1. Preliminary Member Sizing

1.1. Preliminary Flat Plate (without Joists)

1.1.1. Slab Minimum Thickness - Deflection

In lieu of detailed calculation for deflections, ACI 318 Code gives minimum slab thickness for two-way construction without interior beams in Table 8.3.1.1.

For flat plate slab system, the minimum slab thickness per ACI 318-14 are:

Where l_n = length of clear span in the long direction = 33 × 12 – 20 = 376 in.

Use 13 in. slab for all panels (self-weight = 150 pcf × 13 in. / 12 = 162.50 psf)

1.1.2. Slab Shear Strength - One Way Shear

Evaluate the average effective depth (Figure 2):

Where:

Figure 2 – Average Effective Depth for Flat Plate

At an interior column:

Consider a 12-in. wide strip. The critical section for one-way shear is located at a distance d, from the face of support (see Figure 3).

Tributary area for one-way shear is:

where λ = 1 for normal weight concrete, more information can be found in “Concrete Type Classification Based on Unit Density” technical article. 

Slab thickness of 13 in. is adequate for one-way shear.

Figure 3 – Critical Section for One-Way Shear

1.1.3. Slab Shear Strength - Two-Way Shear

Check the adequacy of slab thickness for punching shear (two-way shear) at an interior column (Figure 4):

Tributary area for two-way shear is:

Slab thickness of 13 in. is not adequate for two-way shear. This is expected as the self-weight an applied loads are very challenging for a flat plate system.

Figure 4 – Critical Section for Two-Way Shear

In this case, four options can be considered: 1) to increase the slab thickness further, 2) use headed shear reinforcement in the slab, 3) apply drop panels at columns, or 4) use two-way joist slab system. In this example, the latter option will be used to achieve better understanding for the design of two-way joist slab often called two-way ribbed slab or waffle slab.

Check the applicable joist dimensional limitations as follows:

Figure 5 – Joists Dimensions

Use

Based on the previous discussion, Figure 6 shows the dimensions of the selected two-way joist system.

Figure 6 – Two-Way Joist (Waffle) Slab

1.2. Preliminary Two-Way Joist Slab (Waffle Slab)

For slabs with changes in thickness and subjected to bending in two directions, it is necessary to check shear at multiple sections as defined in the ACI 318-14. The critical sections shall be located with respect to:

1.2.1. Slab Minimum Thickness - Deflection

In lieu of detailed calculation for deflections, ACI 318 Code gives minimum slab thickness for two-way construction without interior beams in Table 8.3.1.1.

For this slab system, the minimum slab thicknesses per ACI 318-14 are:

Where l_n = length of clear span in the long direction = 33 × 12 – 20 = 376 in.

For the purposes of analysis and design, the ribbed slab will be replaced with a solid slab of equivalent moment of inertia, weight, punching shear capacity, and one-way shear capacity.

The equivalent thickness based on moment of inertia is used to find slab stiffness considering the ribs in the direction of the analysis only. The ribs spanning in the transverse direction are not considered in the stiffness computations. This thickness, h_MI, is given by:

Where:

I_rib = Moment of inertia of one joist section between centerlines of ribs (see Figure 7)

b_rib = The center-to-center distance of two ribs (clear rib spacing plus rib width) (see Figure 7)

Since h_MI = 12.00 in. > h_min = 11.39 in., the deflection calculation can be neglected. However, the deflection calculation will be included in this example for comparison with the spSlab software results.

The drop panel depth for two-way joist (waffle) slab is set equal to the rib depth. The equivalent drop depth based on moment of inertia, d_MI, is given by:

Where:

h_rib = 3.00 + 14.00 – 12.00 = 5.00 in.

Figure 7 – Equivalent Thickness Based on Moment of Inertia

Find system self-weight using the equivalent thickness based on the weight of individual components (see the following Figure). This thickness, h_w, is given by:

Where:

V_mod = The Volume of one joist module (the transverse joists are included – 11 joists in the frame strip).

A_mod = The plan area of one joist module = 33 × 36/12 = 99.00 ft².

Self-weight for slab section without drop panel = 150 pcf × 8.00 in. /12 = 100.057 psf

Self-weight for slab section with drop panel = 150 pcf × (14.00 + 3.00 - 8.00) in. /12 = 112.443 psf

Figure 8 – Equivalent Thickness Based on the Weight of Individual Components

1.2.2. Slab Shear Strength - One Way Shear

For critical section at distance d from the edge of the column (slab section with drop panel):

Evaluate the average effective depth:

Where:

Figure 9 – Average Effective Depth for Slab Section with Drop Panel

Consider a 12-in. wide strip. The critical section for one-way shear is located at a distance d, from the edge of the column (see Figure 10).

Tributary area for one-way shear is:

where λ = 1 for normal weight concrete

Slab thickness is adequate for one-way shear for the first critical section (from the edge of the column).

Figure 10 – Critical Section at Distance d from the Edge of the Column for One-Way Shear

For critical section at the edge of the drop panel (slab section without drop panel):

Evaluate the average effective depth:

Where:

Figure 11 – Average Effective Depth for Slab Section without Drop Panel

Consider a 12-in. wide strip. The critical section for one-way shear is located at the face of the solid head (see Figure 12).

Tributary area for one-way shear is:

where λ = 1 for normal weight concrete

Slab thickness of 12 in. is adequate for one-way shear for the second critical section (at the edge of the drop panel).

Figure 12 – Critical Section at the Face of the Drop Panel for One-Way Shear

1.2.3. Slab Shear Strength - Two-Way Shear

For critical section at distance d/2 from the edge of the column (slab section with drop panel):

Check the adequacy of slab thickness for punching shear (two-way shear) at an interior column (Figure 13):

Slab thickness is adequate for two-way shear for the first critical section (from the edge of the column).

Figure 13 – Critical Section at d/2 from the Edge of the Column for Two-Way Shear

For critical section at the edge of the drop panel (slab section without drop panel):

Check the adequacy of slab thickness for punching shear (two-way shear) at an interior drop panel (Figure 14):

Tributary area for two-way shear is:

Slab thickness of 12 in. is adequate for two-way shear for the second critical section (from the edge of the drop panel).

Figure 14 – Critical Section at d/2 from the Edge of the Drop Panel for Two-Way Shear

1.2.4. Column Dimensions - Axial Load

Check the adequacy of column dimensions for axial load:

For live load, superimposed dead load, and self-weight of the slab around an interior column:

q_u = 340.07 psf (on page 14)

For self-weight of additional slab thickness due to the presence of the drop panel around an interior column:

q_u = 475.00 – 340.07 = 134.93 psf (see page 12 and page 14)

Assuming four story building

Assume 20 in. square column with 12 – No. 11 vertical bars with design axial strength, ϕP_n,max of

Column dimensions of 20 in. × 20 in. are adequate for axial load.