One-way shear/beam shear is concerned with resistance by a singular plane such as beams and slabs. The punching/two-way shear resistance is across peripheral portions of the member. For example, this could be a column resting flat slab or column supporting a footing. One-way shear is resisted critically at a distance equivalent to depth and not the support of the member. Along corresponding planes, the beam shear tends to shear off members in the direction of the force for shearing. A two-way shear, on the other hand, can punch into the member slab, triggering failure. Punching/ two-way shears attain criticality at distances of around half the member’s depth (slab divided by footing). With respect to one-way and punching shear for flat-plate slabs, the following research findings have been observed.
2.2.1 One-Way Shear Resistance of Flat Plate
Shear failures have been actively researched since decades. Since the first model for beams with stirrups and the beam shear studied by Talbot between 1904 and 1909, researchers have examined the failure of flat-plate slabs at ambient temperatures from various angles. Models for one-way shear resistance range across the following. Compression field theory proposed by Collins (1978 in Lantsoght et al., 2015) utilizes a stress-strain bond for cracked concrete. Once the concrete cracks, it does not carry tension, resulting in diagonal compression fields, as per the theory. However, concrete transmits stresses in various ways such as existing and new cracks, aggregate interlocking forces and variations of bond stresses (Lansoght et al., 2015).
Another popular model is the Critical Shear Crack Theory by Muttoni (2003 in Lansoght et al., 2015) assuming one-way shear strength of members sans transverse reinforcement is governed by roughness and wideness of shear cracks. The strain is proportion to the critical shear crack’s width, and it is influenced by spacing and aggregate size between reinforcement layers. Yet another approach is based on the upper-bound plasticity theory, where yield line for shear critical cracks is studied (Nielsen & Hoang, 2011 in Lansoght et al., 2015). In a lower bound perspective, strut-and-tie models are represented. As members become larger and more lightly reinforced, codes like ACI require modification.
Essentially, one-way shear occurs at a zone, where transverse extent is measured from the column’s face, till a distance equivalent to a certain preestablished and measured value. Shear reinforcement serves to make structures safe. The failure in one-way shear occurs from edge-to-edge through action of shearing off. In members without shear reinforcement, according to the ACI 318-19, shear is assumed to be resisted by the concrete. In those with shear reinforcement, portion of the shear strength is held to be provided by concrete and the remaining by shear reinforcement.
2.2.2 Two-Way Shear Resistance of Flat Plate
Typical flat plate punching, or two-way shear failure is characterized by slab failing at the column’s intersection point. This enables the column to emerge through the surrounding slab’s portion. While assessing thickness of flat plates at column-slab meeting point, accurate prediction of punching shear strength is critical
Figure. Punching shear failures within flat plate slabs
Punching shear is the result of concentrated support interactions within a flat slab causing its top surface to be perforated. While this follows flexural failure usually, punching shear triggers progressive collapse of the structural design’s safety. Punching shear resistance is enhanced by boosting concrete through structural strength reinforcements. But in high strength concrete, brittleness and spalling can be issues. Arna’ot et al. (2017).
Flat plate systems of RC slabs supported on columns sans the use of supporting beams, drop panels or capitals due to affordability, construction ease and architectural flexibility afforded from absence of the beam, often experience increased slab shear stresses in column connections. As flat plates can be susceptible to brittleness and governing modes of failure if not designed properly on account of punching shear failures, Goh and Hyrnyk (2018) have examined how numerous studies have investigated punching shear resistance of flat plates. Considerable research tests isolated reinforced-concrete slab-column connections at singular column locations spanning perimeters of slab assemblies or slabs supported within edges and loaded at the point of the column regions. In contrast to this, multibay specimens are difficult to test. However, the research has erroneously concluded isolated slab specimens are indicative of real-world flat plate systems (Sherif, 1996 as cited in Goh and Hyrnyk 2018). Other studies have found the need to examine compressive membrane action and moment redistribution (Einpaul et al., 2016).
2.2.2 Flexural Resistance
Flexural strength/resistance impacts punching resistance without shear reinforcement (Sacramento et al., 2012). The flexural reinforcement ratio is the ratio between the area of tensile flexural reinforcement and the area of concrete, which is the outcome of the product of the effective slab depth by width. Increasing the flexural reinforcement ratio raises the compression zone, lowering the chances in cracks within the slab-column connection on account of bending. It facilitates the formation of mechanisms for transmission of shear forces. Thickness of bending cracks is lowered, facilitating the transfer of forces via aggregates interlocks, increasing the dowel effect. Considerable research found in studies on flat slabs 150 mm thick that punching strength increased when flexural reinforcement ratio was raised (Marzouk & Hussein, 1991 in Chetchotisak et al., 2018). Sherif and Dilger (2000) also conducted research which suggests punching resistance is influenced by a function proportional to the tensile flexural reinforcement ratio’s cube root (Sacramento et al., 2012). figure below is the minimum length of flexural reinforcement in a flat-plate slab (Darwin et al., 2010).
Minimum length of reinforcement in slab without beams (Darwin et al., 2010)
2.3 ACI Code Methods of Analysis for Flat Plate Slab at Ambient Temperature
The ACI Building Code recognizes two methods: Direct Design Method and Equivalent Frame Method. These calculate an equivalent statical moment for the entire slab’s width and then distribute that moment over the width and length of the slab. The analysis is conducted in both directions for service and strength loads (Dolan & Hamilton, 2019). Live and full dead loads are utilized in each analysis, whereby full loads in each direction assume mid-slab structural compatibility deflections are maintained (Dolan & Hamilton, 2019).
2.3.1 Direct Design Method
Moments in two-way flat plate slabs can be calculated using this semi-empirical method but there are some restrictions. Panels need to be regular in size. Span ratios of longer to shorter ones should not exceed 2. Furthermore, there should be at least 3 continuous spans in each direction (Darwin et al., 2010). Span lengths in directions should not vary consecutively by a value greater than 25% of the wider span. Columns offset percentage should be 10 percent of the span (Darwin et al., 2010). It should be in the direction of the offset from either axis to centerlines at maximum values for successive columns. Loads on the basis of gravity as well as unfactored live loads should not be more than unfactored dead loads by a value greater than twice. According to the ACI Code 318-19. Such a method, therefore, limits itself to slab structures with equally distributed loading and columns that are spaced equidistantly.
2.3.2 Equivalent Frame Method
This method is derived based on the assumption that analysis would be carried out using the moment distribution method. Through this method, the structure is divided for analysis into frames continuously centered on the column lines and longitudinally, as well as transversely extending as shown in Figure.
Figure. Two-way Flat-Plate Floor (Darwin et al., 2010)
In the above two-way flat-plate floor example, the structure is the same in every direction, permitting the design for one direction to be used for both. At moments of distribution the moments can be calculated. Flat plate structures should be calculated for stiffness assuming prismatic members, cutting down on heightened joint region stiffness, for this affects shears and design moments in minimal ways. Following this, slab spans and column stiffness would be calculated. The torsional deformation of transverse slab strips working as supporting beams forms the basis of equivalent column stiffness calculations.
Effective depth for all panels will be the same, therefore larger negative moments found for panels B will control the design. Moments would also be distributed laterally across the slab width. Crucial design aspects regarding flat plates constitute designs for punching shears per column requiring additional shear reinforcement. Such a transfer to columns of the unbalanced moments needs more flexural bars in column strips’ negative bending regions, or adjusting negative steel spacing, more so at corners and external columns as well (Darwin et al., 2010).
Figure. Design moments and shear for flat plate floor interior panel C:
(a) moments and (b) shear (Darwin et al., 2010).
The equivalent frame method comprises torsional stiffness of slab edges and edge beams distributing the statical moment to adjust for torsional edge stiffness. Furthermore, equivalent frame analysis results in column strips, portions of the slab 1/4th of the span length either side of the column, middle strips and the central one half of the slab bounded by strips of columns (Dolan & Hamilton, 2019).
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