Maximum Reinforcement Steel Area per. ACI 318 Calculator

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Maximum Reinforcement Steel Area per. ACI 318 Calculator

Civil Engineering and Design
Strength of Materials Basics and Equations | Mechanics of Materials

Maximum Reinforcement Steel Area per. ACI 318 Equations and Calculator

In a ductile failure mode, the steel yields in tension before the concrete crushes in compression. (The opposite would be a brittle failure mode.) The area of tension steel for which steel yielding occurs simultaneously with concrete crushing in compression is known as the balanced steel area, A_sb. An under-reinforced beam is defined as one having a steel area less than A_sb. An over-reinforced beam has more steel than A_sb.

The balanced steel area is

Eq. 1
A_sb = ρ_sb · b · d = 0.85 · f '_c · A_cb / f_y = 0.85 · f '_c · a_b · b / f_y

A_sb = 0.85 · f '_c · β₁ c_b b / f_y

Figure 1 illustrates that the compression area for balanced conditions, A_cb, extends from the most compressed fiber to a depth equal to a_b.

Eq. 2(a), SI
a_b = β₁ ( 600 / ( 600 + f_y ) ) d

Eq. 2(b), US
a_b = β₁ ( 87,000 / ( 87,000 + f_y ) ) d

Figure 1, Beam at Balanced Condition

The factor β₁ is defined as

Eq. 3(a), SI
β₁ = 0.85 [ f '_c ≤ 27.6 MPa]

Eq. 3(b), US
β₁ = 0.85 [ f '_c ≤ 4,000 lbf/in²]

Eq. 3(c), SI
β₁ = 0.85 - 0.05 (( f '_c - 27.6 ) / 6.9 ) ≥ 0.65 MPa

Eq. 3(d), US
β₁ = 0.85 - 0.05 (( f '_c - 4,000 ) / 1,000) ≥ 0.65 lbf/in²

Combining Eq. 1 and Eq. 2 with A_cb = a_bb, for rectangular sections, the balanced steel area is

Eq. 4(a), SI
A_sb = ρ_sb · b · d = ( ( 0.85 ·β₁ · f '_c ) / f_y · (600 / ( 600 + f_y ) ) · b · d

Eq. 4(b), US
A_sb = ρ_sb · b · d = ( ( 0.85 · β₁ · f '_c ) / f_y · (87,000 / ( 87,000 + f_y ) ) · b · d

Previous versions of ACI 318 ensured the steel would fail in tension before the concrete failed in compression (i.e., ductile failure) by limiting the maximum reinforcement to 75% of the balanced reinforcement. (This provision is still present in ACI 318 Sec. B10.3.3 of the alternative provisions.)

Eq. 5
A_s,max = 0.75 · A_sb [alternative provisions only]

For unified design, the ductile failure is ensured by specifying a minimum tensile steel strain, ε_t,min of 0.004.

The unified design provisions limit the maximum reinforcement in a flexural member (with factored axial load less than 0.1f'_cA_g) to that which would result in a net tensile strain, ε_t , not less than 0.004 [ACI 318 Sec. 10.3.5]. The fraction of the balanced reinforcement ratio equivalent to a steel strain of 0.004 depends on both the concrete ultimate compressive strength and the steel tensile yield strength. For a singly reinforced section with grade 60 reinforcement, this is equivalent to a maximum reinforcement ratio of 0.714ρ_b. (By comparison, the ACI 318 App. B limit of ρ_max = 0.75ρ_b results in a net tensile strain of 0.00376.) At the net tensile strain limit of 0.004, the strength reduction factor of ACI 318 Sec. 9.3.2, Φ, is reduced to 0.817.

It is almost always advantageous to limit the net tensile strain in flexural members to a minimum of 0.005, which is equivalent to a maximum reinforcement ratio of 0.63ρ_b [ACI 318 Sec. R9.3.2.2], even though the code permits higher amounts of reinforcement that produce lower net tensile strains. Where member size is limited and extra strength is needed, it is best to use compression reinforcement to limit the net tensile strain so that the section is tension controlled.

Where:

A_sb = Area, in² (mm²)
a_b = depth from equation 2, in (mm)
b = width, in (mm)
c_b = distance, in (mm)
d = Beam effective depth (tension to steel centroid), in, (m)
β₁ = stress constant, equation 3, lbf/in² (MPa)
f_y = Tension steel yield strength, lbf/in² (MPa)
f '_c = Compressive strength of concrete, lbf/in² (MPa)
b_w = Width of web, ft, (m)
ρ_sb = reinforment steel ratio

Reference

ACI 318 Building Code Requirements for Structural Steel, paragraph 10.5 Minimum reinforcement of flexural members
Civil Engineering Reference Manual, Fifteenth Edition, Chapter 48, p. 50.6, Paragraph 8 (Maximum Steel Area), Michael R. Lindeburg, PE