3　Thermal stress analysis

In order to measure the development of thermal stress in the concrete interior，temperature stress tests were carried out to obtain the measured thermal stresses using the temperature stress test machine（TSTM）.The shape and working principle of TSTM are shown in Figures 4 and 5.TSTM is a closed-looped，uniaxial restrained testing device，which has three main functions：the measurement of load；the measurement and control of temperature；the measurement and control of deformation；and the measurement of reinforcement stain.For the concrete specimen，one of the cross-heads is restrained by the steel claw and the other is controlled by a stepper motor according to the specimen deformation.The load on the specimen is monitored by a load cell with accuracy of 1N placed at the adjustable cross-head（Shi，Ouyang，Zhang & Huang，2014）.

Thermal stresses of the structure were calculated on the basis of the reproduced temperature field and the effect of time-dependent elastic modulus.In addition，concrete creep was incorporated in the stress analysis（Amin et al.，2009）.

The detailed description of thermal stress analysis is shown in the part two of Figure 2.A general relationship for the thermal stress calculation is as follows（Amin et al.，2009）.

where E（τ） is the transient elastic modulus；σ（τ）is the transient tensile stress；C（t，τ）is the creep degree；γis the thermal dilation coefficient；ΔTdenotes the temperature change；Kr[Kr=（0～100%）]is the degree of restraint；if the deformation is totally free，Kr=0；if the deformation is fully restrained，Kr=100%.In this paper，good agreement of stresses was noted between the calculated values and the experimental values in Figure 6.

In creep analysis，numerous models[e.g.the CEB-FIP Model（CEB-FIP Model code for concrete structures，1978]，the ACI Model（ACI Committee 209，1978），the BP Model（Bazant & Panula，1982）and the Exponential Model（Bazant & Wu，1974））are used to calculate the thermal stress of concrete.In view of memory usage，the Exponential Model was selected in the code（Zhu，2009）.Besides，the creep degree can be calculated as follows：

where C（t，τ） denotes the creep coefficient；tis the time；τis the loading age.The creep parameters x1～x8 are constant，which can be determined by the optimising algorithm based on the experimental data（Zhu，2003a）.

Moreover，the time-dependent elastic modulus can be calculated by Equation 13.

where τis the concrete age（day）；E0 is the ultimate elastic modulus[E0=1.5E28（Zhu，2003a）]；E28 is the elastic modulus of concrete in 28 days.In this study，E28 was measured as 20GPa；both Aand Bare constants based on the measured values（Zhu，2003a）.

Therefore，the thermal stress can be formulated by：

where Kr=100%，E0=30 GPa.However，other parameters are unknown，such as the creep parameters x1～x8 and elastic modulus parameters A，B.

The back analysis was employed to calculate these unknown parameters.The specific values of a“black box”strategy are as：the design variables are chosen as x1，x2，x3，x4，x5，x6，x7，x8，A，B，and the objective function Er，σcan be described as Equations（15）and（16）.

where the parameters satisfied the constraints：xi≥0，i=1～8；A≥0，and B≥0.

In practice，the design parameters are constrained in ranges，which can be given by Equation （17）