5　Numerical example

5.1　Model parameters

This section illustrates the efficiency of the proposed method.A sluice pier was investigated as an example in the paper.Temperature field was analysed by the program FZFX3D.General preprocessing of the software is developed based on the finite element software ANSYS.Moreover，FZFX3D coded in FORTRAN 90 was used to calculate the temperature field and the stress field，which considered the heat generation and the cooling water pipe cooling process.

The thermal parameters are identified by the GA with the following characteristics：the initial value and the initial range of the（α，β，θ0，a，b）are set as（0.086 m2/d，1450 kJ/m2d℃，27℃，0.33，0.92）and（0.07～0.15 m2/d，1000～2000 kJ/m2d ℃，25～50℃，0.25～0.70，0.5～1.8），respectively.The population size，mutation method and stopping criteria are designed as 20，Gaussian and Er≤50，respectively.And the final results of the（α，β，θ0，a，b）are computed using the GA tool of the MATLAB 7.0 as（0.1032 m2/d，1500 kJ/m2d℃，32℃，0.65，1）by 116 generations（in Table 1）.

The parameters of creep and elastic modulus are identified by the GA with the following characteristics：all of the initial value of（x1，x2，x3，x4，x5，x6，x7，x8，A，B）are zero and the initial range of the ten parameters are selected as（0～10，0～60，0～10，0～10，0～10，0～60，0～10，0～10，0～1，0～1），respectively；The population size，mutation method and stopping criteria are designed as 50，Gaussian and Er，σ≤2，respectively；and the final results of the（x1，x2，x3，x4，x5，x6，x7，x8，A，B）are computed as Table 2.

5.2　Model details and boundary conditions

The geometry size and finite element model of sluice pier are shown in Figure 8（a）and（b），respectively.In the structure，the points DT01 and DT02 are selected in the back analysis.Meanwhile，the points 01，02 and 03 are adopted in the temperature and thermal stress analysis.The finite element model is divided into space hexahedral element.Different colour denotes the different kind of concrete.Moreover，the FEM is used to simulate the process of concrete layered casting.Each layer thickness is 0.75 metre along the longitudinal direction，and total 60924 elements and 69372 nodes were divided in the model.The concrete and reinforcement can be modelled by Solid 65 and Link 8，respectively.Besides，in this model，adiabatic boundary condition and fully constrained mechanical boundary are adopted in the surrounding rock.

In the temperature field analysis，the third-type boundary condition is adopted and concrete is adiabatic and fully restraint around the surrounding rock.To verify the accuracy of proposed method，comparisons between calculation results and experimental results are performed，and excellent agreements with measured results are obtained for the example structure（shown in Figure 9）.The proposed model can be used to estimate the generation of the cracking and crack size in massive RC structure.

In the example，concrete nearby the bedrock was chosen as the research object，in which original reinforcement scheme is 5Φ36（means five Φ36 diameter reinforcing steel per metre）.The maximum principal stress value is between -1.59 and 5.76 MPa.

Moreover，the temperature histories of points 01，02 and 03 are shown in Figure 10.After temperature rise and drop，the concrete temperature changes with the air temperature.The maximum temperatures of surface point（01）and internal point（03）are 32.53 and 45.09℃，respectively.Meanwhile，they reached their maximum temperatures in 2.5 and 4.0 days，respectively.The maximum temperature of concrete can reach 45.09℃ due to the high cement content.Besides，the sluice pier structure is thin，thus the time of the highest temperature is also relatively early.The thermal stress histories of point 01，02 and 03 are shown in Figure 11.In the early time，concrete was in the expansion state and produced about 0.4～0.8 MPa compressive stress.The surface point 01 began to produce tensile stress at the age of 6.74 day.After that，the tensile stress increased with temperature reduction.The maximum tension stress was 3.59 MPa when the temperature dropped to the lowest value.

The whole calculation was divided into 17 load steps.In step 5，the maximum tensile stress exceeds the tensile strength of concrete and then crack appears.Crack width increased with the increase in stress，and crack width had a sudden increase value.Meanwhile，reinforcement stresses had a sudden increase when cracks appeared.The variation curves of crack width and reinforcement stress were shown in Figure 12.Finally，the calculation value of the maximum crack width can be achieved as 0.22 mm.Meanwhile，the corresponding maximum reinforcement stress value is 83.17 MPa.Both of the crack width and reinforcement stress conform to the design requirements.

However，in the structure，all the reinforcement stress values are very small except the stress in the cracking area.Most of the reinforcement did not have enough effect.Consequently，four reinforcement design schemes were carried out to improve the effect of reinforcement，and the calculation results were shown in Table 3.

5.3　Reinforcement optimisation

The cracking patterns and stress distribution with different reinforcement schemes were shown in Table 3.Considering the decrease in the reinforcement，and meanwhile satisfied the objective function Equations 17 and 18，the optimisation scheme of reinforcement configuration should choose Scheme 3（336）.But the maximum crack width of Scheme 3 is slightly bigger than limiting value in GB 50496—2009.Consequently，the Scheme 5（432）was selected as the best in the end，in which the total amount of reinforcement was reduced by 38% comparing with the Scheme 1.Meanwhile，both of the increase in crack width and depth were very small.Reinforcement stressi ncreased by 31.55%，in other words，reinforcement utilisation rate was enhanced.