## 3 Numerical Model for the Simulation of Time-Dependent Effects in Concrete  
### Linear Creep of Concrete  
Creep can be defined as the progressive accumulation of the plastic strain in the concrete element under constant stresses (external load) over a long period of time. It occurs under a long-term (permanent) load when the element is exposed to stresses at high levels but below the ultimate material strength. The deformation rate corresponds to the material properties, time and temperature and the level of load applied to the structure. Figure 4: Coordinate systems of 3D concrete elements with embedded reinforcement required for the numerical interpretation. Figure 3: Development of the yield surface—triaxial presentation (**a**); hardening and softening functions in correlation with total plastic strains (**b**). Total strain at time t, in basic form, can be determined as:  
\[\varepsilon(t)=\varepsilon_{m}+\varepsilon_{c}\big{(}t,t^{\prime}\big{)}+ \varepsilon_{sh}(t,t_{s}), \tag{1}\]  
where:  
\(\varepsilon(t)\) --total strain at some specific time \(t\),  
\(\varepsilon_{m}\) --average immediate mechanical (elastic) strain recorded at the moment after loading,  
\(\varepsilon_{c}(t,t^{\prime})\) --creep strain at some specific time \(t\), for element loaded at time \(t^{\prime}\),  
\(\varepsilon_{sh}(t,t_{s})\) --total strain caused by shrinkage at time \(t\), determined on an unloaded specimen that begun to dry at time \(t_{s}\). The strains in a concrete member over time are illustrated in Figure 5.  
Total creep strain at a specific time is the sum of the basic and the drying creep components. For the sealed concrete without any moisture exchange with the ambient medium, only the basic creep component exists. If the moisture exchange with the surrounding medium is possible, drying creep also occurs. For the calculation of the uniaxial creep deformation, the method of [PERSON] and Dischinger, based on the assumption that the creep rate function is in correlation with the current concrete uniaxial stress and time t, was used. In the most general form, this can be expressed as:  
\[\frac{d\varepsilon_{c}(t,t^{\prime})}{dt}=f(\sigma,t). \tag{2}\]  
For practical purposes, time is often divided into discrete time intervals \(\Delta t\), with \(t_{n}=t\) and \(t_{n+1}=t+\Delta t\), so the incremental version of Equation (2) becomes:  
\[\Delta\varepsilon_{c}\big{(}t_{n+1},t^{\prime}\big{)}=\varepsilon_{m}\big{(}t _{n+1},t^{\prime}\big{)}\cdot\Delta\varphi(t_{n+1},t_{n})=\varepsilon_{m} \big{(}t_{n+1},t^{\prime}\big{)}\cdot\big{[}\varphi\big{(}t_{n+1},t^{\prime} \big{)}-\varphi\big{(}t_{n},t^{\prime}\big{)}\big{]}, \tag{3}\]  
where:  
\(\Delta\varepsilon_{c}(t_{n+1},t^{\prime})\) --increase in the creep deformation between time \(t_{n}\) and \(t_{n+1}\), for the element loaded at time \(t^{\prime}\),  
\(\varepsilon_{m}(t_{n+1},t^{\prime})\) --current mechanical deformation of concrete at time \(t_{n+1}\) (loaded at time \(t^{\prime}\)), which can be a nonlinear function of stress \(\sigma\),  
\(\Delta\varphi(t_{n+1},t_{n})\) --the difference in the creep coefficient between time \(t_{n}\) and \(t_{n+1}\),  
\(\varphi(t_{n+1},t^{\prime})\) --creep coefficient at time \(t_{n+1}\),  
\(\varphi(t_{n},t^{\prime})\) --creep coefficient at time \(t_{n}\). According to this method, the creep deformation increment \(\Delta\varepsilon_{c}(t_{n+1},t^{\prime})\) is calculated based on the conditions at the time of the upcoming time increment \(t_{n+1}\). This method, basically very simple, describes well the history of deformation even in the case of irregular and sudden strain changes. Since the increment of creep deformation is based only on  
Figure 5: Strains in a concrete member over time. the current strain (or stress) and the current time, this method is very appealing from a computational point of view.