Axial strength or Combined Flexural and Axial Strength per ACI 318-19 with ideCAD

How does ideCAD calculate combined flexural and axial strength according to ACI 318-19?

Notation

A_g = gross area of the concrete section, in²
A_s = area of non-prestressed longitudinal tension reinforcement, in²
A_st = total area of non-prestressed longitudinal reinforcement, in²
α = depth of equivalent rectangular stress block, in.
b_w = width of compression face of the member, in.
c = distance from extreme compression fiber to the neutral axis, in.
C_c = concrete compressive force, lb
C_s = reinforcement tension force, lb
f_c^' = specified compressive strength of concrete, psi
f_y = specified yield strength for non-prestressed reinforcement, psi
M_n = nominal flexural strength at section, in.-lb
P_n = nominal axial compressive strength of member, lb
P_n,max = maximum nominal axial compressive strength of a member, lb
P_nt = nominal axial tensile strength of member, lb
P_nt,max = maximum nominal axial tensile strength of member, lb
P_o = nominal axial strength at zero eccentricity, lb
ϕ = strength reduction factor
ε_t = net tensile strain in the extreme layer of longitudinal tension reinforcement at nominal strength, excluding strains due to effective prestress, creep, shrinkage, and temperature
β₁ = factor relating depth of equivalent rectangular compressive stress block to depth of neutral axis

Maximum axial compressive strength

Nominal axial compressive strength P_n is limited to a value of P_n,max =0.8P_o for non-prestressed members.

For non-prestressed concrete members, P_o shall be calculated by:

According to ACI Table 22.4.2.1 Maximum axial compressive strength for a non-prestressed concrete member with closed tie;

The maximum design compressive strength for a non-prestressed concrete member with a closed tie;

Strength reduction factors ϕ is determined according to using ACI Table 21.2.2. Since the section is compression controlled, a ϕ factor for compression control is used.

Maximum axial tensile strength

Nominal axial tensile strength P_nt is limited to a value of P_nt,max in ACI 22.4.3.1.

The maximum tensile axial load for a non-prestressed concrete member;

Strength reduction factors ϕ is determined according to using ACI Table 21.2.2. Since the section is tension controlled, a ϕ factor for tension control is used.

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Design assumptions

The flexural and axial strength of a member calculated by the strength design method, two basic conditions should be satisfied:

1. equilibrium

2. compatibility of strains

Equilibrium means balancing of forces acting on the element cross section at nominal strength. Stress-strain relationship for the concrete and the reinforcement at nominal strength is established within the design assumptions described following:

• Equilibrium is satisfied in each section.

• It is assumed that strain in concrete and reinforcement is proportional to the distance from the neutral axis.

Design strength is calculated by using these assumptions together with design assumptions for concrete described following:

• Maximum strain at the extreme concrete compression fiber is assumed to equal 0.003.

• The tensile strength of concrete is neglected.

• The equivalent rectangular concrete stress distribution method represents the relationship between concrete compressive stress and strain.

• Concrete stress of 0.85f_c^' is assumed to be uniformly distributed over. Equivalent rectangular concrete stress zone bounded by edges of the cross-section and a line parallel to the neutral axis located a distance α from the fiber of maximum compressive strain, as calculated by:

• The distance between the fiber of maximum compressive stress and the neutral axis, c, is perpendicular to the neutral axis.

• The value of β₁ is determined using ACI Table 22.2.2.4.3.

Combined Flexural and Axial Strength

Nominal flexural strength M_n and axial strength are calculated using Design Assumptions. While finding the flexural design strength, combined with axial force ϕM_n, it should be found in which control zone the cross-section is. When the section is tension controlled, a ϕ factor for tension control is used. A ϕ factor for compression control is used when the section is compression controlled. When the section is within the transition region, ϕ is linearly interpolated between the two limit values.

Nominal flexural strength M_n with zero compression is calculated as described in the title of Flexural Strength per ACI 318-19 with ideCAD. Similarly, with the same design assumptions combined nominal flexural and axial strength M_n and P_n are calculated as shown below.

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From the equation of equilibrium:

Nominal flexural strength M_n: