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Overview

1 Introduction

2 Theoretical Background

2.7.10.5 Cross-section properties (cracked and uncracked state)

Cross-section properties (cracked and uncracked state)

The cross-section properties depend on the governing side and the reinforcement direction φ₁. The minimum values are used for the reinforcement surfaces a_s2,φ1, a_s1,φ2, and a_s2,φ2.

The following cross-section properties for the uncracked and cracked state are to be calculated in order to be able to assemble the stiffness matrix of the material D.

Centroid

The distance of the centroid of the ideal cross-section from the concrete surface in compression is calculated directly for the uncracked state.

$z_{l, ϕ_{1}} = \frac{\frac{b \cdot h^{2}}{2} + α (a_{s_{1}, ϕ_{1}} \cdot d_{1, ϕ_{1}} + a_{s_{2}, ϕ_{1}} \cdot d_{2, ϕ_{1}})}{b \cdot h + α \cdot (a_{s_{1}, ϕ_{1}} + a_{s_{2}, ϕ_{1}})} = = \frac{\frac{1000 \cdot 200^{2}}{2} + 6.061 \cdot (1000 \cdot 150 \cdot 15 \cdot 50)}{1000 \cdot 200 \cdot 6.061 \cdot (1000 + 15)} = 101.4 mm$

For the cracked state, the depth χ_II,φ1 of the zone in compression must be calculated with the iterative method. Then the distance of the centroid of the ideal cross-section from the surface in compression is calculated for the cracked state.

Ideal cross-section area A_c,d

The effective cross-section area in the uncracked state without the influence of creep is:

$A_{l, ϕ_{1}} = b \cdot h + α \cdot (a_{s_{1}, ϕ_{1}} + a_{s_{2}, ϕ_{1}}) = 1000 \cdot 200 + 6.061 \cdot (1000 + 15) = 2061.5 {cm}^{2}$

The effective cross-section area in the cracked state is determined with the influence of creep.

$A_{I l, ϕ_{1}} = b \cdot χ_{I I, ϕ_{1}} + α \cdot (a_{s_{1}, ϕ_{1}} + a_{s_{2}, ϕ_{1}}) = 1000 \cdot 68.3 + 18.182 \cdot (1000_{} +_{} 15) = 867.19 {cm}^{2}$

The coefficient α is the ratio of the moduli of elasticity of steel and concrete with or without the influence of creep.

Ideal moment of inertia to ideal centroid I_c,d

The effective moment of inertia to the ideal centroid in the uncracked state without the influence of creep is:

$I_{I, ϕ_{1}} = \frac{1}{12} \cdot b \cdot h^{3} + b \cdot h \cdot {(z_{I, ϕ_{1}} - \frac{h}{2})}^{2} + α \cdot a_{s_{1}, ϕ_{1}} \cdot {(d_{1, ϕ_{1}} - z_{I, ϕ_{1}})}^{2} + α \cdot a_{s_{2}, ϕ_{1}} \cdot {(z_{I, ϕ_{1}} - d_{2, ϕ_{1}})}^{2} = = \frac{1}{12} \cdot 1000 \cdot 200^{3} + 1000 \cdot 200 \cdot {(101.4 - \frac{200}{2})}^{2} + 6.061 \cdot 1000 \cdot (150 - 101.4)^{2} + 6.061 \cdot 15 \cdot (101.4 - 50)^{2} = = 68 161.30 {cm}^{4}$

The effective moment of inertia to the ideal centroid in the cracked state is determined with the influence of creep.

$I_{I I, ϕ_{1}} = \frac{1}{12} \cdot b \cdot {χ^{3}}_{I I, ϕ_{1}} + b \cdot χ_{I I, ϕ_{1}} \cdot {(z_{I I, ϕ_{1}} - \frac{χ_{I I, ϕ_{1}}}{2})}^{2} + + α \cdot a_{s_{1}, ϕ_{1}} \cdot {(d_{1, ϕ_{1}} - z_{I I, ϕ_{1}})}^{2} + α \cdot a_{s_{2}, ϕ_{1}} \cdot {(z_{I I, ϕ_{1}} - d_{2, ϕ_{1}})}^{2} = = \frac{1}{12} \cdot 1000 \cdot 68 . 3^{3} + 1000 \cdot 68.3 \cdot {(58.5 - \frac{68.3}{2})}^{2} + + 18.182 \cdot 1000 \cdot {(150 - 58.5)}^{2} + 18.182 \cdot 15 \cdot {(58.5 - 50)}^{2} = = 21 928.70 {cm}^{4}$

Ideal moment of inertia to the geometric center of the cross-section I_0,c,d

The ideal moment of inertia to the geometric center of the cross-section in the uncracked state without the influence of creep is:

$I_{0, I, ϕ_{1}} = \frac{1}{12} \cdot b \cdot h^{3} + α \cdot a_{s_{1}, ϕ_{1}} \cdot {(d_{1, ϕ_{1}} - \frac{h}{2})}^{2} + α \cdot a_{s_{2}, ϕ_{1}} \cdot {(\frac{h}{2} - d_{2, ϕ_{1}})}^{2} = = \frac{1}{12} \cdot 1000 \cdot 200^{3} + 6.061 \cdot 200 \cdot {(150 - \frac{200}{2})}^{2} + 6.061 \cdot 15 \cdot {(\frac{200}{2} - 50)}^{2} = = 68 204.50 {cm}^{4}$

The ideal moment of inertia to the geometric center of the cross-section in the cracked state is determined with the influence of creep.

$I_{0, I I, ϕ_{1}} = \frac{1}{12} \cdot b \cdot {χ_{I I, ϕ_{1}}}^{3} + b \cdot χ_{I I, ϕ_{1}} \cdot {(\frac{h}{2} - \frac{χ_{I I, ϕ_{1}}}{2})}^{2} + α \cdot a_{s_{1}, ϕ_{1}} \cdot {(d_{1, ϕ_{1}} - \frac{h}{2})}^{2} + α \cdot a_{s_{2}, ϕ_{1}} \cdot {(\frac{h}{2} - d_{2, ϕ_{1}})}^{2} = = \frac{1}{12} \cdot 1000 \cdot 68 . 3^{3} + 1000 \cdot 68.3 \cdot {(\frac{200}{2} - \frac{68.3}{2})}^{2} + 18.182 \cdot 1000 \cdot {(150 - \frac{200}{2})}^{2} + + 18.182 \cdot 15 \cdot {(\frac{200}{2} - 50)}^{2} = = 36 881.50 {cm}^{4}$

Eccentricity of centroid e_c,d

The eccentricity of the ideal centroid of the cross-section is determined as follows:

$e_{c, ϕ_{1}} = z_{c, ϕ 1} - \frac{h}{2}$

uncracked state:

$e_{ϕ_{1}, l} = 101.4 - \frac{200}{2} = 1.4 mm$

cracked state:

$e_{ϕ_{1}, I I} = 58.5 - \frac{200}{2} = - 41.5 mm$

Image 2.120 Cross-sectional properties in reinforcement direction 1

Image 2.121 Cross-sectional properties in reinforcement direction 2

Parent Chapter

2.7.10 Example