Concrete Slab Design - Two Way Slab Direct Design Method
ACI 318 provides two alternative methods for designing two-way slabs for concrete, The equivalent frame method? (EFM) and the direct design method (DDM). This section will explain how the direct design method is used.
For the Direct Design Method moments are found using a simplified procedure similar to analyzing a One-Way Slab.
The Conditions:
The following conditions must be met to use the DDM:
- Panels must be rectangular in shape with a ratio of the long side to the short side of no more than 2 (this ensures that the slab acts as a two-way slab).
- The loading consists of uniformly distributed gravity loads.
- The live load does not exceed two times the dead load
- There are a minimum of three spans.
- If there are beams are present, the relative stiffness in two perpendicular directions, 0.2 < α_{1}l_{2}^{2}/α_{2}l_{1}^{2} < 5.0
- Successive span lengths do not differ by 1/3 of the longest span.
- And Columns are not offset by more than 10% of the span in the direction of the offset. While designing two-way slabs, column offsets will not be considered, this is why the offset must remain small (L.T. 10%). For large column offsets neither the DDM or EFM can be used, instead a finite element model must be used to calculate the moment in the slab.
Direct Design Method Steps:
Step 1:
Divide the slab into wide beams (Similar to a tributary area method but how the equations are set up I believe basically voids this idea, it is just good for visualization of the problem).
Step 2:
Calculate the total moment in each span using ACI 13.6.2.2^{[1]}
w_{u} = The total factored distributed load (See Concrete LRFD to understand the required loading factors)
l_{2} = the width of the wide beam
l_{n} = face to face of the columns or other supports (note that l_{n} ≥ 0.65l_{1})
Note: The idea is to find a maximum moment in a beam spanning l_{n} carrying a load w_{u}l_{2}.
Step 3:
The Moment (M_{O}) for each span must be distributed up into positive and negative moments according to the tables below:
Table 1: Distribution of Moments in Exterior Spans | |||||
Slabs that contain no beams b/w interior supports | |||||
Type of Moment | exterior edge unrestrained | slab w/ beams b/w all supports | without edge beam | with edge beam | exterior edge fully restrained |
interior negative moment (factored) | 0.75 | 0.70 | 0.70 | 0.70 | 0.65 |
positive moment (factored) | 0.63 | 0.57 | 0.52 | 0.50 | 0.35 |
exterior negative moment (factored) | 0 | 0.16 | 0.26 | 0.30 | 0.65 |
Table 2: Distribution of Moments in Interior Spans | |
Type of Moment | Factor |
negative moment (factored) | 0.65 |
positive moment (factored) | 0.35 |
The factor is multiplied by the total moment to find the positive and negative moments (e.g. a positive interior factored moment will be 0.35xM_{O})
Step 4:
The width of the wide beam will now be divided into column-strip and middle-strip regions.
where:
Middle Strip = a design strip bounded by two column strips (the leftovers)
Step 5:
The column strip will now take the fractions of the moment designated in Table 3 which has been provided below.
Table 3: Distribution of Moments into Column Strips^{[2], a}
→(3.1) Positive Factored Moment | |||
(l_{2} / l_{1}) | 0.5 | 1.0 | 2.0 |
\alpha\left(l_2\over l_1\right) = 0 \hbox{ (no beams)} | 0.60 | 0.60 | 0.60 |
\alpha\left(l_2\over l_1\right)\ge 1 | 0.90 | 0.75 | 0.45 |
→(3.2) Interior Negative Factored Moment | |||
(l_{2} / l_{1}) | 0.5 | 1.0 | 2.0 |
\alpha\left(l_2\over l_1\right) = 0 \hbox{ (no beams)} | 0.75 | 0.75 | 0.75 |
\alpha\left(l_2\over l_1\right)\ge 1 | 0.90 | 0.75 | 0.45 |
→(3.3) Exterior Negative Factored Moment | |||||
(l_{2} / l_{1}) | β_{t}^{b} | 0.5 | 1.0 | 2.0 | |
\alpha\left(l_2\over l_1\right) = 0 | β_{t} = 0 | 1.00 | 1.00 | 1.00 | |
\alpha\left(l_2\over l_1\right) = 0 | β_{t} ≥ 2.5 | 0.75 | 0.75 | 0.75 | |
\alpha\left(l_2\over l_1\right)\ge 1 | β_{t} = 0 | 1.00 | 1.00 | 1.00 | |
\alpha\left(l_2\over l_1\right)\ge 1 | β_{t} ≥ 2.5 | 0.90 | 0.75 | 0.45 |
where:
Notes:
(a) Linear interpolation can be used when α(l_{2}/l_{1}) is between 0 and 1.
(b) β_{t} is a torsional stiffness calc. for the edge beams.
Step 6:
Middle strips will be designed for the fraction of the moment not assigned to the column strip (which has been computed using the factors from Table 3 above).
Therefore if section 5 gave a factor (for the two-way slab in question) of 0.35 then the moment for the middle strip will be 1-0.35 or 0.65.
References:
- American Concrete Institute, "ACI 318", 2005
- This equation can be found in ACI 318 13.6.2.2
- American Concrete Institute, "ACI 318", 2005
- This information can be found in §13.6.4.1 through §13.6.4.3