Local slab behavior

By using the results of the jconc calculations, one can analyze the local slab behavior at the vicinity of the support area. Besides the in-plane stresses obtained by jconc, the moment was calculated by the integration of the horizontal stresses along several vertical sections. Additionally, the stresses in the flexural reinforcement and the strains in the shear reinforcement were analyzed. Figure 6.12 to Figure 6.14 show the analysis of the local slab behavior for load levels of 60% VR, 75% VR, and 90% VR.

With respect to the general behavior it can be noted that at the load level of 60% VR, the stresses

are generally low, only at the outermost fiber at the compression zone and at the anchorage zone of the first and second stud the stresses are close to the concrete strength. With further increase of the load, the stresses at these zones grow. Additionally, a clear path of the load transfer can be seen. A portion of the load is taken by the first stud and directly transferred to the column face. The load transfer of the second row of studs can be described as a compression strut that is deviated at the first row of studs. In other words, the load descends from the second row in a rather steep angle to the first row of studs. At this point, a portion of the load is transferred to the first row of studs. The rest of the load is transferred by a slightly inclined compression strut to the column face. This is in agreement with the strain measurements in the studs discussed in Chapter 3. It was presented that the sum of the forces calculated based on the strain measurements on the top part of first and second row of studs exceeds the total measured shear force. This indicates that a certain amount of the load in the second row of studs has to be transferred to the first row of studs.

At the load levels of 75% VR and 90% VR, the general behavior does not significantly change.

However, the plot of the relative stresses clearly shows the development of the failure zone at the compression strut between the first stud and the column. The maximum relative stresses are first reached at the intersection of the first stud and the flexural reinforcement before the failure zone propagates towards the column face. Additionally, large compressive stresses occur at the bottom side of the slab near the column face.

Figure 6.12: Local behavior at the load level of 60% VR: (a) shear transfer, (b) radial moment, (c) stresses in the flexural reinforcement, and (d) strains in the shear reinforcement

With respect to the stresses in the top reinforcement and to the moment calculated by the horizontal concrete stresses and the forces in the bottom reinforcement, it can be noted that the yielding strength of the flexural reinforcement is reached between load level 60% and 75%. However, more interesting to note is that yielding of the flexural reinforcement occurs only between the studs in the first row. Since the compression strut is rather steep, the moment decreases rapidly between the column face and the first stud. This effect is not considered in the NLFEA leading to a flatter moment curve. At the intersection point of the first stud and the flexural reinforcement, the stresses in the flexural reinforcement decrease. This effect can be explained by the compression strut that adds forces to the inner part of the reinforcement. Therefore, the stresses outside the first row of stud are reduced.

0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 r [m] z [m] σ s [MPa] m r [kN] ε s [‰] ε s [‰] ε s [‰] ε s [‰] 0 200 400 600 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 −400 −300 −200 −100 0 jconc (in-plane) NLFEA adjusted (bending) NLFEA non−adjusted (bending) (a)

(b)

(c)

Figure 6.13: Local behavior at the load level of 75% VR: (a) shear transfer, (b) radial moment, (c) stresses in the flexural reinforcement, and (d) strains in the shear reinforcement

Unlike the analysis of the flexural reinforcement, the calculated strains of the transverse reinforcement have to be regarded with reservations. Since the strains in the studs mainly depend on the shear cracks and the bond conditions, which both are not properly considered in the calculation, the results do not resemble the effective strain distribution accurately. Nevertheless, the calculated values show a qualitative behavior leading to certain general conclusions. It can be noted that the calculated average strains are much less than the yielding strains (εy≈2.5‰). Additionally, it can be seen that the first and the second row of studs exhibit

nearly the same strains. Thus, they transfer nearly the same amount of force. This corresponds to the strain measurements on the top end of the first and second row of studs (refer to Figure 3.20). 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 r [m] z [m] σ s [MPa] m r [kN] ε s [‰] ε s [‰] ε s [‰] ε s [‰] 0 200 400 600 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 −400 −300 −200 −100 0 jconc (in-plane) NLFEA adjusted (bending) NLFEA non−adjusted (bending) (a)

(b)

(c)

Figure 6.14: Local behavior at the load level of 90% VR: (a) shear transfer, (b) radial moment, (c) stresses in the flexural reinforcement, and (d) strains in the shear reinforcement

The herein discussed shear transfer at the column vicinity shows also a good agreement with the experimentally obtained cracking pattern. Figure 6.15 illustrates the cracking pattern observed at the saw-cut after failure overlaid by the results of the calculation at a load level of 90% VR.

Especially at the right-hand side, the cracks indicate a similar behavior as predicted by the numerical calculation. The calculated compression strut from the second row of studs seems to follow the cracks in this area. Similarly, the calculated concrete strut from the first stud to the column face corresponds to the experimentally observed cracking pattern. Moreover, the numerical analysis predicts basically two failure areas: one at the bottom surface close to the column and one close to the top surface at the intersection of the stud and the flexural reinforcement. Again, both failure zones can be seen at the cracking pattern. At the bottom surface, the concrete is spalling. However, it has to be noted that although the calculated radial strains are not small, the spalling seen in the test results from the large tangential strains near the column face, which are not modeled in jconc. At the intersection between the flexural reinforcement and the first stud, crushing of concrete occurs at a large area due to lateral tensile strains. This corresponds to the predicted behavior of the numerical calculation. Thus, the local model helps to understand qualitatively the behavior of the slab in the vicinity of the column.

0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 r [m] z [m] σ s [MPa] m r [kN] ε s [‰] ε s [‰] ε s [‰] ε s [‰] 0 200 400 600 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 −400 −300 −200 −100 0 jconc (in-plane) NLFEA adjusted (bending) NLFEA non−adjusted (bending) (a)

(b)

(c)

Figure 6.15: Calculated stresses at a load level of 90% VR overlaid by the experimentally observed cracking pattern

Since the global and the local model showed good agreement to the experimentally observed and measured behavior, the results help to understand the actual response of the slab. However, the calculations are based on rather coarse simplification and thus the application of this approach is limited to the investigation of tested slab specimens. For the prediction of the slab response, a more applicable approach is desired. In the following chapter, an analytical model will be presented with which the load-rotation response of slabs can be easily predicted. Additionally, formulations for failure criteria enable the estimation of the punching strength and rotation capacity.