Consideration of Shrinkage

Consideration of Shrinkage

Shrinkage is defined as a time-dependent change in volume without the influence of external loads or temperature. The further subdivision of the shrinkage problem into individual manifestations (drying shrinkage, autogenous shrinkage, plastic shrinkage, and carbonation shrinkage) is not discussed in detail here.
The essential influencing factors of shrinkage are relative humidity, effective member thickness, aggregate, concrete strength, water-cement ratio, temperature, as well as the type and duration of curing. The quantity used to capture shrinkage is the total shrinkage strain ε_cs at the considered time t.

According to EN 1992-1-1, Section 3.1.4, ([2] Eq. (3.8)) the total shrinkage strain ε_cs consists of the components drying shrinkage ε_cd and autogenous shrinkage ε_ca, as summarized in the following equation.

Drying Shrinkage

The component from drying shrinkage ε_cd is determined according to [2] Eq. (3.9) as follows.

with

• $${\mathrm{\beta }}_{\mathrm{ds}}(\mathrm{t}, {\mathrm{t}}_{\mathrm{s}}) =(\mathrm{t} - {\mathrm{t}}_{\mathrm{s}})/\left(\right(\mathrm{t} - {\mathrm{t}}_{\mathrm{s}}) + 0.4 · \sqrt{{{\mathrm{h}}_0}^3})$$

  t	Age of concrete at relevant point of time in days
  ts	Age of concrete when shrinkage starts in days
  h0	Effective thickness of the structural component [mm] (for surfaces: h0 = h)
  kh	Coefficient according to [1], Table 3.3, depending on the effective cross-section thickness h0
  εcd,0	Basic value according to [1] Table 3.2, or Annex B, Eq. (B.11)

  Factor β_ds (t,t_s)
• $${\mathrm{h}}_0 = \frac{2 · {\mathrm{A}}_{\mathrm{c}}}{\mathrm{u}}$$

  Ac	Cross-sectional area
  u	Cross-section perimeter

  Effective Thickness of Structural Component [mm] (for Surfaces: h₀ = h)
• $${\mathrm{\epsilon }}_{\mathrm{cd}, 0} = 0.85 · \left[(220 + 110 · {\mathrm{\alpha }}_{\mathrm{ds}1}) · \exp (-{\mathrm{\alpha }}_{\mathrm{ds}2} · \frac{{\mathrm{f}}_{\mathrm{cm}} }{{\mathrm{f}}_{\mathrm{cmo}} })\right] · {10}^{-6} · {\mathrm{\beta }}_{\mathrm{RH}}$$

  αds1, αds2	Factors for considering the type of cement
  fcm	Mean cylinder compressive strength of concrete in [N/mm²]
  fcmo	= 10 N/mm²

  Basic Value for Drying Shrinkage According to [1], Table 3.2, or Annex B, Eq. (B.11)
• $${\mathrm{\beta }}_{\mathrm{RH}} = 1.55 · \left[1 - \frac{\mathrm{R} {\mathrm{H}}^3}{\mathrm{R} {\mathrm{H}}_0}\right]$$

  RH	Relative humidity of environment [%]
  RH0	= 100%

  Formula β_RH

Autogenous Shrinkage Strain

The autogenous shrinkage strain ε_ca is determined according to [2] Eq. (3.11) as follows.

with

• $${\mathrm{\epsilon }}_{\mathrm{ca}}(\infty ) = 2.5 · ({\mathrm{f}}_{\mathrm{ck}} - 10) · {10}^{-6}$$

  Formula ε_ca(∞)
• $${\mathrm{\beta }}_{\mathrm{as}}\left(\mathrm{t}\right) = 1 - {\mathrm{e}}^{-0.2\sqrt{\mathrm{t}}}$$

  Formula β_as

Consideration of Shrinkage in Concrete Design (with Consideration of Reinforcement)

The shrinkage strain is specified in the material dialog in the section Time-Dependent Characteristics of the Concrete. There, the concrete age at the considered time and at the beginning of shrinkage, the relative humidity, and the type of cement must be entered. The program then determines the shrinkage strain ε_cs from these specifications.

The shrinkage strain ε_cs(t,t_s) can also be specified manually independently of the standard.
The shrinkage strain is applied only to the concrete layers; the reinforcement layers are not considered. Thus, there is a difference compared with the classical temperature load, which also acts on the reinforcement layers. The shrinkage model used in the program therefore takes into account the restraint of the shrinkage strain ε_sh caused by the reinforcement or the cross-section curvature in the case of asymmetric reinforcement. The resulting loads from the shrinkage strain are automatically applied and calculated as virtual loads on the surfaces. Depending on the structural system, the shrinkage strain leads to additional stresses (statically indeterminate system) or additional deformations (statically determinate system). The program therefore considers the influence of the static boundary conditions in different ways for the application of shrinkage.

Shrinkage depends on the correct distribution of stiffness in the cross-section. Therefore, consideration of Tension Stiffening is recommended for the tensile zone of the concrete. The 1D model shown in the following figure illustrates how shrinkage is captured in the program.

Schematic Shrinkage Diagram in 1D Model

For simplification, four layers are considered:

• The dark orange layers represent the less damaged concrete,
• the light orange layers represent the more damaged concrete.
• The blue layer corresponds to the reinforcement.
• Each concrete layer is characterized by the actual modulus of elasticity E_c,i, each cross-sectional area by A_c,i.
• The reinforcement is characterized by the actual modulus of elasticity E_s and the cross-sectional area A_s.
• Each layer is described by the coordinate z_i.