Four-element model of concrete modulus and creep under long-term cyclic loading
By using a four-element model and a phased calibration method, the complex relationship between concrete modulus and creep under long-cycle cyclic loading was solved, enabling accurate simulation and prediction of concrete creep behavior, which is applicable to the time-dependent deformation analysis of large structures.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- HUANENG LANCANG RIVER HYDROPOWER CO LTD
- Filing Date
- 2024-12-20
- Publication Date
- 2026-07-24
AI Technical Summary
Existing models cannot effectively describe the complex relationship between concrete modulus and creep under long-term cyclic loading, and long-term experimental or measured data for specific objects are scarce, making parameter calibration difficult.
A four-element model was adopted, including an instantaneous elastic deformation element, a recoverable creep viscoelastic element, an unrecoverable compressive creep element, and an unrecoverable tensile creep element. Through phased calibration analysis, the model parameters were gradually determined using experimental data from a large sample set and multiple small sample sets.
It enables accurate description and prediction of concrete modulus and creep, is applicable to different engineering objects, improves the simulation accuracy and applicability of the model, and provides a scientific basis for the analysis of the aging deformation of large structures.
Smart Images
Figure CN119849136B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of engineering structural analysis, and specifically discloses a four-element model of concrete modulus and creep under long-cycle cyclic loading and a method for determining its parameters. Background Technology
[0002] Creep in concrete refers to the time-dependent deformation of concrete under long-term loading. It typically manifests as a gradual increase in strain over time, followed by a gradual decrease in the increment and eventual stabilization. Macroscopically, it represents the structural deformation increasing year by year. Creep has a profound impact on the stability and durability of large structures such as dams. Excessive creep can cause structural deformation to exceed the design safety range, leading to serious problems such as cracks and settlement. In many cases, creep is also a harmless deformation, sometimes even helping to neutralize localized stress concentrations. When significant time-dependent deformation occurs in engineering projects, concerns arise regarding the presence of plastic deformation that could lead to progressive failure. Therefore, it is crucial to accurately separate creep and other factors from time-dependent deformation to precisely assess the structural safety.
[0003] Large-volume concrete structures such as dams often use low-heat cement, which has a low hydration rate. The concrete strength and modulus continue to increase after pouring, extending from the construction period to the initial operational phase for a considerable time. Combined with the creep properties of concrete under long-term cyclic loading (the main loads, such as reservoir water and temperature, exhibit interannual quasi-periodic variations), the constitutive relationship of concrete in the initial operational phase exhibits complex time-varying nonlinear characteristics. Current models only focus on the increase in strength or modulus with age, or are limited to creep models under continuous loading. No effective physical and mechanical model can fully and accurately express the aforementioned constitutive relationship. Therefore, a new "modulus growth + creep" model is urgently needed that can encompass the increase in concrete modulus and is applicable to any loading and unloading process.
[0004] Concrete of different uses and types exhibits significant differences in modulus growth and creep characteristics, with even different specimens showing some dispersion under the same conditions. Obtaining these distinct characteristics requires long-term, multi-sample experimental or measured data for specific applications to calibrate the parameters. However, the high cost of acquiring long-term experimental or measured data for a single object leads to a scarcity of data samples, making it difficult to calibrate the more than 10 parameters of the model proposed in this invention. Therefore, a phased calibration method is proposed. By screening and fully utilizing existing "modulus growth / creep" models and experimental results with similar backgrounds, and based on calibration analysis of these large sample sets and multiple sample sets, the parameter range is gradually defined and narrowed. Parameters are reduced by consolidation, and then combined with experimental data and operational measured data for specific applications, the concrete modulus and creep model parameters for the specific object are finally calibrated. Summary of the Invention
[0005] The purpose of this invention is to provide a four-element model of concrete modulus and creep under long-cycle cyclic loading and a method for determining its parameters. This model integrates characteristics such as modulus growth, instantaneous elasticity, viscoelasticity (recoverable creep), and non-recoverable creep, and can effectively describe the creep behavior of large concrete structures under complex loading and unloading conditions such as long-cycle loading.
[0006] To achieve the above objectives, the technical solution provided by this invention includes the following:
[0007] The technical solution adopted by this invention to solve the technical problem is as follows: a four-element model and parameter determination method for concrete modulus and creep under long-cycle cyclic loading. This model quantitatively and effectively expresses the complex changes in concrete modulus and creep by connecting instantaneous elastic deformation elements with time-dependent stiffness, recoverable creep viscoelastic elements, irrecoverable compressive creep elements, and irrecoverable tensile creep elements in series. The method employs a progressive calibration analysis approach, starting with a large sample set with similar backgrounds, then moving to multiple small sample sets from similar projects, and finally to a small sample set for a specific object, to overcome the difficulty of limited available sample size for the specific object itself.
[0008] The four-element model of concrete modulus and creep under long-cycle cyclic loading consists of an instantaneous elastic deformation element, a recoverable creep viscoelastic element, an unrecoverable compressive creep element, and an unrecoverable tensile creep element connected in series to form an overall modulus and creep model.
[0009] The instantaneous elastic deformation element is composed of a spring whose stiffness increases over time; the recoverable creep viscoelastic element is composed of a spring with increasing stiffness and a damper connected in parallel; the unrecoverable compressive creep element is composed of a spring with increasing stiffness, a damper and a friction device connected in parallel; the unrecoverable tensile creep element is composed of a spring with increasing stiffness, a damper and a friction device connected in parallel.
[0010] The four-element model is as follows:
[0011] (1) The instantaneous elastic deformation element consists of a stiffness that increases with time from 0. # It consists of springs;
[0012] 0 # The spring modulus growth model is as follows:
[0013]
[0014] In the formula, E e 0 # The instantaneous elastic modulus of the spring, where t0 is the design age and E0 is 0.# The instantaneous elastic modulus of the spring at its design age, c e 0 # Undetermined parameters in the spring modulus growth model;
[0015] Instantaneous strain increment Δε of an instantaneous elastic deformation element e Calculated using the following formula:
[0016]
[0017] Δσ is the stress increment of the entire assembly consisting of four elements, and the stress increment of each element is Δσ.
[0018] (2) A recoverable creep viscoelastic element consists of a stiffness that can increase by 1 # Spring and a 1 # The dampers are connected in parallel;
[0019] 1 # The spring modulus growth model is as follows:
[0020]
[0021] In the formula, E1 is 1 # The elastic modulus of the spring, t0 is the design age, E 1_0 1 # The elastic modulus of the spring at its design age, c1 is 1. # Undetermined parameters in the spring modulus growth model;
[0022] The stress σ of a recoverable creep viscoelastic element is 1 # Spring and 1 # The relationship between the damper's load σ and the strain ε1 of this element is as follows:
[0023]
[0024] Iterate over time steps, assuming that σ and ε1 remain approximately constant within each time step, and derive the strain rate from the above equation. Therefore, the strain increment of this element within this time step is:
[0025]
[0026] In the formula, η1 is 1 # Damping parameters of the damper;
[0027] (3) The non-recoverable compressive creep element is composed of a No. 2 spring with increasing stiffness, a No. 2 damper and a friction device connected in parallel;
[0028] 2 # The spring modulus growth model is as follows:
[0029]
[0030] In the formula, E2 is 2 # The elastic modulus of the spring, t0 is the design age, E 2_0 2 # The elastic modulus of the spring at its design age, c2 is 2. # Undetermined parameters in the spring modulus growth model;
[0031] As compressive strain increases, the stress σ of the non-recoverable compressive creep element changes from 2 # Spring and 2 # The relationship between the damper's load σ and the strain ε2 of this element is as follows:
[0032]
[0033] Iterate over time steps, assuming that σ and ε² remain approximately constant within each time step, and derive the strain rate from the above equation. Therefore, the increment of compressive strain of this element within this time step is obtained as follows:
[0034]
[0035] In the formula, η2 is 2 # Damping parameters of the damper;
[0036] When the compressive strain attempts to decrease, the friction completely prevents the compressive strain from decreasing, so that the compressive strain of this element can only increase and not decrease, and the compressive strain is irreversible;
[0037] (4) Irreversible tensile creep element consists of a stiffness that can increase by 3 # Spring, a 3 # It consists of a damper and a friction device connected in parallel;
[0038] 3 # The spring modulus growth model is as follows:
[0039]
[0040] In the formula, E3 is 3 # The elastic modulus of the spring, t0 is the design age, E 3_0 3 # The elastic modulus of the spring at its design age, c3 is 3. # Undetermined parameters in the spring modulus growth model;
[0041] As tensile strain increases, the tensile stress σ of the non-recoverable tensile creep element changes from 3 # Spring and 3 # The relationship between the damper's share of strain, σ, and the tensile strain ε3 of this element is as follows:
[0042]
[0043] The iteration is performed step by step, and σ and ε3 are approximately considered constant within each time step. The strain rate is then derived from the above formula. Therefore, the increase in tensile strain of this element within this time step as the tensile strain increases is:
[0044]
[0045] In the formula, η3 is 3 # Damping parameters of the damper;
[0046] When the tensile strain attempts to decrease, the friction completely prevents the tensile strain from decreasing, causing the tensile strain of this element to only increase and never decrease, and the tensile strain is irreversible.
[0047] The calibration method for the relevant parameters of the four-element model is carried out in a classified and phased manner, specifically as follows:
[0048] For instantaneous modulus and its growth parameters, the sample data are relatively reliable, the sample size is large, and the coupling with other factors is weak. Therefore, separate calibration should be carried out for these parameters first.
[0049] The remaining parameters are calibrated in stages. Initial calibration is performed using creep models with similar backgrounds and samples with abundant creep experimental data to define the parameter range and examine whether parameters can be reduced in number, such as c. e The parameters c1, c2, and c3 are reduced to 1-2 parameters, and η1, η2, and η3 are reduced to 1-2 parameters. Then, the parameters are further calibrated and narrowed down by combining the short-term and long-term experimental data and creep experimental data of the application object. Finally, the concrete modulus and creep model parameters are determined by using the measured data of the application object during its operation period.
[0050] In this invention, the four springs in the four components have different initial stiffnesses; the stiffness growth model uses the same form but different model parameters, and determines the same type of parameters (e.g., c) in different springs based on a stepwise calibration analysis from a large sample to multiple small samples. eThe values of c1, c2, and c3 are determined by the rules, and similar parameters are correlated to reduce the number of parameters. The three dampers in the four components can use different parameters, or they can be correlated. For specific application projects, model parameter calibration is conducted in three stages: Stage 1: Existing experimental results and creep models with similar background conditions are selected. Using large samples and multiple small samples with different characteristics, calibration samples and test samples are divided to conduct preliminary parameter calibration analysis, define the reasonable range of parameter values, study the sensitivity of different parameters and their correlation, and connect and shrink parameters to meet the needs of subsequent small sample calibration, and verify the simulation accuracy of the model after connection and shrinkage; Stage 2: Using short-age and long-age experiments on concrete specimens of the specific application object and creep experiment results, the parameters are classified and calibrated based on the characteristics of abundant short-age sample data and scarce long-age sample data, further defining and narrowing the range of parameter values and verifying the simulation effect of the model; Stage 3: Using measured data of large concrete structures such as dams, as many usable data samples as possible are extracted, and finally the parameters of the concrete modulus and creep model are calibrated to verify the model's simulation effect and prediction ability on the measured data.
[0051] This invention, by introducing instantaneous elastic deformation elements with stiffness increasing over time, recoverable creep viscoelastic elements, unrecoverable compressive creep elements, and unrecoverable tensile creep elements, considers the modulus growth, elasticity, viscoelasticity, and creep deformation of concrete under different time scales and arbitrary loading and unloading conditions. It overcomes the shortcomings of existing models with relatively simple functions and can effectively simulate the time-dependent deformation law of structures under long-term loads and predict their development and change trends.
[0052] The four-element model of this invention can accurately reflect the growth of concrete modulus and creep behavior. The model has a large number of parameters and can also extend the one-dimensional model to the spatial stress state, making the model have good applicability and ensuring the simulation accuracy of the model for different engineering objects.
[0053] This invention provides a calibration method for the parameters of each component, enabling effective calibration and verification using long-term experimental and measurement results with very limited data for specific applications. By leveraging data from similar backgrounds and projects, calibration analysis is performed in stages, moving from large sample sets to multiple smaller sample sets. This gradually reduces the number of parameters and narrows the range of parameter values, ultimately achieving the goal of calibrating complex multi-parameter models with good simulation results using a small sample set of data from a specific object.
[0054] This invention, by accurately simulating the creep behavior of concrete, can provide a scientific basis for the analysis of time-dependent deformation of large structures such as dams, and better predict and assess the impact of time-dependent deformation on structural safety. Attached Figure Description
[0055] Figure 1 This is a schematic diagram of a four-element creep model of concrete under long-term cyclic loading.
[0056] Figure 2 This is a diagram illustrating the stress-strain variation process of concrete under annual cyclic loading.
[0057] Figure 3 The diagram illustrates the process of increasing strain and rope-like change in concrete under annual cyclic loading. Detailed Implementation
[0058] The present invention will now be described in further detail through specific embodiments. However, this should not be construed as limiting the scope of the above-described subject matter of the present invention to the following embodiments; all technologies implemented based on the content of the present invention fall within the scope of the present invention.
[0059] Example 1: A four-element model of concrete modulus and creep under long-term cyclic loading. The four-element model consists of an instantaneous elastic deformation element, a recoverable creep viscoelastic element, an unrecoverable compressive creep element, and an unrecoverable tensile creep element connected in series to form an overall modulus and creep model.
[0060] Each element describes the deformation characteristics of concrete under arbitrary loading and unloading conditions based on different physical mechanisms.
[0061] The calibration method for the four-element model and its related parameters is as follows:
[0062] (1) Instantaneous elastic deformation element
[0063] The instantaneous elastic deformation element adopts a spring model and is 0. # Spring with elastic modulus E e Stiffness increases with time and is used to describe the elastic modulus of concrete under instantaneous load; E e The initial value is determined by the instantaneous elastic modulus E0 at the design age, and the growth model is used to reflect the characteristic of the concrete modulus increasing with age.
[0064] 0 # The spring modulus growth model is as follows:
[0065]
[0066] In the formula, E e 0 # The instantaneous elastic modulus of the spring, where t0 is the design age and E0 is 0. # The instantaneous elastic modulus of the spring at its design age, c e 0 # Undetermined parameters in the spring modulus growth model;
[0067] For large-scale concrete projects, the instantaneous elastic modulus E0 at the design age is obtained directly from experimental results. Then, all parameters in the component model are derived by using the instantaneous modulus measurements at one or more other ages.
[0068] For a certain project, the instantaneous elastic modulus of concrete at the 90-day design age is 30 GPa, according to multi-sample statistics. The instantaneous modulus at 365 days increases by 10%. Therefore, E0 = 30, t0 = 90, and t = 365. Substituting these values into the growth model, we get:
[0069]
[0070] Find c e =0.071424;
[0071] The concrete modulus growth model for this project is as follows:
[0072]
[0073] Because the sample data for instantaneous modulus is reliable, the sample size is large, and the coupling with other factors is weak, this type of parameter was calibrated separately in advance.
[0074] Since the four elements are connected in series, under the stress increment Δσ, the stress increment of each element in the entire assembly is Δσ, and the instantaneous strain increment Δε of the instantaneous elastic deformation element is... e Calculated using the following formula:
[0075]
[0076] The above equation is the strain correction formula for the instantaneous elastic deformation element under the stress increment Δσ when iterating by time step.
[0077] (2) Recoverable creep viscoelastic element
[0078] The recoverable creep viscoelastic element consists of 1 # Spring and 1 # The dampers are connected in parallel, and their stiffness increases over time.
[0079] 1 # The spring modulus growth model is as follows:
[0080]
[0081] In the formula, E1 is 1 # The elastic modulus of the spring, t0 is the design age, E 1_0 1 # The elastic modulus of the spring at its design age, c1 is 1. # Undetermined parameters in the spring modulus growth model.
[0082] The stress σ of the recoverable creep viscoelastic element is shared by the spring and the damper. The relationship between σ and the strain ε1 of the element is as follows:
[0083]
[0084] Iterate over time steps, assuming that σ and ε1 remain approximately constant within each time step, and derive the strain rate from the above equation. Therefore, the strain increment of this element within this time step is:
[0085]
[0086] In the formula, η1 is 1 # From the perspective of parameter calibration, the damping parameters of this element have a strong coupling relationship with the non-recoverable compressive creep element, and will be combined with the next element for parameter calibration.
[0087] (3) Irreversible compression creep element
[0088] Non-recoverable compressive creep element consists of 2 # Spring, 2 # The damper and a one-way friction device are connected in parallel, and the stiffness increases with time.
[0089] 2 # The spring modulus growth model is as follows:
[0090]
[0091] In the formula, E2 is 2 # The elastic modulus of the spring, t0 is the design age, E 2_0 2 # The elastic modulus of the spring at its design age, c2 is 2. # Undetermined parameters in the spring modulus growth model;
[0092] When the compressive strain increases, the one-way friction device does not function; when the compressive strain attempts to decrease, the one-way friction device of this element completely prevents the recovery of the compressive creep of this element, making the compressive creep of this element irreversible.
[0093] As compressive strain increases, the stress σ of the non-recoverable compressive creep element changes from 2... # Spring and 2 # The relationship between the damper's load σ and the strain ε2 of this element is as follows:
[0094]
[0095] The iteration is performed step by step, and σ and ε2 are approximately considered constant within each time step. The strain rate is then derived from the above equation. Therefore, the increment of compressive strain of this element within this time step is obtained as follows:
[0096]
[0097] In the formula, η2 is 2 # Damping parameters of the damper;
[0098] When the compressive stress attempts to decrease, the compressive strain increment of this element is zero.
[0099] In the aforementioned recoverable creep viscoelastic elements and non-recoverable compressive creep elements, the concrete design age t0 is a determined parameter for a specific application; therefore, the parameters to be determined are:
[0100] E 1_0 E 2_0 There are a total of 6 parameters: c1, c2, η1, and η2.
[0101] The above six parameters were determined through calibration analysis on large and multiple sample sets, thereby reducing the number of parameters and enabling categorized and phased calibration. Specifically:
[0102] By collecting creep models and experimental data with similar backgrounds, and based on the analysis and comparison of calibration results, the differences and similarities of spring parameters of several components in the model are analyzed, and the corresponding parameters are connected and condensed; according to the actual situation of specific cases, such as letting c1 = c2 = c e At this point, the stiffness growth patterns of the three springs are exactly the same, and the relationship between the three springs is expressed using two multiple parameters β1 and β2: E1(t) = β1E e (t), E2(t)=β2E e (t), which will have 4 parameters E 1_0 E 2_0 c1 and c2 are merged into two parameters β1 and β2, reducing the number of parameters.
[0103] Setting 0 # Spring, 1 # Spring, 2 # The stiffness growth pattern of the springs is exactly the same, using the calibrated c. e Given the parameters, we have c1 = c2 = c e =0.071424, using multiple parameters β1 and β2 to express the relationship between the three springs:
[0104] E 1_0 =β1E0;
[0105] E1(t)=β1E e (t);
[0106] E 2_0 =β2E0;
[0107] E2(t)=β2E e (t);
[0108] Furthermore, for specific applications, the following concrete specimen creep tests are conducted:
[0109] A compressive stress of 9 MPa was applied for one year starting from the design age. The compressive strains measured in the middle and at the end of the first year were 328 × 10⁻⁶. -6 and 365×10 -6 Then, at the beginning of the second year, the load was completely unloaded. After another six months and one year, i.e., in the middle and at the end of the second year, the residual compressive strains were measured to be 113 × 10⁻⁶. -6 and 100×10 -6 Based on these four strain measurements, conventional inversion analysis software, such as parameter inversion algorithms based on least squares and optimization methods, or based on some open-source MATLAB inversion analysis code, was used to perform parameter inversion in this case. The calibrated parameters were found to be the multiple parameter β1≈β2≈2 and the damping parameter η1≈η2≈100 (GPa·year).
[0110] The parameters of the concrete modulus creep model can also be calibrated using neural network methods and Gaussian regression models. The specific process of calibrating mechanical component parameters using neural network methods includes multiple steps such as data collection and preprocessing, model design, training and optimization, and evaluation and verification. The core of this method is to learn the patterns of the parameters to be determined from experimental or simulation data samples through a neural network model, thereby achieving accurate simulation of the mechanical component performance against the sample characteristics. The specific process of calibrating mechanical component parameters using Gaussian regression models includes steps such as data preparation, model building, training and optimization, and prediction and evaluation. Gaussian regression is a nonparametric regression method based on Bayesian theory, suitable for solving regression problems with complex nonlinear relationships, especially suitable for prediction of small sample data. It describes the relationship between input and output by constructing a joint Gaussian distribution model, and can provide uncertainty estimation during prediction, making it of significant application value in mechanical parameter calibration.
[0111] Based on the above parameter calibration, the strain increment formula for the recoverable creep viscoelastic element applicable to this embodiment over one iteration time step is:
[0112]
[0113] In the formula, 365 is the unit conversion factor for converting "GPa·year" to "GPa·day".
[0114] For the non-recoverable compressive creep element applicable to this embodiment, when the compressive strain attempts to decrease, the compressive strain increment in one iteration time step is zero; when the compressive strain increases, the formula for the compressive strain increment in one iteration time step is:
[0115]
[0116] The aforementioned classification and phased calibration first defines the parameter range and reduces the number of parameters through calibration analysis using large and multiple sample sets in the early stages, creating conditions for parameter calibration under small sample set conditions for specific objects. The above calibration assumes an ideal scenario, i.e., that the experimental specimens are highly representative and the four strain measurements are highly accurate; in this case, at least four samples are needed to calibrate four parameters. Based on the aforementioned sample representativeness and the dispersion of experimental results, to calibrate four parameters, more sample data should be collected as much as possible, and strain values at more time points should be acquired during the experiment. Concrete experiments typically use a group of three specimens, performing the same experiment, and taking the average of the three as the adopted value. Measurements with significant deviations from the average are discarded. To improve sample diversity, different load values and loading / unloading time points can be set within the same group of specimens, and specimen data with significant deviations from the model can be discarded during the calibration process.
[0117] If the available sample size is small in this stage, it is necessary to utilize the calibration analysis of the large sample set in the previous stage to further reduce the number of parameters in this stage. For example, β1 = β2, η1 = η2, or a certain proportional relationship can be set to further reduce the 4 parameters to 2 parameters. At the same time, based on the reasonable range of parameter values obtained from the calibration analysis in the previous stage, it can be ensured that the calibration based on the small sample set in this stage will not have a large deviation.
[0118] Where possible, utilize available collected data and concrete experimental results from specific applications to calibrate model parameters. Then, apply the model to large concrete structures in design or actual operation to predict aging deformation caused by effects such as creep, or to reveal the underlying mechanisms of aging deformation in existing buildings. If the sample data relied upon for initial calibration is poor or the sample size is insufficient, resulting in unreliable model parameters, inversion analysis can be performed using field measurement data from specific applications to correct the model parameters.
[0119] (4) Irreversible tensile creep element
[0120] The non-recoverable tensile creep element is similar to the non-recoverable compressive creep element, except that the unidirectional friction device is reversed. When the tensile strain increases, the unidirectional friction device has no effect; when the tensile strain attempts to decrease, the unidirectional friction device of this element completely prevents the decrease of the tensile strain, making the tensile creep of this element non-recoverable.
[0121] Non-recoverable tensile creep element consists of 3# Spring, 3 # The damper and a one-way friction device are connected in parallel, and the stiffness increases with time.
[0122] 3 # The spring modulus growth model is as follows:
[0123]
[0124] In the formula, E3 is 3 # The elastic modulus of the spring, t0 is the design age, E 3_0 3 # The elastic modulus of the spring at its design age, c3 is 3. # Undetermined parameters in the spring modulus growth model;
[0125] When the tensile strain increases, the unidirectional friction device does not function; when the tensile strain attempts to decrease, the unidirectional friction device completely prevents the tensile strain of this element from decreasing.
[0126] As tensile strain increases, the tensile stress σ of the non-recoverable tensile creep element changes from 3 # Spring and 3 # The relationship between the damper's share of strain, σ, and the tensile strain ε3 of this element is as follows:
[0127]
[0128] The iteration is performed step by step, and σ and ε3 are approximately considered constant within each time step. The strain rate is then derived from the above formula. Therefore, the increase in tensile strain of this element within this time step as the tensile strain increases is:
[0129]
[0130] In the formula, η3 is 3 # Damping parameters of the damper.
[0131] When the tensile stress attempts to decrease, the tensile strain increment of this element is zero.
[0132] The tensile strength of concrete is much lower than its compressive strength, and concrete in concrete structures is primarily subjected to compression. It is generally believed that when the tensile stress is less than the design tensile strength, the deformation characteristics under tension are similar to those under low-stress compression. Therefore, the model parameters of an irreversible compressive creep element can be borrowed as the model parameters for this element. In the absence of available tensile creep calibration samples, the parameters of a compressive creep element can be directly borrowed, letting E... 3_0 =E 2_0 c3 = c2, β3 = β2, η3 = η2, thus obtaining the formula for the tensile strain increment of the unrecoverable tensile creep element in one iteration time step in this embodiment:
[0133]
[0134] If the parameters of the compression creep element are not directly used, a "tension-unloading" creep test can be carried out to measure the strain value. After the parameters of the first three elements are calibrated, the parameters of this element can be calibrated.
[0135] The tensile-unloading creep test of concrete specimens is as follows: A tensile stress of 1 MPa is applied from the design age for one year. The tensile strains measured in the middle and at the end of the first year are assumed to be 37 × 10⁻⁶. -6 and 42×10 -6 Then, at the beginning of the second year, the load was completely unloaded. After another six months and one year, that is, in the middle and at the end of the second year, the residual tensile strain was assumed to be 13 × 10⁻⁶. -6 and 12×10 -6 Based on these four strain measurements, and assuming the same modulus growth parameter as that used for the instantaneous elastic deformation element, that is, E... 3_0 C3 and C2 are merged into a single parameter β3. Using conventional inversion analysis software, the spring parameter β3≈2 and the damping parameter η3≈90 (GPa·year) of this element can be calibrated. Therefore, the strain increment formula for the unrecoverable tensile creep element in this embodiment over one iteration time step is:
[0136]
[0137] Based on the aforementioned model with calibrated parameters, the stress-strain variation process of concrete under annual cyclic loading (0–10 MPa varying according to a sine function) is shown in the attached figure. The stress-strain variation curves exhibit a rope-like variation pattern with deformation increasing year by year, revealing the underlying mechanism of the peculiar aging deformation phenomenon in some large-scale projects under long-term cyclic loading.
Claims
1. A four-element model for concrete modulus and creep under long-cycle cyclic loading, characterized in that... The four-element model consists of an instantaneous elastic deformation element, a recoverable creep viscoelastic element, an unrecoverable compressive creep element, and an unrecoverable tensile creep element connected in series to form an overall modulus and creep model. The instantaneous elastic deformation element consists of a spring whose stiffness increases over time; the recoverable creep viscoelastic element consists of a spring with increasing stiffness and a damper connected in parallel; the non-recoverable compressive creep element consists of a spring with increasing stiffness, a damper, and a one-way friction device connected in parallel. When the compressive strain attempts to decrease, the one-way friction device completely prevents the decrease in compressive strain, so that the compressive strain of this element only increases and does not decrease, and the compressive strain is irrecoverable; the non-recoverable tensile creep element consists of a spring with increasing stiffness, a damper, and a one-way friction device connected in parallel. When the tensile strain attempts to decrease, the one-way friction device completely prevents the decrease in tensile strain, so that the tensile strain of this element only increases and does not decrease, and the tensile strain is irrecoverable.
2. The four-element model for concrete modulus and creep under long-cycle cyclic loading as described in claim 1, characterized in that... The four-element model is as follows: (1) The instantaneous elastic deformation element consists of a stiffness that increases with time of 0. # It consists of springs; 0 # The spring modulus growth model is as follows: ； In the formula, 0 # The instantaneous elastic modulus of the spring. For the design age, 0 # The instantaneous elastic modulus of the spring at its design age. 0 # Undetermined parameters in the spring modulus growth model; Instantaneous strain increment of an instantaneous elastic deformation element Calculated using the following formula: ； Let represent the stress increment of the entire assembly consisting of four elements, where the stress increment of each element is . ; (2) A recoverable creep viscoelastic element consists of a stiffness that can increase by 1 # Spring and a 1 # The dampers are connected in parallel; 1 # The spring modulus growth model is as follows: ； In the formula, 1 # The elastic modulus of a spring. For the design age, 1 # The elastic modulus of a spring at its design age. 1 # Undetermined parameters in the spring modulus growth model; Stress recovery of creep viscoelastic elements From 1 # Spring and 1 # Damper sharing, With strain of this element The relationship is as follows: ； Iterate by time step, within a time step and Assuming it remains constant, the strain rate can be derived from the above formula. Therefore, the strain increment of this element within this time step is: ； In the formula, 1 # Damping parameters of the damper; (3) The non-recoverable compressive creep element is composed of a No. 2 spring with increasing stiffness, a No. 2 damper and a one-way friction device connected in parallel; 2 # The spring modulus growth model is as follows: ； In the formula, 2 # The elastic modulus of a spring. For the design age, 2 # The elastic modulus of a spring at its design age. 2 # Undetermined parameters in the spring modulus growth model; As compressive strain increases, the stress in an irreversible compressive creep element increases. From 2 # Spring and 2 # Damper sharing, With strain of this element The relationship is as follows: ； Iterate by time step, within a time step and Assuming the compressive strain remains constant, when the compressive strain increases, the increment of the compressive strain of this element within that time step is: ； In the formula, 2 # Damping parameters of the damper; (4) The non-recoverable tensile creep element consists of a stiffness that can be increased by 3 # Spring, a 3 # It consists of a damper and a one-way friction device connected in parallel; 3 # The spring modulus growth model is as follows: ； In the formula, 3 # The elastic modulus of a spring. For the design age, 3 # The elastic modulus of a spring at its design age. 3 # Undetermined parameters in the spring modulus growth model; When tensile strain increases, the tensile stress in an unrecoverable tension creep element... By 3 # Spring and 3 # Damper sharing, With the tensile strain of this component The relationship is as follows: ； Iterate by time step, within a time step and Assuming it remains constant, the strain rate can be derived from the above formula. Therefore, the increase in tensile strain of this element within this time step as the tensile strain increases is: ； In the formula, 3 # Damping parameters of the damper; The calibration method for the relevant parameters of the four-element model is carried out in a classified and phased manner, specifically as follows: For instantaneous modulus and its growth parameters, the sample data are relatively reliable, the sample size is large, and the coupling with other factors is weak. Therefore, separate calibration should be carried out for these parameters first. The remaining parameters were calibrated in stages. Initial calibration was performed using creep models with similar backgrounds and samples with abundant creep experimental data to define the parameter range and examine whether parameters could be reduced in number. Reduce to 1-2 parameters. The parameters are reduced to 1-2; then, combined with the short-term and long-term experimental data of the application object and the creep experimental data, further calibration is carried out and the parameter range is narrowed; finally, the concrete modulus and creep model parameters are determined by using the measured data of the application object during its operation period.