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    Calculates locally in your browser · ACI 318 slenderness
    Concrete / second-order effects

    Slender column calculator

    A slender column does not just carry the moment you apply: the axial load acting on the deflected shape adds more. This calculator computes the effective stiffness, the critical buckling load and the ACI moment magnifier, and shows how quickly the design moment runs away as the load approaches the buckling limit.

    FrameBracedCm1.00δ1.38
    Magnified moment
    276 kN·m
    1.38× the applied 200
    Critical load Pc
    1.455e+4 kN
    0.75Pc = 1.091e+4 kN
    Slenderness klu/r
    26.0
    Magnification required
    Status
    Stable
    Within the stability limit
    Magnification steps
    ACI 318 effective stiffness, critical load and the δ factor.
    δ 1.38
    Magnification steps
    TermValue
    Ec2.487e+4 MPa
    Ig2.133 ×10⁹ mm⁴
    0.4EcIg/(1+βdns)13.264 ×10¹² N·mm²
    Pc = π²EI/(klu)²1.455e+4 kN
    Cm1.00
    δns or δs1.379
    Mc276 kN·m

    When δ is capped at 1.0 the section is stocky enough that magnification does not increase the moment.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Magnification against axial load
    δ rises steeply as the load approaches 0.75Pc, then becomes infinite.
    Magnification against axial load0.00016.8033.6050.4067.2084.00001076107621532153322932294306430653825382645864587535753586118611968796871.076e+41.076e+4δ
    Applied vs magnified moment
    The gap is the second-order effect.
    Applied vs magnified moment0.00065.10130.21195.31260.41325.52200.00Applied M2Applied M2275.86Design McDesign MckN·m
    Canonical equations
    The exact relationships implemented by this engine.

    EI = 0.4EcIg/(1 + βdns) · ACI 318 §6.6.4.4.4.

    Pc = π²EI/(klu)² · Euler buckling of the effective section.

    δ = Cm/(1 − Pu/0.75Pc) ≥ 1 · Cm = 0.6 + 0.4(M1/M2), 1.0 for sway.

    engineering methodology

    A clear path from inputs to results.

    A slender column does not just carry the moment you apply: the axial load acting on the deflected shape adds more. This calculator computes the effective stiffness, the critical buckling load and the ACI moment magnifier, and shows how quickly the design moment runs away as the load approaches the buckling limit.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Enter the section and concrete strength.
    2. Set the factored axial load and both end moments.
    3. Choose braced or sway and the sustained load ratio.
    4. Read the magnified moment and the stability check.

    The formula

    δ=Cm1−Pu/(0.75Pc)≥1,Pc=π2EI(klu)2\delta = \frac{C_m}{1 - P_u/(0.75P_c)} \ge 1, \qquad P_c = \frac{\pi^2 EI}{(kl_u)^2}

    Reference: ACI 318-19 §6.6.4 (moment magnification), §6.6.4.4.4 (effective stiffness).

    Worked example

    400 × 400 mm, 3,000 kN, 200 kN·m

    Inputs
    • 400 × 400 mm, f′c 28 MPa.
    • Pu 3,000 kN, M2 200 kN·m, single curvature.
    • lu 3,000 mm, k = 1, βdns 0.6.
    Calculation path
    1. 1

      Ec = 4700√28 = 24,870 MPa; EI = 0.4EcIg/1.6 = 1.33 × 10¹³ N·mm².

    2. 2

      Pc = π²EI/(klu)² = 14,546 kN.

    3. 3

      Cm = 1.0 for single curvature.

    4. 4

      δ = 1/(1 − 3,000/(0.75 × 14,546)) = 1.379.

    5. 5

      Design moment: 200 × 1.379 = 276 kN·m.

    Result

    Pc 14,546 kN · δ 1.379 · Mc 276 kN·m.

    Interpreting the result

    The magnifier is benign until it is not: at 60% of the buckling load δ is about 1.4, but at 90% it is over 3. That is why the code stops the calculation at 0.75Pc rather than letting the formula run to infinity — a column that close to buckling has no reserve for imperfections.

    Assumptions & limitations

    • First-order analysis with ACI moment magnification.
    • Rectangular section, k entered by the user.
    • Sustained load ratio βdns entered directly.
    • Full second-order (P-Δ) analysis is preferable for tall frames.

    Frequently asked questions

    When can I ignore slenderness?

    Below klu/r of 22 for a braced column, magnification is not required.

    Why 0.75Pc?

    It provides a margin against buckling and covers the stiffness uncertainty in the effective EI.

    What is Cm?

    A factor for the end-moment pattern: 0.6 + 0.4(M1/M2), never below 0.4, and 1.0 for sway.

    Methodology last reviewed 2026-10-21. Results are informational and not financial or engineering advice — see our disclaimer.