2.7.7 Cross-Section Properties for Deformation Analysis

Cross-Section Properties for Deformation Analysis

For the material stiffness matrix D for the deformation analysis, the program requires the cross-section properties dependant on the cracked state that are available in every reinforcement direction. These are in detail

• the moment of inertia to the ideal center of gravity I_Φ,
• the moment of inertia to the geometric center of the cross-section I_0,Φ,
• the cross-section area A_Φ,
• the eccentricity of the ideal center e_Φ to the geometric center.

The mean strain ε_Φ and mean curvature Κ_Φ are interpolated from the cracked and uncracked state according to EN 1992-1-1, Equation (7.18):

$$\begin{gathered} {\epsilon }_{\varphi } = {\zeta }_{\varphi } · {\epsilon }_{\varphi ,II} + \left(1 - {\zeta }_{\varphi }\right) · {\epsilon }_{\varphi ,1} \\ {\kappa }_{\varphi } = {\zeta }_{\varphi } · {\kappa }_{\varphi ,II} + \left(1 - {\zeta }_{\varphi }\right) · {\kappa }_{\varphi ,1 } \end{gathered}$$

The strains in the cracked state c (state I and II) are calculated according to the following equations:

$$\begin{gathered} {\epsilon }_{\varphi ,c} = \frac{n_{\varphi }}{E · A_{\varphi ,c}} \\ {\kappa }_{\varphi ,c} = {\kappa }_{sh,\varphi ,c} · \frac{m_{\varphi } - n_{\varphi } · e_{\varphi ,c}}{E · I_{\varphi ,c}} \end{gathered}$$

The influence of shrinkage is therefore considered with the factor k_sh,φ,c.

When no axial forces n_Φ act, such as with the model type 2D - XY (u_Z / φ_X / φ_Y), only the ideal cross-section properties that relate to the ideal center of the cross-section are relevant:

$$\begin{gathered} A_{\varphi } = \frac{A_{\varphi ,I} · A_{\varphi ,II}}{{\zeta }_{\varphi } · A_{\varphi ,I} · k_{sh, \varphi ,II} + \left(1 - {\zeta }_{\varphi }\right) · A_{\varphi ,II} · k_{sh,\varphi ,I}} \\ {I}_{\varphi } = \frac{I_{\varphi ,I} · I_{\varphi ,II}}{{\zeta }_{\varphi } · I_{\varphi ,I} · k_{sh,\varphi ,II} + \left(1 - {\zeta }_{\varphi }\right) · I_{\varphi ,II} · k_{sh,\varphi ,l}} \end{gathered}$$

If axial forces are available, the cross-section properties are related to the geometric center of the cross-section:

$$\begin{gathered} A_{\varphi } = \frac{n_{\varphi }}{A · {\epsilon }_{\varphi }} \text{where} {\epsilon }_{\varphi } = \frac{m_{\varphi } - {\kappa }_{\varphi } · E · I_{\varphi }}{n_{\varphi }} \\ {I}_{\varphi ,0} = I_{\varphi } + A_{\varphi } · e_{\varphi }^2 \text{where} I_{\varphi } \text{as }\text{per}\text{Equation 2.87} \end{gathered}$$

In the course of the calculation of the cross-section properties, the initial value of poisson's ratio ν_init is reduced according to the following equation:

$\nu = \left(1 - \underset{\varphi \in \{1,2\}}{\text{max}} \left({\zeta }_{\varphi }\right)\right) · {\nu }_{\text{init }}$