Structural Concrete - Hassoun

618 Chapter 17 Design of Two-Way Slabs
Two ACI Code procedures are based on the results of elastic analysis of the structure as a
whole using factored loads. A modified approach to the direct design method was presented in the
commentary of the 1989 Code as the modified stiffness method, or MSM. It is based on specific
distribution factors introduced as a function of the stiffness ratio, α ec , for proportioning the total
static moment in an end span. This method is explained later.
In addition to the ACI Code procedures, a number of other alternatives are available for the
analysis and design of slabs. The resulting slabs may have a greater or lesser amount of reinforcement.
The analytical methods may be classified in terms of the basic relationship between load and
deformation as elastic, plastic, and nonlinear.
1. In elastic analysis, a concrete slab may be treated as an elastic plate. The flexure, shear,
and deflection may be calculated by the fourth differential equation relating load to deflection
for thin plates with small displacements, as presented by Timoshenko and Krieger [6].
Finite difference as well as finite element solutions have been proposed to analyze slabs and
plates [7, 8]. In the finite element method, the slab is divided into a mesh of triangles or
quadrilaterals. The displacement functions of the nodes (intersecting mesh points) are usually
established, and the stiffness matrices are developed for computer analysis.
2. For plastic analysis, three methods are available. The yield line method was developed by
Johansen [9] to determine the limit state of the slab by considering the yield lines that occur
in the slab as a collapse mechanism. The strip method was developed by Hillerborg [10]. The
slab is divided into strips, and the load on the slab is distributed in two orthogonal directions.
The strips are analyzed as simple beams. The third method is optimal analysis. There has been
considerable research into optimal solutions. Dhir, and others Munday [11] presented methods
for minimizing reinforcement based on plastic analysis. Optimal solutions are complex
in analysis and produce complex patterns of reinforcement.
3. Nonlinear analysis simulates the true load deformation characteristics of a reinforced concrete
slab when the finite element method takes into consideration the nonlinearity of the
stress–strain relationship of the individual elements [11, 12]. In this case, the solution becomes
complex unless simplified empirical relationships are assumed.
Waffle slab with light fixtures at the centers of the squares.