CN116976166A - Fatigue performance analysis method for double-hole beam based on fatigue damage constitutive model - Google Patents
Fatigue performance analysis method for double-hole beam based on fatigue damage constitutive model Download PDFInfo
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Abstract
The invention relates to a fatigue performance analysis method of a double-hole beam based on a fatigue damage constitutive model, which comprises the following steps: respectively establishing a concrete model and a steel bar model of the double-hole beam according to the design scheme of the double-hole beam; carrying out a loading test on the prepared test piece of the double-hole beam to obtain test data; building a concrete static constitutive model, a concrete fatigue constitutive model, a steel bar static constitutive model and a steel bar fatigue constitutive model based on test data; setting a fatigue failure criterion; and carrying out loading simulation test on the established concrete model and the reinforced bar model, and carrying out simulation calculation on the performance of the double-hole beam after fatigue by utilizing the established constitutive model to obtain a simulation calculation result. Based on a test, the invention introduces a fatigue constitutive model of concrete and steel bars, and uses finite element software to carry out numerical simulation on the hole-opened beam so as to predict the performance of the hole-opened beam after fatigue loading.
Description
Technical Field
The invention relates to the technical field of fatigue performance analysis of reinforced concrete structures, in particular to a fatigue performance analysis method of a double-hole beam based on a fatigue damage constitutive model.
Background
The opening in the beam inevitably has a negative effect on the structure itself: on the one hand, the existence of the holes enables the section of the beam not to keep continuous, the rigidity of the section changes, obvious stress concentration phenomenon occurs, and the component is damaged in advance. On the other hand, the ductility and the energy consumption capacity of the large-opening frame beam are reduced because the beam section is weakened to different degrees, and the whole synergistic earthquake-resistant effect is not beneficial to be exerted in a whole structure system.
Fatigue failure of concrete structures tends to be more abrupt and dangerous than static failure. With the development and the deep development of static force research of a concrete structure, the understanding and the research of structural static load damage are quite mature, and the research of double-hole girder structure failure caused by fatigue load is not mature.
The most accurate and visual research data can be obtained by carrying out the fatigue test generally, but the fatigue test is time-consuming and labor-consuming.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, provides a double-hole beam fatigue performance analysis method based on a fatigue damage constitutive model, and solves the problems that the existing fatigue load-induced hole beam structure failure research is still immature, and the fatigue test is time-consuming and labor-consuming.
The technical scheme for achieving the purpose is as follows:
the invention provides a fatigue performance analysis method of a double-hole beam based on a fatigue damage constitutive model, which comprises the following steps:
respectively establishing a concrete model and a steel bar model of the double-hole beam according to the design scheme of the double-hole beam;
carrying out a loading test on the prepared test piece of the double-hole beam to obtain test data;
establishing a concrete static constitutive model based on test data;
analyzing a degradation rule of the double-hole beam test piece after fatigue based on test data and establishing a concrete fatigue constitutive model;
establishing a static constitutive model of the reinforcing steel bar and a fatigue constitutive model of the reinforcing steel bar;
setting a fatigue failure criterion;
carrying out loading simulation test on the established concrete model and the reinforced bar model, and carrying out simulation calculation on the performance of the double-hole beam after fatigue by utilizing the established concrete static constitutive model, the established concrete fatigue constitutive model, the reinforced bar static constitutive model and the reinforced bar fatigue constitutive model to obtain a simulation calculation result;
and judging whether the simulation calculation result reaches a set fatigue failure criterion, if not, updating parameters of the concrete fatigue constitutive model and the reinforcing steel bar fatigue constitutive model by using the simulation calculation result, and continuously carrying out a loading simulation test to obtain a simulation calculation result corresponding to the fatigue times.
According to the fatigue performance analysis method for the double-hole beam based on the fatigue damage constitutive model, based on the test, the fatigue constitutive model of concrete and steel bars is introduced, and numerical simulation is carried out on the hole beam by using finite element software so as to predict the performance of the hole beam after fatigue loading.
The invention relates to a further improvement of a double-hole girder fatigue performance analysis method based on a fatigue damage constitutive model, which is characterized in that the established concrete static constitutive model comprises the following steps:
stress strain curve equation under uniaxial stress of concrete:
σ＝(1-d c )E c epsilon-type of 1-1,
in the formulae 1-1 to 1-5, a c For the concrete pressure drop section parameter epsilon c,r The peak strain corresponding to the concrete when the uniaxial pressure reaches a representative value is obtained; f (f) c,r Is the representative value of the uniaxial compressive strength of the concrete, d c Is the damage factor of concrete material, E c Is the elastic modulus of the concrete in the state of rigidity degradation after entering the plastic deformation stage, f c For the initial compressive strength of the concrete, ε is the strain under the uniaxial stress of the concrete, σ is the stress under the uniaxial stress of the concrete, x is the ratio of the compressive strain of the concrete to the peak strain corresponding to the peak strain when the uniaxial stress of the concrete reaches a representative value, n is the ratio of the initial stiffness to the loss value of the stiffness, ρ c Is the ratio of stiffness at peak load under compression to the initial stiffness;
stress strain curve equation of concrete under uniaxial tension:
σ t ＝(1-d t )E c epsilon type 1-6，
In the formulae 1 to 6 to 1 to 11, a t For the parameters of the concrete pull-down section epsilon t,r For the peak strain corresponding to the concrete single axis when being pulled down to reach the representative value, f t,r Is the representative value of the tensile strength of the concrete, sigma t Is the stress of the concrete under the uniaxial tension.
The invention relates to a method for analyzing the fatigue performance of a double-hole beam based on a fatigue damage constitutive model, which is further improved in that the established concrete fatigue constitutive model comprises the following steps:
the formula of the change of the elastic modulus of the concrete along with the fatigue times:
in formula 2-1, E N The elastic modulus of the concrete after the concrete is subjected to N times of fatigue loading, N is the number of times of fatigue loading of the concrete, N f For the total fatigue life of the concrete at the corresponding stress ratio E 0 The initial elastic modulus of the concrete before the concrete is not subjected to fatigue loading;
concrete fatigue S-N curve:
S max ＝1-0.0576lg N f ,S max >0.75 2-2 of the total number of the components,
S max ＝1.0505-0.0656lg N f ,S max less than or equal to 0.75 of the formula 2-3,
in the formulas 2-2 and 2-3, S max Is the maximum stress level of concrete fatigue, N f Is the fatigue life of the concrete material at that stress level;
concrete compression fatigue structure:
in the formulas 2-4, sigma (N) is stress when concrete is stressed and fatigued, E c (N) is the elastic modulus of the concrete after the Nth fatigue load is acted, epsilon is the fatigue residual strain of the concrete, and ρ is c … …, n is … …,is equivalent peak strain after concrete fatigue, f c (N) is the compressive strength of the concrete after the Nth fatigue load is acted, alpha c Is a parameter of a descending section of a uniaxial stress strain curve of the concrete,
residual strength of concrete after uniaxial tension fatigue:
in the formulae 2 to 5, f t,N For the residual tensile strength of the concrete after N times of fatigue loading, f t The initial tensile strength of the concrete before fatigue loading is not developed.
The invention relates to a further improvement of a double-hole girder fatigue performance analysis method based on a fatigue damage constitutive model, which is characterized in that the built reinforcing steel static constitutive model comprises the following steps:
in formula 3-1, σ s Stress to which the steel bar is subjected E s For the elastic modulus, epsilon, of the reinforcing bars s Is the strain of the steel bar in the stress state, f y Is the maximum stress value before the steel bar yields, f u For the maximum stress value epsilon after the steel bar yields and before the ultimate strength u Is the stress f u Corresponding strain, ε y Is the stress f y Corresponding strain.
The invention relates to a fatigue performance analysis method of a double-hole beam based on a fatigue damage constitutive model, which is further improved in that the built reinforcing steel bar fatigue constitutive model comprises the following steps:
steel bar S-N curve:
lg N f = 28.625 to 9.542lg (Δσ) formula 3 to 3,
in the formula 3-2 and the formula 3-3, f y (N) is the residual strength after the steel bar is fatigued, f y N is the number of times of fatigue load action undergone by the steel bar and is the maximum stress value before the steel bar yields f The fatigue life of the steel bar is ten thousand times, delta sigma is the stress amplitude of the steel bar, and delta sigma=sigma max -σ min ，σ max For the tensile stress sigma of the longitudinal bar when the hole beam is loaded to the upper limit of fatigue min The longitudinal bar is stressed by the tensile stress when the open beam is loaded to the lower limit of fatigue.
The invention relates to a fatigue damage constitutive model-based double-hole beam fatigue performance analysis method, which is further improved in that the setting of fatigue failure criteria comprises the following steps:
the failure criteria of the concrete after the fatigue load is set as follows:
△ε r ≥0.4ε 0 4-1 of the total number of the components,
the failure criteria of the steel bar after fatigue loading are set as follows:
f y (N)≤σ max 4-2 of the total number of the components,
in the formula 4-1 and the formula 4-2, deltaε r Is the residual strain of the concrete after the cyclic load is subjected to 0.4 epsilon 0 Is the ultimate compressive strain of concrete, f y (N) is the residual strength after the steel bar is fatigued, sigma max The longitudinal bar is stressed by tension when the open beam is loaded to the upper limit of fatigue.
The invention further improves the fatigue performance analysis method of the double-hole beam based on the fatigue damage constitutive model in that the loading simulation test comprises a static loading stage, a fatigue loading stage and a damage stage;
in the static load stage, loading the concrete model and the steel bar model to be destroyed so as to determine the limit load, the loading upper limit value and the loading lower limit value of the hole beam;
in the fatigue loading stage, carrying out N times of fatigue loading on the concrete model and the reinforcing steel bar model, updating the constitutive parameters of the concrete fatigue constitutive model and the reinforcing steel bar fatigue constitutive model after fatigue, loading the concrete model and the reinforcing steel bar model to the upper limit of fatigue, judging whether damage occurs according to a set fatigue damage criterion, if so, entering the damage stage, and if not, continuing to carry out fatigue loading;
and in the breaking stage, loading the concrete model and the reinforcing steel bar model to be broken, and obtaining the residual bearing capacity after fatigue.
The fatigue performance analysis method for the double-hole beam based on the fatigue damage constitutive model is further improved in that the established concrete model and the reinforced bar model are restrained by utilizing parameters output by the concrete fatigue constitutive model and the reinforced bar fatigue constitutive model before the fatigue failure criterion is reached, so that the fatigue loading process of the double-hole beam is simulated.
Drawings
FIG. 1 is a schematic structural diagram of a concrete model in a fatigue performance analysis method of a double-hole beam based on a fatigue damage constitutive model.
Fig. 2 is a schematic structural diagram of a steel bar model in the fatigue performance analysis method of the double-hole beam based on the fatigue damage constitutive model.
Fig. 3 and fig. 4 are diagrams of concrete CDP models in the fatigue performance analysis method of the double-hole beam based on the fatigue damage constitutive model of the invention.
FIG. 5 is a diagram showing the degradation process of the elastic modulus of the concrete material with the number of fatigue load actions in the method for analyzing the fatigue performance of the double-hole beam based on the fatigue damage constitutive model.
FIG. 6 is a graph of the residual strength envelope of the coagulated fatigue in the method for analyzing the fatigue performance of a double-hole beam based on the fatigue damage constitutive model of the present invention.
FIG. 7 shows a compressive fatigue constitutive model in a method for analyzing fatigue performance of a double-hole beam based on a fatigue damage constitutive model according to the invention.
Fig. 8 is a schematic diagram of the static constitutive relation of the steel bars in the fatigue performance analysis method of the double-hole beam based on the fatigue damage constitutive model.
Fig. 9 is a graph comparing a test curve of a static load open beam test piece with a mid-span load displacement curve obtained by simulation calculation.
Detailed Description
The invention will be further described with reference to the drawings and the specific examples.
Referring to fig. 1, the invention provides a fatigue performance analysis method of a double-hole beam based on a fatigue damage constitutive model, aiming at developing fatigue research through a numerical simulation method, improving research efficiency and reducing fatigue test cost. By comparing and analyzing the existing finite element simulation software, the ABAQUS finite element software is adopted to simulate and verify the expansion numerical value of the reinforced concrete hole beam fatigue test, and based on a verification model, the influence of fatigue load action times on the residual bearing capacity degradation rule of the hole beam is further expanded and analyzed. The method for analyzing the fatigue performance of the double-hole beam based on the fatigue damage constitutive model is described below with reference to the accompanying drawings.
Referring to fig. 1, a schematic structural diagram of a concrete model in the fatigue performance analysis method of a double-hole beam based on a fatigue damage constitutive model is shown. Referring to fig. 2, a schematic structural diagram of a steel bar model in the fatigue performance analysis method of the double-hole beam based on the fatigue damage constitutive model is shown. The method for analyzing the fatigue performance of the double-hole beam based on the fatigue damage constitutive model is described below with reference to fig. 1 and 2.
As shown in fig. 1 and 2, the fatigue performance analysis method of the double-hole beam based on the fatigue damage constitutive model comprises the following steps:
respectively establishing a concrete model and a steel bar model of the double-hole beam according to the design scheme of the double-hole beam;
carrying out a loading test on the prepared test piece of the double-hole beam to obtain test data;
establishing a concrete static constitutive model based on test data;
analyzing a degradation rule of the double-hole beam test piece after fatigue based on test data and establishing a concrete fatigue constitutive model;
establishing a static constitutive model of the reinforcing steel bar and a fatigue constitutive model of the reinforcing steel bar;
setting a fatigue failure criterion;
carrying out loading simulation test on the established concrete model and the reinforced bar model, and carrying out simulation calculation on the performance of the double-hole beam after fatigue by utilizing the established concrete static constitutive model, the established concrete fatigue constitutive model, the reinforced bar static constitutive model and the reinforced bar fatigue constitutive model to obtain a simulation calculation result;
and judging whether the simulation calculation result reaches a set fatigue failure criterion, if not, updating parameters of the concrete fatigue constitutive model and the reinforcing steel bar fatigue constitutive model by using the simulation calculation result, and continuously carrying out a loading simulation test to obtain a simulation calculation result corresponding to the fatigue times.
Specifically, according to specific parameters of the double-hole beam, a model is built for the double-hole beam in ABAQUS, a concrete model and a reinforcement cage are modeled separately, wherein the concrete model adopts a C3D8R unit, all reinforcements in the double-hole beam are modeled by adopting a T3D2 unit, and the reinforcement cage and the concrete adopt an 'embed' mode in an interaction module, so that bonding slip between the concrete and the reinforcements is temporarily not considered. The concrete model and the reinforcement cage model are shown in fig. 1 and 2. The double-hole beam is long-strip-shaped, the total length is 4.85m, the height is 490mm, the beam width is 210mm, two holes are formed in the middle of the double-hole beam, the size of each hole is 700mm wide, the height is 140mm, the hole spacing is 490mm, the beam structure on the upper portion of each hole is an upper beam, the beam structure on the lower portion of each hole is a lower beam, the double-hole beam is in a C40 concrete pouring shape, hoops and longitudinal bars are HRB400 finish rolling screw threads, and the thickness of a protective layer is 20mm. At the position of the hole, the top and the bottom of the hole are additionally provided with hole longitudinal ribs, the hole longitudinal ribs are connected with the corresponding beam longitudinal ribs at the top and the bottom through reinforcing hoops, oblique ribs are arranged on two sides of the hole, the oblique ribs are arranged on the side parts of the hole in a crossing mode, and part of the oblique ribs extend in the horizontal direction at the end parts.
Manufacturing test pieces according to specific parameters of the double-hole beam and performing fatigue performance test to obtain corresponding test data, specifically manufacturing and forming a first test piece to a fourth test piece according to a design scheme of the double-hole beam, wherein the outer contour of a hole on the first test piece to the third test piece in the four manufactured test pieces is in a round angle rectangle, the outer contour of a hole on the fourth test piece is in a right angle rectangle, and a strain gauge is arranged on a steel bar at the hole when the test pieces are manufactured; a comparison is formed by the rounded rectangle and the right-angle rectangle, so that the performance influence of the hole shape on the double-hole beam is checked by the comparison analysis. Setting a test loading scheme, wherein the set loading scheme comprises three stages of static loading, fatigue loading and destructive loading, loading the first test piece to the fourth test piece according to the set loading scheme, and collecting test data of the first test piece to the fourth test piece in the loading process; and in the static loading stage, three-point concentrated force is adopted, the three-point concentrated force is applied to the distribution beam through the top of the MTS electrohydraulic servo actuator and then acts on the test piece, the test loading is carried out according to the test method standard of the concrete structure, the test loading is carried out according to the selected fatigue load upper limit value, the test loading is carried out to the fatigue upper limit according to the stage, after each stage is maintained for 5 minutes, the load generating cracks and the positions, the widths and the heights of the cracks are observed and recorded, and mark is formed on the test piece by a Mark pen. In the fatigue loading stage, three-point concentrated force loading is adopted, a constant-amplitude sine wave form is adopted to apply load to the double-hole beam, and the loading frequency is 5Hz according to the upper limit value of the fatigue load. When the fatigue action times reach the preset fatigue action times of 1 ten thousand, 5 ten thousand, 8 ten thousand, 13 ten thousand and 25 ten thousand, stopping fatigue loading, performing a static loading test, continuing the fatigue loading test after the static loading test is completed, and stopping after the preset fatigue action times are finally reached. And in the stage of destructive loading, after the upper limit of the predetermined fatigue loading times of 8 ten thousand, 13 ten thousand, 25 ten thousand and 41 ten thousand is finished, checking whether the test beam is damaged, if the test beam can still work normally, carrying out a static load destructive test on the test piece of the hole beam, adopting force control before the upper limit of the fatigue loading is reached, carrying out load classification the same as that in the stage of static loading, and adopting a displacement control mode after the upper limit of the fatigue loading is reached, loading until the load of the test beam is reduced to 85% of the peak load or the test beam is seriously deformed, and stopping the test. Determining a loading device, wherein an MTS electrohydraulic servo structure fatigue testing machine is adopted in the fatigue test, and a dynamic and static resistance strain acquisition system is used for acquiring data such as concrete strain, steel bar strain, test piece deflection and the like; the crack width is measured by using a high-precision crack width measuring instrument, the measuring precision is 0.02mm, and the maximum crack width is measured to be 20mm. The test device consists of four parts, namely a support, a distribution beam, a loading control system and an MTS actuator. The hydraulic control is a main control system of the actuator, the upper limit load and the lower limit load of the loading process of the actuator are recorded in real time, and the fatigue times of the test are automatically recorded by a computer. The maximum achievable static and dynamic loads of the MTS actuator are 500kN. The load is applied through MTS actuator, and the distribution beam distributes the concentrated force to the inside of the openings on two sides, and simultaneously welds the side guard plates on the distribution beam to prevent the distribution beam from shifting in the fatigue process.
In one embodiment of the invention, a concrete static constitutive model is established:
in finite element computation, the accuracy and correctness of subsequent numerical computation are determined by the choice of the material structure. Aiming at concrete materials, three models of a brittle fracture model, a dispersion crack model and a concrete plastic damage model are provided in ABAQUS software, and the three models have proper use conditions for different research purposes, wherein the concrete plastic damage model (CDP) can reflect the rigidity degradation of the concrete materials caused by plastic damage, can well reflect the irreversible damage of the concrete materials in the compression and tension process, can well simulate and calculate nonlinear problems while ensuring better convergence, and is widely applied to finite element simulation of building structures.
In the CDP model, the concrete material, after entering the plastic deformation phase, will be irreversibly damaged, while the resulting stiffness degradation is calculated by the following formula:
E c ＝(1-d c )E 0
wherein E is 0 Is the elastic modulus d of the concrete material in a nondestructive state c Is a damage factor of concrete material, the value of the parameter is between 0 and 1, d c When=0, the material is in a nondestructive state, d c Material reached a fully destroyed state at =1. As shown in fig. 3 and 4, a concrete CDP model is shown, wherein,is inelastically strained in the stressed state of the concrete, < + >>Is elastic strain without considering damage under the state of concrete compression, +.>For the elastic strain taking into account the damage in the stressed state of the concrete, < +.>For the plastic strain of the concrete in the stressed state, taking into account the damage, c and t represent the stressed and tensioned state of the concrete, respectively.
The invention adopts a uniaxial compressive stress-strain constitutive equation, and corrects the used concrete static load constitutive according to the measured data of the material property test.
Stress strain curve equation under uniaxial stress of concrete:
σ＝(1-d c )E c epsilon-type of 1-1,
in the formulae 1-1 to 1-5, a c For the concrete pressure drop section parameter epsilon c,r The peak strain corresponding to the concrete when the uniaxial pressure reaches a representative value is obtained; f (f) c,r Is the representative value of the uniaxial compressive strength of the concrete, d c Is the damage factor of concrete material, E c Is the elastic modulus of the concrete in the state of rigidity degradation after entering the plastic deformation stage, f c For the initial compressive strength of the concrete, epsilon is the strain of the concrete under the uniaxial pressure, sigma is the stress of the concrete under the uniaxial pressure, x is the ratio of the compressive strain of the concrete to the peak strain corresponding to the time when the uniaxial pressure of the concrete reaches a representative value,n is the ratio of the initial stiffness to the loss of stiffness, ρ c Is the ratio of stiffness at peak load under compression to the initial stiffness; the purpose of equations 1-2 is to determine the damage to the concrete under strain from the concrete strain, and then determine the stress of the concrete under strain.
Stress strain curve equation of concrete under uniaxial tension:
σ t ＝(1-d t )E c epsilon-type of 1-6,
in the formulae 1 to 6 to 1 to 11, a t For the parameters of the concrete pull-down section epsilon t,r For the peak strain corresponding to the concrete single axis when being pulled down to reach the representative value, f t,r Is the representative value of the tensile strength of the concrete, sigma t Is the stress of the concrete under the uniaxial tension.
The plastic damage model parameters of the concrete materials used in the invention are shown in the following table 1:
| expansion angle | Eccentricity ratio | f b0 /f c0 | Yield constant | Coefficient of viscosity |
| 30 | 0.1 | 1.16 | 0.667 | 0.0005 |
Table 1 concrete plastic damage model parameters.
Further, the built concrete fatigue constitutive model:
the formula of the change of the elastic modulus of the concrete along with the fatigue times:
in formula 2-1, E N The elastic modulus of the concrete after the concrete is subjected to N times of fatigue loading, N is the number of times of fatigue loading of the concrete, N f For the total fatigue life of the concrete at the corresponding stress ratio E 0 The initial elastic modulus of the concrete before the concrete is not subjected to fatigue loading;
concrete fatigue S-N curve:
S max ＝1-0.0576lg N f ,S max >0.75 2-2 of the total number of the components,
S max ＝1.0505-0.0656lg N f ,S max less than or equal to 0.75 of the formula 2-3,
in the formulas 2-2 and 2-3, S max Is the maximum stress level of concrete fatigue, N f Is the fatigue life of the concrete material at that stress level;
concrete compression fatigue structure:
in the formulas 2-4, sigma (N) is stress when concrete is stressed and fatigued, E c (N) is the elastic modulus of the concrete after the Nth fatigue load is acted, epsilon is the fatigue residual strain of the concrete, and ρ is c … …, n is … …,is equivalent peak strain after concrete fatigue, f c (N) is the compressive strength of the concrete after the Nth fatigue load is acted, alpha c Is a parameter of a descending section of a uniaxial stress strain curve of the concrete,
residual strength of concrete after uniaxial tension fatigue:
in the formulae 2 to 5, f t,N For the residual tensile strength of the concrete after N times of fatigue loading, f t The initial tensile strength of the concrete before fatigue loading is not developed.
The reduction of the bearing capacity of the reinforced concrete open beam structure after fatigue is mainly caused by the degradation of the concrete material, so the open beam can be unfolded and analyzed according to the strength degradation rule of the concrete material after fatigue. Fatigue of concrete is a process of changing a material characteristic component, and in the fatigue process, generation, development and expansion of concrete microcracks can cause the change of the overall performance of the concrete, including the elastic modulus, the residual strength and the residual strain of the concrete, and the degradation of the mechanical properties can also directly influence the performance of a concrete structure.
The degradation process of the elastic modulus of the concrete material along with the fatigue load action times is obtained through analysis of test data, as shown in fig. 5, the degradation of the elastic modulus shows an obvious three-stage rule, in the first stage, the damage development of the concrete material is faster, the rate of decrease is gradually reduced, and the stage accounts for about 10% of the total fatigue life of the concrete; in the second stage, the damage speed of the concrete material is basically kept unchanged, the damage is in a slow descending trend, the damage grows linearly along with the increase of the fatigue load, and the damage speed accounts for 75 to 80 percent of the total fatigue life; in the third stage, the deformation damage rate of the concrete material is rapidly increased, and finally the concrete material is destroyed, wherein the stage accounts for about 10% of the total service life. The empirical formula of the concrete elastic modulus changing along with the fatigue times is obtained by fitting test data, and is shown in the formula2-1. Further, by referring to the S-N double logarithmic curve of the concrete, the formula 2-1 is improved to obtain an elastic modulus degradation equation of the concrete:
when the fatigue analysis is carried out by using the concrete elastic modulus degradation model, the total fatigue life of the concrete material needs to be predicted in advance, and the fatigue life of the concrete can be predicted through a concrete S-N equation. The S-N curve of the concrete material refers to the relationship between the fatigue strength and the fatigue life of a standard concrete test block under the specified cyclic load, and is generally called a stress-fatigue life curve. For concrete materials, the tensile cracks do not affect the fatigue life, the maximum stress applied during fatigue affects the fatigue life of the concrete the most, and in fatigue research, the ratio of the maximum stress applied to the peak strength of the concrete is generally expressed, and the relation between the maximum stress ratio and the fatigue life is established, as shown in the formulas 2-2 and 2-3.
Under the fatigue action, the concrete material has irreversible plastic damage and residual strain due to the generation, accumulation and development of microcracks in the material and the same load action, and the residual strain is accumulated continuously along with the increase of the fatigue load action times, so that the strength of the concrete is reduced continuously due to the degradation of the material. The maximum strain of the concrete under the action of tensile and compressive fatigue load when in fatigue failure is equivalent to the strain corresponding to the maximum stress of the uniaxial loading softening section. Therefore, the envelope curve of the concrete under the compression condition obtained through calculation can be used for describing the degradation rule of the compressive bearing capacity of the concrete under the fatigue action, and a concrete uniaxial fatigue strength degradation model is established based on the degradation rule.
And analyzing and calculating according to the collected test data to obtain an envelope curve of the concrete under the condition of compression, and describing the degradation rule of the concrete compression bearing capacity under the fatigue effect by using the envelope curve. The equation of the descending section of the uniaxial compressive stress-strain curve of the concrete provided in the concrete specifications of China is as follows:
in the formulas 2 to 6, sigma is a stress value under the uniaxial cyclic load of the concrete, and epsilon is a strain value under the uniaxial cyclic load of the concrete.
According to the solution form of equations 2-6, the relevant parameters in FIG. 6 are introduced, assuming that the concrete compressive strength envelope curve equation under fatigue loading is approximately as follows:
by analyzing the definition of x (N) and the range of values thereof, the following can be found:
x| N＝1 ＝x(1)＝1,σ r,c (1)＝f c the method comprises the steps of carrying out a first treatment on the surface of the Namely, the residual stress is the compressive strength of the concrete axle center when the concrete is loaded for only 1 time;
when the fatigue load acting times N reach the fatigue life, the residual stress corresponds to the upper limit stress of the coagulation fatigue.
Define x (N) as a function of the number of fatigue loading events:
the boundary condition is substituted into 1-2 to obtain x (N) f ) The method comprises the following steps:
in the formulae 2 to 9, f c The initial compressive strength of the concrete; sigma (sigma) max Stress of the concrete at the upper limit of fatigue; alpha c The parameters of the descending section of the compressive stress strain curve are 1.94.
In summary, the residual compressive strength of the concrete after undergoing N fatigue loading events can be calculated by the following formula:
in the formulae 2 to 10, f c (N) is the compressive residual strength of the concrete after N times of fatigue loading, alpha c The parameters of the single-shaft pressure drop section of the concrete can be taken according to the specification.
The tensile property of the concrete has limited influence on the evaluation of the structural bearing capacity, so that a tensile fatigue model of the concrete can be further simplified, and a calculation formula after the tensile strength fatigue of the concrete is obtained according to a relevant test that the tensile strength of a single shaft of a concrete material is degraded along with the fatigue times:
in the formulae 2 to 5, f t,N The residual tensile strength of the concrete after being subjected to N times of fatigue load; f (f) t The initial tensile strength of the concrete before fatigue loading is not developed.
The deformation of the concrete under fatigue load is nonlinear, the residual strain of the concrete after fatigue is continuously accumulated along with the fatigue process, and the concrete also shows a three-section change rule, and the initial rapid growth stage accounts for 10% of the fatigue life, the stable rise stage accounts for 80% and the fatigue failure stage accounts for 10% similar to the rigidity degradation. Residual strain reflects the development of concrete damage and can be used to measure one parameter of fatigue damage and fatigue life of a concrete material under various loading conditions. The concrete fatigue residual strain calculation formula taking acceleration frequency and stress level as parameters is as follows: epsilon=129 σ m (1+3.87△)t 1/3 In the formula 5-1, epsilon is residual strain after concrete fatigue and sigma in the formula 5-1 m Is the fatigue stress level of the concrete, and delta is the stress level difference. On the basis of the formula 5-1, the influence of fatigue times on the residual strain of the concrete is added, and the improved formula is as follows: delta epsilon (N) =129×s m ×t 1/3 +17.8×S m △N 1/3 In the formula 5-2, delta epsilon (N) is the residual strain of the concrete after being subjected to N times of fatigue; s is S m Is the fatigue average stress level; delta is the stress level difference. In practical application, the stress borne by the concrete is difficult to detect, the stress can only be calculated by detecting the stress of the concrete, and a concrete residual strain model considering the fatigue strain and the cycle number is obtained through test data analysis: in formula 5-3, ε is represented by formula 5-3 min 、ε max The strain corresponding to the lower fatigue limit and the upper fatigue limit in fatigue loading is respectively; epsilon r (1) Concrete residual strain once fatigue loaded according to epsilon r (1)＝0.25(ε max /ε unstab ) Calculating; epsilon unstab The strain corresponding to the peak stress of the concrete in the failure curve is directly loaded for one static load.
The concrete constitutive model after fatigue is as follows: the tensile structure of the concrete is based on the fact that the tensile strength of the concrete is far lower than the compressive strength, and the concrete is provided with cracks to work in the fatigue process of the hole-forming beam, so that the concrete is considered to be linear elastic before reaching the tensile strength, the concrete cracks after reaching the tensile strength, meanwhile, the tensile stress is reduced to 0, and the tensile strength of the concrete after being fatigued can represent the tensile structure of the concrete, and the structural formula is shown as the formula 2-5. The stress strain curve of the concrete can gradually displace the straight line under the action of high fatigue, so that the peak strain under uniaxial compression after the concrete is fatigued can be approximately calculated by the following formula:formula 5-4. The obtained concrete compression fatigue constitutive model curve is shown in figure 7, the fatigue constitutive model is shown as 2-4, and the equivalent peak strain after fatigue is +.>ε tot (N) =ε (1) +Δε (N), ε (1) is the strain that would be produced when initially loaded to the upper limit. Alpha c The parameters of the descending section of the uniaxial stress strain curve of the concrete are obtained. ρ c N is calculated by the formula->
Still further, the built static constitutive model of the reinforcing steel bar comprises:
in formula 3-1, σ s Stress to which the steel bar is subjected E s For the elastic modulus, epsilon, of the reinforcing bars s Is the strain of the steel bar in the stress state, f y Is the maximum stress value before the steel bar yields, f u For the maximum stress value epsilon after the steel bar yields and before the ultimate strength u Is the stress f u Corresponding strain, ε y Is the stress f y Corresponding strain. As shown in FIG. 8, the stress-strain curve of the steel bar has an obvious yield platform, the steel bar is in an elastic state before yield, the hardening rigidity of the steel bar at the tempering stage from the post yield to the pre-ultimate strength is 0.1 time of the elastic modulus of the steel bar, E 0 The elastic modulus was 200GPa.
Still further, the built reinforcing steel bar fatigue constitutive model comprises:
steel bar S-N curve:
lg N f = 28.625 to 9.542lg (Δσ) formula 3 to 3,
in the formula 3-2 and the formula 3-3, f y (N) is the residual strength after the steel bar is fatigued, f y N is the number of times of fatigue load action undergone by the steel bar and is the maximum stress value before the steel bar yields f The fatigue life of the steel bar is ten thousand times, delta sigma is the stress amplitude of the steel bar, and delta sigma=sigma max -σ min ，σ max For the tensile stress sigma of the longitudinal bar when the hole beam is loaded to the upper limit of fatigue min The longitudinal bar is stressed by the tensile stress when the open beam is loaded to the lower limit of fatigue.
In one embodiment of the present invention, setting the fatigue failure criteria includes:
the failure criteria of the concrete after the fatigue load is set as follows:
△ε r ≥0.4ε 0 4-1 of the total number of the components,
the failure criteria of the steel bar after fatigue loading are set as follows:
f y (N)≤σ max 4-2 of the total number of the components,
in the formula 4-1 and the formula 4-2, deltaε r Is the residual strain of the concrete after the cyclic load is subjected to 0.4 epsilon 0 Is the ultimate compressive strain of concrete, f y (N) is the residual strength after the steel bar is fatigued, sigma max The longitudinal bar is stressed by tension when the open beam is loaded to the upper limit of fatigue.
Under the action of uniaxial cyclic compressive stress, the residual strain of the concrete is accumulated continuously, and when the ultimate compressive strain of the concrete reaches 0.4 times, the damage of the concrete can be considered to reach the upper limit and the concrete cannot work continuously, so that the failure criterion of the concrete is shown as formula 4-1. When the fatigue residual bearing capacity of the steel bar reaches the fatigue upper limit value, the steel bar is broken, and the breaking criterion is shown as 4-2.
In one embodiment of the invention, the loading simulation test comprises a static loading stage, a fatigue loading stage and a breaking stage;
in the static load stage, loading the concrete model and the steel bar model to be destroyed so as to determine the limit load, the loading upper limit value and the loading lower limit value of the hole beam;
in the fatigue loading stage, carrying out N times of fatigue loading on the concrete model and the reinforcing steel bar model, updating the constitutive parameters of the concrete fatigue constitutive model and the reinforcing steel bar fatigue constitutive model after fatigue, loading the concrete model and the reinforcing steel bar model to the upper limit of fatigue, judging whether damage occurs according to a set fatigue damage criterion, if so, entering the damage stage, and if not, continuing to carry out fatigue loading;
and in the breaking stage, loading the concrete model and the reinforcing steel bar model to be broken, and obtaining the residual bearing capacity after fatigue.
The invention simplifies the numerical simulation of the whole process of the fatigue test of the hole beam into three stages:
1. and (3) a static load stage. And constructing a static load model in software according to the test piece component parameters, inputting material static load constitutive parameters, loading the model to damage, determining the limit load of the hole beam, and determining the upper and lower limits of fatigue loading.
2. And a fatigue loading stage. And for the beam subjected to N times of fatigue loading, updating constitutive parameters of concrete and steel bars after N times of fatigue, loading the model to the upper limit of fatigue by adopting static force, analyzing whether the structure is damaged according to a damage criterion, and entering the next cycle if the damage occurs.
3. And (3) a destruction stage. And updating parameters of the concrete and the steel bars after the model is subjected to N times of fatigue loading for the model which is not damaged after the set target N times of fatigue loading is reached, and loading static force of the model to damage to obtain the residual bearing capacity after the fatigue.
And before the fatigue failure criterion is reached, the parameters output by the concrete fatigue constitutive model and the steel bar fatigue constitutive model are utilized to restrain the established concrete model and the steel bar model, so that the fatigue loading process of the double-hole beam is simulated.
The fatigue model is based on a static load model, so that the accuracy of the static load model has great influence on the development of a subsequent fatigue model, the static load model of the hole beam is required to be verified, a static load hole beam test piece test curve is selected to be compared with a mid-span load displacement curve obtained by simulation calculation, as shown in fig. 9, the load displacement curve obtained by numerical simulation has the same trend with a test actual curve, the two curves are well matched integrally, the obtained limit load is not greatly different from a test value, the curve grows linearly before concrete cracking, the rigidity of the curve after cracking is reduced but basically kept unchanged, the difference exists between the two curves, the simulated cracking load is higher, because the simulation condition is more ideal, the concrete is in reality nonuniform, the position and the size of cracks are in certain randomness in the test process, the test is basically consistent with the simulated load displacement curve, and the error is still in a reasonable range. Meanwhile, the damage modes are compared, the damage mode under simulation is basically consistent with the damage mode in practice, the damage is finished when the girder is put on the opening, and in conclusion, the model can be considered to be capable of reliably predicting the static load mechanical behavior of the opening girder through analysis, so that the model can be used for the subsequent fatigue test numerical analysis.
Residual bearing capacity contrast:
according to the method, concrete and steel bar constitutive parameters are updated according to fatigue times experienced by each test piece, a static force damage test is carried out on the concrete and steel bar constitutive parameters, a finite element model calculation result is extracted, a load displacement curve of the hole beam after the specific fatigue times is obtained is shown in the following graph, and the load displacement curve is compared with a test result.
The calculated residual bearing capacity curve after fatigue has the same trend as the actual measurement curve of the test, the rigidity of the finite element model curve is slightly larger than that of the curve obtained by the test before the upper limit of the fatigue is set, and the obtained residual bearing capacity after fatigue is basically consistent and has the maximum difference of 10.4 percent. At the same time, the simulated curve and the experimental measured curve
Compared with the prior stage of peak stress, the method has the advantages that the simulation calculation is carried out in an ideal state, the fatigue performance degradation of a test piece is simulated in a static calculation mode by updating the material structure after specific fatigue times, and the finite element model is a crack-free beam by default before calculation, so that a crack part of the concrete is still pulled by static force in the calculation, the difference from the actual situation is a little, obvious concrete crack inflection points exist in a simulated curve in the figure, and the deflection calculation value is slightly smaller than the test value under the same load, which is a reasonable phenomenon.
Steel bar strain-graded load contrast:
the strain of the longitudinal bars in the initial loading stage of the simulation curve is slowly increased, the steel bars bear more tensile force along with the concrete cracking of the tensile region, and the strain increase speed of the steel bars is accelerated, because the model beam is defaulted to be a nondestructive beam during loading, the concrete cracking stage exists, the simulated curve after cracking is the same as the actual measurement curve in trend, the difference of the increase speed is not great, the difference of the simulated strain of the steel bars fatigue and the actual measurement strain of the test is not more than 10% at the maximum, and the reliability of the model for calculating the fatigue residual bearing capacity of the hole beam is further verified.
Residual load capacity extension analysis:
because of the limitation of test conditions, the highest fatigue test of the fatigue test piece is only carried out for 41 ten thousand times of cyclic loads, and the comparison result of the static load model and the fatigue model shows that the finite element model established by the invention can effectively simulate and calculate the residual bearing capacity of the hole-opened beam after the fatigue load, so that the fatigue frequency is further expanded to be greater than 41 ten thousand times after the fatigue frequency is further improved.
The residual performance of the hole beam is limited by the damage criteria of two materials, namely concrete and steel bars, and when one of the materials is reached, the hole beam can be judged to be damaged. And calculating residual strain of the concrete and stress after the steel bar is fatigued according to the stress on the steel bar and the concrete under the upper limit of fatigue acquired by the test, and calculating the residual bearing capacity if the steel bar is not destroyed.
The test pieces of the open-hole beam can be subjected to fatigue loading for 200 ten thousand times and are free from fatigue damage, the reliability of the double-open-hole concrete beam structure under fatigue load is proved, the residual bearing capacity of the double-open-hole concrete beam structure after 200 ten thousand times of fatigue is 117.5kN and 114.6kN respectively, and compared with the bearing capacity of the double-open-hole concrete beam structure in an unbeard state, the bearing capacity of the double-open-hole concrete beam structure is reduced by 29.0% and 28.5%, and the impact of the fatigue load on the shearing bearing capacity of the open-hole concrete beam is obvious.
The beneficial effects of the invention are as follows:
1. and a standard concrete structure is adopted, a static load model of the open beam is established and verified according to a lumber property test result, a simulation curve is well fitted with a test curve, and the feasibility of the simulation method is verified.
2. Based on a fatigue simplification calculation method, an established concrete fatigue constitutive model and a reinforcement strength degradation model are input into ABAQUS, a material fatigue failure criterion is determined, simulation calculation is carried out on the residual performance of the hole beam after fatigue, a calculation curve is compared with a test result, the difference between the calculation curve and the test result is found to be not large, the development trend of the curve is the same, the simulation failure mode is consistent with the test phenomenon, and the accuracy of the hole beam fatigue model is proved.
3. On the premise of good calculation based on a fatigue simulation model, the fatigue performance degradation rule of the open-hole beam under the stress of 0.5Pu is further analyzed, and the residual bearing capacity of the open-hole beam concrete after 80 ten thousand, 120 ten thousand, 160 ten thousand and 200 ten thousand fatigue loads is expanded and analyzed.
The present invention has been described in detail with reference to the embodiments of the drawings, and those skilled in the art can make various modifications to the invention based on the above description. Accordingly, certain details of the illustrated embodiments are not to be taken as limiting the invention, which is defined by the appended claims.
Claims (8)
1. The method for analyzing the fatigue performance of the double-hole beam based on the fatigue damage constitutive model is characterized by comprising the following steps of:
respectively establishing a concrete model and a steel bar model of the double-hole beam according to the design scheme of the double-hole beam;
carrying out a loading test on the prepared test piece of the double-hole beam to obtain test data;
establishing a concrete static constitutive model based on test data;
analyzing a degradation rule of the double-hole beam test piece after fatigue based on test data and establishing a concrete fatigue constitutive model;
establishing a static constitutive model of the reinforcing steel bar and a fatigue constitutive model of the reinforcing steel bar;
setting a fatigue failure criterion;
carrying out loading simulation test on the established concrete model and the reinforced bar model, and carrying out simulation calculation on the performance of the double-hole beam after fatigue by utilizing the established concrete static constitutive model, the established concrete fatigue constitutive model, the reinforced bar static constitutive model and the reinforced bar fatigue constitutive model to obtain a simulation calculation result;
and judging whether the simulation calculation result reaches a set fatigue failure criterion, if not, updating parameters of the concrete fatigue constitutive model and the reinforcing steel bar fatigue constitutive model by using the simulation calculation result, and continuously carrying out a loading simulation test to obtain a simulation calculation result corresponding to the fatigue times.
2. The method for analyzing fatigue performance of the double-opening beam based on the fatigue damage constitutive model according to claim 1, wherein the established concrete static constitutive model comprises the following steps:
stress strain curve equation under uniaxial stress of concrete:
σ＝(1-d c )E c epsilon-type of 1-1,
in the formulae 1-1 to 1-5, a c For the concrete pressure drop section parameter epsilon c,r The peak strain corresponding to the concrete when the uniaxial pressure reaches a representative value is obtained; f (f) c,r Is the representative value of the uniaxial compressive strength of the concrete, d c Is the damage factor of concrete material, E c Is the elastic modulus of the concrete in the state of rigidity degradation after entering the plastic deformation stage, f c For the initial compressive strength of the concrete, ε is the strain under the uniaxial stress of the concrete, σ is the stress under the uniaxial stress of the concrete, x is the ratio of the compressive strain of the concrete to the peak strain corresponding to the peak strain when the uniaxial stress of the concrete reaches a representative value, n is the ratio of the initial stiffness to the loss value of the stiffness, ρ c Is the ratio of stiffness at peak load under compression to the initial stiffness;
stress strain curve equation of concrete under uniaxial tension:
σ t ＝(1-d t )E c epsilon-type of 1-6,
in the formulae 1 to 6 to 1 to 11, a t For the parameters of the concrete pull-down section epsilon t,r For the peak strain corresponding to the concrete single axis when being pulled down to reach the representative value, f t,r Is the representative value of the tensile strength of the concrete, sigma t Is the stress of the concrete under the uniaxial tension.
3. The method for analyzing fatigue performance of the double-open-hole beam based on the fatigue damage constitutive model according to claim 1, wherein the established concrete fatigue constitutive model comprises the following steps:
the formula of the change of the elastic modulus of the concrete along with the fatigue times:
in formula 2-1, E N The elastic modulus of the concrete after the concrete is subjected to N times of fatigue loading, N is the number of times of fatigue loading of the concrete, N f To coagulate under corresponding stress ratioTotal fatigue life of earth, E 0 The initial elastic modulus of the concrete before the concrete is not subjected to fatigue loading;
concrete fatigue S-N curve:
S max ＝1-0.0576lgN f ,S max >0.75 2-2 of the total number of the components,
S max ＝1.0505-0.0656lgN f ,S max less than or equal to 0.75 of the formula 2-3,
in the formulas 2-2 and 2-3, S max Is the maximum stress level of concrete fatigue, N f Is the fatigue life of the concrete material at that stress level;
concrete compression fatigue structure:
in the formulas 2-4, sigma (N) is stress when concrete is stressed and fatigued, E c (N) is the elastic modulus of the concrete after the Nth fatigue load is acted, epsilon is the fatigue residual strain of the concrete, and ρ is c … …, n is … …,is equivalent peak strain after concrete fatigue, f c (N) is the compressive strength of the concrete after the Nth fatigue load is acted, alpha c Is a parameter of a descending section of a uniaxial stress strain curve of the concrete,
residual strength of concrete after uniaxial tension fatigue:
in the formulae 2 to 5, f t,N For the residual tensile strength of the concrete after N times of fatigue loading, f t The initial tensile strength of the concrete before fatigue loading is not developed.
4. The method for analyzing fatigue performance of the double-open-hole beam based on the fatigue damage constitutive model according to claim 1, wherein the built reinforcement static constitutive model comprises:
in formula 3-1, σ s Stress to which the steel bar is subjected E s For the elastic modulus, epsilon, of the reinforcing bars s Is the strain of the steel bar in the stress state, f y Is the maximum stress value before the steel bar yields, f u For the maximum stress value epsilon after the steel bar yields and before the ultimate strength u Is the stress f u Corresponding strain, ε y Is the stress f y Corresponding strain.
5. The method for analyzing fatigue performance of the double-open-hole beam based on the fatigue damage constitutive model of claim 1, wherein the built reinforcing steel bar fatigue constitutive model comprises the following steps:
steel bar S-N curve:
lgN f = 28.625 to 9.542lg (Δσ) formula 3 to 3,
in the formula 3-2 and the formula 3-3, f y (N) is the residual strength after the steel bar is fatigued, f y N is the number of times of fatigue load action undergone by the steel bar and is the maximum stress value before the steel bar yields f The fatigue life of the steel bar is ten thousand times, delta sigma is the stress amplitude of the steel bar, and delta sigma=sigma max -σ min ，σ max For the tensile stress sigma of the longitudinal bar when the hole beam is loaded to the upper limit of fatigue min The longitudinal bar is stressed by the tensile stress when the open beam is loaded to the lower limit of fatigue.
6. The method for analyzing fatigue performance of a double-hole beam based on a fatigue damage constitutive model according to claim 1, wherein setting a fatigue failure criterion comprises:
the failure criteria of the concrete after the fatigue load is set as follows:
△ε r ≥0.4ε 0 4-1 of the total number of the components,
the failure criteria of the steel bar after fatigue loading are set as follows:
f y (N)≤σ max 4-2 of the total number of the components,
in the formula 4-1 and the formula 4-2, deltaε r Is the residual strain of the concrete after the cyclic load is subjected to 0.4 epsilon 0 Is the ultimate compressive strain of concrete, f y (N) is the residual strength after the steel bar is fatigued, sigma max The longitudinal bar is stressed by tension when the open beam is loaded to the upper limit of fatigue.
7. The method for analyzing the fatigue performance of the double-hole beam based on the fatigue damage constitutive model according to claim 1, wherein the loading simulation test comprises a static load stage, a fatigue loading stage and a damage stage;
in the static load stage, loading the concrete model and the steel bar model to be destroyed so as to determine the limit load, the loading upper limit value and the loading lower limit value of the hole beam;
in the fatigue loading stage, carrying out N times of fatigue loading on the concrete model and the reinforcing steel bar model, updating the constitutive parameters of the concrete fatigue constitutive model and the reinforcing steel bar fatigue constitutive model after fatigue, loading the concrete model and the reinforcing steel bar model to the upper limit of fatigue, judging whether damage occurs according to a set fatigue damage criterion, if so, entering the damage stage, and if not, continuing to carry out fatigue loading;
and in the breaking stage, loading the concrete model and the reinforcing steel bar model to be broken, and obtaining the residual bearing capacity after fatigue.
8. The method for analyzing the fatigue performance of the double-hole beam based on the fatigue damage constitutive model according to claim 1, wherein the established concrete model and the reinforced bar model are constrained by utilizing parameters output by the concrete fatigue constitutive model and the reinforced bar fatigue constitutive model before the fatigue failure criterion is reached, so that the process of simulating the fatigue loading of the double-hole beam is realized.
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| CN118364542A (en) * | 2024-04-12 | 2024-07-19 | 西南交通大学 | Fatigue life analysis method of concrete bridge deck under moving load and related equipment |
| CN118427491A (en) * | 2024-07-03 | 2024-08-02 | 石家庄学院 | Calculation method of shear stiffness of studs in composite bridges under fatigue load |
| CN118627346A (en) * | 2024-06-26 | 2024-09-10 | 河北交通职业技术学院 | Calculation method and storage medium of stiffness in negative moment zone of steel-concrete composite beams under random fatigue load |
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN118364542A (en) * | 2024-04-12 | 2024-07-19 | 西南交通大学 | Fatigue life analysis method of concrete bridge deck under moving load and related equipment |
| CN118627346A (en) * | 2024-06-26 | 2024-09-10 | 河北交通职业技术学院 | Calculation method and storage medium of stiffness in negative moment zone of steel-concrete composite beams under random fatigue load |
| CN118427491A (en) * | 2024-07-03 | 2024-08-02 | 石家庄学院 | Calculation method of shear stiffness of studs in composite bridges under fatigue load |
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