Columns and Slenderness in TS 500: As the Column Bends, the Moment Grows

1. Second order: as the column bends, the moment grows

When a column carrying axial load bends, the load acts with an eccentricity relative to the column axis and produces an additional moment of its own. This second-order moment increases the bending, and the increased bending enlarges it further. In a short column the effect is small; in a long, slender column it can govern. TS 500 calls this the slenderness effect and gives two routes in Clause 7.6: a general method based on second-order analysis and a moment magnification method.

2. Dimensions and reinforcement first

TS 500 Clause 7.4.1 sets the basic conditions for columns: width at least 250 mm for rectangular sections (diameter at least 300 mm for circular ones); N_d ≤ 0.9·f_cd·A_c (Eq. 7.7); longitudinal reinforcement ratio at least 1% (down to 0.5% if at least 1.3 times the required steel is provided) and at most 4% (6% in lap zones); longitudinal bars at least Φ14; tie diameter not less than one third of the largest longitudinal bar, spacing not more than 12 times the smallest longitudinal bar and 200 mm.

3. When is a column slender?

The measure is the slenderness ratio: effective length over radius of gyration, ℓ_k/i. For rectangular sections i may be taken as 30% of the dimension in the bending direction. The effective length is the clear length times a factor k reflecting the end restraint (ratio α of column to beam stiffnesses) and whether storey sway is prevented (Eqs. 7.14–7.16).

Slenderness may be neglected if: in a braced storey ℓ_k/i ≤ 34 − 12·(M₁/M₂) ≤ 40 (Eq. 7.17); in an unbraced storey ℓ_k/i ≤ 22 (Eq. 7.18). When the end moments bend the column in single curvature (M₁/M₂ = +1) the limit drops to 22; in double curvature it rises to 40. Beyond a slenderness ratio of 100 the approximate method may not be used; a second-order analysis is required (7.6.1).

4. Moment magnification

In the approximate method the larger end moment is magnified by a factor. For a braced column:

β = C_m / (1 − 1.3·N_d/N_k) ≥ 1 (7.24),   C_m = 0.6 + 0.4·(M₁/M₂) ≥ 0.4 (7.25),   N_k = π²(EI)/ℓ_k² (7.19),   (EI) = 0.4·E_c·I_c/(1 + R_m) (7.21)

R_m is the sustained-load share of the axial force: creep under sustained load reduces stiffness. The design moment is M_d = β·M₂, and M₂ may not be less than the moment from the minimum eccentricity (e_min = 15 + 0.03h mm, Eq. 6.16). In unbraced storeys a storey factor βs is also computed, ΣN_d ≤ 0.45·ΣN_k must hold, and for very slender columns the two factors are multiplied (Eqs. 7.27–7.29).

An example (the site’s new function, verified by hand): 400×400 mm, C30, clear length 4 m, braced, k = 1, N_d = 1500 kN, R_m = 0.6, M₁ = 60, M₂ = 100 kNm (single curvature). ℓ_k/i = 33.3 > 34 − 12·0.6 = 26.8: the column is slender. E_c = 31,801 MPa, N_k = 10,462 kN, C_m = 0.84, β = 1.032, M_d = 103.2 kNm.

The figure shows how sensitive β is to the axial load level: for a column with n = N_d/(f_ck·A_c) = 0.35, β = 1.43 at ℓ_k/i = 40 and 3.09 at 60. As load grows, the column approaches instability at a much shorter length.

5. What we found in the site’s engine

• There was no slenderness check. The site’s column presizing tool said "slenderness is checked separately", but no function did so. A new function implementing all of TS 500 Clause 7.6.2 was added (the example and figure above come from it).
• Detailing rules are not checked: Φ14, tie diameter and spacing, 300 mm between restrained bars. These are outside the engine’s scope; the designer must check them separately.
• Two relaxations are not used: ρ ≥ 0.5% (with the 1.3 condition) and ρ ≤ 6% at laps. The engine stays on the safe side.
• Pure axial capacity: TS 500 gives no explicit equation for it; the engine uses the gross section and counts the concrete displaced by steel twice (about 0.8% high at 1% steel). Since the Eq. 7.7 limit is far below this value in design, results are not affected.
• A notation in the source: in the upper bound of Eq. 7.14 TS 500 writes "0.85 + 0.05·α₁"; in the ACI 318 counterpart this term uses the smaller of the two end values of α. The function applies the TS 500 text as printed.

6. Rules

1. Compute the slenderness ratio first. In most columns it can be neglected — but the calculation shows that, not an assumption.
2. Single curvature is the worst case. If the end moments bend in the same direction the neglect limit drops to 22.
3. Sustained load reduces stiffness. As R_m grows, N_k falls and β rises.
4. Take sway seriously. In an unbraced storey the limit is 22 and the storey as a whole must satisfy ΣN_d ≤ 0.45·ΣN_k.