# Weekly Course Distribution

DESIGN OF TWO-WAY SLABS BY DIRECT DESIGN METHOD 4 DESIGN OF TWO-WAY SLABS BY THE ACI CODE Direct Design Method The Code (13.6) provides a procedure with which a set of moment coefficients can be determined. The method, in effect, involves a single-cycle moment distribution analysis of the structure based

on (a) the estimated flexural stiffnesss of the slabs, beams (if any), and columns and (b) the torsional stiffnesss of the slabs and beams (if any) transverse to the direction in which flexural moments are being determined. Same types of moment coefficients have been used satisfactorily for many years for slab design. They do not, however, give very satisfactory results for slabs with unsymmetrical dimensions and loading patterns. 5

COLUMN AND MIDDLE STRIPS 6 DIRECT DESIGN METHOD FOR COLUMN-SUPPORTED SLABS 7 COLUMN AND MIDDLE STRIPS After the design moments have been determined by either the direct

design method or the equivalent frame method, they are distributed across each panel. The panels are divided into column and middle strips, as will be shown in the next figure, and positive and negative moments are estimated in each strip. The column strip is a slab with a width on each side of the column centerline equal to one-fourth the smaller of the panel dimensions l1 or l2. It includes beams if they are present. The middle strip is the part of the slab between the two column strips. The part of the moments assigned to the column and middle strips may be assumed to be uniformly spread over the strips. As will be described later, the percentage of the moment assigned to a column strip depends

on the effective stiffness of that strip and on its aspect ratio l2/l1 (where l1 is the length of span, center to center, of supports in the direction in which moments are being determined and l2 is the span length, center to center, of supports in the direction transverse to l1). 8 COLUMN AND MIDDLE STRIPS 9

SHEAR RESISTANCE OF SLABS 10 SHEAR RESISTANCE OF SLABS For two-way slabs supported by beams or walls, shears are calculated at a distance d from the faces of the walls or beams. The value of Vc is, as for beams, Shear is not usually a problem for these types of

slabs. For flat slabs and flat plates supported directly by columns, shear may be the critical factor in design. In almost all tests of such structures, failures have been due to shear or perhaps shear and torsion. These conditions are particularly serious around exterior columns. Two kinds of shear must be considered in the design of flat slabs and flat plates. These are the same two that will be considered in column footingsone-way and two-way shears (that is, beam shear and punching shear). For beam shear analysis, the slab is considered to act as a wide beam running between the supports. The critical sections are taken at a distance d from the face of the column or capital. For

punching shear, the critical section is taken at a distance d/2 from the face of the column capital, or drop panel and the shear strength, as usually used in footings, is 11 SHEAR RESISTANCE OF SLABS If shear stresses are too large around interior columns, it is possible to increase the shearing strength of the slabs by as much as 75% by using shear heads. A shear head, as defined in Section 11.12.4 of the Code, consists of four steel I or channel shapes

fabricated into cross arms and placed in the slabs, as will be shown in the next figure. The Code states that shear head designs of this type do not apply at exterior columns. Thus special designs are required, and the Code does not provide specific requirements. Shear heads increase the effective bo for two-way shear, and they also increase the negative moment resistance of the slab, as described in the Code (11.12.4.9). The negative moment reinforcing bars in the slab are usually run over the top of the steel shapes, while the positive reinforcing is normally stopped short of the shapes.

13 SHEAR RESISTANCE OF SLABS Another type of shear reinforcement permitted in slabs by the Code (11.12.3) involves the use of groups of bent bars or wires. One possible arrangement of such bars is also shown in the next figure. The bars are bent across the potential diagonal tension cracks at 45 angles, and they are run along the bottoms of the slabs for the distances needed to fully develop the bar strengths. Another type of bar arrangement that might be used is shown,

when bars (or wires) are used as shear reinforcement, the Code (11.12.3.2) states that the nominal two-way shear strength permitted on the critical section at a distance d/2 from the face of the column may be increased from 14 SHEAR RESISTANCE OF SLABS 15

SHEAR RESISTANCE OF SLABS The main advantage of shear heads is that they push the critical sections for shear farther out from the columns, thus giving a larger perimeter to resist the shear, as illustrated in next figure. In this figure, l v is the length of the shear head arm from the centroid of the concentrated load or reaction, and c1 is the dimension of the rectangular or equivalent rectangular column or capital or bracket, measured in the direction in which moments are being calculated. The Code (11.12.4.7) states that the critical section for shear shall cross the shear head arm at a distance

equal to from the column face. Although this critical section is to be located so that its perimeter is a minimum, it does not have to be located closer to the column face or edges of capitals or drop panels than d/2 at any point. 16 SHEAR RESISTANCE OF SLABS

17 SHEAR RESISTANCE OF SLABS When shear head reinforcing is provided with reinforcing bars or steel I or channel shapes, the maximum shear strength can be increased to at a distance d/2 from the column. According to the Code (11.12.4.8), this is only permissible if the maximum computed shear does not exceed along the dashed critical

section for shear shown in previous figure. In later section the subject of shear stresses may be continued with a consideration of the transfer of moments and shears between slabs and columns. Remember the maximum load that a two-way slab can support is often controlled by this transfer strength. 18 DEPTH LIMITATIONS AND

STIFFNESS REQUIREMENTS 19 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS It is obviously very important to keep the various panels of a twoway slab relatively level (that is, with reasonably small deflections). Thin reinforced two-way slabs have quite a bit of moment resistance, but deflections are often large. As a consequence, their depths are very carefully controlled by the ACI Code so as to limit these deflections. This is accomplished by

requiring the designer to either (a) compute deflections and make sure they are within certain limitations or (b) use certain minimum thicknesses as specified in Section 9.5.3 of the Code. Deflection computations for two-way slabs are rather complicated, so the average designer usually uses the minimum ACI thickness values, presented in the next few slides. 20 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS

Slabs without Interior Beams For a slab without interior beams spanning between its supports and with a ratio of its long span to short span not greater than 2.0, the minimum thickness can be taken from [Table 9.5(c) in the Code]. The values selected from the table, however, must not be less than the following values (ACI 9.5.3.2): 1. Slabs without drop panels 5 in.

2. Thickness of those slabs with drop panels outside the panels 4 in. In Table 16.1 some of the values are given for slabs with drop panels. To be classified as a drop panel, according to Sections 13.3.7.1 and 13.3.7.2 of the Code, a panel must (a) extend horizontally in each direction from the centerline of the support no less than one-sixth the distance, center to center, of supports in that direction and (b) project vertically below the slab a distance no less than one-fourth the

thickness of the slab away from the drop panel. 21 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS Slabs without Interior Beams In this table, ln is the length of the clear span in the long direction of two-way construction, measured face to face of the supports in slabs without beams and face to face of beams or other supports in other cases. Very often slabs are built without interior beams between the

columns but with edge beams running around the perimeter of the building. These beams are very helpful in stiffening the slabs and reducing the deflections in the exterior slab panels. The stiffness of slabs with edge beams is expressed as a function of , which follows. 22 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS Slabs without Interior Beams

Throughout this chapter the letter is used to represent the ratio of the flexural stiffness (EcbIb) of a beam section to the flexural stiffness of the slab (EcsIs) whose width equals the distance between the centerlines of the panels on each side of the beam. If no beams are used, as for the flat plate, will equal 0. For slabs with beams between columns along exterior edges, for the edge beams may not be < 0.8 as specified in a footnote to Table 16.1. 23

DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS Slabs without Interior Beams 24 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS Example 16.1 SOLUTION

25 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS 26 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS

28 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS Slabs with Interior Beams To determine the minimum thickness of slabs with beams spanning between their supports on all sides, Section 9.5.3.3 of the Code must be followed. Involved in the expressions presented there are span lengths, panel shapes, flexural stiffness of beams if they are used, steel yield stresses, and so on. In these equations the following terms are used:

29 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS Slabs with Interior Beams 30 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS Slabs with Interior Beams For panels with discontinuous edges, the Code (9.5.3.3d) requires that edge beams

be used, which have a minimum stiffness ratio equal to 0.8, or else that the minimum slab thicknesses, as determined by ACI Equations 9-12 and 9-13, must be increased by 10%. The designer may use slabs of lesser thicknesses than those required by the ACI Code as described in the preceding paragraphs if deflections are computed and found to be equal to or less than the limiting values given in Table 9.5(b) of the ACI Code (Table 6.1 in this text). Should the various rules for minimum thickness be followed but the resulting slab be insufficient to provide the shear capacity required for the particular column size, column capitals will probably be required. Beams running between the columns

may be used for some slabs where partitions or heavy equipment loads are placed near column lines. A very common case of this type occurs where exterior beams are used when the exterior walls are supported directly by the slab. Another situation where beams may be used occurs where there is concern about the magnitude of slab vibrations. Example 16.2 illustrates the application of the minimum slab thickness rules for a two-way slab with beams. 31 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS Example 16.2

SOLUTION 32 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS 33 DEPTH LIMITATIONS AND STIFFNESS REQUIREMENTS

34 DISTRIBUTION OF MOMENTS IN SLABS 35

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