5. Two-Way Slab Deflection Control (Serviceability Requirements)

Since the slab thickness was selected based on the minimum slab thickness tables in ACI 318-14, the deflection calculations are not required. However, the calculations of immediate and time-dependent deflections are covered in this section for illustration and comparison with spSlab model results.

5.1. Immediate (Instantaneous) Deflections

The calculation of deflections for two-way slabs is challenging even if linear elastic behavior can be assumed. Elastic analysis for three service load levels (D, D + L_sustained, D+L_Full) is used to obtain immediate deflections of the two-way slab in this example. However, other procedures may be used if they result in predictions of deflection in reasonable agreement with the results of comprehensive tests.

ACI 318-14 (24.2.3)

The effective moment of inertia (I_e) is used to account for the cracking effect on the flexural stiffness of the slab. I_e for uncracked section (M_cr > M_a) is equal to I_g. When the section is cracked (M_cr < M_a), then the following equation should be used:

The values of the maximum moments for the three service load levels are calculated from structural analysis as shown previously in this document. These moments are shown in the following Figure.

Figure 13 – Maximum Moments for the Three Service Load Levels

The calculations shown below are for the design strip (frame strip). The values of these parameters for column and middle strips are shown in Table 9.

As calculated previously, the exterior span frame strip near the interior support is reinforced with 17 #4 bars located at 1.25 in. along the section from the top of the slab. Figure 14 shows all the parameters needed to calculate the moment of inertia of the cracked section transformed to concrete.

Figure 14 – Cracked Transformed Section

The effective moment of inertia procedure described in the Code is considered sufficiently accurate to estimate deflections. The effective moment of inertia, I_e, was developed to provide a transition between the upper and lower bounds of I_g and I_cr as a function of the ratio M_cr/M_a. For conventionally reinforced (nonprestressed) members, the effective moment of inertia, I_e, shall be calculated by Eq. (24.2.3.5a) unless obtained by a more comprehensive analysis.

I_e shall be permitted to be taken as the value obtained from Eq. (24.2.3.5a) at midspan for simple and continuous spans, and at the support for cantilevers.

ACI 318-14 (24.2.3.7)

For continuous one-way slabs and beams. I_e shall be permitted to be taken as the average of values obtained from Eq. (24.2.3.5a) for the critical positive and negative moment sections.

ACI 318-14 (24.2.3.6)

For the exterior span (span with one end continuous) with service load level (D + LL_full):

Where I_e⁺ is the effective moment of inertia for the critical positive moment section (midspan).

Since midspan stiffness (including the effect of cracking) has a dominant effect on deflections, midspan section is heavily represented in calculation of I_e and this is considered satisfactory in approximate deflection calculations. The averaged effective moment of inertia (I_e,avg) is given by:

For the interior span (span with both ends continuous) with service load level (D + LL_full):

The following Table provides a summary of the required parameters and calculated values needed for deflections for exterior and interior equivalent frame. It also provides a summary of the same values for column strip and middle strip to facilitate calculation of panel deflection.

Deflections in two-way slab systems shall be calculated taking into account size and shape of the panel, conditions of support, and nature of restraints at the panel edges. For immediate deflections two-way slab systems the midpanel deflection is computed as the sum of deflection at midspan of the column strip or column line in one direction (Δ_cx or Δ_cy) and deflection at midspan of the middle strip in the orthogonal direction (Δ_mx or Δ_my). Figure 15 shows the deflection computation for a rectangular panel. The average Δ for panels that have different properties in the two direction is calculated as follows:

Figure 15 – Deflection Computation for a Rectangular Panel

To calculate each term of the previous equation, the following procedure should be used. Figure 16 shows the procedure of calculating the term Δ_cx. same procedure can be used to find the other terms.

Figure 16 – Δ_cx Calculation Procedure

Where LDF_c is the load distribution factor for the column strip. The load distribution factor for the column strip can be found from the following equation:

And the load distribution factor for the middle strip can be found from the following equation:

For the end span, LDF for exterior negative region (LDF_L¯), interior negative region (LDF_R¯), and positive region (LDF_L^＋) are 1.00, 0.75, and 0.60, respectively (From Table 6 of this document). Thus, the load distribution factor for the column strip for the end span is given by:

I_c,g = The gross moment of inertia (I_g) for the column strip for service dead load = 2,401 in.⁴

Following the same procedure, Δ_mx can be calculated for the middle strip. This procedure is repeated for the equivalent frame in the orthogonal direction to obtain Δ_cy, and Δ_my for the end and middle spans for the other load levels (D+LL_sus and D+LL_full).

Assuming square panel, Δ_cx = Δ_cy= 0.0747 in. and Δ_mx = Δ_my= 0.0386 in.

The average Δ for the corner panel is calculated as follows:

From the analysis in the transverse direction the deflection values below are obtained:

For DL loading case:

For DL+LL_sust loading case:

For DL+LL_full loading case:

These values for the x-direction are shown in the previous Table. Then, the total midpanel deflection is calculated by combining the contributions of the column and middle strip deflections from the X and Y directions:

5.2. Time-Dependent (Long-Term) Deflections (Δ_lt)

The additional time-dependent (long-term) deflection resulting from creep and shrinkage (Δ_cs) may be estimated as follows:

The following Table shows long-term deflections for the exterior and interior spans for the analysis in the x-direction, for column and middle strips.