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6.2.1

Overview

The cross-sectional dimensions of a flexural member and the required amount of flexu- ral reinforcement at critical sections are determined using the strength and serviceability requirements of the Code. Typically, the member depth is determined first on the basis of the deflection requirements of ACI 9.5, which have been presented in Section 4.4 for one-way construction. The minimum thickness requirements given in ACI Table 9.5(a) are applicable to beams and one-way slabs that are not attached to partitions or other construction that is likely to be damaged by relatively large deflections. Section 6.5 contains methods to calculate deflections in any situation.

Once the depth of a member has been established, the width of the member and the required flexural reinforcement are determined using the basic requirement for flexural design strength:

φMn≥ Mu (6.1)

In this equation, φMn is the design strength of the member at a particular section,

which consists of the strength reduction factorφ (Section 4.3) and the nominal flexural strength Mnthat is determined in accordance with the provisions and assumptions of

the strength design method (Section 5.4). The required strength Muis calculated at a

section by combining the bending moments obtained from the analysis of the structure, using nominal loads in accordance with the load combinations of ACI Chap. 9 (Section 4.2).

The following sections provide the fundamental requirements and methods for sizing the cross-section and determining the required amount of flexural reinforcement for a reinforced concrete flexural member.

6.2.2

Sizing the Cross-Section

Establishing the dimensions of the cross-section is typically the initial step in the design of a reinforced concrete flexural member. The depth h is usually determined first on the basis of deflection requirements. This depth is sometimes modified for constructability, economy, or architectural reasons, to name a few. For a rectangular beam, the width b is subsequently determined on the basis of strength requirements, assuming that the section is tension-controlled, whereas for a one-way slab, the design width is commonly taken as 12 in. The following steps can be utilized to determine the dimensions of a reinforced concrete flexural member.

Step 1: Determine the depth of the member.As noted earlier, it is common to determine the depth of the member first on the basis of the serviceability requirements for deflection given in ACI 9.5 for one-way construction (i.e., members that bend in primarily one direction, such as beams and one-way slabs).

Consider the reinforced concrete beam depicted in Fig. 6.1. Although the following discussion focuses on beams, it applies equally to one-way slabs. If this member is not attached to partitions or other construction that is likely to be damaged by relatively large deflections, the minimum depth h that will satisfy the deflection requirements of

FIGURE6.1 Minimum thickness requirements for continuous beams.

ACI 9.5.2 is determined by ACI Table 9.5(a). These requirements are summarized in Fig. 4.3. Alternatively, the member depth can be established on the basis of calculated deflections and the maximum permissible deflections given in ACI Table 9.5(b) (see Section 6.5).

It is important to consider economical formwork when choosing a member thick- ness from ACI Table 9.5(a). For the usual case of continuous construction and assuming normal-weight concrete and Grade 60 reinforcement, it is permitted to use a minimum thickness of2/21 for the interior spans and a minimum thickness of 1/18.5 for the end spans. More than one beam depth along the same line of beams results in formwork that is not economical. Thus, the minimum depth of all of the beams should be determined on the basis of the span that yields the largest minimum h because this thickness will satisfy deflection criteria for all spans. In the case of equal end and interior spans or where2< 1(see Fig. 6.1), the minimum h based on the end span governs, whereas in cases where2> 1.141, the minimum h based on the interior span governs. Recall that deflections need not be computed where a thickness equal to at least the minimum is provided.

The beam or one-way slab depth h that is actually constructed is specified in whole- or half-inch increments. For beams, whole-inch increments are usually used; however, this is not mandatory (in joist systems, half-inch increments for beam depths are com- mon). An approximate value of d can be calculated as follows:

r _{Beams with one layer of reinforcement: d= h − 2.5 in} r _{One-way slabs: d= h − 1.25 in}

The values of 2.5 and 1.25 in for beams and one-way slabs, respectively, are based on cover requirements and other reinforcement details, which are covered later in this chapter.

Step 2: Assume that the section is tension-controlled.The graph shown in Fig. 6.2 illustrates the effect of the strength reduction factorφ on the design strength φMn

for the case of 4,000 psi concrete and Grade 60 reinforcement. In particular, it shows what happens to the design strength when the limit for tension-controlled sections (φ = 0.9) is passed. Similar curves can be generated for other material strengths. The reinforcement ratio corresponding to tension-controlled sections (εt= 0.0050) is ρt, and

FIGURE6.2 Design strength curve for 4,000 psi concrete and Grade 60 reinforcement.

the reinforcement ratio corresponding to the maximum permitted reinforcement (εt =

0.0040) is ρmax.

The strength reduction factorφ for tension-controlled sections is equal to 0.9. If a section contains reinforcement greater than that corresponding toρt, thenφ < 0.9 and

the strength gain is minimal up toρmax(see Fig. 6.2). Any gain in strength with higher reinforcement ratios is canceled by the reduction inφ when net strains are less than 0.0050. Thus, for overall efficiency, flexural members should be designed as tension- controlled sections whenever possible.

Step 3: Determine the width of the member.The width of a beam is determined by settingφMn= Mufor an assumed reinforcement ratioρ. A range for ρ is established as

follows. Because a minimum amount of flexural reinforcement is required at any section, the assumed value ofρ must be greater than or equal to the minimum value prescribed in ACI 10.5 (see Section 5.4). Flexural members should be designed as tension-controlled sections whenever possible (see Step 2). As such, the assumed value ofρ should be less than or equal toρt.

The reinforcement ratioρtcorresponding to tension-controlled sections withεt =

0.0050 can be derived using the basic principles and assumptions of the strength de- sign method presented in Chap. 5. Figure 5.6 illustrates the strain condition in a rect- angular, tension-controlled section with a single layer of tension reinforcement where

εt= 0.0050. It is evident from the figure that the depth to the neutral axis ct= 0.375dt.

Because at= β1ct, at = 0.375β1dt. Substituting atinto Eq. (5.9) and solving for Asresult

in the following:

As=

0.319β1fcbdt

fy

(6.2)

Substituting As = ρtbdtinto Eq. (6.2) and solving forρtgive

ρt=

0.319β1fc

fy