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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 Examples

10 Literature

2.2.6 Creep and Shrinkage

Creep and Shrinkage

Determination of initial values

This chapter gives an overview of the time-dependent stresses and strains due to creep and shrinkage. The influence of creep and shrinkage is used in the analytical serviceability limit state design for the determination of the deformation. The approach of creep and shrinkage in the nonlinear calculation is described in Chapter 2.4.6.

Creep is the time-dependent deformation of concrete under loading over a specific period of time. The essential influence values are similar to those of shrinkage, with the so-called creep-producing stress having considerable effects on the creep deformation. Special attention must be paid to the load duration, the point of time of load application, as well as to the extent of actions. The creep determining value is the creep coefficient φ (t, t₀) at the relevant point of time t.

Shrinkage describes a time-dependent modification of volume without influence due to external loads or temperature. We will not elaborate on further expansion of the shrinkage problem into individual types (drying shrinkage, autogenous shrinkage, plastic shrinkage, and carbonation shrinkage). Significant influence values of shrinkage are relative humidity, effective thickness of structural components, aggregate, concrete strength, water-cement ratio, temperature, as well as the type and duration of curing. The shrinkage determining value is the shrinkage strain ε_c,s (t, t_s) at the relevant point of time t.

Hereafter, the determination of the creep coefficient φ (t, t₀) and shrinkage strain ε_c,s (t, t_s) according to EN 1992-1-1, clause 3.1.4 and annex B is described.

Creep coefficient φ (t, t₀)

Using the following formulas requires the creep-producing stress σ_c of the acting permanent load to not exceed the following value:

$σ_{c} \leq 0.45 \cdot f_{c k j}$

where

f_ckj : cylinder compressive strength of concrete at point of time when creep-producing stress is applied

Under the assumption of a linear creep behavior (σ_c ≤ 0.45 ⋅ f_ckj), the concrete's creep can be determined by a reduction of the modulus of elasticity for concrete.

$E_{c, e f f} = \frac{E_{c m}}{1 + φ_{e f f} (t, t_{0})}$

where

E_cm : mean modulus of elasticity according to EN 1992-1-1, Table 3.1
φ_eff (t, t₀) : effective creep coefficient, φ_eff (t, t₀) = φ (t, t₀) ⋅ M_QP / M_Ed
t : age of concrete in days at relevant point of time
t₀ : age of concrete in days when load application starts

The creep coefficient φ (t, t₀) at the analyzed point of time t may be calculated as follows:

$φ (t, t_{0}) = φ_{R H} \cdot β (f_{c m}) \cdot β (t_{0}) \cdot β (t, t_{0})$

where

$φ_{R H} = [1 + \frac{1 - \frac{R H}{100}}{0.1 \cdot \sqrt[3]{h_{0}}} \cdot α_{1}] \cdot α_{2}$

RH : relative humidity in [%]
h₀ : effective thickness of structural component in [mm]
- h₀ = 2 ∙ A_c /u
- A_c : cross-sectional area
- u : cross-section perimeter
α₁, α₂ : adjustment factors
- α₁ = (35 / f_cm)^0.7
- α₂ = (35 / f_cm)^0.2
- f_cm : mean value of cylinder compressive strength

$β (f_{c m}) = \frac{16.8}{\sqrt{f_{c m}}}$

f_cm : mean value of cylinder compressive strength of concrete in [N/mm²]

$β_{(t_{0})} = \frac{1}{0.1 + t_{0}^{0.20}}$

t₀ : age of concrete in days when load application starts

$β (t, t_{0}) = {[\frac{t - t_{0}}{β_{H} + t - t_{0}}]}^{0.3}$

t : age of concrete in days at relevant point of time
t₀ : age of concrete in days when load application starts
β_H = 1.5 ⋅ [1 + (0.012 ⋅ RH)¹⁸] ⋅ h₀ + 250 ⋅ α₃ ≤ 1500 ⋅ α₃
- RH : relative humidity in [%]
- h₀ : effective thickness of structural component [mm]
- α₃ : adjustment factor
- α₃ = (35 / f_cm)^0.5 ≤ 1.0

The following input is required to calculate the creep coefficient:

RH : relative humidity in [%]
t₀ : age of concrete in days when load application starts
t : age of concrete in days at relevant point of time (optionally ∞)

The influence of high or low temperature ranging from 0°C to 80°C on the concrete's maturity can be taken into account by correcting the concrete age with the following equation:

$t_{T} = \sum_{i = 1}^{n} e^{- [\frac{4000}{273 + T (∆ t_{i})} - 13.65]} \cdot ∆ t_{i}$

where

Table 2.1
n	number of periods with identical temperature
T(Δt_i)	temperature in [°C] during time period Δt_i
Δt_i	number of days with this temperature T

The influence of the type of cement on the concrete's creep coefficient can be taken into account by modifying the load application age t₀ with the following equation:

$t_{0} = t_{0, T} \cdot {(1 + \frac{9}{2 + {(t_{0, T})}^{1.2}})}^{α} \geq 0.5$

where

Table 2.1
t_0,T = t_T	effective age of concrete when load application starts considering influence of temperature
α	exponent, depends on type of cement, see Table 2.2

Table 2.2 Exponent α
α	Type of Cement
−1	slow-hardening cements of class S
0	normal- or rapid-hardening cements of class N
1	rapid-hardening, high-strength cements of class R

Example

Table 2.2 Cross-section
Figure 2.6	concrete C25/30 cement CEM 42.5 N RH: 50% Two temperature changes: 6 days - temperature 15 °C 8 days - temperature 7 °C Considered age of concrete t_k: 365 days

Age of concrete when creeping starts:

$t_{τ} = \sum_{i = 1}^{n} e^{- [\frac{4000}{273 + τ (Δ t_{i})} - 13.65]} \cdot Δ t_{i} = e^{- [\frac{4000}{273 + τ (Δ t_{i})} - 13.65]} \cdot 6 + e^{- [\frac{4000}{273 + τ (Δ t_{i})} - 13.65]} \cdot 8 = 8.96 days$

Age of concrete under influence of type of cement:

$t_{0} = t_{0, τ} \cdot {(1 + \frac{9}{2 + {(t_{0, τ})}^{1.2}})}^{α} = 8.96 \cdot {(1 + \frac{9}{2 + {(8.96)}^{1.2}})}^{0} = 8.96 days$

Effective structural component thicknesses:

$h_{0} = \frac{2 \cdot A_{c}}{u} = \frac{2 \cdot 0.3 \cdot 0.5}{2 \cdot (0.3 + 0.5)} = 0.1875 c m$

Creep coefficient:

$φ (t, t_{0}) = φ_{R H} \cdot β (f_{c m}) \cdot β (t_{0}) \cdot β_{c} (t, t_{0}) = 1.933 \cdot 2.923 \cdot 0.606 \cdot 0.758 = 2.595$

where

$φ_{R H} = [1 + \frac{1 - \frac{R H}{100}}{0.1 \cdot \sqrt[3]{h_{0}}} \cdot α_{1}] \cdot α_{2} = [1 + \frac{1 - \frac{50}{100}}{0.1 \cdot \sqrt[3]{187.5}} \cdot 1.042] \cdot 1.012 = 1.933 α_{1} = {(\frac{35}{f_{c m}})}^{0.7} = {(\frac{35}{33})}^{0.7} = 1.042 α_{2} = {(\frac{35}{f_{c m}})}^{0.2} = {(\frac{35}{33})}^{0.2} = 1.012$

$β (f_{c m}) = \frac{16.8}{\sqrt{f_{c m}}} = \frac{16.8}{\sqrt{33}} = 2.923$

$β_{(t_{0})} = \frac{1}{0.1 + t_{0}^{0.2}} = \frac{1}{0.1 + 8 . 96^{0.2}} = 0.606$

$β_{c} (t, t_{0}) = {[\frac{t - t_{0}}{β_{H} + t - t_{0}}]}^{0.3} = {[\frac{365 - 8.96}{538.779 + 365 - 8.96}]}^{0.3} = 0.758$

$β_{H} = 1.5 [1 + (0.012 \cdot R H)^{18}] \cdot h_{0} + 250 \cdot α_{3} = = 1.5 \cdot [1 + {(0.012 \cdot 50)}^{18}] \cdot 187.5 + 250 \cdot 1.030 = 538.779$

$β_{H} \leq 1500 \cdot α_{3} = 1500 \cdot 1.030 = 1545 α_{3} = {(\frac{35}{33})}^{0.5} = 1.030$

Coefficient of shrinkage ε (t, t_s)

When determining the coefficient of shrinkage ε (t, t_s) according to EN 1992-1-1, clause 3.1.4, the shrinkage strain ε_cs (t) can be calculated from the sum of the components of the autogenous shrinkage ε_ca (t) and drying shrinkage ε_cd (t, ts).

$ε_{c s} (t) = ε_{c a} (t) + ε_{c d} (t, t_{s})$

The autogenous shrinkage strain ε_ca at the relevant point of time (t) is determined as follows:

$ε_{c a} (t) = β_{a s} (t) \cdot ε_{c a} (\infty)$

where

${β_{a s}}_{} (t) = 1 - e^{- 0.2 \cdot \sqrt{t}}$

$ε_{c a} (\infty) = 2.5 \cdot (f_{c k} - 10) \cdot 10^{- 6} f_{c k} {in [N/mm}^{2}]$

The component from drying shrinkage ε_cd is determined as follows:

$ε_{c d} (t, t_{s}) = β_{d s} (t, t_{s}) \cdot k_{h} \cdot ε_{c d, 0} (f_{c m})$

where

$β_{d s} (t, t_{s}) = \frac{t - t_{s}}{t - t_{s} + 0.04 \cdot \sqrt{h_{0}^{3}}}$

t age of concrete in days at relevant point of time
t_s age of concrete in days when shrinkage starts
h₀ effective cross-section thickness in [mm] : h₀ = 2 ⋅ A_c / u

$ε_{c d, 0} = 0.85 \cdot [(220 + 110 + α_{d s 1}) \cdot e^{- α_{d s 2} \frac{f_{c m}}{f_{c m 0}}}] \cdot 10^{- 6} \cdot β_{R H}$

f_cm : mean cylinder compressive strength of concrete in [N/mm²]
f_cm0 : 10 N/mm²

Table 2.3 α_ds1 and α_ds2
Cement class	Property	α_ds1	α_ds2
S	slow-hardening	3	0.13
N	normal-hardening	4	0.12
R	rapid-hardening	6	0.11

$β_{R H} = 1.55 \cdot [1 - {(\frac{R H}{R H_{0}})}^{3}]$

RH relative humidity of environment in [%]
RH₀ 100 %

Example

concrete C25/30

cement CEM 42.5 N

RH: 50 %

Age of concrete t_s when shrinkage starts: 28 days

Considered age of concrete t: 365 days

Effective cross-section thickness:

$h_{0} = \frac{2 \cdot A_{c}}{u} = \frac{2 \cdot 0.3 \cdot 0.5}{2 \cdot (0.3 + 0.5)} = 0.1875 m$

Autogenous shrinkage:

$ε_{c a} (t) = β_{a s} (t) \cdot ε_{c a} (\infty) = 0.978 \cdot 0.0000375 = 0.0000367$

where

$β_{a s} (t) = 1 - e^{- 0.2 t^{0.5}} = 1 - e^{- 0.2 \cdot \sqrt{365}} = 0.978 ε_{c a} (\infty) = 2.5 \cdot (f_{c k} - 10) \cdot 10^{- 6} = 2.5 \cdot (25 - 10) \cdot 10^{- 6} = 0.0000367$

Drying shrinkage:

$ε_{c d} (t, t_{s}) = β_{d s} (t, t_{s}) \cdot k_{h} \cdot ε_{c d, 0} = 0.766 \cdot 0.87 \cdot 0.000512 = 0.000341$

where

$β_{d s} (t, t_{s}) = \frac{t - t_{s}}{t - t_{s} + 0.04 \cdot \sqrt{h_{0}^{3}}} = \frac{365 - 28}{365 - 28 + 0.04 \cdot \sqrt{187 . 5^{3}}} = 0.766$

$h_{0} = 187.5 mm \Rightarrow k_{h} = 0.87$

$ε_{c d, 0} = 0.85 \cdot [(220 + 110 \cdot α_{d s 1}) \cdot e^{- α_{d s 2} \frac{f_{c m}}{f_{c m 0}}}] \cdot 10^{- 6} \cdot β_{R H} = = 0.85 \cdot [(220 + 110 \cdot 4) \cdot e^{- 0.12 \frac{33}{10}}] \cdot 10^{- 6} \cdot 1.356 = 0.000512$

$β_{R H} = 1.55 \cdot [1 - {(\frac{R H}{R H_{0}})}^{3}] = 1.55 \cdot [1 - {(\frac{50}{100})}^{3}] = 1.356$

Cement class N ⇒ α_ds1 = 4; α_ds2 = 0.12

Total coefficient of shrinkage:

$ε_{} (t, t_{s}) = ε_{c d} (t, t_{s}) + ε_{c a} (t) = 0.0000367 + 0.000341 = 0.000378 = 0.378 ‰$

Parent Chapter

2.2 Serviceability Limit State Design