pile. So, the lateral ultimate bearing capacity of the structure does not depend on the bearing capacity of the foundation soil.   WANG Yuan-zhan, HE Lin-lin China Ocean Eng., 2017, Vol. 31, No. 2, P. 182-191 185 3.2 Criterion of the ultimate bearing capacity The calculation result shows that the loading safety factor K of the impact load is 3.34, which is much smaller than that of the wave load. So the impact load is the lateral controlled load for the above wharf, and it is taken as the design load in the next analysis. The bending moment of the pile is calculated by the FEM, and when the impact load is taken as the design load and the ultimate load, the calculated results are shown in Table 4 with the piles numbered as #1, #2, #3, and #4.
The formula of the ultimate bending moment for piles in mechanics of materials is as follows: where σ s is the yield stress of the pile; R and r are the internal and external radius of the pile, respectively. The yield stress σ s of the steel pile is 345 MPa for the structure in the paper. The internal and external radius of the pile are 0.878 m and 0.9 m, respectively. In the limit state under the impact load, the ultimate moment M u of the pile body is 23995.37 kN·m in Eq. (2), which is in agreement with the calculated result by the FEM shown in Table 4. And it indicates that the maximum moment of the pile body reaches the ultimate moment when the above wharf fails under impact load. Fig. 7 shows the equivalent plastic deformation distribution of the pile body in the lateral limit state under the impact loading. The figure shows that the apparent plastic deformation zone occurs for the pile body at a certain depth below the mud surface, and the plastic deformation zone almost extends to the whole cross section. The 'plastic hinge' is formed. So, it is suggested that the lateral limit bearing state of the all-vertical-piled wharf in offshore deep water is determined based on the 'plastic hinge' failure mode.
3.3 Verifications of the simplified static calculation method P-Y curve method and the finite element method were all used to calculate the displacement of the all-verticalpiled wharf in the offshore deep water. Comparisons of the calculated results by the two methods are shown in Tables 5 and 6. It can be seen that the results are in good agreement with each other, indicating that P-Y curve method can be used to evaluate the bearing behavior of the all-verticalpiled wharf in the offshore deep water.
In addition, the maximum bending moment M of the pile body calculated by P-Y curve method is 6659.5 kN·m    under the design impact load for the above wharf, and the ultimate bending moment M u calculated by Eq. (2) is 23995.37 kN·m, then the safety factor K, which is the ratio of the ultimate bending moment M u to the maximum bending moment M, is equal to 3.6. It can be seen that it is basically in agreement with the safety factor 3.34 calculated by the FEM, which verifies the validity of the simplified static calculation method.

Simplified calculation method of the dynamic response
Firstly, modal analyses were conducted by use of the embedded fixed model and the structure-foundation interaction model of the all-vertical-piled wharf in offshore deep water.

Modal analysis
Modal analyses were conducted by using the Lanczos method, and analysis results of the first five order modal are shown in Table 7. When the all-vertical-piled wharf is analyzed by the embedded fixed model and the structure-foundation interaction model, it can be found that both the basic frequencies are relatively small, both the basic periods which approach 3 seconds are relatively large. However, the period of the storm surge is ranging from a few seconds to tens of seconds, which is close to the period of the structure.
Figs. 8 and 9 are the first order vibration modes of the structure. The figures show that the displacement mode of the first order vibration mode is all lateral oscillation along the horizontal direction.

Dynamic response analysis of the structure
According to the analysis results, it can be seen that the basic period of the all-vertical-piled wharf approaches the period of the load and the dynamic response of the structure may be very remarkable. So, the dynamic response analysis should be conducted for the structure. The displacement dynamic magnification factor of the structure is defined as follows: where S 1 and S 2 are the displacements of the structure under the static load and the dynamic load, respectively. The displacement mode of the first order vibration mode of the structure is the lateral oscillation along the horizontal direction, which is similar to the vibration characteristics of the single degree of freedom (SDOF) system, gathered the quality on the top of the pile. It is assumed that the dynamic response of the structure can be calculated using the theory formula based on the displacement dynamic amplification coefficient of the SDOF system.

Dynamic response analysis of structure under wave loads
According to the structural dynamics, the displacement dynamic amplification coefficient of the SDOF system with the damping under simple harmonic loads is determined by Eq. (4) (Tang, 2002): where α d is the dynamic amplification coefficient, γ is the frequency ratio of the load and the structure which is equal to the period ratio T 1 /T 2 of the structure and the load, and ξ is the damping ratio of the structure.
The upper structure quality of the basic model is m for the embedded fixed model. The basic period of the structure is varied by the change of the upper structure quality. The upper structure qualities are taken as 0.1m, 0.5m, m, 2m, 3m, 4m, 7m and 10m. The period of wave loads is a constant, 10.5 s.
The basic period of the structure can be obtained by the model analysis. Then, the period ratio T 1 /T 2 of the structure and the load can be determined. The displacement dynamic amplification coefficient corresponding to each model is calculated by the theory formula based on the displacement  dynamic amplification coefficient of the SDOF system under simple harmonic loads. The displacement dynamic amplification coefficient is determined by Eq. (4) on the basis of the FEM results. Comparisons of both the calculated results are shown in Fig. 10.
The upper structure quality m is a constant. The periods of the wave load are taken as 6.5 s, 8.5 s, 10.5 s, 12.5 s, and 14.5 s to analyze the dynamic response. The displacement dynamic amplification coefficients are calculated by the theory formula based on the dynamic amplification coefficient of the SDOF system under simple harmonic loads and the FEM. Comparisons of both calculation results are shown in Fig. 11.
Curves obtained from the above two calculation methods are basically coincident, which indicates that for the allvertical-piled wharf in offshore deep water under wave loads, the dynamic response characteristics determined by the FEM using the embedded fixed model is similar to those determined by the SDOF system under simple harmonic loads.
The upper structure qualities are taken as 0.1m, 0.5m, m, 1.5m, 2m and 2.5m for the structure-foundation interaction model. The periods of wave loads are taken as 8 s and 10.5 s. The displacement dynamic amplification coefficients are calculated by the theory formula based on the dynamic amp-lification coefficient of the SDOF system under simple harmonic loads and the FEM. Comparisons of both calculation results are shown in Figs. 12 and 13.
Curves obtained from the above two calculation methods are basically coincident, which indicates that for the allvertical-piled wharf in offshore deep water under wave loads, the dynamic response characteristics determined by the FEM using the structure-foundation interaction model is similar to those determined according to the SDOF system under simple harmonic loads.
In summary, the displacement dynamic amplification coefficient of the all-vertical-piled wharf in offshore deep water under periodic loads, such as wave loads, can be calculated by the theory formula of the SDOF system under simple harmonic loads.

Dynamic response analysis of structure under impact
loads It is generally assumed that the ship impact load is the triangular or semi-sinusoidal load. In the present paper, the impact load is the semi-sinusoidal pulse load. The time under the ship impact loading is different for structures with different stiffnesses and different types of the rubber fenders. So, the dynamic response should be analyzed for the different time under the impact loading.
According to the structural dynamics, the frequency ratio γ is defined as the ratio of the load frequency ω to the ba-    sic frequency λ of the structure. The analysis theory of the SDOF system under impact loads are as follows.
When γ is smaller than 1, the maximum dynamic response appears in the forced vibration stage. The time when the maximum dynamic response occurs and the displacement dynamic amplification coefficient are calculated by (Tang, 2002): When γ is equal to 1, the displacement dynamic amplification coefficient is calculated by (Tang, 2002): When γ is larger than 1, the maximum response appears in the free vibration stage. The displacement dynamic amplification coefficient is calculated by: The upper structure quality is taken as m for the embedded fixed model. The time t 1 under the impact loading is taken as 0.5 s, 1 s, 2 s, 2.5 s, 3 s, 4 s and 5 s. The displacement dynamic amplification coefficients are calculated by the theory formula based on the dynamic amplification coefficient of the SDOF system under impact loads and by the finite element method. Comparisons of both calculation results are shown in Fig. 14.
The upper structure quality is taken as 0.1m, 0.5m, m, 2m, 3m, 4m, 7m and 10m for the embedded fixed model. The time t 1 under the impact loading is taken as 1 s. The displacement dynamic amplification coefficients are calculated by the theory formula of the SDOF system under impact loads and by the finite element method. Comparisons of both calculation results are shown in Fig. 15.
Curves obtained from the above both methods are basically in agreement with each other, which indicates that for the all-vertical-piled wharf in offshore deep water under impact loads, the dynamic response characteristics determined by the FEM using the embedded fixed model is similar to those determined by the SDOF system under impact loads.
The upper structure quality is taken as 0.1m, 0.5m, m, 1.5m, 2m and 2.5m for the structure-foundation interaction model and the time t 1 under the impact loading is taken as 1 s. In addition, the upper structure quality is taken as m for the structure-foundation interaction model and the time t 1 under the impact loading is taken as 0.    al-piled wharf in the offshore deep water under impact loads, the dynamic response characteristics determined by the FEM using the structure-foundation interaction model is similar to those determined according to the SDOF system under impact loads.
In summary, the displacement dynamic amplification coefficient of the all-vertical-piled wharf in offshore deep water under impact loads can be calculated by the theory formula of the SDOF under impact loads.
4.3 Verification of the dynamic response simplified calculation method According to the above analysis, the displacement dynamic amplification coefficient of the all-vertical-piled wharf in offshore deep water can be calculated by the theory formula of the SDOF system. On this basis, the simplified calculation method of the dynamic response for the all-vertical-piled wharf is developed in combination with P-Y curve method. It is that, P-Y curve method can be used to calculate the static displacement of the wharf, and then α d , the displacement dynamic amplification coefficient, can be calculated by the theory formula of the SDOF system under wave loads or impact loads. Therefore, the dynamic response of the structure can be obtained through the static calculation results multiplied by the displacement dynamic amplification coefficient α d .
The displacement of the pile is calculated by the dynamic simplified calculation method proposed above. Comparisons of the calculated results by the dynamic simplified calculation method and the FEM are shown in Tables 8 and 9. It is shown that both results are in good agreement, which verifies the feasibility of the established dynamic simplified calculation method.

Conclusions
All-vertical-piled wharf is a new type of high-piled wharf which is built in offshore deep water. However, its bearing behavior and failure mode are unclear, and there is lack of practical calculation methods. For the above problems, some research works were conducted. And conclusions are drawn follows.
(1) Research results show that the plastic failure of the pile body is the controlled factor for the failure of the allvertical-piled wharf in offshore deep water under the lateral loading. And it was suggested that the lateral limit bearing capacity of the all-vertical-piled wharf in offshore deep water was determined based on the 'plastic hinge' failure mode.
(2) Combined with the P-Y curve method, the simplified calculation method of the static ultimate bearing capacity was established for the all-vertical-piled wharf under the lateral static load. The rationality of the simplified method was verified by comparison with the results of the finite element method.
(3) Through calculations, it was found that for the all-vertical-piled wharf under wave loads and impact loads, the displacement dynamic amplification coefficient can be calculated by the theory formula of the SDOF system. On this basis, the dynamic simplified calculation method was proposed in combination with the P-Y curve method. That is, the dynamic response of the structure can be obtained through the static calculation results multiplied by the displacement dynamic amplification coefficient. The feasibility of the dynamic simplified calculation method was verified by comparison with the results of the FEM.