TWO-WAY SLAB ANALYSIS BY COEFFICIENTS—DIRECT DESIGN METHOD For gravity loads, ACI Chapter 13 provides two analysis methods for two-way slab systems: the Direct Design

Method (ACI 13.6) and the Equivalent Frame Method (ACI 13.7). The Equivalent Frame Method, using member stiffnesses and complex analytical procedures, is not suitable for hand calculations. Only the Direct Design Method, using moment coefficients, will be presented in this Chapter.

The Direct Design Method applies when all of the conditions illustrated in Fig. 4-5 are satisfied (ACI 13.6.1):

• There must be three or more continuous spans in each direction.

• Slab panels must be rectangular with a ratio of longer to shorter span (c/c of supports) not greater than 2.

• Successive span lengths (c/c of supports) in each direction must not differ by more than one-third of the longer span.

• Columns must not be offset more than 10% of the span (in direction of offset) from either axis between centerlines of successive columns.

• Loads must be due to gravity only and must be uniformly distributed over the entire panel.

• The unfactored live load must not be more than 2 times the unfactored dead load (L/D ≤ 2).

• For two-way slabs, relative stiffnesses of beams in two perpendicular directions must satisfy the minimum and maximum requirements given in ACI 13.6.1.6.

• Redistribution of moments by ACI 8.4 must not be permitted.

Simplified Design • EB204

˜1

Uniformly Distributed Loading (L/D ≤ 2)

˜1 (≥ 2/3) ˜1

Three or More Spans

˜2

Column offset Rectangular slab

panels (2 or less: 1)

≤ ˜2/10

In essence, the Direct Design Method is a three-step analysis procedure. The first step is the calculation of the total design moment M_ofor a given panel. The second step involves the distribution of the total moment to the negative and positive moment sections. The third step involves the assignment of the negative and positive moments to the column strips and middle strips.

For uniform loading, the total design moment M_o for a panel is calculated by the simple static moment expression, ACI Eq. (13-4):

M_o= q_u˜₂˜_n²/ 8

where q_u is the factored combination of dead and live loads (psf), q_u = 1.2 w_d + 1.6 w_˜. The clear span ˜_n is defined in a straightforward manner for slabs supported on columns or other supporting elements of rectangular cross section (ACI 13.6.2.5). Circular or regular polygon shaped supports must be treated as square supports with the same area (see ACI Fig. R13.6.2.5). The clear span starts at the face of support and must not be taken less than 65% of the span center-to-center of supports (ACI 13.6.2.5). The span ˜₂is simply the span transverse to ˜_n; however, when the panel adjacent and parallel to an edge is being considered, the distance from edge of slab to panel centerline is used for ˜₂in calculation of M_o(ACI 13.6.2.4).

Division of the total panel moment M_o into negative and positive moments, and then into column and middle strip moments, involves direct application of moment coefficients to the total moment M_o. The moment coefficients are a function of span (interior or exterior) and slab support conditions (type of two-way slab system). For design convenience, moment coefficients for typical two-way slab systems are given in Tables 4-2 through 4-6. Tables 4-2 through 4-5 apply to flat plates or flat slabs with various end support conditions.

Table 4-6 applies to two-way slabs supported on beams on all four sides. Final moments for the column strip and middle strip are computed directly using the tabulated values. All coefficients were determined using the appropriate distribution factors in ACI 13.6.3 through 13.6.6.

End Span Interior Span

Slab Moments

Note: All negative moments are at face of support.

2

1 3 4 5

Interior Span End Span

Table 4-2 Flat Plate or Flat Slab Supported Directly on Columns

Simplified Design • EB204

End Span Interior Span

Slab Moments Note: All negative moments are at face of support.

Table 4-4 Flat Plate or Flat Slab with End Span Integral with Wall

1 2 3 4 5

Interior Span End Span

End Span Interior Span

Slab Moments Note: (1) All negative moments are at face of support.

(2) Torsional stiffness of spandrel beams βt βt β_t

> 2.5. For values of less than 2.5, exterior negative column strip moment increases to (0.30 – 0.03 ) Mo.

Table 4-3 Flat Plate or Flat Slab with Spandrel Beams

1 2 3 4 5

Interior Span End Span

End Span Interior Span

(2) Interpolated between values shown for different "₂/ "₁ ratios.

(3) All negative moments are at face of support.

(4) Concentrated loads applied directly to beams must be accounted for separately.

End Span Interior Span

Slab Moments

Note: All negative moments are at face of support.

Table 4-5 Flat Plate or Flat Slab with End Span Simply Supported on Wall

1 2 3 4 5

Interior Span End Span

Table 4-6 Two-Way Beam-Supported Slab

1 2 3 4 5

Interior Span End Span

αf1˜2/˜1 ≥ 1.0 and βt≥ 2.5

Simplified Design • EB204

CL CL

CL Slab, Is

Slab, Is

Beam,

Ib

Beam_, Ib

b

h h

a

a

b + 2 (a - h) ≤ b + 8h

b + (a - h) ≤ b + 4h

˜2/2+ c1/2

˜2

The moment coefficients of Table 4-3 (flat plate with spandrel beams) are valid for β_t≥ 2.5, the coefficients of Table 4-6 (two-way beam-supported slabs), are applicable when α_f1˜₂^/˜₁ ≥ 1.0 and β_t ≥ 2.5 (β_t, and α_f1 are stiffness parameters defined below). Many practical beam sizes will provide beam-to-slab stiffness ratios such that α_f1˜₂^/˜₁^and β_twould be greater than these limits, allowing moment coefficients to be taken directly from the tables. However, if beams are present, the two stiffness parameters α_f1 and β_twill need to be evaluated.

For two-way slabs, the stiffness parameter α_f1is simply the ratio of the moments of inertia of the effective beam and slab sections in the direction of analysis, α_f1= I_b/I_s, as illustrated in Fig. 4-6. Figures 4-7 and 4-8 can be used to determine α_f.

Relative stiffness provided by a spandrel beam is reflected by the parameter β_t= C/2I_s, where I_sis the moment of inertia of the effective slab spanning in the direction of ˜₁and having a width equal to ˜₂^{, i.e., I}_s⁼ ˜₂^h3/12.

The constant C pertains to the torsional stiffness of the effective spandrel beam cross section. It is found by dividing the beam section into its component rectangles, each having smaller dimension x and larger dimension y, and summing the contribution of all the parts by means of the equation.

The subdivision can be done in such a way as to maximize C. Figure 4-9 can be used to determine the torsional constant C.

C= 1− 0.63x y

⎛

⎝⎜ ⎞

⎠⎟

∑

2.8

Figure 4-7 Beam to Slab Stiffness Ratio α_f(Interior Beam)

f 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1

1.0 1.5 2 3 4 5 6 7 8 9 10 2.4

b/h = 0.4

0.6

0.8

b/h = 1.0

1.6 2.0 1.2 1.4

5.0

10.0 20.0 b/h = 3.0 Simplified Design • EB204

a

b

˜2/2 + b/2 CL b + (a – h) ≤ b + 4h

h

αf= b a h

3

f

˜2

x*

*Small side of a rectangular cross section with dimensions x and y.

Values of torsion constant, C = (1 – 0.63 x/y)(x³ y/3)

Figure 4-9 Design Aid for Computing Torsional Section Constant C

Simplified Design • EB204

The column strip and middle strip moments are distributed over an effective slab width as illustrated in Fig. 4-10. The column strip is defined as having a width equal to one-half the transverse or longitudinal span;

whichever is smaller (ACI 13.2.1). The middle strip is bounded by two column strips.

Middle strip

1/2 middle strip Interior column strip Exterior column strip

˜₂

˜₂

˜₂/2 ˜₂/2 ˜₂/2

˜₂/4 ˜₂/4 ˜₂/4

˜1

(a) Column strip for˜₂ < ˜₁

Middle strip

1/2 middle strip Interior column strip Exterior column strip

˜₂

˜₂

˜₂/2 ˜₂/2 ˜₂/2

˜₁/4 ˜1/4 ˜₁/4

˜1

(b) Column strip for˜₂ > ˜₁ Figure 4-10 Definition of Design Strips