Calculation method of wave load on superstructure of offshore box girder bridge

Abstract

The application discloses a wave load calculation method for offshore box girder bridge superstructure and belongs to the field of bridge wave load calculation. The method comprises the following steps: obtaining the wave surface time history and the flow field element time history according to the numerical simulation of the wave field where the box girder bridge is located by iSquares, and determining the submerged state according to the position elevation; calculating the vertical wave force of the top plate and the bottom plate of a rectangular beam with the same width and height as the box girder according to the flow field element time history and the submerged depth respectively, and superimposing to obtain the total vertical wave force of the rectangular beam; calculating the relative width coefficient and the relative height coefficient according to the current box girder size characteristics, and then calculating the length vertical and horizontal wave force cross-section effect coefficient of the box girder; and obtaining the vertical wave force and the horizontal wave force of the box girder in combination with the flow field element time history. The application breaks through the limitation of the traditional wave load static extreme value model, provides a time-varying calculation method suitable for the wave load of offshore box girder bridge superstructure, and effectively improves the accuracy and calculation efficiency of wave force prediction under extreme sea conditions.

Description

Technical Field

[0001] This application relates to a wave load calculation method for the superstructure of a near-shore box girder bridge, belonging to the technical field of wave load calculation for near-shore bridges. Background Technology

[0002] The superstructures of nearshore bridges are constantly exposed to strong nonlinear long waves and complex nearshore hydrodynamic environments during extreme marine events, making them susceptible to damage such as overhang, slippage, or capsizing, significantly threatening their safety and operational reliability. In recent years, against the backdrop of rising sea levels and intensifying tropical cyclones, the uncertainty of the wave environment in nearshore shallow waters has further increased. Numerous hurricanes and the Great East Japan Earthquake have recorded severe damage to bridge superstructures, making a refined assessment of wave loads on superstructures an urgent necessity.

[0003] Compared to the substructure, the wave load mechanism of bridge superstructures is more complex, and current research is still in the stage of exploring the underlying mechanisms. Existing wave load formulas for superstructures have limited accuracy, and conservative empirical formulas are often used in engineering practice. Theoretical research shows that the wave force on the superstructure is mainly controlled by factors such as wave height, submersion degree, and cross-sectional shape, and the relationship is quite complex. Current methods for calculating wave loads on superstructures face two major challenges: first, the applicability of existing empirical formulas is limited, making it difficult to reflect the load differences and load time history evolution characteristics of box girders with different cross-sections; second, although high-precision numerical simulation methods (such as CFD methods) can obtain relatively accurate results, they are difficult to use for rapid engineering evaluation due to low computational efficiency and complex parameter verification.

[0004] Furthermore, existing horizontal force estimation formulas mostly characterize the structural geometric effects using the vertical projected area in the direction of flow, failing to explicitly consider the influence of web inclination changes on the resultant force, which easily leads to overestimation of the horizontal force. Simultaneously, most existing methods rely on the assumption of regular waves or single isolated waves, making it impossible to directly use experimental waveforms or field wavefront reconstruction sequences as input, thus limiting their application in practical engineering. Therefore, there is an urgent need for a more widely applicable, computationally efficient method for calculating wave loads on the superstructure of near-shore box girder bridges that can reflect the characteristics of load time-history evolution. Summary of the Invention

[0005] This application provides a method for calculating wave loads on the superstructure of a near-shore box girder bridge, addressing the limitations of existing empirical formulas in terms of applicability and their inability to reflect load differences and load time-history evolution characteristics of box girders with different cross-sections. The specific solution is as follows: A method for calculating wave loads on the superstructure of a near-shore box girder bridge, comprising the following steps: S1: Based on the wave field of the box girder in iSquares numerical simulation, the wave surface time history and flow field element time history of the box girder are obtained respectively, and its submersion state is determined according to the elevation of the box girder's location. S2: Based on the current flow field time history and submersion depth of the box girder, calculate the vertical wave forces of the top and bottom plates of the rectangular beam with the same width and height as the box girder, and then superimpose them to obtain the total vertical wave force of the rectangular beam. S3: Calculate the relative width coefficient and relative height coefficient of the box girder according to the current dimensional characteristics; calculate the vertical wave force section effect coefficient of the box girder based on the relative height coefficient; calculate the horizontal force section effect coefficient of the box girder based on the relative width coefficient; S4: Based on the vertical wave force section effect coefficient and the horizontal wave force section effect coefficient calculated in S3, the vertical wave force time history and the horizontal wave force of the box girder are obtained by combining the flow field element time history.

[0006] Preferably, in S2, the total vertical wave force of the rectangular beam is obtained based on the superposition of the vertical wave force of the bottom plate and the vertical wave force of the top plate, and is expressed as follows: ; In the formula, Vertical wave force on a rectangular beam; The vertical wave force on the bottom plate is the instantaneous static pressure at each monitoring point of the rectangular beam, which is determined by the hydrostatic pressure integral term, the bottom plate overpressure term, and its spatiotemporally separable correction term. The vertical wave force on the top plate is determined by the hydrostatic pressure integral term and the non-static correction factor.

[0007] Preferably, the spatiotemporally separable correction term is represented in a separable form using time harmonic basis and spatial polynomial basis; the spatiotemporally separable correction term is used to correct the hydrostatic pressure term of the bottom plate of the rectangular beam, thereby obtaining the corrected pressure time history of the bottom plate of the rectangular beam; The overpressure term of the bottom plate is constructed considering the overpressure effect of the bottom plate; the overpressure effect is used to characterize the upward compression of the bottom plate by the incoming water body in the confined space between the bottom plate and the bottom of the channel. The overpressure effect is related to the width of the box girder bottom plate, the bottom elevation of the girder, the water depth and the lateral relative position of the measuring points.

[0008] Preferably, the spatiotemporally separable correction term is constructed based on time harmonic basis functions and spatial polynomial basis functions. The coefficients of the spatiotemporally separable correction term are determined by least squares fitting, and stable convergence is achieved using low-order time harmonics and low-order spatial polynomials. The calculation formula for the vertical wave force of the bottom plate is obtained by interpolating and integrating the pressure at each monitoring point: ; In the formula, W This refers to the width of the bottom of the beam; ρ For fluid density; g It is the acceleration due to gravity; h For water depth; The time-varying wavefront elevation at various locations is given for cases where the top slab is not covered by water. Simplified to ; t For time; h gb This is the elevation of the bottom of the beam;  The relative lateral position of the measuring point; L eq The equivalent length at which pressure disturbances exert their effect in a local water body; The rate of change of flow velocity at the measuring point over time; H Wave height; This is a spacetime separability correction term.

[0009] Preferably, in S2, the calculation of the vertical wave force on the top plate of the rectangular beam includes both cases where the top plate is covered with water and cases where it is not covered with water. When the roof is covered with water, a non-static correction factor κ should be introduced to correct the vertical wave force of the roof when calculating the vertical wave force, in order to compensate for the non-static effect caused by the significant vertical acceleration near the free surface. In this case, the formula for calculating the vertical wave force of the roof is: ; In the formula, The vertical wave force on the top plate; For non-static water correction factors; h gt This is the elevation of the top of the beam.

[0010] Preferably, the vertical wave force section effect coefficient of the box girder is obtained by performing multiple linear regression on simulation data of different section combined parameters, using the relative width coefficient and relative height coefficient of the section as independent variables; The horizontal wave force section effect coefficient of the box girder is based on the ratio of the top plate width to the bottom plate width. The variables are obtained by linear regression of simulated data with different cross-sections and parameters.

[0011] Preferably, in S4, the vertical wave force section effect coefficient C vsec This is used to characterize the influence of the box girder's cross-sectional geometry on the peak value of the vertical wave force; when the superstructure is a box girder, the formula for calculating the vertical wave force of the box girder is: ; In the formula, This is a vertical wave force; C vsec The vertical wave force section effect coefficient is determined by the submersion condition of the box girder. The vertical wave force is for the rectangular beam.

[0012] Preferably, wherein:C vsec The strategy for determining the value is: Under total submersion conditions, the vertical wave force section effect coefficient C vsec Taking 1, the peak value of the vertical wave force of the box girder is approximately equal to the peak value of the vertical wave force of the rectangular beam with the same width and height. Under semi-submerged conditions C vsec It is an additive function related to the relative width factor of the cross section, the relative width factor of the flange, and the outward inclination characteristic of the web.

[0013] Preferably, in S4, when calculating the horizontal wave force, the McPherson formula is used as the benchmark, and the horizontal force section effect coefficient Chsec is introduced to modify the horizontal force formula, resulting in the modified formula for calculating the horizontal wave force of the box girder: ; In the formula, F h For the horizontal wave force of the box girder; F static_front and F static_front These are the horizontal components of the hydrostatic pressure on the wave-facing side and the horizontal components of the hydrostatic pressure on the wave-avoiding side, respectively, both of which are calculated directly using the McPherson formula.

[0014] Compared with the prior art, the present invention has the following beneficial effects: This invention no longer relies on the assumption of regular waves or single isolated waves. It can directly use experimental waveforms or field wavefront reconstruction sequences as input, and obtain the wavefront time history and flow field element time history at the structure through iSquares numerical simulation. It has a wider range of applications and is more in line with the complex wave conditions in actual engineering. This invention replaces the direct solution of the full three-dimensional transient by using the hydrostatic pressure main control term plus the physically interpretable overpressure term and its spatiotemporally separable correction term. This not only preserves the key physical mechanisms, such as the overpressure effect of the bottom plate and the non-hydrostatic effect, but also significantly reduces the computational cost and improves the computational efficiency. This invention introduces vertical wave force section effect coefficients and horizontal force section effect coefficients, which can quantitatively characterize the influence of different box girder cross-sectional geometries, including bottom plate width, top plate width, flange height, web inclination angle, etc., on wave loads. It can provide the time history of vertical wave loads for rectangular beams and box girders under different submersion coefficients, and can predict the peak values ​​of vertical and horizontal forces relatively accurately, providing a reliable load input for the wave-resistant design of the superstructure of near-shore box girder bridges. Attached Figure Description

[0015] Figure 1 This is a general framework diagram of the iSquares flow field simulation and wave load method for the superstructure of box girder bridges of the present invention; Figure 2 This is a schematic diagram of the theoretical model of wave load on the thin plate structure of the present invention; Figure 3 This is a schematic diagram of the rectangular cross-section with variable dimensions of the present invention; Figure 4 This is a schematic diagram of the variable-size box girder cross-section of the present invention; Figure 5 This is a comparison chart of the calculated and reference values ​​of the vertical force time history at section BOX1 of this invention; Figure 6 This is a comparison chart of the calculated and reference values ​​of the vertical force time history at section BOX7 of this invention; Figure 7 This is a comparison chart showing the results of calculating and referencing the vertical wave load values ​​for different types of box girder sections using the formula of this invention. Figure 8 A comparison of calculated and reference values ​​of vertical wave loads for different types of box girder sections calculated using the McPherson formula; Figure 9 This is a comparison chart showing the results of calculating and referencing horizontal wave loads for different types of box girder sections using the formula of this invention. Figure 10 A comparison of the calculated and reference values ​​of horizontal wave loads for different types of box girder sections calculated using the McPherson formula; Figure 11 Let Leq change over time (REC1-N6-1.33); Figure 12 To compare the static pressure results of the base plate before and after considering overpressure (N6-1.33); Figure 13 The change in base plate pressure correction term over time (REC1-N6-1.33); Figure 14 Error heatmaps for different harmonic orders and space polynomial degrees. Detailed Implementation

[0016] The wave load calculation method for the superstructure of a near-shore box girder bridge provided in this application specifically includes the following steps: S1: Based on the wave field of the box girder bridge in iSquares numerical simulation, the wave surface time history and flow field element time history at the box girder structure are obtained respectively, and the submersion state of the structure is determined according to the elevation of the box girder structure. It should be noted that in this application: The wavefront time history includes the time-varying wavefront elevation η(x, t) at various locations on the box girder structure; the flow field element time history includes the flow velocity u(x, t) at various measuring points near the bottom plate of the box girder structure and its rate of change over time ∂u / ∂t. The wavefront time history and the flow field element time history are obtained by simulating the wave-bridge interaction scenario to be analyzed within the iSquares numerical framework to acquire the time-varying wavefront elevation and the rate of change of flow velocity over time at the target location.

[0017] The specific process for determining the submersion status of a box girder based on its elevation is as follows: The submersion coefficient is defined based on the ratio of the distance from the bottom of the box girder to the still water surface to its height at the initial moment. Cs ,when Cs When ≤0, it is defined as the total flooding condition; when 0 < Cs When < 1, it is defined as a semi-submerged condition. Cs When ≥1, it is defined as a non-submerged condition.

[0018] S2: Based on the time history of the flow field elements and the current submersion depth of the box girder, calculate the vertical wave force of the top plate and bottom plate of the rectangular beam with the same width and height as the box girder, and obtain the total vertical wave force of the rectangular beam by superimposing the vertical wave forces of the top plate and bottom plate of the rectangular beam. In this application, since the refined model of vertical wave load on rectangular beams specifies that the positive direction of the vertical wave force on the rectangular beam is vertically upward, the vertical wave force on the rectangular beam is the vertically upward wave force of the bottom plate minus the vertically downward wave force of the top plate. The total vertical wave force of the rectangular beam obtained by superimposing the vertical wave forces of the bottom plate and the top plate is expressed as follows:

[0019] In the formula, Vertical wave force on a rectangular beam; The vertical wave force on the bottom plate is the instantaneous static pressure at each monitoring point of the rectangular beam, which is determined by the hydrostatic pressure integral term, the bottom plate overpressure term, and its spatiotemporally separable correction term. The vertical wave force on the top plate is determined by the hydrostatic pressure integral term and the non-static correction factor.

[0020] In this application, for the base plate of the rectangular beam, due to the wave force in the vertical direction of the base plate during wave propagation, the local water body at the bottom of the structure is constrained by the space below the structure, resulting in upward compression when the wave crest passes, thus triggering an overpressure effect. Therefore, this application introduces a base plate overpressure term. Simultaneously, since the functional form of pipe flow has certain deviations when applied to wave-structure interaction scenarios, a spatiotemporal separability correction term is added to correct the calculation deviations after introducing the overpressure term.

[0021] The spatiotemporally separable correction term is represented in a separable form using time harmonic basis and space polynomial basis. This term is used to correct the hydrostatic pressure term of the rectangular beam's base plate, thereby obtaining the corrected pressure time history of the rectangular beam's base plate. The spatiotemporally separable correction term is constructed based on time harmonic basis functions and space polynomial basis functions, and its expression is as follows:

[0022] In the formula, m The time harmonic order, q Let the degree be the space polynomial. , T During the period of wave action, a mq and b mq These are the fitting coefficients. ξ The relative lateral position of the measuring point; the coefficients of the spatiotemporally separable correction term are determined by least squares fitting, and stable convergence is achieved by using low-order time harmonics and low-order spatial polynomials (P=3, M=4 is the preferred truncation combination).

[0023] The overpressure term of the bottom plate is constructed considering the overpressure effect of the bottom plate; the overpressure effect is used to characterize the upward compression of the bottom plate by the incoming water body in the confined space between the bottom plate and the bottom of the channel. The overpressure effect is related to the width of the box girder bottom plate, the bottom elevation of the girder, the water depth and the lateral relative position of the measuring points.

[0024] The formula for calculating the vertical wave force on the bottom plate of the rectangular beam is obtained by interpolating and integrating the pressure at each monitoring point:

[0025] In the formula, W This refers to the width of the bottom of the beam; ρ For fluid density; g It is the acceleration due to gravity; h For water depth; The time-varying wavefront elevation at various locations is given for cases where the top slab is not covered by water. Simplified to ; t For time; h gb This is the elevation of the bottom of the beam;  The relative lateral position of the measuring point; L eq The equivalent length at which pressure disturbances exert their effect in a local water body; The rate of change of flow velocity at the measuring point over time; H Wave height; This is a spacetime separability correction term.

[0026] In this application, for the top plate of a rectangular beam, since the calculation of vertical wave force on the top plate may involve both water surfacing and non-water surfacing situations, this application introduces a non-static correction coefficient κ to quantify the amplification or attenuation of the vertical pressure on the top plate by non-static effects such as vertical acceleration near the free surface. The formula for calculating the vertical wave force on the top plate is then:

[0027] In the formula, The vertical wave force on the top plate; The non-static water correction coefficient is obtained by fitting the data output from the iSquares numerical model using the least squares method. h gt This is the elevation of the top of the beam; t For time; For cases where the top slab is covered by water and not covered by water, when the top slab is not covered by water, the vertical wave force on the top slab of the rectangular beam is zero.

[0028] S3: Calculate the relative width coefficient and relative height coefficient of the box girder structure according to the current structural dimensional characteristics; calculate the vertical wave force section effect coefficient of the box girder based on the relative height coefficient; calculate the horizontal force section effect coefficient of the box girder based on the relative width coefficient; In this application, the total vertical wave force on the actual box girder and the total vertical wave force on an equivalent rectangular beam of the same width and height are calculated separately, and the vertical wave force section effect coefficient is defined. C vsec The ratio of the two is used to quantify the impact of cross-sectional shape differences on vertical wave forces. Parametric simulations with different cross-sectional geometric parameters are conducted using the iSquares numerical framework to extract the vertical wave force ratio data for each working condition. Specifically, this is achieved by changing the relative width coefficient. λ w = W T / W B and relative height coefficient λ h = h f / h g Multi-condition parametric numerical simulations were conducted to extract the vertical wave force ratio under each condition. The least squares method was then used for multiple linear regression to determine the... C vsec With relative width coefficient λ w Relative height coefficient λ h The correlation was used to obtain the vertical wave force section effect coefficient of the box girder. Cvsec The expression is: C vsec = 0.4 λ w + 0.24 λ h +0.49 In the formula, λ w This is the relative width coefficient. λ h This is a relative height coefficient. W T The width of the top plate. W B The width of the base plate. h f For the wing edge height, h g This represents the total height of the cross-section.

[0029] In this application, the horizontal force section effect coefficient C hsec Using the horizontal component of hydrostatic pressure calculated by the McPherson formula as a benchmark, the ratio of the actual horizontal wave force of the box girder obtained from numerical simulation to the benchmark value is defined as... C hsec Parametric simulations of different cross-sectional geometric parameters were conducted using the iSquares numerical framework, based on different relative width coefficients. λ w Numerical experimental results analysis revealed that Chsec is mainly affected by λ w The influence of horizontal force section effect coefficient of box girder was obtained by linear regression. C hsec The expression is: C hsec = 0.36 + 0.58 λ w In the formula, λ w This refers to the relative width coefficient. λ w = W T / W B .

[0030] S4: Based on the vertical wave force section effect coefficient and horizontal force section effect coefficient of the box girder calculated in S3, the vertical wave force and horizontal wave force of the box girder are obtained by combining the time history information of the flow field elements.

[0031] In this application, for the calculation of vertical wave forces on box girders, under the condition of total submersion, the vertical wave forces on box girders with different cross-sectional shapes exhibit a synchronous evolution trend over time during the initial wave action phase, and the peak force differences are not significant. Therefore, equivalent rectangular beams with the same width and height can be used to simplify the calculation of various box girder cross-sections. For the semi-submerged state, a dimensionless vertical force section effect coefficient is introduced. C vsec The formula for calculating the vertical wave force of a box girder, reflecting the degree to which box girders with different cross-sectional geometries reduce the wave force compared to rectangular beams of the same height and width, is as follows:

[0032] In the formula, The vertical wave force on the box girder; For the vertical wave force on the rectangular beam, C vsec This is the vertical wave force section effect coefficient, and its value is determined by the submersion condition of the box girder. C vsec The strategy for determining the value is: Under total submersion conditions, the vertical wave force section effect coefficient C vsec Taking 1, the peak value of the vertical wave force of the box girder is approximately equal to the peak value of the vertical wave force of the rectangular beam with the same width and height. Under semi-submerged conditions C vsec This is an additive function related to the relative width factor of the cross section, the relative width factor of the flange, and the outward inclination characteristic of the web, i.e.: C vsec = 0.96-0.07 λ w + 0.23 λ h - 0.15 α In the formula, λ w This is the relative width coefficient. λ h This is a relative height coefficient. α The characteristic coefficient for web inclination is... α = cos θ , θ It is the angle between the web and the vertical direction.

[0033] In this application, the McPherson formula is used to calculate the horizontal load on the superstructure of the box girder for the calculation of the horizontal wave force, and a horizontal section effect factor is introduced. C hsec The baseline results are corrected. This is the relative width coefficient. λ w The linear function yields the corrected formula for calculating the horizontal wave force of the box girder:

[0034] In the formula, F h For the horizontal wave force of the box girder; F static_front and F static_front These are the horizontal components of the hydrostatic pressure on the wave-facing side and the wave-avoiding side, respectively, both of which are calculated directly using the McPherson formula. The specific formula is as follows: In the formula, η max The peak elevation; h For water depth; h girders This is the elevation of the bottom of the beam; H bridge This refers to the vertical height of the bridge. L bridge This refers to the longitudinal length of the bridge. γ This refers to the unit weight of water. h deck This refers to the bridge deck elevation. C hsec This is the horizontal force section effect coefficient.

[0035] To further illustrate the technical means and effects adopted by the present invention to achieve the intended purpose, the following embodiments, in conjunction with the accompanying drawings and specific implementation methods, provide a detailed explanation of the wave load calculation method for the superstructure of a near-shore box girder bridge proposed in accordance with the present invention.

[0036] Example 1: This embodiment provides a specific example illustrating the wave load calculation method for the superstructure of a near-shore box girder bridge described in this application. Please refer to [link / reference]. Figure 1 , Figure 1 This is the overall framework diagram of the iSquares flow field simulation and wave load method for the superstructure of a box girder bridge in this embodiment. Figure 1 As shown, the wave field was simulated using iSquares numerical simulation to obtain the wave surface time history and current velocity time history at the structure, and the submersion status of the structure was determined based on the elevation of the structure's location.

[0037] Based on the time history information of the aforementioned flow field elements and the current submersion depth of the main girder, the time history of the vertical wave force of the rectangular beam with the same width and height as the box girder is calculated. The relative width coefficient and relative height coefficient are calculated based on the structural dimensional characteristics of the box girder, and then the section effect coefficient of the vertical force of the box girder is calculated to obtain the vertical wave force of the box girder.

[0038] The relative width coefficient is calculated based on the structural dimensional characteristics of the box girder, and then the horizontal force section effect coefficient of the box girder is calculated. The peak value of the horizontal wave force of the box girder is obtained through the flow field element information.

[0039] 1. Calculation of pressure at any point on a thin plate under ideal conditions: Please see Figure 2 , Figure 2 This is a schematic diagram of the theoretical model of the wave load on the thin-plate structure of this invention. (See diagram below.) Figure 2 As shown, under ideal fluid conditions, if the free water surface fluctuation can be described by the simple harmonic wave equation and satisfies the following basic assumptions, then the analytical solution proposed in this invention can be used to directly calculate the pressure time history at any point on the structural surface. The basic assumptions are as follows: (1) The fluid is considered to be inviscid and incompressible. The wave type is a one-dimensional linear harmonic wave, which acts perpendicularly to the cross section of the structure. Considering that extreme waves usually have instantaneous characteristics, the non-inertial term in the pressure is ignored in the theoretical model, that is, the static pressure and dynamic pressure are considered to be independent of each other.

[0040] (2) The structure has no displacement or deformation during the wave action. Since the height of the structure is much smaller than the wavelength, it is assumed that the cross-sectional shape of the wave does not change when it passes through the cross section of the structure.

[0041] According to hydrostatic pressure theory, the hydrostatic pressure at any point in a fluid is directly proportional to the perpendicular distance from that point to the still water surface. Extending this principle to wave fields, since the wave surface dynamically changes over time, the hydrostatic pressure at any point on the structure also changes accordingly, and is directly proportional to the perpendicular distance from that point to the instantaneous wave surface. Taking the measuring points at the top plate and the wave-facing plate as examples, denoted as... P top ( x top , y top )and P yf ( x yf , y yf According to hydrostatic pressure theory, the expressions for its hydrostatic pressure are as follows: In the formula, A wAmplitude; Angular frequency; t For time; k Wave number; τ For Liang Gao; h g The elevation of the geometric center of the structure; h For water depth; A ( x yf , t (This is the point of contact with the wave-facing plate) P top ( x yf , y yf Time-varying wavefront elevation at ( ) location; A ( x top , t ) is the top plate point P top ( x top , y top Time-varying wavefront elevation at ( ) location; t st This represents the moment when static pressure begins; t end This represents the moment when static pressure is completed; This represents the time function of pressure application.

[0042] 2. Calculation of vertical time-varying wave force on a rectangular beam under actual sea conditions: In actual working conditions, waves deform when passing through structures, causing changes in wave surface shape. In this case, it is necessary to use the iSquares numerical model to simulate and obtain the real-time wave surface elevation at the structural location, and then substitute it into the calculation formula.

[0043] In this embodiment, the following input information is first obtained: Please see Figure 3 , Figure 3 This is a schematic diagram of the cross-sectional dimensions of the rectangular beam of the present invention. For example... Figure 3 As shown, the structural geometric parameters and specific dimensional information include beam height, beam width, and height-to-width ratio. λ (The ratio of beam height to beam width).

[0044] Structural location parameters include: beam bottom elevation and bridge deck elevation.

[0045] Wave and flow field parameters include: local free surface elevation time history, flow velocity near the bottom plate and its rate of change over time, and submersion coefficient. In this invention, the submersion coefficient is defined as the ratio of the distance between the bottom of the beam and the still water surface at the initial moment to the beam height.

[0046] This invention calculates the vertical wave force of a rectangular beam by dividing it into two parts: the vertical wave force of the bottom plate and the vertical wave force of the top plate.

[0047] For the vertical wave force on the base plate, substitute the input parameters into the following formula for calculation: In the formula, For the vertical force of the rectangular beam; For the vertical force of the base plate; This refers to the vertical force on the top plate; t For time; W The width of the beam bottom plate; x The relative lateral position of the measuring point; ρ For fluid density; g It is the acceleration due to gravity; h For water depth; The time-varying wavefront elevation at various locations is given for cases where the top slab is not covered by water. Simplified to ; h gb This is the elevation of the bottom of the beam; L eq ( x The effective length of the pressure disturbance in the local water body is given by the formula, which is obtained by fitting the data from the numerical simulation using the least squares method. The rate of change of flow velocity at the measuring point over time; This is a spacetime separability correction term. C s A is the inundation coefficient; t ), B( t ), C( t ), D( t () is a coefficient function that varies with time; m For time harmonic indexing; For the first m Angular frequency, according to calculate, T Take the period of wave action; a q0 ( q =0, 1, 2, 3) are the fitting coefficients, representing... xq The average time contribution of the item; The fitting coefficients characterize... x q Item, No. m The amplitude of the first harmonic. In the formula, this invention uses a unified basis function system to fit over all operating conditions, obtaining a sample set of each coefficient. For each coefficient component... j Give the median. and the 5%~95% quantile range [ Q 0.05 ( ), Q 0.95 ( )], the specific values ​​are shown in Table 1: Table 1. Statistics of space-time separable Fourier-polynomial expansion coefficients

[0048] For the vertical wave force on the top plate, substitute the input parameters into the following formula for calculation: In the formula, This refers to the vertical force on the top plate; t For time; For non-static water correction factors; C s denoted as the flooding coefficient. The coefficients in the formula are obtained by fitting the data output from the iSquares numerical model using the least squares method.

[0049] 3. Calculation of vertical time-varying wave force for box girders under actual sea conditions: Based on the formula for the vertical force of a rectangular beam, the vertical time-varying wave force of a box girder can be further derived. Similar to the case of a rectangular beam, the calculation of the vertical wave force of a box girder also depends on the structural geometric parameters and wave and flow field parameters.

[0050] When implementing the method of the present invention, the following input information is first obtained: Please see Figure 4 , Figure 4 This is a schematic diagram of the variable-size box girder cross-section of the present invention. (See diagram below.) Figure 4 As shown, the structural geometric parameters and specific dimensional information include: base plate width, flange width, flange height, web inclination angle, and web height.

[0051] This invention simplifies the characterization of complex cross-sectional geometric features by introducing two types of dimensionless parameters. These two parameters are the relative height coefficient and the relative height coefficient. With relative width coefficient Its definition is as follows: In the formula, h f This refers to the wing flange height. h g This is the total height of the cross-section; W B The width of the base plate; W T This refers to the width of the top plate.

[0052] Structural location parameters include: beam bottom elevation and submersion coefficient.

[0053] Wave and flow field parameters include: water depth, local free surface elevation time history of the rectangular cross-section with equal height and width of the target box girder, flow velocity near the bottom plate and its rate of change over time.

[0054] about The formula for the vertical wave force on a rectangular beam is: In the formula, For the vertical force of the box girder; C vsec This is the vertical force section effect coefficient; For the vertical force of the rectangular beam; This is the relative width coefficient; This is the relative height coefficient; C s This is the flooding coefficient.

[0055] Please see Figures 5-6 , Figure 5 This is a comparison chart of the calculated and reference values ​​of the vertical force time history at section BOX1 of this invention. Figure 6 This is a comparison chart of the calculated and reference values ​​of the vertical force time history at section BOX7 of this invention. (See figure below.) Figures 5-6 The figure shows the calculated force time history results for BOX1 and BOX7. As can be seen from the figure, the vertical force calculation formula for the box girder proposed in this invention has better synchronization with the numerical simulation results during the rising-falling phase of the first wave and the oscillation phase after the first wave, and can more accurately reproduce the main evolution characteristics of the wave force time history.

[0056] Please see Figures 7-8 , Figure 7 This is a comparison chart of the calculated and reference values ​​of the vertical wave loads on different types of box girder sections calculated by the formula of this invention. Figure 8 This is a comparison chart of the calculated and reference values ​​for vertical wave loads on different types of box girder sections calculated using the McPherson formula. (Example:) Figures 7-8As shown, the calculation results of the vertical wave force peak value of the box girder using the formula proposed in this invention are compared with those calculated using the McPherson formula. The scatter plots of the formula proposed in this invention generally approximate the baseline. y = x The dispersion is relatively minimal, with samples generally falling within the 0%–20% error band, and no trend of systematic deviation increasing with force amplitude is observed. For the total submersion case, the scatter points are concentrated near the baseline. For the McPherson formula, except for the total submersion case, the scatter points are generally biased below the baseline and exceed the -20% error line. Although this formula has a high degree of agreement under the total submersion case, it exhibits a systematic underestimation under the semi-submersion condition. This is because the formula does not explicitly account for the contribution of the web to the vertical force.

[0057] 4. Calculation of peak horizontal wave force on box girders under actual sea conditions: Under wave action, shear layer separation easily occurs on the wave-facing side of box girders, inducing backflow and vortex structures. This separation-entrainment process causes the wave-facing pressure to exhibit amplitude and phase differences at different sections and times, making it difficult to obtain a robust time-varying function expression under a unified parameter system. Meanwhile, the peak vertical force is generally more than twice the peak horizontal force, and compared to the vertical force, the horizontal force is a secondary factor in controlling the wave moment. Therefore, based on considerations of physical mechanisms and engineering applications, the horizontal wave force formula proposed in this invention is applicable to the prediction of peak horizontal force dominated by quasi-static pressure under total / partial submerged conditions.

[0058] When implementing the method of the present invention, the following input information is first obtained: Please see Figure 4 , Figure 4 This is a schematic diagram of the variable-size box girder cross-section of the present invention. (See diagram below.) Figure 4 As shown, the structural geometric parameters and specific dimensional information include the base plate width, flange width, flange height, web inclination angle, and web height. This invention considers seven typical cross-sectional types. This invention introduces a relative height coefficient... With relative width coefficient Two types of dimensionless parameters simplify and characterize complex cross-sectional geometric features.

[0059] Structural location parameters: including the elevation of the bottom of the beam and the elevation of the bridge deck.

[0060] Wave and flow field parameters: including water depth and wave crest elevation.

[0061] Existing horizontal force estimation formulas often characterize structural geometric effects using the vertical projected area in the direction of flow, without explicitly considering the influence of web inclination changes on the resultant force, which easily leads to overestimation of the horizontal force. Therefore, this invention introduces a horizontal force section effect coefficient based on the McPherson formula. C hsecAs a comprehensive correction factor for the influence on cross-sectional geometry, the relative width coefficient is further selected. As the independent variable, C hsec A univariate linear regression was performed, considering envelope fitting on a one-sided Chebyshev scale. The final formula for calculating the peak horizontal wave force of the box girder is as follows: In the formula, η max The peak elevation; h For water depth; h girders This is the elevation of the bottom of the beam; H bridge This refers to the vertical height of the bridge. L bridge This refers to the longitudinal length of the bridge. γ This refers to the unit weight of water. h deck This refers to the bridge deck elevation. C hsec This is the horizontal force section effect coefficient; This is the relative width coefficient; This is the cosine of the web inclination angle.

[0062] Please see Figures 9-10 , Figure 9 This is a comparison chart of the calculated values ​​and reference values ​​of horizontal wave loads for different types of box girder sections calculated by the formula of this invention. Figure 10 This is a comparison chart of the calculated and reference values ​​of horizontal wave loads for different types of box girder sections calculated using the McPherson formula. Figures 9-10 As shown, this is an example of considering both approaches mentioned above. C hsec The revised calculation results are compared with those calculated using the formula proposed by McPherson. As shown in the figure, the scatter points calculated by the formula of this invention are basically along the contour lines. y = x The distribution of the calculated values ​​is relatively large, and the error is generally within the 20% error threshold, indicating that the formula has good applicability within the scope of this invention. In contrast, the McPherson formula's scatter points are all distributed above the 20% error threshold, suggesting an overly conservative estimate.

[0063] Example 2 This embodiment provides a specific example illustrating the derivation process of the overpressure effect in this application. Because the physical mechanisms of the pressure at the detection points of the top and bottom plates of the rectangular beam are different, the area above the top plate is a free liquid surface, and its stress scenario conforms to the assumptions of hydrostatic pressure theory; while the water below the bottom plate will exert an additional squeezing effect on the bottom plate during wave propagation. This results in the bottom plate of the box girder and the bottom of the water tank forming a space similar to pipe flow during wave propagation. When the local water moves forward, it exerts upward squeezing on the bottom plate, thus generating an additional pressure component.

[0064] Therefore, this application defines the additional pressure on the base plate during the interaction between waves and the structure as overpressure (OP). The formula for calculating the overpressure of the base plate under wave action is as follows: In the formula, x The relative lateral position of the measuring point; x The horizontal coordinate position of the measuring point; W This refers to the width of the bottom of the beam; ρ For fluid density; L eq The equivalent length at which pressure disturbances exert their effect in a local water body; Let be the rate of change of flow velocity at the measuring point over time. The overpressure directed inwards along the vertical wall is defined as positive. This overpressure formula compensates for the shortcomings of hydrostatic pressure theory by incorporating the squeezing and extraction effects of the water below as waves pass through the bottom plate.

[0065] To verify the applicability of this calculation model, this embodiment first selected typical working conditions for preliminary verification. Considering the equivalent length... L eq The physical meaning of the pressure is necessarily related to the location of the measuring point and the operating conditions. However, in the initial stage of model validation, the focus is on examining whether the model can reasonably reflect the fluctuation law of overpressure. Therefore, given the known rate of change of velocity and fluid density, the hydrostatic pressure term is derived by subtracting the reference value. L eq , L eq The results of the changes over time are shown in Figure 11 .Depend on Figure 11 It can be seen that, L eq It exhibits certain fluctuation characteristics over time. In the initial stationary phase, because the rate of change of velocity is close to zero and fluctuates around zero, it leads to... L eq The calculated values ​​showed several abrupt changes. To obtain more stable parameters, this application only selected the period after the pressure began to rise for back-calculation. The results show that... L eqWithin the selected time period, the fluctuation amplitude is relatively small, and it can be simplified as a time invariant. Therefore, the preliminary calculation model for the wave load on the bottom plate of the rectangular beam is obtained as follows: In the formula, The time-varying wavefront elevation at various locations is given for cases where the top slab is not covered by water. Simplified to x The point at =0 represents the time-varying wave surface elevation on the wave-facing side of the model. h For water depth; h gb This is the elevation of the bottom of the beam.

[0066] Example 3 This embodiment provides a specific example illustrating the derivation of the spatiotemporal separability correction term in this application.

[0067] When there is water accumulation on the top slab, the wave force on the bottom slab can be calculated using the liquid level elevation at the same longitudinal position on the top slab. (The formula is missing from the original text.) This allows for the calculation of the pressure time history at the base plate measuring points, taking overpressure effects into account. Taking the REC1 section N6-1.33 working condition as an example, a comparison of the calculated pressure results at the base plate measuring points considering and not considering overpressure is shown below. Figure 12 . Figure 12 In this table, HPT represents the theoretical calculation result of hydrostatic pressure; OP represents the calculation result considering the overpressure effect; and Fluent represents the reference value of hydrostatic pressure obtained from numerical simulation.

[0068] Depend on Figure 12 It can be seen that the calculated hydrostatic pressure results differ from the reference values ​​in terms of time history, characterized by the calculated values ​​generally lagging behind the reference values. After introducing the overpressure effect (OP), the pressure exerted by the bottom water body in advance is reflected, and the calculated results agree well with the reference values ​​in terms of time, but differences still exist in terms of amplitude. For further analysis, this paper defines the width-to-height ratio of the beam bottom space. ζ This is the ratio of the length of the beam's base along the wave propagation direction to the height of the water below the base. In the subjects studied in this paper... ζ The range is [0.38, 3.38]. Analogous to pipe flow, ζ Corresponding to the ratio of pipe length to pipe diameter, common engineering pipelines ζ The values ​​range from 10 to 100. Due to the significant difference in magnitude between the two, the formula proposed in this paper is not entirely applicable to the calculation of the base pressure when waves pass over a structure. In other words, when there is a semi-open space below the base, the overpressure effect formula will overestimate the actual pressure. Therefore, this paper introduces a correction term. R ( x , t Further corrections are made to the pressure. The formula for the vertical wave force on the base plate with the correction term is: The difference between the reference value and the calculated value considering overpressure is denoted as Δ. P c Observe Δ P c The specific functional form of the correction term is determined by the time history variation. Taking the N6-1.33 working condition at section REC1 as an example, the results of the correction amount of the bottom plate pressure at different measuring points after dimensionless transformation are shown in the figure. Figure 13 .

[0069] Depend on Figure 13 It can be seen that Δ at different measuring points P c The variation with time is similar to that of a sine function, and can be fitted using a Fourier expansion of a sine function. Simultaneously, Δ at different measurement points... P c The amplitudes are different, and Δ is observed at different times. P c The relationship between Δ and position shows that P c The relationship between position and location can be fitted using a polynomial function.

[0070] Correction term Δ P c It is a function that is related to both time and spatial location, and uses a separable two-basis expansion: the time dimension uses a finite-term Fourier basis, and the spatial dimension uses 0- P ξ A polynomial basis of degree n is represented as: In the formula, M t and P x These represent the time harmonic order and the spatial polynomial degree, respectively. ; ; Then, at all sampling points, the model estimate is... for In the formula, That is, the values ​​of the time base at all points in time; That is, the values ​​of the spatial basis at all points. C The coefficient matrix was determined using least squares estimation. To mitigate the Gibbs phenomenon caused by discontinuities in the time endpoint splicing, a one-sided half-cosine window was used for mild smoothing at the endpoints. Two operating conditions were selected to determine the time harmonic order and the spatial polynomial degree. M t For 3~6 P x The fitting results for various combinations of orders from 2 to 5 are shown in Table 2. Most M t and Px The overall correlation of the fitted results is good, and the coefficient of determination is [missing information]. R 2 A value greater than 0.8 indicates that the constructed time-space basis function can effectively characterize the time history and spatial distribution characteristics of pressure.

[0071] Table 2 Comparison of fitting results for different time harmonic orders and space polynomial degrees

[0072] Further comparison of the fitting effects of different orders and degrees at different times and relative positions, taking REC3-N9-1.33 as an example, shows that... M t For 4~5 P x Error distribution for values ​​of 3-4 is shown in the figure. Figure 14 The error amplitude is relatively small in most intervals, mainly concentrated in the 2.2~2.8s interval after the pressure trough at the first measuring point. Increasing either the time harmonic order or the spatial polynomial order has limited effect on improving the error value. Considering both accuracy, robustness, and model complexity, P x =3、 M t The truncation combination with a value of 4 can effectively control the error, and it is chosen as the truncation order for the final correction term.

[0073] The above description is merely a few embodiments of this application and is not intended to limit this application in any way. Although this application discloses preferred embodiments as described above, it is not intended to limit this application. Any changes or modifications made by those skilled in the art without departing from the scope of the technical solution of this application using the disclosed technical content are equivalent to equivalent implementation cases and all fall within the scope of this technical solution.

Claims

1. A method for calculating wave loads on the superstructure of a near-shore box girder bridge, characterized in that, The calculation method includes the following steps: S1: Based on the wave field of the box girder in iSquares numerical simulation, the wave surface time history and flow field element time history of the box girder are obtained respectively, and its submersion state is determined according to the elevation of the box girder's location. S2: Based on the current flow field time history and submersion depth of the box girder, calculate the vertical wave forces of the top and bottom plates of the rectangular beam with the same width and height as the box girder, and then superimpose them to obtain the total vertical wave force of the rectangular beam. S3: Calculate the relative width coefficient and relative height coefficient of the box girder according to the current dimensional characteristics; calculate the vertical wave force section effect coefficient of the box girder based on the relative height coefficient; calculate the horizontal force section effect coefficient of the box girder based on the relative width coefficient; S4: Based on the vertical wave force section effect coefficient and the horizontal wave force section effect coefficient calculated in S3, the vertical wave force time history and the horizontal wave force of the box girder are obtained by combining the flow field element time history.

2. The wave load calculation method for the superstructure of a near-shore box girder bridge according to claim 1, characterized in that, In S2, the total vertical wave force on the rectangular beam is obtained by superimposing the vertical wave forces on the bottom plate and the top plate, and is expressed as follows: ； In the formula, Vertical wave force on a rectangular beam; The vertical wave force on the bottom plate is the instantaneous static pressure at each monitoring point of the rectangular beam, which is determined by the hydrostatic pressure integral term, the bottom plate overpressure term, and its spatiotemporally separable correction term. The vertical wave force on the top plate is determined by the hydrostatic pressure integral term and the non-static correction factor.

3. The wave load calculation method for the superstructure of a near-shore box girder bridge according to claim 2, characterized in that, The spatiotemporally separable correction term is represented in a separable form using time harmonic basis and spatial polynomial basis; the spatiotemporally separable correction term is used to correct the hydrostatic pressure term of the bottom plate of the rectangular beam, thereby obtaining the corrected pressure time history of the bottom plate of the rectangular beam. The overpressure term of the bottom plate is constructed considering the overpressure effect of the bottom plate; the overpressure effect is used to characterize the upward compression of the bottom plate by the incoming water body in the confined space between the bottom plate and the bottom of the channel. The overpressure effect is related to the width of the box girder bottom plate, the bottom elevation of the girder, the water depth and the lateral relative position of the measuring points.

4. The wave load calculation method for the superstructure of a near-shore box girder bridge according to claim 3, characterized in that, The spatiotemporally separable correction term is constructed based on time harmonic basis functions and spatial polynomial basis functions. The coefficients of the spatiotemporally separable correction term are determined through least-squares fitting, and stable convergence is achieved using low-order time harmonics and low-order spatial polynomials. The calculation formula for the vertical wave force on the bottom plate is obtained by interpolating and integrating the pressure at each monitoring point: ； In the formula, W This refers to the width of the bottom of the beam; ρ For fluid density; g It is the acceleration due to gravity; h For water depth; The time-varying wavefront elevation at various locations is given for cases where the top slab is not covered by water. Simplified to ; t For time; h gb This is the elevation of the bottom of the beam;  The relative lateral position of the measuring point; L eq The equivalent length at which pressure disturbances exert their effect in a local water body; The rate of change of flow velocity at the measuring point over time; H Wave height; This is a spacetime separability correction term.

5. The wave load calculation method for the superstructure of a near-shore box girder bridge according to claim 2, characterized in that, In S2, the calculation of the vertical wave force on the top plate of the rectangular beam includes both cases where the top plate is covered with water and cases where it is not covered with water. When the roof is covered with water, a non-static correction factor κ should be introduced to correct the vertical wave force of the roof when calculating the vertical wave force, in order to compensate for the non-static effect caused by the significant vertical acceleration near the free surface. In this case, the formula for calculating the vertical wave force of the roof is: ； In the formula, The vertical wave force on the top plate; For non-static water correction factors; h gt This is the elevation of the top of the beam.

6. The wave load calculation method for the superstructure of a near-shore box girder bridge according to claim 1, characterized in that, The vertical wave force section effect coefficient of the box girder is obtained by performing multiple linear regression on simulation data of different section combined parameters, using the relative width coefficient and relative height coefficient of the section as independent variables. The horizontal wave force section effect coefficient of the box girder is based on the ratio of the top plate width to the bottom plate width. The variables are obtained by linear regression of simulated data with different cross-sections and parameters.

7. The wave load calculation method for the superstructure of a near-shore box girder bridge according to claim 2, characterized in that, In S4, the vertical wave force section effect coefficient is used to characterize the influence of the box girder section geometry on the peak value of the vertical wave force; When the superstructure is a box girder, the formula for calculating the vertical wave force of the box girder is: ； In the formula, This is a vertical wave force; C vsec The vertical wave force section effect coefficient is determined by the submersion condition of the box girder. The vertical wave force is for the rectangular beam.

8. The wave load calculation method for the superstructure of a near-shore box girder bridge according to claim 7, characterized in that, in: C vsec The strategy for determining the value is: Under total submersion conditions, the vertical wave force section effect coefficient C vsec Taking 1, the peak value of the vertical wave force of the box girder is approximately equal to that of the vertical wave force of the rectangular beam with the same width and height. Under semi-submerged conditions C vsec It is an additive function related to the relative width factor of the cross section, the relative width factor of the flange, and the outward inclination characteristic of the web.

9. The wave load calculation method for the superstructure of a near-shore box girder bridge according to claim 1, characterized in that, In S4, when calculating the horizontal wave force, McPherson formula is used as a benchmark and the horizontal force section effect coefficient C is introduced hsec The horizontal force formula is modified, and the modified calculation formula of the horizontal wave force of the box girder is: ； In the formula, F h For the horizontal wave force of the box girder; F static_front and F static_front These are the horizontal components of the hydrostatic pressure on the wave-facing side and the horizontal components of the hydrostatic pressure on the wave-avoiding side, respectively, both of which are calculated directly using the McPherson formula.

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