Online Manuals RFEM 6 | Time-Dependent Analysis (TDA)

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Dr. rer. nat. Marc Gebhardt

2026-06-01

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Overview

Time-Dependent Analysis (TDA)

Theoretical Principles

Input

Results

Calculation

General

The time-dependent analysis of the Long-term effects is performed using the time-step method. For this, the time period to be analyzed is incrementally decomposed into time points, between which a linearized calculation of the stress and deformation state is carried out. The time points to be calculated can be distributed linearly or logarithmically. However, due to the general nature of time-dependent behavior (decaying exponential function), a logarithmic distribution is recommendable. Its definition is discussed in more detail in Chapter Input. The approach is performed according to the definition in Input of the time-dependent parameters. The shrinkage strain is applied as loading according to the time-vector values and, depending on the system, generates strains and/or constraint stresses. For aging, an increase or decrease of the material parameters (modulus of elasticity and strength) is applied according to the time-vector definition per time increment. A somewhat more complex approach is required for creep and relaxation, which is why the following point discusses this in more detail. In simplified terms, a material law with memory is applied based on the input.

Viscoelastic Behavior

Within the framework of the finite element analysis for creep and relaxation, the models described in Chapter Rheological Models are time-discretized from the differential equation form. For this purpose, the time-dependent kernel functions are numerically implemented in each time step via internal variables, stress, strain, and retardation time. This results in a sum of exponentially decaying functions per time step. Thus, in each step, the strain of the FE element is determined from the displacement field. Subsequently, the internal variables are updated, and the stresses are determined from the material law. If geometric or material nonlinearities exist, the nonlinear equation system must be solved iteratively.

Creep of Concrete Using the Creep Coefficient According to Eurocode

According to EN 1992-1-1, Section 3.1.4, the creep coefficient φ(t,t₀) is a dimensionless quantity that describes the ratio of creep to instantaneous strain:

φ (t, t_{0}) = \frac{ε_{c r}}{ε_{e l}} = \frac{ε_{c} (t) - ε_{c 0}}{ε_{c 0}}

φ(t,t0)	Creep coefficient
εcr	Creep strain
εel	Elastic strain
εc(t)	Strain of Concrete over Time
εc0	Instantaneous strain of concrete

Creep Coefficient

The total strain under constant stress can be calculated using the following equation:

ε_{c} (t) = ε_{c 0} \cdot (1 + φ (t, t_{0})) = \frac{σ_{0}}{E_{c m}} \cdot (1 + φ (t, t_{0}))

εc(t)	Strain of Concrete over Time
εc0	Instantaneous strain of the concrete
φ(t,t0)	Creep coefficient
σ0	Stress (constant)
Ecm	Mean modulus of elasticity of concrete

Total Strain (Creep Coefficient)

By comparing the creep coefficient formulation and the Kelvin chain (without free damper), the total creep function (creep compliance) can be represented as a function of the creep coefficient:

J (t) = \frac{ε_{c} (t)}{σ_{0}} = \frac{(1 + φ (t, t_{0}))}{E_{c m}}

εc(t)	Strain of Concrete over Time
σ0	Stress (constant)
φ(t,t0)	Creep coefficient
Ecm	Mean modulus of elasticity of the concrete

Creep Compliance in Dependence of Creep Coefficient

By adapting the creep coefficient as a Prony or Dirichlet series, a minimization of this compared to the analytical creep coefficients can be performed:

min_{α_{k}} \sum_{i = 1}^{m} {(φ (t_{i}) - φ_{∞} \sum_{k = 1}^{n} α_{k} (1 - e^{- t_{i} / τ_{k}}))}^{2}

αk	Weight factor (boundary conditions: greater than 0 and their sum must be 1)
i	Time Increments
m	Number of Time Increments
φ(ti)	Creep coefficient at time i
φ∞	Creep coefficient (infinite)
k	Current Element
n	Number of elements
e	Euler's number
ti	Time at time i
τk=ηk/Ek	Retardation time of the current element

Adjustment Kriechzahl Prony Series

In comparison to the Kelvin chain, the spring stiffnesses of the elements can now be derived from the adapted weights, and the damper constants can be derived from the retardation times:

E_{k} = \frac{E_{c m}}{φ_{∞} \cdot α_{k}}

η_{k} = E_{k} \cdot τ_{k} = \frac{E_{c m} \cdot τ_{k}}{φ_{∞} \cdot α_{k}}

Spring Stiffness and Damping Constant From Adjusted Weights

For a simplified creep approach, the creep of concrete can be applied as a reduction of the mean to the effective modulus of elasticity through a time-independent calculation (no time-step method). It should be noted that this simplification is only valid for proportional, monotonically increasing loading and is not equivalent to the Kelvin chain. It underestimates the creep deformation under variable load and does not account for relaxation.

E_{c, eff} = \frac{E_{c m}}{1 + φ (∞, t_{0})}

Ec,eff	Effective modulus of elasticity of concrete
Ecm	Mean modulus of elasticity of concrete
φ(∞,t0)	Creep coefficient (infinity, related to start time)

Effective Modulus of Elasticity

Info

This variant is implemented in the Concrete Design add-on with the analysis type Creep and Shrinkage (linear).

References

Bazant, Z.P. & Wu, S.T. (1974) "Rate-type creep law of aging concrete based on Maxwell chain." RILEM Materials and Structures, 7(37), S. 45–60.
Bazant, Z.P. (1988) "Mathematical Modeling of Creep and Shrinkage of Concrete." John Wiley & Sons, Chichester.
Bazant, Z.P. & Prasannan, S. (1989) "Solidification Theory for Concrete Creep." Journal of Engineering Mechanics, ASCE, 115(8), S. 1691–1725.
Bazant, Z.P. & Cedolin, L. (1991) "Stability of Structures." Oxford University Press.
Carol, I. & Bazant, Z.P. (1993) "Viscoelasticity with aging caused by solidification of nonaging constituent." Journal of Engineering Mechanics, ASCE, 119(11), S. 2252–2269.
European Committee for Standardization (CEN). (2010). Eurocode 2: Design of Concrete Structures – Part 1‑1: General Rules and Rules for Buildings, EN 1992‑1‑1:2004 + AC:2010.
fib bulletin 65 Model Code 2010 (2012). International Federation for Structural Concrete (fib), Lausanne, Switzerland.

Parent Chapter

Theoretical Principles