Diseño de sedimentadores

Courtesy of Symons Corporation.

Water Tower Place, Chicago, Illinois, tallest reinforced concrete building in the United States (74 stories, 859 ft).

member by the strength reduction factor,φ, which is less than 1. These values generally vary from 0.90 for bending down to 0.65 for some columns.

In summary, the strength design approach to safety is to select a member whose computed ultimate load capacity multiplied by its strength reduction factor will at least equal the sum of the service loads multiplied by their respective load factors.

Member capacities obtained with the strength method are appreciably more accurate than member capacities predicted with the working-stress method.

3.4 Derivation of Beam Expressions

Tests of reinforced concrete beams confirm that strains vary in proportion to distances from the neutral axis even on the tension sides and even near ultimate loads. Compression stresses vary approximately in a straight line until the maximum stress equals about 0.50f_c^. This is not the case, however, after stresses go higher. When the ultimate load is reached, the strain and stress variations are approximately as shown in Figure 3.1.

The compressive stresses vary from zero at the neutral axis to a maximum value at or near the extreme fiber. The actual stress variation and the actual location of the neutral axis vary somewhat from beam to beam, depending on such variables as the magnitude and history of past loadings, shrinkage and creep of the concrete, size and spacing of tension cracks, speed of loading, and so on.

If the shape of the stress diagram were the same for every beam, it would be possible to derive a single rational set of expressions for flexural behavior. Because of these stress variations, however, it is necessary to base the strength design on a combination of theory and test results.

Although the actual stress distribution given in Figure 3.2(b) may seem to be important, in practice any assumed shape (rectangular, parabolic, trapezoidal, etc.) can be used if the

²s ≥ ²yield

²c

F I G U R E 3.1 Nonlinear stress distribution at ultimate conditions.

resulting equations compare favorably with test results. The most common shapes proposed are the rectangle, parabola, and trapezoid, with the rectangular shape used in this text as shown in Figure 3.2(c) being the most common one.

If the concrete is assumed to crush at a strain of about 0.003 (which is a little conservative for most concretes) and the steel to yield at f_y, it is possible to make a reasonable derivation of beam formulas without knowing the exact stress distribution. However, it is necessary to know the value of the total compression force and its centroid.

Whitney¹replaced the curved stress block [Figure 3.2(b)] with an equivalent rectangular block of intensity 0.85f_c^ and depth α = β1c, as shown in Figure 3.2(c). The area of this rectangular block should equal that of the curved stress block, and the centroids of the two blocks should coincide. Sufficient test results are available for concrete beams to provide the depths of the equivalent rectangular stress blocks. The values ofβ₁given by the code (10.2.7.3) are intended to give this result. For f_c^ values of 4000 psi or less,β₁ = 0.85, and it is to be reduced continuously at a rate of 0.05 for each 1000-psi increase in f_c^ above 4000 psi. Their value may not be less than 0.65. The values of β1 are reduced for high-strength concretes primarily because of the shapes of their stress–strain curves (see Figure 1.1 in Chapter 1).

For concretes with f_c^> 4000 psi, β1 can be determined with the following formula:

β₁= 0.85 −

f_c^− 4000 psi 1000



(0.05) ≥ 0.65

a = β1c c

(a) (b) (c)

T = A_sf_y T = A_sf_y

f'_c 0.85f'_c

F I G U R E 3.2 Some possible stress distribution shapes.

1Whitney, C. S., 1942, “Plastic Theory of Reinforced Concrete Design,” Transactions ASCE, 107, pp. 251–326.

McCormac c03.tex V2 - January 9, 2013 5:01 P.M. Page 69

3.4 Derivation of Beam Expressions 69

In SI units, β₁ is to be taken equal to 0.85 for concrete strengths up to and including 30 MPa. For strengths above 30 MPa, β₁ is to be reduced continuously at a rate of 0.05 for each 7 MPa of strength in excess of 30 MPa but shall not be taken less than 0.65.

For concretes with f_c^> 30 MPa, β1 can be determined with the following expres-sion:

β1 = 0.85 − 0.008(fc^− 30 MPa) ≥ 0.65

Based on these assumptions regarding the stress block, statics equations can easily be written for the sum of the horizontal forces and for the resisting moment produced by the internal couple. These expressions can then be solved separately for a and for the moment, M_n. A very clear statement should be made here regarding the term M_nbecause it otherwise can be confusing to the reader. M_nis defined as the theoretical or nominal resisting moment of a section. In Section 3.3, it was stated that the usable strength of a member equals its theoretical strength times the strength reduction factor, or, in this case,φMn. The usable flexural strength of a member, φMn, must at least be equal to the calculated factored moment, M_u, caused by the factored loads

φMn ≥ Mu

For writing the beam expressions, reference is made to Figure 3.3. Equating the horizontal forces C and T and solving for a, we obtain

0.85f_c^ab= Asf_y a= A_sf_y

0.85f_c^b = ρfyd

0.85f_c^, whereρ = A_s

bd = percentage of tensile steel

Because the reinforcing steel is limited to an amount such that it will yield well before the concrete reaches its ultimate strength, the value of the nominal moment, M_n, can be written as

M_n = T

and the usable flexural strength is

φMn= φAsf_y If we substitute into this expression the value previously obtained for a (it wasρf_yd/0.85f_c^), replace A_s withρbd, and equate φMn to M_u, we obtain the following expression:

φMn = Mu = φbd²f_yρ

F I G U R E 3.3 Beam internal forces at ultimate conditions.

Replacing A_s withρbd and letting R_n = M_u/φbd², we can solve this expression for ρ (the percentage of steel required for a particular beam) with the following results:

ρ = 0.85f_c^ f_y

 1−



1− 2R_n 0.85f_c^



(Eq. 3-3) Instead of substituting into this equation for ρ when rectangular sections are involved, the reader will find Tables A.8 to A.13 in Appendix A of this text to be quite convenient.

(For SI units, refer to Tables B.8 and B.9 in Appendix B.) Another way to obtain the same information is to refer to Graph 1 in Appendix A. The user, however, will have some difficulty in reading this small-scale graph accurately. This expression for ρ is also very useful for tensilely reinforced rectangular sections that do not fall into the tables. An iter-ative technique for determination of reinforcing steel area is also presented later in this chapter.