A method for calculating displacement response of reinforced concrete beam under near-blast load

Abstract

The present application relates to a kind of reinforced concrete beam displacement response calculation method under near explosive load, belong to protective engineering and structural dynamics technical field, solve the problem that the overall dynamic process is affected by ignoring local damage in prior art, leading to the problem that the dynamic response of reinforced concrete beam under near explosion cannot be accurately described.It includes obtaining charge parameters and beam structural parameters, and performing local damage feature prediction to obtain local damage size;Based on the charge parameters of near explosion load, the local response of reinforced concrete beam is calculated to obtain the initial state of the overall response of the local dynamic response, and then the cross response;Based on the structural parameters of reinforced concrete beam, local damage size, local overall coupling parameter correction is carried out to obtain the correction relationship, and then the conversion coefficient is obtained, and then the dynamic response is calculated by using elastic-plastic SDOF difference motion equation to obtain the cross displacement response result.The high-precision displacement response calculation of reinforced concrete beam is realized.

Description

Technical Field

[0001] This invention relates to the fields of protective engineering and structural dynamics, and in particular to a method for calculating the displacement response of reinforced concrete beams under near-explosion loads. Background Technology

[0002] Reinforced concrete (RC) structures are the primary load-bearing form in protective engineering. In the event of a close-in blast attack, the explosive is extremely close to the structural members (typically a proportional distance Z < 1.0). The load is characterized by extremely high peak value, extremely short duration of action, and extremely uneven spatial distribution.

[0003] Under this condition, the explosive load energy is highly concentrated. The structure first experiences localized brittle failure, such as crushing of the blast-facing surface, spalling of the sides, and collapse of the back surface, before gradually developing into overall plastic deformation. Its damage mechanism is fundamentally different from that of a long-distance explosion. The dynamic response process of the RC beam exhibits significant stages and coupling. In the local response stage, the explosive shock wave first acts on the local area of ​​the blast-facing surface, causing concrete crushing and collapse of the back surface, accompanied by high-frequency local shear motion, forming initial cross-sectional defects and velocity fields. In the overall response stage, as the shock wave decays, the localized high-frequency vibrations gradually transform into low-frequency bending deformation of the entire component (similar to the vibration of a single-degree-of-freedom system).

[0004] Existing computational methods exhibit significant disconnects when handling the aforementioned process: while the finite element method can simulate the entire process, its computational cost is extremely high, and the modeling is complex, making it difficult to meet the needs of rapid evaluation in engineering. The traditional equivalent single degree of freedom (SDOF) method is a commonly used simplification method in engineering, but it has two inherent drawbacks when dealing with proximity bombing problems: First, it ignores the transmission of the initial state; traditional SDOF typically assumes the structure begins its response from a static state. In reality, before the overall bending begins, the local response stage already endows the beam with certain initial deformation and initial velocity (kinetic energy). Second, it ignores the abrupt change in stiffness / mass; traditional methods often calculate stiffness and mass based on the intact state of the component. However, local concrete spalling and collapse caused by proximity bombing directly lead to a reduction in the effective height of the cross-section and a decrease in mass, thus significantly altering the natural frequency and resistance characteristics of the component in subsequent overall motion.

[0005] Therefore, there is an urgent need for a coupled calculation method that can effectively transfer the local damage effect and local motion state to the overall response analysis, so as to achieve accurate and rapid prediction of the dynamic response of reinforced concrete beams under near-explosion conditions. Summary of the Invention

[0006] Based on the above analysis, the present invention aims to provide a method for calculating the displacement response of reinforced concrete beams under near-blast loads, in order to solve the problem that existing methods neglect the influence of local damage on the overall dynamic process, resulting in an inability to accurately describe the dynamic response of reinforced concrete beams under near-blast conditions.

[0007] This invention provides a method for calculating the displacement response of a reinforced concrete beam under near-explosion load, comprising the following steps: The charge parameters of the near-explosive load and the structural parameters of the reinforced concrete beam are obtained, and the local failure characteristics of the reinforced concrete beam are predicted to obtain the local failure size. Based on the charge parameters of the near-explosion load, the local response of the reinforced concrete beam is calculated to obtain the local dynamic response. Then, based on the local failure size, the initial state of the mid-span response of the overall response is obtained. Based on the structural parameters and local failure dimensions of the reinforced concrete beam, local and overall coupling parameters are corrected to obtain the correction relationship, and then the conversion coefficient is obtained; among them, the correction relationship includes the relationship between bending moment and curvature at the mid-span section, the relationship between displacement and curvature at the mid-span section, and the relationship between resistance at the mid-span section and bending moment at the mid-span section; Based on the initial state, correction relationship, and transformation coefficient of the mid-span response of the overall response, the dynamic response is calculated using the elastoplastic SDOF differential motion equation to obtain the mid-span displacement response result.

[0008] Compared with the prior art, the present invention can achieve at least one of the following beneficial effects: This invention provides a method for calculating the displacement response of reinforced concrete beams under near-blast loads. For the first time, it achieves the correction of the dual impact of local damage on the overall dynamic response of reinforced concrete beams under near-blast loads in a simplified calculation model. By calculating the local response, the initial kinetic energy is accurately transferred to the overall stage. Furthermore, the cross-sectional stiffness and mass distribution are dynamically corrected based on the local damage characteristics. This solves the inherent defects of ignoring the initial state and stiffness degradation, significantly improving the calculation accuracy. At the same time, it retains the calculation efficiency of the simplified model. Only basic geometric and explosive parameters need to be input to quickly assess near-blast damage. This provides a high-precision and highly applicable engineering tool for the rapid design of protective engineering and the analysis of defense effectiveness.

[0009] In this invention, the above-described technical solutions can be combined with each other to achieve more preferred combinations. Other features and advantages of this invention will be set forth in the following description, and some advantages may become apparent from the description or be learned by practicing the invention. The objects and other advantages of this invention can be realized and obtained from the description and drawings, which are particularly pointed out. Attached Figure Description

[0010] The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Throughout the drawings, the same reference numerals denote the same parts.

[0011] Figure 1 This is a flowchart illustrating the method for calculating the displacement response of a reinforced concrete beam under near-explosion load provided in Embodiment 1 of the present invention. Figure 2 This is a schematic diagram showing the position of the near-explosion load and the reinforced concrete beam provided in Embodiment 1 of the present invention; Figure 3 This is a schematic diagram of the layered structure of a reinforced concrete beam provided in Embodiment 1 of the present invention; Figure 4(a) is a schematic diagram of the simulation results under the working condition provided in Embodiment 2 of the present invention; Figure 4(b) is a schematic diagram of the simulation results under working condition 2 provided in Embodiment 2 of the present invention. Detailed Implementation

[0012] Preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings, which form part of this application and are used together with the embodiments of the present invention to illustrate the principles of the present invention, but are not intended to limit the scope of the present invention.

[0013] Example 1 A specific embodiment of the present invention discloses a method for calculating the displacement response of a reinforced concrete beam under near-explosion load, such as... Figure 1 As shown, it includes the following steps: S1. Obtain the charge parameters of the near-explosion load and the structural parameters of the reinforced concrete beam, and predict the local failure characteristics of the reinforced concrete beam to obtain the local failure size.

[0014] Specifically, the charge parameters include charge mass, blast height, charge height and diameter; the structural parameters of the reinforced concrete beam include beam length, beam width, beam thickness, reinforcement parameters, and reinforced concrete parameters; among them, blast height is the vertical distance from the charge centroid to the beam surface.

[0015] It should be noted that the near-blast load in this embodiment is a cylindrical charge, and the near-blast load is located above the mid-span of the reinforced concrete beam, such as... Figure 2 As shown.

[0016] In practice, the local damage dimensions include the lateral spalling length and the crushing depth, which are expressed as follows: ; ; in, ; ; In the formula, , These represent the lateral spalling length and crushing depth, respectively. , These represent the thickness and width of the reinforced concrete beam, respectively. Indicates the first The dimensionless quantity of local damage characteristics , , , , , , , They represent the first The first, second, third, fourth, fifth, sixth, seventh, and eighth correlation coefficients of the dimensionless quantities of local damage characteristics. Indicates the quality of the explosive charge. Indicates the density of concrete. Indicates the detonation height of the near-explosion load. , These represent the height and diameter of the propellant column, respectively. , , These represent the reinforcement ratios of stirrups, compression longitudinal bars, and tension longitudinal bars, respectively.

[0017] In practice, hour, , , , , , , , The values ​​are 10.79, -0.185, -0.484, -0.004, 0.066, 0.015, 0.724, and 0.822, respectively. hour, , , , , , , , The values ​​are 4.26, -0.239, -0.225, -0.010, 0.050, 0.010, 0.083, and 0.416, respectively. Understandably, setting these correlation coefficients allows for the rapid and accurate quantification of the local failure characteristics of reinforced concrete beams under near-blast loads, providing reliable initial damage input for subsequent local-global coupled response calculations, thereby significantly improving the accuracy of overall displacement prediction.

[0018] S2. Based on the charge parameters of the near-explosion load, perform local response calculations on the reinforced concrete beam to obtain the local dynamic response, and then, based on the local failure size, obtain the initial state of the mid-span response of the overall response.

[0019] During implementation, the initial state of the overall response at mid-span includes the initial mid-span displacement, the initial mid-span velocity, and the initial mid-span acceleration; wherein, the initial mid-span displacement... Represented as: ; in, ; In the formula, This indicates the length of a reinforced concrete beam. , Let represent the instantaneous position and velocity of the transition hinge motion in the local response, respectively. This represents the initial mid-span velocity in the local response.

[0020] Specifically, the initial mid-span velocity can be obtained from the initial mid-span displacement using the first and second derivatives. Initial mid-span acceleration , represented as: ; ; In specific implementation, the velocity of the transition hinge motion in the local response Represented as: ; in, ; In the formula, This indicates the duration of the blast shock wave's effect on the reinforced concrete beam. This indicates the cross-sectional mass of the reinforced concrete beam when no localized damage has occurred. The yield moment of a reinforced concrete beam is represented as follows: ; ; ; ; ; ; In the formula, Indicates the local dynamic enhancement factor. Indicates the uniaxial tensile strength of concrete. , These represent the elastic modulus of steel reinforcement and the elastic modulus of concrete, respectively. , These represent the areas of the bottom and top reinforcing bars of the reinforced concrete beam, respectively. , These represent the distances from the compression longitudinal reinforcement and tension longitudinal reinforcement to the fracture surface, respectively. The area of ​​the bottom reinforcement in the reinforced concrete beam is also included. The total cross-sectional area of ​​the longitudinal reinforcing bars arranged in the tension zone of a beam (usually on the back side); the area of ​​the top reinforcing bars in a reinforced concrete beam. It refers to the total cross-sectional area of ​​the longitudinal reinforcing bars arranged in the compression zone of the beam (usually on the side facing the explosion).

[0021] Specifically, in this embodiment, the local dynamic enhancement factor Setting it to 1.3 can better reflect the strain rate effect of reinforced concrete beams under explosive loads.

[0022] In practical implementation, the initial mid-span velocity in the local response Represented as: ; In the formula, This represents the horizontal coordinate along the span of the reinforced concrete beam, with the projection point of the explosive charge on the beam as the origin. This represents the peak specific impulse of a near-explosive load explosion. The initial effective span of a reinforced concrete beam is expressed as: ; in, .

[0023] Specifically, This represents the horizontal coordinate along the span of the reinforced concrete beam, with the origin at the mid-span section (directly below the projection point of the charge), and the positive direction pointing towards the support.

[0024] It should be noted that in step S2, based on the charge parameters of the near-explosion load, the local response of the reinforced concrete beam is calculated to obtain the local dynamic response. Then, based on the local failure size, the initial state of the mid-span response of the overall response is obtained through the following derivation: First, based on the charge parameters of the near-explosion load, the equivalent uniformly distributed specific impulse is calculated, specifically: The spatial distribution of specific impulse is described using an exponential decay function: ; in, ; ; In the formula, Indicates the incident angle of the shock wave. This indicates that at the incident angle of the shock wave is Specific impulse at that time.

[0025] Considering the relationship between angle, blast height, and position: ; The equivalent uniformly distributed specific impulse can be obtained by combining the equations. : .

[0026] In the formula, This indicates the distribution of specific impulse.

[0027] Secondly, the local dynamic response is obtained through local response calculation of the reinforced concrete beam; the local dynamic response includes local dynamic response displacement, local dynamic response velocity, and local dynamic response acceleration; specifically: For the transition process from local response to overall structural response in a concrete beam, since the explosive load has an extremely short duration, it is generally assumed that this process mainly occurs after the explosive load has ended and no external forces are acting. Therefore, this process can be described using the theory of transitional hinges and the law of conservation of momentum. The following assumptions are made for this: (1) Decouple the brittle local failure and dynamic response of reinforced concrete beams under near-explosion load. Assume that the brittle local failure is mainly caused by the overpressure of the shock wave, while the dynamic response of the structure is caused by the impulse. (2) Although the concrete beam will experience compressive-shear failure during the explosive load, since the duration of the explosive load is extremely short, it is assumed that during this stage, the concrete blocks that are locally damaged by the explosive load still move in the same direction as the main body of the beam and do not separate; therefore, the concrete beam can still be regarded as having a uniform mass distribution during the initial local response. (3) Since the explosive load of the cylindrical charge is mainly concentrated in a limited area below the end face of the charge, especially when significant local damage is induced by a close-range explosion, its action can be regarded as a rapid impact on the mid-span of the beam; therefore, it is assumed that the total impulse acting on the span of the concrete beam is completely concentrated in the initial mid-span length. Within the region, and let the effective length be equal to the initial value of the plastic hinge length; this assumption reflects both the spatial concentration of the load and facilitates the subsequent coupling of the plastic hinge mechanism with the impact response; (4) Based on the theory of transitional hinges, it is assumed that under the action of bending stress waves, the effective span of the concrete beam is Two transition hinges are formed at the boundary and move towards the support along the span of the beam. As the transition hinges gradually move towards the support, assuming the effective span... The concrete beam inside can be regarded as a fixed beam with constantly changing boundaries, while the beam members outside the transition hinge remain in a stable state.

[0028] Based on the principle of conservation of momentum, the explosive impulse of a cylindrical charge is entirely converted into the initial effective span. The momentum of the internal concrete beam is expressed as: ; ; In the formula, This represents the displacement distribution function in the local response. This represents the velocity distribution function in the local response.

[0029] Considering that the length of the plastic hinge is much smaller than the beam length L The shock wave load from a cylindrical charge explosion is mainly concentrated in a very small region at the mid-span. Therefore, in the initial disturbance region at the mid-span... Within this region, it is typically assumed that the initial velocity of the concrete beam is uniformly distributed. That is, the initial velocity at every point within this region is a constant. If the value exceeds this range, it becomes 0. Therefore, when the explosive load ends, the transition hinge is located at... The mathematical expression for the initial velocity distribution of the right half of the beam can be written as: ; Initial mid-span velocity in local response It can be represented as: ; In the formula, It means to fry high.

[0030] When a plastic hinge forms at the mid-span (plastic hinge length) After that, the transition hinges at both ends of the plastic hinge... The initial position begins to move towards the support. Since the explosive load has ended and no external force is acting, the transition hinge moves from... Move to any position At that time, the concrete beams within the two transition hinges always maintain momentum conservation. Taking the right half-span concrete beam at mid-span as an example, when the transition hinge moves to... At that time, we can obtain: ; The local dynamic response speed in the local response for: ; consider and The effect of plastic hinges on the effective response span of concrete beams ( The moment balance is achieved. It should be noted that the bending moment at the mid-span section is... At the transition hinge position ( The bending moment is and shear force It is 0, that is: ; In the formula, This represents the local dynamic response acceleration of a concrete beam.

[0031] According to the law of conservation of momentum, it can be converted to: ; In the formula, This represents the local dynamic response acceleration in the local response.

[0032] right By performing definite integrals, we can obtain: ; The terms obtained after integration are simplified algebraically and combined. According to the rules of calculus, this long string contains... , The algebraic expressions of its derivatives can be equivalently combined into a composite function relating the time of the blast shock wave on the reinforced concrete beam. The total derivative form, simplified, is: ; According to the principle of the transition hinge, the bending moment at the mid-span section... This can be expressed as the yield moment of a concrete beam. : ; in, ; ; In the formula, The moment of inertia representing the equivalent cross section is used in calculating the dynamic response of flexural members (such as reinforced concrete beams). Since the cross section is composed of two materials with different moduli, concrete and steel, it is necessary to consider the ratio of their elastic moduli for ease of mechanical calculation. The area of ​​the reinforcing steel bars is converted into an equivalent area of ​​concrete. That is, the moment of inertia of this equivalent cross section relative to its neutral axis; The neutral axis depth represents the depth of the converted section. In the elastic stage, after the reinforcement is converted into an equivalent concrete area according to the elastic modulus ratio, the vertical distance from the edge of the compression zone of the entire composite section to its physical centroidal axis (neutral axis) is the vertical distance from the edge of the compression zone to the physical centroidal axis (neutral axis) of the composite section.

[0033] Further integration of the differential equation derived from the condition that the shear force is zero yields: ; right Differentiate to obtain the velocity of the transition hinge motion in the local response. ,Right now The velocity of the transition hinge motion in the local response can be obtained using the numerical software MATLAB. .

[0034] Assume the transition hinge reaches any point on the beam surface. Time is expressed as The displacement at that point is then obtained by integrating the velocity over time: ; In the formula, Indicates the position on the beam In time The vertical displacement.

[0035] The local dynamic response displacement can be obtained by integrating. : ; Use numerical software to perform a time-related calculation on the above equation. Taking the derivative of , we can obtain the velocity and acceleration distribution functions. The above formula is mainly applicable to The range, because here we assume The concrete beam within the span is the direct area of ​​action of the equivalent impulse, and the velocity is taken as... .

[0036] Finally, based on the local failure size, the initial state of the mid-span response of the overall response is obtained as follows: After obtaining the local failure dimensions in step S1, the length of the plastic hinge at the end of the local response can be considered equal to the length of the lateral spalling, i.e. Thus, the local dynamic response displacement in the local response is obtained. Local dynamic response speed and local dynamic response acceleration , respectively represented as: ; ; .

[0037] When the transition hinge moves At this time, the local response phase ends, and the middle ( The displacement, velocity, and acceleration of the sample represent the initial state of the mid-span response during the overall response phase, i.e., the initial mid-span displacement. Initial mid-span velocity Initial mid-span acceleration : ; ; ; in, ; .

[0038] S3. Based on the structural parameters and local failure dimensions of the reinforced concrete beam, perform local-to-overall coupling parameter correction to obtain the correction relationship, and then obtain the conversion coefficient; wherein, the correction relationship includes the relationship between bending moment and curvature of the mid-span section, the relationship between mid-span displacement and curvature, and the relationship between mid-span resistance and bending moment of the mid-span section.

[0039] During implementation, the relationship between bending moment and curvature at the mid-span section is obtained in the following way: a1. Divide the mid-span section of the reinforced concrete beam after localized failure into compression zones along its height. Layers and stretching zones Layers; wherein, compression distinguishes layer thickness. Stretching the layer thickness , Indicates the depth of the neutral axis.

[0040] Specifically, such as Figure 3 As shown, the compression and tension zones are defined by numbering each layer sequentially from the neutral axis along the outermost fiber direction. The compression and tension zones are independently divided, and the numbering direction is from the neutral axis outwards. Specifically, from the neutral axis to the outermost fiber direction of the beam compression zone... The coordinates of a layer can be represented as , =1, 2, 3…, From the neutral axis to the outermost fiber direction in the beam tensile zone The coordinates of a layer can be represented as .

[0041] a2. Based on the curvature of the mid-span section in the current iteration, calculate the stress of each layer of concrete and the stress of the reinforcing steel on the mid-span section.

[0042] Specifically, the stress in steel reinforcement includes compressive longitudinal reinforcement stress and tensile longitudinal reinforcement stress.

[0043] In practical implementation, the stress in the compressive longitudinal reinforcement is expressed as: ; in, ; ; ; ; ; ; In the formula, This indicates that the strain of the longitudinal reinforcement under compression is The compressive stress of the longitudinal reinforcement at that time , These represent the compression strain and strain rate of the longitudinal reinforcement, respectively. , These represent the curvature and rate of change of curvature of the mid-span section of the reinforced concrete beam in the current iteration, respectively. This indicates the cross-sectional area of ​​the compressed longitudinal reinforcement. Indicates the spacing of the stirrups. Represents the plastic moment of the steel reinforcement. Indicates the diameter of the reinforcing bar. Indicates the yield stress of the steel reinforcement. This represents the compressive strain rate enhancement factor of steel reinforcement. , represent the first and second reference strain rate constants, respectively. , , The unit is seconds.

[0044] In practical implementation, the stress in the tensile longitudinal reinforcement is expressed as follows: ; in, ; ; ; In the formula, This indicates that the strain of the longitudinal reinforcement is... The tensile stress of the longitudinal reinforcement at that time , These represent the strain and strain rate of the tensile longitudinal reinforcement, respectively. This represents the tensile strain rate enhancement factor of steel reinforcement. This represents the tangent modulus of the steel reinforcement after it yields. , These represent the yield strain and fracture strain of the steel reinforcement, respectively. It should be noted that, to simplify the calculation process, the strain rate of both the steel reinforcement and the concrete is taken as 1×10⁻⁶ in the calculation. 3 s -1 .

[0045] Specifically, the stress in each layer of concrete includes the stress in the compression zone and the stress in the tensile zone.

[0046] In practical implementation, the concrete stress in the compression zone of each layer is expressed as follows: ; in, ; ; ; ; ; ; ; In the formula, Indicates the first The strain of the concrete in the compression zone is Concrete stress in the compression zone at that time , They represent the first Strain and strain rate of concrete in the compression zone of the layer. Indicates the first Uniaxial compressive damage variables of concrete in the compression zone. Indicates the first The compressive strain rate enhancement factor of concrete in the compression zone of the layer. Indicates the uniaxial compressive strength of concrete. This represents the strain corresponding to the peak compressive stress in concrete. This represents the concrete compression softening parameter.

[0047] In practical implementation, the concrete stress in the tensile zone of each layer is expressed as follows: ; in, ; ; ; ; ; ; ; ; ;; ; In the formula, Indicates the first The strain of the concrete in the tensile zone is The tensile stress in the concrete at that time , They represent the first The strain and strain rate of the concrete in the tensile zone of the layer, Indicates the first Uniaxial compressive damage variables of concrete in the tensile zone. Indicates the first Tensile strain rate enhancement factor of concrete in the tensile zone of the layer. Indicates the uniaxial tensile strength of concrete. This represents the strain corresponding to the peak tensile stress in concrete. This represents the tensile softening parameter of concrete. This indicates the static tensile strength of concrete.

[0048] a3. Based on the stress of each layer of concrete and the stress of the steel reinforcement at the mid-span section, the resultant force and bending moment at the mid-span section are obtained.

[0049] In practical implementation, the resultant force on the mid-span section Represented as: ; in, ; ; In the formula, , These represent the resultant force in the compression zone and the resultant force in the tension zone, respectively. This indicates the cross-sectional area of ​​the compressed longitudinal reinforcement; This indicates the cross-sectional area of ​​the longitudinal reinforcement bars.

[0050] In practical implementation, the bending moment at the mid-span section is expressed as: .

[0051] a4. Determine if the absolute value of the resultant force at the mid-span section is less than the preset tolerance threshold. If it is less, then the curvature of the mid-span section in the current iteration corresponds to the bending moment at the mid-span section; otherwise, adjust the neutral axis depth using the bisection method. Repeat steps a1 to a4.

[0052] Specifically, the tolerance threshold is set to 0.001, which ensures that the bending moment at the mid-span section has sufficiently high accuracy while also taking into account computational efficiency.

[0053] Specifically, given the curvature of the mid-span section, a neutral axis depth (initially T) is first assumed within the interval [0,T]. ), calculate the resultant force at the mid-span section ;like According to The value of the value (positive indicates high pressure, and the median value in the current compression zone is used as the updated neutral axis depth; negative indicates high tension, and the median value in the current tension zone is used as the updated neutral axis depth) is continuously narrowed down and recalculated until... The neutral axis depth at this point is the neutral axis depth under this curvature.

[0054] a5. Increase the curvature of the mid-span section by the set curvature step size, and repeat steps a1 to a5 to obtain the mid-span section bending moment for different mid-span section curvatures, i.e., the relationship between the bending moment and curvature of the mid-span section. It can be understood that, given the curvature of the mid-span section, the corresponding mid-span section bending moment can be obtained through the relationship between the bending moment and curvature of the mid-span section.

[0055] It should be noted that the derivation of the relationship between bending moment and curvature at the mid-span section in this embodiment is based on the following assumptions: the damage to the concrete beam's blast-facing surface caused by proximity load is due to the reduction in the depth of the cross-sectional compression zone. To indicate, and Determined based on initial input parameters (charge parameters and component parameters); the effect of back collapse failure is not considered; the mid-span section of the reinforced concrete component always remains planar; the stress and strain of each component are obtained from the distance between the component and the neutral axis of the section; the shear deformation of the mid-span section is ignored, and the bond slip between the reinforcement and the concrete is not considered.

[0056] In practice, the relationship between the mid-span displacement and curvature is expressed as follows: ; in, ; ; In the formula, This represents the mid-span displacement in the overall response. This represents the yield curvature of a reinforced concrete beam.

[0057] Specifically, the yield curvature of reinforced concrete beams For the first time the strain of the tensile longitudinal reinforcement reaches the yield strain The corresponding cross-sectional curvature at that time.

[0058] It should be noted that the relationship between mid-span displacement and curvature is derived as follows: Since localized damage affects the stiffness and yielding behavior of the cross section, the displacement-curvature relationship needs to be corrected.

[0059] During the overall response phase, the shape of the deformation remains constant; only the magnitude of the deformation changes over time. The deformation shape of the beam can be represented by a fixed deformation function. The description is expressed as: ; In the formula, This represents the effective span, i.e., the distance between the outermost stirrups at both ends of the beam. The normalized distribution of curvature over half the span of the reinforced concrete beam during the overall structural response is as follows: ; It should be noted that, and Approximately equal, a common practice in engineering calculations.

[0060] Due to the dynamic response of the beam It can be represented as: ; In the formula, This represents the time history of mid-span displacement.

[0061] Therefore, curvature The distribution function can be expressed as: ; During the elastic phase, the mid-span curvature With mid-span elastic displacement component The relationship can be represented as: ; When a reinforced concrete beam enters a state of plastic deformation, a plastic hinge appears at the mid-span, exhibiting plastic rotation. : ; In the formula, This represents the curvature of the mid-span section of a reinforced concrete beam.

[0062] The additional mid-span displacement caused by the rotation of the plastic hinge, i.e., the plastic displacement component, is expressed as: ; Mid-span displacement of reinforced concrete beam in overall response It can be represented as: ; In the formula, , These represent the elastic displacement component and the plastic displacement component, respectively.

[0063] In practice, the relationship between the mid-span resistance and the mid-span section bending moment is expressed as follows: ; In the formula, Indicates the overall resistance during the cross-section of the response. This represents the bending moment at the mid-span section.

[0064] It should be noted that the relationship between the mid-span resistance and the bending moment at the mid-span section is derived as follows: Structural resistance of continuous beams With bending moment at any cross section Existence Relationship: ; Overall Response Phase Take as The span of the beam It can be expressed as the bending moment at the mid-span section. With bending moment distribution function η ( x The composite function of ), i.e.: ; Then structural resistance With bending moment at any cross section The relationship can be represented as: ; Among them, the bending moment distribution function obtained by fitting the experimental relationship for: ; Will , Substituting and integrating, we get: ; In the formula, This represents the mid-span resistance in the overall response, i.e., the overall structural resistance. This represents the bending moment at the mid-span section.

[0065] In practice, the conversion factors include mass conversion factors and stiffness conversion factors, which are expressed as follows: ; ; in, ; ; In the formula, , These represent the mass conversion factor and the stiffness conversion factor, respectively. , These represent the initial stiffness of the reinforced concrete beam before local failure and the residual stiffness after local failure, respectively. To determine the slope of the elastic segment in the moment-curvature relationship of the undamaged section, in this embodiment, the initial tangent slope of the moment-curvature relationship at the mid-span section can be taken as... ; The slope of the elastic segment represents the relationship between the bending moment and curvature at the mid-span section after local damage occurs in this embodiment.

[0066] Understandably, mass conversion coefficients and stiffness conversion coefficients can simplify complex continuum structures (beams) into simple single-degree-of-freedom systems (mass-spring models), thereby enabling rapid calculation of dynamic response.

[0067] It should be noted that the conversion coefficients are derived as follows: Concrete beam section mass without local failure It can be represented as: ; The mass distribution of a concrete beam experiencing localized failure at mid-span can be described as follows: ; For concrete beams experiencing localized failure, the mass conversion factor K M Mass distribution needs to be considered m ( x The effect of ) is expressed as: ; .

[0068] To ensure that the strain energy in an equivalent single-degree-of-freedom system is equal to the internal force energy in an actual continuum structure, a stiffness conversion factor is defined. If a concrete beam experiences localized failure, resulting in uneven cross-sectional distribution across the span, the bending stiffness of the reinforced concrete beam will change, thus affecting the stiffness conversion factor. Assuming that the localized failure depth in the concrete beam is uniformly distributed, the bending stiffness of the remaining cross-section within the length of the localized failure remains unchanged, while the intact sections retain their initial bending stiffness. Therefore, the stiffness conversion factor... It can be represented as: ; ; In the formula, , These represent the initial stiffness of the reinforced concrete beam before local failure and the remaining stiffness after local failure, respectively.

[0069] S4. Based on the initial state, correction relationship and conversion coefficient of the mid-span response of the overall response, the dynamic response is calculated using the elastoplastic SDOF differential motion equation to obtain the mid-span displacement response result.

[0070] During implementation, the mid-span displacement response results are obtained in the following manner: Based on the elastoplastic SDOF differential motion equation, the time curves of the overall response time and mid-span displacement under near-explosion load were obtained by iterative calculation. Based on the time curves of the overall response time and mid-span displacement under near-explosion load, the maximum mid-span displacement during the overall response and the residual mid-span displacement at the end of the overall response are obtained. The ratio of the residual mid-span displacement at the end of the overall response to the maximum mid-span displacement during the overall response is taken as the mid-span displacement response result.

[0071] In practical implementation, the elastic-plastic SDOF differential motion equation is expressed as: ; in, ; ; ; ； ; ; ; In the formula, , , These represent the mid-span displacements at the current, previous, and next moments in the overall response, respectively. This represents the external load at the current moment in the overall response. Indicates equivalent mass. Indicates the damping coefficient. This represents the equivalent resistance at the current moment in the overall response; Indicates the time step. Indicates the limit of plastic displacement. Indicates the elastic displacement limit. Indicates unloading stiffness. This indicates the current historical maximum mid-span displacement; This indicates the loading or unloading flag at the current moment in the overall response, where "1" indicates loading and "-1" indicates unloading. The external load at the current moment in the overall response... It is usually set to 0 after an explosion.

[0072] Specifically, in the iterative calculation, the mid-span displacement of the previous time step before the initial time step. Represented as: .

[0073] Specifically, the iterative calculation terminates when the displacement response tends to remain still or reaches a preset number of time steps.

[0074] Specifically, based on the relationship between mid-span displacement and curvature, the critical displacement at which the beam structure begins to enter plasticity can be obtained, i.e., when the curvature reaches the yield curvature. The displacement corresponding to this time is the elastic displacement limit. Specifically, let the relationship between mid-span displacement and curvature be... At this point, the plastic term is zero, and only the elastic term is retained, resulting in... .

[0075] Specifically, based on the relationship between mid-span displacement and curvature, the maximum displacement when the structure reaches its ultimate failure state can be obtained, i.e., when the curvature reaches its ultimate curvature. The displacement corresponding to this time is the plastic displacement limit. Specifically, let the mid-span displacement be related to the curvature. At this point, both the elastic and plastic terms are active, resulting in... .

[0076] Specifically, yield curvature From the relationship between bending moment and curvature at the mid-span section, take the curvature corresponding to the first yielding of the tensile reinforcement; ultimate curvature. From the relationship between bending moment and curvature at the mid-span section, take the curvature corresponding to the concrete crushing or steel reinforcement fracture.

[0077] Specifically, Indicates the mid-span displacement as Trans-temporal resistance, based on The curvature of the mid-span section of the reinforced concrete beam is obtained by relating the mid-span displacement to the curvature. Then, the bending moment at the mid-span section is obtained through the relationship between the bending moment and curvature at the mid-span section. Furthermore, based on the relationship between mid-span resistance and mid-span section bending moment, we can obtain... .

[0078] It should be noted that the elastoplastic SDOF differential equation of motion is derived through the following derivation: The elastic-plastic SDOF equation after conversion factor correction is expressed as: ; In the formula, This indicates the total mass of the reinforced concrete beam. This represents the equivalent load time history of the external load.

[0079] Discretization is performed using the central difference method to determine the time step. and time series ;Will and Transform it into a central difference scheme, i.e.: and ; the difference format and Substituting the modified elastoplastic SDOF equation, we get: ; Among them, based on the initial velocity and acceleration Sure .

[0080] Compared with existing technologies, the displacement response calculation method for reinforced concrete beams under near-blast load provided in this embodiment is the first to realize the dual influence correction of local damage to reinforced concrete beams on the overall dynamic response under near-blast load in a simplified calculation model. By calculating the local response, the initial kinetic energy is accurately transferred to the overall stage, and the cross-sectional stiffness and mass distribution are dynamically corrected based on the local failure characteristics. This solves the inherent defects of ignoring the initial state and stiffness degradation, significantly improving the calculation accuracy. At the same time, it retains the calculation efficiency of the simplified model. Only basic geometric and explosive parameters need to be input to quickly evaluate near-blast damage. This provides a high-precision and highly applicable engineering tool for rapid design of protective engineering and analysis of defense effectiveness.

[0081] Example 2 To further verify the model's engineering applicability, two typical working conditions from field tests were selected for comparative analysis: Condition 1: The component dimensions are 244cm×25cm×12.5cm, with a uniaxial compressive strength of 38.7MPa; the charge parameters are 1kg TNT (length-to-diameter ratio 0.78) and a blast height of 57cm; the reinforcement configuration is as follows: tie bars with a diameter of 16mm and a yield strength of 435MPa (reinforcement ratio 1.29), compression bars with a diameter of 8mm and a yield strength of 540MPa (reinforcement ratio 0.32%), stirrups with a diameter of 6mm and a stirrup spacing of 15cm (reinforcement ratio 0.3%); the failure mode is minor local failure and bending response.

[0082] Condition 2: The component dimensions are 244cm×25cm×12.5cm, with a uniaxial compressive strength of 38.7MPa; the charge parameters are 0.5 kg TNT (length-to-diameter ratio 0.39), and the blast height is 45 cm; the reinforcement configuration is as follows: tie bars with a diameter of 16 mm and a yield strength of 435 MPa (reinforcement ratio 1.29); compression bars with a diameter of 8 mm and a yield strength of 540 MPa (reinforcement ratio 0.32%); stirrups with a diameter of 6 mm and a stirrup spacing of 15cm (reinforcement ratio 0.3%); the failure mode is minor local failure and bending response.

[0083] When the ratio is 0.57 Under the explosive loads of 1kg TNT and 0.5kg TNT, the concrete beam only suffered side spalling and back collapse, without front crushing.

[0084] The calculation method proposed in Example 1 was used to predict the displacement response for these two working conditions, and the results were compared with the measured results. The mid-span displacement response results are shown in Figure 4; Figure 4(a) and Figure 4(b) show the model prediction curves and experimental curves for working conditions 1 (1 kg TNT, 0.57 m blast height) and 0.5 kg TNT (0.45 m blast height), respectively. It can be seen that the calculated maximum displacement and residual displacement agree well with the experimental results, and the maximum displacement deviation is as follows: 8.3% and +5.0%. The results show that the model can predict the dynamic response of the component relatively accurately and has good engineering applicability.

[0085] Those skilled in the art will understand that all or part of the processes of the methods described in the above embodiments can be implemented by a computer program instructing related hardware, and the program can be stored in a computer-readable storage medium. The computer-readable storage medium may be a disk, optical disk, read-only memory, or random access memory, etc.

[0086] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for calculating the displacement response of a reinforced concrete beam under near-explosion load, characterized in that, Includes the following steps: The charge parameters of the near-explosive load and the structural parameters of the reinforced concrete beam are obtained, and the local failure characteristics of the reinforced concrete beam are predicted to obtain the local failure size. Based on the charge parameters of the near-explosion load, the local response of the reinforced concrete beam is calculated to obtain the local dynamic response. Then, based on the local failure size, the initial state of the mid-span response of the overall response is obtained. Based on the structural parameters and local failure dimensions of the reinforced concrete beam, local and overall coupling parameters are corrected to obtain the correction relationship, and then the conversion coefficient is obtained; among them, the correction relationship includes the relationship between bending moment and curvature at the mid-span section, the relationship between displacement and curvature at the mid-span section, and the relationship between resistance at the mid-span section and bending moment at the mid-span section; Based on the initial state, correction relationship, and transformation coefficient of the mid-span response of the overall response, the dynamic response is calculated using the elastoplastic SDOF differential motion equation to obtain the mid-span displacement response result.

2. The method for calculating the displacement response of reinforced concrete beams under near-explosion load according to claim 1, characterized in that, The mid-span displacement response results were obtained in the following manner: Based on the elastoplastic SDOF differential motion equation, the time curves of the overall response time and mid-span displacement under near-explosion load were obtained by iterative calculation. Based on the time curves of the overall response time and mid-span displacement under near-explosion load, the maximum mid-span displacement during the overall response and the residual mid-span displacement at the end of the overall response are obtained. The ratio of the residual mid-span displacement at the end of the overall response to the maximum mid-span displacement during the overall response is taken as the mid-span displacement response result.

3. The method for calculating the displacement response of a reinforced concrete beam under near-explosion load according to claim 1, characterized in that, The near-explosive charge is a cylindrical charge; the local damage dimensions include the lateral spalling length and the crushing depth, respectively expressed as: ； ； in, ； ； In the formula, , These represent the lateral spalling length and crushing depth, respectively. , These represent the thickness and width of the reinforced concrete beam, respectively. Indicates the first The dimensionless quantity of local damage characteristics , , , , , , , They represent the first The first, second, third, fourth, fifth, sixth, seventh, and eighth correlation coefficients of the dimensionless quantities of local damage characteristics. Indicates the quality of the explosive charge. Indicates the density of concrete. Indicates the detonation height of the near-explosion load. , These represent the height and diameter of the propellant column, respectively. , , These represent the reinforcement ratios of stirrups, compression longitudinal bars, and tension longitudinal bars, respectively.

4. The method for calculating the displacement response of a reinforced concrete beam under near-explosion load according to claim 3, characterized in that, The near-explosive load is located above the mid-span of the reinforced concrete beam; the initial state of the mid-span response of the overall response includes the initial mid-span displacement, the initial mid-span velocity, and the initial mid-span acceleration; wherein, the initial mid-span displacement... Represented as: ； in, ; In the formula, This indicates the length of a reinforced concrete beam. , Let represent the instantaneous position and velocity of the transition hinge motion in the local response, respectively. This represents the initial mid-span velocity in the local response.

5. The method for calculating the displacement response of a reinforced concrete beam under near-explosion load according to claim 4, characterized in that, The velocity of the transition hinge motion in the local response Represented as: ； in, ； In the formula, This indicates the duration of the blast shock wave's effect on the reinforced concrete beam. This indicates the cross-sectional mass of the reinforced concrete beam when no localized damage has occurred. This represents the yield moment of a reinforced concrete beam.

6. The method for calculating the displacement response of a reinforced concrete beam under near-explosion load according to claim 4, characterized in that, The initial mid-span velocity in the local response Represented as: ； In the formula, This represents the horizontal coordinate along the span of the reinforced concrete beam, with the projection point of the explosive charge on the beam as the origin. This represents the peak specific impulse of a near-explosive load explosion. This indicates the initial effective span of the reinforced concrete beam.

7. The method for calculating the displacement response of a reinforced concrete beam under near-explosion load according to claim 5, characterized in that, The relationship between bending moment and curvature at the mid-span section is obtained as follows: a1. Divide the mid-span section of the reinforced concrete beam after localized failure into compression zones along its height. Layers and stretching zones Layers; wherein, compression distinguishes layer thickness. Stretching layer thickness , Indicates the depth of the neutral axis; a2. Based on the curvature of the mid-span section in the current iteration, calculate the stress of each layer of concrete and the stress of the reinforcing steel on the mid-span section; a3. Based on the stress of each layer of concrete and the stress of the steel reinforcement at the mid-span section, the resultant force and bending moment at the mid-span section are obtained. a4. Determine if the absolute value of the resultant force at the mid-span section is less than the preset tolerance threshold. If it is less, then the curvature of the mid-span section in the current iteration corresponds to the bending moment at the mid-span section; otherwise, adjust the neutral axis depth using the bisection method. Repeat steps a1 to a4; a5. Increase the curvature of the mid-span section by the set curvature step size, and repeat steps a1 to a5 to obtain the mid-span section bending moment with different mid-span section curvatures.

8. The method for calculating the displacement response of a reinforced concrete beam under near-explosion load according to claim 6, characterized in that, The relationship between the mid-span displacement and curvature is expressed as follows: ； in, ； ； In the formula, This represents the mid-span displacement in the overall response. This represents the yield curvature of a reinforced concrete beam. This represents the curvature of the mid-span section of a reinforced concrete beam.

9. The method for calculating the displacement response of a reinforced concrete beam under near-explosion load according to claim 8, characterized in that, The relationship between the mid-span resistance and the mid-span section bending moment is expressed as follows: ； In the formula, Indicates the overall resistance during the cross-section of the response. This represents the bending moment at the mid-span section.

10. The method for calculating the displacement response of a reinforced concrete beam under near-explosion load according to claim 9, characterized in that, The conversion factors include the mass conversion factor and the stiffness conversion factor, which are expressed as follows: ； ； in, ； ； In the formula, , These represent the mass conversion factor and the stiffness conversion factor, respectively. , These represent the initial stiffness of the reinforced concrete beam before local failure and the remaining stiffness after local failure, respectively.

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