Evaluation of punching shear design criteria to prevent progressive collapse of RC flat slabs

Abstract

This study focuses on a RC building with flat slabs that were designed according to current standards. A scenario of a slab with failed connections that is detached from its supporting columns and is falling downward is considered, and the impact results with the slab underneath are assessed. The suitability of the standards design criteria to provide safe design against impact loading is evaluated. It was found that larger span slabs experience heavier damage. Falling from a floor height and from a quarter floor height are analyzed and the damage results of these impacts are examined. At the lower impact velocity, the concrete slab surrounding the column undergoes major damage and shear deformations; In the case of a relatively short span slab, the rebars undergo large plastic deformations and almost reach the ultimate strain, such that a slightly higher impact velocity would cause rebars fracture and total failure of the slab-column connection. In the case of high impact velocity, the concrete in the slab around the column is fully crashed and the longitudinal and the bent up rebars are ruptured. The yield of the rebars occurs within a few milliseconds. During this extremely short time the impacted slab hardly starts developing its downward displacement. The impacted slab responds like a rigid body with severe damage concentration at the slab-column connection region. Different parameters affecting the slabs dynamic response are examined, and new insight is gained on the complex impact response of flat RC slabs. This study finds that the current design standards that are based on static loading considerations do not provide resilience to flat slab connections that are subjected to impact loading and therefore cannot prevent a progressive collapse scenario.

Keywords

Dynamic punching flat slabs reinforced concrete column-slab connection shear reinforcement impact progressive collapse

Introduction

The problem

Reinforced concrete buildings with flat slabs are very common in residential and office buildings due to their many advantages, such as: fast and easy construction, efficient net story height, flexibility in interior floor design, cost effectiveness, etc. The sensitivity of the flat slab-column joint to punching shear is well known. Consider the simple common case of an evenly distributed static loading: each column carries part of the total floor load that depends on the spans and the column location in the slab layout. This load is transferred from the slab to the column through shear in the slab at the column circumference. The shear stresses in the slab increase at shorter distance from the column, therefore rather relatively high shear stresses are developed at the slab-column connection. These high shear stresses may lead to punching shear failure at the connection zone. This is a brittle mode failure that disintegrates the connection between the slab and the supporting column without warning. In the case of a connection failure, the slab load that cannot be carried by this column, is rapidly redistributed to the neighboring columns. If these columns cannot sustain the excessive transferred load additions, punching shear failure occurs at the connections of these columns to the slab as well, and a progressive collapse of the slab-column connections at the slab plane occurs.

To avoid this catastrophic event, the research in recent years has expanded to deal with post punching shear behavior of flat slabs, in an attempt to contribute to the structure robustness and reduce the likelihood of this failure propagation in the slab plane. It aims at providing post punching strength that inhibits the possible fall of the slab and preventing failure propagation (Broms, 2005; Kai, 2013; Qian and Bing, 2015; Qian and Li, 2013; Ruiz et al., 2013; Sasani et al., 2011a, 2011b; Yi et al., 2014). This is mainly done by including special integrity reinforcement. These are straight rebars placed at the soffit of the slab over the area that is supported by the column. These rebars are activated after the occurrence of punching shear, where the slab tends to deflect considerably with respect to this column. The aim of the integrity reinforcement is to hang the impacted slab on the column by the integrity rebars that are bonded to the slab, thus supporting the slab and bridging over the failed connection zone. With aid of this action, post punching shear static equilibrium may be obtained. The contribution of the integrity reinforcement has been demonstrated in static load tests (Ruiz et al., 2013; Wieczorek, 2013).

The mechanism of static punching shear failure has been studied widely, both experimentally and analytically (Einpaul et al., 2016; Hallgren and Bjerke, 2002; Hueste and Wight, 2002; Micallef et al., 2014; Muttoni, 2008; Muttoni et al., 2013; Yamada et al., 1992; Yankelevsky and Leibowitz, 1999). The results of many investigations are implemented in design codes, for example, Eurocode 2 (EC2, 2004), Model code 2010 (2011), CSA A23.3 (CAN/CSA-A23.3-04, 2004), and ACI 318 (2014). Although the recent codes explicitly aim at design of flat slabs such that they will not be damaged even by extreme events like impact and blast, the included design requirements stem from static loading considerations. The different standards provide detailed instructions and recommendations for the proper design of a flat slab and include guidelines for the slab thickness and special reinforcement in order to avoid punching shear failure. A review of the major requirements is outlined in section “Summary of standards requirements for static design,” and their adequacy will be examined later.

Despite the standards attention to punching shear in concrete flat slabs and the provided design instructions, punching shear failures are reported from time to time. There are numerous reasons for the occurrence of such failures even under static conditions, such as excessive or unexpected loads, foundations settlements, deficiencies in design, poor reinforcement quantities and detailing, poor concrete quality, lack of quality control etc., and their combinations. A typical punching shear collapse result of a parking garage is shown in Figure 1. This four stories parking garage progressively collapsed in Tel Aviv in 2016, at an advanced stage of its construction. After the collapse, the columns remained at their full height with observed slope variation at the column-slab connection level, indicating of a weak, almost hinge like, connection at the slab level.

Figure 1.

Punching shear collapse of a parking garage-Tel Aviv 2016.

The collapse scenario started with punching shear failure at the top slab which caused disintegration of this slab from the columns and was followed by the slab free fall downwards. The falling slab impacted the slab underneath and caused dynamic punching shear failures to all its slab-column connections. This initiated a progressive collapse event in the vertical direction, which affected all the other slabs below. There are numerous examples of disastrous impact punching shear failures, which ended with a pile of slabs resting on the ground, one on top of the other, and the columns of several stories height protrude through as described above (Lew et al., 1982; Ltd et al., 2006; Ruiz et al., 2013; Wieczorek, 2013; Yankelevsky et al., 2020).

Regardless of the cause of failure of the first connection in the first damaged floor, which initiates the progressive collapse of all connections at this slab level, the following stage is common: this slab lost its connections with its supporting columns and accelerates downward at an increasing velocity, until its impact with the slab underneath. This impact event between the two slabs is critical to the survivability of the entire building. If the impacting slab is arrested by the impacted slab, the failure scenario ends and progressive collapse in the vertical direction is prevented. Otherwise, a progressive collapse process continues over the entire building with more dynamic punching shear failures at the lower slabs, causing their loss of supports and leading to failure of the entire building. These two entirely different scenarios depend on the impact resistance of the impacted slab: if the impacted slab survives the impact event it saves the building from the outcomes of a progressive collapse. The survivability of the building or its destruction has major implications on the number of casualties and on the total direct and indirect damage.

Literature review

Dynamic punching shear due to impacting slabs has gained very little attention in the literature and there is much need of its investigation. Dynamic punching has been investigated in problems of projectiles penetration into concrete both regarding to low velocity impact (Yankelevsky, 1997) and high velocity impact (Dancygier and Yankelevsky, 2002; Dancygier et al., 2007). Experimental investigations on concrete impact also include low velocity drop tests (Chen and May, 2009; Kumar and Mittal, 2017; Zineddin and Krauthammer, 2007). In these scenarios, the impacting masses also produce dynamic local punching shear, that has some similarity to the present problem and therefore may be analyzed with similar computational tools, however the two problems are different in many parameters, therefore available information on drop mass or penetration impact cannot be deduced to evaluate the behavior of impacting slabs. Several experimental and numerical studies on RC flat slabs under static and dynamic loading showed that shear mechanisms dominate their behavior. They showed that higher loading rates increase the failure load and affect the deformation modes and the failure modes (Miyamoto et al., 1991a, 1991b). These studies provided some background to several new models. In Micallef et al. (2014) a model for predicting the dynamic response of RC slabs subjected to drop impact loading is developed based on the Critical Shear Crack Theory (CSCT). This model evaluates the shear demand considering the strain-rate effects and the initial increased stiffness of the structure due to inertial effects. In Sagaseta et al. (2017), an analytical approach for punching shear analysis under blast loading is developed where the dynamic punching shear capacity is obtained using the same methodology as in Micallef et al. (2014).

These testing and analysis studies on dynamic punching shear have gained much attention, however the case of a dropped mass is considerably different from the present problem where a flat slab is impacting a similar slab underneath, and therefore better and more suitable models are needed to provide insight regarding the problem under investigation.

A limited number of studies referred to impact punching failure of a flat slab-column connection, mostly by numerical analysis, using simplified approaches (Olmati et al., 2017; Vlassis et al., 2009) or finite element analysis techniques (Genikomsou and Polak, 2015; Terranova et al., 2017, 2018). These studies showed that there is a low chance of survivability in the case of a progressive collapse due to falling debris. These studies use some uniform “equivalent” domains and do not take into consideration the reinforcement details.

The impact of slabs creates a nonlinear inelastic dynamic process where a complex time dependent state of strain develops in the slab domain. The entire action is characterized by its short durations and high rates of loading. Therefore, it cannot be simulated by simplified models. A realistic model is required to properly simulate the complex behavior of the slab-column connection and its local components such as the longitudinal and shear reinforcements including its details and locations that take part in the local resistance to the impacting slab.

The great importance of the problem on the one hand and the limited available data and insights and lack of advanced suitable tools for its analysis on the other hand, triggered our research program aiming at modeling the slabs impact under such conditions, and at investigating the response and evaluating the effect of major parameters.

In the first stage of this study, a numerical model was formulated (Yankelevsky et al., 2020). It simulates the slab and the rebars using the finite element software AUTODYN and conducts the impact analysis of the slabs, accounting for the inelastic wave propagation, large strains and major damage development. The proposed analysis tool and model have been validated with dynamic low velocity impact tests on reinforced concrete plates, and very good correspondence was obtained. A model of the impacting slabs problem was then formulated. It is composed of a single column connected to a surrounding flat slab with a tributary area, and a similar slab at a floor height above that is isolated from the column to represent its post punching shear state. Release of the upper (impacting) slab affects its later impact with the connected (impacted) slab. Precise simulation of all the involved steel rebars, including the top and bottom reinforcing meshes and the bent up diagonal shear reinforcement, allows a follow up of the dynamic response and the detailed development of damage and failure. In the first stage of the study, general features were examined such as the effect of the slab size, the height of the falling slab, the effect of bent up shear reinforcement, rebar properties, etc. The dynamic local punching shear mode of failure was identified in the impacted slab. It was found that slab falling from a floor height (3.0 m) produces a high impact velocity that causes dynamic punching shear failure. The local severe damage caused to the concrete and rebars was observed within a short time after impact. Enhancing the reinforcement amount and properties in an attempt to increase shear resistance and prevent punching shear failure turned the local punching shear mode into a global large area severe bending damage. With these analysis abilities of the proposed model and with the insights gained from examination of different parameters effects, a general understanding of the slabs impact behavior has been gained.

Research goals

The present stage of the study aims at examination of the current standards provisions for static punching shear and their suitability to provide resistance and resilience under the described impact scenario.

The results of the present study will determine whether the impacted slab, that is designed according to the present standards requirements survives the impact and prevents a continuing progressive collapse or whether a progressive collapse of the slab underneath is unavoidable.

A priori it is supposed that the present standards are based on the accumulated studies and understanding of the slab-column behavior under static loading, and as no impact studies of such slabs are available, the present standards requirements that satisfy static conditions have not been yet examined under impact conditions. Thus, it is not obvious that the current provisions provide the expected dynamic resistance to avoid the dynamic failure in punching shear. Also, the impact scenario may introduce other different effects or conditions that are not accounted for at all in the static solution. This issue will be clarified in the present research.

Summary of standards requirements for static design

Slab thickness

The flat slab thickness design is usually governed by the deflection criterion at the serviceability limit state (SLS). According to Section 7.4 in the Eurocode (EC2, 2004), the deformation of a member or structure shall not be such that it adversely affects its proper functioning or appearance.

Usually it is related to the calculated sag of a beam or slab subjected to quasi-permanent loads (not exceeding 1/250 (0.4%) of the span, and even 1/500 (0.2%) for quasi-permanent loads to prevent the damage of sensitive parts of the structure). Other limits may be considered, depending on the sensitivity of adjacent parts. Generally, it is not necessary to calculate the deflections explicitly, and simple rules, in terms of span/depth ratio may be applied, to avoid excessive deflections under normal conditions. The limiting span/depth ratio may be estimated using expressions 7.16 in EC2 (2004) which depends on the type of the structural system (number of spanning directions and the type of the support), the reinforcement properties (ratio and strength) and the compressive strength of concrete. The recommended span/depth ratio for concrete flat slabs commonly varies between 17 and 24 (EC2, 2004).

Slab’s reinforcement

Steel properties

For static design, either of the following assumptions may be made with respect to the stress-strain curve (Figure 2): (1) an inclined top branch with a strain limit of $ϵ_{u d} .$ (2) A horizontal top branch without the need to check the strain limit. The recommended value of $ϵ_{u d}$ is $90 %$ of $ε_{u k}$ which is the characteristic strain at maximum force. The $ε_{u k}$ value depends on the steel class (Table C.1; EC2, 2004) and its lower bound values vary between 2.5% and 7.5%, depending on the class (cold worked/hot rolled/seismic). Although ductility is not specified for the longitudinal reinforcement in a slab, it should be generally considered in designing bent up rebars or considering the post punching possible inclination angle of the integrity reinforcement. The characteristic yield strength $f_{y k}$ (Figure 2) usually varies between 400 and 600 MPa.

Figure 2.

Idealized and design stress-strain diagrams for reinforcing steel (EC2, 2004).

Longitudinal reinforcement

The design of the longitudinal reinforcement is performed at the ultimate limit state (ULS). The required reinforcement cross section area (or number of rebars per unit width and their diameter) is determined based on the bending moment diagram when the loads are multiplied by partial safety factors (Section 6.1 in EC2, 2004).

Integrity reinforcement

Additionally, according to (ACI 318, 2014; CAN/CSA-A23.3-04, 2004), integrity rebars are added at the slab bottom to connect between the slab and the column as mentioned above. The code ACI 318 (2014) requires at least two integrity rebars in each direction without specifying the rebars size. The Joint ACI-ASCE Committee 352 (ACI 352.1R) provides further guidance: It requires that at interior connections, bottom reinforcement in each direction passing through column cage have a minimum area of: s

A_{s} = \frac{0.5 ω_{u} l_{1} l_{2}}{ϕ f_{y}}

(1)

where $ω_{u}$ is the factored uniformly distributed load, fy is the yield strength of steel, ϕ = 0.9, and ℓ1 and ℓ2 are the center-to-center spans in normal directions. The Canadian code for concrete structures design (CAN/CSA-A23.3-04, 2004) assumes that in the post punching failure, the integrity reinforcement forms an angle of 30° from its original horizontal position (in a vertical plane) and the vertical component of the force in these rebars support the slab at this location. The required cross section area of the integrity rebars is:

\sum A_{s b} = \frac{2 V_{s e}}{f_{y}}

(2)

Where $f_{y}$ is the yield strength of steel and $V_{s e}$ is the shear transmitted to the column due to specified loads but not less than the shear corresponding to twice the self-weight of the slab. Fib Model Code 2010 (2011) expresses the post punching resistance that is provided by the integrity reinforcement in accordance with the reinforcement ductility class, which determines the angle of the integrity rebar with respect to the slab plane at failure. For straight bars, it varies between 0° (i.e. no contribution) for low ductile reinforcement to 25° for ductile reinforcement.

Shear reinforcement

Additional punching shear reinforcement is required when the shear resistance of the concrete is less than the required design shear resistance. For that purpose, different reinforcement types may be used: links, studs or bent bars near the column perimeter (EC2, 2004). Where punching shear reinforcement is required, it should be placed between the column and 1.5 d (where d is the slab effective depth) inside the control perimeter, beyond which shear reinforcement is no longer required, that is, where the shear resistance of the concrete is equal to the required design shear resistance. If links are used, at least two perimeters of link legs should be provided around the column. The spacing of the link leg perimeters should not exceed 0.75 d. If bent bars are used, one perimeter of link legs may be considered sufficient. The use of large diameter reinforcement is limited for thin slabs since the links and the bent bars need to meet the requirements of minimum mandrel diameter to avoid damage to reinforcement (4 D for D < 16 mm and 7 D for D > 16 mm (EC2, 2004), where D is the diameter of the bar).

The model for the slabs impact analysis

General geometry

Consider a layout of a typical reinforced concrete (RC) flat slab that is supported by RC columns having a square cross section, as shown in Figure 3(a).

Figure 3.

Typical RC slab plan and a typical internal field.

The distance between columns is “a” (meters) in each direction. The tributary area of a typical interior column, that is relatively far from the slab edges, is therefore a × a, (Figure 3(b)). In the following we shall focus on this representative column and connected slabs, as an idealized model representing the entire slab-clumn system (Yankelevsky et al., 2020).

The designed slab

The slab span

The present study analyses two slab spans representing a relatively short span of 400 cm which is common in residential buildings and a relatively large span of 800 cm which is common in office and industrial buildings where open space flexible interior design is important. It aims at examination of the severity of damage occurred in both cases and considers the results of the entire range of possible spans.

The assumed net floor height is 3 m and the column’s cross section dimensions are 20 × 20 cm for the smaller span case and 40 × 40 cm for the larger span case. These columns cross sections may suffice low-rise buildings or the upper floors of a high rise buildings.

Analysis of these representative cases will shed light on the damage severity and on the resilience of the impacted slab. Further analysis may examine the sensitivity of this behavior to different parameters, such as the amount and type of shear reinforcement.

The slab thickness

According to existing codes, the minimum required thickness of such slab is 14 cm for the shorter span and 24 cm for the larger span. These minimum thicknesses require punching shear reinforcement as the design values of the punching shear resistance of the concrete slab without added shear reinforcement are insufficient. Without shear reinforcement the shear resistance for the shorter span is 179.5 kN in case of a 20 × 20 cm column cross section (or 222.2 kN for in the case of a 40 × 40 cm column cross section). The shear resistance for the larger span is 396.6 kN for a column cross section of 40 × 40 cm. The required design shear loads are higher: 230.7 kN for the shorter span and 1083.6 kN for the larger span. To satisfy the standards shear resistance demand the slab thickness should increase to at least 16 cm for the short span case, or shear reinforcement should be added.

Reinforcement details

The reinforcement of the analyzed slab is based on static analysis according to a relevant code (Israeli Standard 466, 2013). The reinforcement is made from S400 grade steel according to (ISO, 2019) with minimum elongation at fracture of 12% and elongation at maximum force of 8%. The latter may be considered to provide stable resistance, whereas the fracture ductility is associated with the descending branch of the stress-strain curve.

Slab longitudinal reinforcement

The reinforcement detailing of the slab is based on the bending moment diagrams in both directions, x and y, that is calculated by FE analysis. For the shorter span of 400 cm the bottom reinforcement consists of 8 mm diameter rebars with spacing of 150 mm in both directions over the entire slab. The top reinforcement detailing in the x direction is illustrated in Figure 4. Due to symmetry, the reinforcement in the y direction is identical.

Figure 4.

Top reinforcement in the x direction for 400 cm span slab.

This top reinforcement is arranged in three regions (strips). The strips are considered as shallow beams having the slab depth. In strip 1 (near the column), 16 mm diameter bars are placed at 100 mm spacing in both (x and y) directions. In strip 2, the rebars are of the same diameter (16 mm) but at a spacing of 200 mm in both directions. Strip 3 consists of 8 mm diameter rebars at a spacing of 200 mm in both directions. The thickness of the rebar concrete cover is 20 mm.

Two 10 mm diameter S400 integrity rebars in each direction were inserted, at the bottom of the slab, passing through the column cross-section (ACI 318, 2014). For the given type of concrete (cylindrical compression strength of 35 MPa), the length of the integrity rebars is 110 cm to ensure sufficient anchorage.

For the larger span of 800 cm, the top and the bottom reinforcement detailing in the x direction (which is identical to the y direction due to symmetry) is shown in Figure 5. The thickness of the rebar concrete cover is also 20 mm.

Figure 5.

Top (T) and bottom (B) reinforcements in x direction for 800 cm span slab.

Similarly, to the case of the shorter span, 110 cm length S400 integrity rebars were inserted at the bottom of the slab, passing through the column cross-section. For this case of the larger span, three 18 mm diameter bars in each direction are required (ACI 318, 2014).

The column

The RC column reinforcement is illustrated in Figure 6.

Figure 6.

Columns reinforcement details: (a) 400 cm span and (b) 800 cm span.

For the shorter span slab (Figure 6(a)) the column reinforcement cage includes four longitudinal 12 mm diameter ribbed bars and 10 mm diameter stirrups spaced at 120 mm from each other along the column height, while for the larger span slab (Figure 6(b)) it includes six 20 mm diameter longitudinal ribbed bars with similar stirrups as in the other column. The concrete cover is 25 mm for all the cases.

Punching shear reinforcement

For the shorter span of 400 cm with a limited slab height (14 cm), punching shear reinforcement is required to satisfy the required punching shear resistance, as either bent-up bars or links,. The former option includes three bent-up ribbed bars of 10 mm diameter in each direction as illustrated in Figure 7.

Figure 7.

Bent-up bars detailing for 400 cm span slab. (a) top view and (b) side view.

In the second option, three contours of links located at 50, 100, and 150 mm from the column contour is required (Figure 8).

Figure 8.

Links detailing for 400 cm span slab.

The first two contours include four ribbed 10 mm diameter two legs links, one at each side of the column. The third outer contour includes eight links because of the requirement of a maximal distance of 1.5 d between the legs, where d is the effective depth of the slab.

For the 800 cm span slab we shall examine only bent-up bars. To satisfy the shear resistance requirement nine bent-up ribbed bars of 12 mm diameter in each direction are provided, as illustrated in Figure 9.

Figure 9.

Bent-up bars detailing for 800 cm span slab: (a) top view and (b) side view.

The computational model

The model geometry

A single typical interior column with the tributary slab area around, that is shown in Figure 3(b), represents a typical unit of the building system, and the analysis is carried with regard to this representative structural system. The computational model isolates the column and the tributary slab area from the rest of the building. Each edge of the representative slab is laid along the mid distance between two adjacent parallel columns rows. Referring to the coordinate system attached to Figure 3(b), the boundary conditions of the isolated tributary area may be assumed as ∂w/∂x = 0 along the edges x = 0 and x = b and ∂w/∂y = 0 along y = 0, and y = a (“clamped–guided” or “symmetrical” boundary conditions, e.g. ANSYS AUTODYN, 2005).

An elevation view of the idealized representative model is shown in Figure 10. It shows the RC column that is originally connected to the impacted slab. The column extends below the slab to the level of a floor underneath, and for the sake of simplicity it is connected to a fixed foundation at its bottom. The column extends one level above as well, to the original level of the falling slab. It is assumed that the upper (impacting) slab had undergone a punching shear failure in all its connections prior to its downward falling with an increasing velocity until its impact with the slab underneath. The behavior of the impacted slab is in the focus of our study as failure of the impacted slab connection will initiate a progressive collapse. To simulate this scenario in the model, the upper slab in the model is ideally detached from the column as shown in the figure and is about to start its accelerated motion downwards. Its velocity V(t) is continuously increasing due to gravity, and upon reaching the slab underneath its velocity is denoted V₀ which depends on the falling height.

Figure 10.

Elevation of the computational model.

It is assumed that the slab travels parallel to its original position and that its velocity field is uniform. This paper examines the dynamic response of a column-slab connection for two falling heights: V₀ = 7.75 m/s (that is the free-fall velocity from the typical floor height of 3 m) and of 3.875 m/s (i.e. 0.5 V₀) that simulates the effect of a reduced impact velocity during the upper slab’s fall height that is equal to a quarter of the floor height. While the full floor height fall represents the real case and a measure for the upper limit of the kinetic energy affecting the impact, the latter is aimed for comparison at a relatively very low fall height that represents 25% of the kinetic energy of the former case. The following analysis will show that even under this moderate impact energy, severe conditions already develop, and this adds to our understanding of the examined problem. These velocities are assigned to the falling slab just prior to impact, while the impacted slab is at rest.

Due to the short duration response (see section “Reference case”), perfect bond is assumed between the rebars and the concrete in both slab and column. During the very short time until severe damage of the surrounding concrete is developed there is hardly any slip development, and right after there is hardly any connection and rebar-concrete interfacial interaction as will be shown below.

The analysis

The entire investigation has been performed by an explicit FEM analysis using commercial software ANSYS AUTODYN (2005), implementing a Lagrange approach for all elements of the problem. A fine mesh of three-dimensional solid 8 nodes elements was used for the concrete slab domain, and beam elements were used for the longitudinal rebars and for the shear reinforcement. The elements sizes were examined to obtain a converged solution. The slab reinforcement mesh, the shear reinforcement bars and the columns reinforcement are not connected to each other, and load transfer is carried through the rebars interaction with the concrete. AUTODYN models were applied for both concrete and steel, as described in more detail below. The erosion procedure uses AUTODYN built-in algorithm and is defined as “effective plastic strain” for the concrete and as “on failure” for the steel rebars. The constitutive relationships for concrete that were used in the analysis are the P~α model for the equation of state and the RHT model for the deviatoric behavior. The implementation of these models for concrete in the present analysis is justified in (Riedel, 2004; Riedel et al., 2008). The linear EOS and a bi-linear elastic plastic deviatoric model with failure at given effective plastic strain were used for the steel rebars. Both slabs were fully modeled and their responses were analyzed. The velocity of the falling slab was assigned to it prior to impact. The type of contact between the slabs (Lagrange/Lagrange) is the “External gap” (ANSYS AUTODYN, 2005) with automatically calculated gap size of ~0.707 mm.

Material properties

Table 1 presents the data used for concrete and for steel in most simulations (ANSYS AUTODYN, 2005).

Table 1.

Material properties for numerical simulations.

Concrete^a		Steel^b
Property	Data	Property	Data
Reference density	2750 kg/m³	Reference density	7860 kg/m³
Bulk modulus	35.27 GPa	Bulk modulus	167 GPa
Shear modulus	16.7 GPa	Shear modulus	77 GPa
Compressive strength	35 MPa	Yield stress	400 MPa
Tensile to compressive strengths ration	0.1	Modulus of linear hardening	0.667 MPa
Shear to compressive strengths ration	0.18	Hardening	Isotropic
Damage constant	0.04	Failure	Plastic strain
Damage constant	1.00	Plastic strain on failure	8%
Erosion strain	2	Erosion	On failure

35 MPa cylindrical strength.

Type S400.

Effect of the mesh size and the erosion value

The effect of the FE mesh size on the solution convergence is examined through analysis of several mesh sizes at low impact velocity V₀ = 3.875 m/s (fall height of 0.75 m). This study is carried out on the 14 cm thick 400 cm span slab with the reinforcement that is specified above. This case shows damage of the slab-column connection without failure (see below) and is therefore suitable for this analysis. Figure 11 presents the sensitivity of the peak deflection at point A and its dependence on the mesh size. Point A is located at the center of a slab field, at mid distance between two adjacent column rows (see Figure 3(a)), that is the corner of the computational domain (Figure 3(b)).

Figure 11.

The mesh size effect on the deflection: (a) solution convergence and (b) limit cases time histories.

Five different cases were studied to examine the mesh size issue:

(1) Two coarse meshes with 40 × 40 × 28 mm solid elements in the concrete slab domain (4 or 5 elements along the slab thickness), and 50 × 50 × 50, or 40 × 40 × 28 mm solid elements for the concrete in the column domain, and respectively 50 or 40 mm long beam elements for the rebars in the slab and the column.

(2) Three fine meshes with 25 × 25 × 14, 20 × 20 × 14, and 10 × 10 × 7 mm solid elements in a 80 × 80 cm zone of the slab centered at the column axis (including the column within the slab depth), where the rest of the slab and column the elements mesh in the slab gradually increases with increasing distance from the column axis up to 70 × 70 × 14 mm for the 25 and 20 mm meshes at the central part and up to 50 × 50 × 7 mm for the 10 mm central mesh. The entire slab is meshed with 10 or 20 elements along the slab thickness respectively. The rebars were meshed accordingly to the slab and column meshing.

Figure 11(a) shows that the calculated peak deflection strongly depends on the coarse mesh size, but shows a converging solution with the fine mesh size. Using a fine mesh of 20 × 20 × 14 mm shows a converged solution, that is not different from a more refined mesh of 10 × 10 × 7 mm. Therefore, the 20 × 20 × 14 mm mesh was chosen for the following analysis.

Figure 11(b) compares the deflection time history for a representative 50 × 50 × 35 mm coarse mesh with the response using a 20 × 20 × 14 mm fine mesh. The major difference is observed in the peak deflection magnitude and the time of development as well as the entire post peak vibratory motion, that is considerably nilder with the fine mesh analysis.

The erosion effect has been examined for the concrete elements in the selected “fine mesh” configuration, and it shows that selection of an erosion effective strain that is lower than “2” (the value that was adopted for this analysis) is not affecting the solution. Increasing the erosion effective strain beyond “2,” yields non-physical results and when this parameter equals to “2.5” the analysis is terminated prematurely due to appearance of “corrupted” elements in the slab nearby the column.

Validation of the computational model

In order to validate the method of analysis, a numerical analysis was carried out in comparison with available test data. Unfortunately, there are no reported studies on monitored tests where a falling slab impacts a slab underneath. Therefore, a close problem of a drop weight on a reinforced concrete plate at similar impact velocities is analyzed, which produces local dynamic punching shear, to demonstrate the capability and the good predictions obtained by the proposed analysis method in the case of a similar problem.

To achieve this goal, impact tests performed on RC plates are selected (Bhatti et al., 2011; Muttoni et al., 2013). The impactor (hammer) having a cylindrical steel flat nose hit the central part of the plate at velocities that are similar to the impact velocities of the collapsing slab under discussion. The objective of this study is to validate the analysis method and demonstrate its ability to calculate the local dynamic punching behavior similarly to the reported test results.

Considers a 150 mm thick, 1400 × 1650-mm net rectangular one-way clamped RC slab. A 60 mm diameter 300 kg impactor impacted the slab at the velocity of 5.0 m/s. The slab was reinforced with a single bottom mesh, including 9 rebars of 13 mm diameter in each direction. The concrete average cylindrical compressive strength, shear to compression strength ratio and elastic modulus were 24 MPa, 0.25 and 13.9 GPa respectively. The steel yield strength was 354.5 MPa. Other material parameters for the analysis are presented in the section “Material properties.” For the central part of the slab mesh we used 7 × 7 × 14 mm solid elements (10 elements along the slab thickness); with increasing distance from the slab center the mesh became gradually coarser up to 45.5 × 45.5 × 14 mm at the slab boundaries.

Figure 12 shows the experimental measured deflection time history during ~30 ms (Bhatti et al., 2011) in comparison with the present model calculated mid-span deflection.

Figure 12.

Comparison of mid-span deflection (Bhatti et al., 2011).

Very good agreement is obtained between the calculated and measured peak deflection with a difference that is smaller than 2%. This may be considered excellent considering the fact that the measured results for that comparison is based on digitized data taken from a small size figure in Bhatti et al. (2011). This impact event ended with the development of shear planes that could be uncovered in post-test autopsy of the test specimen (Figure 13(a)).

Figure 13.

Comparison of the measured and analyzed specimen punching shear damage: (a) observed cross-section damage and (b) predicted cross-section damage.

Figure 13(b) compares the punching shear planes in the test specimen with the calculated punching shear damage. Very good agreement is obtained for this complex mode of failure. This comparison demonstrated the capabilities of the proposed model to analyze dynamic punching shear problems such as occur in the event of impacting slabs, that is investigated in detail in the following sections.

Analysis of two impacting flat slabs—Low impact velocity

General

Refer to a low impact velocity of 3.875 m/s that is half of the reference velocity 7.75 m/s. This relatively low impact velocity corresponds to slab falling height of 0.75 m only. This is an introductory stage to the following analysis referring to slab falling from a floor height. It aims at providing indications on the damage that already develops at a considerably lower impact. Falling from a floor height represents the real scenario and provides an upper limit of the impact velocity, which is determined disregarding drag and constraints that may attenuate the velocity during the slab motion downwards. The considerably lower impact velocity in this section may represent an enormous energy loss (75%) in case of falling from a floor height, which is intuitively below a realistic lower limit, yet it demonstrates that already at such lower impact velocity severe damage at the verge of failure already develops.

Effect of the slab span

Consider a 14 cm thick flat slab with a 400 cm span that is supported by columns with a cross section of 20 × 20 cm, and for comparison consider a 24 cm thick flat slab with a 800 cm span that is supported by columns with a cross section of 40 × 40 cm. For the latter case of a large slab the 40 × 40 × 24 mm (10 elements along the slab thickness) size elements were used in the column and in part of the slab surrounding the column (160 × 160 cm zone centered at the column axis). Beyond that region the mesh size gradually increases up to 140 × 140 × 24 mm. The rebars were meshed accordingly to the slab and column meshing.

Figure 14 shows the calculated mid-span (point A) velocities and deflections time histories for these two cases. A major difference between the two cases is observed. The shorter span slab reached a final stable state of damage characterized by reaching a zero velocity (Figure 14(a)) and a large permanent deflection (Figure 14(b)). The slab connection region reached a severe damaged state that is close to the limit state of failure, nevertheless, complete punching shear failure has not developed.

Figure 14.

Span effect on damage form a low velocity impact: (a) velocity and (b) deflection.

In the case of the larger span slab, a severe dynamic punching shear occurred and the connection completely failed; this is observed in the relatively low residual velocity (of about 0.5 m/s (Figure 14(a)) and the increasing large displacement (Figure 14(b)). Figure 15 shows the local complete failure of the 800 cm span slab at the connection (Figure 15(a)).

Figure 15.

Slab failure for 800 cm span slab: (a) cross-section (damage) and (b) bent-up rebars (effective plastic strain).

Figure 15(b) shows the rapture of all the bent-up rebars, while in the case of the shorter span slab the bent up rebars strongly deform however maintain their continuity and are able to support the concrete slab after the impact event without rapture, as will be described in detail below.

These comparative results demonstrate that the shorter span slab undergoes a less severe damage compared to the large span slab upon the same low velocity impact. Therefore, in the following we shall focus on the shorter span slab, with the logics that if it shows severe damage, it means that larger span slabs will experience even larger damage. However, if the shorter span impacted slab will survive the impact and will be found safe, the larger span slab will be then examined, to determine whether its larger expected damage keeps the slab safe as well, or not. This strategy refers to the high velocity impact which represents real conditions of a floor falling and obviously heavier damage, while the low velocity impact is examined for comparison, but cannot serve for a conclusive opinion on the standards suitability to guarantee the connections resilience.

The effect of major parameters

This section refers to a slab with a 400 cm span and 14 cm thick slab that is supported by 20 × 20 cm columns. In addition to the top and bottom reinforcing meshes, three local reinforcing types are examined. In the shorter span slab, local shear reinforcement at the connection is provided either in the shape of three 45° bent up rebars in each direction with/without integrity rebars (Figure 7) or alternatively three contours of links having the shape of stirrups (Figure 8); in addition, in either case integrity bars are provided as detailed above. The bent up rebars and the contour links are equivalent in their static shear resistance. Figure 16 shows the calculated mid-span (point A) time histories of the velocities and deflections for these three cases.

Figure 16.

Motion time histories for point A—14-cm thickness span: (a) velocities and (b) deflections.

This comparison shows that the impacted slab velocity at point A sharply drops to zero within 50 to 70 ms (milliseconds) and then fluctuates around zero and attenuates after some 200 ms. In all three variants large peak deflections are developed that later attenuate to a somewhat lower permanent deflection. For the bent up rebars we also examine the contribution of the integrity rebars by comparison the slab response with or without the integrity rebars.

The positive effect of the integrity reinforcement is not observed during the first 45 ms (where a deflection of ~100 mm is already developed), but is clearly observed afterwards. The alternative shear reinforcement using contour links is inferior compared to the bent up rebars option having the same contribution to the shear resistance. Using of integrity rebars with the bent-up bars decrease the permanent deflection by about 10%, while, in contrary, using of contour links increase the permanent deflection by about 20%.

For all the examined cases a final stable state is reached with large permanent deflections (~70% of the slab thickness) accompanied with severe local damage, however no complete punching shear failure is noticed at this low impact velocity. Thus, the standards requirements for static design cannot avoid the severe damage but marginally prevent the complete dynamic punching failure of the column-slab connection.

The final state of the slab-column damage field for the case of 3 bent up rebars in each direction with integrity bars is shown in Figure 17(a) and (b). The damage legend shown in Figure 17(a) is common to all the following damage plots unless otherwise is specified. The size of the damaged zone is limited, and there is no damage in the slab area beyond this zone. This is clearly shown in the cross section (Figure 17(c)) where the pronounced deflection results from the shear deformation in the narrow-damaged zone.

Figure 17.

Damaged zone for low impact velocity: (a) top view, (b) bottom view, and (c) cross-section.

Figure 18 shows the deflection-time history at different gauge locations in the slab and around the column, from which it is clear that the slab surrounding the column is not undergoing any bending but rather undergoes a rigid body displacement with respect to the column.

Figure 18.

Slab motion under low-velocity impact: (a) Gauges location on the slab and (b) slab points deflection.

The longitudinal reinforcement in the slab meshes strongly deforms and local plastic strains are developed around the column contour. The rebars are not raptured though (Figure 19(a)), although the effective plastic strain almost reaches the ultimate strain capacity of 8%. This means that a slightly higher impact velocity may cause these rebars fracture.

Figure 19.

The effective plastic strain in the reinforcement at the final state: (a) longitudinal reinforcement and (b) shear reinforcement.

This is even more pronounced with regard to the bent up rebars (Figure 19(b)) which have undergone large plastic strains and almost all bent up rebars have been raptured. The legend of the effective plastic strain shown in Figure 19(b) is common to all of the following plots unless otherwise is specified. It is interesting to notice that the large plastic strains in the bent up rebars develop along the diagonal and the top parts of these rebars. At this low impact velocity, a few of the shear bent up rebars are fractured and the others are close to fracture. It is interesting to note that the column reinforcement for such low-velocity impact remains elastic.

Analysis of two impacting flat slabs—High impact velocity

Reference case

Consider now the case of the high impact velocity of 7.75 m/s, which represents the impact of the falling slab from a typical floor height of 3 m.

We start with the reference case of a 14 cm thick 400 cm span slab, supported by a column with a 20 × 20 cm cross section, that is designed to satisfy the standards design requirements as specified above. The slab is reinforced with top and bottom reinforcing meshes, three bent up shear rebars in each direction and two integrity rebars in each direction, as described above.

Although this solution fully satisfies the standard requirements for punching shear design, the impact with a falling slab from a floor height results in a total failure that is characterized by full damage in shear of the concrete slab in a limited area surrounding the column and within the column core, where the rest of the slab remains undamaged (Figure 20).

Figure 20.

Connection failure and surrounding damage: (a) top view of the concrete at 20 ms, (b) cross-section at 20 ms, (c) top view of the concrete at 65 ms, and (d) cross-section at 65 ms.

The total failure develops within a short duration of less than 20 ms after impact. Figure 20(a) and (b) show the fully developed damage in the slab around the column and the relative displacement with respect to the column that is fully due to the slab displacement at the column vicinity.

Figure 20(c) and (d) present the position of the slab with respect to the column 65 ms after impact, when the slab has been displaced downwards more than twice the slab thickness. The concrete at the column core has been fully crushed and damage in the column concrete is observed both above the slab top surface and at the column bottom. Damage is also observed at the bottom of the column. Note the upper part of the column is not aligned with the lower part of the column, that is similar to the observations from real cases (Figure 1). The damaged zone in the slab extends over a somewhat larger area compared to the case of the low velocity impact, however beyond the damaged zone the slab remains flat and undamaged. The severe damage of the connection includes the column core in the slab’s domain, in which the concrete is fully crushed and removal of the 100% crushed concrete in the core shows the column reinforcing cage skeleton only (Figure 20(d)).

Isolating the reinforcement from the entire slab allows a close inspection of the reinforcement damage. Figure 21(a) shows the fractured longitudinal rebars, both the reinforcing meshes close to the column circumference and the integrity rebars. Figure 21(b) shows the fractured bent up rebars, where the diagonally inclined part is the most stressed zone where fracture finally occurred. The column reinforcement undergoes plastic strain up to 3% in the connection zone with the slab (Figure 21(c)).

Figure 21.

Reinforcement damage and failure at the connection: (a) slab longitudinal reinforcement, (b) slab shear reinforcement, and (c) column reinforcement.

Figure 22 shows the velocity time history of point A, where a residual velocity of ~6.6 m/s characterizes the post failure stage of the 800 cm span slab, and a somewhat lower residual velocity of ~ 6.2 m/s in the case of a 400 cm span slab.

Figure 22.

Time histories for 400 cm and 800 span slabs.

It is worth noting that in the case of the 800 cm span slab, where the connection is heavily damaged in shear, bending damage (~10%) in strip 1 was also observed in both directions (Figure 23). Such bending damage has not been developed in the shorter span slab case.

Figure 23.

Failure of 800 cm span slab: (a) concrete damage (top view), (b) bent-up bars (eff. plast. strain), and (c) cross-sectional damage.

To summarize the high velocity impact results, we find that in both shorter and larger span flat slabs total failure occurs at the slab-column connection. It is characterized by severe concrete damage and crushing accompanied with rupture of all the reinforcing bars, both longitudinal and bent up rebars. This yields a complete destruction of the slab-column connection and detachment of the slab from the column. The impacted slab loses its column support and begins its downward falling.

The clear conclusion is that the standards requirements for punching shear design aim at providing safe design under specified static loads, but cannot avoid the dynamic punching failure due to slabs impact, and a total failure of the column-slab connection occurs under such impact conditions.

Let’s examine the evolution of this failure in more detail and follow the development of stresses and strain in both concrete and rebars in the connection zone from the instant of impact. We refer to a 400 cm span flat slab and focus on key stages of failure/fracture development (Figure 24). Figure 24(a) shows the growth of the effective plastic strain in the rebars with time from the instant of impact until yield of the different reinforcing bars. The bent up rebars reach the yield stress level after 1 ms. During this very short time, the surrounding concrete reached a damage level of about 60% and disintegration of the rebars and concrete occurs. The slab downward displacement then was 0.4 cm. Full concrete damage occurred ~1.5 ms from the instant of impact.

Figure 24.

Effective plastic strain of the reinforcement: (a) rebars’ yielding and (b) rebars’ failure.

At 3 ms after impact the bottom mesh rebars and the integrity rebars yield, where the slab displacement was about 2 cm. The top layer reinforcement yield somewhat later after ~7.2 ms.

It turns out that force transfer from the rebars to the concrete through bond occurs during the time before yielding only. This is an extremely short time during which the tensile stress in a rebar elevates from zero to its yield stress magnitude, and the bond stresses are transferred at a correspondingly high strain rate, while the slab deflection is very small. The bond-slip stiffness is assumed very high and an ideal bond connection is justified.

The rupture of the bent up rebars occurs ~14 ms after impact (Figure 24(b)). The longitudinal reinforcement in both bottom and top meshes as well as the integrity rebars fracture at about 19 ms after impact.

Thus, within an extremely short time of a few milliseconds, the rebars reach the yield state at the critical points and their rupture develops shortly afterwards. During that short time the impacted slab deflection is very small.

These results indicate that the integrity reinforcement, that is inserted in order to support the slab in the post punching stage, when it develops a significant displacement, is not surviving the impact stage and yields within 3 ms and ruptures within less than 20 ms. Therefore, it is unable to provide a stable support to the slab and fulfil its expected function. This is a key finding, among others, that demonstrates that the impact behavior is entirely different than the static behavior. The clear conclusion is that the static load design guidelines are inadequate to the impact scenario.

It is interesting to examine the impact event and the momentum transfer between the impacting slab to the impacted slab. Although the slabs material is brittle-plastic, the analysis shows that an almost perfectly elastic collision takes place between the slabs when a total impulse transfer occurs within less than 200 ms as shown in Figure 25.

Figure 25.

Collision between the slabs and momentum transfer.

Following this stage, the impacted slab’s velocity is identical to the falling slab velocity prior to impact. It will be shown later that the energy absorbed by the impacted slab during its response to this impact is relatively small with respect to the imparted kinetic energy imparted during impact, and the residual velocity of the detached slab is still high (about 75% of the initial velocity). This means that the impacted slab that is fully detached from the column, starts its falling downward with a significant initial velocity, and therefore will reach the slab underneath at a considerably higher velocity compared to the velocity that caused its failure. The catastrophic outcome result is obvious.

The effect of major parameters

General

Following the analysis of the reference case and observing the total failure caused, it is interesting to examine possible design alternatives of several key parameters and analyze their effect on the overall response of the impacted slab and on the connection resilience. The comparison will focus on the velocity-time history of point A of the slab (Figure 3). The major parameters that will be examined in the 14 cm thick slab are: the number of bent up rebars, the length and ductility of the bent up rebars, the contribution of the integrity rebars, the contribution of the contour links, the effect of the column cross section dimensions and the effect of the slab thickness.

Comparison of different solutions based on the 3 bent up rebars alternative

In this comparison the following cases are examined (Figure 26):

The reference case (including 3 bent up rebars + integrity reinforcement)

Three bent up rebars without integrity reinforcement

Three long bent up rebars with and without integrity reinforcement (the long rebar is obtained by 2 bent up bars which are shifted with respect to each other by half of the length (Figure 27).

Three bent up rebars and integrity reinforcement made of a ductile steel (with an ultimate strain of 20%)

Figure 26.

Variations of the 3 bent up rebars solution.

Figure 27.

Long bent up rebars: (a) general view and (b) side view.

Figure 26 shows that all the examined cases exhibit a rather similar time history with relatively small differences. The signal starts with the transferred velocity, at the magnitude of the impacting slab’s velocity. Velocity is increasing at the midpoint and after ~10 ms it is somewhat decreasing, and then stabilizes at the velocity level between ~5.8 and 6.3 m/s, which occurs at a time that correspond to time when full failure of the connection occurs.

The integrity rebars do not contribute to the resistance, due to their early fracture. Increasing length or ductility of the bent up rebars has no pronounced effect.

The failure of the slab-column connection for the ductile rebars case (20% ultimate strain) is presented in Figure 28. The observed damage is very similar to the damage in the reference case, although no rupture occurs in the rebars and the slab is hung on the column.

Figure 28.

Connection failure and surrounding damage—case of ductile rebars: (a) top view of the concrete at 20 ms, (b) cross-section at 20 ms, (c) top view of the concrete at 66 ms, and (d) cross-section 66 ms.

Comparison of the reference shear solution with alternative solutions

In this comparison the following cases are examined:

The reference case (3 bent up rebars)

A larger number of bent up rebars (7 rebars in each direction with integrity rebars).

Comparison with the solution of contour links and integrity rebars.

Figure 29 shows that increasing the number of bent up rebars from 3 to 7 in each direction, has minor effect on the response. Replacing the bent up rebars with contour links (stirrups) does not improve the response either. It even causes a velocity increase during the time interval between 9 to 13 ms, and does not affect the residual velocity (6.1 m/s compared to 6.3 m/s).

Figure 29.

Different shear solutions.

Examination of a 16 cm thick slab

This examination stems from the fact that according to the standards in such thick slab the concrete satisfies the full shear capacity demand and there is no need of any special shear reinforcement (Figure 30). This case is examined and compared with the following alternatives:

Shear bent up rebars without integrity reinforcement

Integrity reinforcement only without shear bent up rebars

Shear bent up rebars and integrity reinforcement.

Variation of the reference case (no shear reinforcement): the same slab is supported by a 40 × 40 cm column.

Figure 30.

16 cm thick slab.

For this case we used 20 × 20 × 16 mm (10 elements along the slab thickness) elements in the column and in part of the slab surrounding the column were used (80 × 80 cm zone centered at the column axis). Beyond that region the mesh size gradually increases up to 70 × 70 × 16 mm. The rebars were meshed accordingly to the slab and column meshing.

Figure 30 shows similar trends for all the examined alternatives with the 16 cm thick slabs. The shear reinforcement as well as the integrity bars do not affect significantly the velocity-time history and do not affect at all the residual velocity after 19 ms. It may be noted that the residual velocity (~7.1 m/s) is slightly higher than for the 14 cm thick slab.

This behavior of the 16 cm thick slab is unexpected, because the static design criteria for punching shear indicate that no shear reinforcement is required for a safe design and the increased thickness is expected to enhance the punching shear resistance. The obtained results however clearly demonstrate the weakness of this slab to withstand the impact load, and the total failure that is developed as a result, with no advantage of the 16 cm thick slab compared to the 14 cm thick slab. These conclusions are also valid when bent up rebars and integrity rebars are added to this slab.

Following these results, a following analysis has been carried out with a 21 cm thick slab, and similar results were obtained. This demonstrates once again the major differences between the static approach presented in the modern standards, and the present impact scenario. In the static case, increasing the slab thickness considerably contributes to the shear resistance and despite the added dead weight, the slab shear capacity and its resistance to punching increases. On the contrary, increasing slab thickness does not improve the impacted slab resistance to the imparted impact. This is due to the fact that increasing slab thickness means increasing slabs masses, and therefore the impact of two thicker slabs means a larger momentum and a higher kinetic energy that is transferred to the impacted slab and is proportional to the increasing slab thickness. Therefore, increasing thickness cannot improve the impact punching shear resistance whatsoever.

The failure of the slab-column connection for the 16 cm slab with 3 bent up and 2 integrity bars in both directions is presented in Figure 31. The damage in this case is very similar in general to the damage developed in the 14 cm slab.

Figure 31.

Connection failure and surrounding damage in a 16 cm slab: (a) top view of the concrete at 20 ms, (b) cross-section at 20 ms, (c) top view of the concrete at 50 ms, and (d) cross-section at 50 ms.

All the above examinations show that the shear reinforcement contributes to the slab resistance and even to its survivability only at low impact levels. However, when high impact is considered, the shear reinforcement is incapable of significantly contributing to the resistance and prevent failure.

The effect of the column size

This examination compares the behavior observed for the reference column (20 × 20 cm) with a wider supporting column (40 × 40 cm). The following cases are examined and compared with the reference case:

Shear bent up rebars and integrity reinforcement—14 cm thick slab, 40 × 40 cm column.

Contour links and integrity rebars in both cases (20 × 20 and 40 × 40 cm columns)—14 cm thick slab.

Integrity rebars only in both cases (20 × 20 and 40 × 40 cm columns)—16 cm thick slab.

Figure 32 shows that the column size almost does not affect the residual velocity for the thicker slab (16 cm) and slightly reduces the permanent velocity for the 14 cm thickness slabs: from 6.5 to 5.8 m/s (12%) for bent up shear bars and from 6.1 to 5.7 m/s (7%) for the contour links (stirrups). This indicate a positive effect of the column size, however rather large size columns are needed to have a considerable effect on reducing the residual velocity and in the examined scenario with a considerable residual velocity, this direction does not seem realistic.

Figure 32.

Column size effect.

At a glance it may be concluded that at about ~20 ms the residual velocity is almost stabilizing. For all the examined cases the slab-column connection has failed with some difference of the residual velocity; at about 20 ms the residual velocity varies between ~5.7 and 7.2 m/s. This means that this is the initial velocity of the impacted slab, and as it has been detached from the supporting column, it starts its motion downwards with this initial velocity. This will result a considerably higher impact velocity compared to the impact velocity which caused total failure of the impacted slab connection, as already mentioned above.

Conclusion

This paper investigates the dynamic punching shear scenario of two impacting slabs and examines the standards design criteria and their suitability to prevent progressive collapse. The occurrence of a punching shear failure of slab-column connections in a flat slab leads to detachment of the slab from its supporting columns. The detached slab starts its falling downwards and leads to its impact with the slab underneath. This stage of slabs impact is critical to the survivability of the structure: if the impacted slab is unable to withstand the impact, a progressive collapse failure follows, accompanied with major structural damage and casualties. Modern standards provisions for safe design to prevent punching shear are based on static load considerations. This paper examines the suitability of these standards design criteria to the resilience of the slabs structure in the case of an impacting slabs scenario. An advanced detailed numerical model is developed to represent the impacting slabs and the column details including their reinforcement. The nonlinear inelastic impact analysis calculates the response and evaluates the extent of damage to the concrete and the rebars. Several major parameters affecting the dynamic response of this connection are examined.

Analysis of low impact velocity resulting from a falling height of a quarter of the floor height has been conducted, and relatively short span and large span slabs were analyzed. The damage in the larger span slabs was found greater. In both cases it was found that the concrete surrounding the column undergoes major damage and shear deformations, and the rebars undergo large plastic deformations but remain intact and able to carry the slab and prevent the total collapse. It was concluded that the standards requirements for static design marginally avoid the dynamic punching failure of the column-slab connection at that limited impact velocity for that slab span.

Falling from a floor height level, yields severe impact conditions causing total failure of the slab-column connection. The concrete is completely damaged and crushed and the longitudinal and the bent up rebars rupture. The connection fails and the slab starts falling downward. The connection damage results for 400 and 800 cm span slabs are not much different, however, the residual velocity of the larger span slab is somewhat higher. In addition to the local punching failure, bending damage (~10%) is observed in the larger slab, whereas no bending damage occurs in the shorter span slab.

The rebars reach the yield point within a few milliseconds. During this extremely short time, the impacted slab hardly starts developing its displacement. Concrete at these critical zone develops major damage within 1 ms. As force transfer from the rebars to the concrete through bond occurs only during that short time, a perfect bond assumption may be adopted. During the entire event the slab beyond the connection zone responds almost like a rigid body.

The severe damage of the connection includes the column core in the slab’s domain, in which the concrete is fully crushed in the column reinforcing cage and plastic strains develop in the reinforcing cage in this domain. This critically weakens the column capacity to support higher level slabs in cases where the impacting slab is located at an intermediate level, and it may cause a catastrophic collapse of the entire building.

The entire failure of the reinforcement in the connection zone occurs within less than 20 ms. As the integrity reinforcement ruptures during this early time, it turns out that this reinforcement, is not surviving the impact stage and therefore cannot carry the slab load in the post punching shear stage. The bent up rebars, fail at the diagonally inclined part and therefore extended anchorage cannot enhance its performance.

Increasing the number of bent up rebars in attempt to increase shear resistance, has a minor effect which is not affecting the connection failure. Replacing the bent up rebars with contour links (stirrups) does not improve the response.

The analysis results showing the short duration to reach local failure, during which minor displacements are developed, and most of the slab remains flat and undamaged, cannot support flexural based models to analyze dynamic punching shear failure.

All the examined cases of the different parameters show a rather similar time history with relatively small differences that end with a relatively high residual velocity. This residual velocity turns to be the initial velocity of the impacted slab upon its downwards motion. This will result in a considerably higher impact velocity of the impacted slab with the slab underneath. This is a clear scenario of catastrophic progressive collapse, with higher velocity impact occurrences with the progress of the process.

Analyzing the cases of 16 cm thick and 21 cm thick slabs shows similar results. Opposed to the case of a slab subjected to static loading, where a thicker slab provides higher shear capacity to withstand punching shear, the impact analysis shows that increasing thickness has no contribution to reduce the damage and prevent failure. This is because increase of the slab thickness means increase of the slab mass, and therefore the impact of two thicker slabs means a larger momentum and a higher kinetic energy that is imparted to the impacted slab.

The present study provides new insight into the complex impact response of slabs and examines different parameters affecting the response. The new findings indicate that the standards that are based on static loading considerations cannot provide the resilience of flat slabs connection to impact loading. Further analysis is ongoing to examine ways to improve the slabs resistance under these conditions.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by a joint grant from the Ministry of Defence and the Centre for Absorption in Science of the Ministry of Immigrant Absorption and the Committee for Planning and Budgeting of the Council for Higher Education under the framework of the KAMEA Program.

ORCID iDs

David Z Yankelevsky

Vladimir R Feldgun

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