Calculation of concrete shrinkage effects

Analysis of concrete shrinkage, the induced self‑stresses, the differences with thermal effects, and the conditions for applying EC2 formula (7.21).

This article examines the mechanical behaviour of reinforced concrete subjected to shrinkage, highlighting the fundamental differences between shrinkage and thermal effects, and introducing the notion of self‑stresses that develop within the section.
It then analyses how the constitutive laws of concrete and steel are modified and how the mechanical diagrams of a reinforced‑concrete section (geometry, strains, stresses, internal forces) evolve under shrinkage.
Finally, the article clarifies the conditions under which Eurocode 2 formula (7.21)—used to estimate the curvature of a flexural member due to shrinkage—can be validly applied.
This contribution forms the third part of the series “Axial behaviour of flexural reinforced‑concrete elements” (3/4). 

 

 

Back to the previous article: Axial analysis, concrete shrinkage and thermal expansion in flexural structures – thermal expansion and thermal gradient (2/4)

 

 

The concrete shrinkage phenomenon

 

Concrete is a mixture of water, aggregates and cement. Cement, in the presence of water, undergoes a chemical exothermic reaction leading to its setting (a few hours) and then hardening (a few days), thereby forming a continuous and monolithic matrix linking the aggregates.

The chemical reaction occurs with a volume reduction, a phenomenon called autogenous shrinkage.

For practical mixing purposes, water is added in excess compared with the stoichiometric needs of the chemical reaction. This excess water must eventually leave the material over time, causing an additional volume reduction called desiccation shrinkage or drying shrinkage (spanning months or years).

Beyond ambient temperature and humidity conditions, the kinetics of shrinkage depend on the path length of water inside the element before evaporating at the surface. It therefore depends on the geometry of the element via its “mean radius”.

 

Shrinkage effects and thermal effects: completely different mechanics

 

In mechanical modelling terms, shrinkage is handled differently from thermal effects because it applies exclusively to concrete and not to steel. Practically, this means that shrinkage effects cannot rigorously be modelled by a simple temperature drop.

 

Independently of any stress related to bending moment and axial force, concrete shrinkage produces compressive stresses in the reinforcement, which resists shortening. Because of this resistance, the concrete ends up in tension. These self‑stresses reduce the “visible” structural effects induced by concrete shrinkage at the scale of global analysis.

Conversely, a concrete element that is freely axially dilatable or isostatic in bending is not necessarily free from “visible” structural deformations under shrinkage.

 

Considering concrete shrinkage at the local level

 

Modification of material constitutive laws

 

The effect of shrinkage is treated locally through constitutive laws. The principle is similar to that of thermal effects (see previous article), with two major differences:

• The steel constitutive law is unaffected.
• The concrete constitutive law shifts only to the right. Shrinkage necessarily induces “shortening.” We write σ = f(ε − ε_c), with ε_c > 0.

 

Generation of self‑stresses

 

Below is a bi‑symmetric, fully compressed section, lightly reinforced, freely axially dilatable, subjected to a shrinkage strain of 3×10^‑4. We observe that the concrete does not shorten by 3×10^‑4, but only by 2.6×10^‑4, because part of the shrinkage is restrained by the reinforcement, whose compressive contribution increases, while concrete compression decreases.

 

 

The curling phenomenon under a shrinkage gradient

 

Consider the same concrete element, now permanently immersed on its upper surface and exposed to air on the underside, while remaining freely axially dilatable and isostatic in bending. The concrete cannot undergo shrinkage on the immersed surface. Assuming a linear shrinkage distribution, we obtain a curling phenomenon, the element bending toward the side where shrinkage is prevented.

 

 

Note: The small moment of 1 kNm is a second‑order effect resulting from the slender column model used for this example. For an isostatic element in bending, a shrinkage gradient does not produce a first‑order bending moment.

 

Evaluating curvature due to shrinkage

 

Simplifying the study of shrinkage effect

 

Let us return to the assumption of uniform shrinkage across the height of a section: ε_c(h) = ε_c = 3×10^‑4.

In what follows, we assume that structural analysis has provided the bending moment M and axial force N acting on the reinforced concrete section, and that {M, N} is not affected by shrinkage‑induced curvature or elongation. In other words, the member is isostatic in bending, freely axially dilatable, and free from second‑order effects: assumption {H1}.

If the steel remains elastic after shrinkage (assumption {H2}), we can isolate shrinkage and return to the usual RC‑section study by performing a change of reference frame.

Consider a simply supported beam subjected to N = 400 kN (compression), M = 5 kNm at midspan, and shrinkage. Let us examine the midspan section:

 

 

We define ε^fictitious = ε^real − ε_c.

In this fictitious frame:

• For concrete: σ = f(ε^fictitious) as if no shrinkage existed
• For steel: σ = E_s(ε^fictitious + ε_c) = E_sε^fictitious + E_sε_c

Thus, it is as if the steel had been prestressed (in compression) before casting, by an amount E_sε_c.

The corresponding effects are:

N_cs = Σ A_si E_s ε_c > 0

M_cs = Σ(h/2 − y_i) A_si E_s ε_c

Let A_s,tot = Σ A_si, S_s,tot = Σ(h/2 − y_i) A_si, α_e = E_s/E_b.

We obtain:

N_cs = A_s,tot E_s ε_c

M_cs = S_s,tot α_e E_c ε_c

The study of the RC section subjected to {M, N} + shrinkage becomes equivalent to studying the section under the fictitious loading {M − M_cs, N − N_cs} without shrinkage.

We note that shrinkage:

• reduces concrete compression (−N_cs < 0)
• increases bending in concrete (−M_cs > 0)

After solving the fictitious section, we return to real values using:

Concrete:

• σ_c^real(h) = σ_c^fictitious(h)
• ε^real(h) = ε^fictitious(h) + ε_c

Steel:

• σ_s,i^real = σ_s,i^fictitious + A_s,i α_e E_c ε_c
• ε_s,i^real = ε_s,i^fictitious + ε_c

We must also check ε_s,i^real < f_yk/E_s to validate {H2}.

 

Numerical example

 

Consider a section of a simply supported beam under N = 0, M = 58 kNm.

We apply shrinkage: ε_c = 3×10^‑4.

 

 

Now place ourselves in the “fictitious” case:

• N_real = 0 → N_fictitious = −37.8 kN
• M_real = 58 kNm → M_fictitious = 63.7 kNm
• ε_c,real = 3×10^‑4 → ε_c,fictitious = 0

Results match exactly (up to rounding):

 

 

In the above example, additional curvature due to shrinkage is 9.6×10^‑4 m^‑1.

 

The Eurocode 2 formula for shrinkage‑induced curvature

 

Shifting the static moment to the centroid

 

Consider the following bending‑compression case.

Section 20×40 cm, reinforcement 2HA20 at d = 35 cm, E_cm = 28 GPa (thus α_e = 7.1), subjected to N = 500 kN and M = 63 kNm before shrinkage.

As seen earlier, shrinkage is represented by the complementary force set −{N_cs, M_cs}. We separate the two load sets and place −{N_cs, M_cs} at the cracked‑section centroid “CG”.

The centroid is located at y_CG = 0.15 m.

For shrinkage effects, S_s,tot becomes:

S_s,tot = Σ(y_CG − y_i)A_si

 

Consider the effect of −N_cs alone applied at the centroid. Two common cases arise:

 

 

Placing the load at the centroid minimizes the curvature influence of N_cs. Therefore, −M_cs is the only component significantly affecting curvature.

Since curvature = M / (E_c I), shrinkage curvature is:

(1/r)_cs = S_s,tot α_e / I · ε_c

This is EC2 formula (7.21).

Applying it yields (1/r)_cs = 9.4×10^‑4 m^‑1, matching the earlier result.

 

What if −N_cs or −M_cs causes cracking?

 

If shrinkage loads cause the section to crack, EC2 formula (7.21) no longer applies.

Cracking causes curvature to increase far beyond the formula’s estimation.

EC2 recommends fictitiously considering:

• fully uncracked beam → (1/r)_cs,I
• fully cracked beam → (1/r)_cs,II

Then:

(1/r)_cs = ζ(1/r)_cs,II + (1−ζ)(1/r)_cs,I

Thus shrinkage‑induced curvature is underestimated in zones where shrinkage causes cracking.

 

Conditions for applying the formula

 

The conditions for applying EC2 formula (7.21) are:

• Uniform shrinkage across the section height
• Global loading {M, N} must remain unaffected by shrinkage curvature (assumption {H1}):
  • formula cannot be used for continuous beams
  • cannot be used for second‑order effects

• Steel must remain elastic (assumption {H2}), therefore formula cannot be used at ULS.

• Shrinkage effects −{N_cs, M_cs} must not change the cracking state established under {M, N}.

Thus, EC2 formula is only strictly valid if shrinkage does not increase cracking along the beam.

Let us revisit assumption {H1}:

Consider a 20×40 cm beam with reinforcement 2HA20 top, 4HA20 bottom, under shrinkage 3×10^‑4.

Case 1: simply supported → isostatic → formula valid → constant curvature → downward bending.

Case 2: add a middle support → hyperstatic → compatibility imposes M = 7 kNm at support → curvature is not constant → formula invalid.

 

 

In the next article, we examine situations where shrinkage must be considered, the cases of concomitance with thermal effects, and the influence of cracking:Axial analysis, concrete shrinkage and thermal expansion in flexural structures - Concomitance and impact of cracking (4/4)