Basic equations for the analysis of the HC slab subjected to fire are written separately for the hygro-thermo-chemical analysis and mechanical analysis. The coupling between both phases is performed in a weak manner, i.e. through computational procedure, where the results from the hygro-thermo-chemical analysis are used as an input data for the mechanical analysis. In case of fire, the HC slab is usually in the hot zone, meaning that air temperature along the HC slab does not change significantly. This implies that temperature and moisture field does not change along the HC slab and hygro-thermo-chemical analysis can be determined over a characteristic cross-section of the HC slab. Therefore, basic equations of hygro-thermo-chemical model are written for the 2D domain (Fig. 1). The considered precast element is characterized as a slab, although, in principle, it behaves as a beam element, since load is transmitted to the support only in the longitudinal direction. For this reason, basic equations describing mechanical model are formed on the 1D beam theory.

### 2.1 Basic Equations of Coupled Hygro-Thermo-Chemical Model

Numerical model for coupled hygro-thermo-chemical analysis is based on the model of Davie et al. (2006) with modifications given in Kolšek et al. (2014). The model is suitable for the 2D hygro-thermo-chemical analysis across the concrete beam cross-section, where the influence of tendon on the distribution of temperatures within the cross-section is neglected. Further on, the model takes into account the evaporation of free water, the liquefaction of water vapour and the dehydration of chemically bond water. In addition, the capillary pressure and the part of the free water co-existing in an absorbed state is also taken into account. The model is described by the system of mass conservation equations for each phase separately and by the energy conservation equation.

Water conservation:

$$\begin{aligned} {{\frac{\partial (\varepsilon _{FW}\rho _{FW})}{ \partial t}}}={{-\nabla \cdot {\mathbf {J}}_{FW}}}- {{{\dot{E}}_{FW}}}+{{\frac{ \partial (\varepsilon _{D}\rho _{FW})}{\partial t}}}, \end{aligned}$$

(1)

water vapour conservation:

$$\begin{aligned} \frac{\partial (\varepsilon _{G}{\tilde{\rho }}_{V})}{\partial t}=-\nabla \cdot {\mathbf {J}}_{V}+{\dot{E}}_{FW}, \end{aligned}$$

(2)

air conservation:

$$\begin{aligned} \frac{\partial (\varepsilon _{G}{\tilde{\rho }}_{A})}{\partial t}=-\nabla \cdot {\mathbf {J}}_{A}, \end{aligned}$$

(3)

energy conservation:

$$\begin{aligned} {(\underline{\rho C})\frac{\partial T}{\partial t}}= {-\nabla \cdot (-k\nabla T)}-{ (\underline{\rho C{\mathbf {v}}})\cdot \nabla T}-{ \lambda _{E}{\dot{E}}_{FW}}-{\lambda _{D} \frac{\partial (\varepsilon _{D}\rho _{FW})}{\partial t}}. \end{aligned}$$

(4)

In Eqs. (1–4) \(\rho _{FW}\), \({\tilde{\rho }}_{V}\) and \({\tilde{\rho }}_{A}\) is the mass of free water, water vapour and dry air per unit volume of gaseous mixture. \(\varepsilon _{G}{\tilde{\rho }}_{A}\), \(\varepsilon _{G}{\tilde{\rho }}_{V}\) and \(\varepsilon _{FW}\rho _{FW}\) denotes the mass of air, water vapour and free water per unit volume of concrete. \({\mathbf {J}}_{i}\) refers to the mass flux of phase *i* (\(i=FW,V,A\)), \({\dot{E}}_{FW}\) is the rate of evaporation of free water (desorption included), *t* is time and \(\nabla\) is nabla operator. Furthermore, *k* and \(\underline{\rho C}\) are conductivity and heat capacity of concrete, \(\underline{\rho C{\mathbf {v}}}\) is energy transferred by the fluid flow, \(\lambda _{E}\) and \(\lambda _{D}\) are the specific heat of evaporation and dehydration, \(\varepsilon _{D}\rho _{FW}\) is the mass of bound water released by dehydration per unit volume of concrete, and *T* is the temperature.

By summing Eqs. (1) and (2) three partial differential equations are obtained describing the transfer of dry air, moisture and energy conservation. This three partial differential equations are the rearranged so that the basic unknowns of heat and moisture transfer problem are temperature *T*, pore pressure \(P_{G}\) and water vapour content \({\tilde{\rho }}_{V}\) (Kolšek et al. 2014). Temperature variation of thermal properties of concrete, air and water vapour are taken according to CEN (2005) and Davie et al. (2006).

The cooling phase of fire is also addressed in the paper. Therefore, certain assumptions are introduced in order to describe the behaviour of concrete in cooling phase. First of all, when chemically bond water is adsorbed into capillary pores it can no longer return to the previous state, i.e. dehydration of chemically bond water is irreversible process and thus simply neglected in the cooling phase. In addition, it is assumed that porosity and permeability of concrete does not change in the cooling phase and are determined at the maximum temperature in each point of concrete section. Similarly it is applied, that thermal properties of concrete remain constant in the cooling phase and are equal to the values reached at the maximum temperature in each point of the cross-section.

In order to account for the heat and mass exchange between concrete body and the surrounding, the boundary conditions at the boundary layer are prescribed. The heat surface flux at the boundary is given as:

$$\begin{aligned} \mathbf{n }\cdot \nabla T = \frac{h_{qr}}{k}(T_{\infty }-T) \end{aligned}$$

(5)

where \(\mathbf{n }\) is normal unit vector of the boundary surface, \(T_{\infty }\) is the surrounding temperature and \(h_{qr}\) is the heat transfer coefficient composed of convective part \(h_{q}\) and radiative part \(h_{r}\). The mass transfer at the boundary is determined as:

$$\begin{aligned} \mathbf{n }\cdot \mathbf{J }_{V}=-\beta ({\tilde{\rho }}_{V,\infty }-{\tilde{\rho }}_{V}) \end{aligned}$$

(6)

where \({\tilde{\rho }}_{V,\infty }\) is the concentration of water vapour in the surrounding and \(\beta\) is mass transfer coefficient determined in Cengel (1998). Further on, it is assumed that the pressure in concrete pores at the boundary is equal to the environmental pressure \(P_{G,\infty }\), hence:

$$\begin{aligned} P_{G}=P_{G,\infty } \end{aligned}$$

(7)

#### 2.1.1 Air, Temperature and Moisture Exchange in the Opening

When analysing the HC slab, the opening in the geometry has an influence on the temperature and moisture distribution within the HC slab. For this reason, the cross-section is divided on two subsections, first is represented by the concrete part and second represents the air within the opening. The exchange of heat through the contact boundary layer of these two subsystems, is formally expressed by the heat flux density. In general, both radiative and convective heat transfer has to be considered. However, simplification is introduced, assuming only convective heat exchange between the body and the opening. In view, the heat flux density between the two subsections.

$$\begin{aligned} q_{B}=h_{q,B}({{T}_{B}}-{{T}_{op}}), \end{aligned}$$

(8)

where \(T_{B}\) denotes temperature of the boundary surface, \(T_{op}\) is the air temperate inside the opening and \(h_{q,B}\) is the convective coefficient at the boundary surface. Assuming the uniform air temperature inside the opening, the heat flux through the boundary surface size of \(1\times ds\), can be written as:

$$\begin{aligned} dQ_{B}=h_{q,B}({{T}_{B}}-{{T}_{op}})ds. \end{aligned}$$

(9)

where *ds* denotes the partial arc-length of the boundary surface. The entire heat flux at the contact between the two subsystems can be obtained by integrating heat flux density along internal edge of opening:

$$\begin{aligned} Q_{B}=\int \limits _{0}^{L_{op}}h_{q,B} T_{B} ds - T_{op} \int \limits _{0}^{L_{op}}h_{q,B} ds, \end{aligned}$$

(10)

where \(L_{op}\) denotes the perimeter of the opening. The internal energy of the air is dependent on the air mass *m*, specific heat at constant volume \(c_{v}\) and on the change of air temperature \(dT_{op}\) within the opening.

$$\begin{aligned} Q_{B}dt=m c_{v}dT_{op}. \end{aligned}$$

(11)

Air mass can be expressed as product of air density \(\rho _{A}\) and total air volume \(V_{A}\) within the opening. After the arrangement the heat flux can be written as:

$$\begin{aligned} Q_{B}=\rho _{A}V_{A}c_{v}\frac{dT_{op}}{dt}=\int \limits _{0}^{L_{op}}h_{q,B}T_{B}ds-T_{op}\int \limits _{0}^{L_{op}}h_{q,B}ds. \end{aligned}$$

(12)

Equation (12) represent the basis to determine air temperature inside the opening and is solved iteratively, based on the temperature of the boundary surface from the previous time step.

For the exchange of moisture between the two subsystems Eq. (6) is applied.

### 2.2 Basic Equations of Mechanical Model

In this section the second phase of the fire analysis of the HC slab is presented, where the mechanical response of the HC slab simultaneously exposed to external mechanical and fire load is determined. Mathematical model to describe the deformation of concrete part of the HC slab is based on the Reissner exact beam theory (Reissner 1972), while the behaviour of prestressing tendon is simplified with the wire model. To interconnect these two systems (concrete part and tendon) constraining equations are introduced, allowing for the tangential slip at the contact between concrete and tendon, while neglecting the normal separation. Further on, mechanical properties of the contact between concrete and tendon are temperature dependent. Mechanical and rheological properties of concrete and prestressing steel are temperature dependent as well. Another important feature of the model is considering additive decomposition of the increment of the extensional strain of random concrete fiber, being composed of the increment of mechanical, temperature, creep and transient strain: \(\varDelta \varepsilon _{\text{c}}=\varDelta \varepsilon _{\sigma ,{\text {c}}}+\varDelta \varepsilon _{\text{th,c}}+\varDelta \varepsilon _{\text{cr,c}}+\varepsilon _{\text{tr,c}}\). Similarly, increment of extensional strain of tendon consists of the increment of mechanical, temperature and thermal-creep strain which occur due to the viscous creep of steel at elevated temperatures: \(\varDelta \varepsilon _{\text{p}}=\varDelta \varepsilon _{\sigma ,{\text {p}}}+\varDelta \varepsilon _{\text{th,p}}+\varDelta \varepsilon _{\text{cr,p}}\). Although the mechanical model is based on a physically highly precise description of the problem, there are two assumptions that somewhat simplify model a bit. First one is neglecting tension strength of concrete since it does not have a significant influence on load-bearing capacity of the HC slab. Second one is disregarding the influence of shear strain on the deformation of the HC slab given the fact that the span of the HC slab is usually by factor of 25 or more greater than the height of the HC slab, meaning that shear strain does not have considerable influence.

To better understand model derivation and basic quantities involved, undeformed and deformed configuration of prestressed beam is given in Fig. 2. Concrete beam with the initial length *L* and constant cross-section \(A_{\text{c}}\) is prestressed by \(n_{\text{p}}\) tendons, that are symmetrically distributed over the cross-section. Cross section of each tendon is labelled as \(A_{\text{p}}^{k}\) \((k=1,2,\ldots,n_{\text{p}})\), \(\phi _{\text{p}}^k\) is the tendon diameter, while \(N_{\text{p,0}}^k\) denotes the initial prestressing force of individual tendon. The reference axis of the concrete part of beam is in the centre of the cross-section of the beam, while the reference axis of the *k*th tendon is in the centre of the cross-section of each tendon.

When setting the governing equations of the mathematical model the derivation can be simplified by introducing a material coordinate \(x_{\text{p}}^{*k}\) which determines a particle of the *k*th tendon in undeformed configuration (point \(Q_{\text{p}}^k\)) that is in contact with a particle of concrete in deformed configuration whose material coordinate is \(x_{\text{c}}\) (point \(T_{\text{c}}^k\)) (see Fig. 2). This means that the particles of concrete (point \(T_{\text{c}}^k\)) and *k*th tendon (point \(T_{\text{p}}^k\)), which in undeformed state coincide, occupy different points in space in deformed configuration, i.e. slip \(\varDelta ^k(x_{\text{c}})\) between them occurs. Since the tendon dimension are relatively small it is simplified that \(z_{\text{p}}^k = z_{\text{c}}^k\). Considering the fact that the slips between concrete and tendon are relatively small, it is further assumed:

$$\begin{aligned} (\bullet )_{\text{p}}^{*k} \approx (\bullet )_{\text{p}}^{k} \quad \text{and} \quad \int _{0^*}^{L^*} (\bullet )_{\text{p}}^{*k} \text{d}x^{*} \approx \int _{0}^{L} (\bullet )_{\text{p}}^{k} \text{d}x, (k=1,2,\ldots,n_{\text{p}}). \end{aligned}$$

(13)

which means that differences between functions determined at \(x^{*}\) and *x* are insignificant for any \(k = 1,\ 2,\ \ldots,\ n_{p}\).

#### 2.2.1 The Stress–Strain Field

The basic equations of prestressed concrete beam subjected to fire are composed of kinematic, constraining and equilibrium equations. In addition, constitutive equations and corresponding constitutive laws are needed to consider properties of each material.

In accordance to the Reissner planar beam model, the kinematic equations written separately for concrete part of the beam and tendon are as follows (see Markovič et al. 2013):

$$\begin{aligned}{\text {concrete part of the beam:}}\,1+u_{\text{c}}^{\prime} - \left( 1+\varepsilon _{\text{c}0}\right) \cos \varphi _{\text{c}}=0, \end{aligned}$$

(14)

$$\begin{aligned}w_{\text{c}}^{\prime}+ \left( 1+\varepsilon _{\text{c}0}\right) \sin \varphi _{\text{c}}=0, \end{aligned}$$

(15)

$$\begin{aligned}\varphi _{\text{c}}^{\prime}-\kappa _{\text{c}}=0, \end{aligned}$$

(16)

$$ {\text {tendon:}}\,1+{u_{\text{p}}{^{k}}^{\prime}}- \left( 1+\varepsilon _{\text{p}}^{k}\right) \cos \varphi _{\text{p}}^{k}=0, \quad (k=1,2,\ldots,n_{\text{p}}). $$

(17)

Here \((\bullet )^{\prime}\) represents a derivative of selected quantity with the respect to material coordinate \(x_{\text{c}}\) (see Fig. 2), quantities \(u_{\text{c}}\) and \(w_{\text{c}}\) denote horizontal and vertical displacement of a random point on the reference axis of concrete part of the beam, similarly, \(u_{\text{p}}^{k}\) is horizontal displacement of the reference axis of *k*th tendon. \(\varepsilon _{\text{c}0}\) and \(\kappa _{\text{c}}\) are extensional strain and pseudo-curvature of the reference axis of the concrete part of the beam, while \(\varphi _{\text{c}}\) represent the rotation of the concrete cross-section. The geometrical strain of concrete part of the section is thus determined as: \(\varepsilon _{\text{c}} = \varepsilon _{\text{c}0} + z_{c} \kappa _{\text{c}}\). The extensional strain of \(k {^{\mathrm{th}}}\) tendon and the rotation of the \(k {^{\mathrm{th}}}\) tendon cross-section are denoted as \(\varepsilon _{\text{p}}^{k}\) and \(\varphi _{\text{p}}^{k}\), respectively.

The constraining equations provide the description of the interaction at the contact between concrete and \(k {^{\text{th}}}\) tendon. Considering the simplification that \(z_{\text{p}}^k = z_{\text{c}}^k\), than vectors describing the deformed position of the chosen particle of concrete beam (point \(T_{\text{c}}^k\)) and tendon (point \(Q_{\text{p}}^k\)) at the contact become the same.

$$\begin{aligned} {\varvec{R}}_{\text{c}}={\varvec{R}}_{\text{p}}^{k} \end{aligned}$$

(18)

In component form, Eq.(18) can be written:

$$\begin{aligned} x_{\text{c}}+u_{\text{c}}+z_{\text{c}}^{k}\sin \varphi _{\text{c}}= x_{\text{p}}^{*k}+u_{\text{p}}^{*k} \approx x_{\text{p}}^{*k}+u_{\text{p}}^{k}, \end{aligned}$$

(19)

$$\begin{aligned} w_{\text{c}}+z_{\text{c}}^{k}\cos \varphi _{\text{c}}= w_{\text{p}}^{*k} \approx w_{\text{p}}^{k}. \end{aligned}$$

(20)

where \(w_{\text{p}}^{k}\) represents vertical displacement of the reference axis of *k*th tendon. Slip at the contact between concrete and *k*th tendon (see Fig. 2) is given by the following equation:

$$\begin{aligned} \varDelta ^k(x_{\text{c}}) = \int _{x_{\text{p}}^{*k}}^{x_{\text{c}}} (1+\varepsilon _{\text{p}}^k) \text{d}x, \quad (k=1,2,\ldots ,n_{\text{p}}). \end{aligned}$$

(21)

Based on the constraining equations (Eqs. 19 and 20) and kinematic equations (Eqs. 14–17) and taking into account the assumptions (13), the following relationships can be given:

$$\begin{aligned} \varphi _{\text{p}}^{k} = \varphi _{\text{c}} \quad \text{{and}} \quad \kappa _{\text{p}}^k=\frac{1+\varepsilon _{\text{p}}^{k}}{1+\varepsilon _{\text{c}0}+z_{\text{c}}^{k}\kappa _{\text{c}}}\kappa _{\text{c}}. \end{aligned}$$

(22)

The above equations simplifies the rotation of tendon cross-section, making it equal to the rotation of concrete part of the cross-section while the pseudo-curvature of the reference axis of the *k*th tendon \(\kappa _{\text{p}}^k\) can be determined via trivial expression.

The equilibrium equations deliver the relationship between beam internal forces and external load. Internal forces in the concrete part of the beam are represented by equilibrium axial and shear force (\({\mathscr {N}}_{\text{c}}\), \({\mathscr {Q}}_{\text{c}}\)) and by equilibrium bending moment \({\mathscr {M}}_{\text{c}}\), \({\mathscr {N}}_\text{p}^k\) is the equilibrium axial force of the *k*th tendon. It is assumed that the external load acts along the beam reference axis and is presented by conservative line traction *q*\(_{\text{c}}\) = \(q_{X,{\text {c}}}\) *E*\(_{X}+q_{Z,{\text {c}}}\) *E*\(_{Z}\) and moment traction *m*\(_{\text{c}}\) = \(m_{Y,{\text {c}}}\) *E*\(_{Y}\) (see Fig. 2). At the contact between concrete and tendon contact load is considered. Loads *p*\(_{\text{c}}^{k}\) = \(p_{X,{\text {c}}}^{k}\) *E*\(_{X}+p_{Z,{\text{c}}^{k}}\) *E*\(_{Z}\) and *h*\(_{\text{c}}^{k}\) = \(p_{X,{\text {c}}}^{k}z_{\text{c}}^{k}\) *E*\(_{Y}\) represent the impact of tendon on the concrete part of the beam. On the other hand, the influence of concrete on tendon is, due to the wire model, expressed only by line load *p*\(_{\text{p}}^{k}\) = \(p_{X,{\text {p}}}^{k}\) *E*\(_{X}+p_{Z,{\text {p}}}^{k}\) *E*\(_{Z}\), where the equilibrium of contact forces must be met, namely:

$$\begin{aligned}p_{X,{\text{p}}}^k+p_{X,{\text {c}}}^k=0 \end{aligned}$$

(23)

$$\begin{aligned}p_{Z,{\text{p}}}^k+p_{Z,{\text {c}}}^k=0, \quad (k=1,2,\ldots,n_{\text{p}}). \end{aligned}$$

(24)

The equilibrium equations are:

$$\begin{aligned} {\text {concrete part of the beam:}}\,{\mathscr {R}}_{{X,{\text {c}}}^{\prime}}+q_{X,{\text {c}}}+\sum _{k=1}^{n_{\text{p}}} p_{X,{\text {c}}}^k=0, \end{aligned}$$

(25)

$$\begin{aligned} {\mathscr {R}}_{{Z,{\text {c}}^{\prime}}}+q_{Z,{\text {c}}}+\sum _{k=1}^{n_{\text{p}}} p_{Z,{\text {c}}}^k=0, \end{aligned}$$

(26)

$$\begin{aligned}{\mathscr {M}}_{\text{c}}^{\prime}-(1+\varepsilon _{\text{c}0}){\mathscr {Q}}_{\text{c}}+m_{Y,{\text {c}}}+\sum _{k=1}^{n_{\text{p}}} p_{X,{\text {c}}}^k z_{\text{c}}^k=0, \end{aligned}$$

(27)

$${\text {tendon:}}\,{{\mathscr {N}}_{\text{p}}{^{k}}^{\prime}}+p_{\text{t,p}}^{k}=0, \qquad (k=1,2,\ldots,n_{\text{p}}), $$

(28)

$$\begin{aligned}p_{\text{n,p}}^{k}-{{\mathscr {N}}_{\text{p}}^{k}}\kappa _{\text{p}}^k=0, \quad (k=1,2,\ldots,n_{\text{p}}). \end{aligned}$$

(29)

In Eqs. (25) and (26) \({\mathscr {R}}_{X,{\text {c}}}\) and \({\mathscr {R}}_{Z,{\text {c}}}\) denotes the equilibrium horizontal and vertical component of the internal forces \({\mathscr {N}}_{c}\) and \({\mathscr {Q}}_{c}\). They are related as: \({\mathscr {R}}_{X,{\text {c}}}={\mathscr {N}}_{c} \cos \varphi _{\text{c}}+ {\mathscr {Q}}_{c} \sin \varphi _{\text{c}}\) and \({\mathscr {R}}_{Z,{\text {c}}}=-{\mathscr {N}}_{c} \sin \varphi _{\text{c}}+ {\mathscr {Q}}_{c} \cos \varphi _{\text{c}}\). \(p_\text{n,p}^{k}\) in \(p_{\text{t,p}}^{k}\) are so-called normal and tangential components of the contact load for the *k*th tendon defined in the material basis (\(x_{\text{p}}^{k}\), \(z_{\text{p}}^{k}\)). They are linked to the components of contact load in a fixed Cartesian coordinate system as: \(p_{\text{n,p}}^{k}=p_{X,\text p}^{k}\sin \varphi _{\text{c}}+p_{Z,\text p}^{k}\cos \varphi _{\text{c}}\) and \(p_{\text{t,p}}^{k}=p_{X,\text p}^{k}\cos \varphi _{\text{c}}-p_{Z,\text p}^{k}\sin \varphi _{\text{c}}\).

#### 2.2.2 Constitutive Equations and Constitutive Laws

Constitutive equations interconnect equilibrium quantities with the deformation quantities:

$$\begin{aligned} \text {concrete part of the beam:}\,
\\
{\mathscr {N}}_{\text{c}}={\mathscr {N}}_{\text{c,c}}=\int _{A_{\text{c}}}\sigma _{\text{c}} \text{d}A_{\text{c}}, \end{aligned}$$

(30)

$$\begin{aligned}{\mathscr {M}}_{\text{c}}={\mathscr {M}}_{\text{c,c}}=\int _{A_{\text{c}}} z_{\text{c}}\sigma _{\text{c}} \text{d}A_{\text{c}}, \end{aligned}$$

(31)

$$ \text {tendon:}\,{\mathscr {N}}_{\text{p}}^{k}={\mathscr {N}}_{\text{c,p}}^{k}=\sigma _{\text{p}}^{k} A_{\text{p}}^{k}\quad (k=1,2\ldots,n_{\text{p}}). $$

(32)

Above, \(\sigma _{\text{c}}(\varepsilon _{\sigma ,{\text {c}}}, T)\) and \(\sigma _{\text{p}}^{k}(\varepsilon _{\sigma ,\text p}^k, T)\) denote normal physical stresses in concrete and \(k^{\text {th}}\) tendon, which are function only of the mechanical strain and temperature. The stress–strain relationship is given by the chosen constitute law. In addition, the constitutive law of slip must be defined in order to determine the relationship between slip \(\varDelta ^k\) and shear stress \(p_{\text{t,c}}^k\) at the contact between concrete and *k*th tendon. In what follows the chosen constitute laws for concrete, prestressing steel and contact are presented.

The behaviour of concrete in compression at elevated temperatures is described by the constitutive law found in EN 1992-1-2 (see Fig. 3a). Load bearing capacity of concrete in tension is neglected. Temperature dependent material parameters are compressive strength \(f_{\text{c,T}}\), the corresponding strain \(\varepsilon _{\text{c1,T}}\) and the ultimate compressive strain \(\varepsilon _{\text{cu1,T}}\). Further on, the constitutive law given in EN 1992-1-2 (see Fig. 3b) is used to represent the behaviour of tendon in tension and compression. Here the temperature dependent material parameters of prestressing steel are the maximum stress level \(f_{\text{py,T}}\), the corresponding strain \(\varepsilon _{\text{pt,T}}\), the proportional limit \(f_\text{pp,T}\), the modulus of elasticity \(E_{\text{p,T}}\) and the ultimate strain \(\varepsilon _{\text{pu,T}}\). In constitutive models for both materials, the isotropic hardening is accounted for. The unloading slope is linear and equal to the current modulus of elasticity. For concrete this is a secant modulus of elasticity \(E_\text{cm,T}=0.4\,f_{\text{c,T}}/(0.2693\,\varepsilon _{\text{c1,T}})\), while for prestressing steel this is \(E_{\text{p,T}}\).

The idealized multilinear law introduced by Keuser and Mehlhorn (1983) describes the relationship between the shear stress \(p_\text{t,c}^{k}\) and slip \(\varDelta ^k\) at the contact between concrete and *k*th tendon (see Fig. 3c). It is applied that \(p_{\text{t,c}}^k=\pi \, \phi _{\text{p}}^k \, \tau ^k\) (\(k=1,2,\ldots ,n_{\text{p}}\)), where \(\tau ^k\) represents the bond stress at concrete-tendon contact. Temperature dependent material parameters are bond stresses \(\tau _{1,T}\), \(\tau _{2,T}\) and the bond strength \(\tau _{\text{u,T}}\) (see Khalaf and Huang 2016).

In the cooling phase it is assumed that the concrete does not recover the compressive strength, the same is applied for the bond strength, while prestressing steel fully recovers its strength. Additional assumption is made by considering temperature strains of steel as fully reversal. Creep of concrete is the same in heating and cooling phase, while the viscous creep of steel in cooling phase is disregarded. Likewise, the transient strain of concrete is not accounted for in the cooling phase (Bratina et al. 2007).

#### 2.2.3 Boundary Conditions

A set of kinematic, equilibrium and constitutive equations is supplemented by the corresponding static and kinematic boundary conditions:

$$ \text {concrete part:}\,S_{1\text{,c}}+{\mathscr {R}}_{\text{X,c}}(0)=0 \quad \text {or} \quad u_{1\text{,c}}=u_{\text{c}}(0), $$

(33)

$$\begin{aligned}S_{2\text{,c}}+{\mathscr {R}}_{\text{Z,c}}(0)=0 \quad \text {or} \quad u_{2\text{,c}}=w_{\text{c}}(0), \end{aligned}$$

(34)

$$\begin{aligned}S_{3\text{,c}}+{\mathscr {M}}_{\text{c}}(0)=0 \quad \text {or} \quad u_{3\text{,c}}=\varphi _{\text{c}}(0),\end{aligned}$$

(35)

$$\begin{aligned}S_{4\text{,c}}-{\mathscr {R}}_{\text{X,c}}(L)=0 \quad \text {or} \quad u_{4\text{,c}}=u_{\text{c}}(L), \end{aligned}$$

(36)

$$\begin{aligned}S_{5\text{,c}}-{\mathscr {R}}_{\text{Z,c}}(L)=0 \quad \text {or} \quad u_{5\text{,c}}=w_{\text{c}}(L), \end{aligned}$$

(37)

$$\begin{aligned}S_{6\text{,c}}-{\mathscr {M}}_{\text{c}}(L)=0 \quad \text {or} \quad u_{6\text{,c}}=\varphi _{\text{c}}(L), \end{aligned}$$

(38)

$$ \text {tendon:}\,{\mathscr {N}}_{\text{p}}^k(0)=0 \quad \text {or} \quad u_{1\text{,p}}^k=u_{\text{p}}^k(0), \quad (k=1,2\ldots,n_{\text{p}}), $$

(39)

$$\begin{aligned}{\mathscr {N}}_{\text{p}}^k(L)=0\ \text {or}\ u_{4\text{,p}}^k=u_{\text{p}}^k(L), \quad (k=1,2\ldots,n_{\text{p}}). \end{aligned}$$

(40)

In Eqs. (33–38), \(u_{i,{\text {c}}}\) (\(i=1,\ldots,6\)) and \(u_{j,\text p}^k\) (\(j=1 \, \text{or} \, 4\)) denote the given values of the boundary displacements, similarly \(S_{i,{\text {c}}}\) (\(i=1,\ldots,6\)) mark given forces at the begining and at the end of the concrete part of the beam.