Proceedings of the International rilem conference Materials, Systems and Structures in Civil Engineering 2016

44

International RILEM Conference on Materials, Systems and Structures in Civil Engineering 

Conference segment on Service Life of Cement-Based Materials and Structures 

22-24 August 2016, Technical University of Denmark, Lyngby, Denmark 

 

 

(3)

In this equation,    is the concrete age and   the concrete age at loading.  ,    and    are 

model parameters that define the magnitude and the shape of the creep curves. The assumed 

values of  ,   and   are listed in table 3. 

 

Table 3: Parameters for DPL 

1

 

m n 

0.37 0.10 0.17 

 

 

3. Numerical 

simulations 

 

3.1  Finite element model and calculation procedure 

The temperature and stress evolution in the tunnel structure has been analysed using the finite 

element software TNO Diana. The calculations were carried out as staggered analyses, which 

means that the temperatures are calculated first and act as loading in the subsequent structural 

analysis. For the thermal analysis the measured hydration heat from fig. 2 was implemented 

as heat source. The heat exchange with the surrounding air has been modelled with 

convective boundary elements that take into account the formwork or other surface treatments 

through an adaption of the film coefficient. For the ambient temperature, the following 

scenario was assumed: The concrete casting is performed during mild winter weather (10°C). 

The fresh concrete is assumed to have a constant temperature of 20°C. After five days of 

curing, a sudden start of winter occurs and the ambient temperature decreases linearly to -7°C. 

This scenario can be treated as worst case scenario, because the cooling down of the air leads 

to an additional contraction of the concrete that increases the magnitude of restraint stresses. 

To take into account the phased construction, separate models were used to perform the 

temperature and stress calculations. In each model, the experimentally determined material 

properties of the early age concrete were implemented directly into the simulation. The 

material properties of the previously cast segments were assumed to be constant and equal to 

the values at 28 days, because the intervals between the construction phases are relatively 

long. 

The simulations do not take into account cracking even if the calculated stress exceeds the 

tensile strength. This simplification is accepted because a description of the strain-softening 

behaviour needs complex experiments and its implementation into numerical simulations is 

computationally expensive. Thus, the stress evolution calculated after the exceeding of the 

tensile strength is not realistic because cracking leads to a redistribution of the stresses. 

Nevertheless the calculated stress distributions can be used to identify zones with a high risk 

of cracking, which was shown in a comparative study in [16].  

Because of the symmetry, only a quarter of the total system has to be modelled for the finite 

element analyses. Fig. 5 shows the finite element mesh for the construction phases of the 

middle wall and the outer walls together with the ceiling slab. All construction parts are 

assumed to be monolithically connected. The nodes at the lower surface of the underwater 

concrete slab are assumed to be fixed in each direction. 

 

---

45

International RILEM Conference on Materials, Systems and Structures in Civil Engineering 

Conference segment on Service Life of Cement-Based Materials and Structures 

22-24 August 2016, Technical University of Denmark, Lyngby, Denmark 

 

 

Figure 5: Finite element mesh for the casting of the middle wall (left) and the casting of the 

outer walls and the ceiling slab (right) 

 

3.2 Temperature evolution 

Fig. 6 shows the temperature distribution for the two investigated construction phases at the 

time when the maximum temperature is reached. The maximum temperature in the middle 

wall occurs 2 days after casting and reaches a value of 44.2°C. In the outer walls, the 

maximum temperature is 49.0°C which occurs at an age of 2 days and 6 hours. The maximum 

overall temperature occurs in the ceiling slab directly above the middle wall, because this part 

of the construction has the largest thickness. A maximum value of 57°C after 2 days and 20 

hours was calculated for the ceiling slab. 

 

 

Figure 6: Temperature distribution in the middle wall (left) and the outer walls and ceiling 

slab (right) at the time when the maximum temperature is reached. 

 

3.3 Stress evolution 

The restraint of the thermal deformations by already hardened parts of the construction leads 

to the formation of compressive stresses in the warming phase and tensile stresses in the 

cooling phase. In addition, the temperature gradients between the core and the surface lead to 

---

46

International RILEM Conference on Materials, Systems and Structures in Civil Engineering 

Conference segment on Service Life of Cement-Based Materials and Structures 

22-24 August 2016, Technical University of Denmark, Lyngby, Denmark 

 

eigenstresses in the cross sections. The largest values of tensile stresses that may cause 

cracking are thus expected during or after the cooling down of the structure. 

Fig. 7 shows the distribution of stresses in the longitudinal direction in the middle wall at an 

age of 28 days. 

 

 

Figure 7: Stress distribution (longitudinal direction) in the middle wall 28 days after casting 

 

Due to the restraint by the already hardened slab, the highest tensile stresses occur in the 

symmetry plane near to the joint. The large contraction caused by the cooling down and the 

shrinkage of the concrete leads to the excess of the tensile strength in big parts of the wall. 

From the results of the stress calculation, it has to be assumed that vertical cracks from the 

joint between slab and wall up to more than half of the height of the wall will occur.  

A similar stress distribution can be observed in the outer walls, see fig. 8. The ceiling slab is 

restrained both in longitudinal and transverse direction. The highest tensile stresses in 

longitudinal direction occur directly above the walls, because these areas correspond to the 

areas with the maximum temperature and are at the same time highly restrained due to the 

monolithical connection with the walls. The part between the walls shows a lower tensile 

stress level that does not exceed the tensile strength. In the transverse direction, high tensile 

stresses that exceed the tensile strength occur nearly in the whole ceiling slab. These stresses 

are mainly caused by the rigid connections between the slab and the outer walls. 

 

 

Figure 8: Stress distribution in the outer walls and ceiling slab 28 days after casting 

 

The results of the stress simulations reveal that all parts of the tunnel construction are 

subjected to a high risk of cracking due to restraint. To limit the crack width and ensure the 

serviceability of the structure, the resulting stresses must be taken into account for the design 

---