Determination of Minimum Reinforcement for Centric Restraint on Thick Structural Components According to EN 1992-1-1

Technical Article

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In general, avoiding cracking in concrete structures is neither possible nor necessary. However, cracking must be limited in a way that the proper use, appearance and durability of the structure are not affected. Therefore, limiting the crack width does not mean preventing crack formation, but restricting the crack width to harmless values.

The crack width w is the width of the cracks on the component surface, because the crack width decreases with increasing distance from the surface. The allowable size of the crack width depends on the environmental conditions, the function of the structural component, and the corrosion sensitivity of the reinforcing steel [1] .

Causes of cracking due to early restraint

A significant effect in the formation of cracks is the so-called early constraint, where the most important constraint-producing variables are the temperature changes during the drain of the hydration heat, the shrinkage of the concrete and soil movements. Especially with massive components, the cracks occur at a young age, usually a few days after stripping. In the case of thick walls, the flow of hydration heat can also lead to the development of residual stresses caused by temperature differences over the cross-section and resulting in shell cracks on the wall surface.

Figure 01 - Internal Stresses, Neutral Axis Position and Crack Depth in Case of Cooling Slab on Both Sides [2]

Up to three days old concrete is called new concrete. After this time, the new concrete achieves a degree of hydration of 60 to 90%, depending on the type of cement, the ambient temperature and the water-cement value. The new concrete is characterized by the following properties:

  • strong heat generation and thus heat exchange with the environment,
  • large change in volume due to heat generation,
  • rapid change of mechanical properties due to progressive hydration.

By flowing away the heat of hydration, residual stress states occur, especially in massive components, which lead to compressive stresses within and to tensile stresses in the edge areas of the cross-section. Due to the differences, this stress state leads to the formation of large cracks without the corresponding countermeasures.

Countermeasures

In general, it is possible to minimize or delay the development of imposed stresses by concrete technology measures, a suitable after-treatment and, if necessary, the arrangement of expansion joints. However, since it is not possible to completely avoid cracks, they must be limited and distributed by a suitable reinforcement.

Determination of the Minimum Reinforcement

To ensure the limitation of crack widths, a minimum reinforcement must be inserted to limit the crack width. The calculation of the minimum reinforcement according to DIN EN 1992-1-1 will be compared with the results in RF-CONCRETE Surfaces.

Input values:
Wall thickness: h = 100 cm
Concrete cover: c nom = 40 mm for exposure class XC4
Concrete: C30/37
Reinforcing steel: B 500 S (A)
allowable crack width: w k = 0.2 mm
Selected member diameter: ds = 14 mm

${\mathrm a}_{\mathrm s,\min}\;=\;{\mathrm k}_\mathrm c\;\cdot\;\mathrm k\;\cdot\;\frac{\displaystyle{\mathrm f}_{\mathrm{ct},\mathrm{eff}}}{{\mathrm\sigma}_\mathrm s}\;\cdot\;{\mathrm a}_\mathrm{ct}$
where
k c = 1.0 (pure tension)
k = 0.65 ∙ 0.8 = 0.52 (with modification for residual stresses)
f ct, eff = 0.5 ∙ f ctm = 1.45 N/mm²
a ct = h/2 ∙ b = 5000 cm²/m

σ s is determined on the basis of the limit diameter d s *
$\begin{array}{l}{\mathrm\sigma}_\mathrm s\;=\;\sqrt{{\mathrm w}_\mathrm k\;\cdot\;\frac{3,48\;\cdot\;10^6}{\mathrm d_\mathrm s^\ast}}\;=\;185,41\;\mathrm N/\mathrm{mm}²\\\mathrm d_\mathrm s^\ast\;=\;{\mathrm d}_\mathrm s\;\cdot\;\frac{8\;\cdot\;(\mathrm h\;-\;\mathrm d)}{{\mathrm k}_\mathrm c\;\cdot\;\mathrm k\;\cdot\;{\mathrm h}_\mathrm{cr}}\;\cdot\;\frac{\displaystyle2,9}{{\mathrm f}_{\mathrm{ct},\mathrm{eff}}}\;\leq\;{\mathrm d}_\mathrm s\;\cdot\;\frac{\displaystyle2,9}{{\mathrm f}_{\mathrm{ct},\mathrm{eff}}}\end{array}$
20.2 mm ≤ 28.0 mm
where
d = h - (c nom + d s/2) = 95.3 cm
h h cr = h = 100 cm

${\mathrm a}_{\mathrm s,\min}\;=\;1,0\;\cdot\;0,52\;\cdot\;\frac{\displaystyle1,45\;\mathrm N/\mathrm{mm}²}{185,41\;\mathrm N/\mathrm{mm}²}\;\cdot\;5.000\;\mathrm{cm}²/\mathrm m\;=\;20,33\;\mathrm{cm}²/\mathrm m$

Figure 02 - First Calculated Value of Minimum Reinforcement

For thicker structural components, it is allowed to perform the calculation of the minimum reinforcement taking into account an effective edge zone A c, eff , whereby no more reinforcement has to be inserted than was determined in the previous calculation [3] .

${\mathrm a}_{\mathrm s,\min}\;=\;{\mathrm f}_{\mathrm{ct},\mathrm{eff}}\;\cdot\;\frac{\displaystyle{\mathrm a}_{\mathrm c,\mathrm{eff}}}{{\mathrm\sigma}_\mathrm s}\;\geq\;\mathrm k\;\cdot\;{\mathrm f}_{\mathrm{ct},\mathrm{eff}}\;\cdot\;\frac{\displaystyle{\mathrm a}_\mathrm{ct}}{{\mathrm f}_\mathrm{yk}}$
where
k = 0.52
f ct, eff = 0.5 ∙ f ctm = 1.45 N/mm²
a c, eff = h c, eff ∙ b = 19.4 cm ∙ 100 cm/m [according to Figure 7.1d) DE]
a ct = h/2 ∙ b = 5,000 cm²/m
f yk = 500 N/mm²
σ s is determined on the basis of the limit diameter d s *
$\begin{array}{l}{\mathrm\sigma}_\mathrm s\;=\;\sqrt{{\mathrm w}_\mathrm k\;\cdot\;\frac{3,48\;\cdot\;10^6}{\mathrm d_\mathrm s^\ast}}\;=\;157,66\;\mathrm N/\mathrm{mm}²\\\mathrm d_\mathrm s^\ast\;=\;{\mathrm d}_\mathrm s\;\cdot\;\frac{\displaystyle2,9}{{\mathrm f}_{\mathrm{ct},\mathrm{eff}}}\;=\;28,0\;\mathrm{mm}\end{array}$

${\mathrm a}_{\mathrm s,\min}\;=\;1,45\;\mathrm N/\mathrm{mm}²\;\cdot\;\frac{\displaystyle1.940\;\mathrm{cm}²/\mathrm m}{157,66\;\mathrm N/\mathrm{mm}²}\;=\;17,84\;\mathrm{cm}²/\mathrm m\;\geq\;0,52\;\cdot\;1,45\;\mathrm N/\mathrm{mm}²\;\cdot\;\frac{\displaystyle5.000\;\mathrm{cm}²/\mathrm m}{500\;\mathrm N/\mathrm{mm}²}\;=\;7,54\;\mathrm{cm}²/\mathrm m$

Figure 03 - Second and Third Calculated Value of Minimum Reinforcement

Literature

[1]   Avak, R .: Reinforced concrete structures in examples, DIN 1045 and European Standardization, Part 1: Building materials, Basics, Design of beams. Dusseldorf: Werner, 1991
[2]  Rostásy, F. S; Henning, W .: Restraint and cracking in walls on foundations. DAfStb Issue 407. Berlin: Beuth, 1990
[3]  Eurocode 2: Design of reinforced concrete and prestressed concrete structures - Part 1-1: General rules and rules for buildings; EN 1992-1-1: 2004

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