Analysis versus Design
Two different types of problems arise in the study of reinforced concrete:
1. Analysis.Given a cross section, concrete strength, reinforcement size and loca- tion, and yield strength, compute the resistance or strength. In analysis there should be one unique answer.
2. Design.Given a factored design moment, normally designated as select a suitable cross section, including dimensions, concrete strength, reinforcement, and so on.
In design there are many possible solutions.
Although both types of problem are based on the same principles, the procedure is different in each case. Analysis is easier, because all of the decisions concerning reinforce- ment, beam size, and so on have been made, and it is only necessary to apply the strength- calculation principles to determine the capacity. Design, on the other hand, involves the choice of section dimensions, material strengths, and reinforcement placement to produce a cross section that can resist the moments due to factored loads. Because the analysis problem is easier, this chapter deals with section analysis to develop the fundamental concepts before considering design in the next chapter.
Required Strength and Design Strength The basic safety equation for flexure is:
(4-1a) or for flexure,
(4-1b) where is the moment due to the factored loads, which commonly is referred to as the factored design moment. This is a load effect computed by structural analysis from the governing combination of factored loads given in ACI Code Section 9.2. The term refers to the nominal moment strengthof a cross section, computed from the nominal di- mensions and specified material strengths. The factor in Eq. (4-1b) is a strength- reduction factor(ACI Code Section 9.3) to account for possible variations in dimensions and material strengths and possible inaccuracies in the strength equations. Since the mid 1990s, the ACI Code has referred to the load factors and load combinations developed by ASCE/SEI Committee 7, which is responsible for the ASCE/SEI Standard for Minimum Design Loads for Buildings and Other Structures[4-1]. The load factors and load combinations given in ACI Code Section 9.2 are essentially the same as those
f
Mn Mu
fMn Ú Mu
Reduced nominal strength Ú factored load effects
Mu,
D
B
A E
G C
F
P
Fig. 4-1
One-way flexure.
b⫽ be
b⫽ bw
Fig. 4-2
Cross-sectional dimensions.
developed by ASCE/SEI Committee 7. The strength-reduction factors given in ACI Code Section 9.3 are based on statistical studies of material properties [4-2] and were selected to give approximately the same level of safety as obtained with the load and strength-re- duction factors used in earlier editions of the code. Those former load and strength-re- duction factors are still presented as an alternative design procedure in Appendix C of the latest edition of the ACI Code, ACI 318-11 [4-3]. However, they will not be dis- cussed in this book.
For flexure without axial load, ACI Code Section 9.3.2.1 gives for what are called tension-controlled sections. Most practical beams will be tension-controlled sections, and will be equal to 0.90. The concept of tension-controlled sections will be discussed later in this chapter. The product, commonly is referred to as the reduced nominal moment strength.
Positive and Negative Moments
A moment that causes compression on the top surface of a beam and tension on the bottom surface will be called a positive moment. The compression zones for positive and negative moments are shown shaded in Fig. 4-2. In this textbook, bending-moment diagrams will be plotted on the compression side of the member.
Symbols and Notation
Although symbols are defined as they are first used and are summarized in Appendix B, several symbols should essentially be memorized because they are commonly used in dis- cussions of reinforced concrete members. These include the terms and (defined earlier) and the cross-sectional dimensions illustrated in Fig. 4-2. The following is a list of common symbols used in this book:
• is the area of reinforcement near the tension face of the beam, tension rein- forcement,
• is the area of reinforcement on the compression side of the beam, compres- sion reinforcement,
• bis a general symbol for the width of the compression zone in a beam, in. This is illustrated in Fig. 4-2 for positive and negative moment regions. For flanged sections this symbol will normally be replaced with beor bw.
in.2. Asœ
in.2. As
Mn Mu
fMn, f
f = 0.90
• is the effective width of a compression zone for a flanged section with com- pression in the flange, in.
• is the width of the web of the beam (and may or may not be the same as b), in.
• dis the distance from the extreme fiber in compression to the centroid of the longitudinal reinforcement on the tension side of the member, in. In the posi- tive-moment region (Fig. 4-2a), the tension steel is near the bottom of the beam, while in the negative-moment region (Fig. 4-2b) it is near the top.
• is the distance from the extreme compression fiber to the centroid of the longitudinal compression steel, in.
• is the distance from the extreme compression fiber to the farthest layer of tension steel, in. For a single layer of tension reinforcement, as shown in Fig. 4-2b.
• is the specified compressive strength of the concrete, psi.
• is the stress in the concrete, psi.
• is the stress in the tension reinforcement, psi.
• is the specified yield strength of the reinforcement, psi.
• his the overall height of a beam cross section.
• jdis the lever arm, the distance between the resultant compressive force and the resultant tensile force, in.
• jis a dimensionless ratio used to define the lever arm,jd.It varies depending on the moment acting on the beam section.
• is the assumed maximum useable compression strain in the concrete.
• is the strain in the tension reinforcement.
• is the strain in the extreme layer of tension reinforcement.
• is the longitudinal tension reinforcement ratio,