## 3 Numerical Model for the Simulation of Time-Dependent Effects in Concrete
### Linear Creep of Concrete
Some of the most used internationally recommended types of models for the prediction of creep strains without a need for specific laboratory tests are fib Model Code 2010 (2010) [23], ACI 209R-92 (1992) [8], AASHTO LRFD 2010 (2010), AS 3600, EC-2, among many others. These models are all empirical, and they vary widely in their techniques. As input, they all require main material properties and concrete age at the time of loading as well as certain intrinsic and/or extrinsic variables, for example, ambient moisture and mixture proportion. In this paper, the fib Model Code 2010 was used [23]. The analytical form of the creep function is given with the following expression:  
\[J\big{(}t,t^{\prime}\big{)}=\frac{1}{E(t^{\prime})}+\frac{1}{E_{28}}\cdot \big{[}\varphi_{bc}\big{(}t,t^{\prime}\big{)}+\varphi_{dc}\big{(}t,t^{\prime} \big{)}\big{]}, \tag{4}\]  
with parameters [23]:  
\(\varphi_{bc}(t,t^{\prime})\) --part of the creep coefficient that represents basic creep,  
\(\varphi_{dc}(t,t^{\prime})\) --part of the creep coefficient that represents drying creep,  
\(E(t^{\prime})\) --concrete elasticity modulus at load time,  
\(E_{28}\) --concrete elasticity modulus after 28 days. The total creep coefficient is equal to the sum of its components:  
\[\varphi\big{(}t,t^{\prime}\big{)}=\varphi_{bc}\big{(}t,t^{\prime}\big{)}+ \varphi_{dc}\big{(}t,t^{\prime}\big{)}, \tag{5}\]  
The creep coefficient components that appear in expressions (4) and (5) are determined according to expressions (6) and (7), [23]:  
\[ \begin{gathered}\varphi_{bc}(t,t^{\prime})=\beta_{bc}(f_{cm}) \cdot\beta_{bc}(t,t^{\prime}),\\ \beta_{bc}(f_{cm})=\frac{18}{(f_{cm})^{0.2}},\ \ \ \beta_{bc}(t,t^{\prime})=ln \left[\left(\frac{30}{f_{all}}+0.035\right)^{2}\cdot(t-t^{\prime})+1\right],\\ \varphi_{dc}(t,t^{\prime})=\beta_{dc}(f_{cm})\cdot\beta(R\dot{H}) \cdot\beta_{dc}(t^{\prime})\cdot\beta_{dc}(t,t^{\prime}),\\ \beta_{dc}(f_{cm})=\frac{412}{(f_{cm})^{1.4}},\ \ \ \beta(R\dot{H})= \frac{1-\frac{800}{100}}{\sqrt{\frac{1}{100}}},\\ \beta_{dc}(t^{\prime})=\frac{1}{0.1+\left(t^{\prime}_{adj}\right) ^{0.2}},\ \ \ \beta_{dc}(t,t^{\prime})=\left[\frac{(t-t^{\prime})}{\beta_{n}+(t-t^{\prime})} \right]^{\gamma(t^{\prime})},\\ \gamma(t^{\prime})=\frac{1}{23+\frac{85}{\sqrt{f_{all}}}},\\ \beta_{n}=1.5\cdot h+250\cdot\alpha_{f_{cm}}\leq 1500\cdot\alpha_{f_{cm }},\\ \alpha_{f_{cm}}=\left(\frac{35}{f_{cm}}\right)^{0.5},\ \ \ h=\frac{2\cdot\lambda_{c}}{u}.\end{gathered}  \tag{6}\]  
In expressions (6) and (7) \(f_{cm}\) is the mean compressive strength of concrete in MPa after 28 days, coefficient \(RH\) is the relative humidity in %, \(u\) is the perimeter of the member in contact with the atmosphere [mm] and \(A_{c}\) is the area of the cross-section [mm\({}^{2}\)]. The variable \(t^{\prime}_{adj}\) is the adjusted concrete age at the time of loading that can be calculated according to [23] as:  
\[t^{\prime}_{adj}=t^{\prime}_{T}\cdot\left[\frac{9}{2+\left(t^{\prime}_{T} \right)^{1.2}}+1\right]^{\alpha_{c}}\geq 0.5\;days, \tag{7}\]  
with \(t^{\prime}_{T}\) --temperature adjusted concrete age that can be calculated from:  
\[t^{\prime}_{T}=\sum_{i=1}^{n}\Delta t_{i}\cdot e^{13.65-\frac{800}{273+1( \Delta t_{i})}}. \tag{8}\]In the above expressions, \(\Delta t_{i}\) is the number of days where temperature \(T\) (in \({}^{\circ}\)C) prevails, while the constant \(\alpha_{c}\) in (8) depends on the cement class, e.g., for cement class 32.5 N \(\alpha_{c}=-1\), for cement classes 32.5 R and 42.5 N \(\alpha_{c}=0\) and for cement classes 42.5 R, 52.5 N and 52.5 R \(\alpha_{c}=1\)[23]. The basic creep function is not limited: \(lim_{t\rightarrow\infty}\beta_{bc}(t,t^{\prime})=\infty\), while there is a limit for the drying creep function: \(lim_{t\rightarrow\infty}\beta_{dc}(t,t^{\prime})=1\)[23].