Creep and shrinkage theory

Mathematical background on the age-adjusted effective modulus method (AEMM) for computing time-dependent stress redistribution in concrete sections.

Introduction

Concrete is a viscoelastic material: under sustained stress it deforms progressively (creep), and it contracts as internal moisture evaporates (shrinkage). These time-dependent behaviours redistribute stresses between concrete and reinforcement, increase long-term deflections, and contribute to prestress losses.

ACS uses the Age-Adjusted Effective Modulus Method (AEMM) to compute these effects. AEMM is the standard engineering method for time-dependent analysis of reinforced and prestressed concrete sections, recommended by all three major code families.

Creep

Creep coefficient

The creep coefficient $\varphi(t, t_0)$ defines the ratio of creep strain to elastic strain at time $t$ for concrete loaded at age $t_0$ :

\varepsilon_{cc}(t, t_0) = \varphi(t, t_0) \cdot \frac{\sigma_c}{E_{c,28}}

Where:

$\varepsilon_{cc}$ = creep strain
$\sigma_c$ = sustained concrete stress
$E_{c,28}$ = elastic modulus at 28 days

Final creep coefficients typically range from 1.5 to 3.0, depending on:

Factor	Effect on creep	Why
Age at loading $t_0$	Earlier loading $\rightarrow$ higher creep	Younger concrete has less developed microstructure
Relative humidity (RH)	Lower RH $\rightarrow$ higher creep	Moisture loss accelerates creep
Notional size $h_0$	Thinner members $\rightarrow$ higher creep	Greater surface-to-volume ratio increases drying
Concrete strength $f'_c$	Higher strength $\rightarrow$ lower creep	Denser matrix resists deformation
Cement type	Rapid hardening $\rightarrow$ lower creep	Faster strength gain means more mature concrete at loading

Code models

Each design code provides its own creep prediction model:

EN 1992-1-1 (Annex B):

\varphi(t, t_0) = \varphi_0 \cdot \beta_c(t, t_0)

Where:

$\varphi_0$ = notional creep coefficient (depends on RH, $h_0$ , $f_{cm}$ , cement type)
$\beta_c(t, t_0)$ = time development function

The notional creep coefficient:

\varphi_0 = \varphi_{RH} \cdot \beta(f_{cm}) \cdot \beta(t_0)

AS 3600:2018 (Cl. 3.1.8):

\varphi_{cc} = k_2 \cdot k_3 \cdot k_4 \cdot k_5 \cdot \varphi_{cc.b}

Where $k_2$ accounts for duration, $k_3$ for age at loading, $k_4$ for environment, $k_5$ for member size, and $\varphi_{cc.b}$ is the basic creep coefficient.

ACI 209R-92:

\varphi(t, t_0) = \frac{(t - t_0)^{0.6}}{10 + (t - t_0)^{0.6}} \cdot \varphi_u

Where $\varphi_u$ is the ultimate creep coefficient, adjusted for loading age, humidity, volume-to-surface ratio, and other factors.

Shrinkage

Shrinkage strain

Shrinkage strain $\varepsilon_{cs}$ is the free contraction of concrete due to moisture loss. It is independent of applied stress.

EN 1992-1-1 separates shrinkage into two components:

\varepsilon_{cs} = \varepsilon_{cd} + \varepsilon_{ca}

Where:

$\varepsilon_{cd}$ = drying shrinkage (depends on RH, $h_0$ , cement type)
$\varepsilon_{ca}$ = autogenous shrinkage (depends on concrete strength, independent of environment)

Total shrinkage strains at 50 years typically range from 300 to 600 microstrain ( $\mu\varepsilon$ ), depending on the environment and member geometry.

Factors affecting shrinkage

Factor	Effect on shrinkage	Why
Relative humidity	Lower RH $\rightarrow$ higher shrinkage	Greater moisture gradient drives more drying
Notional size $h_0$	Thinner members $\rightarrow$ faster shrinkage	Moisture escapes more quickly
Cement type	Rapid hardening $\rightarrow$ higher autogenous shrinkage	Finer pore structure
Concrete strength	Higher $f'_c$ $\rightarrow$ higher autogenous, lower drying	Denser matrix traps more moisture internally

Age-adjusted effective modulus method (AEMM)

The effective modulus concept

The simplest approach to creep is the Effective Modulus Method (EMM), which replaces the elastic modulus with a reduced modulus:

\bar{E}_c = \frac{E_c}{1 + \varphi(t, t_0)}

This is exact only when the stress has been constant since first loading. In practice, stress changes continuously due to creep redistribution, so EMM overestimates creep effects.

Age adjustment

AEMM corrects this by introducing an aging coefficient $\chi$ that accounts for the fact that load increments applied at later ages experience less creep than the original loading:

\bar{E}_{c,adj} = \frac{E_c}{1 + \chi \cdot \varphi(t, t_0)}

Where:

$\chi$ = aging coefficient (typically 0.6—0.9)
For most practical cases, $\chi \approx 0.8$ is a good approximation
The exact value depends on the loading history and concrete maturity

The age-adjusted modulus $\bar{E}_{c,adj}$ is used in place of $E_c$ when computing the response to stress increments that develop gradually (creep redistribution, relaxation).

Analysis procedure

ACS follows this procedure for each time step:

Compute creep and shrinkage — calculate $\varphi(t, t_0)$ and $\varepsilon_{cs}(t)$ using the code model for the given environment and section
Compute restraining forces — shrinkage and creep cause strains that are restrained by the reinforcement, generating internal forces:

\Delta N_{cs} = -E_c \cdot \varepsilon_{cs} \cdot A_c

\Delta N_{cc} = -\varphi \cdot \sigma_{c,0} \cdot A_c

Solve for stress increment — using the age-adjusted modulus, compute the stress change required to maintain equilibrium between concrete and reinforcement:

\Delta\sigma_c = \frac{\Delta N_{cs} + \Delta N_{cc}}{A_c + n_{adj} \cdot A_s}

Where $n_{adj} = E_s / \bar{E}_{c,adj}$ is the age-adjusted modular ratio.

Update stresses — the final concrete and steel stresses at time $t$ are:

\sigma_c(t) = \sigma_{c,0} + \Delta\sigma_c

\sigma_s(t) = \sigma_{s,0} - \frac{A_c}{A_s} \cdot \Delta\sigma_c

Where the sign convention ensures that as concrete stress decreases (relaxes), steel stress increases to maintain equilibrium.

Compute curvature — the section curvature at time $t$ is:

\kappa(t) = \frac{M}{(\bar{E}_{c,adj} \cdot I_c) + (E_s \cdot I_s)}

Physical interpretation

The AEMM analysis captures a physically intuitive behaviour:

Creep relaxation: Under sustained load, concrete deforms and sheds stress. The reinforcement, which does not creep, picks up the shed load. Over time, the concrete stress reduces and the steel stress increases.
Shrinkage restraint: Free shrinkage would shorten the entire section uniformly, but the reinforcement restrains this contraction. The result is compressive stress in the steel and tensile stress in the concrete — which can cause cracking in lightly reinforced members.
Combined effect: The net stress change is the superposition of creep relaxation and shrinkage restraint. In a typical reinforced concrete beam, the bottom steel stress increases by 20—40% over the first few years.

Limitations

AEMM assumes a single loading event at age $t_0$ . Multiple loading events at different ages are not currently supported (each would require superposition of creep effects).
The aging coefficient $\chi = 0.8$ is a common approximation. For more precise analysis, $\chi$ can be computed from the relaxation function, but the improvement is typically marginal for design purposes.
The analysis assumes uncracked behaviour. If the section is cracked under service loads, the effective section properties should account for the cracked state.

Creep and shrinkage theory

Introduction

Creep

Creep coefficient

Code models

Shrinkage

Shrinkage strain

Factors affecting shrinkage

Age-adjusted effective modulus method (AEMM)

The effective modulus concept

Age adjustment

Analysis procedure

Physical interpretation

Limitations

Further reading

Related pages