Precipitación química

14 in.

21 in.

c

d – c

d – c

²t = c (0.003) 0.003

3 #9 bars (3.00 in.²)

F I G U R E 3.4 Beam cross section for Example 3.1.

c= a

β₁ = 5.04 in.

0.85 = 5.93 in.

_t= d− c

c (0.003)=

21 in.− 5.93 in.

5.93 in.



(0.003)= 0.00762

This value of strain is much greater than the yield strain of 0.002. This is an indication of ductile behavior of the beam, because the steel is well into its yield plateau before concrete crushes.

3.6 Balanced Sections, Tension-Controlled Sections, and Compression-Controlled or Brittle Sections

A beam that has a balanced steel ratio is one for which the tensile steel will theoretically just reach its yield point at the same time the extreme compression concrete fibers attain a strain equal to 0.003. Should a flexural member be so designed that it has a balanced steel ratio or be a member whose compression side controls (i.e., if its compression strain reaches 0.003 before the steel yields), the member can suddenly fail without warning. As the load on such a member is increased, its deflections will usually not be particularly noticeable, even though the concrete is highly stressed in compression and failure will probably occur without warning to users of the structure. These members are compression controlled and are referred to as brittle members. Obviously, such members must be avoided.

The code, in Section 10.3.4, states that members whose computed tensile strains are equal to or greater than 0.0050 at the same time the concrete strain is 0.003 are to be referred to as tension-controlled sections. For such members, the steel will yield before the compression side crushes and deflections will be large, giving users warning of impending failure. Furthermore, members witht ≥ 0.005 are considered to be fully ductile. The ACI chose the 0.005 value fort to apply to all types of steel permitted by the code, whether regular or prestressed. The code further states that members that have net steel strains ortvalues betweenyand 0.005 are in a transition region between compression-controlled and tension-controlled sections. For Grade 60 reinforcing steel, which is quite common,_y is approximated by 0.002.

3.7 Strength Reduction or φ Factors

Strength reduction factors are used to take into account the uncertainties of material strengths, inaccuracies in the design equations, approximations in analysis, possible variations in dimen-sions of the concrete sections and placement of reinforcement, the importance of members in

the structures of which they are part, and so on. The code (9.3) prescribesφ values or strength reduction factors for most situations. Among these values are the following:

0.90 for tension-controlled beams and slabs 0.75 for shear and torsion in beams 0.65 or 0.75 for columns

0.65 or 0.75 to 0.9 for columns supporting very small axial loads 0.65 for bearing on concrete

The sizes of these factors are rather good indications of our knowledge of the subject in question. For instance, calculated nominal moment capacities in reinforced concrete members seem to be quite accurate, whereas computed bearing capacities are more questionable.

For ductile or tension-controlled beams and slabs where t ≥ 0.005, the value of φ for bending is 0.90. Shouldt be less than 0.005, it is still possible to use the sections if t is not less than certain values. This situation is shown in Figure 3.5, which is similar to Figure R.9.3.2 in the ACI Commentary to the 2011 code.

Members subject to axial loads equal to or less than 0.10f_c^A_g may be used only when

t is no lower than 0.004 (ACI Section 10.3.5). An important implication of this limit is that reinforced concrete beams must have a tension strain of at least 0.004. Should the members be subject to axial loads≥ 0.10fc^A_g, thentis not limited. Whentvalues fall between 0.002 and 0.005, they are said to be in the transition range between tension-controlled and compression-controlled sections. In this range,φ values will fall between 0.65 or 0.70 and 0.90, as shown in Figure 3.5. When t ≤ 0.002, the member is compression controlled, and the column φ factors apply.

The procedure for determiningφ values in the transition range is described later in this section. You must clearly understand that the use of flexural members in this range is usually uneconomical, and it is probably better, if the situation permits, to increase member depths and/or decrease steel percentages untilt is equal to or larger than 0.005. If this is done, not only willφ values equal 0.9 but also steel percentages will not be so large as to cause crowding of reinforcing bars. The net result will be slightly larger concrete sections, with consequent smaller deflections. Furthermore, as you will learn in subsequent chapters, the bond of the reinforcing to the concrete will be increased as compared to cases where higher percentages of steel are used. Grade 60 reinforcement and for prestressing steel.

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3.7 Strength Reduction orφ Factors 73

We have computed values of steel percentages for different grades of concrete and steel for whicht will exactly equal 0.005 and present them in Appendix Tables A.7 and B.7 of this textbook. It is desirable, under ordinary conditions, to design beams with steel percentages that are no larger than these values, and we have shown them as suggested maximum percentages to be used.

The horizontal axis of Figure 3.5 gives values also for c/d_t ratios. If c/d_t for a particular flexural member is≤ 0.375, the beam will be ductile, and if it is > 0.600, it will be brittle. In between is the transition range. You may prefer to compute c/d_t for a particular beam to check its ductility rather than computingρ or t. In the transition region, interpolation to determine φ using c/dt instead oft, when 0.375< c/dt < 0.600, can be performed using the equations

φ = 0.75 + 0.15

The equations forφ here and in Figure 3.5 are for the special case where f_y = 60 ksi and for prestressed concrete. For other cases, replace 0.002 with_y = f_y/E_s. Figure 10.25 in Chapter 10 shows Figure 3.5 for the general case, where_y is not assumed to be 0.002.

The resulting general equations in the range_y< _t < 0.005 are φ = 0.75 + (t− y) 0.15

(0.005 − _y) for spiral members and

φ = 0.65 + (t− y) 0.25

(0.005 − y) for other members

The impact of the variable φ factor on moment capacity is shown in Figure 3.6. The two curves show the moment capacity with and without the application of theφ factor. Point A corresponds to a tensile strain,_t, of 0.005 and ρ = 0.0181 (Appendix A, Table A.7). This is the largest value ofρ for φ = 0.9. Above this value of ρ, φ decreases to as low as 0.65 as shown by point B, which corresponds to_t of_y. ACI 10.3.5 requires _t not be less than 0.004 for flexural members with compressive axial loads less than 0.10 f_m^A_g. This situation corresponds to point C in Figure 3.6. The only allowable range forρ is below point C. From the figure, it is clear that little moment capacity is gained in adding steel area above point A.

The variableφ factor provisions essentially permit a constant value of φMn when t is less

0

0 0.005 0.01 0.015 0.02 0.025 0.03

′

than 0.005. It is important for the designer to know this because often actual bar selections result in more steel area than theoretically required. If the slope between points A and C were negative, the designer could not use a larger area. Knowing the slope is slightly positive, the designer can use the larger bar area with confidence that the design capacity is not reduced.

For values of f_y of 75 ksi and higher, the slope between point A and B in Figure 3.6 is actually negative. It is therefore especially important, when using high-strength reinforcing steel, to verify your final design to be sure the bars you have selected do not result in a moment capacity less than the design value.

Continuing our consideration of Figure 3.5, you can see that whentis less than 0.005, the values ofφ will vary along a straight line from their 0.90 value for ductile sections to 0.65 at balanced conditions wheretis 0.002. Later you will learn thatφ can equal 0.75 rather than 0.65 at this latter strain situation if spirally reinforced sections are being considered.