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004743

Dipl.-Ing. Georg Dlubal

2025-01-07

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Overview

1 Introduction

2 Theoretical Background

3 Input Data

4 Calculation

5 Results

6 Result Evaluation

7 Printout

8 General Functions

9 Literature

2.6.4.7 Limitation of Concrete Compressive Stress

Limitation of concrete compressive stress

In window 1.3 Surfaces, the concrete compressive stress is limited to σ_c,max = 0.45 ⋅ f_ck and the steel stress to σ_s,max = 0.80 ⋅ f_yk.

Figure 2.90 Limiting Concrete Compressive and Steel Stresses in Stress Check Tab of Window 1.3 Surfaces

For concrete C30/37, the maximum (negative) concrete stress σ_c,max is thus determined:
σ_c,max = 0.45 ∙ f_ck = 0.45 ∙ (-30.0) = -13.5 N/mm²

The provided concrete compressive stress is determined under the assumption of a linear stress distribution because the multitude of iterations for determining the suitable direction of the concrete compression strut would be too time-consuming. A linear distribution is sufficiently accurate because in the serviceability state, there are normally concrete compressive strains of at most 0.3 to 0.5 ‰.
The maximum stress σ_c,max is to be compared with the provided stress of the concrete compression zone for both reinforcement directions.
The provided concrete compressive stress σ_c is determined as follows:

σ_{c} = \frac{m_{Ed}}{I_{i, II}} \cdot x

mEd	Applied moment
Ii , I I	Ideal moment of inertia in state II

Concrete Compressive Stress

I_{i, II} = 13 \cdot b \cdot x^{3} + α_{e} \cdot a_{s} \cdot {(d - x)}^{2}

b	Width of element (always 1m for plates)
αe	Effective modular ratio
as	Provided tension reinforcement
d	Effective depth

Ideal Moment of Inertia in State II

x = \frac{α_{E} \cdot a_{s}}{b} \cdot - 1.0 + \sqrt{1.0 + \frac{2.0 \cdot b \cdot d}{α_{E} \cdot a_{s}}}

Depth of Concrete Compression Zone

Für die Bewehrungsrichtung φ1 ergibt sich somit folgende Druckzonenhöhe x_-z,φ1:

x_{- z, ϕ 1} = \frac{6.061 \cdot 11.31}{100} \cdot - 1.0 + \sqrt{1.0 + \frac{2.0 \cdot 100 \cdot 17}{6.061 \cdot 11.31}} = 4.19 cm

Depth of Neutral Axis x_-z,φ1

The same value and the related intermediate values can also be found in the details table.

Figure 2.91 Depth of Concrete Compression Zone for Reinforcement Direction 1

For the reinforcement direction φ₂, the neutral axis depth x_-z,φ2 is obtained:

x_{- z, ϕ_{2}} = \frac{6.061 \cdot 11.31}{100} \cdot - 1.0 + \sqrt{1.0 + \frac{2.0 \cdot 100 \cdot 15.8}{6.061 \cdot 11.31}} = 4.02 cm

Depth of Neutral Axis x_-z,φ2

This value and the related intermediate values can also be found in the details.

Figure 2.92 Depth of Concrete Compression Zone for Reinforcement Direction 2

The ideal moments of inertia I_i,II in state II (cracked section) are determined as follows for the two directions of reinforcement:
I_{i,II,-z,ϕ₁} = 1/3 · 100.0 · 4.19³ + 6.061 · 11.31 · 17 - 4.19² = 13 701 cm⁴
I_{i,II,-z,ϕ₂} = 1/3 · 100.0 · 4.02³ + 6.061 · 11.31 · 15.8 - 4.02² = 11 678 cm⁴

Für die beiden Bewehrungsrichtungen φ₁ und φ₂ ergeben sich damit gemäß Gleichung 2.69 folgende Betondruckspannungen σ_c in der Betondruckzone (d. h. an Flächenoberseite):

σ_{c, o, ϕ_{1}} = \frac{3676 \cdot 4.19}{13701} = - 11.24 N / {mm}^{2}

Concrete Compressive Stresses for Reinforcement Directions φ₁

σ_{c, o, ϕ_{2}} = \frac{2773 \cdot 4.02}{11678} = - 9.41 N / m m^{2}

Concrete Compressive Stresses for Reinforcement Directions φ₂

These values are also shown in Figure 2.92 (the program takes more decimal places into account).
Die vorhandenen Druckspannungen σ_c,+z,φ1 und σ_c,+z,φ2 sind somit kleiner als die maximale Betonspannung σc,max (siehe Bild 2.90). Der maßgebende Quotient von vorhandener zu zulässiger Betondruckspannung liegt in die Bewehrungsrichtung φ₁ vor. The design is fulfilled.

Figure 2.93 Analysis of Concrete Compressive Stress

Parent Chapter

2.6.4 Serviceability Limit State Designs