Many scholars have carried out a series of studies on bridge structure damage caused by near-fault earthquakes. Jiang and Yang [5,6] used the vulnerable surface model to study the sensitive regions under different damage states and discuss the mechanism of structural brittleness changes caused by earthquake damage. Furthermore, many scholars have studied the influence of vertical seismic action on bridge structures. Sung [7] took the Santa Monica bridge as the research object and compared the experiment with the Zeus-NL hybrid simulation method to study the influence of axial force changes on the bridge structure. Rodrigues [8] applied different two-way horizontal loads and variable axial forces to reinforced concrete columns through experiments, and the test results showed that the horizontal deformation of concrete columns under irregular axial forces was less than under constant pressures. Quantitative studies have been carried out on the effects of tomography on complex structural anomalies [9–11]. Zuo and Yang [12,13] studied and developed a near-fault pulse seismic ductility spectrum based on machine learning and put forward suggestions for the seismic design of structures in near-fault areas. Zhong [14–16] analyzed the vulnerability of bridges through algorithms considering a nearfield earthquake, especially pulse excitation. The above studies considered the situation of bridge impairment under bidirectional excitation for different models and obtained some results. However, for vertical seismic excitation, the numerical value specified in the code is more used for analysis, ignoring the large vertical excitation in the near-fault area.

In China, most girder bridges adopt rubber bearings, which lack tensile capacity. A near-fault vertical earthquake can cause the separation of the girder and the support. For the bridge collision, the current stage mainly considers the transverse adjacent beam collision [17], while the research on the vertical collision is less. The vertical collision is

**Citation:** An, W.; Zhou, L.; Kang, M.; Yang, H.; Mo, L. Failure Mode Analysis of Bridge Pier Due to Eccentric Impact Based on Separation of Pier and Beam. *Sustainability* **2023**, *15*, 1435. https://doi.org/10.3390/ su15021435

Academic Editors: Kai Wei, Mingjin Zhang, Jian Zhong and Yutao Pang

Received: 7 December 2022 Revised: 5 January 2023 Accepted: 11 January 2023 Published: 12 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

close to the beam rod collision. Ruan [18] analyzed the dynamic behavior of a moving free beam impacting the tip of a cantilever beam using the modal approximation technique; however, this collision condition belongs to a single collision, and the actual situation is that the pier and beam have multiple separation contacts. Yang [19] used theoretical methods to solve the vertical impact force of the pier beam under multiple separation conditions. However, Yang's study only considered the vertical excitation and ignored the horizontal displacement of the bridge caused by the separation, and the analysis of collision process takes the structure as a whole. When the compressive stiffness of the bearing is low, the calculated force is unreasonable. The authors of [20,21] analyzed the influence of separation on the failure of a bridge pier under vertical and longitudinal excitation, but did not provide a qualitative discussion on separation conditions and only considered the flexural failure of the bridge pier.

In this study, the influences of pier–beam separation on bridge force and the displacement response are calculated by establishing a reliable theoretical method for dynamic bridge response. By using the mode superposition method [22], we calculate the limit solution of the vertical impact force of the pier beam and the longitudinal deformation of the pier top after the first pier–beam separation. Through calculation, we analyze the bridge separation under different excitation frequencies and amplitudes, and study the influence of dynamic response changes caused by separation on the failure mode of the bridge pier.

### **2. Structural Model**

This paper mainly studies the separation of a pier and a beam and the impact of the separation on pier failure, which lays a foundation for the study of the combined influence of multispan structures in the future. Therefore, a double-span prestressed box girder bridge was selected in this paper, as shown in Figure 1. Both ends of the main beam are hinged with the ground, and the pier bottom is fixed. The displacements of horizontal excitation and vertical excitation are *D*(t) and *B*(t), respectively. The axial stiffness of the bearing is *KC* and the shear stiffness is *KV*. Three assumptions have been made, as follows:

	- (3) Vertical and longitudinal dynamic responses are calculated separately, ignoring the coupling effect between them.

**Figure 1.** Theoretical solution model of the bridge structure. **Figure 1.** Theoretical solution model of the bridge structure.

#### **3. Bridge Dynamic Response Calculation 3. Bridge Dynamic Response Calculation**

*dt*

tion, *B* ( ) τis

#### *3.1. Vertical Dynamic Response 3.1. Vertical Dynamic Response*

1 *p*

The vertical dynamic response of the bridge refers to [20,21]. Near-fault earthquakes may contain pulse-type earthquakes, which transfer most of the energy of the ground mo-The vertical dynamic response of the bridge refers to [20,21]. Near-fault earthquakes may contain pulse-type earthquakes, which transfer most of the energy of the ground

 rectangular excitation sinusoidal excitation triangle excitation **Figure 2.** Acceleration curves of different pulse excitations.

ratio of the peak force is 111 *ppp* : : 2 :1: 4 =

α

 ×

−

τ

tion to structures in a short time. The impulse excitation is usually expressed as the superposition of two or more simple excitations, including rectangular excitation, sinusoidal excitation, and triangular excitation, as shown in Figure 2. To study the influence of dif-

3 *p*

When the impulse generated by the three kinds of pulse excitation is the same, the

( ) 8 triangle excitation,0< 0.5

= − ≤

*B t*

8 ( ) triangle excitation,0.5 <

−− ≤

Figure 3 shows the relationship between pier–beam vertical contact force and excitation conditions. The range of excitation duration is 0.25~2ଵ, and the range of excitation

 π

2 rectangular excitation

sin sinusoidal excitation

(1)

*d dd*

*t tt*

π

max

 ωτ

max 0 max

α τ

π

π

α

*d*

τ

*t*

max

*d*

*t*

α

π

*dt*

. For different levels of pulse excita-

τ

 τ *d*

pulse excitation, the impulse of three kinds of pulse excitation is the same.

*dt*

2 *p*

D(t) B(t)

motion to structures in a short time. The impulse excitation is usually expressed as the superposition of two or more simple excitations, including rectangular excitation, sinusoidal excitation, and triangular excitation, as shown in Figure 2. To study the influence of different forms of pulse excitation on the vertical dynamic response of a bridge under vertical pulse excitation, the impulse of three kinds of pulse excitation is the same. tion to structures in a short time. The impulse excitation is usually expressed as the superposition of two or more simple excitations, including rectangular excitation, sinusoidal excitation, and triangular excitation, as shown in Figure 2. To study the influence of different forms of pulse excitation on the vertical dynamic response of a bridge under vertical pulse excitation, the impulse of three kinds of pulse excitation is the same.

B(t)

D

**Figure 1.** Theoretical solution model of the bridge structure.

C

*U*, ξ

**3. Bridge Dynamic Response Calculation**

*3.1. Vertical Dynamic Response*

*Sustainability* **2023**, *15*, x FOR PEER REVIEW 3 of 17

y

L L

A O B

H

*Kc*

*Kv*

D(t) B(t)

x

*d*

D(t)

The vertical dynamic response of the bridge refers to [20,21]. Near-fault earthquakes may contain pulse-type earthquakes, which transfer most of the energy of the ground mo-

**Figure 2.** Acceleration curves of different pulse excitations. **Figure 2.** Acceleration curves of different pulse excitations. and amplitude of excitation acceleration. When the excitation duration is 2ଵ and the ex-

When the impulse generated by the three kinds of pulse excitation is the same, the ratio of the peak force is 111 *ppp* : : 2 :1: 4 = π π . For different levels of pulse excitation, *B* ( ) τ is When the impulse generated by the three kinds of pulse excitation is the same, the ratio of the peak force is *p*1 : *p*1 : *p*1 = 2/*π* : 1 : 4/*π*. For different levels of pulse excitation, .. *B*(*τ*) is citation acceleration amplitude is 1.0 g, the minimum contact force is 4.3 MN. Under sinusoidal pulse excitation, the length of the pulse excitation and the amplitude of the excitation acceleration have more influence on the vertical dynamic response than rectangular and triangular pulse excitation. When the pulse excitation duration is

response of the pier beam increases monotonically with the increase in excitation duration

$$\ddot{B}(\tau) = \begin{cases} -\frac{2}{\pi}a\_{\text{max}} & \text{rectangular excitation} \\ a\_{\text{max}} \times \sin \omega\_0 \tau & \sin \text{usual excitation} \\ -\frac{8a\_{\text{max}}\tau}{\pi t\_d} & \text{triangle excitation}, 0 < \tau \le 0.5t\_d \\ -\frac{8a\_{\text{max}}}{\pi t\_d}(t\_d - \tau) & \text{triangle excitation}, 0.5t\_d < \tau \le t\_d \end{cases} \tag{1}$$

max max ( ) 8 triangle excitation,0< 0.5 8 ( ) triangle excitation,0.5 < *d d dd d B t t t tt t* τ α τ τ π α τ τ π = − ≤ −− ≤ (1) Figure 3 shows the relationship between pier–beam vertical contact force and excitation conditions. The range of excitation duration is 0.25~2ଵ, and the range of excitation Figure 3 shows the relationship between pier–beam vertical contact force and excitation conditions. The range of excitation duration is 0.25 ∼ 2*T*1, and the range of excitation acceleration amplitude is 0.25~1.0 g. The value *T*1 is the vertical first-order natural period of the bridge. Under the rectangular pulse excitation, the contact force changes the least. When the pulse excitation time exceeds *T*1, the contact force remains unchanged, and the minimum contact force is 8.6 MN. Under triangular pulse excitation, the vertical dynamic response of the pier beam increases monotonically with the increase in excitation duration and amplitude of excitation acceleration. When the excitation duration is 2*T*1 and the excitation acceleration amplitude is 1.0 g, the minimum contact force is 4.3 MN. pier beam occurs at this time. According to the vertical dynamic responses of different pulse excitation forms, it can be concluded that the vertical separation is most easily caused by sinusoidal pulse excitation. However, a single sinusoidal pulse causes a significant excitation acceleration requirement for the pier–beam separation, which exceeds the specification. At the same time, the pulse excitation lasts for a short time, while the periodic excitation can last for a long time. When the excitation frequency is close to the natural vertical frequency, there is a large dynamic response. Next, the dynamic response of the bridge is analyzed theoretically by simple harmonic excitation.

**Figure 3.** The minimum force of pier beam under different vertical pulse excitations. **Figure 3.** The minimum force of pier beam under different vertical pulse excitations.

For the near-fault seismic response, the maximum value of the acceleration response spectrum is concentrated in the range T = 0.1~0.5 s. The high-order frequency has little influence on the excitation response of the structure. According to the conventional engi-

Figure 4 shows the separation of the piers and beams under different excitation frequencies and accelerations. When the ଵ (vertical first-order natural frequency) is close to the (excitation frequency), a slight vertical acceleration results in pier–beam separation. When the deviates from the ଵ, the vertical excitation acceleration required for the separation increases significantly. When = 40 rad/s, the vertical excitation acceleration needs to reach 2.2 g to cause pier–beam separation. When the is close to the ଶ

are taken.

Under sinusoidal pulse excitation, the length of the pulse excitation and the amplitude of the excitation acceleration have more influence on the vertical dynamic response than rectangular and triangular pulse excitation. When the pulse excitation duration is *T*1, the vertical contact force is the lowest, which is because the vertical pulse excitation frequency is equal to the natural vertical frequency at this time. The resonance causes the vertical dynamic response of the bridge to surge. When the pulse excitation duration is *T*1, the vertical contact force is −0.58 MN, which means that the vertical separation of the pier beam occurs at this time.

According to the vertical dynamic responses of different pulse excitation forms, it can be concluded that the vertical separation is most easily caused by sinusoidal pulse excitation. However, a single sinusoidal pulse causes a significant excitation acceleration requirement for the pier–beam separation, which exceeds the specification. At the same time, the pulse excitation lasts for a short time, while the periodic excitation can last for a long time. When the excitation frequency is close to the natural vertical frequency, there is a large dynamic response. Next, the dynamic response of the bridge is analyzed theoretically by simple harmonic excitation.

For the near-fault seismic response, the maximum value of the acceleration response spectrum is concentrated in the range T = 0.1~0.5 s. The high-order frequency has little influence on the excitation response of the structure. According to the conventional engineering calculation, the first five orders of the modal truncation of the dynamic response are taken.

Figure 4 shows the separation of the piers and beams under different excitation frequencies and accelerations. When the *ω*1 (vertical first-order natural frequency) is close to the *ω*0 (excitation frequency), a slight vertical acceleration results in pier–beam separation. When the *ω*0 deviates from the *ω*1, the vertical excitation acceleration required for the separation increases significantly. When *ω*0 = 40 rad/s, the vertical excitation acceleration needs to reach 2.2 g to cause pier–beam separation. When the *ω*0 is close to the *ω*2 (vertical second-order natural frequency), the smaller excitation acceleration also causes pier–beam separation. Compared with the separation interval near the *ω*1, the separation interval of the second-order vertical natural frequency is narrower. The excitation acceleration required for the separation is greater, essentially exceeding the standard value. The separation of the bridge piers and beams mainly occurs when *ω*0 ≈ *ω*1. *Sustainability* **2023**, *15*, x FOR PEER REVIEW 5 of 17 (vertical second-order natural frequency), the smaller excitation acceleration also causes pier–beam separation. Compared with the separation interval near the ଵ, the separation interval of the second-order vertical natural frequency is narrower. The excitation acceleration required for the separation is greater, essentially exceeding the standard value.

> When pier and beam separation occurs, the natural frequency of the structure changes, resulting in a different dynamic response from when there is no separation. When the pier and beam collide in the vertical direction, there may be a large deviation in

> lyze the extreme value of longitudinal pier deformation under the condition of pier–beam

For the calculation of the longitudinal dynamic response of the bridge, refer to [20,21], where the time of the longitudinal separation is taken as the time of the vertical separation. In actual earthquakes, there is a time difference between horizontal and vertical earthquakes due to their different transmission wave velocities. In addition, many separation–collision phenomena occur during the seismic excitation time. By referring to the longitudinal displacement solution in [20], the theoretical solution of the longitudinal displacement response of a bridge pier after separation can be obtained through calculation, as shown in Equation (2). Because the natural frequency of the pier decreases after separation, the longitudinal deformation of the structure increases. Considering the phase difference, the extreme value of the longitudinal deformation of the bridge pier after separation is the sum of the extreme value of the initial state (the first two terms) and the excita-

The separation of the bridge piers and beams mainly occurs when ≈ ଵ.

**Figure 4.** Vertical separation conditions of pier beam. **Figure 4.** Vertical separation conditions of pier beam.

*3.2. Longitudinal Dynamic Response*

tion state (the third term).

separation.

When pier and beam separation occurs, the natural frequency of the structure changes, resulting in a different dynamic response from when there is no separation. When the pier and beam collide in the vertical direction, there may be a large deviation in the longitudinal direction of the pier top. The longitudinal deformation of the pier top directly affects the degree of eccentric vertical collision. Therefore, it is necessary to analyze the extreme value of longitudinal pier deformation under the condition of pier–beam separation.

#### *3.2. Longitudinal Dynamic Response*

For the calculation of the longitudinal dynamic response of the bridge, refer to [20,21], where the time of the longitudinal separation is taken as the time of the vertical separation. In actual earthquakes, there is a time difference between horizontal and vertical earthquakes due to their different transmission wave velocities. In addition, many separation–collision phenomena occur during the seismic excitation time. By referring to the longitudinal displacement solution in [20], the theoretical solution of the longitudinal displacement response of a bridge pier after separation can be obtained through calculation, as shown in Equation (2). Because the natural frequency of the pier decreases after separation, the longitudinal deformation of the structure increases. Considering the phase difference, the extreme value of the longitudinal deformation of the bridge pier after separation is the sum of the extreme value of the initial state (the first two terms) and the excitation state (the third term).

$$
\begin{split}
\overline{\mathcal{W}}\_{d}(\mathbf{x},t) &= \begin{bmatrix}
\sum\_{n=1}^{5} \overline{\boldsymbol{\sigma}}\_{nr}(\boldsymbol{\xi})\boldsymbol{q}\_{nr}(t\_{1}) + \sum\_{n=1}^{5} \frac{\boldsymbol{A}\_{nr}\boldsymbol{B}\_{nr}\omega\_{0}^{2}\boldsymbol{B}\_{0}}{(\omega\_{0}^{2} - \omega\_{nr}^{2})} \sin\omega\_{0}t\_{1} \\
+ \sum\_{n=1}^{5} \overline{\boldsymbol{\sigma}}\_{nr}(\boldsymbol{\xi})\boldsymbol{q}\_{nr}(t\_{1}) + \sum\_{n=1}^{5} \frac{\boldsymbol{A}\_{nr}\boldsymbol{B}\_{nr}\omega\_{0}^{2}\boldsymbol{B}\_{0}}{\omega\_{0}\omega\_{0}\omega\_{0}} \omega\_{0}\cos\omega\_{0}t\_{1} \\
\end{bmatrix}
\end{split}
\tag{2}
$$

Here, *ϕnr* is the pier wave mode function in the separation phase, *qnt* is the pier time function in the separation phase, *ω*0 is the excitation frequency, *ωnr* is the pier natural frequency in the separation phase, and *Anr* and *Bnr* are the correlation coefficients of the wave mode function.

The first part vibrates at the natural frequency of the separated pier, and the amplitude is not affected by the frequency. By calculation, the maximum absolute deformation value is 16.7 mm. The second part vibrates with the natural frequency of the separated pier, and the amplitude is affected by the frequency. By calculation, the maximum absolute deformation value is 32.6 mm. The maximum deformation obtained by the first two calculations is 34.5 mm. The third part vibrates at the excitation frequency. The amplitude is affected by the frequency, and the maximum value is 8.2 mm. Considering the different vibration frequencies of the first, second, and third terms, when considering the separation, the absolute value of the maximum deformation at the top of the pier increases to 42.7 mm, an increase of 120%. This situation is caused by the numerical difference between the *ωnr* and the *ωn* after separation.

Figure 5 shows the longitudinal deformation of the pier top under different excitation frequencies. When the peak excitation acceleration is 0.2 g, the pier and beam are always in contact, and the maximum deformation of the pier top occurs in the interval *ω*1 = *ω*0. When the peak excitation acceleration of the excitation is 0.5 g and 0.8 g, vertical excitation causes pier–beam separation. When *ω*0 ≈ *µ*1, the peak value of the pier roof deformation appears; *µ*1 is the first longitudinal natural frequency of the structure.

5 5 2

*n n nr*

5 5 2

*n n nr nr*

2 2 0

ω

) *nr*

*d nr nr nr*

*AB B W xt q t t t t*

1 1 0

1 1 0

2 0 0

ω

= =

*nr nr*

= =

ϕξ

ϕξ

*nr nr*

*n nr*

ω ω

*AB B*

*nr*

1 0

the and the after separation.

−

ω

(

−

5

=

of the wave mode function.

0 0 1 01 1 2 2

( , ) ( ) ( ) sin cos ( ) ( )

ω ω

 +

*nr nr*

=+ − −

( ) ( ) cos ( ) sin ( )

ωω

*nr*

 ω*t*

ω

sin

appears; ଵ is the first longitudinal natural frequency of the structure.

*nr nr*

*AB B qt t*

− + −

ω

ω

0 0 1 0 01 2 2

Here, ത is the pier wave mode function in the separation phase, ௧ is the pier time function in the separation phase, is the excitation frequency, is the pier natural frequency in the separation phase, and and are the correlation coefficients

The first part vibrates at the natural frequency of the separated pier, and the amplitude is not affected by the frequency. By calculation, the maximum absolute deformation value is 16.7 mm. The second part vibrates with the natural frequency of the separated pier, and the amplitude is affected by the frequency. By calculation, the maximum absolute deformation value is 32.6 mm. The maximum deformation obtained by the first two calculations is 34.5 mm. The third part vibrates at the excitation frequency. The amplitude is affected by the frequency, and the maximum value is 8.2 mm. Considering the different vibration frequencies of the first, second, and third terms, when considering the separation, the absolute value of the maximum deformation at the top of the pier increases to 42.7 mm, an increase of 120%. This situation is caused by the numerical difference between

Figure 5 shows the longitudinal deformation of the pier top under different excitation frequencies. When the peak excitation acceleration is 0.2 g, the pier and beam are always in contact, and the maximum deformation of the pier top occurs in the interval ଵ = . When the peak excitation acceleration of the excitation is 0.5 g and 0.8 g, vertical excitation causes pier–beam separation. When ≈ ଵ, the peak value of the pier roof deformation

 ω  ω

 ω  ω

 ω

1

(2)

*t t*

*nr*

ω

**Figure 5.** Longitudinal relative displacements of pier beams during different excitation periods. **Figure 5.** Longitudinal relative displacements of pier beams during different excitation periods. *3.3. Dynamic Response during a Collision*

#### *3.3. Dynamic Response during a Collision* In the previous study, the vertical impact force of the structure was calculated using

In the previous study, the vertical impact force of the structure was calculated using the model in Figure 6. In the actual collision process, the pier is in a longitudinal bending state. The eccentric distance between the vertical collision position of the pier beam and the pier axis is ∆*x*. To simplify the calculation, the structural collision model is simplified as Figure 6b. In the process of a vertical collision, the structure is affected by three forces: the vertical collision force, the bearing shear force, and the inertial force of the forced vibration of the piers. The indirect mode method [22] is adopted for the calculation, and the change in the bending strain energy of the bridge pier caused by the initial offset of the bridge pier during collision is ignored. the model in Figure 6. In the actual collision process, the pier is in a longitudinal bending state. The eccentric distance between the vertical collision position of the pier beam and the pier axis is ∆. To simplify the calculation, the structural collision model is simplified as Figure 6b. In the process of a vertical collision, the structure is affected by three forces: the vertical collision force, the bearing shear force, and the inertial force of the forced vibration of the piers. The indirect mode method [22] is adopted for the calculation, and the change in the bending strain energy of the bridge pier caused by the initial offset of the bridge pier during collision is ignored.

**Figure 6.** Vertical collision calculation model. **Figure 6.** Vertical collision calculation model.

at the time of collision are

*L*

ρ

the bending strain energy of the pier.

When the longitudinal deformation of the pier reaches its extreme value, the inertial force of the pier is very small; therefore, it is not urgent to ignore it here. During eccentric collision, the kinetic energy and strain energy of the main beam are converted into the axial compression strain energy of the bearing, and to the bending strain energy of the When the longitudinal deformation of the pier reaches its extreme value, the inertial force of the pier is very small; therefore, it is not urgent to ignore it here. During eccentric collision, the kinetic energy and strain energy of the main beam are converted into the axial compression strain energy of the bearing, and to the bending strain energy of the main

1

2 1

Where ଵ is the initial time of the collision between the main beam and the support.

The vertical collision diagram of the bridge is shown in Figure 7. During the collision, the strain changes of the bridge include the bending strain energy of the main beam and

2 5

*x*

ϕ

main beam and the pier. The kinetic energy and bending strain energy of the main beam

(3)

*b bb n L n*

1 () () 2

=

 ϕ

*T A x q t dx*

*b b nb n L n*

− =

− =

1

beam and the pier. The kinetic energy and bending strain energy of the main beam at the time of collision are

$$\begin{aligned} T\_b &= \frac{1}{2} \int\_{-L}^{L} \rho A\_b \left[ \sum\_{n=1}^{5} \overline{\varphi}\_{nb}(\mathbf{x}) \dot{q}\_n(t\_1) \right]^2 d\mathbf{x} \\\mathcal{U}\_b &= \frac{1}{2} \int\_{-L}^{L} E\_b I\_b \left[ \sum\_{n=1}^{5} \frac{\partial^2 \overline{\varphi}\_{nb}(\mathbf{x})}{\partial \mathbf{x}^2} q\_n(t\_1) \right]^2 d\mathbf{x} \end{aligned} \tag{3}$$

where *t*1 is the initial time of the collision between the main beam and the support.

The vertical collision diagram of the bridge is shown in Figure 7. During the collision, the strain changes of the bridge include the bending strain energy of the main beam and the bending strain energy of the pier. *Sustainability* **2023**, *15*, x FOR PEER REVIEW 8 of 17

**Figure 7.** Deformation of vertical impact bridge. **Figure 7.** Deformation of vertical impact bridge.

process is

stiffness of the pier.

The strain energy generated by the self-weight of the main beam during the collision process is The strain energy generated by the self-weight of the main beam during the collision process is

$$\mathcal{W}\_b = \frac{L^3}{E\_b I\_b} \left( \frac{2q^2 L^2}{15} - \frac{F\_p^{\,^2}}{24} \right) \tag{4}$$

15 24 *b b E I* The strain energy generated by the self-weight of the main beam during the collision process is

$$\mathcal{W}\_{\rm r} = \frac{H}{6E\_{\rm r}I\_{\rm r}} \left( 3F\_p \, ^2 \eta^2 \Delta \text{x}^2 - F\_v \, ^2 H^2 \right) \tag{5}$$

( ) 22 2 2 2 3 6 *r pv H W F x FH E I* = Δ− η (5) where *η* is the eccentric impact amplification coefficient; for the coefficient value, we refer to [21]. The value *Kv* is the bearing shear force.

*r r* According to the energy conservation in the collision process, it can be concluded that

$$T\_b + \mathcal{U}\_b + \mathcal{W}\_b = \frac{1}{2} \mathcal{K} \Delta y^2 + \mathcal{W}\_r \tag{6}$$

that 1 2 where *K* = *K*1*K*2 *K*1+*K*2 . *K*1 is the bearing axial compression stiffness and *K*2 is the axial stiffness of the pier.

2 *T U W Ky W bbb r* + + = Δ+ (6) where 1 2 1 2 *K K K K K* = + . ଵ is the bearing axial compression stiffness and ଶ is the axial Under specific excitation conditions, piers and beams are separated and vertical collisions occur. The first collision is selected here for study. Figure 8 is the calculation flow chart of the vertical displacement response of the bridge under vertical excitation. When pier–beam separation does not occur, the calculation ends when the calculation time is 2 s. When the vertical contact force of the pier beam is zero, the analysis ends.

Under specific excitation conditions, piers and beams are separated and vertical collisions occur. The first collision is selected here for study. Figure 8 is the calculation flow chart of the vertical displacement response of the bridge under vertical excitation. When

**Figure 8.** Flow chart of the bridge vertical displacement calculation program. **Figure 8.** Flow chart of the bridge vertical displacement calculation program.

To ensure the accuracy of the calculation, an appropriate time-step increment must be selected. For the choice of time-step increment, since it is necessary to clearly express the transmission characteristics of the bending wave and the longitudinal wave in the main beam and pier, the time-step increment should not be too large. The longitudinal wave velocity of the pier is = ඥ⁄ = 3492 m/s, and the bending wave velocity of the pier is = ඥ⁄ = 1060 m/s. The maximum time-step increment is less than the time taken by the bending wave and the longitudinal wave to pass through the whole To ensure the accuracy of the calculation, an appropriate time-step increment must be selected. For the choice of time-step increment, since it is necessary to clearly express the transmission characteristics of the bending wave and the longitudinal wave in the main beam and pier, the time-step increment should not be too large. The longitudinal wave velocity of the pier is *cr* = p *Er*/*ρr* = 3492 m/s, and the bending wave velocity of the pier is *ar* = p *ErIr*/*ρrAr* = 1060 m/s. The maximum time-step increment is less than the time taken by the bending wave and the longitudinal wave to pass through the whole pier. A time-step increment of 0.001 s was selected for the calculation.

pier. A time-step increment of 0.001 s was selected for the calculation. Figure 9 shows the change in the vertical collision force in the process of the first impact of the pier calculated by the indirect mode superposition method [2]. The contact force fluctuates with the first peak of 28.92 MN and the approximate maximum impact force of 30.84 MN. The maximum value does not appear in the first-wave peak because Figure 9 shows the change in the vertical collision force in the process of the first impact of the pier calculated by the indirect mode superposition method [2]. The contact force fluctuates with the first peak of 28.92 MN and the approximate maximum impact force of 30.84 MN. The maximum value does not appear in the first-wave peak because the collision process is also affected by the vertical excitation force vibration.

the collision process is also affected by the vertical excitation force vibration. Figure 10 shows the vertical impact force under different levels of vertical peak excitation acceleration and excitation frequency. From the analysis, we concluded the following: (1) the maximum impact force of the pier beam increases monotonically with the increase in vertical excitation acceleration, and the slight fluctuation in the middle is due to the fact that the upthrow velocity in the middle of the main beam has a certain contingency at the vertical separation moment; (2) the collision mainly occurs when *ω*0 ≈ *ω*1, and the closer the two frequencies, the greater the collision force; (3) when the excitation frequency/natural vertical frequency is <0.8 or >1.2, the change in excitation frequency has little influence on the maximum impact force of the pier beam.

*Sustainability* **2023**, *15*, x FOR PEER REVIEW 10 of 17

30.84MN

**Figure 9.** The vertical contact force of pier beam under the first impact. **Figure 9.** The vertical contact force of pier beam under the first impact. little influence on the maximum impact force of the pier beam.

**Figure 10.** Vertical contact force of pier beam. **Figure 10.** Vertical contact force of pier beam.

#### **4. Pier Failure**

15

20

25

30

28.92MN

35

**4. Pier Failure**  The previous research only considered the influence of the dynamic response change of the pier top and the increase in the vertical contact force on the bending failure of the pier under separation—collision coupling conditions. However, pier failure includes compression failure, bending failure, and shear failure. Nevertheless, the influence of the change in the dynamic response of bridges caused by separation on the failure mode of The previous research only considered the influence of the dynamic response change of the pier top and the increase in the vertical contact force on the bending failure of the pier under separation—collision coupling conditions. However, pier failure includes compression failure, bending failure, and shear failure. Nevertheless, the influence of the change in the dynamic response of bridges caused by separation on the failure mode of bridge piers still requires research.

#### bridge piers still requires research. *4.1. Compression Failure*

**Figure 10.** Vertical contact force of pier beam. **4. Pier Failure**  The previous research only considered the influence of the dynamic response change Under longitudinal excitation, a pier is always under eccentric compression. The previous studies indicate that when the excitation frequency is close to the natural vertical frequency of the bridge, the vertical impact force of the pier beam increases monotonically with the increase in the vertical excitation peak value. Moreover, the separation may increase the longitudinal deformation at the top of the pier and affect the allowable pressure

of the pier top and the increase in the vertical contact force on the bending failure of the pier under separation—collision coupling conditions. However, pier failure includes com-

change in the dynamic response of bridges caused by separation on the failure mode of

bridge piers still requires research.

of the pier. Three typical seismic waves from Imperial Valley, Northridge, and Kobe were used to analyze the time history of bridge pier damage. The excitation acceleration in the vertical direction is the excitation acceleration in the horizontal direction multiplied by the ratio parameter α(V/H). To simplify the calculation, α = 1 is taken here. The maximum and allowable pressure at the bottom of the pier under different excitation amplitudes and the maximum pressure at the collision stage adopt the limit solution. In the case of eccentric compression, the bearing capacity of the bridge pier should conform to the requirements of Equation (7), as specified in the code [23].

the ratio parameter α(V/H). To simplify the calculation, α = 1 is taken here.

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*4.1. Compression Failure*

The maximum and allowable pressure at the bottom of the pier under different excitation amplitudes and the maximum pressure at the collision stage adopt the limit solution. In the case of eccentric compression, the bearing capacity of the bridge pier should conform to the requirements of Equation (7), as specified in the code [23]. 0 *N d u cd* γ ≤ *n Af* (7) Figure 11 shows the changes in pier bottom pressure under different excitation am-

$$
\gamma\_0 \mathbf{N}\_d \le n\_u \mathbf{A} f\_{cd} \tag{7}
$$

Under longitudinal excitation, a pier is always under eccentric compression. The previous studies indicate that when the excitation frequency is close to the natural vertical frequency of the bridge, the vertical impact force of the pier beam increases monotonically with the increase in the vertical excitation peak value. Moreover, the separation may increase the longitudinal deformation at the top of the pier and affect the allowable pressure of the pier. Three typical seismic waves from Imperial Valley, Northridge, and Kobe were used to analyze the time history of bridge pier damage. The excitation acceleration in the vertical direction is the excitation acceleration in the horizontal direction multiplied by

Figure 11 shows the changes in pier bottom pressure under different excitation amplitudes. When the peak excitation acceleration is less than 0.55 g, the vertical contact force increases monotonically. Nevertheless, the longitudinal deformation of the pier top changes little because there is no pier–beam separation. When the excitation peak exceeds 0.55 g, vertical separation occurs, the extreme value of the longitudinal deformation at the top of the pier increases significantly, and the allowable pressure of the pier decreases. With the increase in excitation amplitude, the maximum values of the vertical contact force of the pier beam and the longitudinal deformation of the pier top increase monotonically. When the peak excitation acceleration is 0.65 g, the pier is destroyed under compression. changes little because there is no pier–beam separation. When the excitation peak exceeds 0.55 g, vertical separation occurs, the extreme value of the longitudinal deformation at the top of the pier increases significantly, and the allowable pressure of the pier decreases. With the increase in excitation amplitude, the maximum values of the vertical contact force of the pier beam and the longitudinal deformation of the pier top increase monotonically. When the peak excitation acceleration is 0.65 g, the pier is destroyed under compression.

**Figure 11.** Bottom pressure of pier when = 30 rad/s. **Figure 11.** Bottom pressure of pier when *ω*0 = 30 rad/s.

Figure 12 shows the change in the maximum and allowable pressure under different excitation peaks when = 50 rad/s. When is far away from ଵ, the vertical contact force increases monotonically with the decrease in the excitation peak. Since the structure does not separate, and is far away from ଵ, the allowable value of the pier pressure remains constant. The maximum vertical contact force of the pier beam is always less than Figure 12 shows the change in the maximum and allowable pressure under different excitation peaks when *ω*0 = 50 rad/s. When *ω*0 is far away from *ω*1, the vertical contact force increases monotonically with the decrease in the excitation peak. Since the structure does not separate, and *ω*0 is far away from *µ*1, the allowable value of the pier pressure remains constant. The maximum vertical contact force of the pier beam is always less than the allowable value, and the structure is in the safe range.

the allowable value, and the structure is in the safe range.

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**Figure 12.** Bottom pressure of pier when = 50 rad/s. **Figure 12.** Bottom pressure of pier when *ω*0 = 50 rad/s. Figure 13 shows the changes in the maximum and allowable values of pier bottom pressure under different excitation amplitudes when = 15 rad/s. At this point, the ex-

Figure 13 shows the changes in the maximum and allowable values of pier bottom pressure under different excitation amplitudes when = 15 rad/s. At this point, the excitation frequency deviates from the natural vertical frequency of the bridge but approaches the longitudinal excitation frequency of the bridge. When the peak seismic acceleration is less than 1.2 g, the maximum pressure at the pier bottom increases monotonically with the increase in excitation. By contrast, the extreme deformation value at the pier top changes little. When the peak excitation acceleration exceeds 1.2 g, the structure separates and undergoes a small mutation, leading to a further decrease in the allowable Figure 13 shows the changes in the maximum and allowable values of pier bottom pressure under different excitation amplitudes when *ω*0 = 15 rad/s. At this point, the excitation frequency deviates from the natural vertical frequency of the bridge but approaches the longitudinal excitation frequency of the bridge. When the peak seismic acceleration is less than 1.2 g, the maximum pressure at the pier bottom increases monotonically with the increase in excitation. By contrast, the extreme deformation value at the pier top changes little. When the peak excitation acceleration exceeds 1.2 g, the structure separates and undergoes a small mutation, leading to a further decrease in the allowable pier pressure. When the peak excitation acceleration exceeds 1.5 g, the pier is destroyed under compression. citation frequency deviates from the natural vertical frequency of the bridge but approaches the longitudinal excitation frequency of the bridge. When the peak seismic acceleration is less than 1.2 g, the maximum pressure at the pier bottom increases monotonically with the increase in excitation. By contrast, the extreme deformation value at the pier top changes little. When the peak excitation acceleration exceeds 1.2 g, the structure separates and undergoes a small mutation, leading to a further decrease in the allowable pier pressure. When the peak excitation acceleration exceeds 1.5 g, the pier is destroyed under compression.

To calculate the maximum bending moment at the bottom of the first impact pier, the

**Figure 13.** Bottom pressure of pier when = 15 rad/s. **Figure 13.** Bottom pressure of pier when *ω*0 = 15 rad/s.

calculation model is simplified as follows:

*4.2. Bending Failure*

*4.2. Bending Failure*

**Figure 13.** Bottom pressure of pier when = 15 rad/s.

## *4.2. Bending Failure*

To calculate the maximum bending moment at the bottom of the first impact pier, the calculation model is simplified as follows: 1. The maximum deformation time at the top of the pier is selected for the calculation. At this time, the bending moment generated by the horizontal vibration inertia force of the pier is zero;

The allowable bending moment calculation of an eccentric compression pier is shown in Equation (8), and the specific parameters are shown in the specification [23]: sin 2 (1 ) ( ) 2 *cd r t sd s N fA fA* πα α αα πα= − +−

$$\begin{array}{l} N = \alpha f\_{cd} A\_r (1 - \frac{\sin 2\pi a}{2\pi a}) + (\alpha - a\_t) f\_{sd} A\_s\\ M\_{\rm{ud}} = \frac{2}{3} f\_{cd} A\_r \frac{\sin^3 \pi a}{\pi} + f\_{sd} A\_s r\_s \frac{\sin \pi a + \sin \pi a\_l}{\pi} \end{array} \tag{8}$$

(8)

Figure 14 shows the changes in the total and acceptable bending moments under different excitation amplitudes. With the increase in the excitation amplitude, the allowable bending moment continues to decrease. By contrast, the total bending moment continues to increase and surges when the amplitude is 0.55 g. At this point, the sudden change in the maximum bending moment is due to the sharp increase in the longitudinal displacement caused by the separation. Figure 14 shows the changes in the total and acceptable bending moments under different excitation amplitudes. With the increase in the excitation amplitude, the allowable bending moment continues to decrease. By contrast, the total bending moment continues to increase and surges when the amplitude is 0.55 g. At this point, the sudden change in the maximum bending moment is due to the sharp increase in the longitudinal displacement caused by the separation.

**Figure 14.** Bending moment at the bottom of the bridge pier under different levels of vertical excitation acceleration. **Figure 14.** Bending moment at the bottom of the bridge pier under different levels of vertical excitation acceleration.

#### *4.3. Shear Failure 4.3. Shear Failure*  The change in a pier's longitudinal displacement and vertical force may also cause

The change in a pier's longitudinal displacement and vertical force may also cause pier shear failure. See Equation (9) for the check calculation of the shear strength, and see the specification [23] for the meaning of each parameter. pier shear failure. See Equation (9) for the check calculation of the shear strength, and see the specification [23] for the meaning of each parameter. ( ) *c cs V VV* φ≤ +

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$$\begin{array}{c} V\_{\mathcal{L}} \le \phi(V\_{\mathcal{L}} + V\_{\mathcal{s}})\\ V\_{\mathcal{c}} = 0.1 v\_{\mathcal{L}} A\_{\mathcal{e}} \end{array} \tag{9}$$

Since the elastic model is adopted in this paper and the plastic hinge interval is ignored, H is the height from the top to the bottom of the pier. In order to study the shear condition at the bottom of the pier when the bending moment is at its maximum, P is selected as the maximum pressure at the bottom of the pier. According to Equation (9), the shear strength of the bridge pier increases monotonically with the increase in the pier bottom pressure. The increase in pressure can not only increase the shear strength of the pier, but also lead to the increase in the bending moment at the bottom of the pier. Figure 15 shows the shear changes of the bridge pier under different vertical excitation amplitudes. When there is no separation between the main beam and the pier, both the shear design value and the allowable shear value of the pier increase with the increase in the vertical excitation amplitude, and the pier is always in the safe range. When the structure is separated vertically, the bending moment caused by the horizontal excitation and the eccentric collision increases greatly due to the increase in the pier deformation caused by the separation. However, the increase in the pier's axial force can improve the pier's shear design value. However, this increase is less than the shear force generated by the seismic excitation, and the bottom of the pier is damaged by the shear. nored, H is the height from the top to the bottom of the pier. In order to study the shear condition at the bottom of the pier when the bending moment is at its maximum, P is selected as the maximum pressure at the bottom of the pier. According to Equation (9), the shear strength of the bridge pier increases monotonically with the increase in the pier bottom pressure. The increase in pressure can not only increase the shear strength of the pier, but also lead to the increase in the bending moment at the bottom of the pier. Figure 15 shows the shear changes of the bridge pier under different vertical excitation amplitudes. When there is no separation between the main beam and the pier, both the shear design value and the allowable shear value of the pier increase with the increase in the vertical excitation amplitude, and the pier is always in the safe range. When the structure is separated vertically, the bending moment caused by the horizontal excitation and the eccentric collision increases greatly due to the increase in the pier deformation caused by the separation. However, the increase in the pier's axial force can improve the pier's shear design value. However, this increase is less than the shear force generated by the seismic excitation, and the bottom of the pier is damaged by the shear.

**Figure 15.** Bottom shear of pier under different excitation amplitudes. **Figure 15.** Bottom shear of pier under different excitation amplitudes.

#### *4.4. Influence of Dynamic Response on the Failure Mode of Bridge Pier 4.4. Influence of Dynamic Response on the Failure Mode of Bridge Pier*

Under the conditions of pier–beam separation, there may be multiple modes of failure under high-amplitude excitation. However, the limit solution is adopted in the calculation of pier failure in the front, while the dynamic response of the bridge under the actual excitation is within the limit solution. Therefore, it is necessary to analyze the first failure mode of the bridge pier under the condition of separation. Under the conditions of pier–beam separation, there may be multiple modes of failure under high-amplitude excitation. However, the limit solution is adopted in the calculation of pier failure in the front, while the dynamic response of the bridge under the actual excitation is within the limit solution. Therefore, it is necessary to analyze the first failure mode of the bridge pier under the condition of separation.

The dynamic response of the bridge after pier–beam separation is calculated and compared with the standard formula. Two parameters affecting the failure of the pier are The dynamic response of the bridge after pier–beam separation is calculated and compared with the standard formula. Two parameters affecting the failure of the pier are proposed: the vertical impact force of the pier beam and the longitudinal deformation of

the pier top. Figure 16 shows the failure of the pier under different parameters. When the pier top is slightly deformed, the increase in the vertical collision force of the pier beam leads first to the compression failure of the structure, followed by shear failure, and, finally, bending failure. The failure modes of the bridge pier change with the increase in the maximum deformation. The failure gradually changes from compression failure to shear failure, and the pier bottom pressure required for failure gradually decreases. When large longitudinal deformation occurs at the pier top, bending failure occurs first, and only a small axial force is needed to cause it. cal frequency, the pier–beam contact force is low, and the pier is mainly subjected to bending failure. When the excitation frequency deviates significantly from the longitudinal frequency of the bridge and is close to the natural vertical frequency of the bridge, the deformation at the top of the pier gradually decreases, the pressure gradually increases, and shear failure gradually occurs. When ≈ ଵ, the pier is subjected to a large vertical impact force. Through the analysis of Figure 16, it can be found that compression failure is the most likely failure to be caused at this time. The failure modes of the pier are different with different excitation frequencies.

Regarding the bridge model in Figure 2, when ≈ μଵ, the top of the pier is deformed significantly, but the excitation frequency deviates greatly from the natural verti-

proposed: the vertical impact force of the pier beam and the longitudinal deformation of the pier top. Figure 16 shows the failure of the pier under different parameters. When the pier top is slightly deformed, the increase in the vertical collision force of the pier beam leads first to the compression failure of the structure, followed by shear failure, and, finally, bending failure. The failure modes of the bridge pier change with the increase in the maximum deformation. The failure gradually changes from compression failure to shear failure, and the pier bottom pressure required for failure gradually decreases. When large longitudinal deformation occurs at the pier top, bending failure occurs first, and only a

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small axial force is needed to cause it.

**Figure 16.** Failure of bridge pier under different axial forces. **Figure 16.** Failure of bridge pier under different axial forces.

**5. Conclusions**  In this paper, a double-span continuous beam bridge is selected to analyze the influence of near-field vertical earthquake on pier failure. Analysis of the longitudinal displacement and vertical contact force response of the bridge is undertaken. Different types of possible damage of the pier under seismic excitation are solved. It is calculated that: (1) Under simple harmonic excitation, there is a risk of mound separation when the excitation frequency is close to the first-order vertical intrinsic frequency of the firstorder of the bridge, and it is mainly focused on the excitation frequency being close to the first-order vertical intrinsic frequency interval. (2) The vertical separation of the mounds increases the longitudinal deformation on the Regarding the bridge model in Figure 2, when *ω*0 ≈ µ1 , the top of the pier is deformed significantly, but the excitation frequency deviates greatly from the natural vertical frequency, the pier–beam contact force is low, and the pier is mainly subjected to bending failure. When the excitation frequency deviates significantly from the longitudinal frequency of the bridge and is close to the natural vertical frequency of the bridge, the deformation at the top of the pier gradually decreases, the pressure gradually increases, and shear failure gradually occurs. When *ω*0 ≈ *ω*1, the pier is subjected to a large vertical impact force. Through the analysis of Figure 16, it can be found that compression failure is the most likely failure to be caused at this time. The failure modes of the pier are different with different excitation frequencies.

#### top of the piers, and the frequency difference is larger. At the same time, the collision **5. Conclusions**

that the separation would cause would greatly increase the mound vertical contact In this paper, a double-span continuous beam bridge is selected to analyze the influence of near-field vertical earthquake on pier failure. Analysis of the longitudinal displacement and vertical contact force response of the bridge is undertaken. Different types of possible damage of the pier under seismic excitation are solved. It is calculated that:

(3) With pier and beam separation, the increase in excitation amplitude causes a monotonic increase in dock longitudinal deformation. The dock will be damaged by compression, bending, and shearing. With the longitudinal deformation value of the pier top as the failure basis, when the vertical contact force of the pier beam is low, the flexural failure of the structure occurs first. With the increase in pier bottom pressure, the first failure mode gradually changes into shear failure. When the vertical impact force of the pier beam is large, only a small longitudinal deformation is required to cause compression failure of the pier beam.

**Author Contributions:** Formal analysis, W.A.; Data curation, M.K.; Writing—original draft, L.Z.; Writing—review & editing, L.M.; Supervision, H.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Jiangxi Provincial Department of Education Science and Technology [CJJ202915] And The APC was funded by [CJJ202908].

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The datasets used and analysed during the current study are available from the corresponding author on reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

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## *Article* **Research on Elevation Survey Method of Sea-Crossing Bridge under Adverse Conditions**

**Jun Xiao 1,2, Jianping Xian 1,2, Song Li 1 and Shuai Zou 2,3,\***

**Abstract:** Aiming to survey scenarios of offshore projects with difficult horizontal elevation transmission and long-distance, all-weather elevation monitoring operations, a long-distance, total station, trigonometric leveling based on dynamic compensation is proposed. The feasibility of this method was verified by an outdoor survey experiment, and the range of transverse coverage and accuracy reached by this method was quantitatively analyzed. The results indicate that this method shows a good correction effect on the survey results of test points under different environmental conditions, which proves that this method is feasible. The correction effect of this method is affected by the distance between the test point and the datum point; within the range of 60 m horizontally from the datum point, an assurance rate of about 90% can be achieved for the error range of 20 mm. Combining with the built-in ATR (Automatic Target Recognition) technology of the total station, this method can make the elevation survey result reach the accuracy of millimeter level under the range of about 1000 m, by obtaining multiple groups of data and then calculating the mean value. This paper provides a new method for the elevation transfer of sea-crossing bridges under long-distance conditions and harsh environmental conditions.

**Keywords:** sea-crossing bridges; triangle elevation; leveling method; dynamic compensation; total station; harsh environment; long distance

## **1. Introduction**

In recent years, there has been an upsurge in sea-crossing traffic engineering construction around the world, and the length of sea-crossing bridges has been refreshed again and again. The earliest sea-crossing bridge in China is the Xiamen Bridge, which was built in 1987. The total length of the bridge is 6.695 km. It was finally completed in 1991 and opened to traffic in May. Since then, the curtain of China's sea-crossing bridge construction has been lifted [1]. The following Table 1 lists the top 10 typical sea-crossing bridges that have been built or are under construction in China.

To meet the increasing traffic demand, the construction of the sea-crossing bridge is gradually developing towards the strong-wave, deep-water area. Its length ranges from several kilometers to tens of kilometers. Large-scale sea-crossing projects under harsh environmental conditions will inevitably cause many severe technical problems in construction and maintenance [2,3]. In the maintenance of large-scale sea-crossing bridges, to meet the high-standard testing requirements, it has become an important research direction in this field to innovate the testing methods by combining the traditional non-destructive testing (NDT) methods with advanced techniques [4,5]. Khedmatgozar Dolati et al. [4] compiled in one place all the NDT techniques, including the application of drones, sensors, or robots for rapid and efficient assessment of damage on small and large scales, which kept researchers up-to-date with existing methods and paved the way for further innovations in this regard. In the construction of sea-crossing bridges, long-distance

**Citation:** Xiao, J.; Xian, J.; Li, S.; Zou, S. Research on Elevation Survey Method of Sea-Crossing Bridge under Adverse Conditions. *Sustainability* **2022**, *14*, 11641. https://doi.org/10.3390/ su141811641

Academic Editors: Kai Wei, Mingjin Zhang, Jian Zhong and Yutao Pang

Received: 28 July 2022 Accepted: 15 August 2022 Published: 16 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

elevation surveys in a complex marine environment is one of the important technical problems. In the project, some offshore survey platforms are often established through offshore test pile projects, and some bridge piers are under priority construction so that the survey distance can be shortened to less than 2 km, which provides practical conditions for using high-precision total stations to transfer the elevation data to offshore buildings or structures and to realize the direct connection of the sea-crossing elevation [6,7]. Affected by waves and fluctuating tides, the offshore survey platform and piers are always subject to slight shaking, and the vertical angle of the total station placed on the platform or pier can be changed for tens of seconds or even minutes [8,9]. To ensure the survey effect, the stiffness and size of the offshore survey platforms are often designed to be very large, and the distance between the survey platforms will be increased based on the consideration of construction cost.

**Table 1.** The top 10 typical sea-crossing projects that have been built or are under construction in China.

For high-precision total station trigonometric leveling, the main factors affecting the accuracy of the survey are the size of the zenith distance, the observation error, and the measurement error of the vertical atmospheric refraction coefficient, if the stability of the station is guaranteed [10,11]. The influence degree of each factor gradually increases with the increase in the distance between the survey station and the target prism. Therefore, it is of great significance to improve the accuracy of the survey by either improving the survey method or compensating and correcting the survey results to reduce the deviation caused by various influencing factors the survey results.

Qiu Yang et al. [12] used auxiliary equipment to measure the height of the total station and the target plate, appropriately increased the number of angular survey rounds to improve the accuracy of vertical angle, and carefully planned the survey steps to reduce the vertical atmospheric refraction error and other means to effectively eliminate and reduce the impact of the main error source. Yaming Xu et al. [13] proposed a linear structure survey method based on the traditional method, combined with relatively strict synchronous opposite observation, which reduced the complexity and observation amount of the original method. References [14–16] improved and optimized the survey method proposed in reference [13] and carried out long-distance, sea-crossing elevation transmission in the Chuandao area of Jiangmen Taishan City. Zhongping Wang et al. [17] improved the point layout structure based on the trigonometric elevation observation method proposed in reference [13] and eliminated and screened the data of each survey section in combination with the specification and reasonable restriction of the elevation difference between each group. The requirements of the national second-class leveling accuracy were met through project testing.

Heng Zhang et al. [18] have greatly weakened the influence of atmospheric refraction, vertical deviation, and Earth curvature by using two high-precision total stations of the same model to observe the opposite direction of the high–low biprism group at the same time. Based on the least square theory, Mingbo Liu et al. [19] gave a new method to calculate the refraction coefficient of the survey area. It was demonstrated in production practice that using this method can produce triangular elevations with accuracy close to

the second-class level. Jianzhou Li et al. [20] deduced the error formula of the survey result through the error propagation law according to the rigorous calculation formula of precision trigonometric leveling and proved that controlling the vertical angle and survey distance of the instrument can improve the survey accuracy and achieve second-class leveling accuracy. Jianzhou Li et al. [21] developed TriLevel, a precision triangulation elevation survey system that realizes the automatic measurement and quality control of precision triangulation elevation. Refs. [22–24] weakened the deviation caused by atmospheric refraction on elevation survey results by avoiding measuring the total station height and realizing a high-precision elevation survey. Peibing Yang et al. [25] designed a special observation mark to solve the problem of long-distance surveys, aiming at the need for high-precision elevation transmission in wide sea areas, and also responded to the impact of the shaking of the survey platform on the survey accuracy by carrying out rigorous and rapid observation operations under favorable survey conditions.

At present, there are many pieces of research on the improvement of the survey method of total station trigonometric leveling and the methods of compensation and correction of the survey results, which can make the accuracy of the survey results reach a high level, but to ensure the accuracy requirements of the survey results, the improved survey method or the method of correction of the survey results are generally cumbersome. However, there is relatively little research on developing targeted, simple, and efficient survey methods or correction methods under some specific conditions and harsh environments.

Because of this, this paper proposes a long-distance total station elevation survey method based on dynamic compensation, which is aimed at survey scenarios of offshore projects with difficult horizontal elevation transmission under long-distance, all-weather elevation monitoring operations and harsh environmental conditions. The feasibility of the proposed elevation survey method was verified by an outdoor survey experiment. At the same time, through the processing and analysis of a large number of surveyed data at different test points, a quantitative study was carried out within the scope that this method can cover in the horizontal direction and the accuracy that the survey results can reach. This method can be used as an efficient and high-precision method for the elevation survey of sea-crossing bridges under adverse conditions.

### **2. Principles and Methods**

### *2.1. Engineering Background*

The elevation survey method proposed in this paper mainly serves the elevation survey operation in the construction of the Peljesac Bridge. The Peljesac Bridge is located at the southern end of Croatia, bordering Bosnia and Herzegovina. It is located in the Malostonski Bay protection area on the west side of the Adriatic Sea, with the Peljesac Peninsula and mainland on either side of the bridge. The main Peljesac Bridge is a lowtower, cable-stayed bridge with a total length of 2404 m, which has a central single cable plane and a steel box girder, and its span group is (84 + 108 + 108 + 189.5 + 5 × 285 + 189.5 + 108 + 108 + 84) m. The bridge has 14 piers and abutments, of which 4 piers and abutments are arranged on land on both banks, and the remaining 10 pier towers are in the water. The layout of the Peljesac Bridge is shown in Figure 1.

During the construction of the Peljesac Bridge, no sea survey platform was set up. By setting survey stations on both banks, the elevations of the bridge piers and tower columns in the sea were surveyed. The survey operations are shown in Figure 2.

#### *2.2. Basic Principle of Total Station Trigonometric Leveling*

Trigonometric leveling is a method to determine the height difference between two points by surveying the horizontal distance and vertical angle of the two control points. The basic formula for the calculation of one-way survey height difference is shown in Formula (1), which is generally derived from Figure 3:

$$h\_{AB} = Scos\alpha + \frac{1-K}{2R}S\_0^2 + i\_A - v\_B \tag{1}$$

where *S* is the oblique distance between two points *A* and *B*, m; α is the vertical angle between the observation point of the total station and the target prism, (◦ ); *R* is the average curvature radius of the earth, *R* ≈ 6371 km; *K* is the local atmospheric refraction coefficient (related to temperature, pressure, and atmospheric density), which can be taken as 0.10~0.15 for general mountainous areas and 0.5 for flat areas near the ground; *S*0 is the horizontal distance between two points, m, *S*0 = *Scos*α; *iA* is the height of the total station, m; *vB* is the height of the prism, m.

**Figure 1.** The layout of the Peljesac Bridge.

**Figure 2.** The process of survey operations.

The calculation formula of the mean square error in the elevation difference of a one-way survey can be derived from the total differential of Formula (1) [26] and then from the error propagation rate, as shown in Formula (2):

$$M\_{\rm li\_{AB}}^2 = S^2 \cos^2 \alpha \frac{m\_{\alpha}^2}{\rho^2} + \sin^2 \alpha m\_s^2 + \frac{1}{4R^2} S\_0^4 m\_{K\_i}^2 + m\_{i\_A}^2 + m\_{\upsilon\_B}^2 \tag{2}$$

where *m*α is the mean square error of vertical angle observation; *mS* is the mean square error of the oblique distance survey; *mK* is the mean square error of the atmospheric refraction coefficient measurement, generally ±0.03~±0.05 [27]; *miA* is the mean square error of the total station height measurement; *mvB* is the mean square error of the target height measurement; *ρ* is the constant used for angle unit conversion, *ρ* = 206265.

**Figure 3.** Principle of triangle elevation.

#### *2.3. Principle of Dynamic Compensation*

According to the calculation formula of the mean square error of the survey height difference shown in Formula (2), the high-precision total station needs to be compensated and corrected due to the influence of atmospheric refraction, topographic conditions, the earth's curvature, etc., during long-distance trigonometric leveling. As shown in Figure 4, the line of sight *e1* of the datum point in the figure and the line of sight *e2* of the test point to be surveyed should be the same at the same time because the distance between the datum point and the test point is short.

**Figure 4.** The principle of dynamic compensation.

If the accurate elevation data of the datum point is known, the elevation can be transmitted to the datum point by densifying the datum point on land or structure under the meteorological conditions required in the survey specification, such as night or morning with good weather conditions, and finally determined by taking the mean value through multiple surveys; when the survey operation conditions allow, the accurate elevation value of the datum point can also be obtained through leveling. When the elevation survey operation is carried out, the elevation of the datum point is surveyed first, and the difference between its accurate elevation value and the current elevation survey value is taken as the dynamic compensation correction value under the current condition. Then, the elevation of the test point is surveyed, and the obtained dynamic compensation correction value is used to correct the elevation survey result of the test point, to reduce or eliminate the influence of external factors. The correction formula is as follows:

Dynamic compensation value:

$$
\mu = H\_0 - H\_0'\tag{3}
$$

Correction value of current survey result:

$$H\_i' = H\_i + \mu \tag{4}$$

where *H*0 is the precise elevation value of the datum point; *H*0 0 is the elevation survey result of the datum point during the current survey; *µ* is the dynamic compensation value of the current elevation survey result; *Hi* is the elevation survey result of the point to be surveyed in the current survey; *H*0 *i* is the elevation correction value of the point to be surveyed in the current survey.

### *2.4. Introduction of Experiment*

To verify and carry out quantitative research on the trigonometric leveling method based on dynamic compensation proposed in this paper, an outdoor field survey experiment was carried out. The specific contents of the experiment are as follows.

### 2.4.1. Scheme Design

Considering the ultimate purpose of this experiment, the selection of the experimental site should meet the following requirements:

To meet the requirements of the above experimental site, an open construction site was selected for this experiment, and the layout of the site is shown in Figure 5. To quantify the range that the method can cover horizontally over long distances, the distance between the datum point (JZ01) and the survey station (CZ01) was selected as 1000 m. According to the different distances between the datum point and the test point, a total of seven test points were proposed, as shown in Table 2. Due to the long-distance survey, the efficiency of the target prism aiming manually is low, and large observation errors may be caused by human factors. Therefore, the built-in ATR function of the total station was used for target collimation in this experiment. At the same time, to verify the applicability of this method under different environmental conditions, this experiment chose to continuously observe for a day, and the observation frequency was 1 group every 10 min.

**Table 2.** Test point details.

**Figure 5.** The layout of the site.

## 2.4.2. Experiment Content

In order to carry out the experiment orderly and ensure the reliability of the survey results, the experiment should be carried out according to the following operating steps:

The test procedure is shown in Figure 6.

The following points should be paid attention to during the experiment to ensure the accuracy of the experimental results.

**Figure 6.** The procedure of the outdoor survey test.

### 2.4.3. Experimental Equipment

The orderly conduct of the experiment is closely related to the preliminary preparations. Therefore, the test equipment shall be prepared in advance and shall be within the qualification period. The equipment required for this experiment includes a high-precision level, a high-precision total station (Leica TS60), a thermometer, a barometer, a humidity meter, some measuring nails, a manual painting, a test record book, etc. A Leica TS60 total station with an angle survey accuracy of 0.500 and a distance survey accuracy of 0.500 was used for this experimental survey, with a survey range of 3500 m in ATR mode, to ensure the testing effect.

Before the experiment, the specific position of the datum point and each test point should be determined through the distance survey function of the observation instrument, and the datum point and the test points should be marked by measuring nails and manual painting.

## **3. Results**

Because of the constant changes in external factors (temperature, atmospheric refraction, and so on), the results of each survey varied to varying degrees. According to the basic principle of the elevation survey method proposed in this paper, the difference between the elevation survey result of the datum point and the accurate elevation value of the datum point was taken as the current dynamic compensation value *µ*, which was used to correct the elevation survey results of each test point to be surveyed at that time.

Through the analysis of multiple groups of data at different test points, the assurance rate of different survey accuracies and the overall correction effect is used to evaluate the applicability of the method under different distances between the datum point and the test point.

For simplicity of expression, the following applicable scope refers to the scope covered by the distance between the datum point and the test point, which can be used to make a relatively ideal correction of the elevation survey results. This scope is the circular area with the datum point as the center and the distance between the datum and the test point as the radius.

## *3.1. Experimental Result*

According to the basic principle of the elevation survey method proposed in this paper, the accuracy of the survey results was evaluated by the following Formula (5):

$$
\Delta H = \left(H\_i' - H\_{i0}\right) \times 1000\tag{5}
$$

where ∆*H* is the deviation between the corrected survey result and the precise elevation value of the test point, mm; *Hi*0 is the precise elevation survey value of the test point, m; and *H*0 *i* is the elevation correction value of the test point in the current survey, m.

Figure 7 shows the changes in accuracy curves before and after the correction of elevation survey results for each test point at different times. The figure shows:

#### *3.2. Data Process*

3.2.1. The Accuracy Assurance Rate

The quantitative statistical analysis of the survey accuracy of multiple groups of survey data at the same test point can be completed by the assurance rate under the corresponding accuracy conditions, and the accuracy assurance rate of survey results *η* is calculated with the following formula:

$$
\eta = \frac{N}{n} \times 100\% \tag{6}
$$

$$N = number(|\Delta H| \le \mathfrak{a})\tag{7}$$

or

$$N = number(|\Delta H| > \alpha) \tag{8}$$

where *n* is the number of s data groups; *N* is the number of groups for which the absolute value of the the survey result's correction value accuracy meets a certain condition; and α is the accuracy condition of the survey results, mm.

Figures 8 and 9 show the distribution of different accuracy assurance rates of the corrected survey results and the variation trend of the same accuracy ratio with the increase in the distance between the test point and the datum point, respectively.

**Figure 7.** *Cont.*

137

Figures 8 and 9 show that:

**Figure 8.** Accuracy assurance rate distribution.

**Figure 9.** Accuracy ratio trend.

3.2.2. Data Dispersion

The mean value of the data sample reflects the centralized trend of the data. Table 3 shows the corrected mean value of multiple groups of survey results at all test points.

It can be seen from the data in Table 3 that after the elevation survey method provided in this paper is used to correct the survey results, the survey results can better approximate the accurate value by taking the mean value of multiple groups of corrected elevation data and finally reach millimeter-level accuracy. Moreover, the mean value of the corrected data is less affected by the distance between the test point and the datum point and the change in the survey environment.

### **4. Discussion**

Because of its great convenience, the use of a high-precision total station for elevation surveys has been widely concerned and used by the engineering community. In a longdistance elevation survey, its accuracy is significantly affected by atmospheric refraction, the Earth's curvature, temperature, and other factors, and the survey results may have a large deviation [10,11]. It is therefore very necessary to compensate for and correct the survey results.

A method of conducting an elevation survey for long distances based on dynamic compensation is proposed in this paper, which makes use of the principle that the compensation amount of two points that are close to each other shall be the same in a short time to correct the elevation survey results. Through the above experimental analysis, this method had a good correction effect on the elevation survey results within the experimental range, and its correction effect was affected by the distance between the test point and the datum point, thus showing the change in the corresponding accuracy assurance rate under different distances. In the actual project, the elevation survey method proposed in this paper can be reasonably selected according to the accuracy requirements of the project. In the above experiments, the uncorrected total station elevation survey results showed an obvious window period for the impact of environmental changes. In the morning, when the environmental conditions were suitable, the survey results were relatively stable, and the survey accuracy was relatively high. Therefore, it is recommended to adjust the long-distance elevation survey to the morning or night with good weather conditions, to reach a more ideal survey effect. Since the elevation survey method can maintain good adaptability under any environmental conditions, if combined with the ATR function of the total station and the corresponding data-processing program for the total station is developed, the method of taking the mean value after multiple surveys can also be used as a high-precision elevation survey method suitable for harsh environmental conditions.

To sum up, the new method proposed in this paper for elevation surveys under adverse conditions can provide sufficient technical support for elevation surveys in the construction of large-scale sea-crossing bridges due to its efficiency, high precision, and strong environmental adaptability. Limited by space, this paper only presents experimental research on the application of this method under the condition of a 1000 m range and does not involve research on the accuracy that this method can achieve under different range conditions. The relevant research will be carried out in the follow-up to clarify the change rules for the applicability, coverage area, and accuracy of measurement results with the distance of range.

#### **5. Conclusions**

By experimenting, this paper verified the feasibility of a total station elevation survey method based on dynamic compensation. At the same time, it made a quantitative study on the horizontal coverage of this method and further analyzed the practical situation of this method by counting the accuracy of the survey results under the condition of different distances from the datum point. The main conclusions are as follows:

(1) The analysis shows that the total station elevation survey method based on dynamic compensation proposed in this paper shows a good correction effect on the survey results within 100 m between the test point and the datum point, making the survey results fluctuate within a certain range near the accurate value, which proves that the survey method proposed in this paper is feasible.

 All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Science and Technology Project of CCCC (grant number: No. 2020-ZJKJ-QNCX04) and the Science and Technology Special Major Project of CCCC (grant number: No. 2019-ZJKJ-07).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data are contained within the article.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**

## *Article* **Design and Working Performance Evaluation of a Combined Survey Platform under Strong Wave and Deep-Water Conditions**

**Jun Xiao 1,2,3, Jianping Xian 1,2, Shuai Zou 2,4,\* , Song Li 1 and Yongshui Zhang 4**

**Abstract:** To meet the engineering requirements of long-distance and high-precision elevation transfer under adverse sea conditions, a new structure for a combined survey platform based on the basic principle of vibration reduction and isolation is designed in this paper. The combined survey platform uses the barrier effect of the external support system on the wave-current load to reduce the influence of the wave-current load on the internal support system and the platform, so it can maintain good performance, even in harsh sea conditions, under the premise of no collision between the internal and external support systems. The expected working performance of the structure was verified by numerical simulation, and the influence of the layout, structure size, waves, and water flow on the working performance of the structure was quantitatively analyzed. The results show that: (1) the external support system can better realize the barrier effect of the wave-current load and significantly reduce the influence of the wave-current load on the internal support system and platform; (2) for the independent combined survey platform, when the pile diameters of the internal and external support systems are 0.8 m and 1.2 m and the wall thicknesses are 11.0 mm and 12.0 mm, respectively, the period of the wave is 8.0 s, the water depth is 20.0 m, the speed of water flow is 1.0 m/s, and the wave height is 4.0 m, then the maximum variation of the vertical angle of the platform is only 19.3"; (3) for the attached combined survey platform, the lateral stiffness of the external support system is increased and the displacement of the external support system is significantly reduced because the external support system is connected with the cushion cap through the attachments; therefore, the structure size of the survey platform can be greatly reduced.

**Keywords:** sea-crossing elevation; combined survey platform; numerical simulation; strong wave and deep-water conditions; wave-current load; working performance

## **1. Introduction**

During the construction of large-scale sea-crossing bridge projects, it is necessary to carry out a high-precision sea-crossing elevation transfer to unify the elevation data on both sides of the bridge and accurately transfer the elevation on both sides to the pier and main beam in the sea. Combined with the many sea-crossing bridges that have been built, the precision of the elevation control survey is generally required to meet national secondclass survey standards [1]. The corresponding control parameters are that the accidental mean square error of elevation difference per kilometer is ±1 mm and the total mean square error of elevation difference per kilometer is ±2 mm. However, the span of seacrossing bridges is generally long, as some of these bridges reach several kilometers or even tens of kilometers; therefore, it is extremely difficult to carry out a sea-crossing elevation transfer survey using the conventional leveling method [2,3]. In the project, the elevation control network is generally densified by setting a survey platform in the sea to shorten

**Citation:** Xiao, J.; Xian, J.; Zou, S.; Li, S.; Zhang, Y. Design and Working Performance Evaluation of a Combined Survey Platform under Strong Wave and Deep-Water Conditions. *Sustainability* **2022**, *14*, 14360. https://doi.org/10.3390/ su142114360

Academic Editors: Kai Wei, Mingjin Zhang, Jian Zhong and Yutao Pang

Received: 14 September 2022 Accepted: 31 October 2022 Published: 2 November 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the survey distance. With the continuous progress of the project, the survey platform is often set on the foundation works, such as the test pile and some piers under priority construction, which greatly shortens the distance measurement [4], creating conditions for the sea-crossing elevation transfer survey using the total station. However, in the actual construction process, due to the influence of the wind, waves, tide, and other factors, the survey platform is always subject to slight shaking, resulting in the vertical angle change range of the total station installed on the survey platform reaching tens of seconds or even minutes [5,6], and the problem is particularly prominent for engineering projects located in strong wave and deep-water area.

At present, the main challenge to realize long-distance and high-precision elevation transfer on the sea is that, under complex sea conditions, the traditional survey platform will have micro-amplitude shaking, which makes the accuracy of elevation difficult to meet the high standard requirements of relevant specifications or standards.

In the previous engineering construction process, the main methods to reduce the impact of external wave-current load (the wave-current refers to waves and currents, which are independent of each other) on the survey were: (1) reducing the shaking amplitude of the survey platform through the designed structure, and (2) reducing the survey error by combining the method based on taking the average value of multiple repeated observations [5,7]. In order to keep the offshore survey platform stable under the impact of wave-current load, the most straightforward method is to increase the stiffness of the platform by increasing the geometric size of the platform. In references [5,8], the sea-crossing elevation survey under the influence of current and waves is carried out directly by increasing the structural stiffness of the survey platform and setting the forced centering observation stand. Practical tests show that the maximum shaking amplitude of the survey platform is still more than 10000 under the influence of sea conditions, such as waves and currents. At the same time, the construction cost of a survey platform designed with this concept is huge and even unacceptable when it adapts to the environment of strong waves and deep water. Although the survey error can also be reduced by taking the average value of multiple repeated observations, the method is different from the traditional single survey, and the survey methods and the data processing systems require special design, which is inefficient. In essence, this method assumes that the impact of waves in the survey window period is stable to ensure the statistical stability of the survey data. In fact, waves are mostly unsteady [9], and their impact on the survey platform and the survey reference point is also unsteady, so the statistical survey data itself changes with time. To ensure the accuracy of the survey data, it is necessary to observe for a long time or select an appropriate survey window period. The elevation survey is generally carried out at night [10], which causes obvious interference with the construction.

In view of this, to meet the engineering requirements of a long-distance high-precision elevation transfer under severe wave-current conditions, this paper proposes a combined survey platform suitable for strong wave and deep-water conditions based on the idea of vibration reduction and isolation. A reasonable design is carried out according to engineering practice to ensure the stability of the platform under harsh surge conditions. At the same time, through numerical simulation, the influence of different structural sizes and different sea conditions on the working performance of the combined survey platform is quantitatively studied. The main purpose of the combined survey platform proposed in this paper is to properly reduce the project investment. At the same time, it can make the offshore elevation survey free from the influence of the survey window period, and it can realize fast and accurate real-time offshore elevation monitoring [11]. It is of great engineering and practical significance.

## **2. Principles and Methods**

*2.1. Research Background*

2.1.1. Project Case

This paper is based on the projects of the Peljesac Bridge in Croatia and the Shenzhen-Zhongshan Bridge in China.

The main bridge of the Peljesac Bridge is a six-tower central single cable plane steel box girder low tower cable-stayed bridge. The total length of the bridge is 2404 m, and its span group is (84 + 108 + 108 + 189.5 + 5 × 285 + 189.5 + 108 + 108 + 84) m. The bridge has 14 piers and abutments in total, including 4 piers and abutments arranged on the land on both banks and the remaining 10 piers and towers arranged in the water, as shown in Figure 1.

**Figure 1.** The layout of the Peljesac Bridge.

The Peljesac Bridge crosses Mariston Bay and the European Union Nature Reserve. It has strict control over the setting of temporary structures in the sea, and the average water depth of the bay at the bridge location is 27 m. The bridge adopts a high pile cap foundation with a soft foundation. During the construction of the bridge, no marine survey platform was set. The elevation transfer is carried out by arranging the survey station on the cushion cap, as shown in Figure 2.

**Figure 2.** The offshore survey layout of the Peljesac Bridge.

The Shenzhen-Zhongshan Bridge is located in the middle of the Pearl River in the core area. It is a bridge under construction that connects Shenzhen City and Zhongshan City in Guangdong Province, and it has a total length of 24 km. To meet the needs of the construction survey, two permanent survey platforms and two temporary survey platforms are set up, as shown in Figure 3.

**Figure 3.** The survey platform layout of the Shenzhen-Zhongshan Bridge. **Figure 3.** The survey platform layout of the Shenzhen-Zhongshan Bridge. **Figure 3.** The survey platform layout of the Shenzhen-Zhongshan Bridge.

#### 2.1.2. Working Performance Test 2.1.2. Working Performance Test

Peljesac Bridge.

Due to the influence of the waves, the water flow, and the tide, the survey instruments placed on the pier, the cushion cap, and the separately set survey platform each shake greatly during the working period. Figures 4 and 5, respectively, show the changes in the instrument coordinates and the vertical angles of a total station set on a cushion cap in the sea that were caused by the sea conditions present during the construction of the Peljesac Bridge. Due to the influence of the waves, the water flow, and the tide, the survey instruments placed on the pier, the cushion cap, and the separately set survey platform each shake greatly during the working period. Figures 4 and 5, respectively, show the changes in the instrument coordinates and the vertical angles of a total station set on a cushion cap in the sea that were caused by the sea conditions present during the construction of the Peljesac Bridge. 2.1.2. Working Performance Test Due to the influence of the waves, the water flow, and the tide, the survey instruments placed on the pier, the cushion cap, and the separately set survey platform each shake greatly during the working period. Figures 4 and 5, respectively, show the changes in the instrument coordinates and the vertical angles of a total station set on a cushion cap in the sea that were caused by the sea conditions present during the construction of the

**Group number**

**Figure 4.** The coordinate changes in the survey instrument. **Figure 4.** The coordinate changes in the survey instrument. **Figure 4.** The coordinate changes in the survey instrument.

**Figure 5.** The vertical angle changes in the survey instrument. **Figure 5.** The vertical angle changes in the survey instrument. **Figure 5.** The vertical angle changes in the survey instrument.

The changes in the coordinates and the vertical angles of the measuring instrument under the conditions of waves and no waves show that: The changes in the coordinates and the vertical angles of the measuring instrument under the conditions of waves and no waves show that: The changes in the coordinates and the vertical angles of the measuring instrument under the conditions of waves and no waves show that:

## *2.2. The Principle of Design*

For the previous survey platform, the wave-current load on the platform is borne by the pile foundation. When the sea conditions are relatively bad, the survey platform will shake to some extent, which affects the survey accuracy of the total station erected on it.

Based on the basic principle of vibration reduction and isolation, this paper designs a new structure for a combined survey platform, and its principle of design is shown in Figure 6. The combined survey platform is composed of internal and external support systems and platforms, and the internal and external support systems are independent of each other in terms of load. The external support system directly bears the wave-current load, while the internal support system is not directly affected by the wave and water flow. It only bears the load imposed by the top platform and the load transmitted by the water and the soil between the internal and external support systems so the combined survey platform can maintain stability even under adverse sea conditions.

**Figure 6.** The design principle of the combined survey platform.

According to the different functions of the internal and external support systems, the corresponding structural dimensions need to be designed separately. The larger the inner diameter of the external support system is, the greater the rigidity. Additionally, the larger the gap between the external support system and the internal support system is, the less likely it is that the internal support system will touch the external support system, but this will increase the investment of the entire survey platform. At the same time, with the increase in the inner diameter of the external support system, the wave-current load on the external surface increases correspondingly. Selecting the appropriate stiffness and geometric size while meeting the needs of the working performance is an important measure to achieve the dual goals of applicability and economy.

#### *2.3. Design of the Layout Scheme*

According to the different layouts, the design of the combined survey platform can be divided into two schemes: independent and attached.

## 2.3.1. The Independently Set Combined Survey Platform

Since the construction of sea-crossing bridges is mostly in wide sea areas, the survey platform needs to work as an independent structure. Therefore, the loads caused by waves and water flow are all borne by the survey platform. The rigidity of the traditional survey platform should be strong enough to meet the high standard requirements of survey instruments for the stability of the survey platform. The direct problem is a large amount of investment, and the economy of engineering construction cannot be guaranteed. For the combined survey platform proposed in this paper, because its external wave-current load is completely loaded onto the external support system and the internal support system only bears the load generated by the upper platform and the load transmitted by the soil and the water between the internal and external support systems, its stability does not need to be ensured by greatly increasing the rigidity of the structure like the traditional survey platform, but the high standard working performance requirements of the survey platform are realized.

For the upper platform part of the independently set combined survey platform, two HN-600 load-bearing beams can be set on top of the internal support system. UB-25b distribution beams with a spacing of 488 mm are placed on the load-bearing beams. Fifteen mm thick steel plates are laid on the distribution beams. The design drawing is shown in Figure 7.

**Figure 7.** The design drawing of the independently set combined survey platform. (**a**) The elevation view of the survey platform. (**b**) The side view of the survey platform. (**c**) The vertical view of the survey platform.

*D*1 and *D*2 are the diameters of the external and internal support systems, respectively; *D*3 and *D*4 are the diameters of the external lateral connection and the internal lateral

connection, which are 0.8 m and 0.4 m, respectively, and their thicknesses are 0.01 m; *d*0 is the penetration depth of the internal and external support systems in the soil, which are 30 m and 25 m, respectively; *d* is the height from the still water surface to the seabed, taken as 20 m; *d* 0 is the distance from the top of the internal and external support systems to the still water surface, taken as 5 m and 4 m, respectively; *S*0 is the spacing between the distribution beams, taken as 0.448 m; *S*1 and *S*2 are the length and the width of the platform, which are both 5 m; and ∆1 and ∆2 are the thicknesses of the external and internal support systems, respectively.

## 2.3.2. The Attached Combined Survey Platform

The survey and positioning accuracy of the pile foundation and the cushion cap is lower. In the early stage of the project, cross-water surveys can be carried out by GNSS and other methods. When the construction of the cushion cap is completed, the external support system of the combined survey platform can be attached to the cushion cap through attachments (a steel pipe or other structures with sufficient rigidity). The stiffness of the cushion cap can make up for the deficiency in the stiffness of the external support system so the external support system will not shake significantly under the wave-current load. The cushion cap will shake to a certain extent under the wave-current load, and if the internal support system is also connected with the cushion cap, the shaking will have a negative impact on the working performance of the independent combined survey platform; therefore, the internal support system will not be connected with the cushion cap. At the same time, the attached combined survey platform can be a single-pile platform. The gap between the external support system and the internal system can be properly reduced. The design drawing is shown in Figure 8.

**Figure 8.** The design drawing of the attached combined survey platform.

Where the parameters in the figure have the same meanings as those in Figure 7.

### 2.3.3. Project Cost Comparison

In order to make an approximate evaluation of the project cost required for the combined survey platform designed in this paper, the materials required for the construction of the independently set combined survey platform and the permanent survey platform (as shown in Figure 9) in project case 2 are compared. The steel pipe piles of the platform shown in Figure 9 are all ∅1500 × 12 mm, and the size of the concrete platform is 10 × 10 × 2 m. The structural dimensions of the external and internal support systems of the independently set combined survey platform are: ∆1 = 12.0 mm, ∆2 = 11.0 mm, *D*1 = 1.2 m, *D*2 = 0.8 m.

**Figure 9.** The permanent survey platform. (**a**) The elevation view. (**b**) The side view. (**c**) The vertical view. (**d**) The general arrangement plan. Note: All dimensions in the figure are in millimeters.

The comparison of material consumption is shown in Figure 10. It can be seen from the figure that the traditional survey platform needs to provide enough stiffness to resist the impact of the external wave current load, so its structure size is often large, and the required engineering investment is more than that of the combined survey platform proposed in this paper.

#### *2.4. Dynamic Equations and Boundary Conditions*

This section describes the dynamic equations and boundary conditions used in the numerical simulation in Section 2.6.

#### 2.4.1. Governing Equation of Water

It is assumed that the fluid is a compressible, small-disturbance fluid without rotation and viscosity. The governing equation of water, represented by the velocity potential of water, *ϕ*, is

$$\frac{\partial^2 \varphi}{\partial x^2} + \frac{\partial^2 \varphi}{\partial y^2} + \frac{\partial^2 \varphi}{\partial z^2} = \frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2} \tag{1}$$

where *c* is the speed of sound in water, *c* = p *K*/*ρw*, and *c* → ∞ when the compressibility of the water is neglected; *K* is the bulk modulus of water [12]; *ρw* is the water density; and *T* is the time.

**Figure 10.** The comparison of material consumption.

2.4.2. Governing Equations of Soil and Pile

According to the basic equations of elastodynamics, the governing equations of soil and pile can be expressed as

$$(\lambda + G)\nabla\nabla\cdot u + G\nabla^2 u = \rho\ddot{u} \tag{2}$$

$$\nabla \nabla \cdot \boldsymbol{u} = \begin{cases} \frac{\partial^2 u\_x}{\partial x^2} + \frac{\partial^2 u\_y}{\partial x \partial y} + \frac{\partial^2 u\_z}{\partial x \partial z} \\ \frac{\partial^2 u\_x}{\partial x \partial y} + \frac{\partial^2 u\_y}{\partial y^2} + \frac{\partial^2 u\_z}{\partial y \partial z} \\ \frac{\partial^2 u\_x}{\partial x \partial z} + \frac{\partial^2 u\_y}{\partial y \partial z} + \frac{\partial^2 u\_z}{\partial z^2} \end{cases} \tag{3}$$

$$\nabla^2 \boldsymbol{\mu} = \begin{cases} \nabla^2 \boldsymbol{\mu}\_x \\ \nabla^2 \boldsymbol{\mu}\_y \\ \nabla^2 \boldsymbol{\mu}\_z \end{cases} \tag{4}$$

where ∇ = n *∂ ∂x ∂ ∂y ∂ ∂z* o*T* is the vector differential operator of rectangular coordinates; *λ* and *G* are the lame constant; and *ux*, *uy*, and *uz* are the displacements of soil and pile in *x*, *y*, and *z* directions respectively.

#### 2.4.3. Surface Boundary Conditions of Water

1

The pressure on the free surface of still water is equal to the atmospheric pressure. For simplicity of the following calculation,

$$p\_{z=H\_{\mathcal{L}}} = 0 \tag{5}$$

the hydrodynamic pressure at any time and space can be expressed as

$$p(x, y, z) = -\rho\_w \frac{\partial \varphi}{\partial t} \tag{6}$$

where *p*(*x*, *y*, *z*) is the hydrodynamic pressure of water.

If a small amplitude wave is formed on the free surface of the water, the gravity wave included in the analysis shall be considered through the linear surface wave condition [13]. The pressure on the surface can be expressed as

$$p\_{z=H\_L} = \rho\_w g h \tag{7}$$

The above equation can be expressed as

$$
\rho\_w g h = -\rho\_w \frac{\partial \varphi}{\partial t} \tag{8}
$$

### *2.5. Calculation of Wave-Current Load*

In addition to the load generated by the wave, the offshore structure is often acted upon by the current load generated by the water flow. Therefore, when analyzing the response of the structure under the action of the sea conditions, the wave load and the current load should be considered together. In this section, the wave parameter calculation model and the model of current velocity varying with water depth used in the numerical calculation in Section 2.6 are described.

#### 2.5.1. Wave Parameter

Waves can be divided into regular and irregular waves according to their shapes. The waves that propagate freely after leaving the wind zone are close to regular waves, while the wind waves in the ocean are irregular. The regular linear wave theory is selected as the wave theory in this paper, and the expression of the wave surface equation is

$$
\eta = \frac{H}{2} \cos(kx - \omega t) \tag{9}
$$

where *H* is the wave height and *K* is the wave number.

The strong wave and deep-water conditions described in this paper are defined according to the international wave level table (as shown in Table 1) and the micro amplitude wave theory.

According to the micro amplitude wave theory, wave types can be divided as follows. The deep-water wave:

$$\frac{d}{L} \ge 0.5\tag{10}$$

The finite depth wave:

$$0.5 > \frac{d}{L} > 0.05\tag{11}$$

The shallow water wave:

$$\frac{d}{L} \le 0.05\tag{12}$$

where *L* is the wavelength and *d* is the water depth. In engineering, *d* = 0.5 L is often used as the boundary between infinite water depth and finite water depth.

The dispersion equation of linear waves is

$$
\mu = \frac{\mu\_0}{\tanh(\mu)}\tag{13}
$$

where *µ* = 2*π L d* and *µ*0 = 4*π* 2 *gT*2 *d*.

After the period is determined, the wavelength can be solved through the above dispersion equation, and vice versa. Many scholars have put forward some approximate methods for a direct solution to the above equations. In this paper, the following calculation (Formula 14 ), proposed by Vatankhah and Aghashariatmadari [14] through numerical analysis, is used to solve them.

$$\mu = \frac{\mu\_0}{\sqrt{\tanh(\mu\_0)}} \left( 1 + \mu\_0 e^{-(1.835 + 1.225\mu\_0^{1.35})} \right) \tag{14}$$

Under any water depth condition, the maximum relative error of the formula is less than 0.019%.

#### 2.5.2. The Flowing Speed Model of Seawater

In the direction perpendicular to the sea level, the speed of the seawater flow changes with the depth of the seawater. Generally, the speed changes from the sea level to the seabed in a regular manner of "large up and small down." In this paper, the 1/7 exponential rate distribution model [15], proposed by Soulsby, is adopted. The model divides the vertical distribution of the flow speed into two layers for calculation. The upper layer adopts a linear distribution, and the lower layer adopts an exponential distribution. The schematic diagram of the model is shown in Figure 11. The vertical distribution of flow speed is given by

$$u(z) = \begin{cases} u\_0 \left(\frac{z}{0.32h}\right)^{\frac{1}{7}} & 0 \le \frac{\frac{\tau}{h}}{h} \le 0.5\\ 1.07u\_0 & 0.5 < \frac{\tau}{h} < 1 \end{cases} \tag{15}$$

where *u*0 is the velocity of the seawater surface, m/s.

**Figure 11.** The flowing speed model of seawater in the direction of water depth.

## 2.5.3. Wave-Current Load

There are many methods used to calculate the wave-current load [16,17]. In this paper, the nonlinear wave force [18] is not considered. The Morison formula with current participation is used to calculate the wave-current load on the pile. The wave-current load is given by

$$p\_H = \frac{1}{2} \frac{\gamma}{g} \mathbf{C}\_D D u\_{\rm inv} |u\_{\rm inv}| + \frac{\gamma}{g} \mathbf{C}\_M A \frac{\partial u}{\partial t} \tag{16}$$

$$
\mu\_{\rm wc} = \mu\_{\rm w} + \mu \tag{17}
$$

where *pH* is the wave-current load on a unit length micro-segment at any height *z* of the pile, N/m; *A* is the sectional area of the pile, m2 ; *CD* is the drag coefficient; *CM* is the inertia force coefficient; *γ* is the weight of the water, kN/m3 ; *∂u ∂t* is the horizontal acceleration of the orbital motion of the water particles, m/s2 ; *uwc* is the total horizontal velocity of the water particles, m/s; *uw* is the flow speed of the water, m/s; and *u* is the flow speed of the water particles generated by waves, m/s.

#### 2.5.4. Hydrodynamic Coefficient

The values of the two hydrodynamic coefficients in the Morison formula, namely the drag coefficient *CD* and the inertia force coefficient *CM*, are related to the accuracy of the final calculation results. For the values of these two coefficients, local scholars and scholars abroad have done a lot of experimental research, and they have given many recommended values in relevant papers, as well as specifications for reference in engineering practice. In general, if we want to get more accurate results, we must carry out hydrodynamic experiments in the actual water area to determine the value of the correlation coefficient. In this paper, the *CD* and *CM* values are determined according to references [19,20]. Finally, the *CD* is taken as 1.2, and the *CM* is taken as 2.0.

When calculating the wave-current load on the pile foundation of offshore structures, if the relative pile distance *S*/*D* (*S* is the pile center distance and *D* is the pile diameter) is small, the blocking effect of the pile array on the wave-current will increase the flow speed at the pile foundation, and the pile group effect cannot be ignored. When *S*/*D* ≥ 3.0, the pile group effect is weak, and the impact can be ignored [21].

#### *2.6. Numerical Simulation*

### 2.6.1. Method

In this paper, the Coupled Acoustic-Structure approach (*CAS*) is used for numerical simulation analysis in the finite element software ABAQUS. The acoustic wave control equation expressed by the acoustic pressure *p* as an independent variable is also called the Helmholtz equation, which is given as

$$\frac{\partial^2 p}{\partial t^2} - c^2 \nabla^2 p = 0 \tag{18}$$

where *c* is the speed of sound in the liquid, *c* = p *K*/*ρ*, and *k* is the bulk modulus of the liquid.

The advantage of using the *CAS* approach for modeling is that it is relatively simple and effective for numerical processing because it assumes no material flow, so no mesh deformation occurs. In addition, since the acoustic element has only pressure freedom at each node, it significantly reduces the computational time required for simulation [22].

The impedance boundary condition is defined on the free surface of the acoustic element using linear wave theory. The boundary impedance relates the pressure of the acoustic element to the normal motion at the acoustic structure interface. The speed ( . *uout*) of acoustic particles in the direction of the external normal of the fluid surface is related to the pressure and the rate of change in the pressure over time, which is given as

$$
\mu\_{out}^{\cdot} = \frac{1}{k\_1}\dot{p} + \frac{1}{c\_1}p\tag{19}
$$

where *p* is the acoustic pressure; . *p* is the change rate of sound pressure with time; 1 *k*1 is the ratio coefficient between the acoustic pressure and the displacement in the normal direction of the surface; and 1 *c*1 is the ratio coefficient between the acoustic pressure [23].

## 2.6.2. The Finite Element Model

In the modeling and numerical analysis of the combined survey platform shown in Figure 7, acoustic elements are used to simulate the internal and external support systems and the water between the support systems [24–26]. The internal and external support systems are simulated by shell elements, and the seabed is simulated by solid elements. The simulation of the water-pile and water-soil interfaces is realized by "tie" constraints, and the simulation of the pile-soil interface is realized by "embedded" constraints. The interaction between the platform and the distribution beam above the internal support system is defined by "tie" constraints in the software [27], and the analysis program uses dynamic implicit analysis. The parameters of pile and soil in the model are shown in Table 2.

**Table 2.** The parameters of the model.

\* Note: To eliminate the boundary effect, the size of the seabed in the depth direction is 1.5 times the embedded depth of the pile, and it is 20 times the pile diameter in the direction perpendicular to the depth [28].

During the shaking process in the water between the internal and external support systems, gravity is essential for resilience, but it is not allowed to directly define the uniformly distributed gravity load in the acoustic elements. In order to solve this problem, the acoustic impedance is introduced [22]. In the "geostatic step" of the model, the hydrostatic pressure corresponding to the water depth should be applied to the soil, and the treatment of the stress balance should be carried out by introducing the stress field. The wave-current load caused by the wave and water flow on the external support system is loaded by using the time history load obtained by the calculation method described in Section 2.5.3. The modeling process is shown in Figure 12.

**Figure 12.** The flow chart of the numerical simulation modeling. \* Note: *Uz* is the displacement in the depth direction of the soil in the geostatic step.

### 2.6.3. The Development and Verication of Subroutine

The "AQUA" module in ABAQUS can only calculate the wave-current load on smallscale structures, and the model element can only be calculated by the beam element. To load the combined survey platform proposed in this paper with the wave-current load, the load on the beam element is converted to the shell element by the principle applied in reference [29], and then the subroutine "Utracload" is compiled in the Fortran language to load the corresponding load.

The principle applied in this process is: assuming that the pressure *P* is uniformly distributed on the half circumference of the pile, which is the surface facing the direction of the wave, the resultant force *Fx* of the pile in the *x* direction is the total wave load on the pile, and the resultant force *Fy* of the pile in the *y* direction is *0* [29]. The calculation principle is shown in Figure 13. Reference [29] has studied the rationality of this method. The research results show that the method can be used to simplify the wave load on small-scale structures, and the accuracy of the calculation results fully meets the engineering requirements.

**Figure 13.** The conversion principle of the wave-current load on the shell element.

The expression of the wave-current load on the shell element is given as

$$\text{F}\_{\text{H}} = \int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} d\text{F}\_{\text{x}} = 2 \int\_{0}^{\frac{\pi}{2}} \text{Pcos}\theta d\text{s} = 2 \int\_{0}^{\frac{\pi}{2}} \text{PRcos}\theta d\theta = 2 \text{PR} = \text{PD} \tag{20}$$

where *θ* represents the angle between the *X* axis and the line connected by the wave propagation direction and the center of the steel pipe. Its function is to decompose the wave force of the steel pipe pile along the *X* axis and the *Y* axis, respectively.

To verify the accuracy of the load subroutine, the beam element and the shell element are used to establish models for the steel pipe pile with a diameter of 1 m and a pile length of 20 m in ABAQUS, respectively, and contrast analysis of the two models in the wave height of 1 m, a period of 8 s wave conditions, and the steel pipe piles by the total difference of wave load. Among them, the beam element model uses the "AQUA" module to load the wave load, and the shell element model uses the above-mentioned load subroutine to load the wave load calculated by the theoretical formula. The total wave load received by the beam element model is the support reaction force received by the pile bottom in the wave propagation direction, while the shell element model is the sum of the support reaction forces received by each node at the pile bottom in the wave propagation direction. The comparison of the results is shown in Figure 14.

direction. The comparison of the results is shown in Figure 14.

**Figure 14.** The comparison of the results. **Figure 14.** The comparison of the results.

 = ∫ 2 − 2

It can be seen from the comparison of the two models shown in Figure 14 that the results obtained by the shell element and the beam element are the same. This shows that the principle of the method used to convert the wave load on the beam element into the shell element is reasonable, and it also verifies the correctness of the Fortran subroutine written in this paper. It can be seen from the comparison of the two models shown in Figure 14 that the results obtained by the shell element and the beam element are the same. This shows that the principle of the method used to convert the wave load on the beam element into the shell element is reasonable, and it also verifies the correctness of the Fortran subroutine written in this paper.

The expression of the wave-current load on the shell element is given as

wave force of the steel pipe pile along the axis and the axis, respectively.

= 2 ∫ 2 0

where represents the angle between the axis and the line connected by the wave propagation direction and the center of the steel pipe. Its function is to decompose the

To verify the accuracy of the load subroutine, the beam element and the shell element are used to establish models for the steel pipe pile with a diameter of 1 m and a pile length of 20 m in ABAQUS, respectively, and contrast analysis of the two models in the wave height of 1 m, a period of 8 s wave conditions, and the steel pipe piles by the total difference of wave load. Among them, the beam element model uses the "AQUA" module to load the wave load, and the shell element model uses the above-mentioned load subroutine to load the wave load calculated by the theoretical formula. The total wave load received by the beam element model is the support reaction force received by the pile bottom in the wave propagation direction, while the shell element model is the sum of the support reaction forces received by each node at the pile bottom in the wave propagation

= 2 = (20)

= 2 ∫ 2 0

#### **3. Analysis of Working Performance 3. Analysis of Working Performance**

Based on the basic principle of vibration reduction and isolation, a new structure for a combined survey platform is proposed in this paper. Two kinds of layout designs are carried out according to the different positions of the survey platform: independent and attached. Quantitative research is carried out to understand the specific effects of the structural size and the sea conditions on the working performance of the combined survey platform. For the convenience of expression, the pile diameter and the wall thickness men-Based on the basic principle of vibration reduction and isolation, a new structure for a combined survey platform is proposed in this paper. Two kinds of layout designs are carried out according to the different positions of the survey platform: independent and attached. Quantitative research is carried out to understand the specific effects of the structural size and the sea conditions on the working performance of the combined survey platform.

tioned below refer to the combination of the pile diameter and the wall thickness of the external support system, and the platform described is the survey platform on the top of the internal support system. For the convenience of expression, the pile diameter and the wall thickness mentioned below refer to the combination of the pile diameter and the wall thickness of the external support system, and the platform described is the survey platform on the top of the internal support system.

### *3.1. The Working Performance of the Independent Layout Scheme*

According to the different roles played by the internal and external support systems in the process of quantitative research on the independently set combined survey platform, six combination schemes are designed according to the different structure sizes, as shown in Table 3.

**Table 3.** The schemes of size combinations.

Note: In the process of construction, the verticality of steel pipe piles is often difficult to ensure. Therefore, during the driving of the steel pipe piles of the internal and external support systems, limit devices should be used to keep their inner wall surfaces parallel to the maximum extent and to provide more guarantees for avoiding collisions between the internal and external support systems under the wave-current load.

### 3.1.1. The Structural Response of the Independently Set Combined Survey Platform

In order to verify the working performance of the independently set combined survey platform under the wave-current load, the wave height, the wave period, the speed of the water flow, and the water depth are set as 1.0 m, 8.0 s, 1.0 m/s, and 20.0 m, respectively. The time history of the wave-current load is loaded on the survey platform mentioned in Scheme 2 in Table 3. Figure 15a shows the displacement variation time history of the free end of the internal and external support systems. As the figure shows, under the action of the wave-current load, the variation amplitude of the displacement of the external support system is large, with a maximum value of up to 65.1 mm, while the variation amplitude of the displacement of the internal support system is significantly reduced compared to the external support system, and the maximum displacement is only 5.4 mm.

**Figure 15.** The time history of structural response. (**a**) The displacement of the internal and external support systems. (**b**) The vertical angle of the platform.

The gap between the internal and external support systems can be expressed as

$$
\Delta \mathcal{U} = (D\_1 - D\_2) \times 1000 - \mathcal{U}\_1 + \mathcal{U}\_2 \tag{21}
$$

where ∆*U* is the gap between the internal and external support systems, mm; *D*1 is the diameter of the external support systems, m; *D*2 is the diameter of the internal support systems, m; *U*1 is the displacement of the external support system, mm; and *U*2 is the displacement of the internal support system, mm.

The subgraph in Figure 15a shows the time-history curve of the gap between the internal and external support systems. The large difference in displacement between the internal and external support systems indicates that: (1) the external support system achieves the expected barrier effect to the wave-current load, and the internal support system only bears the load transferred by soil and water quality between the internal and external support systems, which can significantly reduce the wave-current load on the influence of the internal support system; and (2) since the displacement of the internal support system is small, whether the internal and external support systems collide is mainly affected by the displacement of the external support system. Figure 15b shows the variation time history of the vertical angle of the independently set combined survey platform. As the figure displays, under the action of the wave-current load, the variation amplitude of the vertical angle of the platform is small, and the maximum value of the vertical angle is only 5.8", which proves that the independently set combined survey platform designed in this paper still has a good working performance under the influence of the wave and the current.

The prerequisite for the good working performance of the combined survey platform proposed in this paper is that the internal and external support systems do not collide. Additionally, since the displacement of the internal support system is small compared to the displacement of the external support system, the quantitative research carried out in the following is evaluated by the displacement of the external support system and the vertical angle of the top platform.

## 3.1.2. The Influence of Sea Conditions