Shrinkage effects in nonlinear analysis
When concrete structures are designed, a linear analysis is usually not fully sufficient. There are several other aspects influencing the behavior of the structure mainly from the serviceability point of view. These effects are mainly:
 Concrete cracking
 Creep and shrinkage
The cracking of concrete is usually an irreversible effect which appears when the tensile strength of the concrete is exceeded. The structure could however still be durable and reliable with cracks if those are below the code given limits. This article is mainly about the notification of these effects during design with a focus on shrinkage.
Shrinkage effect
Creep and shrinkage phenomena are the most important timedependent material characteristics of concrete. When concrete is placed in situ, the cement starts to hydrate using the free water in the mixture. Usually, the water exits the concrete, and the process of drying shrinkage is starting. This is the reason why the concrete must be cured especially in the early stages after casting. Without a special curing process, additional stresses are presented inside of the concrete which can lead to shrinkage cracking.
The second part of shrinkage is autogenous shrinkage when the concrete consumes its own water during the hardening process. Here the size of the autogenous shrinkage depends mainly on the water cement ratio (w/c). When the w/c ratio is lower, then the autogenous shrinkage is higher.
There are several aspects affecting the progress of the shrinkage as follows:
 Beginning and duration of curing
 Amount and position of reinforcement
 Dimension of the structure
 Relative humidity and temperature
 Cement type
 And others
Theoretical background
As has been mentioned above, the total shrinkage (ε_{cs(t,ts)}) strain is composed from two parts:
 Drying shrinkage (ε_{cd(t,ts)})
 Autogenous shrinkage (ε_{ca(t)})
ε_{cs(t,ts)} = ε_{cd(t,ts)} + ε_{ca(t)}
The calculation of both strains is described directly in EN199211[1].
SCIA Engineer [2] considers both parts of shrinkage effects automatically by default. (This can be disabled if needed in Concrete settings / Concrete member data). The value of shrinkage is calculated based on the predefined time of curing and time of the loading.
The deflections caused by shrinkage are calculated based on the strain and curvatures caused by the shrinkages for the uncracked and fully cracked crosssection. The whole procedure can be expressed in three steps:
Calculation of shrinkage forces
The forces caused by shrinkage are calculated according to the formulas below based on the shrinkage strain.
 N_{shr} = ε_{cs(t,ts)}·Coef_{Reinf}Σ(E_{si}·A_{si})
 M_{shr,y} = N_{shr}·e_{shr,z}
 M_{shr,z} = N_{shr}·e_{shr,y}
where

e_{shr,y} = Σ(E_{si}·A_{si})/ Σ(E_{si}·A_{si}·y_{si})  t_{iy}

e_{shr,z} = Σ(E_{si}·A_{si})/ Σ(E_{si}·A_{si}·z_{si})  t_{iz}

E_{si}  is modulus of elesticity of ith bar of reinforcement

A_{si}  is area of reinforcement of ith bar of reinforcement

y_{si} – position of ith bar of reinforcement from centre of gravity of crosssection in ydirection

z_{si} – position of ith bar of reinforcement from centre of gravity of crosssection in zdirection

t_{iy} – distance between centre of gravity transformed uncracked/cracked crosssection and centre of gravity of concrete crosssection in ydirection

t_{iz} – distance between centre of gravity transformed uncracked/cracked crosssection and centre of gravity of concrete crosssection in ydirection
Calculation of strain and curvature caused by shrinkage
Strain and curvature caused by shrinkage are calculated for each element and these values are calculated for both states (uncracked and cracked crosssection). Calculation of strain caused by shrinkage:
 ε_{sh} = ε_{cs(t,ts)}·Coef_{Reinf}·Σ(E_{si}·A_{si})/(E_{ceff}·A_{i})
Calculation curvature around y and zaxis caused by shrinkage
 (1/r_{y}) = ε_{cs(t,ts)}·Coef_{Reinf}·Σ(E_{si}·A_{si}·(t_{iz}z_{si}))/(E_{ceff}·I_{iy})
 (1/r_{z}) = ε_{cs(t,ts)}·Coef_{Reinf}·Σ(E_{si}·A_{si}·(t_{iy}y_{si}))/(E_{ceff}·I_{iz})
Calculation of stiffnesses for shrinkage
The stiffness of the uncracked / cracked crosssection for shrinkage is calculated from the strain and curvatures caused by shrinkage by using the total level of load (total load combination) and then used in the finite element calculation

bending stiffness around yaxis EI_{y} = M_{tot,y}/(1/r_{y})

bending stiffness around z axis EI_{z} = M_{tot,z}/(1/r_{z})

axial stiffness EA = N_{tot}/ε_{sh}
Note: As indicated above, as a simplification the total forces are used for the calculation of the stiffnesses for the FEM analysis instead of the forces caused by shrinkage.
Results presentation
The output of the deflection results can be seen graphically in the 3D window and also numerically via the Brief, Standard or Detailed output. The following picture shows an example of the Brief table which represents the following values:
 creep factor φ_{(t,t0)}
 shrinkage strain ε_{(t,ts)}
 deflection caused by creep (δ_{creep})
 deflection caused by shrinkage (δ_{shr})
Case study
The effect of shrinkage is illustrated on a mediumsized project. The structure has a slab thickness of 270 mm from concrete C30/37, which is loaded by a permanent load of 2,5 kN/m2 and a variable load of 3,0 kN/m2. The reinforcement in both surfaces and directions is f12/200 mm in the span and above supports f12/100 with B500B. The concrete cover is 30 mm. The time of concrete curing is 7 days, the time of load application is 28 days. The time of investigation is 50 years. The relative ambient humidity is 50%.
A comparison is made between the results of the deflections coming from a linear and those of a nonlinear analysis, including concrete cracking and time dependent effects as a creep and shrinkage.
 Nonlinear deflection with cracking
 Nonlinear deflection with cracking and creep
 Nonlinear deflection with cracking, creep and shrinkage
The previous graphical results can be summarized:

Linear deflection: maximal deformation of 4.2 mm

Nonlinear deflection with cracking: maximal deformation of 7.8 mm

Nonlinear deflection with cracking and creep: maximal deformation of 16.8 mm

Nonlinear deflection with cracking, creep and shrinkage: maximal deformation of 26.0 mm
Conclusions
As can be seen from the results, using merely a linear analysis is not sufficient for reinforced concrete structures. The effect of cracking gives almost 2x higher values compared to the linear deflection. Additionally, it is essential to consider also creep and shrinkage effects to obtain the real behavior of the structure. In case of creep only this leads to 4x higher results and when considering all effects together almost 7x. Neglecting such effects during design could lead to significant issues during the life cycle of the structure, especially a lower serviceability due to high deflections.
References
[1] EN199211  Eurocode 2: Design of concrete structures  Part 11: General rules and rules for buildings
[2] www.scia.net