Method for calculating effective modulus of concrete and inherent frequency of beam bridge in consideration of water phase change

Abstract

A concrete effective modulus and beam bridge inherent frequency calculation method considering water phase change relates to the field of engineering structure health monitoring, and comprises the following steps: S1, regarding concrete as a multiphase composite material comprising a homogeneous matrix material and various spherical inclusion particles (including air, water or ice); measuring the porosity, the water content and the ice content, and calculating the volume percentage of air, water and ice; s2, calculating the elastic modulus of the base material based on the porosity, and determining the elastic modulus of each inclusion material; s3, calculating the effective modulus of the concrete according to the volume percentages of the air, the water and the ice, the elastic modulus of the base material and the elastic modulus of each inclusion material; s4, calculating the inherent frequency of the bridge according to the effective modulus; s5, calculating the upper limit and the lower limit of the effective modulus; and S6, calculating the upper limit and the lower limit of the inherent frequency of the beam bridge. According to the method, the phase change of water in concrete pores is considered, the effective modulus of the concrete at different temperatures can be calculated, and the inherent frequency and the value limit of the beam bridge can be calculated.

Description

Technical Field

[0001] The present invention relates to engineering structure health monitoring, and in particular to a calculation method for the effective modulus of concrete and the natural frequency of a beam bridge considering water phase change. Background Art

[0002] The dynamic characteristics of bridges, including natural frequencies and modal shapes, are fundamental mechanical features directly determined by the stiffness and mass of the structure. They have long been widely used in structural dynamic analysis and design. In recent years, these parameters have also been increasingly used in bridge condition assessment and structural health monitoring. Modal analysis-based structural health monitoring methods have made significant progress and have become a non-destructive means for engineers to detect structural deterioration, damage, and degradation during the service life of bridges. However, the reliability of such technologies relies heavily on a deep understanding of the various factors that affect the modal parameters in the working environment.

[0003] Among numerous environmental factors, temperature variation is one of the most significant influencing factors on bridge dynamic characteristics. Numerous studies have demonstrated a close correlation between temperature variations and changes in bridge modal frequencies, highlighting the importance of fully considering environmental effects in condition assessment. Temperature-induced frequency fluctuations often mask subtle frequency changes caused by structural damage, thereby weakening the effectiveness of structural health monitoring techniques and increasing the difficulty of determining bridge health status. Therefore, accurately distinguishing frequency changes caused by structural damage from those induced by environmental factors has become a key challenge in bridge monitoring practice.

[0004] Temperature factors have received extensive attention and in-depth research from numerous researchers. To date, researchers have systematically explored the changes in modal parameters caused by temperature through numerical simulations, laboratory tests, and field monitoring. Among temperature-related factors, freezing is an extreme case that has a significant impact on the structural properties of concrete bridges. Long-term monitoring of bridges located in cold regions (such as the Z24 Bridge in Switzerland, the Kishwaukee Bridge, and the Dowling Hall Pedestrian Bridge in the United States) has shown that when the temperature drops from slightly above freezing (0°C) to slightly below freezing (-5°C), the natural frequency of the bridge increases significantly. Similar phenomena have also been documented in experimental studies, emphasizing that freezing of pore water inside concrete is the main mechanism causing the frequency change.

[0005] Previous studies have shown that freezing primarily alters concrete structural stiffness through two mechanisms: volume expansion caused by freezing, and changes in its elastic modulus. Zhou et al. and Sun et al. studied concrete and mortar specimens with varying mix proportions and saturations and found that the expansion deformation caused by freezing was generally less than 0.1%. Lv et al. also noted that even after numerous freeze-thaw cycles, concrete dimensional changes remained minimal. Therefore, these minor dimensional changes were generally considered to have negligible effects on the natural frequency of bridges. In contrast, scholars have focused on the significant changes in concrete's elastic modulus caused by freezing. Shoukry et al. noted that when the temperature rises from below freezing to above freezing, the elastic modulus of concrete decreases significantly, with the most dramatic changes occurring in the 0°C to -10°C range. Gong et al. and Wang et al. further confirmed through micropore mechanics simulations and experiments that ice-filled pores can enhance the mechanical properties of concrete. Quantitative analysis by Xie et al. also established a clear relationship between subfreezing temperature, the ice content in concrete pores, and its mechanical properties.

[0006] Despite numerous studies, a widely accepted, sufficiently accurate, and practical theoretical model to quantitatively assess the effects of pore water phase transitions (freezing or thawing of pore water) on the natural frequencies of concrete beam bridges remains lacking. Current methods often rely on empirical observations or provide only qualitative descriptions, which are insufficient for practical bridge engineering management and assessment. Summary of the Invention

[0007] To address the above problems, the present invention proposes a calculation method for the effective modulus of concrete and the natural frequency of beam bridges considering water phase change, which is used to quantitatively analyze and evaluate the influence of concrete pore water phase change on the effective modulus of concrete and the natural frequency of concrete beam bridges. This method is a new theoretical framework based on composite material mechanics, which regards concrete as a composite material composed of a uniform and isotropic cement slurry-aggregate matrix material and spherical inclusions (pores filled with air, water, ice or a combination thereof). Taking into account the multiphase composite properties of concrete before and after the pore water phase change, the calculation formulas for the effective modulus of concrete and the natural frequency of concrete beam bridges under different temperature conditions (including freezing conditions) are derived through the composite material mechanics theory and the Mori-Tanaka method, and practical value limits of the effective modulus of concrete and the natural frequency of concrete beam bridges are proposed. This method has broad engineering application prospects in the field of structural health monitoring.

[0008] The calculation method for the effective modulus of concrete and the natural frequency of beam bridges considering water phase transition is as follows:

[0009] S1, measuring the porosity, moisture content and ice content of the concrete, and calculating the volume percentages of air, water and ice in the concrete based on the porosity, moisture content and ice content of the concrete;

[0010] S2, calculating the elastic modulus of a matrix material based on the porosity of the concrete, and determining the elastic modulus of each inclusion material; the matrix material is the solid part of the concrete;

[0011] S3, calculating the effective modulus of concrete based on the volume percentages of air, water, and ice in the concrete, the elastic modulus of the matrix material, and the elastic modulus of each inclusion material;

[0012] S4, calculate the natural frequency of concrete beam bridge based on the effective modulus of concrete.

[0013] Preferably, the elastic modulus of the matrix material is expressed as:

[0014]

[0015] Among them, K s (T) represents the bulk modulus of the concrete matrix material; G s (T) represents the shear modulus of the concrete matrix material; φ represents the porosity of the concrete, ν represents the Poisson's ratio of the concrete, and E 20℃ It represents the elastic modulus of dry concrete at 20℃; T represents the concrete temperature.

[0016] Preferably, the elastic modulus of the inclusion material includes bulk modulus and shear modulus; wherein, the bulk modulus of water is in the range of 2.0 to 2.3 GPa; the shear modulus of water is 0; the bulk modulus of ice is in the range of 8.6 to 9.0 GPa; the shear modulus of ice is in the range of 3.8 to 4.0 GPa; the bulk modulus of air is 0; and the shear modulus of air is 0.

[0017] Preferably, the effective modulus of concrete is calculated based on the volume percentages of air, water, and ice in the concrete, the elastic modulus of the matrix material, and the elastic modulus of each inclusion material. Specifically, the effective modulus of concrete is calculated based on the composite material mechanics theory and the classic Mori-Tanaka method, and is expressed as:

[0018] P=(3K p ,2G p )

[0019]

[0020] in, represents the effective modulus of concrete, The compressed expression of the fourth-order stiffness tensor of an isotropic material is (,) means simplifying high-dimensional tensors into two-dimensional diagonal matrix operations. represents the effective bulk modulus of concrete, represents the effective shear modulus of concrete; C srepresents the elastic modulus of the concrete matrix material, C r represents the elastic modulus of the r-phase inclusion material, R represents the total R-phase inclusion material; P represents the Eshelby polarization tensor; K p G represents the volume positivity factor; p represents the shear polarization factor; c s represents the volume percentage of concrete matrix material, c r represents the volume percentage of the r-th phase inclusion; [] -1 Indicates inversion; G s (T) represents the shear modulus of the concrete matrix material; K s (T) represents the bulk modulus of the concrete matrix material.

[0021] Preferably, the method further comprises: calculating the effective Young's modulus of the concrete according to the effective modulus of the concrete, which is expressed as:

[0022]

[0023] in, represents the effective Young's modulus of concrete; represents the effective bulk modulus of concrete, represents the effective shear modulus of concrete.

[0024] Preferably, after S4, the method further includes:

[0025] S5, calculate the upper and lower limits of the effective modulus of concrete;

[0026] S6. Calculate the upper and lower limits of the natural frequency of the concrete beam bridge based on the upper and lower limits of the effective modulus of concrete.

[0027] Preferably, the calculation process of the upper limit of the effective modulus of concrete is as follows:

[0028] When the concrete temperature is 0℃≤T≤50℃, the effective modulus of concrete pores when saturated with water is taken as the upper limit of the effective modulus of concrete; it is expressed as:

[0029]

[0030] Where, T represents the concrete temperature; φ represents the porosity of concrete; K1 represents the bulk modulus of water; K s (T) represents the bulk modulus of the concrete matrix material; G s (T) represents the shear modulus of the concrete matrix material; Indicates the upper limit of the effective bulk modulus of concrete; represents the upper limit of the effective shear modulus of concrete; G1 represents the shear modulus of water;

[0031] When the concrete temperature is -50°C ≤ T ≤ -5°C, the effective modulus when the concrete pores are saturated with ice is taken as the upper limit of the concrete effective modulus; it is expressed as:

[0032]

[0033] Among them, K2 represents the bulk modulus of ice; G2 represents the shear modulus of ice;

[0034] When the concrete temperature is -5°C < T < 0°C, the upper limit value of the concrete effective bulk modulus is obtained by linear interpolation based on the two upper limit values of the concrete effective bulk modulus obtained when the concrete temperature is 0°C ≤ T ≤ 50°C and when the concrete temperature is -50°C ≤ T ≤ -5°C; similarly, the upper limit value of the concrete effective shear modulus is obtained by linear interpolation based on the two upper limit values of the concrete effective shear modulus obtained when the concrete temperature is 0°C ≤ T ≤ 50°C and when the concrete temperature is -50°C ≤ T ≤ -5°C.

[0035] Preferably, the calculation process of the lower limit of the concrete effective modulus is as follows:

[0036] When the concrete temperature is -50°C ≤ T ≤ 50°C, the effective modulus when the concrete pores are saturated with air is taken as the lower limit of the concrete effective modulus, and it is expressed as:

[0037]

[0038] Among them, represents the lower limit value of the concrete effective bulk modulus; K3 represents the bulk modulus of air; T represents the concrete temperature; φ represents the porosity of the concrete; K s (T) represents the bulk modulus of the concrete matrix material; G s (T) represents the shear modulus of the concrete matrix material; represents the lower limit value of the concrete effective shear modulus; G3 represents the shear modulus of air.

[0039] Preferably, the lower limit of the concrete effective modulus is substituted into the calculation formula of the natural frequency of the beam bridge to obtain the lower limit of the natural frequency of the concrete beam bridge; the upper limit of the concrete effective modulus is substituted into the calculation formula of the natural frequency of the beam bridge to obtain the upper limit of the natural frequency of the concrete beam bridge.

[0040] Compared with the prior art, the present invention has the following beneficial effects:

[0041] (1) The calculation method proposed in the present invention can take into account the phase change of pore water and calculate the effective modulus of concrete at different specific temperatures (especially sub-zero temperatures). This solves the problem that the existing theoretical model can only analyze the effective modulus of concrete at room temperature, but cannot analyze the sudden change of the effective modulus of concrete in cold areas caused by the phase change (freezing or melting) of pore water near zero degrees;

[0042] (2) The present invention can be used to calculate the natural frequency of concrete beam bridges under actual operating conditions. The proposed natural frequency limit can effectively evaluate the impact of pore water freezing on the natural frequency of concrete beam bridges. It has good applicability in the current situation in my country where small and medium-span bridges account for more than 80%. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] The present invention will be described in further detail below with reference to the accompanying drawings;

[0044] Figure 1 This is a flow chart of a method for calculating the effective modulus of concrete and the natural frequency of a beam bridge considering water phase change according to an embodiment of the present invention;

[0045] Figure 2 A schematic diagram of a concrete composite structure of a method for calculating the effective modulus of concrete and the natural frequency of a beam bridge considering water phase change according to an embodiment of the present invention;

[0046] Figure 3 Schematic diagram of the test block and experimental device for the method for calculating the effective modulus of concrete and the natural frequency of a beam bridge considering water phase change according to an embodiment of the present invention; wherein (a) represents the strain acquisition device; (b) represents the temperature monitoring device; (c) represents the thermal insulation measures; (d) represents the test block in the freezer; and (e) represents the experimental system.

[0047] Figure 4 The stress history of the primary elastic modulus measurement of the method for calculating the effective modulus of concrete and the natural frequency of a beam bridge considering water phase change according to an embodiment of the present invention;

[0048] Figure 5 A comparison chart of experimental results and theoretical results of elastic modulus of concrete effective modulus and beam bridge natural frequency calculation method considering water phase change according to an embodiment of the present invention;

[0049] Figure 6 A schematic diagram of the Swiss Z24 Bridge, illustrating a method for calculating the effective modulus of concrete and the natural frequency of a beam bridge taking into account water phase change according to an embodiment of the present invention;

[0050] Figure 7 This is a comparison chart between the measured results and theoretical results of the first-order frequency of the Z24 bridge of the calculation method of the effective modulus of concrete and the natural frequency of beam bridges considering water phase change in an embodiment of the present invention. DETAILED DESCRIPTION

[0051] The present invention is further described below through specific embodiments.

[0052] like Figure 1 As shown in Figure 2, the calculation method for the effective modulus of concrete and the natural frequency of beam bridges considering water phase change is as follows:

[0053] S1, measures the porosity, moisture content and ice content of concrete, and calculates the volume percentages of air, water and ice in the concrete based on the porosity, moisture content and ice content of the concrete.

[0054] See also Figure 2 As shown in

[15] , concrete is regarded as a multiphase particle inclusion composite material, and the influence of the pore part on the elastic constants of concrete is focused on: the solid part of the cementitious material and the aggregate is regarded as a homogeneous matrix material, while the pore part (containing air, water, ice or their combination) is regarded as a variety of spherical particle inclusion materials.

[0055] By measuring the porosity, water content, and ice content of concrete, the volume percentage of each phase can be determined. The specific measurement methods for concrete porosity and water content are as follows:

[0056] For concrete specimens or concrete bridges with conditions for sampling, indoor experiments can be used for accurate measurement: the porosity is generally measured accurately using the mercury intrusion porosimetry (MIP) or the beta-electrochemical (BET) method; the water content is generally measured accurately using the drying method.

[0057] For concrete specimens or concrete bridges without conditional sampling, only non-destructive measurement techniques can be used: porosity can be measured using ultrasonic pulse velocity (UPV) or resistivity imaging; moisture content can be measured using buried moisture probes, microwave hygrometers, or infrared thermography (IRT). However, it is worth noting that the reliability and accuracy of these measurement methods have yet to be verified.

[0058] Furthermore, measuring the ice content of concrete at a specific temperature currently presents significant challenges. Differential scanning calorimetry (DSC) can be used in the laboratory to measure the temperature-dependent ice content of concrete specimens. This allows for the development of a mathematical model that correlates the ice content of concrete with temperature, which can be used to analyze the effective modulus of concrete and the natural frequency of concrete beam bridges. However, non-destructive measurement of ice content in concrete remains at the research stage.

[0059] S2, calculating the elastic modulus of a matrix material based on the porosity of the concrete, and determining the elastic modulus of each inclusion material; the matrix material is the solid part of the concrete.

[0060] According to the fib Model Code for Concrete Structures 2010 and the theory of porous media mechanics, the bulk modulus K of the concrete matrix material is s and shear modulus G s The relationship with temperature T can be expressed as:

[0061]

[0062] Where, φ is the porosity of concrete, v is the Poisson's ratio of concrete (v can be taken as 0.18), E 20℃ is the elastic modulus of dry concrete at 20°C.

[0063] Existing studies have shown that within the commonly used temperature range of this embodiment (-50°C ≤ T ≤ 50°C), the effect of temperature on the elastic constants of each phase of inclusion materials (including water, ice, and air) is negligible and can be determined according to Table 1.

[0064] Table 1: Elastic constants of various phase inclusion materials (including water, ice and air).

[0065]

[0066] S3, calculating the effective modulus of the concrete based on the volume percentages of air, water, and ice in the concrete, the elastic modulus of the matrix material, and the elastic modulus of each inclusion material.

[0067] For this multi-phase inclusion composite material, a representative unit is taken as the research object, whose size is much smaller than the structure or component size, and much larger than the inclusion material size. It can be defined as the linear transformation relationship between the average stress and the average strain in the element:

[0068]

[0069] Where σ is the stress tensor, ε is the strain tensor, It represents the average of a physical quantity within a representative unit with a volume of V.

[0070] Effective modulus of composite materials It can also be expressed as:

[0071]

[0072] in, is the effective bulk modulus of concrete (to be determined here), This is the effective shear modulus of concrete (to be determined here). The representation is called the eigenmodulus decomposition of the isotropic elastic tensor, which condenses the six degrees of freedom of the fourth-order tensor into two scalars and (,) means simplifying high-dimensional tensors into two-dimensional diagonal matrix operations.

[0073] Similarly, the elastic modulus C of the matrix material s and the elastic modulus C of the r-phase inclusion material r Expressed as:

[0074] C s =(3K s ,2G s )

[0075] C r =(3K r ,2G r )

[0076] Among them, K r is the bulk modulus of the r-th phase inclusion, G r is the shear modulus of the r-th phase inclusion, G s That is G s (T), K s K s (T).

[0077] Based on the general composite material mechanics theory and the classical Mori-Tanaka method, the effective modulus of the composite material can be calculated using the following formula:

[0078] P=(3K p ,2G p )

[0079]

[0080] Among them, c s is the volume percentage of concrete matrix material, c r is the volume fraction of the r-th phase inclusion; P represents the Eshelby polarization tensor; K p G represents the volume positivity factor; p represents the shear polarization factor.

[0081] In particular, the effective Young's modulus of concrete can be obtained as follows:

[0082]

[0083] S4, calculate the natural frequency of concrete beam bridge based on the effective modulus of concrete.

[0084] By incorporating the effective elastic constants of concrete in a multiphase inclusion state into the classical calculation theory and formula for the natural frequency of simply supported or continuous beams, the natural frequency of concrete beam bridges can be calculated. Specifically:

[0085] For a Timoshenko simply supported beam (considering the moment of inertia and shear deformation), its natural frequency can be calculated by the following formula:

[0086]

[0087] Where n is the frequency order, L is the bridge length, I is the cross-sectional bending moment of inertia, m is the unit mass, r is the radius of gyration, A is the cross-sectional area, and k is the cross-sectional shape coefficient.

[0088] For a simply supported Euler-Bernoulli beam (ignoring the moment of inertia and shear deformation), its natural frequency can be calculated by the following formula:

[0089]

[0090] For a Timoshenko continuous beam (considering the moment of inertia and shear deformation), its natural frequency can be calculated using the following formula:

[0091]

[0092] Q j L j M j-1 -(P j L j +P j+1 L j+1 )M j +Q j+1 L j+1 M j+1 =0

[0093]

[0094]

[0095] Among them, M j It represents the internal force of the bending moment at the right support section of the continuous beam in the jth span, M j-1 represents the bending moment internal force of the cross section at the left support end of the j-th span of the continuous beam. M is the column vector of the bending moment internal force of each support cross section, and M is the time variable during the vibration process of the continuous beam. That is, M must be a non-zero vector, that is, the homogeneous linear equation system C·M=0 has a non-zero solution; C is called the coefficient matrix or characteristic matrix; L j It represents the span of the j-th span of the continuous beam. The continuous beam has a total of N spans.

[0096] For an Euler - Bernoulli continuous beam (ignoring the moment of inertia and shear deformation), its natural frequency can be calculated by the following formula:

[0097]

[0098] V j L j M j-1 -(U j L j +U j+1 L j+1 )M j +V j+1 L j+1 M j+1 =0

[0099]

[0100] It should be noted that the calculation of the natural frequency of the concrete beam bridge is also applicable to the calculation of other simply - supported beams or continuous beams, which can be specifically set according to needs and are not restricted in this embodiment.

[0101] S5. Calculate the upper and lower limits of the effective modulus of concrete.

[0102] Using the above - mentioned method for calculating the effective modulus of composite materials, if the assumption of completely saturated concrete is adopted, the calculated effective modulus is its upper bound. In particular, when the temperature is 0℃ ≤ t ≤ 50℃, assuming that the concrete is saturated with water, the upper limit of its effective modulus can be simplified and calculated by the following formula:

[0103]

[0104]

[0105] When the temperature is - 50℃ ≤ t ≤ - 5℃, assuming that the concrete is saturated with ice, the upper limit of its effective modulus can be simplified and calculated by the following formula:

[0106]

[0107] When the temperature is - 5℃ < T < 0℃, the saturated pores inside the concrete are in an ice - water mixed state, and the upper limit of its effective modulus can be determined by linear interpolation based on the upper limits of the effective modulus at - 5℃ and 0℃.

[0108] Using the above - mentioned method for calculating the effective modulus of composite materials, if the assumption of absolutely dry concrete is adopted, the calculated effective modulus is its lower bound, which can be simplified and calculated by the following formula:

[0109]

[0110] S6. Calculate the upper and lower limits of the natural frequency of the concrete beam bridge based on the upper and lower limits of the effective modulus of concrete.

[0111] Substituting the calculated upper limit of the concrete effective modulus into the formula for calculating the natural frequency of a beam bridge can determine the upper limit of the natural frequency of the concrete beam bridge. Substituting the lower limit of the concrete effective modulus into the formula for calculating the natural frequency of the concrete beam bridge can also determine the lower limit of the natural frequency of the concrete beam bridge. This method of calculating the limits of the bridge's natural frequency based on the limits of the concrete effective modulus is not limited to the concrete beam bridges analyzed in step S4 but can be applied to the analysis of all concrete bridges.

[0112] Result verification example 1: Experiment on temperature correlation of concrete elastic modulus.

[0113] In order to verify the proposed calculation method and effective modulus limit of concrete, the following experiments were carried out.

[0114] Ordinary Portland cement (PO 42.5) was used as the cementitious material, supplemented with medium sand and manufactured sand (fineness modulus 2.7 and 2.9, respectively) as fine aggregate, crushed granite (particle size 5 to 20 mm) as coarse aggregate, and slag powder, fly ash, and a polycarboxylate superplasticizer were added. The water-binder ratio was 0.54. The detailed mix design is shown in Table 2. After mixing, the concrete was formed into cubic specimens with a side length of 100 mm. The specimens were cured for 28 days in a standard curing room at a temperature of 20 ± 2 degrees Celsius and a relative humidity of no less than 95%. After 28 days, compressive strength tests were conducted on some of the specimens, and the results showed that the compressive strength reached 33 MPa.

[0115] Table 2: Detailed concrete mix proportions.

[0116]

[0117] The remaining test blocks were divided into C1 experimental group and C2 control group. After drying naturally at room temperature for more than three weeks, strain gauges and thermocouples were installed. Figure 3 As shown in the figure, strain gauges were vertically affixed to the center of each side of the specimen and connected to the static strain acquisition system via pure copper wire. Temperature monitoring was performed using K-type thermocouples attached to the specimen surface. Specifically, a 50 mm deep hole with an internal K-type thermocouple was drilled into the other side of the C2 control specimen to monitor the internal temperature of the specimen during the test. All four surfaces of the specimen, except the upper and lower loading surfaces, were wrapped with low-thermal-conductivity insulation blankets to minimize heat exchange during low-temperature testing.

[0118] The two groups of test blocks were placed in a CNC freezer for low-temperature treatment, and the set temperatures were -12, -8, -5, 1, 5, 10, 15 and 22 (room temperature) degrees Celsius. Each temperature point was kept constant for more than 6 hours to ensure that the temperature inside and outside the specimens was uniform. Subsequently, a 30-ton electronic universal testing machine was used to compress and load the C1 group test blocks, and the C2 group test blocks were placed in exactly the same thermal environment as temperature control and temperature compensation test blocks. Before loading, a pre-pressure of 0.5MPa was manually applied to ensure that the C1 group test blocks were reliably positioned. Then, a loading-unloading cycle was performed between 0.5 and 10MPa. The loading and unloading rates were both 0.5MPa / s. Each loading and unloading was maintained for 36 seconds to obtain stable strain data. Four complete loading-unloading cycles were performed at each temperature point, and the average elastic modulus of the last three cycles was taken as the representative value at that temperature. The stress history of the experimental process is shown in the figure. Figure 4 shown.

[0119] After the dry state test was completed, all the specimens were soaked for more than one week, the strain gauges and thermocouples were installed again, and the above low temperature loading procedure was repeated. The test temperatures were adjusted to -20, -13, -5, 1, 5 and 10 degrees Celsius to evaluate the elastic modulus in the water-containing state. The experimental results are as follows Figure 3 As shown in the figure, the elastic modulus of concrete in the dry state decreases linearly with increasing temperature, while in the soaked state, the elastic modulus shows a sudden change between -5 and 0 degrees Celsius, indicating that pore water freezing has a significant effect on the stiffness of concrete.

[0120] When calculating the effective modulus and effective modulus limit of concrete, the elastic modulus of the dry test block at 20 degrees Celsius is taken as 25.5GPa, the porosity of the concrete is taken as 15%, the volume proportion of water after the test block is immersed is 7.5%, and the volume proportion of air is 7.5%. The calculation results are as follows Figure 5 However, the test results after immersion are still somewhat different from the upper limit of the effective modulus proposed in this example. This may be due to insufficient immersion of the test block and the unsaturated pores.

[0121] Result verification example 2: Swiss Z24 bridge application.

[0122] In order to verify the practicality and accuracy of the natural frequency limit of the concrete beam bridge proposed in this embodiment, the Swiss Z24 bridge was selected as a field engineering example for calculation and comparative analysis. Figure 6 Built in 1963, the 63.4-meter-long, three-span continuous box girder bridge features a 30-meter central main span, 14-meter side spans, and 2.7-meter cantilever sections at each end. The box girder is approximately 1.1 meters high, with a top deck width of 8.6 meters and a bottom deck width of 6.08 meters. Its simple and typical structure makes it suitable for environmental sensitivity dynamic characteristics analysis.

[0123] The bridge was previously equipped with a comprehensive structural health monitoring system, including multiple accelerometers and thermocouples, to continuously record the bridge's vibration response and temperature changes in natural environments. Reference [Peeters B, De Roeck G. One-year monitoring of the Z24-Bridge: environmental effects versus damage events. Earthquake Engineering & Structural Dynamics. 2001;30:149-71.] provides a detailed monitoring plan and data processing methods. Based on the existing monitoring data, scattered data on the bridge's first-order natural frequency as a function of temperature were extracted to validate the method used in this embodiment.

[0124] According to public data, the concrete material parameters of the Z24 bridge are set as follows: the elastic modulus in the dry state (20°C) is 32GPa, and the material density is 2650kg / m 3 , the porosity is about 20%. Based on the effective modulus prediction method proposed in this embodiment, under the same material parameters and ambient temperature conditions, the limit of the bridge's natural frequency was calculated and compared with the natural frequency obtained from on-site monitoring of the Z24 bridge, as shown in Figure 2. Figure 7 shown.

[0125] The comparison results show that when the temperature drops to the range of -5°C to 0°C, the natural frequency of the bridge shows a clear upward trend, with an increase of 7.7%. This trend is highly consistent with the upper frequency limit calculated by the prediction method of this embodiment. This example verification shows that the proposed theoretical model can not only accurately describe the trend of changes in the elastic modulus of concrete under freezing conditions, but also effectively reflect the sensitivity of the dynamic characteristics of the bridge to environmental factors (such as temperature and moisture content). The calculated frequency of the Z24 bridge is highly consistent with the measured value, proving that the method of this embodiment has good adaptability and promotion value in actual engineering.

[0126] The above is only a specific implementation of the present invention, but the design concept of the present invention is not limited to this. Any non-substantial changes to the present invention using this concept shall be deemed as an infringement of the protection scope of the present invention.

Claims

1. A method for calculating the effective modulus of concrete and the natural frequency of beam bridges considering water phase change, characterized in that: It includes the following steps: S1. Measure the porosity, moisture content and ice content of the concrete, and calculate the volume percentages of air, water and ice in the concrete based on the porosity, moisture content and ice content of the concrete; S2. Calculate the elastic modulus of the matrix material based on the porosity of the concrete, and determine the elastic moduli of each inclusion material; the matrix material is the solid part of the concrete; S3. Calculate the effective modulus of the concrete according to the volume percentages of air, water and ice in the concrete, the elastic modulus of the matrix material and the elastic moduli of each inclusion material; S4. Calculate the natural frequency of the concrete beam bridge according to the effective modulus of the concrete.

2. The method for calculating the effective modulus of concrete and the natural frequency of beam bridges considering water phase change according to claim 1 is characterized in that: The elastic modulus of the matrix material is expressed as: Among them, K s (T) represents the bulk modulus of the concrete matrix material; G s (T) represents the shear modulus of the concrete matrix material; φ represents the porosity of the concrete, ν represents the Poisson's ratio of the concrete, and E 20℃ It represents the elastic modulus of dry concrete at 20℃; T represents the concrete temperature.

3. The method for calculating the effective modulus of concrete and the natural frequency of beam bridges considering water phase change according to claim 1 is characterized in that: The elastic modulus of the inclusion material includes the bulk modulus and the shear modulus; among them, the value range of the bulk modulus of water is 2.0 - 2.3 GPa; the shear modulus of water is 0; the value range of the bulk modulus of ice is 8.6 - 9.0 GPa; the value range of the shear modulus of ice is 3.8 - 4.0 GPa; the bulk modulus of air is 0; the shear modulus of air is 0.

4. The method for calculating the effective modulus of concrete and the natural frequency of beam bridges considering water phase change according to claim 1, characterized in that: Calculating the effective modulus of the concrete according to the volume percentages of air, water and ice in the concrete, the elastic modulus of the matrix material and the elastic moduli of each inclusion material is specifically to calculate the effective modulus of the concrete based on the composite material mechanics theory and the classical Mori-Tanaka method, which is expressed as: P=(3K p ,2G p ) in, represents the effective modulus of concrete, The compressed expression of the fourth-order stiffness tensor of an isotropic material is (,) means simplifying high-dimensional tensors into two-dimensional diagonal matrix operations. represents the effective bulk modulus of concrete, represents the effective shear modulus of concrete; C s represents the elastic modulus of the concrete matrix material, C r represents the elastic modulus of the r-phase inclusion material, R represents the total R-phase inclusion material; P represents the Eshelby polarization tensor; K p G represents the volume positivity factor; p represents the shear polarization factor; c s represents the volume percentage of concrete matrix material, c r represents the volume percentage of the r-th phase inclusion; [] -1 Indicates inversion; G s (T) represents the shear modulus of the concrete matrix material; K s (T) represents the bulk modulus of the concrete matrix material.

5. The method for calculating the effective modulus of concrete and the natural frequency of beam bridges considering water phase change according to claim 1, characterized in that: It also includes: Calculate the effective Young's modulus of the concrete according to the effective modulus of the concrete, which is expressed as: in, represents the effective Young's modulus of concrete; represents the effective bulk modulus of concrete, represents the effective shear modulus of concrete.

6. The method for calculating the effective modulus of concrete and the natural frequency of beam bridges considering water phase change according to claim 1, characterized in that: After S4, it also includes: S5. Calculate the upper and lower limits of the effective modulus of the concrete; S6. Calculate the upper and lower limits of the natural frequency of the concrete beam bridge according to the upper and lower limits of the effective modulus of the concrete.

7. The method for calculating the effective modulus of concrete and the natural frequency of beam bridges considering water phase change according to claim 6, characterized in that: The calculation process of the upper limit of the effective modulus of the concrete is specifically as follows: When the concrete temperature is 0℃ ≤ f ≤ 50℃, take the effective modulus when the concrete pores are saturated with water as the upper limit of the effective modulus of the concrete; it is expressed as: Where, T represents the concrete temperature; φ represents the porosity of concrete; K1 represents the bulk modulus of water; k s (T) represents the bulk modulus of the concrete matrix material; G s (T) represents the shear modulus of the concrete matrix material; Indicates the upper limit of the effective bulk modulus of concrete; represents the upper limit of the effective shear modulus of concrete; G1 represents the shear modulus of water; When the concrete temperature is -50℃ ≤ T ≤ -5℃, take the effective modulus when the concrete pores are saturated with ice as the upper limit of the effective modulus of the concrete; it is expressed as: Among them, K2 represents the bulk modulus of ice; G2 represents the shear modulus of ice; When the concrete temperature is -5℃ < T < 0℃, the upper limit value of the effective bulk modulus of the concrete is obtained by linear interpolation of the two upper limit values of the effective bulk modulus of the concrete obtained when the concrete temperature is 0℃ ≤ T ≤ 50℃ and when the concrete temperature is -50℃ ≤ T ≤ -5℃; similarly, the upper limit value of the effective shear modulus of the concrete is obtained by linear interpolation of the two upper limit values of the effective shear modulus of the concrete obtained when the concrete temperature is 0℃ ≤ T ≤ 50℃ and when the concrete temperature is -50℃ ≤ T ≤ -5℃.

8. The method for calculating the effective modulus of concrete and the natural frequency of beam bridges considering water phase change according to claim 6, characterized in that: The calculation process of the lower limit of the effective modulus of the concrete is specifically as follows: When the concrete temperature is -50℃ ≤ T ≤ 50℃, take the effective modulus when the concrete pores are saturated with air as the lower limit of the effective modulus of the concrete, which is expressed as: in, represents the lower limit of the effective bulk modulus of concrete; K3 represents the bulk modulus of air; T represents the concrete temperature; φ represents the porosity of concrete; K s (T) represents the bulk modulus of the concrete matrix material; G s (T) represents the shear modulus of the concrete matrix material; It represents the lower limit of the effective shear modulus of concrete; G3 represents the shear modulus of air.

9. The method for calculating the effective modulus of concrete and the natural frequency of beam bridges considering water phase change according to claim 6, characterized in that: Substitute the lower limit of the effective modulus of concrete into the calculation formula of the natural frequency of the beam bridge to obtain the lower limit of the natural frequency of the concrete beam bridge; substitute the upper limit of the effective modulus of concrete into the calculation formula of the natural frequency of the beam bridge to obtain the upper limit of the natural frequency of the concrete beam bridge.

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