Concrete Slab Design

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 < α1l22/α2l12 < 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] MO=8wul2ln2 where: MO = The total moment wu = The total factored distributed load (See Concrete LRFD to understand the required loading factors) l2 = the width of the wide beam ln = face to face of the columns or other supports (note that ln ≥ 0.65l1) l1 = center to center of the columns or supports Note: The idea is to find a maximum moment in a beam spanning ln carrying a load wul2. Step 3: The Moment (MO) 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 0.35xMO) Step 4: The width of the wide beam will now be divided into column-strip and middle-strip regions. where: Column Strip = a strip with a width on each side of the centerline of 0.25l2 or 0.25l1, whichever is less. 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 (l2 / l1) 0.5 1.0 2.0 l1l2=0 (no beams) 0.60 0.60 0.60 l1l21 0.90 0.75 0.45 →(3.2) Interior Negative Factored Moment (l2 / l1) 0.5 1.0 2.0 l1l2=0 (no beams) 0.75 0.75 0.75 l1l21 0.90 0.75 0.45 →(3.3) Exterior Negative Factored Moment (l2 / l1) βtb 0.5 1.0 2.0 l1l2=0 βt = 0 1.00 1.00 1.00 l1l2=0 βt ≥ 2.5 0.75 0.75 0.75 l1l21 βt = 0 1.00 1.00 1.00 l1l21 βt ≥ 2.5 0.90 0.75 0.45 where: =EcsIsEcbIb βt = the torsional stiffness of the edge beam (Computed here) Notes: (a) Linear interpolation can be used when α(l2/l1) is between 0 and 1. (b) βt 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.