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

Technical Article

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 defined as the width of the cracks on the component surface, as 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 susceptibility of the reinforcing steel [1].

Causes of Crack Formation due to Early Restraint

An important effect in the crack formation is the so‑called early restraint, where the most significant values generating the restraint are the temperature changes due to hydration heat formation, concrete shrinkage and building ground motion. Especially in the case of reinforced concrete structural components, the cracks in early‑age concrete occur usually a few days after the stripping. In the case of thick walls, the hydration heat formation can also lead to the development of internal stresses, which are caused by temperature differences over the cross-section and result 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]

Concrete is considered young concrete for up to three days. After this time, the young concrete reaches a degree of hydration of 60 to 90%, depending on cement type, ambient temperature and the water-cement ratio. The young concrete is characterized by the following properties:

  • strong heat development and thus heat exchange with the surroundings,
  • large volume change due to the heat development,
  • rapid change of mechanical properties due to the gradual hydration.

During the hydration heat formation, the internal stress conditions occur especially in reinforced concrete structural components, which lead to compression stresses and tension stresses in the edge areas of the cross‑section. Based on the differences without the corresponding countermeasures, this stress condition leads to the formation of large cracks.

Countermeasures

Generally, it is possible to minimize or slow down the formation of restraint stresses by applying advanced concrete technology measures, appropriate curing or by arranging expansion joints. Since it is not possible to avoid the crack formation completely, the cracks must be limited and distributed by suitable reinforcement.

Determination of Minimum Reinforcement

In order to ensure the limitation of crack widths, it is necessary to create the minimum reinforcement for crack width control. Below, the calculation of the minimum reinforcement according to EN 1992‑1‑1 is compared with the results of RF‑CONCRETE Surfaces.

Initial Values
Concrete  C30/37
Reinforcing steel  B 500 S (A)
Wall thickness   h  100.0 mm
Concrete cover   cnom  40.0 mm  for the exposition class XC4
Allowable crack width   wk  0.2 mm
Selected rebar diameter   ds  14.0 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

kc  =  1.0 (pure tension)
k  =  0.65 ⋅ 0.8 = 0.52 (with modification for internal stresses)
fct,eff  =  0.5 ⋅ fctm = 1.45 N/mm²
act  =  h/2 ⋅ b = 5,000 cm²/m

σs is defined using the limit diameter ds* as follows:

$$\begin{array}{l}\begin{array}{l}{\mathrm\sigma}_\mathrm s\;=\;\sqrt{{\mathrm w}_\mathrm k\;\cdot\;\frac{3.48\;\cdot\;10^6}{\mathrm d_\mathrm s^\ast}}\;=\;185.41\;\mathrm N/\mathrm{mm}²\\\mathrm d_\mathrm s^\ast\;=\;{\mathrm d}_\mathrm s\;\cdot\;\frac{8\;\cdot\;(\mathrm h\;-\;\mathrm d)}{{\mathrm k}_\mathrm c\;\cdot\;\mathrm k\;\cdot\;{\mathrm h}_\mathrm{cr}}\;\cdot\;\frac{\displaystyle2.9}{{\mathrm f}_{\mathrm{ct},\mathrm{eff}}}\;\leq\;{\mathrm d}_\mathrm s\;\cdot\;\frac{\displaystyle2.9}{{\mathrm f}_{\mathrm{ct},\mathrm{eff}}}\end{array}\\20.2\;\mathrm{mm}\;\leq\;28.0\;\mathrm{mm}\end{array}$$

where

 =  h - (cnom + ds / 2) = 95.3 cm
hcr   =  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, you can perform the calculation of the minimum reinforcement considering the effective edge zone Ac,eff. In this case, the reinforcement should not be created anymore as it 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
fct,eff  =  0.5 ⋅ fctm = 1.45 N/mm²
ac,eff  =  hc,eff ⋅ b = 19.4 cm ⋅ 100 cm/m [according to Figure 7.1d)]
act  =  h / 2 ⋅ b = 5,000 cm²/m
fyk  =  500 N/mm²/m

σs is defined using the limit diameter ds* as follows:

$$\begin{array}{l}{\mathrm\sigma}_\mathrm s\;=\;\sqrt{{\mathrm w}_\mathrm k\;\cdot\;\frac{3.48\;\cdot\;10^6}{\mathrm d_\mathrm s^\ast}}\;=\;157.66\;\mathrm N/\mathrm{mm}²\\\mathrm d_\mathrm s^\ast\;=\;{\mathrm d}_\mathrm s\;\cdot\;\frac{\displaystyle2.9}{{\mathrm f}_{\mathrm{ct},\mathrm{eff}}}\;=\;28.0\;\mathrm{mm}\end{array}$$ $$\begin{array}{l}{\mathrm a}_{\mathrm s,\min}\;=\;1.45\;\mathrm N/\mathrm{mm}²\;\cdot\;\frac{\displaystyle1,940\;\mathrm{cm}²/\mathrm m}{157.66\;\mathrm N/\mathrm{mm}²}\;\geq\;0.52\;\cdot\;1.45\;\mathrm N/\mathrm{mm}²\;\cdot\;\frac{\displaystyle5,000\;\mathrm{cm}²/\mathrm m}{500\;\mathrm N/\mathrm{mm}²}\\{\mathrm a}_{\mathrm s,\min}=\;17.84\;\mathrm N/\mathrm{mm}²\;\geq\;7.54\;\mathrm N/\mathrm{mm}²\end{array}$$

Figure 03 - Second and Third Calculated Value of Minimum Reinforcement

Reference

[1]   Avak, R. (1991). Stahlbetonbau in Beispielen, DIN 1045 und Europäische Normung, Teil 1: Baustoffe, Grundlagen, Bemessung von Balken. Düsseldorf: Werner.
[2]   Rostásy, F. & Henning, W. (1990). Zwang und Rißbildung in Wänden auf Fundamenten. DAfStb‑Heft 407. Berlin: Beuth Verlag.
[3]   Eurocode 2: Design of concrete structures - Part 1‑1: General rules and rules for buildings; EN 1992‑1‑1:2004

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