9. LA PROVIDENCIA O RESOLUCIÓN CAUTELAR
10.1. OPINIÓN DE LA DOCTRINA SOBRE EL CONCEPTO DE MEDIDA CAUTELAR
A pilot flat slab was designed using the Eurocode 2 (CEN, 2014b) recommendations. Conventional span length, column width, overall slab depth (The Concrete Society, 2007) and loading (CEN, 2010b) were used. Slab physical characteristics and material properties are as presented in Table 4.2.
Design results and slab strength characteristics are as presented in Punching shear resistance, 𝑉𝑅𝑑,𝑐, of the connections were calculated using Equation 4.1 (CEN, 2014b);
𝑉𝑅𝑑,𝑐. 𝐸𝐶2 = 𝐶𝑅𝑑,𝑐𝑘(100𝜌𝑙𝑓𝑐𝑘)1 3⁄ ≥ 𝑉 𝑚𝑖𝑛 (𝑓𝑐𝑘 𝑖𝑠 𝑖𝑛 𝑀𝑃𝑎)
... eqn. 4.1
where,
𝑘 = 1 + √200 𝑑 ≤ 2.0 (𝑑 𝑖𝑠 𝑖𝑛 𝑚𝑚)... eqn. 4.2
𝜌𝑙 = √𝜌𝑙𝑦. 𝜌𝑙𝑧 ≤ 0.02... eqn. 4.3
and,
𝑉𝑚𝑖𝑛 = 0.035𝑘3/2𝑓1/2... eqn. 4.4
Table 4.3. The flat slab was designed such that the punching shear resistance of the slab was higher than the design load, even without the consideration of punching shear reinforcement. To satisfy the requirements of the Eurocode 2 (CEN, 2014b), 70% of negative moment was apportioned to the column strip of the slab and 50% percent of positive moment apportioned to the column strip. For the flat slab case which was designed without provision of integrity reinforcement, two 10mm bars of bottom nominal reinforcement were passed through the column cores in each orthogonal direction. Detailed information on the design process is as provided in Appendix A. Details of steel reinforcement detailing is as shown in Figure 4.15a. The design approach adopted was similar to that presented in The Concrete Society (2007).
Table 4.2: Slab physical characteristics and material properties
Characteristic Value Unit
Span, 𝐿 6.8 m
Column strip width 3.4 m
Slab overall depth, ℎ 0.275 m
Nominal cover, 𝑐1 0.03 m
Flexural reinforcement bar diameter adopted, ɸ 0.012 m
Average effective depth, 𝑑 0.233 m
Concrete compressive strength, 𝑓𝑐 30 MPa
Concrete tensile strength, 𝑓𝑐𝑡 2.355 MPa
Yield strength of steel reinforcement, 𝑓𝑦 500 MPa
Punching shear resistance, 𝑉𝑅𝑑,𝑐, of the connections were calculated using Equation 4.1 (CEN, 2014b); 𝑉𝑅𝑑,𝑐. 𝐸𝐶2 = 𝐶𝑅𝑑,𝑐𝑘(100𝜌𝑙𝑓𝑐𝑘)1 3⁄ ≥ 𝑉 𝑚𝑖𝑛 (𝑓𝑐𝑘 𝑖𝑠 𝑖𝑛 𝑀𝑃𝑎)
... eqn. 4.1
where,
𝑘 = 1 + √200 𝑑 ≤ 2.0 (𝑑 𝑖𝑠 𝑖𝑛 𝑚𝑚)... eqn. 4.2
𝜌𝑙 = √𝜌𝑙𝑦. 𝜌𝑙𝑧 ≤ 0.02... eqn. 4.3
and,
𝑉𝑚𝑖𝑛 = 0.035𝑘3/2𝑓𝑐𝑘1/2... eqn. 4.4
Table 4.3: Design results and slab strength characteristics
Characteristic Value Unit
Total dead load 8944 Nm-2
Imposed load 3000 Nm-2
Design load 16575 (766.4 per bay) Nm-2 (KN)
Design moment
(middle of interior span)
315.2 KNm
Flexural reinforcement provided, 𝝆𝒔𝒑𝒂𝒏
(interior span)
0.44 %
Design moment (interior support)
525.3 KNm
Flexural reinforcement provided𝝆𝒔𝒖𝒑𝒑𝒐𝒓𝒕,
(interior support)
0.53 %
Punching Shear Strength (Demand-capacity ratio): Eurocode 2 ACI 318 (2011) MC2010: LOA II 873.3 (0.9) 965.3 (0.8) 883.0 (0.9) KN KN KN
The recommended value for 𝐶𝑅𝑑,𝑐 is 0.18/𝛾𝑐. Parameters 𝝆𝑙𝑦 and 𝝆𝑙𝑦 are related to the bonded flexural reinforcement in the two orthogonal directions. They were calculated by taking the mean values of ratio of reinforcement provided within a slab width of (𝑐 + 6𝑑). Where 𝑐 is the column width and 𝑑 is the slab effective depth. Relationships from other codes, Model Code 2010 (fib, 2012a) and ACI 318 (ACI, 2011a), used for the purpose of comparisons in Punching shear resistance, 𝑉𝑅𝑑,𝑐, of the connections were calculated using Equation 4.1 (CEN, 2014b);
𝑉𝑅𝑑,𝑐. 𝐸𝐶2 = 𝐶𝑅𝑑,𝑐𝑘(100𝜌𝑙𝑓𝑐𝑘)1 3⁄ ≥ 𝑉 𝑚𝑖𝑛 (𝑓𝑐𝑘 𝑖𝑠 𝑖𝑛 𝑀𝑃𝑎)
... eqn. 4.1
where,
𝑘 = 1 + √200 𝑑 ≤ 2.0 (𝑑 𝑖𝑠 𝑖𝑛 𝑚𝑚)... eqn. 4.2
𝜌𝑙 = √𝜌𝑙𝑦. 𝜌𝑙𝑧 ≤ 0.02... eqn. 4.3
and,
𝑉𝑚𝑖𝑛 = 0.035𝑘3/2𝑓 𝑐𝑘 1/2... eqn. 4.4
Table 4.3 and Figure 4.15b, are as defined in Equations 3.27 to 3.29 and Equation 4.5 respectively. The control perimeter as adopted in Equation 4.5 of the ACI 318 (ACI, 2011a) was calculated at a distance of 0.5𝑑 from the column face.
Figure 4.15b also shows the overall response of the flat slab connection, when modelled numerically as an isolated slab specimen following procedures explained in Section 3.2 of this thesis. The connection response in flexure, punching shear, residual punching shear and post- punching shear strength were comparable to those obtained analytically. Analytical was determined using the LoAII formulae of the Model Code 2010 (fib, 2012b) of the critical shear crack theory (CSCT). Load rotation curves obtained using the simplified formulae of the CSCT (Equation 2.2 and 2.3) have been shown to provide less stiff responses than those obtained through test and the equations based on the quadrilinear moment-curvature relationship
(Muttoni, 2008). Peak values of punching shear strength were also comparable to those obtained using code based formulae (Equation 2.2 and 2.3) and peak post-punching shear resistances was 30% higher than that obtained using the relationship of Ruiz, Mirzaei and Muttoni (2013) as shown in Figure 4.15b. State of reinforcements present in the slab specimens are as shown in Figure 4.15b, prior to punching shear failure and at the drop in strength during the post-punching shear phase.
(a)
(b)
Figure 4.15: (a) Steel reinforcement detail for slab; (b) Response of column-slab connection (modelled as isolated slab)
Horizontal internal tie force
Flexural reinforcement provided satisfied the 20KNm-1 horizontal internal tie force requirements of the Eurocode 2 (CEN, 2014b), without the consideration of bottom nominal reinforcement around connections or integrity reinforcement. Calculation gave an average value of 566KNm-1. It has been shown that large deformation is required for the activation of tensile membrane action in flat slabs (Sagaseta et al., 2017). Sagaseta, Ulaeto and Russell (2017) also showed flat slab connections tend to fail in punching shear before attaining the large deformations required for activation of tensile membrane action. This is because after punching shear failure, spalling of concrete around these reinforcement makes them ineffective in development of the required tie force (Mitchell and Willliam, 1984; and Ruiz, Mirzaei and Muttoni, 2013).
Integrity reinforcement
For cases with designed integrity reinforcement, the required area of integrity steel reinforcement to be provided was assessed using codes provisions (Mitchell and Willliam, 1984; ACI, 2011b; CSA, 2004; and fib, 2012). These expressions are as defined in Chapter 2 of this Thesis. It must be mentioned that the purpose of integrity reinforcement as stated in codes is to hang the slab over the column in cases of initial damage, hence preventing horizontal or vertical propagation of failure (ACI, 2011b; CSA, 2004; and fib, 2012). The Model code 2010 (fib, 2012a) specifies that the design shear used for calculation of the require area of integrity reinforcement be calculated on the basis of an accidental situation with the objective of preventing progressive collapse. However, Equation 4.6 (ASCE, 2010) requires that the factored uniformly distributed load, 𝑤𝑢, not be less than twice the slab service dead load as explained in Chapter 2 of this Thesis.
𝐴𝑠𝑚 =0.5𝑤𝑢𝑙1𝑙2
The parameter 𝑤𝑢 is the factored uniformly distributed load which is not expected to be less than twice the slab unfactored dead load, 𝑙1 and 𝑙2 are the slab spans in each principal direction and Ф is the reduction factor which has a value of 0.9.
The shear load adopted for the design of integrity reinforcement was 1.25𝑉0. Where 𝑉0 is the shear load imposed on the connection prior to column removal and the factor 1.25 represented the increase in shear load imposed on the connection due to column removal. Comparison of the required area of integrity reinforcement using the various expressions are as shown in Table 4.4, where the parameter V was calculated using the frequent load combination of the Eurocode 0 (CEN, 2010a). For the ACI 352.1R (ACI, 2011b), two times the service dead load, which was found to be greater than 1.25D, was used as the shear load imposed on the connection. As observed in Table 4.4, ACI 352.1R (ACI, 2011b) required the most area of integrity reinforcement (3492mm2). Formulae of the CSA A23.3-04 (CSA, 2004) and Model Code 2010 (fib, 2012a) both required 2683mm2 and 2645mm2 areas of integrity reinforcement respectively. Provision of 2 number 25mm diameter bars per connection face satisfied the requirements of the various code formulae. This was provided to run through the connection in each orthogonal direction. Chapter 6 of this thesis provides a detailed assessment on how code recommendations on integrity reinforcement influences progressive collapse response of flat slab structures.
Table 4.4: Area of designed integrity reinforcements
Source Total area required (mm2) Number of 25mm φ bars required per connection face Mitchell and Willliam (1984), and
ACI 352.1R-11 3492 1.8
CSA A23.3-04 2683 1.3
4.3.4 Numerical model of flat slab system