Numerical analysis of progressive collapse of flat slab systems

Researches by Keyvani, Sasani and Mirzaei (2014), Liu (2014) and Olmati et al. (2017) adopted numerical procedures of assessment of robustness of multi-story flat slab structures. Keyvani, Sasani and Mirzaei (2014) adopted a flat slab system with three bays in each orthagonal direction and designed to the requirements of the ACI 318 (ACI, 2011a). Liu (2014) adopted a four-story prototype structure designed for the purpose of the study and in conformance to the ACI 318 (ACI, 1971) guidelines. Olmati et al. (2017) adopted a Eurocode 2 (CEN, 2014b) design solution of an existing four-story office building provided in Technical Report 64 of The Concrete Society (2007). All slab cases were modelled using two-dimensional layered shell elements.

Keyvani, Sasani and Mirzaei (2014)

A numerical assessment of progressive collapse of a flat slab floor system was carried out by Keyvani, Sasani and Mirzaei (2014). The flat slab system was designed to satisfy the requirements of the ACI 318 (ACI, 2011a). The flat slab bay consisted of three rectangular interior panels (8 m by 6 m as shown in Figure 2.5), and a cantilever panel of length 1m at each end. The slab was 0.28 m thick and was supported on square columns, 0.3 m wide. Design of the

flat slab for the ultimate limit state was based on a design load of 13.1 kNm-2. A reduction factor of 0.59 was applied to the live load for design of the internal columns (Keyvani et al., 2014). Percentages of flexural reinforcement provided over internal columns were 0.51% and 0.3% in the longitudinal and transverse directions respectively. No punching shear reinforcement was provided. Finally, in each orthogonal direction, two integrity reinforcement bars with diameters 22 mm, were designed in compliance to the ACI 352.1R (ACI, 2011b). Sudden removal of the internal column B3 (Figure 2.5) was carried out with a slab load combination of 𝑔_𝑘+ 0.25𝑞_𝑘 (GSA, 2016) which gave a total gravity load of 9.2 kNm-2 _{(Keyvani et al., 2014).}

Figure 2.5: Slab geometry (Keyvani et al., 2014)

Though modelling of this flat slab floor system was implemented on the finite element solver Abaqus, load-deformation models adopted for flexural, punching and post-punching shear responses were obtained analytically. At each connection, punching shear cones were modelled separately from the slab. The slab and each punching cone were attached using eight three- dimensional Cartesian-Cardan connector elements. These connectors contribute 85% of the punching shear strength while 15% was attributed to dowelling action of explicitly modelled rebars also connecting the punching cone to the slab. Post-punching shear response of connections were modelled by assigning concrete breakage strengths calculated analytically, to

Cartesian-Cardan connectors which connected rebars to concrete slab and punching cones at locations expected to depict progression of their post-punching response.

Accuracy of this approach is only as good as its theoretical assumptions. It is also difficult to apply; firstly, due to the need to align connecting nodes of reinforcements, slab and punching shear cone at appropriate depths and distances from the punching shear cone; and secondly due to the calculation and application of break out strength of each connector, based on its distance from the column as well as its position along the slab or punching cone depth. The approach adopted by Keyvani, Sasani and Mirzaei (2014) does not give a basis for interaction between asymmetric punching shear and compressive membrane action. Determination of strength- deformation relationships becomes problematic for the latter stages of the analysis since slab deformation around adjoining connections must be known prior to initiation of analysis for calculation of their punching shear and post-punching shear capacities.

Liu (2014)

Liu (2014) carried out a numerical investigation into the response of a three-story flat slab structure after the sudden loss of exterior and interior columns. The design of the flat slab system was carried out in accordance to the ACI 318 (ACI, 1971). The structure consisted of four square panels, with a 6 m span in each orthogonal direction (Figure 2.6). The characteristic compressive strength for concrete and yield strength of steel used were 27.6 MPa and 427.5 MPa respectively. The dead and live loads used for the ultimate limit state were 5.446 kNm-2 and 2.4 kNm-2 respectively. The percentage of designed flexural reinforcement provided over the internal supports was 0.64%. Compression reinforcement was also provided at the connections but their contribution in post-punching were neglected due to the inadequacy of their anchorage at the connections (Liu, 2014). Slab-column connections were modelled using two connector beam elements at each column side. The failure criterion of the Critical Shear Crack Theory (CSCT) (Muttoni, 2008) was used to determine the rotation at which punching shear failure was expected

to occur. Post-punching shear response of connections were ignored, since it was assumed that this mechanism had negligible influence on the actual response. Results of dynamic analysis using the DoD (2016) guidelines showed punching of adjacent connections to occur after sudden column removal of internal and exterior columns. Failure was also observed to be propagated both vertically and horizontally, leading to the progressive collapse of the modelled structure.

Figure 2.6: Slab geometry (Liu, 2014)

Liu (2014) adopted a slab rotation of 0.022 radians around the interior slab-column connection as the rotation at which punching shear failure was deemed to have occurred. No information was provided on how asymmetric punching shear failure was taken into consideration. Assessment without the consideration of the post-punching shear response could under-estimate slab system response in flat slab systems with well anchored area of integrity reinforcement passing through the column core.

Olmati et al. (2017)

Olmati et al. (2017) adopted the design solution of an existing four-story flat slab office building described in Technical Report 64 of The Concrete Society (2007) in a numerical model to assess the response of a flat slab structure after an initial local failure. The structure was designed using the Eurocode 2 (CEN, 2014b). As shown in Figure 2.7, the slab consisted of irregular spans,

supported on 0.4m wide square columns. The slab had a thickness of 0.3m and a characteristic cylinder compressive strength of 30 MPa at 28days. This gave an average strength of 36 MPa considering a combined strength and aging factor of 1.2. Reinforcing steel bars had characteristic strength of 500 MPa and were provided such that the percentages of reinforcement over the internal column were 0.96% and 1.27% in the x and y directions respectively. The first perimeter of punching shear reinforcement around the column consisted of 12 legs of 0.01 m diameter bars. These gave a total punching shear reinforcement area (𝐴_𝑠𝑤) of 948 mm2 per perimeter. Total dead load adopted for design was 484.5 kN and live load, 226.8 kN. Total design load added up to 947.6 kN. Further details on design characteristics of this flat slab case are available in The Concrete Society (2007).

Numerical model of Olmati et al. (2017) had the slab-shell elements directly connected to the column at the connections. Automatic disconnection of connections at punching was not modelled. After the numerical analysis, load-rotation response of the slab around adjacent connections were obtained. Analyses of the load-rotation responses with the calculated failure criteria (Muttoni, 2008) provided information on whether the adjacent connections had punched, after the sudden column removal. Sudden removal of an internal column under quasi-permanent and frequent load combinations as required by the Eurocode 0 (CEN, 2010a) gave no punching shear failure at the adjacent connections.

Figure 2.7: Slab geometry (The Concrete Society, 2007)

The numerical approach adopted by Olmati et al. (2017) provides limited information on the propagation of failure in a flat slab structure after the sudden removal of an internal column. Information was only obtained on flexural and compressive membrane actions of the flat slab system prior to the punching shear failure of the connection closest to the removed column. After the punching shear failure of the column closest to that which was removed, horizontal and vertical collapse propagation could not be assessed using this approach.

To adequately model flat slab system response numerically after local failure, it is important to adequately take into consideration mechanisms developed during response (flexural, punching shear, post-punching shear, compressive membrane action and tensile membrane action) as well as their possible influence on each other. However, due to limitations in the computing tools and time available, simplified slab system models are adopted which either considers a single stage of response or ignores certain mechanisms (Liu, 2014; and Olmati et al., 2017). In Chapter 4 of this thesis, a numerical model capable of simulating slab system response beyond failure of the connection closest to the removed column is proposed. The proposed numerical model takes into consideration connection flexural, punching shear, post-punching shear and tensile membrane

actions; gravity load redistribution; interaction of the various mechanisms; horizontal and vertical damage propagation.