Calculation

General

The time-dependent analysis of long-term effects is performed using the time step method. For this purpose, the time period to be analyzed is divided into time steps, between which a linearized calculation of the stress and strain states is performed.
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 recommended. Its definition is discussed in more detail in the chapter Input.
The initial conditions are set according to the definition in the input for the time-dependent parameters. The shrinkage strain is applied as a load based on the time-vector values and, depending on the structural system, generates strains and/or constraint stresses. For aging, an increase or decrease in the material properties (modulus of elasticity and strength) is applied per time increment according to the time-vector definition. Creep and relaxation require a slightly more complex approach, which is why this is discussed in more detail in the following section. In simple terms, a material law with memory is applied here based on the input.

Viscoelastic Behavior

In the context of finite element analysis for creep and relaxation, the models described in the chapter Rheological Models are discretized in time from their differential equation form. To do this, the time-dependent kernel functions are numerically implemented at each time step using internal variables: stress, strain, and retardation time. This results in a sum of exponentially decaying functions for each time step. Thus, at each step, the strain of the FE element is determined from the displacement field. The internal variables are then updated, and the stresses are determined from the material laws. If geometric or material nonlinearities are present, it is necessary to solve the nonlinear system of equations iteratively.

Creep of Concrete Using Creep Coefficient According to Eurocode

According to EN 1992-1-1, Section 3.1.4, the creep coefficient φ(t, t < sub > 0 < /sub >) is a dimensionless quantity that describes the ratio of creep strain to initial strain:

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

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

By adjusting the creep coefficient as a Prony or Dirichlet series, it can be minimized compared to the analytical creep coefficients:

Compared to the Kelvin chain, the spring stiffnesses of the elements can now be derived from the adapted weights, and the damping constants can in turn be derived from the retardation times:

For a simplified creep approach, a time-independent calculation (not a time step method) can be used to model concrete creep as a reduction in the effective modulus of elasticity relative to the average modulus of elasticity. It should be noted that this simplification only applies to proportional, monotonically increasing loads and is not equivalent to the Kelvin chain. It underestimates creep deformation under varying loads and does not incorporate relaxation.