RF-CONCRETE Members – Online Manual Version 5

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RF-CONCRETE Members – Online Manual Version 5

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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 εc,s (t, ts) at the relevant point of time t.

Hereafter, the determination of the creep coefficient φ (t, t0) and shrinkage strain εc,s (t, ts) 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 σc of the acting permanent load to not exceed the following value:

σc  0.45 · fckj 

where

    • fckj : cylinder compressive strength of concrete at point of time when creep-producing stress is applied
Figure 2.5 Creep-producing stress

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

Ec,eff = Ecm1 + φeff t,t0 

where

    • Ecm : mean modulus of elasticity according to EN 1992-1-1, Table 3.1
    • φeff (t, t0) : effective creep coefficient, φeff (t, t0) = φ (t, t0) ⋅ MQP / MEd
    • 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:

φt,t0 = φRH · βfcm · βt0 · βt,t0 

where

φRH = 1 + 1 - RH1000.1 · h03 · α1 · α2 

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

βfcm = 16.8fcm

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

βt0 = 10.1 + t00.20 

    • t0 : age of concrete in days when load application starts

βt,t0 = t - t0βH + t - t00.3 

      • t : age of concrete in days at relevant point of time
      • t0 : age of concrete in days when load application starts
      • βH = 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 / fcm)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:

tT = i=1n e-4000273 + Tti - 13.65  · ti 

where

Table 2.1

n

number of periods with identical temperature

T (Δti)

temperature in [°C] during time period Δti

Δti

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:

t0 = t0,T · 1 + 92 + t0,T1.2α  0.5 

where

Table 2.1

t0,T = tT

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 tk: 365 days

Age of concrete when creeping starts:

tτ = i=1n e-4000273 + τti - 13.65 · ti = e-4000273 + τti - 13.65  · 6 + e-4000273 + τti - 13.65 · 8 =     = 8.96 Tage

Age of concrete under influence of type of cement:

t0 = t0,τ · 1 + 92 + t0,τ1.2α = 8.96 · 1 + 92 + 8.961.20 =8.96 Tage

Effective structural component thicknesses:

h0 = 2 · Acu = 2 · 0.3 · 0.52 · 0.3 + 0.5 = 0.1875 cm 

Creep coefficient:

φt,t0 = φRH · βfcm · βt0 · βct,t0 = 1.933· 2.923 · 0.606 · 0.758 = 2.595 

where

φRH = 1 + 1 - RH1000.1 · h03 · α1 · α2 = 1 + 1 - 501000.1 · 187.53 · 1.042 ·1.012 = 1.933           α1 = 35fcm0.7 = 35330.7 = 1.042          α2 = 35fcm0.2 = 35330.2 = 1.012 

βfcm = 16.8fcm = 16.833 = 2.923 

βt0 = 10.1 + t00.2 = 10.1 + 8.960.2 = 0.606 

βc t, t0 = t - t0βH + t - t00.3 = 365 - 8.96538.779 + 365 - 8.960.3 = 0.758 

βH = 1.5 1 + (0.012 · RH)18 · h0 + 250 · α3 =      = 1.5 · 1 + 0.012 · 5018 · 187.5 + 250 · 1.030 = 538.779

βH  1500 · α3 = 1500 · 1.030 = 1545         α3 = 35330.5 = 1.030

Coefficient of shrinkage ε (t, ts)

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

εcs t = εca t + εcd t,ts 

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

εcat = βas t · εca 

where

βas (t) = 1 - e-0.2·t 

εca  = 2.5 · fck - 10 · 10-6       fck in [N/mm2] 

The component from drying shrinkage εcd is determined as follows:

εcd t,ts = βds t,ts · kh · εcd,0 fcm 

where

βdst,ts = t - tst - ts + 0.04 · h03 

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

εcd,0 = 0.85 ·220 + 110 +αds1 · e-αds2 fcmfcm0 · 10-6 · βRH 

      • fcm : mean cylinder compressive strength of concrete in [N/mm2]
      • fcm0 : 10 N/mm2
Table 2.3 αds1 and αds2

Cement class

Property

αds1

αds2

S

slow-hardening

3

0.13

N

normal-hardening

4

 0.12

R

rapid-hardening

6

0.11

βRH = 1.55 · 1 - RHRH03     

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

concrete C25/30

cement CEM 42.5 N

RH: 50 %

Age of concrete ts when shrinkage starts: 28 days

Considered age of concrete t: 365 days

Effective cross-section thickness:

h0 = 2 · Acu = 2 · 0.3 · 0.52 · 0.3 + 0.5 = 0.1875 m

Autogenous shrinkage:

εcat = βast · εca = 0.978 · 0.0000375 = 0.0000367 

where

βast = 1 - e-0.2t0.5 = 1 - e-0.2·365 = 0.978εca = 2.5 · fck - 10 · 10-6 = 2.5 · 25 - 10 · 10-6 = 0.0000367 

Drying shrinkage:

εcdt,ts = βdst,ts · kh · εcd,0 = 0.766 · 0.87 · 0.000512 = 0.000341 

where

βdst,ts = t - tst - ts + 0.04 · h03 = 365 - 28365 - 28 + 0.04 ·187.53 = 0.766 

h0 = 187.5 mm  kh = 0.87 

εcd,0 = 0.85 · 220 + 110 · αds1 · e-αds2fcmfcm0 · 10-6 · βRH =         = 0.85· 220 + 110 ·4 · e-0.123310 · 10-6 · 1.356 = 0.000512

βRH = 1.55 · 1 - RHRH03 = 1.55 · 1 - 501003 = 1.356 

Cement class N ⇒ αds1 = 4; αds2 = 0.12

Total coefficient of shrinkage:

ε t,ts=εcd t,ts+εca t=0.0000367+0.000341=0.000378=0.378