After punching shear failure of slab-column connections, a punching shear cone develops around the column. Due to the action of the shear load at the connection, the slab separates from the punching shear cone as shown in Figure 2.18. This causes a total loss of concrete contribution, both in compression and through aggregate interlock. The only link between the punching shear cone and the slab are flexural and any existing integrity reinforcement (Figure 2.18). To transfer shear forces from slab to column, the slab drops until the reinforcement connecting the cone to the slab are sufficiently activated to resist total vertical collapse at the connection as shown in
Figure 2.18. Several authors including Hawkins and Mitchell (1979), Melo and Regan (1998) and Mirzaei (2010) have reported a small contribution of the flexural reinforcement to post- punching strength. Tests carried out by Melo and Regan (1998), Habibi et al. (2012), Ruiz, Mirzaei and Muttoni (2013) and Peng (2015) show that slabs with flexural reinforcement only, could develop post-punching strength of up to 30% of the attained strength whereas those of slabs with both tensile and integrity reinforcements could be as high as 70%.
Figure 2.18: Post-punching shear
To accurately assess progressive collapse in flat slab structures, models adopted must be capable of effectively predicting post-punching shear response. Existing formulae for prediction of post- punching shear strength have been based on rebar dowelling action, membrane force and breakage of concrete around rebar. Regan (1981) proposed Equation 2.16 for estimation of post- punching resistance of integrity reinforcement provided in slab column connections. This relationship was based on the dowel equation of Rasmussen (1962). In Equation 2.16, 𝑉𝑝𝑝.𝑖𝑛𝑡 is the contribution of integrity reinforcement to post-punching shear resistance, Ф and 𝑓𝑠𝑦 are the diameter and yield stress of integrity reinforcement respectively and 𝑓𝑐 is the concrete
compressive strength.
𝑉𝑝𝑝.𝑖𝑛𝑡. = 1.2 ∑ Ф2√𝑓
𝑠𝑦𝑓𝑐
... eqn. 2.16
Georgopoulos (1986) estimated the post-punching strength of compression reinforcement provided in slab-column connections. His prediction was also based on Rasmussen’s dowel
equation (Rasmussen, 1962), which is as shown in Equation 2.17. He estimated the angle of inclination of the reinforcing bars at failure using Equation 2.18𝑠𝑖𝑛 𝜓𝑖𝑛𝑡 = 1.5√
𝑓𝑐 𝑓𝑠𝑦 ... eqn. 2.18. 𝑉𝐷𝑈 = 𝐵ɸ2√𝑓 𝑐𝑓𝑠𝑦
... eqn. 2.17
𝑠𝑖𝑛 𝜓𝑖𝑛𝑡 = 1.5√𝑓𝑐 𝑓𝑠𝑦... eqn. 2.18
Where, 𝑒 is the eccentricity of the load and;
𝐵 = 𝐶(√1 + (𝜁𝐶)2 − 𝜁𝐶)
𝜁 = 3𝑒 ɸ√
𝑓𝑐
𝑓𝑠𝑦
Melo and Regan (1998) carried out tests on 7 slab specimens after which they concluded that resistance formulae proposed by Georgopoulos (1986) were unrealistic. Melo and Regan (1998) recommended that response of bottom bars be considered like bars embedded in concrete at both ends crossing the critical punching shear crack. Using analytical break out principles provided in the American Concrete Institute code for nuclear safety related concrete structures, ACI 349 (ACI, 1978), Melo and Regan (1998) treated the vertical components of forces in bottom bars as the pull-out load acting at the edge of the slab. This is as shown in Figure 2.19. Expressions for resistance of bottom reinforcement were developed for single and pairs of bars as given in Equations 2.19 and 2.20 respectively.
Figure 2.19: Horizontal projection of conical surfaces 𝑉𝑅.𝑝𝑝 = 0.33√𝑓′𝑐 𝜋𝑑𝑟𝑒𝑠2 2
... eqn. 2.19
𝑉𝑅.𝑝𝑝 = 0.33√𝑓′𝑐 𝜋𝑑𝑟𝑒𝑠2 2 − 𝐴1... eqn. 2.20
Where, 𝐴1 = 𝛥 360𝜋𝑑 2−𝑠 4𝑑𝑟𝑒𝑠𝑠𝑖𝑛 𝛥... eqn. 2.21
𝛥 = 𝑎𝑟𝑐 𝑐𝑜𝑠 ( 𝑠 2𝑑𝑟𝑒𝑠)... eqn. 2.22
Equations 2.19 and 2.20 were found to give accurate predictions of the peak post-punching shear strength when compared to test results (Melo & Regan, 1998). Rupture of bottom reinforcement was also identified as a mechanism capable of limiting the attainment of peak post-punching shear strength. Hence, Equation 2.23 was proposed as the limiting resistance due to rupture of bottom bars. The value, 0.44, in Equation 2.23 corresponds to an angle of inclination of 260 to the horizontal for the reinforcement at fracture.
𝑉𝑅.𝑝𝑝.𝑟𝑢𝑝𝑡 = 0.44 ∑ 𝐴𝑠𝑓𝑦
... eqn. 2.23
Mirzaei (2010) and Ruiz, Mirzaei and Muttoni (2013) carried out 24 tests to study the post- punching shear response of 125mm thick slab-column connections. Tests consisted of three
series. The first series was designed to investigate the effects of tensile reinforcement and the second assessed the contribution of integrity reinforcement and bent-up bars to post-punching shear strength. The third series assessed the influence of anchorage. With the use of these tests, Mirzaei (2010) developed an analytical post-punching model which was based on the break-out concrete strength of both flexural and integrity reinforcement. This model adopted assumed progressive destruction of concrete over reinforcement bars as shown in Figure 2.20. For the section shown in Figure 2.20, the breakage strength of the embedded reinforcement was given by the expression;
𝑉𝑐𝑜𝑛(𝑥) =𝜋
2(𝑥𝑡𝑎𝑛𝛼 𝑐𝑜𝑡𝛾) 2𝑓
𝑐𝑡.𝑒𝑓𝑓
... eqn. 2.24
Where 𝛼 is the angle of inclination of the punching shear cone to the horizontal and 𝛾 is the angle of the breakout cone. The effective concrete tensile strength, 𝑓𝑐𝑡.𝑒𝑓𝑓, was given by 𝑛𝐷𝑓𝑐𝑡, where 𝑛𝐷 is a reduction factor which considers the variation of tensile stress from the edge of the bar to the surface of the slab. For an n number of bars, Equation 2.25 gives the total projected area developed.
𝐴1 = 4 {[(𝛥 +𝑛
2(𝜋 − 2𝛥)] 𝑑𝑟𝑒𝑠
2 +𝑛−1
2 𝑠𝑑𝑟𝑒𝑠𝑠𝑖𝑛 𝛥}
... eqn. 2.25
Figure 2.20: Progressive destruction of concrete during post-punching (adapted from Mirzaei, 2010)
Mirzaei (2010) considered shear transfer from the slab to the column by accounting for the flexural and membrane responses of the reinforcement connecting them. Response at the elastic phase, assumed the development of curvature influenced zones, at the points of entry of the reinforcement into the punching cone and slab (Figure 2.20). This response was based on Equation 2.26;
𝑉𝑠.𝐸𝑙𝑎𝑠. = 𝐴𝑏𝐸𝑠𝑠𝑖𝑛 𝜓 + 𝑉𝐼𝑐𝑜𝑠 𝜓
... eqn. 2.26
where, 𝐴𝑏 and 𝐸𝑠 are the cross-sectional area and elastic modulus of reinforcing bar respectively.
𝑉𝐼 is the shear force at the reinforcing bar section considered. On development of plastic hinges close to the point of entry of rebar into the concrete cone and slab, Equation 2.27 was adopted rather than Equation 2.26.
𝑉𝑠.𝑃𝑙𝑎𝑠. = 𝑁1𝑠𝑖𝑛 𝜓 + Ф3
3𝑙𝑟𝑒𝑓𝑠𝑦(1 −
𝑁12
𝑁𝑦2) 𝑐𝑜𝑠
2𝜓
... eqn. 2.27
The parameters 𝑁1 and 𝑁𝑦 denote the axial forces in the rebar at a phase of response and at yield
respectively. The parameter 𝑙𝑟𝑒 represents the exposed length of rebar. Strain and deflection at
failure were considered using the Equations 2.28 and 2.29 respectively.
𝜀𝑠𝑢 = 1
𝑐𝑜𝑠 𝜓𝑚𝑎𝑥− 1
... eqn. 2.28
𝑡𝑎𝑛 𝜓𝑢 =𝑤𝑢
𝑙𝑟𝑒
... eqn. 2.29
The analytical model proposed by Habibi, Cook and Mitchell (2014) for predicting post- punching response of slab-column connections was similar to that of Mirzaei (2010). Assumption of a punching shear cone at inclinations of 45o to the horizontal from a distance of 2𝑑 3⁄ from the top of the integrity reinforcement, and 14o to the horizontal beyond this point was
adopted. In addition to this, only membrane response of reinforcing bars were considered, hence, limiting the expressions to;
One limitation in the application of analytical models developed by Habibi, Cook and Mitchell (2014) is the fact that the compatibility response is independent of the concrete breakage progression response. This model can only be applied independently if the parameter 𝑙𝑟𝑒 at a particular 𝑤𝑢 is already know, possibly from tests. Application of the model proposed by Mirzaei (2010) faces a similar problem since the axial force in rebar (𝑁1) needs to be assumed to calculate
the shear resistance of rebars.
Ruiz, Mirzaei and Muttoni (2013) proposed Equations 2.31 to 2.37 for the determination of the post-punching shear strength of slab column connections. Contributions of flexural and integrity reinforcements were taken into consideration. Theoretical basis for the development of the expressions were similar to those adopted by in the analytical model of Mirzaei (2010).
𝑉𝑅,𝑝𝑝 = 𝑉𝑅,𝑝𝑝,𝑓𝑙𝑒𝑥. + 𝑉𝑅,𝑝𝑝,𝑖𝑛𝑡
... eqn. 2.31
𝑉𝑅,𝑝𝑝,𝑓𝑙𝑒𝑥. = 𝑛𝑏,𝑓𝑙𝑒𝑥. 𝑓𝑐𝑡. 𝑏𝑒𝑓. 𝑙𝑒𝑓... eqn. 2.32
𝑛𝑏,𝑓𝑙𝑒𝑥. = 4. 𝑛𝑎 = 4.𝑐+2𝑑𝑐𝑜𝑡𝜃 𝑠𝑏... eqn. 2.33
𝑐𝑜𝑡𝜃 = 𝑐𝑜𝑡𝜃𝑡𝑜𝑝+𝑐𝑜𝑡𝜃𝑏𝑜𝑡𝑡𝑜𝑚 2... eqn. 2.34
𝑓𝑐𝑡 ≅ 0.5√𝑓𝑐 (𝑀𝑃𝑎)... eqn. 2.35
𝑏𝑒𝑓 = 𝑚𝑖𝑛(𝑠𝑏 − 𝑑𝑏; 6𝑑𝑏; 4𝑐𝑏)... eqn. 2.36
𝑙𝑒𝑓 = 2𝑑𝑏... eqn. 2.37
In Equation 2.31, 𝑉𝑅,𝑝𝑝 is the post-punching shear strength of the slab-column connection, 𝑉𝑅,𝑝𝑝,𝑓𝑙𝑒𝑥. is the contribution of tensile reinforcement given by Equation 2.32 and 𝑉𝑅,𝑝𝑝,𝑖𝑛𝑡 is the
contribution of integrity reinforcement defined by Equation 2.38. Equations 2.32 to 2.37 express parameters with which 𝑉𝑅,𝑝𝑝,𝑓𝑙𝑒𝑥. is determined. Where 𝑛𝑏,𝑓𝑙𝑒𝑥. is the number of sections of flexural reinforcement activated in the post-punching phase, 𝑓𝑐𝑡 is the tensile strength of
size, 𝑑 is the effect depth of the slab, 𝜃 is the average angle at the point where the flexural bars are activated, 𝑐𝑏 is the concrete cover, 𝑑𝑏 is the diameter of reinforcing bars, and 𝑠𝑏 is the spacing
of reinforcement bars. Ruiz, Mirzaei and Muttoni (2013) noted that a value of 2.8 for 𝑐𝑜𝑡𝜃 gave good estimates of the number of flexural reinforcement activated in the post-punching mechanism.
𝑉𝑅,𝑝𝑝,𝑖𝑛𝑡 ≤ 𝑓𝑐𝑡. 𝐴𝑐,𝑒𝑓
... eqn. 2.38
𝐴𝑐,𝑒𝑓 = 𝑑𝑟𝑒𝑠. 𝑏𝑖𝑛𝑡
... eqn. 2.39
𝑏𝑖𝑛𝑡 = ∑ (𝑠𝑏,𝑖𝑛𝑡+𝜋
2𝑑𝑟𝑒𝑠)
... eqn. 2.40
As explained in the paragraph above, Equation 2.38 expresses the contribution of the integrity reinforcement to post punching, 𝑉𝑅,𝑝𝑝,𝑖𝑛𝑡. The parameter 𝐴𝑐,𝑒𝑓 is the effective concrete area and can be calculated using Equation 2.39; where 𝑑𝑟𝑒𝑠 is the residual effective depth of the slab. The residual effective depth, 𝑑𝑟𝑒𝑠, was described as the distance between centroids of flexural and integrity reinforcement layers. Equation 2.40 expresses 𝑏𝑖𝑛𝑡, which is the control perimeter
activated by the integrity reinforcement. The parameter, 𝑠𝑏,𝑖𝑛𝑡 in Equation 2.40 represents the
width of the group of bars considered in one direction.
Assessment of progressive collapse of flat slab structures requires adequate prediction of the system post-punching shear response of the connections in the flat slab system. Consideration of the post-punching mechanism in the numerical analysis of progressive collapse of flat slabs structures is currently not possible since there exist no numerical approach for determination of post-punching shear response of slab-column connections. Application of existing analytical post-punching shear models to flat slab systems is also not possible because;
• Information needed on some parameters in the analysis can only be obtained from tests or numerical analysis.
• Existing analytical models are based on assumption of a symmetric slab-column response in post-punching. This is not the case in progressive collapse of flat slab systems which is commonly assessed using the sudden column removal scenario. Uneven spans developed after the sudden removal of a column could lead to asymmetric punching shear and post-punching response of the adjoining connections.
Chapter three of this thesis seeks to fill these gaps in literature through the development of numerical and analytical models which can independently predict post-punching response of slab-column connections for both symmetric and asymmetric cases. Results of existing models were also compared to those of the proposed model.