RF-CONCRETE Surfaces – Online Manual Version 5

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RF-CONCRETE Surfaces – Online Manual Version 5

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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 as2,φ1, as1,φ2, and as2,φ2.

The following cross-section properties for the uncracked and the 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.

zl,ϕ1 = b · h22 + α as1,ϕ1 · d1,ϕ1 + as2,ϕ1 · d2,ϕ1b · h + α · as1,ϕ1 + as2,ϕ1 =         = 1000 · 20022 + 6.061 · 1000 · 150 · 15 · 501000 · 200 · 6.061 · 1000 + 15 = 101.4 mm 

For the cracked state, the depth χII,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 Ac,d

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

Al,ϕ1 = b · h + α · as1,ϕ1 + as2,ϕ1 = 1000 · 200 + 6.061 · 1000 + 15 = 2061.5 cm2 

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

AIl,ϕ1 = b · χII,ϕ1 + α · as1,ϕ1 + as2,ϕ1 = 1000 · 68.3 + 18.182 · 1000 + 15 = 867.19 cm2

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 Ic,d

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

II,ϕ1 = 112 · b · h3 + b · h · zI,ϕ1 - h22 + α · as1,ϕ1 · d1,ϕ1 - zI,ϕ12 + α ·as2,ϕ1 · zI,ϕ1 - d2,ϕ12 =       = 112 · 1000 · 2003 +1000 · 200 · 101.4 - 20022 + 6.061 · 1000 · (150 - 101.4) 2 + 6.061 · 15 · (101.4 - 50) 2 =       = 68 161.30 cm4 

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

III,ϕ1 = 112 · b · χ3II,ϕ1  + b · χII,ϕ1  · zII,ϕ1 - χII,ϕ122 +                + α · as1,ϕ1 · d1,ϕ1 - zII,ϕ12 +  α · as2,ϕ1 · zII,ϕ1 - d2,ϕ12 =          = 112 · 1000 · 68.33 + 1000 · 68.3 · 58.5 - 68.322 +               + 18.182 · 1000 · 150 - 58.52 + 18.182 · 15 · 58.5 - 502 =         = 21 928.70 cm4           

Ideal moment of inertia to the geometric center of the cross-section I0,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:

I0,I,ϕ1 = 112 · b · h3 + α · as1,ϕ1 · d1,ϕ1 - h22 + α · as2,ϕ1 · h2 - d2,ϕ12 =           = 112 · 1000 ·2003 + 6.061 ·200 · 150 - 20022 + 6.061 · 15 · 2002 - 502=           = 68 204.50 cm4 

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

I0,II,ϕ1 =112 · b · χII,ϕ13 + b · χII,ϕ1 · h2 - χII,ϕ122 + α · as1,ϕ1 · d1,ϕ1 - h22 + α · as2,ϕ1 · h2 - d2,ϕ12=            = 112 · 1000 · 68.33 + 1000 · 68.3 · 2002 - 68.322 + 18.182 · 1000 · 150 - 20022+                  + 18.182 ·15 · 2002 - 502 =            = 36 881.50 cm4            

Eccentricity of centroid ec,d

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

ec,ϕ1 = zc,ϕ1 - h2 

  • uncracked state:

eϕ1,l = 101.4 - 2002 = 1.4 mm 

  • cracked state:

eϕ1, II = 58.5 - 2002 = - 41.5 mm 

Figure 2.120 Cross-sectional properties in reinforcement direction 1
Figure 2.121 Cross-sectional properties in reinforcement direction 2