Beam-Column Linear Interaction Ratio
Beam-Column Linear Interaction Ratio
Most real structural members do not see pure axial load or pure bending in isolation — a column in a frame typically carries both an axial force from gravity load and a bending moment from frame action or lateral load. A beam-column interaction check combines both effects into a single ratio to judge whether the combined loading is within the member's capacity.Checking axial and bending capacity separately and assuming both pass is not conservative enough, because the two interact: a column already stressed near its axial capacity has much less reserve left for bending than an unloaded one, and vice versa. This is why every steel design code includes an interaction equation as a required check on beam-columns, not an optional refinement.
The simplified linear interaction ratio sums the fractional axial and bending demand-to-capacity ratios, ratio = P/P_allow + M/M_allow, and must not exceed 1.0. where P is the applied axial load, P_allow is the allowable axial capacity, M is the applied bending moment, M_allow is the allowable bending moment capacity, and ratio is the combined interaction value.
Inputs
P := 300 kN=300000 N
P_allow := 800 kN=800000 N
M := 120 kN·m=120000 Nm
M_allow := 250 kN·m=250000 Nm
Adding the fraction of axial capacity used to the fraction of bending capacity used gives a single combined demand-to-capacity ratio for the member.
ratio := P/P_allow + M/M_allow=0.85
Results
A ratio of 0.85, as found here, means the member is adequate but has only about 15% reserve left for either additional axial load or additional moment — a fairly efficiently used section. Because this linear form treats axial and bending demand as fully additive, it is a conservative simplification; production code equations (like AISC H1) use a bilinear form that is somewhat less conservative when axial load is low relative to P_allow. If the ratio exceeds 1.0, increasing the section size affects both P_allow and M_allow simultaneously, which is usually more efficient than adding separate axial or flexural reinforcement.
Assumptions & limitations
- Simplified linear interaction; actual code equations (e.g., AISC H1-1) use separate forms above and below a P/P_allow threshold.
- Single-axis bending assumed — no biaxial bending term included.
- P_allow and M_allow already account for member slenderness and buckling effects, computed separately.
- No second-order (P-delta) moment amplification is included in M.