## Allowable-Stress Design at Service Loads (Alternative Design Method)

Nonprestressed, reinforced-concrete flexural members (Art. 9.63) may be designed for flexure by the alternative design method of the ACI 318 Building Code (working- stress design). In this method, members are designed to carry service loads (load factors and @ are taken as unity) under the straight-line (elastic) theory of stress and strain. (Because of creep in the concrete, only stresses due to short-time loading can be predicted with reasonable accuracy by this method.)

Working-stress design is based on the following assumptions:
1. A section plane before bending remains plane after bending. Strains therefore vary with distance from the neutral axis (Fig. 9.13c).

2. The stress-strain relation for concrete plots as a straight line under service loads within the allowable working stresses (Fig. 9.13c and d), except for deep beams.
3. Reinforcing steel resists all the tension due to flexure (Fig. 9.13a and b).
4. The modular ratio, n = Es /Ec, where Es and Ec are the moduli of elasticity of reinforcing steel and concrete, respectively, may be taken as the nearest whole number, but not less than 6 (Fig. 9.13b).
5. Except in calculations for deflection, n lightweight concrete should be assumed the same as for normal-weight concrete of the same strength.
6. The compressive stress in the extreme surface of the concrete must not exceed 0.45ƒc, where ƒc is the 29-day compressive strength of the concrete.
7. The following tensile stress in the reinforcement must not be greater than the following:
Grades 40 and 50 20 ksi
Grade 60 or greater 24 ksi
For 3⁄8-in. or smaller-diameter reinforcement in one-way slabs with spans not exceeding 12 ft, the allowable stress may be increased to 50% of the yield strength but not to more than 30 ksi.
8. For doubly-reinforced flexural members, including slabs with compression reinforcement, an effective modular ratio of 2Es /Ec should be used to transform the compression-reinforcement area for stress computations to an equivalent concrete area (Fig. 9.13b). (This recognizes the effects of creep.) The allowable stress in the compression reinforcement may not exceed the allowable tension stress.

Because the strains in the reinforcing steel and the adjoining concrete are equal, the stress in the tension steel ƒs is n times the stress in the concrete ƒc. The total force acting on the tension steel then equals nAsƒc. The steel area As, therefore can be replaced in stress calculations by a concrete area n times as large.
The transformed section of a reinforced concrete beam is a cross section normal to the neutral surface with the reinforcement replaced by an equivalent area of concrete (Fig. 9.13b). (In doubly-reinforced beams and slabs, an effective modular  ratio of 2n should be used to transform the compression reinforcement and account for creep and nonlinearity of the stress-strain diagram for concrete.) Stress and strain are assumed to vary with the distance from the neutral axis of the transformed section; that is, conventional elastic theory for homogeneous beams may be applied  to the transformed section. Section properties, such as location of neutral axis, moment of inertia, and section modulus S, may be computed in the usual way for homogeneous beams, and stresses may be calculated from the flexure formula, ƒ = M/S, where M is the bending moment at the section. This method is recommended particularly for T-beams and doubly-reinforced beams.

From the assumptions the following formulas can be derived for a rectangular section with tension reinforcement only. Design of flexural members for shear, torsion, and bearing, and of other types of members, follows the strength design provisions of the ACI 318 Building Code, because allowable capacity by the alternative design method is an arbitrarily specified percentage of the strength.