Cracked Sections

The cracked sections are implemented in the model. The goal is to model non-linearity in a linear model, which can be problematic. The cracked sections are in stage II and will not have the same bending stiffness as the uncracked concrete in stage I. To preserve this effect, the sections must be modified.

As explained in the previous sections in this chapter, the non-linearity approach is executed differently in the two models. The solution in the solid model might be more true to the expected behavior of the beam section, where the crack creates an almost not-existing concrete stiffness in the tension zone.

83 Two situations are analyzed to understand the effect of the non-linearity:

- Free expansion, temperature field 1, and cracked sections - Restrained expansion, temperature 1 and cracked section

Free Expansion

Figure 7.24, 7.25 and figure 7.26 shows the axial force and moments obtained in the frame and solid model.

Figure 7.24: Axial forces in inner beam Free expansion with crack

Figure 7.25: Axial forces in outer beam Free expansion situation with cracks

84 CHAPTER 7. MODELS IN ABAQUS/CEA

Figure 7.26: Total moments Free expansion situation with cracks

The percentage difference in axial force in the inner beams between the frame model the solid model is shown in table 7.12.

Section S7 ZM 7-8 F7 ZM 7-8 S8 ZM 8-9 F8 ZM 9-8 S9 F9

Deviation 3.1 2.7 5.4 9.1 15 7.6 3.9 -3.1 -8.4 -12.7

Table 7.12: Deviations [%] in the frame model compared to the solid model Axial force in inner beam, temperature field 1

The deviations in the axial forces are within a range of approximately 10%, which can be expected due to the many differences. What is interesting is that the frame model now provides higher forces and moments in multiple sections compared to the solid model. The free expansion analysis without cracks has a majority of higher forces in the solid model. This implies that the axial forces are reduced in a higher degree in the solid model compared to the frame model.

Section S7 ZM 7-8 F7 ZM 7-8 S8 ZM 8-9 F8 ZM 9-8 S9 F9

Deviation -13.7 -0.4 21.5 45.5 41 50.5 26.4 -4.9 -1.8 -7.3

Table 7.13: Deviations [%] in the frame model compared to the solid model Total moment, temperature field 1

Table 7.13 presents the deviation in the total moment between the two models.

In areas close to the simulated crack, moment deviations between the two mod-els are increasing in a greater magnitude than the axial forces. This is a result of the differences in simulation in the two models and can be explained by ide-alizing the cross-section to a square cross-section. The axial stiffness is defined by EA = E·bh, whereas bending stiffness depends on EI = E· ^bh₁₂³. If one reduces the total stiffness by reducingE, which is done in the frame model, the reduced stiffness will affect both axial and bending stiffness the same way. On the contrary, if the height is reduced as done in the solid model, the bending

85 stiffness will be reduced by a third exponent compared to the axial stiffness.

Restrained Expansion

The same simulation of cracks is implemented in the analysis with reinforcement giving the resulting axial forces and moments as illustrated in figure 7.27, 7.28 and 7.29.

Figure 7.27: Axial forces in inner beams Restrained expansion with cracks

Figure 7.28: Axial forces in outer beams Restrained expansion with cracks

86 CHAPTER 7. MODELS IN ABAQUS/CEA

Figure 7.29: Total moments Restrained expansion with cracks

The percentage deviations between the axial force in the inner beam exposed to temperature field 1 are reported in table 7.14.

Section S7 ZM 7-8 F7 ZM 7-8 S8 ZM 8-9 F8 ZM 9-8 S9 F9

Deviation 3.5 9.6 2.2 12.9 12.3 4.7 0.3 -2.2 -6.6 -5.9

Table 7.14: Deviations [%] in the frame model compared to the solid model Axial force inner beam, temperature field 1

When reinforcement is included in the analysis, the deviations between the mod-els are in the same range as without reinforcement. These deviations are small but are increasing around the area of the crack.

Section S7 ZM 7-8 F7 ZM 7-8 S8 ZM 8-9 F8 ZM 9-8 S9 F9

Deviation -28.7 38.4 38.2 -5.9 6.6 2.2 29.7 -63.9 -26.4 51.8

Table 7.15: Deviations [%] in the frame model compared to the solid model Total moment, temperature field 1

When the analysis includes reinforcement, the deviation between the two mod-els is smaller compared to the ones reported in table 7.13, especially in the area close to the simulated crack. This is a result of two different types of deviation canceling each other out. The deviations in moment between the two models when reinforced and uncracked show that the frame model obtains about 20-40% lower forces compared to the solid model. When the analysis includes a cracked section, the frame model obtains about 40-50% higher forces.

As the cracks are modeled differently in the models, expected deviations occur as was seen in the free expansion situation. The solid model is modeled with almost zero stiffness in the tension zone, which in a section with small amounts of reinforcement reduces the capacity significantly. Earlier analyses show that

87 the reinforcement is already yielding without the simulation of the crack. When the section is modeled with no tension capacity, the moment is reduced con-siderably since the reinforcement is not able to incorporate additional tension forces. A rearrangement of moment occurs in a much higher degree compared to the frame model, since this model still has a concrete tension capacity. This leads to smaller deviations between the two models compared to the uncracked analysis. The assumption of a crack in the whole web of the beam is conserva-tive and might result in a larger rearrangement in forces compared to the reality.

The deviations in the solid model between temperature field 1 and 2, is corre-sponding with earlier deviations. This implies that the chosen temperature field does not affect the results from the simulation of crack. This also implies that the different temperature fields in the solid model and the frame model should not affect the deviations in the simulation of crack.