5.6 T WO ‐ WAY SLABS
5.6.6 Analysis of Two‐Way Slabs by The Direct Design Method
The direct design method is an approximate method established by the ACI Code to determine the design moments in uniformly loaded two-way slabs. To use this method, some limitations must be met, as indicated by the ACI Code, Section 13.6.1.
Limitations
1. There must be a minimum of three continuous spans in each direction.
2. The panels must be square or rectangular; the ratio of the longer to the shorter span within a panel must not exceed 2.0.
3. Adjacent spans in each direction must not differ by more than one-third of the longer span.
4. Columns must not be offset by a maximum of 10% of the span length, in the direction of offset, from either axis between centerlines of successive columns.
5. All loads must be uniform, and the ratio of the unfactored live to unfactored dead load must not exceed 2.0.
6. If beams are present along all sides, the ratio of the relative stiffness of beams
in two perpendicular directions
,
must not be less than 0.2 nor greater than 5.0.137
Total Factored Static Moment
If a simply supported beam carries a uniformly distributed load w K/ft, then the maximum positive bending moment occurs at midspan and equals , where is the span length. If the beam is fixed at both ends or continuous with equal negative moments at both ends, then the total moment (positive moment at midspan)
+ Mn (negative moment at support) = (Fig.5.6-9). Now if the beam AB carries the load W from a slab that has a width perpendicular to , then , and the total
moment is , where Wu = load intensity in k/ft2 . In this expression, the actual moment occurs when l1 equals the clear span between supports A and B. If the clear span is denoted by ln, then
(ACI Code, Eq. 13.4) (5.67)
Figure 5.6-9 bending moment in a fixed – end beam.
Figure 5.6-10 Critical sections for negative design moments. A-A, section for negative moment at exterior support with bracket.
The face of the support where the negative moments should be calculated is illustrated in Fig.5.6-10. The length l2 is measured in a direction perpendicular to ln and equals the direction between center to center of supports (width of slab). The total moment M0 calculated in the long direction will be referred to here as Mol and that in the short direction, as Mos·
Once the total moment, Mo, is calculated in one direction, it is divided into a positive moment, Mp, and a negative moment, Mn, such that M0 = Mp + Mn. Then each moment, MP and Mn, is distributed across the width of the slab between the column and middle strips, as is explained shortly.
139
Longitudinal Distribution of Moments in Slabs
In a typical interior panel, the total static moment, M0, is divided into two moments, the positive moment, Mp at midspan, equal to 0.35M0, and the negative moment, Mn, at each support, equal to 0.65M0, as shown in Fig. 5.6-10. These values of moment are based on the assumption that the interior panel is continuous in both directions, with approximately equal spans and loads, so that the interior joints have no significant rotation. Moreover, the moment values are approximately the same as those in a fixed-end beam subjected to uniform loading, where the negative moment at the support is twice the positive moment at midspan. In Fig. 5.6-11, if L1 > L2, then the distribution of moments in the long and short directions is as follows:
Figure 5.6-11: Distribution of moments in an interior panel.
If the magnitudes of the negative moments on opposite sides of an interior support are different because of unequal span lengths, the ACI Code specifies that the larger moment should be considered to calculate the required reinforcement.
In an exterior panel, the slab load is applied to the exterior column from one side only, causing an unbalanced moment and a rotation at the exterior joint. Consequently, there will be an increase in the positive moment at midspan and in the negative moment at the first interior support. The magnitude of the rotation of the exterior joint determines the increase in the moments at midspan and at the interior support. For example, if the exterior edge is a simple support, as in the case of a slab resting on a wall (Fig. 5.6-12), the slab moment at the face of the wall there is 0, the positive moment at midspan can be taken as Mp = 0.63M0, and the negative moment at the interior support is Mn = 0.75 M0. These values satisfy the static equilibrium equation
8 , 0.35 0.65 (5.68)
8 , 0.35 0.65 (5.69)
Figure 5.6-12 exterior panel.
According to Section 13.6.3 of the ACI Code, the total static moment M0 in an end span is distributed in different ratios according to Table 5.6-2 and fig 5.6-13.
141
Transverse Distribution of Moments
The transverse distribution of the longitudinal moments to the middle and column strips is a function of the ratios L2/L1.
beam stiffness
slab stiffness (5.70)
2 (5.71)
(5.72)
Table 5.6-2 Distribution of Moments in an End Panel
Where
1 0.63
3 (5.73)
Where x and y are the shorter and longer dimension of each rectangular component of the section. The percentages of each design moment to be distributed to column and middle strips for interior and exterior panels are given in Tables 5.6-3 through Table 5.6-6 In a typical interior panel, the portion of the design moment that is not assigned to the column strip (Table 5.6-2) must be resisted by the corresponding half-middle strips. When no beams are used 0.
Figure 5.6-13 Distribution of total static moment into negative and positive span Moments.
Table 5.6-3 Percentage of Longitudinal Moment in Column Strips, Interior Panels (ACI Code, Section 13.6.4)
For exterior panels, the portion of the design moment that is not assigned to the column strip (Table 5.6-5) must be resisted by the corresponding half-middle strips.
143
From Table 5.6-5 it can be seen that when no edge beam is used at the exterior end of the slab, β1 = 0 and 100% of the design moment is resisted by the column strip.
The middle strip will not resist any moment; therefore, minimum steel reinforcement must be provided.
Figure 5.6-14 Width of the equivalent rigid frame (equal spans in this figure) and distribution of moments in flat plates, flat slabs, and waffle slabs with no beams.
Table 5.6-4 Percentage of Moments in Two-Way Interior Slabs Without Beams (α1 = 0)
Table 5.6-5 Percentage of Longitudinal Moment in Column Strips, Exterior Panels (ACI Code, Section 13.6.4)
Table 5.6-6 Percentage of Longitudinal Moment in Column and Middle Strips, Exterior Panels (For All Ratios of l2/l1 ), Given α1 = β1= 0
Reinforcement Details
After all the percentages of the static moments in the column and middle strips are determined, the steel reinforcement can be calculated for the negative and positive moments in each strip:
∅ 2 (5.74)
Calculate Ru and determine the steel ratio p using the tables in Appendix or use the following equation:
∅ 1
1.7 (5.75)
where ∅ = 0.9. The steel area is As = bd.
145
The spacing of bars in the slabs must not exceed the ACI limits of maximum spacing: 18 in. (450 mm) or twice the slab thickness, whichever is smaller.