• Three-dimensional interaction failure surface of a column 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.
D = dead load
E = earthquake load
fc' = specified compressive strength of concrete, psi
fy = specified yield strength for nonprestressed reinforcement, psi
L = live load
Lr = roof live load
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
R = rain load
S = snow load
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
W = wind load
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

Required strength is calculated in accordance with the factored load combinations in Load Factors and Combinations 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 taken into account.

Design strength

Design strength at all sections should be 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.

Pn and Mn is calculated in accordance with Axial strength or Combined Flexural and Axial Strength title. Combined Axial and Flexural strengths creates 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 shown in the picture above is formed. Nominal flexural strength Mn is calculated according to the assumptions described in Flexural Strength 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. When the section is compression controlled, a ϕ factor for compression control is used.

The Load Combination LC3, which is not within the acceptable region as shown in the picture above, is not suitable for the design. Although the design moment strength for LC2 is lower than LC3, LC2 is suitable for design because it is in the acceptable region.

As described in the title of Axial strength or Combined Flexural and Axial Strength, 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. 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, demand/capacity ratio is calculated by 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 colums capacity is adequate. However, if the point D (Pu,Mu) outside the interaction volume, the column is overstressed. The capacity ratio is calculated using point C obtained by extending the line passing through point O and D to the interaction curve. The capacity ratio is given by the ratio OD/OC.