The three assumptions already made are sufficient to allow calculation of the strength and behavior of reinforced concrete elements. For design purposes, however, the following additional assumptions are introduced to simplify the problem with little loss of accuracy.
1. The tensile strength of concrete is neglected in flexural-strength calculations (ACI Code Section 10.2.5).
The strength of concrete in tension is roughly one-tenth of the compressive strength, and the tensile force in the concrete below the zero strain axis, shown as in Fig. 4-9d, is small compared with the tensile force in the steel. Hence, the contribution of the tensile stresses in the concrete to the flexural capacity of the beam is small and can be neglected.
It should be noted that this assumption is made primarily to simplify flexural calculations.
In some instances, particularly shear, bond, deflection, and service-load calculations for prestressed concrete, the tensile resistance of concrete is not neglected.
2. The section is assumed to have reached its nominal flexural strength when the strain in the extreme concrete compression fiber reaches the maximum useable compres- sion strain,
Strictly speaking, this is an artificial limit developed by code committees to define at what point on the general moment–curvature relationship the nominal strength of the section is to be calculated. As shown in Fig. 4-10, the moment–curvature relationship for a typical beam section is relatively flat after passing the yield point, so the selection of a specific value for will not significantly affect the calculated value for the nominal flexural strength of the section. Thus, design calculations are simplified when a limiting strain is assumed.
The maximum compressive strains, from tests of beams and eccentrically loaded columns of normal-strength, normal-density concrete are plotted in Fig. 4-13a [4-6], [4-7]. Similar data from tests of normal-density and lightweight concrete are compared in Fig. 4-13b. ACI Code Section 10.2.3 specifies a limiting compressive strain, equal to 0.003, which approximates the smallest measured values plotted in Fig. 4-13a and b. In
ecu, ecu,
ecu
ecu.
Tc As. Aœs,
Fig. 4-13
Limiting compressive strain.
(From [4-6] and [4-7].)
Canada, the CSA Standard [4-5] uses for beams and eccentrically loaded columns. Higher limiting strains have been measured in members with a significant moment gradient and in members in which the concrete is confined by spirals or closely spaced hoops [4-8], [4-9]. Throughout this book, however, a constant maximum useable compressive strain equal to 0.003 will be used.
3. The compressive stress–strain relationship for concrete may be based on mea- sured stress–strain curves or may be assumed to be rectangular, trapezoidal, parabolic, or any other shape that results in prediction of flexural strength in substantial agreement with the results of comprehensive tests (ACI Code Section 10.2.6).
Thus, rather than using a closely representative stress–strain curve (such as that given in Fig. 4-8), other diagrams that are easier to use in computations are acceptable, provided they adequately predict test results. As is illustrated in Fig. 4-14, the shape of the
ecu = 0.0035
Fig. 4-14
Mathematical description of compression stress block.
(b) Triangle.
(a) Concrete.
(c) Parabola.
Fig. 4-15
Values of and for various stress distributions.
k2 k1
stress block in a beam at the ultimate moment can be expressed mathematically in terms of three constants:
ratio of the maximum stress, in the compression zone of a beam to the cylinder strength,
ratio of the average compressive stress to the maximum stress (this is equal to the ratio of the shaded area in Fig. 4-15 to the area of the rectangle, ) ratio of the distance between the extreme compression fiber and the resultant of the compressive force to the depth of the neutral axis,c, as shown in Figs. 4-14 and 4-15.
For a rectangular compression zone of width band depth to the neutral axis c, the resultant compressive force is
(4-13a) Values of and are given in Fig. 4-15 for various assumed compressive stress–strain diagrams or stress blocks. The use of the constant essentially has disappeared from the flexural theory of the ACI Code. As shown in Fig. 4-12, a large change in the concrete compressive strength did not cause a significant change in the beam section moment capacity. Thus, the use of either or with typically taken equal to 0.85, is not significant for the flexural analysis of beams. The use of is more significant for col- umn sections subjected to high axial load and bending. Early papers by Hognestad [4-10]
and Pfrang, Siess, and Sozen [4-11] recommended the use of fcfl = k3fcœ when analyzing fcfl
k3 fcfl = k3fcœ, fcœ
k3 k2
k1
C = k1k3fcœbc k2 =
c * k3fcœ k1 =
fcœ
fcfl, k3 =
Fig. 4-16
Equivalent rectangular stress block.
the combined axial and bending strength for column sections. However, the ACI Code does not refer to the use of except for column sections subjected to pure axial load (no bend- ing), as will be discussed in Chapter 11.
Whitney Stress Block
As a further simplification, ACI Code Section 10.2.7 permits the use of an equivalent rec- tangular concrete stress distribution shown in Fig. 4-16 for nominal flexural strength cal- culations. This rectangular stress block, originally proposed by Whitney [4-12], is defined by the following:
1. A uniform compressive stress of 0.85 shall be assumed distributed over an equivalent compression zone bounded by the edges of the cross section and a straight line located parallel to the neutral axis at a distance from the concrete fiber with the maximum compressive strain. Thus, as shown in Fig. 4-16.
2. The distance cfrom the fiber of maximum compressive strain to the neutral axis is measured perpendicular to that axis.
3. The factor shall be taken as follows [4-7]:
(a) For concrete strengths, up to and including 4000 psi,
(4-14a) (b) For 4000 psi 8000 psi,
(4-14b) (c) For greater than 8000 psi,
(4-14c) For a rectangular compression zone of constant width band depth to the neutral axis c, the resultant compressive force is
(4-13b) Comparing Eqs. 4-13a and 4-13b, and setting k3 = 1.0,results in k1 = 0.85b1.
C = 0.85fcœbb1c = 0.85b1fcœbc b1 = 0.65
fcœ
b1 = 0.85 - 0.05fcœ - 4000 psi 1000 psi 6 fcœ …
b1 = 0.85 fcœ,
b1
k2 = b1/2,
a = b1c fcœ fcfl
0.85b1
Fig. 4-17
Values of from tests of concrete prisms. (From [4-7].)
b1
In metric units (MPa), the factor shall be taken as follows:
(a) For concrete strengths, up to and including 28 MPa,
(4-14Ma)
(b) For 28 MPa 56 MPa,
(4-14Mb) (c) For greater than 58 MPa,
(4-14Mc) The dashed line in Fig. 4-17 is a lower-bound line corresponding to a rectangular stress block with a height of and by using as given by Eq. (4-14). This equiva- lent rectangular stress block has been shown [4-6], [4-7] to give very good agreement with test data for calculation of the nominal flexural strength of beams. For columns, the agree- ment is good up to a concrete strength of about 6000 psi. For columns loaded with small eccentricities and having strengths greater than 6000 psi, the moment capacity tends to be overestimated by the ACI Code stress block. This is because Eq. (4-14) for was chosen as a lower bound on the test data, as indicated by the dashed line in Fig. 4-17. The internal moment arm of the compression force in the concrete about the centroidal axisof a rectan- gular column is where c is the depth to the neutral axis (axis of zero strain). If is too small, the moment arm will be too large, and the moment capacity will be overestimated.
To correct this potential error, which can lead to unconservative designs of columns constructed with high-strength concrete, an ACI Task Group [4-13] has recommended the
b1
1h/2 - b1c/22,
b1 b1
0.85fcœ
b1 = 0.65 fcœ
b1 = 0.85 - 0.05fcœ - 28 MPa 7 MPa 6 fcœ …
b1 = 0.85 fcœ,
b1
use of a coefficient, to replace the constant 0.85 as the definition for the height of the stress block shown in Fig. 4-16a. This new coefficient is defined as follows:
(a) For concrete strengths, up to and including 8000 psi,
(4-15a) (b) For 8000 psi between 8000 and 18,000 psi,
(4-15b) (c) For greater than 18,000 psi,
(4-15c) Until the ACI Code adopts a modification of the stress block shown in Fig. 4-16a, the authors recommend the use of this coefficient, when analyzing the flexural strength of columns constructed with concrete strengths exceeding 8000 psi.