Introduction

Often, structural engineers have to deal with the long-term behaviour of concrete structures (modelled either as plate or volume elements). In this context, the concrete mechanical behaviour is characterized by a short-term and a long-term stiffness namely E_short and E_long.

Some users believe that starting the calculation with an elastic Young’s modulus E_short and changing it to E_long once the long-term behaviour of the structural elements is sufficient. It is not! This is due to the fact that the stress relaxation is an irreversible process and should be modelled using a constitutive law accounting for energy dissipation which an elastic model does not have.

This example underneath describes a 5 m deep and 5 m wide excavation considering first a linear elastic material with short-term stiffness E_short = 30 GPa as shown in Figure 1(a), followed by a replacement with another linear elastic material using long-term stiffness E_long = 20 GPa. It can be observed that displacements are identical.

(1a) Short-term stiffness

(1b) Long-term stiffness
Figure 1: Deformed mesh

The same conclusion can be drawn when using plate elements. First, a plate element with short-term properties EA = 15E6 kN/m and EI = 312.5E3 kN.m2/m (so E = 30 GPa and thickness t = 0.5 m) is being used as shown in Figure 2(a). Then the analysis is considering the replacement of the plate property using long-term stiffness with EA = 10E6 kN/m and EI = 208.3E3 kN.m2/m (so E = 20 GPa and t = 0.5 m). Finally, it can be observed that bending moments are identical still.
When comparing the bending moments as shown in Figure 2, no stress relaxation is being observed after changing elastic stiffness properties.

(2a) Short-term stiffness

(2b) Long-term stiffness
Figure 2: Bending moments

Workaround

One possible workaround consists in modelling the structures with short-term stiffness as a combination of a solid element and a plate element, the latest one (so the plate element) being deactivated when the long-term stiffness only should be considered.

In this framework, the solid element will be given the long-term stiffness as a Young’s modulus and the plate's property must be defined such that it represents the difference between the short-term and the long-term stiffness (see figure 3).

Here the solid elements are given an elastic Young’s modulus E = 20 GPa (corresponding to long-term stiffness) and the plate with EA = 5E6 kN/m and EI = 104.1E3 kN.m2/m (so E = E_short - E_long = 10 GPa and t = 0.5 m) is being used.

Figure 3: Modelling strategy solid-plate compound

New results in terms of Deformed mesh are presented in Figure 4.
Now it can be observed that:

1. Deformed mesh after excavation with consideration of both solid and plate elements provides an identical level of deformation compared to figure 1(a) when only using solid elements in combination with short-term stiffness;
2. After deactivation of the plate, signification change of deformation is now being observed compared to figure 1(b) indicating that stress redistribution is happening due to the removal of the plate.

(4a) Short-term stiffness

(4b) Long-term stiffness
Figure 4: Deformed mesh with new modelling strategy

Conclusion

With this specific modelling technique of combining solid and plate elements, stress relaxation associated with creep mechanisms in concrete can effectively be taken into account during plate deactivation.