5. Serviceability Requirements (Deflection Check)

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.

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 Figure 23.

Figure 23 - Maximum Moments for the Three Service Load Levels (No live load is sustained in this example)

For positive moment (midspan) section:

As calculated previously, the positive reinforcement for the end span frame strip is 23 #6 bars located at 1.125 in. along the section from the bottom of the slab. Two of these bars are not continuous and will be conservatively excluded from the calculation of I_cr since they might not be adequately developed or tied (21 bars are used). Figure 24 shows all the parameters needed to calculate the moment of inertia of the cracked section transformed to concrete at midspan.

Figure 24 - Cracked Transformed Section (Positive Moment Section)

For negative moment section (near the interior support of the end span):

The negative reinforcement for the end span frame strip near the interior support is 32 #6 bars located at 1.125 in. along the section from the top of the slab.

Figure 25 - Ig Calculations for Slab Section Near Support

Figure 26 - Cracked Transformed Section (Negative Moment 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.

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.

For the middle span (span with two ends 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. Both the midspan stiffness (I_e⁺) and averaged span stiffness (I_e,avg) can be used in the calculation of immediate (instantaneous) deflection.

The averaged effective moment of inertia (I_e,avg) is given by:

Table 6 provides a summary of the required parameters and calculated values needed for deflections for exterior and interior spans.

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 in 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 27 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 27 - Deflection Computation for a Rectangular Panel

To calculate each term of the previous equation, the following procedure should be used. Figure 28 shows the procedure of calculating the term Δ_cx. Same procedure can be used to find the other terms.

Figure 28 – Δ_cx Calculation Procedure

For this example and like in the spSlab program, the effective moment of inertia at midspan will be used.

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 2 of this document). Thus, the load distribution factor for the column strip for the end span is given by:

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).

Since in this example the panel is squared, Δ_cx = Δ_cy= 0.2170 in. and Δ_mx = Δ_my= 0.1126 in.

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

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:

Table 8 shows long-term deflections for the exterior and interior spans for the analysis in the x-direction, for column and middle strips.