4 Reinforcement design method
4.1 Plane substructure theory
The reinforcement design method of massive RC can be divided into two parts.Firstly，3D elastic finite element model was built to simulate the structure in order to determine the weak parts of structure.Secondly，3D space problem was transformed into 2D plane problem through the “plane substructure theory”.Finally，crack width and reinforcement stress can be obtained under different reinforcement design schemes.
It is feasible that the transformation 3D elastic finite element calculation results are employed as the load boundary condition of plane substructure nonlinearity.In addition，in the“plane substructure theory”，the entire structure stress distribution can be obtained after meshing.Plane substructure is analysed as a part of the overall structure，which satisfies Equation （18）.
The plane subsection element satisfies Equations 19，20 and 21.
“Plane substructure”is a unit thickness plane in the weak part of the structure.The selection of plane substructure had two principles.Firstly，plane substructure should contain the maximum stress area of 3D finite element calculation results.Secondly，plane substructure should include the main reinforcement.There are three steps to obtain the “plane substructure”.Step 1：Through 3D finite element calculation and stresses，results in comparison，the weak parts of the structure can be selected.And unit thickness plane in the weak regions is taken as“plane substructure”；Step 2：Data were transferred from 3D FEM to 2D FEM，and then 2D FEM was calculated；Step 3：2D plane substructure was analysed.And then cracking patterns and reinforcement stress distribution can be obtained.In short，three steps are shown in detail in Figure 7.
4.2 Reinforcement designs of plane substructure
The principle of reinforcement design is to decrease reinforcement on the condition of control crack width.Consequently，the objective function can be written as Equations （22） and （23）.
Figure 7 Flow chart of reinforcement design method
where xopt，iis the ideal rebar diameter；nidenotes the number of reinforcement bars layer；liis the length of rebar；Mmeans the row number of rebar；xirepresents rebar diameter；fy，idenotes design value of rebar stress；and siis the stress values corresponding with xi.
The integrated diagram of thermal stress analysis and reinforcement design can be introduced in Figure 2“Part 1”is about temperature field analysis.Both of the measured temperature and calculated temperature are adopted in the GA，and then the important thermal parameters of site concrete can be obtained.After thermal field calculation of finite element model，all temperature variations were obtained.“Part 2”is the thermal stress analysis，which considers the variation of elastic modulus，concrete creep and tensile strength.“Part 3”is the reinforcement design method of massive RC structures.