Generation of Column Interaction Capacity

RAM Concrete Column uses the dead load and live skip loading results from Concrete Analysis in conjunction with the RAM Frame lateral analysis results, where applicable, to create a set of design points for each column using the defined load combinations. These design points are a set of Axial Forces, Major Moments, Minor Moments, Major Shears, Minor Shears and Torsion at the top and bottom of the column for each load combination and pattern (see Section 2.3.4).

The column capacity is based on a biaxial interaction surface that identifies the Major and Minor Moment limits for any given axial load.

Maximum Column Axial Capacity

The maximum axial compression capacity of the column is limited to:

ϕ_bP_n(max) = 0.85ϕ_c[0.85f'_c(A_g - A_s) + f_yA_s]

Equation ACI 10-1

The maximum axial tension capacity of the column is calculated using only the reinforcement capacity:

ϕ_bP_n = ϕ_bf_yA_s

Column Flexural Capacity

Capacity points on the interaction surface are calculated using the reinforcement force and concrete section compression force assuming a C.S. Whitney Equivalent Rectangular Stress Distribution as outlined in ACI 10.2.7.

P = C_c + C_s + T_s

Equation 4-2

Reinforcement tension force is:

T_s = -ΣF_sA_s

where

F_s

ε_{c} E_{s} \frac{L_{s}}{L_{c}} \leq f_{y}

Equation 4-3

Reinforcement compression force is:

C_s = Σ(F_s - 0.85f'_c)A_s

Equation 4-4

Concrete compression force is:

C_c = 0.85f'_cab

Equation 4-5

where

\frac{A_{s} f_{y}}{0.85 {f^{'}}_{c} b}

Equation 4-6

Section moment capacity is:

M = M_s + M_c

Equation 4-7

M_s = ΣT_sL_s + ΣC_sL_s

Equation 4-8

M_c = C_cL_c

Equation 4-9

Whitney equivalent stress distribution

The above calculation is performed for the major and minor direction of the column section, as well as all intermittent angles at the specified increment. The result is a full 3-D interaction surface. The interaction surface between the major and minor directions is calculated explicitly and is not based on an approximate method like the PCA Method outlined in Reference # 4.

The interaction surface is then reduced by the following factors:

For ϕ_cP_n(max) ≥ P_u > 0.10f'_cA_g

ACI-9.3.2.2

If f_y ≤ 60 ksi and (h - d - d_s)/h ≥ 0.70

ACI-9.3.2.2

then when 0.10f'_cA_g ≥ P_u ≥ 0.0, ϕ is linearly increases from ϕ_c to ϕ_b

Otherwise when:

min(0.10f'_cA_g, ϕ_cP_b) ≥ P_u ≥ 0.0

ACI-10.3.3

ϕ is linearly increases from ϕ_c to ϕ_b

For P_u < 0.0, ϕ = ϕ_b = 0.90

Interaction Diagram Check

Each data point is checked against the interaction surface to confirm that it is inside the surface. The factored moments M_{u major}, M_{u minor} are converted to $M_{u θ} = \sqrt{M_{u major}^{2} + M_{u minor}^{2}}$ and checked against the interaction surface at the angle $θ = \tan^{- 1} (M_{u minor} / M_{u major})$

The load to capacity ratio Ld/Cap is calculated in one of two ways based on the axial load.

When P_u > min (0.1f'_cA_s, ϕ_cP_b)

Ld/Cap = max(P_u/ϕ_cP_{n max}, M_uθ/ϕ_cM_nθ)

When P_u ≤ 0

Ld/Cap = max(P_u/ϕ_bP_{n min}, M_uθ/ϕ_bM_nθ)

when

min(0.1f'_cA_s, ϕ_cP_b) > P_u > 0

ϕ is linearly increases from ϕ_c to ϕ_b

Column Interaction Diagram at Angle θ

Longitudinal Reinforcement Spacing

The longitudinal reinforcement spacing limit is checked using:

s_min = min(1.5d_b, 1.5 in.)

ACI-7.6.3

s_max does not have a code limit other than the requirements for bracing using ties.

The user defined spacing limits may control if they are more restrictive than the code prescribed limits.

Longitudinal Reinforcement Ratio

The area of longitudinal reinforcement is subject to the following limits:

0.01 ≤ ρ ≤ 0.08

ACI-10.9.1

ρ ≤ 0.75ρ_b when ϕP_n ≤ min(0.10f^' _cA_g, ϕP_b)

ACI-10.3.3

where

ρ_b	=	$0.05 β_{1} \frac{{f^{'}}_{c}}{f_{y}} \frac{87, 000}{87, 000 - f_{y}}$ ACI-10.3.2
β₁	=	$0.85 - 0.05 \frac{{f^{'}}_{c} - 4, 000}{1, 000}$ ACI-10.2.7.3 limited by 0.65 ≤ β₁ ≤ 0.85

Note: The design check does not account for the final provision of ACI 318-95 and 99, Section 10.3.3, which allows the reinforcement in compression to not be reduced by 0.75. This will cause some columns to have a Design Warning generated because all the reinforcement in the column is reduced by the 0.75 factor. In most cases the engineer will be able to make a judgment on whether at least one side of the reinforcement will be in compression (which in most cases is a valid assumption), in which case a quick hand calculation will confirm that the provision of 10.3.3 will be met and the Design Warning can be disregarded.