Construction method for mathematical model of critical impact energy of reinforced concrete pier column

Abstract

The invention discloses a method for constructing a critical impact energy mathematical model of a reinforced concrete pier column. The method comprises the following steps: (1) establishing a drop hammer impact two-degree-of-freedom mass-spring-damping motion equation of the reinforced concrete pier column; (2) applying a nonlinear cross-section fiber finite element model to obtain equivalent resistance rigidity and failure displacement of the reinforced concrete pier column with the impact strain rate effect taken into account; (3) solving a two-degree-of-freedom mass-spring-damping motion equation at different impact speeds to obtain a speed-displacement curve and critical impact energy of the pier column under the impact effect; (4) calculating critical impact energy under different axial compression ratios and longitudinal bar ratios; and (5) establishing a mathematical model of the critical impact energy of the reinforced concrete pier column. The mathematical model of the critical impact energy of the reinforced concrete pier column, which is constructed by the method, truly reflects the limit capacity of the reinforced concrete pier column for bearing the impact load, and can provide reference for rapid evaluation of the limit impact resistance bearing capacity of the reinforced concrete.

Description

Technical Field

[0001] The invention relates to a method for constructing a mathematical model of critical impact energy of a reinforced concrete pier column, and belongs to the technical field of impact power. Background Art

[0002] As the main component of the bridge structure that bears and transmits the upper structure load, reinforced concrete piers are subject to various potential impact risks such as vehicles, ships, floating objects and falling rocks during their service. How to effectively evaluate the impact resistance of reinforced concrete piers, especially the evaluation of the ultimate impact resistance, is an important guarantee for maximizing the safety of lifeline projects.

[0003] At present, the critical impact energy is generally used as an evaluation index for the ultimate impact resistance of reinforced concrete piers, which is obtained through experimental research or numerical simulation. The experimental research needs to start from a low speed, with the impact speed as an increment, and load step by step until the pier component is destroyed. The entire implementation process requires multiple impact tests, but considering that a single impact test is expensive, huge impact loads are difficult to achieve, there are many real impacts and the test process is complicated, only theoretical feasibility exists; although numerical simulation can achieve the reshaping of the impact process of pier components with the help of display nonlinear dynamic analysis software, and combine the incremental method with the simulation of multiple impact processes to obtain the critical impact energy, the number of numerical simulation node units is huge, the calculation is time-consuming, the efficiency is low, and it is difficult to consider the impact strain rate effect.

[0004] Therefore, it is particularly important to construct a simplified mathematical model of the critical impact energy of reinforced concrete piers that can take into account material nonlinearity and impact strain rate effects, and truly reflect the ultimate capacity of reinforced concrete piers to withstand impact loads. Summary of the invention

[0005] The purpose of the present invention is to propose a method for constructing a mathematical model of critical impact energy of reinforced concrete piers in order to solve the problems that huge impact loads are difficult to achieve in the process of determining critical impact energy tests, the numerical simulation calculation time is long, the calculation efficiency is low, and the influence of impact strain rate effect on critical impact energy is ignored.

[0006] The technical solution implemented by the present invention is as follows: a method for constructing a mathematical model of critical impact energy of reinforced concrete piers, the steps of which are as follows:

[0007] (1) Establish the two-degree-of-freedom mass-spring-damper motion equation for the drop hammer impact on reinforced concrete piers;

[0008] (2) Using the nonlinear cross-section fiber finite element model and the concrete and steel impact strain rate mathematical model, the equivalent resistance stiffness and failure displacement of the reinforced concrete pier column are obtained;

[0009] (3) Using the finite difference method, the two-degree-of-freedom mass-spring-damper motion equations at different impact velocities are solved step by step, and the velocity-displacement curve and critical impact energy of the pier column under impact are obtained;

[0010] (4) Calculate the critical impact energy of reinforced concrete piers under different axial compression ratios and longitudinal reinforcement ratios;

[0011] (5) Define the mathematical function between critical impact energy, axial compression ratio and longitudinal reinforcement ratio, determine the coefficient values ​​in the mathematical function through binary numerical regression, and establish a mathematical model of critical impact energy of reinforced concrete piers.

[0012] The concrete and steel bar impact strain rate mathematical model includes a mathematical expression of the concrete impact strain rate effect and a mathematical expression of the steel bar impact strain rate effect;

[0013] The mathematical expression of the concrete impact strain rate effect is as follows:

[0014]

[0015]

[0016] In the formula, f co,d It means that when the strain rate is When the compressive strength of concrete is c,d It means that when the strain rate is When the elastic modulus of concrete is co is the ultimate compressive strength of concrete; E c is the elastic modulus of concrete; Represents the impact strain rate.

[0017] The mathematical expression of the steel bar impact strain rate effect is as follows:

[0018]

[0019] In the formula, f sy The strain rate is The yield strength of the steel bar at y is the yield strength of the steel bar.

[0020] The calculation steps of the equivalent resistance stiffness and failure displacement of the reinforced concrete pier are as follows:

[0021] (1) Based on the OPENSEES computing platform, a nonlinear cross-sectional fiber finite element model of a compressed reinforced concrete pier column considering the strain rate effect was established;

[0022] (2) Using the established nonlinear cross-section fiber finite element model, push-over loads are applied step by step to obtain the equivalent resistance-displacement curve;

[0023] (3) According to the equivalent resistance-displacement curve, the secant stiffness of the point corresponding to the limit displacement is taken as the equivalent resistance stiffness k 2 , the limit displacement is the failure displacement D Max .

[0024] The velocity-displacement curve and critical impact energy calculation steps of the pier under impact are as follows:

[0025] (1) Input initial conditions: including the initial velocity V of the impactor 1 and mass m 1 , contact stiffness k 1 and contact damping c 1 , pier column dimensions, material properties, structural mass m 2 , structural damping c 2 , velocity increment ΔV;

[0026] (2) Using the established nonlinear cross-section fiber finite element model, the equivalent resistance stiffness k of the reinforced concrete pier column is obtained. 2 and failure displacement D Max ;

[0027] (3) Using the finite difference method to solve the two-degree-of-freedom mass-spring-damper motion equation, the impact velocity of this stage is obtained as V i The speed of the pier and pier displacement

[0028] (4) When the pier column moves Less than failure displacement D Max , V i+1 =V i +ΔV, go to step (3); if the pier column displacement Greater than or equal to failure displacement D Max , the pier column fails, the pier column velocity-displacement curve is output, and the corresponding kinetic energy is calculated by taking the failure velocity, which is the critical impact energy.

[0029] The calculation of the critical impact energy of the reinforced concrete pier column under different axial compression ratios and longitudinal reinforcement ratios adopts the control variable method, and when analyzing different axial compression ratios or longitudinal reinforcement ratios, other parameters are guaranteed to remain unchanged.

[0030] The steps of establishing the mathematical model of critical impact energy of reinforced concrete piers are as follows:

[0031] (1) Define the quadratic function expression between critical impact energy, axial compression ratio and longitudinal reinforcement ratio:

[0032] E=ax+by+cx 2 +dy 2 +fxy+z 0 ;

[0033] Where E is the critical impact energy, x is the axial compression ratio, y is the longitudinal reinforcement ratio, a, b, c, d, f, z 0 is the function coefficient.

[0034] (2) The critical impact energy under different axial compression ratios and longitudinal reinforcement ratios is calculated. The coefficients in the quadratic function are determined through binary numerical regression, and the mathematical model of the critical impact energy of reinforced concrete piers is obtained.

[0035] The beneficial effect of the present invention is that a simplified mathematical model of the critical impact energy of reinforced concrete pier columns that takes into account material nonlinearity and impact strain rate effects is constructed by the present invention, which truly reflects the ultimate capacity of reinforced concrete pier columns to withstand impact loads and can provide a reference for the rapid evaluation of the ultimate impact bearing capacity of reinforced concrete pier columns. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 A flow chart is constructed for the mathematical model of critical impact energy of reinforced concrete piers of the present invention;

[0037] Figure 2 It is a two-degree-of-freedom mass-spring-damper impact model;

[0038] Figure 3 Velocity-displacement curves of reinforced concrete pier columns under different combinations of axial compression ratios and longitudinal reinforcement ratios;

[0039] Figure 4 Comparison diagram of the critical impact energy between the mathematical model of the present invention and the numerical simulation. DETAILED DESCRIPTION

[0040] like Figure 1 Shown is a flow chart of constructing a mathematical model of critical impact energy of reinforced concrete piers according to an embodiment of the present invention.

[0041] The present embodiment provides a method for constructing a mathematical model of critical impact energy of a reinforced concrete pier column, comprising the following steps:

[0042] S1: This embodiment uses a drop hammer impact test to simulate and construct a mathematical model of the critical impact energy of reinforced concrete piers. The specimen is a reinforced concrete circular pier with a diameter of 200mm and a length of 2200mm. The corresponding protective layer thickness is 20mm. Beams with lengths, widths and heights of 900mm, 250mm and 350mm are set at both ends of the specimen, and an axial force of 200kN is applied through the pre-axial force system. The corresponding axial compression ratio is 0.143. The concrete strength of the specimen is C30, and the measured uniaxial compressive strength is 32.5MPa; the longitudinal reinforcement adopts 12 HRB400 steel bars with a diameter of 8mm, and the stirrups adopt HRB400 steel bars with a diameter of 6mm and a spacing of 55mm. The mass of the drop hammer is 442kg, the initial impact velocity is 4.85m / s, and the radius of the hammer head is 100mm. The two ends of the specimen are constrained by fixed and sliding boundary conditions respectively.

[0043] The above-mentioned mid-span drop hammer impact on the compressed pier is simulated by two mass points with the hammer head mass m 1 and pier mass m 2 , using spring stiffness k 1 and k 2 The local contact stiffness between the hammer head and the pier and the equivalent resistance stiffness of the pier are simulated respectively; the damping c of the linear viscous damper is used 1 and c 2 The local contact damping and structural damping are simulated separately, and the two-free mass-spring-damper impact model of the drop hammer impact on the compressed pier column is obtained as follows: Figure 2 As shown, the corresponding dynamic motion equation is:

[0044]

[0045] Where: M, C, K, P are mass matrix, damping matrix, stiffness matrix and external force matrix respectively; u are the acceleration, velocity and displacement vector of the system respectively.

[0046] in:

[0047]

[0048] g represents the acceleration due to gravity;

[0049] u 1 and u 2 denote the displacements of the hammer head and the pier column respectively; and Represent the speed of hammer head and pier column respectively; and denote the acceleration of the hammer head and the pier column respectively;

[0050] Local contact stiffness k1 Determined by the following formula:

[0051]

[0052] In the formula, E 1 and v 1 are the elastic modulus and Poisson's ratio of the drop hammer head respectively; E 2 and v 2 are the elastic modulus and Poisson's ratio of the pier respectively; R is the radius of the hammer head.

[0053] Substituting the relevant data, we get: k 1 =364797.08 (kN / m).

[0054] Structural damping c 2 =0, local contact damping c 1 It can be determined by the following formula:

[0055]

[0056] Where: ξ is the damping ratio; Substituting the relevant data, we get c 1 =3511.49.

[0057] S2: Based on OPENSEES software, the Concrete02 constitutive simulation of concrete materials and the Steel02 constitutive simulation of steel materials are applied. In combination with the nonlinear beam-column unit, a nonlinear cross-section fiber finite element model of reinforced concrete pier columns considering the strain rate effect is established. By carrying out a push-down analysis, the equivalent resistance stiffness k under different axial compression ratios and longitudinal reinforcement ratio combinations is obtained. 2 and failure displacement D Max , as shown in Table 1.

[0058] Table 1 k under different axial compression ratios and longitudinal reinforcement ratios 2 and D Max

[0059] Working conditions Axial pressure ratio Longitudinal reinforcement ratio (%) <![CDATA[k 2 (kN / m)]]> <![CDATA[D Max (mm)]]> A1 0.1 1.08 11856.08 83 A2 0.2 1.08 16663.86 67 A3 0.3 1.08 21042.06 59 A4 0.1 1.92 16285.72 92 A5 0.2 1.92 20547.89 78 A6 0.3 1.92 24319.24 72 A7 0.1 2.99 21155.65 120 A8 0.2 2.99 24628.36 102 A9 0.3 2.99 27751.57 86

[0060] S3: The finite difference method is used to solve the two-degree-of-freedom motion equations, and combined with step-by-step loading, the velocity-displacement curves of reinforced concrete pier columns under different axial compression ratios and longitudinal reinforcement ratios are obtained, such as Figure 3 shown.

[0061] S4: Calculate the critical impact energy under different combinations of axial compression ratio and longitudinal reinforcement ratio, as shown in Table 2.

[0062] Table 2 Critical impact energy under different axial compression ratios and longitudinal reinforcement ratios

[0063]

[0064] S5: Using binary numerical regression and Origin software, the data in Table 2 are analyzed and fitted, and the mathematical model of the critical impact energy of reinforced concrete piers is obtained as follows:

[0065] E=-2947.44x+4259.47y+30546.17x 2 +951.16y 2 +-13136.18xy+4752.96

[0066] Comparison of the critical impact energy calculation results between the mathematical model and numerical simulation Figure 3 shown.

[0067] Figure 3 The X coordinate represents the axial compression ratio, the Y coordinate represents the longitudinal reinforcement ratio, and the Z coordinate represents the critical impact energy.

[0068] Figure 3 In the above table, the square error SSE = 7.05227E-5, the standard deviation RMSE = 2.35075E-5, and the coefficient of determination R 2 =0.99034, which meets the requirements, indicating that the fitting effect of the mathematical model of critical impact energy of reinforced concrete pier columns is relatively ideal.

Claims

1. A method for constructing a mathematical model of critical impact energy of reinforced concrete piers, characterized in that: The method steps are as follows: (1) Establish the two-degree-of-freedom mass-spring-damper motion equation for the drop hammer impact on reinforced concrete piers; (2) Using the nonlinear cross-section fiber finite element model and the concrete and steel impact strain rate mathematical model, the equivalent resistance stiffness and failure displacement of the reinforced concrete pier column are obtained; (3) Using the finite difference method, the two-degree-of-freedom mass-spring-damper motion equations at different impact velocities are solved step by step, and the velocity-displacement curve and critical impact energy of the pier column under impact are obtained; (4) Calculate the critical impact energy of reinforced concrete piers under different axial compression ratios and longitudinal reinforcement ratios; (5) Define the mathematical function between critical impact energy, axial compression ratio and longitudinal reinforcement ratio, determine the coefficient value in the mathematical function through binary numerical regression, and establish the mathematical model of critical impact energy of reinforced concrete piers; The concrete and steel bar impact strain rate mathematical model includes a mathematical expression of the concrete impact strain rate effect and a mathematical expression of the steel bar impact strain rate effect; The mathematical expression of the concrete impact strain rate effect is as follows: In the formula, f co,d It means that when the strain rate is When the compressive strength of concrete is c,d It means that when the strain rate is When the elastic modulus of concrete is co is the ultimate compressive strength of concrete; E c is the elastic modulus of concrete; represents the impact strain rate; The mathematical expression of the steel bar impact strain rate effect is as follows: In the formula, f sy The strain rate is The yield strength of the steel bar at y is the yield strength of the steel bar.

2. The method for constructing a mathematical model of critical impact energy of reinforced concrete piers according to claim 1, characterized in that: The calculation steps of the equivalent resistance stiffness and failure displacement of the reinforced concrete pier are as follows: (1) Based on the OPENSEES computing platform, a nonlinear cross-sectional fiber finite element model of a compressed reinforced concrete pier column considering the strain rate effect was established; (2) Using the established nonlinear cross-section fiber finite element model, push-over loads are applied step by step to obtain the equivalent resistance-displacement curve; (3) According to the equivalent resistance-displacement curve, the secant stiffness of the point corresponding to the limit displacement is taken as the equivalent resistance stiffness k2, and the limit displacement is the failure displacement D Max .

3. The method for constructing a mathematical model of critical impact energy of reinforced concrete piers according to claim 1, characterized in that: The velocity-displacement curve and critical impact energy calculation steps of the pier under impact are as follows: (1) Input initial conditions: including the initial velocity V1 and mass m1 of the impactor, contact stiffness k1 and contact damping c1, pier size, material properties, structural mass m2, structural damping c2, and velocity increment ΔV; (2) Using the established nonlinear cross-section fiber finite element model, the equivalent resistance stiffness k2 and failure displacement D of the reinforced concrete pier column are obtained. Max ; (3) Using the finite difference method to solve the two-degree-of-freedom mass-spring-damper motion equation, the impact velocity of this stage is obtained as V i The speed of the pier and pier displacement (4) When the pier column moves Less than failure displacement D Max , V i+1 =V i +ΔV, go to step (3); if the pier column displacement Greater than or equal to failure displacement D Max , the pier fails, the pier velocity-displacement curve is output, and the corresponding kinetic energy is calculated by taking the failure velocity, which is the critical impact energy.

4. The method for constructing a mathematical model of critical impact energy of reinforced concrete piers according to claim 1, characterized in that: The calculation of the critical impact energy of the reinforced concrete pier column under different axial compression ratios and longitudinal reinforcement ratios adopts the control variable method, and when analyzing different axial compression ratios or longitudinal reinforcement ratios, other parameters are guaranteed to remain unchanged.

5. The method for constructing a mathematical model of critical impact energy of reinforced concrete piers according to claim 1, characterized in that: The steps of establishing the mathematical model of critical impact energy of reinforced concrete piers are as follows: (1) Define the quadratic function expression between critical impact energy, axial compression ratio and longitudinal reinforcement ratio: E=ax+by+cx 2 +dy 2 +fxy+z0; In the formula, E is the critical impact energy, x is the axial compression ratio, y is the longitudinal reinforcement ratio, a, b, c, d, f, z0 are function coefficients; (2) The critical impact energy under different axial compression ratios and longitudinal reinforcement ratios is calculated. The coefficients in the quadratic function are determined through binary numerical regression, and the mathematical model of the critical impact energy of reinforced concrete piers is obtained.

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