CN106919786A - A kind of concrete fractional order creep model - Google Patents

Abstract

A kind of concrete fractional order creep model, in the case where hydrated reaction of cement is considered.The expression formula of a kind of concrete fractional order creep model concrete fractional order creep model that the present invention is provided isIn formula：C(t_e, τ_e) for loading age be τ_e, hold the lotus time for t_e‑τ_eCreep degree, f₁、g₁、p₁、r₁, β be coefficient, and 0<β≤1.A kind of concrete fractional order creep model that the present invention is provided, fully react concrete creep and load age, hold the related characteristic of lotus time, hydrated reaction of cement, and the new model set up also overcomes the more shortcoming of eight conventional parameter creep model parameters, fitting effect analysis to this model understands, fractional model meets concrete creep rule, with fitting precision higher.

Description

A kind of concrete fractional order creep model

Technical field

The present invention relates to predict concrete creep field, a kind of especially concrete fractional order creep model.

Background technology

Concrete is a kind of Xu's variant material, i.e., under normal stress, extension over time, strain will constantly increase Plus.Concrete creep is not only relevant with the lotus time is held, and relevant with load age etc., and loading is more early, creeps bigger.Not only such as This, temperature also has large effect to concrete creep performance, and temperature is complicated on concrete creep influence relation, not letter Single temperature bigger monotonic relationshi of creeping higher.

Concrete creep has a significant impact to the stress and deformation performance of concrete, how to select one can accurately predict Xu The forecast model of change is always the focus of research.Creep prediction model expression is reflection Creep Coefficient or time deformation with the time The mathematical functional expression of Changing Pattern, the validity and accuracy of creep prediction are heavily dependent on the choosing of creep prediction model Take.For the mathematical function construction of model, the forecast model of domestic and international concrete creep can be roughly divided into three major types：(1) multiply Product module type, will be crept and be expressed as the product of some subitems, and concrete creep is not finely divided, and be described from overall, such as ACI209R series models etc.；(2) and formula model, will creep and be expressed as some subitem sums, such as CEB-FIP series models, B- P series models etc.；(3) mixed model, will product model and with the mixing of formula model, such as GL-2000 models.

Above-mentioned these models are each defective, and such as CEB-FIP (1978) model accuracy is relatively low, creep calculated value and reality of creeping Value degree of agreement is not high；ACI209 model-test datas to meet situation poor, generally underestimate contraction and creep beharior；Eight ginsengs Though exponential model can very well meet rule of creeping, it needs to be determined that parameter it is more.Be given separately below each model expression formula and Major defect, refers to table 1, table 2.

The existing concrete creep model of table 1

For above-mentioned these model limitation and applicability, table 2 below lists the major defect of existing creep model in detail And the scope of application.

The existing model scope of application of table 2 and major defect

For the defect that above-mentioned conventional creep model is present, set up a kind of new using fractional calculus theory below Concrete creep model.

The content of the invention

The technical problems to be solved by the invention are to provide a kind of concrete fractional order creep model, by fractional calculus Rheological model, the creep property of Cemented and aquation topology degree are combined, and establish a kind of concrete fractional order for considering degree of hydration Creep model.Compared with traditional creep model, the new model that the present invention is set up fully has reacted concrete creep with loading age Phase, the related characteristic of lotus time, hydrated reaction of cement is held, and the new model set up also overcomes eight conventional parameters and creeps The more shortcoming of model parameter, the fitting effect analysis to this model understands that fractional model meets concrete creep rule, tool There is fitting precision higher.

In order to solve the above technical problems, the technical solution adopted in the present invention is：

A kind of concrete fractional order creep model,

In the case that stress is constant, according to Riemann-Liouville type fractional calculus operator theories, base is set up It is in the creep model expression formula of fractional calculus

In formula：C (t, τ) be loading age be τ, hold the lotus time be t- τ creep degree, f₁、g₁、p₁、r₁, β be coefficient, and 0 <β≤1。

Concrete creep degree is fitted using the concrete creep model of fractional calculus, unknown ginseng in formula (1) Number has 5, respectively f₁、g₁、p₁、r₁、β；

Each undetermined parameter is designated as X, i.e. X=[x₁,x₂,x₃,x₄,x₅]^T, and Prescribed Properties：0<x₅≤ 1, x_i≥0(i =1~4), then have

Crept with experiment and value and calculate the residual sum of squares (RSS) of value of creeping, as the object function of parametric inversion optimization problem, 5 parameters of expression formula of being crept with the fractional calculus for seeking concrete, i.e.,

In formula：C (t, τ) is creep degree calculated value, and C ' (t, τ) is creep degree experiment value, and F (X) is object function, x_iTo treat Determine parameter,

So as to try to achieve each undetermined parameter.

In the case where hydrated reaction of cement is considered, convolution (1) obtains considering the concrete fraction of hydrated reaction of cement The expression formula of rank creep model is

In formula：C(t_e,τ_e) for loading age be τ_e, hold the lotus time for t_e-τ_eCreep degree, f₁、g₁、p₁、r₁, β be coefficient, And 0<β≤1

In the case where hydrated reaction of cement is considered, convolution (1) obtains considering the concrete fraction of hydrated reaction of cement Rank creep model the step of it is as follows：

Step 1：Degree of hydration described using heat of hydration method, that is, it is the heat that has discharged at certain moment and final to define degree of hydration The ratio of the heat for discharging completely, hydrated cementitious can be accelerated with the rising of temperature,

Based on the solution reaction speed constant β that Arrhenius equations are proposed_TRelational expression with temperature is

U_hIt is aquation activity energy；R is gas constant, is 80315J/K；T is concrete actual temperature；T₀It is reference temperature；

Step 2：Based on the concept of equivalent age, the equivalent age of Arrhenius functional forms is proposed, expression formula is

U in formula_hIt is aquation activity energy；R is gas constant, is 80315J/K；T is concrete actual temperature；T₀It is with reference to temperature Degree, generally takes 293K, i.e., 20 DEG C；

Wherein, the computation model on activation energy is as follows：

Step 3：Regression analysis is carried out by Adiabatic temperature rise of concrete data under existing different temperatures, it is believed thatWith Concrete actual temperature T meets following relation

Using the principle of above-mentioned equivalent age, by t, τ is respectively for t_e,τ_e, convolution (1) may be accounted cement water The expression formula of concrete fractional order creep model for changing reaction is

In formula：C(t_e,τ_e) for loading age be τ_e, hold the lotus time for t_e-τ_eCreep degree, f₁、g₁、p₁、r₁, β be coefficient, And 0<β≤1.

Concrete creep degree is fitted using the concrete creep model of fractional calculus, unknown ginseng in formula (1) Number has 5, respectively f₁、g₁、p₁、r₁、β；

Each undetermined parameter is designated as X, i.e. X=[x₁,x₂,x₃,x₄,x₅]^T, and Prescribed Properties：0<x₅≤ 1, x_i≥0(i =1~4), then have

Crept with experiment and value and calculate the residual sum of squares (RSS) of value of creeping, as the object function of parametric inversion optimization problem, 5 parameters of expression formula of being crept with the fractional calculus for seeking concrete, i.e.,

In formula：C (t, τ) is creep degree calculated value, and C ' (t, τ) is creep degree experiment value, and F (X) is object function, x_iTo treat Determine parameter；

Similarly, with the theory of above-mentioned equivalent age, by t, τ is respectively for t_e,τ_e, convolution (3) can obtain, it is considered to The object function of concrete creep model parameter inverting in the case of hydrated reaction of cement, i.e.,

x_iIt is 5 parameters of fractional model,

So as to try to achieve each undetermined parameter.

The method for trying to achieve each undetermined parameter is：

Step 1：Determine that variable number is 5, apexes of complex number k takes 6, and precision value is 1e-6；

Step 2：Intial compound form is produced, in restriction rangeK random point of middle generation, is constituted just Beginning complex；

Step 3：Convergence is carried out to formula (1) and (9), it is true with reference to specific example according to the convergence of infinite series Determine the value of n；

Step 4：Calculating is iterated by the MATLAB programs of complex method, using each vertex function value size of complex Relation, judges the descent direction of target function value, constantly loses worst point, complex is constantly shunk to optimum point, Zhi Daoman Untill sufficient convergence precision,

So as to obtain each undetermined parameter.

A kind of concrete fractional order creep model that the present invention is provided, by fractional calculus rheological model, concrete slowly Become characteristic and aquation topology degree is combined, establish a kind of concrete fractional order creep model for considering degree of hydration.With traditional Xu Varying model is compared, and the new model that the present invention is set up fully has reacted concrete creep and load age, held lotus time, cement water Change the related characteristic of reaction, and the new model set up also overcomes that eight conventional parameter creep model parameters are more to be lacked Point, the fitting effect analysis to this model understands that fractional model meets concrete creep rule, with fitting essence higher Degree.

Specific embodiment

Embodiment one (does not consider the fractional order concrete creep model of hydrated reaction of cement)

A kind of concrete fractional order creep model,

In the case that stress is constant, according to Riemann-Liouville type fractional calculus operator theories, base is set up It is in the creep model expression formula of fractional calculus

In formula：C (t, τ) be loading age be τ, hold the lotus time be t- τ creep degree, f₁、g₁、p₁、r₁, β be coefficient, and 0 <β≤1。

Concrete creep degree is fitted using the concrete creep model of fractional calculus, unknown ginseng in formula (1) Number has 5, respectively f₁、g₁、p₁、r₁、β；

Each undetermined parameter is designated as X, i.e. X=[x₁,x₂,x₃,x₄,x₅]^T, and Prescribed Properties：0<x₅≤ 1, x_i≥0(i =1~4), then have

Crept with experiment and value and calculate the residual sum of squares (RSS) of value of creeping, as the object function of parametric inversion optimization problem, 5 parameters of expression formula of being crept with the fractional calculus for seeking concrete, i.e.,

In formula：C (t, τ) is creep degree calculated value, and C ' (t, τ) is creep degree experiment value, and F (X) is object function, x_iTo treat Determine parameter,

So as to try to achieve each undetermined parameter.

In order to verify the practicality and accuracy of the above method, divided using Gong's mouth Gravity Dam Foundation part concrete Analysis.For Gong's mouth gravity dam concrete creep degree, its concrete numerical value is given below, see the table below 3.

The Gong's mouth gravity dam concrete creep degree experiment value of table 3

The complex method principle and calculation procedure introduced according to above-mentioned Section 3, calculating journey is write using MATLAB language Sequence, with reference to the concrete creep experiment value of Gong's mouth gravity dam, the expression formula of creeping of the fractional order to being used carries out parametric inversion. During calculating, the present patent application to 180d before Gong's mouth gravity dam concrete load time, hold the lotus time before 720d be fitted.Pass through Convergence is carried out to fractional model expression formula, when n values are more than 40, this model expression is substantially at convergence, is This this patent takes n=40 when MATLAB programs are write, and is found by sensitivity analysis, f₁、r₁, β it is sensitive compared with other two parameters, Initial composite type is debugged, it is 4.87 to obtain the corresponding gentle F of residual error (X).Ginseng inversion result is as shown in table 4 below, fraction Rank fitting formula such as formula (11), fractional model creep degree calculated value the results are shown in Table 5.

The fractional model parametric inversion result of table 4

The Gong's mouth gravity dam concrete creep degree fractional model calculated value of table 5

Inverting is carried out to eight parameter models, is obtained shown in creep degree expression formula such as following formula (12), meanwhile, eight parameter models are slowly The calculated value of variation is as shown in table 6 below.

C (t, τ)=(7+64.4 τ^-0.45)[1-e^-0.3(t-τ)]+(16+27.2τ^-0.45)[1-e^-0.005(t-τ)]

(12)

The parameter model calculated value of 6 Gong's mouth gravity dam concrete creep degree fractional order of table eight

By contrasting concrete creep degree experiment value, fractional model calculated value, eight parameter models：Fractional order mould Type, eight parameter models can preferably fit concrete creep rule, and fractional model fitting precision is in general with eight Parameter model fitting precision is more or less the same, but the parameter determined needed for fractional model is less than the ginseng determined needed for eight parameter models Number.

Precision analysis is carried out to the fractional model that embodiment one determines below, if With f/f₀Value as precision judge standard, by fractional model calculated value and creep degree experiment value Bring f and f into₀In expression formula, f/f can be obtained₀=8.8%<10%, its value illustrates above formula less than the higher limit of defined in engineering (11) accuracy of fitting is high, can be engineering services.

Embodiment two (considers the fractional order concrete creep model of hydrated reaction of cement)

In the case where hydrated reaction of cement is considered, convolution (1) obtains considering the concrete fraction of hydrated reaction of cement The expression formula of rank creep model is

In formula：C(t_e,τ_e) for loading age be τ_e, hold the lotus time for t_e-τ_eCreep degree, f₁、g₁、p₁、r₁, β be coefficient, And 0<β≤1

In the case where hydrated reaction of cement is considered, convolution (1) obtains considering the concrete fraction of hydrated reaction of cement Rank creep model the step of it is as follows：

Step 1：Degree of hydration described using heat of hydration method, that is, it is the heat that has discharged at certain moment and final to define degree of hydration The ratio of the heat for discharging completely, hydrated cementitious can be accelerated with the rising of temperature,

Based on the solution reaction speed constant β that Arrhenius equations are proposed_TRelational expression with temperature is

U_hIt is aquation activity energy；R is gas constant, is 80315J/K；T is concrete actual temperature；T₀It is reference temperature；

Step 2：Based on the concept of equivalent age, the equivalent age of Arrhenius functional forms is proposed, expression formula is

τ_e=∫ β_Tdt (6)

U in formula_hIt is aquation activity energy；R is gas constant, is 80315J/K；T is concrete actual temperature；T₀It is with reference to temperature Degree, generally takes 293K, i.e., 20 DEG C；

Wherein, the computation model on activation energy is as follows：

Step 3：Regression analysis is carried out by Adiabatic temperature rise of concrete data under existing different temperatures, it is believed thatWith Concrete actual temperature T meets following relation

Using the principle of above-mentioned equivalent age, by t, τ is respectively for t_e,τ_e, convolution (1) may be accounted cement water The expression formula of concrete fractional order creep model for changing reaction is

In formula：C(t_e,τ_e) for loading age be τ_e, hold the lotus time for t_e-τ_eCreep degree, f₁、g₁、p₁、r₁, β be coefficient, And 0<β≤1.

Concrete creep degree is fitted using the concrete creep model of fractional calculus, unknown ginseng in formula (1) Number has 5, respectively f₁、g₁、p₁、r₁、β；

Each undetermined parameter is designated as X, i.e. X=[x₁,x₂,x₃,x₄,x₅]^T, and Prescribed Properties：0<x₅≤ 1, x_i≥0(i =1~4), then have

Crept with experiment and value and calculate the residual sum of squares (RSS) of value of creeping, as the object function of parametric inversion optimization problem, 5 parameters of expression formula of being crept with the fractional calculus for seeking concrete, i.e.,

In formula：C (t, τ) is creep degree calculated value, and C ' (t, τ) is creep degree experiment value, and F (X) is object function, x_iTo treat Determine parameter；

Similarly, with the theory of above-mentioned equivalent age, by t, τ is respectively for t_e,τ_e, convolution (3) may be accounted water The object function of concrete creep model parameter inverting in the case of mud hydration reaction, i.e.,

x_iIt is 5 parameters of fractional model,

So as to try to achieve each undetermined parameter.

The method for trying to achieve each undetermined parameter is：

Step 1：Determine that variable number is 5, apexes of complex number k takes 6, and precision value is 1e-6；

Step 2：Intial compound form is produced, in restriction rangeK random point of middle generation, is constituted just Beginning complex；

Step 3：Convergence is carried out to formula (1) and (9), it is true with reference to specific example according to the convergence of infinite series Determine the value of n；

Step 4：Calculating is iterated by the MATLAB programs of complex method, using each vertex function value size of complex Relation, judges the descent direction of target function value, constantly loses worst point, complex is constantly shunk to optimum point, Zhi Daoman Untill sufficient convergence precision,

So as to obtain each undetermined parameter.

The practicality of the checking above method and the content of the test of accuracy are as follows：

1st, experiment of creeping is compressed

This experiment is used to study concrete compression and creeps rule, and contrast has been carried out under many age natural curings and standard curing Lower concrete for hydraulic structure uniaxial compression creep test, is below carried out using the fractional order creep model for considering degree of hydration to test data Analysis.

(1) concrete creep test parameters and instrument and equipment

Concrete in uniaxial compresses creep the concrete prism test specimen that test specimen is 150mm × 150mm × 550mm, stress ratio It is 0.3 (Compression is the 30% of failing load)；Loading device uses the production of Shanghai Hua Jing Trade Co., Ltd.s XBJ-500 type Creep Apparatus；Load transducer model BLR-2 types；Because LVDT and outer patch foil gauge receive outside environmental elements shadow Sound is larger, and S-100 types difference resistive strain gauge is buried using Inner, and strain gauge uses the digital bridge measurements of SO-5.

(2) concrete raw material and match ratio

As shown in table 7, C30 graduation twos are mixed the main physico-mechanical performance of the raw material that the experiment of this concrete creep is selected The match ratio for coagulating soil is as shown in table 8.Wherein, the ratio of mud 0.5, sand coarse aggregate ratio 35%, doping quantity of fly ash 35%.

The basic physical property of the raw material of table 7

The compression creep test concrete mix of table 8 Kg/m³

(3) concrete in uniaxial compression test scheme

2 groups of contrast tests of creeping are devised, as shown in table 9, natural curing is (i.e. outdoor to support in the winter time for battery of tests test specimen Shield), another battery of tests test specimen is conserved (i.e. standard curing) in standard curing room, respectively maintenance 3d, 7d, 14d, 21d, After 28d ages, creep test is compressed in laboratory of creeping, a length of 2 months during loading.3 test specimens of creeping in table 3, wherein 2 Individual to be crept test specimen as loading, 1 used as compensation test specimen in addition；3 standard cube test specimens determine concrete crushing strength.Its In, the temperature range during field curing is 6~12 DEG C, and mean temperature is 9.2 DEG C, and relative humidity is 80% or so；Standard is supported Shield temperature is 20 ± 2 DEG C, relative humidity more than 95%；When conserving load age, be further applied load beginning creep test, reality of creeping The temperature of room is tested between 11 DEG C~12 DEG C, relative humidity is basically stable at more than 90%.

The concrete in uniaxial of table 9 compresses experimental program table of creeping

2nd, parametric inversion is calculated

From table 4, because the creep test time is long, the equipment that is put to the test and place limit, under the conditions of standard curing, plus When carrying age for 21d, only hold lotus 1 month and terminate experiment；And load age does not conform to for the concrete sample result of the test of 28d Reason, therefore related data is not used.

Because the fractional order creep model for considering degree of hydration is an infinite series expression formula, by fractional model table Convergence is carried out up to formula, when n values are more than 50, this model expression is substantially at convergence, is that this this patent is being write During MATLAB programs, n=50 is taken.Found by sensitivity analysis, f₁、r₁, β it is sensitive compared with other two parameters.Initial composite type is entered Row debugging, the gentle F of the fractional model residual error (X) that temperature influence is considered under the standard curing that calculating is obtained is 1.81, and calculating is obtained Consider that the fractional order creep model parameter of degree of hydration is shown in Table 10, considers the fraction of degree of hydration under standard curing under the standard curing for obtaining The creep degree calculated value of rank creep model is shown in Table 12；The fractional order creep model of degree of hydration is considered under the field curing that calculating is obtained Parameter is shown in Table 11, and its corresponding residual sum of squares (RSS) F (X) is 6.46, and the fractional order creep model of degree of hydration is considered under field curing Creep degree calculated value be shown in Table 13.

Fractional model parametric fitting results under the standard curing operating mode of table 10

According to the parametric inversion result of table 10, the fractional order that can obtain the influence of the consideration temperature under the conditions of standard curing is crept Model expression is

Fractional model parametric fitting results under natural curing operating mode outside the Room of table 11

According to the parametric inversion result of table 11, the fractional order of the consideration temperature influence under outdoor natural curing maintenance can be obtained Creep model expression formula is

The fractional order creep degree calculated value of typical time under the standard curing of table 12 each age

Outside the Room of table 13 under natural curing each age typical time fractional order creep degree calculated value

Similarly, parametric inversion is carried out to eight parameter models under the conditions of standard curing, obtains considering temperature on creep influence Eight parameter expressions as shown in following formula 15, while, it is considered to temperature influence eight parameter model creep degrees calculated value such as table 14 below It is shown.

C(t_e,τ_e)=(22.65+15 τ_e ^-300)[1-e^{-0.0405(t-τ)}]+(3.19+92.54τ_e ^-1.42)[1-e^-0.6(t-τ)] (15)

Parametric inversion is carried out to eight parameter models under the conditions of outdoor natural curing, obtains considering the eight of temperature on creep influence Parameter expression as shown in following formula 16, while, it is considered to temperature influence eight parameter model creep degrees calculated value such as table 15 below institute Show.

C(t_e,τ_e)=(15.38+74.18 τ_e ^-1.06)[1-e^-0.06(t-τ)]+(11.18+186.77τ_e ^-1.9)[1-e^-1.83(t-τ)] (16)

The equivalent load age of typical time and eight parameter creep degree calculated values under the standard curing of table 14 each age

The equivalent load age of typical time and eight parameter creep degree calculated values under natural curing each age outside the Room of table 15

Creep degree experiment value, fractional order mould under the conditions of difference comparative analysis standard curing and under the conditions of outdoor natural curing Type calculated value, eight parameter model calculated values understand：Under the conditions of standard curing, when loading age is 3d, 7d, 14d, fractional order is crept Model and 8 parameter creep model fitting effects are all fine, are relatively coincide with experiment value；But when loading age is 21d, fractional order is slowly Varying model is better than 8 parameter creep model fitting effects.

Claims (6)

1. a kind of concrete fractional order creep model, it is characterised in that：

In the case that stress is constant, according to Riemann-Liouville type fractional calculus operator theories, sets up and be based on dividing The creep model expression formula of number rank calculus is

$C(t,\mathrm{\&tau;})=(f_1+g_1{\mathrm{\&tau;}}^{-p_1})\underset{n=0}{\overset{\mathrm{\&infin;}}{\&Sigma;}}\frac{{(-1)}^n{\left(r_1\right)}^{n+1}{(t-\mathrm{\&tau;})}^{(n+1)\mathrm{\&beta;}}}{\mathrm{\&Gamma;}\&lsqb;(n+1)\mathrm{\&beta;}+1)\&rsqb;}---\left(1\right)$

In formula：C (t, τ) be loading age be τ, hold the lotus time be t- τ creep degree, f₁、g₁、p₁、r₁, β be coefficient, and 0<β≤ 1。

2. a kind of concrete fractional order creep model according to claim 1, it is characterised in that：Using fractional calculus Concrete creep model concrete creep degree is fitted, unknown parameter has 5, respectively f in formula (1)₁、g₁、p₁、r₁、 β；

Each undetermined parameter is designated as X, i.e. X=[x₁,x₂,x₃,x₄,x₅]^T, and Prescribed Properties：0<x₅≤ 1, x_i>=0 (i=1~ 4), then have

$C(t,\mathrm{\&tau;})=(x_1+x_2{\mathrm{\&tau;}}^{-x_3})\underset{n=0}{\overset{\mathrm{\&infin;}}{\&Sigma;}}\frac{{(-1)}^n{\left(x_4\right)}^{n+1}{(t-\mathrm{\&tau;})}^{(n+1)x_5}}{\mathrm{\&Gamma;}\&lsqb;(n+1)x_5+1\&rsqb;}---\left(2\right)$

Crept with experiment and value and calculate the residual sum of squares (RSS) of value of creeping, as the object function of parametric inversion optimization problem, to seek The fractional calculus of concrete are asked to creep 5 parameters of expression formula, i.e.,

$\left\{\begin{array}{c}F\left(X\right)=\&Sigma;{\&lsqb;C(t,\mathrm{\&tau;})-C^{\&prime;}(t,\mathrm{\&tau;})\&rsqb;}^2\&RightArrow;\min \\ x_i>0(i=1~4)\\ 0<x_5\&le;1\end{array}\right.---\left(3\right)$

In formula：C (t, τ) is creep degree calculated value, and C ' (t, τ) is creep degree experiment value, and F (X) is object function, x_iFor to be determined Parameter,

So as to try to achieve each undetermined parameter.

3. a kind of concrete fractional order creep model according to claim 1, it is characterised in that considering that hydrated cementitious are anti- In the case of answering, the expression formula that convolution (1) obtains the concrete fractional order creep model for considering hydrated reaction of cement is

$C(t_e,{\mathrm{\&tau;}}_e)=(f_1+g_1{{\mathrm{\&tau;}}_e}^{-p_1})\underset{n=0}{\overset{\mathrm{\&infin;}}{\&Sigma;}}\frac{{(-1)}^n{\left(r_1\right)}^{n+1}{(t_e-{\mathrm{\&tau;}}_e)}^{(n+1)\mathrm{\&beta;}}}{\mathrm{\&Gamma;}\&lsqb;(n+1)\mathrm{\&beta;}+1)\&rsqb;}---\left(9\right)$

In formula：C(t_e,τ_e) for loading age be τ_e, hold the lotus time for t_e-τ_eCreep degree, f₁、g₁、p₁、r₁, β be coefficient, and 0< β≤1

4. a kind of concrete fractional order creep model according to claim 3, it is characterised in that considering that hydrated cementitious are anti- Should in the case of, convolution (1) obtain consider hydrated reaction of cement concrete fractional order creep model the step of it is as follows：

Step 1：Degree of hydration is described using heat of hydration method, that is, it is that the heat for having discharged at certain moment is complete with final to define degree of hydration The ratio of the heat of release, hydrated cementitious can be accelerated with the rising of temperature,

Based on the solution reaction speed constant β that Arrhenius equations are proposed_TRelational expression with temperature is

${\mathrm{\&beta;}}_T=\exp \&lsqb;\frac{U_h}{R}(\frac{1}{T_0}-\frac{1}{T})\&rsqb;---\left(5\right)$

U_hIt is aquation activity energy；R is gas constant, is 80315J/K；T is concrete actual temperature；T₀It is reference temperature；

Step 2：Based on the concept of equivalent age, the equivalent age of Arrhenius functional forms is proposed, expression formula is

τ_e=∫ β_Tdt (6)

U in formula_hIt is aquation activity energy；R is gas constant, is 80315J/K；T is concrete actual temperature；T₀It is reference temperature, Generally take 293K, i.e., 20 DEG C；

Wherein, the computation model on activation energy is as follows：

$E_a=\left\{\begin{array}{cc}33500J\&CenterDot;{\mathrm{mol}}^{-1}& T_c\&GreaterEqual;20\\ 33500+1470(20-T_c)J\&CenterDot;{\mathrm{mol}}^{-1}& T_c<20\end{array}\right.---\left(7\right)$

Step 3：Regression analysis is carried out by Adiabatic temperature rise of concrete data under existing different temperatures, it is believed thatWith concrete Actual temperature T meets following relation

$\frac{U_h}{R}=16.187T^{1.0002}---\left(8\right)$

Using the principle of above-mentioned equivalent age, by t, τ is respectively for t_e,τ_e, convolution (1) may be accounted hydrated cementitious anti- The expression formula of the concrete fractional order creep model answered is

$C(t_e,{\mathrm{\&tau;}}_e)=(f_1+g_1{{\mathrm{\&tau;}}_e}^{-p_1})\underset{n=0}{\overset{\mathrm{\&infin;}}{\&Sigma;}}\frac{{(-1)}^n{\left(r_1\right)}^{n+1}{(t_e-{\mathrm{\&tau;}}_e)}^{(n+1)\mathrm{\&beta;}}}{\mathrm{\&Gamma;}\&lsqb;(n+1)\mathrm{\&beta;}+1)\&rsqb;}---\left(9\right)$

In formula：C(t_e,τ_e) for loading age be τ_e, hold the lotus time for t_e-τ_eCreep degree, f₁、g₁、p₁、r₁, β be coefficient, and 0< β≤1。

5. a kind of concrete fractional order creep model according to claim 4, it is characterised in that：Using fractional calculus Concrete creep model concrete creep degree is fitted, unknown parameter has 5, respectively f in formula (1)₁、g₁、p₁、r₁、 β；

Each undetermined parameter is designated as X, i.e. X=[x₁,x₂,x₃,x₄,x₅]^T, and Prescribed Properties：0<x₅≤ 1, x_i>=0 (i=1~ 4), then have

$C(t,\mathrm{\&tau;})=(x_1+x_2{\mathrm{\&tau;}}^{-x_3})\underset{n=0}{\overset{\mathrm{\&infin;}}{\&Sigma;}}\frac{{(-1)}^n{\left(x_4\right)}^{n+1}{(t-\mathrm{\&tau;})}^{(n+1)x_5}}{\mathrm{\&Gamma;}\&lsqb;(n+1)x_5+1\&rsqb;}---\left(2\right)$

Crept with experiment and value and calculate the residual sum of squares (RSS) of value of creeping, as the object function of parametric inversion optimization problem, to seek The fractional calculus of concrete are asked to creep 5 parameters of expression formula, i.e.,

$\left\{\begin{array}{c}F\left(X\right)=\mathrm{\&Sigma;}{\&lsqb;C(t,\mathrm{\&tau;})-C^{\&prime;}(t,\mathrm{\&tau;})\&rsqb;}^2\&RightArrow;min\\ x_i>0(i=1~4)\\ 0<x_5\&le;1\end{array}\right.---\left(3\right)$

In formula：C (t, τ) is creep degree calculated value, and C ' (t, τ) is creep degree experiment value, and F (X) is object function, x_iFor to be determined Parameter；

Similarly, with the theory of above-mentioned equivalent age, by t, τ is respectively for t_e,τ_e, convolution (3) may be accounted cement water Change the object function of concrete creep model parameter inverting in the case of reacting, i.e.,

$\left\{\begin{array}{c}F\left(X\right)=\&Sigma;{\&lsqb;C(t_e,{\mathrm{\&tau;}}_e)-C^{\&prime;}(t_e,{\mathrm{\&tau;}}_e)\&rsqb;}^2\&RightArrow;\min \\ x_i>0(i=1~4)\\ 0<x_5\&le;1\end{array}\right.---\left(10\right)$

x_iIt is 5 parameters of fractional model,

So as to try to achieve each undetermined parameter.

6. a kind of concrete fractional order creep model according to claim 5, it is characterised in that try to achieve each undetermined parameter Method is：

Step 1：Determine that variable number is 5, apexes of complex number k takes 6, and precision value is 1e-6；

Step 2：Intial compound form is produced, in restriction rangeK random point of middle generation, constitutes initial composite Shape；

Step 3：Convergence is carried out to formula (1) and (9), according to the convergence of infinite series, determines n's with reference to specific example Value；

Step 4：Calculating is iterated by the MATLAB programs of complex method, using the pass of each vertex function value size of complex System, judges the descent direction of target function value, constantly loses worst point, complex is constantly shunk to optimum point, until meeting Untill convergence precision,

So as to obtain each undetermined parameter.