Effect of slab dimensions on the maximum stress and ultimate vertical load of a 15-story flat slab building

The research of flat slab structures requires more investigation into the full-scale analysis of steel flat slab structures under vertical loads. The impact of slab dimensions on the maximum stress of slabs and columns, as well as the ultimate vertical load at the linear elastic stage, has been examined using full-scale finite element models. The particular structures selected for the analysis are 15-story steel flat slab structures with various slab dimensions under the static vertical load and standard earth gravity. This research also involves: (1) the verification of stress singularity at the corner of cubic columns in the finite element model; (2) the discussion about the variation of maximum stress of slabs and columns as the floor changes, and the interaction of columns and slabs; (3) the validation of the vulnerability of slab-column connection.


Introduction
The flat slab building offers a common solution for commercial, office, and residential use in many countries (see Muttoni et al. [1]).The flat slab reduces the floor-to-floor height, accelerates the construction process, and optimizes the arrangement of plumbing and electrical systems (see Al-Katib, Alkhudery and Al-Katib [2]).In China, flat-plate systems have emerged and developed since the 1960s, whereas extensive studies have been carried out in America since the 1900s (see Shao [3]).
Due to the geometrical complexity and high computational cost (see Gao and Guo [4]), most studies merely select small-scale specimens or sub-assemblies, such as the slab-column connection, as their objective (see Coronelli et al. [5]).Only a few researchers have been involved with full-scale modeling tests.The full-scale testing of a two-story flat slab building has been reported by Coronelli et al. [5].The test results show the response with deformations and damage under different loading conditions.Fick, Sozen and Kreger [6] tested a three-story, two-bay, and full-scale flat-plate building.One of the focuses of the experiment is to investigate the vulnerability of the slab-column connections.
Besides, most of the research investigates the materials of reinforced concrete (RC) and explores the structural behavior under lateral loads.An approach to evaluating the deformation capacity of the RC slab-column connection under seismic loads has been proposed by Muttoni et al. [1].Ramos et al. [7] have used HPFRC as a substitute for traditional concrete to reduce punching shear failure under seismic loads.Zhou and Hueste [8] suggested a nonlinear model for nonprestressed and prestressed slab-column connections under lateral load effects.The evaluation of how fiber-reinforced lightweight aggregate concrete can improve the hysteretic behavior of exterior slab-column connections can be found in Scotta and Giorgi [9].However, steel is still a popular material for structures in current design practice due to its excellent performance in strength and ductility (see Işık and Özdemir [10]).The minor failure percentage of steel structures (see Gloncu and Mazzolani [11]) compared with other construction materials may lead to relatively less research about them.
Given the discussion above, there are several topics in flat slab structures that entail further research--full-scale analysis, vertical loads, and materials of steel.Besides, an estimation of the structural behavior with different geometries of components is necessary for structural design.To this aim, we mainly focus on the separate effect of slab thickness and slab side length on the maximum stress and ultimate vertical load of a 15-story steel flat slab structure.The full-scale analysis of this structure is based on numerical FE models, a popular and suitable method for analyzing structures.We establish the 3-D finite element models of the structure with various slab dimensions using the general finite element package ANSYS (2018).To indicate and quantify the relationship between slab dimensions and ultimate vertical load, the linear regression model (least squares) via Python is employed.Also, a comparison of maximum stress is carried out in terms of the different crosssectional shapes of the columns.The linear elastic analysis is adopted.The relationship between the maximum stress of slabs and columns, and the story is also discussed in this paper.

Material properties
In the following analysis, the structure's material properties are defined for Q235 structural steel (235 MPa for yield strength) in Chinese code.In order to make the conclusions more interpretable and analysis more straightforward, we employ the idealized stress-strain relationship (see Trahair et al. [12]) for structural steel in which the strain hardening phase is not considered, as shown in figure 1.The yield strength is given as 235 MPa and the tangent modulus is set as 0 MPa.The value of 2.06×10 5 MPa for Young's modulus and 0.3 for Poisson's ratio are applied.The density of steel is 7.85×10 3 kg/m 3 .

The prototype structural model
The 15-story flat slab building with the structural layout displayed in figure 2 is chosen as the prototype model for this study.The span length in both x-and y-directions is 6000 mm with three bays in all directions.The floor-to-floor height is 3 m.The thickness of flat slabs is constant at 500 mm throughout the layout and floors.The depth and width of the columns' cross-section are set as 1 m.

Processing the model
The structural model is processed through Mechanical APDL.All the contact types for the slabcolumn connection are set as bonded.In order to save computational resources and obtain relatively accurate results simultaneously, the element size of the mesh is defined as 0.4 m.The bottom of the columns on the first floor should be fixed to the ground.Standard earth gravity is inserted.For simplification, the vertical load difference between the roof and the slabs under it is not considered in this investigation, and an identical vertical load is applied on the upper surface of each floor.

Comparison of the maximum stress in terms of different types of columns
We mainly consider two types of models, one of which is the prototype building model.In the other model, the cubic columns in the prototype model are replaced by cylindrical columns whose diameter is 1 m.In this test, the vertical pressure applied on each floor of both models is 5×10 4 Pa.By solving the models in Mechanical APDL, it can be seen that the maximum equivalent stress in both models occurs at the interior slab-column connection in the first story.Especially for cubic columns, the maximum equivalent stress occurs at the corner of the columns, as shown in figure 3. Based on the previous investigation about singularities (see Rössle [13]), the presence of angular corner points generally yields local singularities in the solution.Yamagiwa and Kurahashi [14] have proposed that in the stress analysis based on the FEM, the value of the stress component at the singular point will increase with decreasing mesh size.Meanwhile, in the theory of elasticity, singular solutions at the points of the boundary are associated with infinite stresses (see Korepanova, Matveenko and Sevodina [15]).To validate the above theories, we adjust the max refinement loops to 5 in adaptive mesh refinement and define the allowable change for convergence as 5%.However, the maximum stress in the prototype model still keeps a surging trend after five loops (six times of computation), as shown in figure 4, and during the last refinement, the change is 21.125%, as displayed in table 1.It indicates that stress singularity exists at the angular point in figure 3, and the actual stress value at that point can not be obtained through several times of mesh refinements.
Compared with the prototype model, the maximum equivalent stress in the model with cylindrical columns converges to 84.696 MPa after the second computation.There is a marginal difference of -4.786% in the computational value, as displayed in table 2. Therefore, the first computation, which is not through refinement, can represent the accurate maximum stress of the structure to a certain extent.In order to determine accurate maximum equivalent stress and save computation resources simultaneously, the cylindrical columns are adopted in the following analysis.

Methodology
It is widely agreed that nonlinear analysis can provide a comprehensive assessment of existing structures (see Araújo and Castro [16]).However, the survey conducted by Paret et al. [17] has revealed that linear elastic methods are used and will continue to be used due to their relative simplicity and familiarity with most design practitioners.Also, Toranzo-Dianderas [18] has noted the need for more research on linear analysis.Therefore, this investigation mainly focuses on the maximum stress and ultimate vertical load at the linear elastic stage of the stress-strain relationship.

Linear elastic analysis formula
Due to the linear relationship between the overall vertical loads and the maximum stress, the ultimate vertical load can be calculated through equation ( 1) where G is the weight of each floor.P and Pmax represent the vertical load applied on each floor and its ultimate value, respectively.The value of P should not exceed the calculated value of Pmax.In this paper, the value of P is defined as 5×10 4 Pa for each model in the following analysis to observe the behavior of different structural models under an identical vertical load.A is the area of the slab.s indicates the maximum stress value under the vertical load of 5×10 4 Pa, obtained via Mechanical APDL.fy is the yield strength.

Validation of the formula
The prototype model with cylindrical columns is employed as the validation model.The maximum stress is 88.849 MPa under the vertical pressure of 5×10 4 Pa and gravity.Through equation (1), the calculated value of Pmax is 210296.0532Pa.When we apply the Pmax and gravity load on each floor, the maximum stress is 239.8MPa.The discrepancy of 2% demonstrates the effectiveness of the formula in analyzing elastic ultimate vertical load.

Effect of slab side length on the maximum stress and ultimate vertical load
Three finite element models with a slab length of 18 m, 21 m, and 24 m, respectively, and a slab thickness of 0.5 m are established to observe the maximum stress of slabs and columns and the value of the ultimate vertical load.

The maximum stress of columns.
The maximum stress of the columns on each floor is shown in figure 5. Due to the fact that the lower columns will bear larger loads on the upper floors, the maximum stress generally rises as the number of stories decreases.The maximum stress occurs at the boundary of the slab-column connection.For the top floor in the models with the slab length of 18m and 24m, the corner columns experience the most stress, whereas, for the lower floors, it is the inner columns.The model with the 21-meter-long slab is an outlier; from the 15th to the 13th floors, the corner columns are under the most stress, while the edge columns are under the most stress on the 12th floor and the maximum stress occurs on the interior columns on the lower floors.It indicates that the columns bearing more vertical load on each floor shift from corner to interior columns.This shift may interpret the increase in the maximum stress of columns between the 14th floor and the top floor.It can also be observed from figure 5 that the stress of columns on each floor increases as the side length of slabs increases.

The maximum stress of slabs.
The maximum stress of the slab on each floor is shown in figure 6.The variation in stress through 15 floors can be divided into three phases.The first phase is the almost linear decrease with a higher slope as the story rises from the first floor.Then it follows by a lower slope which is the beginning of the second phase.The third phase is a sharper decrease from the 14th floor to the rooftop.In the third phase, the stress distribution of the 15th and 14th floors show entirely different patterns, as displayed in figure 8 and 9 (take the slab length of 21 m as an example), because there are no columns on the roof and the top-floor columns will directly affect the stress distribution of the slab on the 14th floor.The stress distribution patterns of the slabs on the 13th and 14th floors are also different due to the impact of the "irregular" columns on the 14th floor, as shown in figure 7 and  8.The discrepancy in stress distribution of these slabs has well explained the difference in their maximum stress.From the 13th floor, the stress distribution starts to eliminate the effect from the columns in the 14th story and aligns with the overall trend.The maximum stress in the larger slabs of each floor is greater compared to those with shorter lengths.

The ultimate vertical load.
The ultimate vertical load is closely associated with the maximum stress of the structure.The discussion above shows that the stress of columns is much higher than that of slabs within the same model.The maximum stress at the interior slab-column connection in the first story represents the maximum stress and indicates the most dangerous position of the whole structure.The computed ultimate vertical loads are displayed in table 3. The data are fitted using the linear regression model (least squares), as illustrated in figure 10.The linear fitting equation between the ultimate vertical load and the length of the slab is as follows: y=-20602.62x+584884.28 (2)

Effect of slab thickness on the maximum stress and ultimate vertical load
We establish three models with a slab thickness of 300 mm, 400 mm, and 500 mm, respectively, and a constant side length of 18 m to observe the maximum stress of slabs and columns as well as the value of the ultimate vertical load.

The maximum stress of columns.
The maximum stress of columns on each floor of the three models is shown in figure 12, which decreases as the slabs become thinner because the overall vertical load, which consists of the constant 0.05 MPa applied on each floor and standard earth gravity, reduces and the stress of slabs increases.It can be seen that the maximum stress of the columns is almost the same for the models with the slab thickness of 500 mm and 400 mm, which suggests that under the given vertical pressure, the maximum stress of most columns on the upper floors varies marginally and the slab thickness can be reduced to a certain extent to save the construction materials.

The maximum stress of slabs.
The maximum stress of slabs on each floor of all models is shown in figure 13.As the floor slab gets thinner, the maximum stress of the slab in each story increases.The second phase extends in the models with smaller slab thickness, which indicates that the range of floors affected by the columns on the 14th floor is broadened.The difference between the maximum stress of the slab on the 14th and 15th floors diminishes as the slab thickness decreases.It can be noticed that the maximum stress in the second and third phases tends to increase more than that in the first phase as the slabs get thinner.

The ultimate vertical load.
The ultimate vertical loads are displayed in table 3. The linear fitting result is shown in figure 11, and the ultimate vertical load could be defined by: y=49.34x+185860.40 (3)

Conclusions
Full-scale finite element analyses have been carried out to investigate the effect of slab dimensions on the maximum stress and ultimate vertical load of the 15-story steel flat slab building at the linear elastic stage.By using the prototype model, we tested the stress singularity in the corner of the columns.Six models with different geometries of slabs were established to study the maximum stress of slabs and columns, respectively, and the ultimate vertical load.The linear regression model was employed to fit the relationship between the ultimate vertical load and slab dimensions.
• The results of the maximum stress of columns and slabs confirmed the interaction of columns and slabs.• In the same model, the maximum stress of columns will decrease as the floor rises except for the stress of columns on the top floor.The shift of the components that bear larger vertical loads has been suggested as a potential explanation.The maximum stress of slabs can be divided into three phases as the story varies.• It can be concluded that the increase in slab length will lead to higher maximum stress of slabs and columns on each floor and lower ultimate vertical load; increasing slab thickness will result in lower maximum stress of slabs, higher maximum stress of columns, and higher ultimate vertical load.• More generally, the position of the maximum stress is consistent with research showing the vulnerability of the slab-column connection.The failure mode of the slab-column connection may constitute the object of future studies.• The results on the ultimate vertical load demonstrate a linear relationship between the ultimate vertical load and slab dimensions.

Figure 1 .
Figure 1.Stress-strain relationship for steel.Figure 2. Plan view of the building.

Figure 2 .
Figure 1.Stress-strain relationship for steel.Figure 2. Plan view of the building.

Figure 3 .
Figure 3.The maximum stress at the corner of the cubic column.

Figure 4 .
Figure 4. Maximum equivalent stress of the prototype model for six solutions.

Figure 5 .
Figure 5.The maximum stress of columns on each floor as slab length varies.

Figure 6 .
Figure 6.The maximum stress of slabs on each floor as slab length varies.

Figure 7 .
Figure 7.The stress distribution pattern of the slabs on the 13th floor.

Figure 8 .
Figure 8.The stress distribution pattern of the slabs on the 14th floor.

Figure 9 .
Figure 9.The stress distribution pattern of the slabs on the 15th floor.

Figure 10 .
Figure 10.Linear fitting of the relationship between slab side length and the ultimate vertical load.

Figure 11 .
Figure 11.Linear fitting of the relationship between slab thickness and the ultimate vertical load.

Figure 12 .
Figure 12.The maximum stress of columns on each floor as slab length varies.

Figure 13 .
Figure 13.The maximum stress of slabs on each floor as slab length varies.

Table 1 .
Change, nodes, and elements in mesh refinement of the prototype model.

Table 2 .
Maximum equivalent stress, change, nodes, and elements in mesh refinement of the model with cylindrical columns.

Table 3 .
The ultimate vertical load a .