Analysis and Design of Flat Plates

The flat plate is the simplest form of two-way slab simplest for analysis, design, detailing, bar fabrication and placing, and formwork. A flat plate is defined as a two-way slab of uniform thickness supported by any combination of columns and walls, with or without edge beams, and without drop panels, column capitals, and brackets.
Shear and deflection limit economical flat-plate spans to under about 30 ft for light loading and about 20 to 25 ft for heavy loading. While use of reinforcingsteel or structural-steel shear heads for resisting shear at columns will extend these limits somewhat, their main application is to permit use of smaller columns. A number of other variations, however, can be used to extend economical load and span limits (Arts. 9.60 and 9.61).
The ACI 318 Building Code permits two methods of analysis for two-way construction:
direct design, within limitations of span and load, and equivalent frame
(Art. 9.42). Limitations on use of direct design are:
A minimum of three spans continuous in each direction
Rectangular panels with a ratio of longer to shorter span, center-to-center of supports within a panel, not greater than 2 Successive span ratios, center-to-center of supports in each direction, not to exceed 2:3
Columns offset from centerlines of successive columns not more than 0.10 span in either direction
Specified ratio of live load to dead load (unfactored) does not exceed 2 All loads are due to gravity only and uniformly distributed over the entire panel Design Procedures for Flat Plates

The procedure for either method of design begins with selection of preliminary dimensions for review, and continues with six basic steps.
Step 1. Select a plate thickness expected to be suitable for the given conditions of load and span. This thickness, unless deflection computations justify thinner plates, should not be less than h determined from Table 9.16. With Grade 60 reinforcement, minimum thickness is, from Table 9.16, for an interior panel

Step 4. Distribute panel moments Mu to column and middle strips.
Column strip is a design strip with a width of 0.25L2 <= 0.25L1 on each side of the column centerline, where L1 is the center-to-center span in the direction in which moments are being determined (Fig. 9.28).
Middle strip is the design strip between two column strips (Fig. 9.28).
For flat plates without beams, the distribution of Mu becomes:
For positive moment, column strip 60%, middle strip 40%
For negative moment at the edge column, column strip 100%
For interior negative moments, column strip 75%, middle strip 25%
A factored moment may be modified up to 10% so long as the sum of the positive and negative moments in the panel in the direction being considered is at least that given by Eq. (9.72).

Step 5. Check for shear. Shear strength of slabs in the vicinity of columns or other concentrated loads has to be checked for two conditions: when the slab acts as a wide beam and when the load tends to punch through the slab. In the first case, a  diagonal crack might extend in a plane across the entire width of the slab. Design for this condition is described in Art. 9.47. For the two-way action of the second condition, diagonal cracking might occur along the surface of a truncated cone or pyramid in the slab around the column.
The critical section for two-way action, therefore, should be taken perpendicular to the plane of the slab at a distance d/2 from the periphery of the column, where d is effective depth of slab. Unless adequate shear reinforcement is provided, the

Step 6. When steps 1 through 5 are satisfactory, select flexural reinforcement.

Stiffnesses in Two-Way Construction

The Commentary to the ACI 318 Building Code contains references for a sophisticated procedure for computation of stiffnesses of slabs and equivalent columns to determine moments and shears by an elastic analysis. Variations in cross sections of slab and columns, drop panels, capitals, and brackets are taken into account.
Columns can be treated as infinitely stiff within the joint with the slab. The slab can be considered to be stiffened somewhat within the depth of the column.
In the direct-design method, certain simplifications are permissible in computation of stiffnesses (see Commentary on ACI 318-89).

Transfer of Unbalanced Moments

Design requirements for the transfer of unbalance moment between the slab and columns are included in the ACI 318 Building Code. Consider an exterior-edge column of a flat plate system where the unbalanced moment, Mu, resulting from gravity loads on the slab, must be transferred to the column. The unbalanced moment is transferred by flexure and by eccentricity of shear.

where  specified concrete compressive strength, psi. Æ’’c
Use of this calculation in establishing a preliminary design is a short cut, which will often avoid the need for repeating steps 1 through 5 in Art. 9.59.1, because it gives a close approximation for final design.
The minimum cantilever edge span of a flat plate so that all columns can be considered interior columns and the direct-design method can be used without tedious stiffness calculations is 4â„15 of the length of the interior span (Fig. 9.29). This result is obtained by equating the minimum cantilever moment at the exterior column to the minimum negative-factored moment at the interior column.

Bar Lengths and Details for Flat Plates

The minimum lengths of reinforcing bars for flat plates shown in Fig. 9.30, prescribed by the ACI 318 Building Code, save development (bond) computations.
The size of all top bars must be selected so that the tension development length Ld required for the bar size, concrete strength, and grade of the bar is not greater than the length available for development (see Table 9.8).
The size of top bars at the exterior edge must be small enough that the hook plus straight extension to the face of the column is larger than that required for full embedment (Table 9.9).
Column-strip bottom bars in Fig. 9.30 are shown extended into interior columns so that they lap, and one line of bar supports may be used. This anchorage, which

exceeds ACI 318 Building Code minimum requirements, usually ensures ample development length and helps prevent temperature and shrinkage cracks at the centerline.
Figure 9.24b shows weights of steel and concrete for flat plates of normal-weight concrete carrying a superimposed factored load of 170 psf, for preliminary estimates.
Provisions for structural integrity for two-ways slabs specified in he ACI 318 Building Code require all column-strip bottom bars in each direction to be made continuous or spliced with Class A tension lap splices. At least two of the columnstrip bottom bars must pass within the column core. The bars must be anchored at exterior supports. In slabs with shearheads, at least two of the bottom bars in each direction must pass through the shearhead as close to the column as possible and be continuous or spliced with a Class A tension lap splice. At exterior columns, the bars must be anchored at the shearhead.
Crack Control. The ACI 318 Building Codes requirements (Art. 9.50) apply only to one-way reinforced elements. For two-way slabs, bar spacing at critical sections should not exceed twice the slab thickness, except in the top slab of cellular or ribbed (waffle) construction, where requirements for temperature and shrinkage reinforcement govern.