The design assumptions used in the strength design method are outlined in ACI 10.2. They are applicable in the design of members subjected to flexure, axial loads, or a com- bination of both. The nominal strength of a reinforced concrete member is determined on the basis of these design assumptions.
Design Assumption No. 1: The strains in the reinforcement and the concrete shall be assumed directly proportional to the distance from the neutral axis.
The first design assumption is the traditional assumption made in beam theory: Plane sections that are perpendicular to the axis of bending prior to bending remain plane after bending. This inherently implies that the concrete and the reinforcing steel act together to resist load effects (recall that this is the second of the conditions required in the strength design method; see Sec- tion 5.1).
Strictly speaking, this assumption is not correct for reinforced concrete members after cracking occurs because the strain on the tension side of the neutral axis varies significantly at any given level owing to the presence of cracks. However, many experimental tests have confirmed that the distribution of strain is essentially linear across a reinforced concrete cross-section, even near ultimate strength, when strains are measured across the same gage length on the compressive and tensile faces of a member.2The gage lengths that were used in the tests included several cracks on
the tension face of the member.
For deep beams, which are defined in ACI 10.7, the strain is not linear, and a nonlinear distri- bution of strain must be utilized, or a strut-and-tie model as outlined in Appendix A of the Code may be used.
The strain distribution over the depth of a rectangular reinforced concrete section at ultimate strength is depicted in Fig. 5.1. For illustrative purposes, it is assumed that the strains are compres- sive above the neutral axis and are tensile below it. The strains in the concrete and the reinforcement are directly proportional to the distance from the neutral axis, which is located a distance c from the compression face of the section.
Because the strain distribution is linear, the strain in the concreteεcat the extreme compression
fiber is directly proportional to the strains in the reinforcement. For example, the ratio ofεcto
the strain in the reinforcement farthest from the compression faceεs4can be obtained by similar triangles: εc εs4 = c d4− c (5.1)
Similar relationships can be established betweenεcand the other reinforcement strains and
between the various reinforcement strains.
Note that the largest tensile strain occurs in the reinforcing steel farthest from the compression face. The concrete below the neutral axis is cracked at ultimate strength, and for all intents and purposes, it cannot resist any tensile strains (see design assumption no. 4). That is why no strain is shown on the tension face of the concrete in Fig. 5.1.
Design Assumption No. 2: The maximum usable strain at the extreme concrete compression fiber is 0.0030.
The maximum compressive strain at crushing of concrete has been measured in many ex- perimental tests of reinforced concrete members (beams and eccentrically loaded columns) and eccentrically loaded plain concrete prisms. Test data vary between 0.0030 and 0.0080 (see Ref. 2 for a summary of the test results). A maximum strain ofεc= 0.0030 is a reasonably conservative value
proposed for design (see the compressive stress–strain curves for concrete shown in Fig. 2.6).
Design Assumption No. 3: The stress in the reinforcement fs below its specified yield strength fy
is equal to the modulus of elasticity of the steel Es times the steel strainεs. The stress in the
reinforcement is equal to fyfor strainsεsgreater than or equal to fy/Es.
On the basis of the stress diagram of reinforcing steel (see Fig. 2.15), it is reasonable to assume that there is a linear relationship between stress and strain up to the yield strength fy. As noted in
Chap. 2, the modulus of elasticity can be taken as 29,000,000 psi for all grades of reinforcing steel (ACI 8.5.2).
The second part of the assumption implies that the effect of strain hardening of the steel above the yield point is neglected in strength computations. In other words, the stress in the reinforcement
fsis equal to fyfor any value of steel strainεsthat is greater than the yield strainεy= fy/Es. The
idealized stress–strain curve based on this assumption is illustrated in Fig. 5.2.
Stress
Strain
FIGURE5.2 The idealized stress–strain curve of reinforcing steel used in the strength design method.
Design Assumption No. 4: The tensile strength of concrete is neglected in the axial and flexural calculations of reinforced concrete.
It was discussed in Chap. 2 that the tensile strength of concrete is small compared with the tensile strength of reinforcing steel. Within the tension portion of a reinforced concrete cross-section, the tensile force in the cracked concrete is significantly less than the tensile force in the reinforcing steel. Thus, the tensile strength of concrete is conservatively taken as zero in the axial and flexural calculations of nominal strength.
The tensile resistance of concrete is used in other situations, most notably in serviceability calculations. For example, the modulus of rupture fr, which is related to the tensile strength (see
Chap. 2), is utilized in the determination of the immediate deflection of a reinforced concrete member.
Design Assumption No. 5: The relationship between the concrete compressive stress distribution and the concrete strain shall be assumed to be rectangular, trapezoidal, parabolic, or of any other shape that results in prediction of strength in substantial agreement with the results of comprehensive tests.
Concrete behaves inelastically when subjected to a relatively high compressive stress, as is evident from the stress–strain curves in Fig. 2.6. Nonlinear behavior becomes pronounced after the stress reaches approximately 50% of the compressive strength fc(see Fig. 2.5).
Although a general nonlinear model for compressive stress distribution could be used when determining the nominal strength of a reinforced concrete member, such as the one illustrated in Fig. 5.3 for a flexural member, it is simpler to make use of a less complicated distribution as long as the simpler model yields results close to those from tests. Note that the shape of the stress distribution in the figure follows that of a stress–strain curve in compression where, as expected, zero stress occurs at the level of the neutral axis. The tension force T in the reinforcing steel must be equal to the resultant force C of the compressive stress in the concrete so that equilibrium is satisfied.
Numerous compressive stress distributions have been proposed through the years, and a summary of these can be found in Chap. 6 of Ref. 3. Additional information on the historical background of the distributions and a review of the tests that were performed to support the proposed stress distributions are given in Ref. 2.
Research has shown that models using rectangular, parabolic, trapezoidal, and other-shaped compressive stress distributions can adequately predict test results. The assumption given in ACI 10.2.7 permits the use of an equivalent rectangular concrete stress distribution, which is covered under design assumption no. 6.
Design Assumption No. 6: The requirements of ACI 10.2.6 are satisfied by an equivalent rectangular concrete stress distribution, which is defined in ACI 10.2.7.
FIGURE5.3 Stress conditions at nominal strength.
FIGURE5.4 The equivalent rectangular concrete stress distribution.
The Code permits the use of the equivalent rectangular concrete stress distribution defined in ACI 10.2.7, which is illustrated in Fig. 5.4. Although he was not the first to propose the use of a rectangular stress block, C. S. Whitney is best known in the United States for advocating it.4
A uniform stress equal to 85% of the concrete compressive strength fcis distributed over the depth a , which is equal to the factorβ1times the depth to the neutral axis c. Although this assumed
stress distribution does not represent the actual compressive stress distribution in the concrete at the ultimate state, it does provide basically the same results as those obtained from experimental investigations2; as noted previously, this is a requirement of the strength design method (see design
assumption no. 5).
The need for the factorβ1is due to the variation in shape of the stress–strain curves for different
concrete strengths. It is evident that the stress–strain curves of higher-strength concretes are more linear and exhibit less inelastic behavior than those of lower-strength concretes (see Fig. 2.6). Up to compressive strengths of 4,000 psi, the ratio of the rectangular stress block depth a to the neutral axis depth c that best approximates the actual concrete stress distribution is equal to 0.85, that is,
β1= 0.85. For compressive strengths greater than 4,000 psi, β1must be less than 0.85 in order to
produce adequate results. ACI 10.2.7.3 requires thatβ1be reduced linearly at the rate of 0.05 for
each 1,000 psi in excess of 4,000 psi for compressive strengths up to 8,000 psi; above 8,000 psi,
β1= 0.65. The following equations define β1: r For 2,500 psi≤ f c≤ 4,000 psi, β1= 0.85. r For 4,000 psi< f c≤ 8,000 psi, β1= 1.05 − 0.00005 fc. r For f c> 8,000 psi, β1= 0.65.
The lower limit of 0.65 was introduced in the 1976 supplement to the 1971 Code on the basis of the results of experiments that were performed on concrete specimens with compressive strengths exceeding 8,000 psi.5,6