# Column Combined Flexural and Axial Design per ACI 318-19 with ideCAD

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

• The three-dimensional interaction failure surface of a column is calculated automatically.

• Demand/Capacity Ratio of Columns is calculated automatically.

Notation

Ag = gross area of concrete section, in2
As = area of nonprestressed longitudinal tension reinforcement, in2
Ast = total area of nonprestressed longitudinal reinforcement, in2
α = depth of equivalent rectangular stress block, in.
bw = width of compression face of member, in.
c = distance from extreme compression fiber to neutral axis, in.
Cc = concrete compressive force, lb
Cs = reinforcement tension force, lb
d = distance from extreme compression fiber to centroid of longitudinal tension reinforcement, in.
fc' = specified compressive strength of concrete, psi
fy = specified yield strength for nonprestressed reinforcement, psi
Mn = nominal flexural strength at section, in.-lb
Mu = factored moment at section, in.-lb
Pn = nominal axial compressive strength of member, lb
Pn,max = maximum nominal axial compressive strength of a member, lb
Pnt = nominal axial tensile strength of member, lb
Pnt,max = maximum nominal axial tensile strength of member, lb
Po = nominal axial strength at zero eccentricity, lb
Pu = factored axial force; to be taken as positive for compression and negative for tension, lb
Tn = nominal torsional moment strength, in.-lb
Tu = factored torsional moment at section, in.-lb
Vn = nominal shear strength, lb
Vu = factored shear force at section, lb
ϕ = strength reduction factor
εt = net tensile strain in 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

### Required strength

The required strength is calculated in accordance with the factored load combinations in Load Factors and Combinations per ACI 318-19 with ideCAD title. Combined Axial and Flexural Required strengths of a column Pu and Mu occur simultaneously for a column. Therefore, for each applicable factor load combination specified in ACI Table 5.3.1, the most unfavorable condition of Pu and Mu occurring simultaneously is considered.

### Design strength

Design strength in all sections should satisfy all conditions given below;

• ϕPn ≥ Pu

• ϕMn ≥ Mu

• ϕVn ≥ Vu

• ϕTn ≥ Tu

Strength reduction factors ϕ is determined according to using ACI Table 21.2.2.

Strain, εt

Section Classification

ϕ

εtεty

Compression Controlled Moment

0.65

εty < εt < (εty + 0.003)

Transition region

0.65 + 0.25[(εt - εty)/0.003]

εt ≥ (εty + 0.003)

Compression Controlled Moment

0.90

Pn and Mn are calculated in accordance with Axial strength or Combined Flexural and Axial Strength per ACI 318-19 with ideCAD title. Combined Axial and Flexural strengths create a three-dimensional interaction failure surface. In addition to axial compression and biaxial bending, the formulation allows for axial tension and biaxial bending considerations. An interaction surface of a column is shown below.

Pn means nominal axial compressive strength, and Mn means nominal flexural strength. Mn takes different values for each axial force level. Therefore, a three-dimensional interaction failure surface is formed in the picture above. Nominal flexural strength Mn is calculated according to the assumptions described in Flexural Strength per ACI 318-19 with ideCAD title. Flexural design strength ϕMn is obtained by calculating Mn and strength reduction factor ϕ at each axial force level. While finding the flexural design strength, combined with axial force ϕMn, 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. The ϕ factor for compression is used when the section is compression controlled.

As described in the title of Axial strength or Combined Flexural and Axial Strength per ACI 318-19 with ideCAD, for nonprestressed concrete members nominal axial strength Po and the maximum design compressive strength ϕPn,max values are calculated as given below;

Nominal flexural strength Mn 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 Mn and Pn are calculated as shown below.

From the equation of equilibrium:

Nominal flexural strength Mn:

### Demand/Capacity Ratio of Columns

Because of the axial force and biaxial bending interaction in the columns, the demand/capacity ratio is calculated using the interaction curve. The point (Pu,Mu) is placed in the interaction space shown as point D in the picture below. If point D (Pu,Mu) is within the acceptable range, the column capacity is adequate. However, if the point D (Pu,Mu) is outside the interaction volume, the column is overstressed. The capacity ratio is calculated using point C obtained by extending the line passing through points O and D to the interaction curve. The capacity ratio is given by the ratio OD/OC.