{
    "name": "Euler Buckling Load Calculator",
    "url": "https://calculators.geekmidst.com/calculators/euler-buckling-load-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/euler-buckling-load-calculator.md",
    "category": "Science & engineering",
    "description": "Find the Euler critical buckling load π²EI ÷ (KL)² of a slender column, its slenderness ratio and critical stress, and check it against yield strength.",
    "summary": "A slender column under compression bows sideways and fails at the Euler load, π²EI ÷ (KL)², long before the material is crushed. The end conditions enter through the effective-length factor K.",
    "also_called": [
        "Column buckling calculator",
        "Critical load calculator"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "e",
            "label": "Elastic modulus E",
            "type": "number",
            "default": "200",
            "unit": "GPa"
        },
        {
            "name": "i",
            "label": "Least second moment of area I",
            "type": "number",
            "default": "500",
            "unit": "cm⁴"
        },
        {
            "name": "length",
            "label": "Unbraced length L",
            "type": "number",
            "default": "5",
            "unit": "m"
        },
        {
            "name": "k",
            "label": "Effective-length factor K",
            "type": "number",
            "default": "1"
        },
        {
            "name": "area",
            "label": "Cross-sectional area A",
            "type": "number",
            "default": "20",
            "unit": "cm²"
        },
        {
            "name": "fy",
            "label": "Yield strength (optional)",
            "type": "number",
            "default": "250",
            "unit": "MPa",
            "optional": true
        }
    ],
    "outputs": [
        {
            "key": "pcr",
            "label": "Critical buckling load",
            "format": "number"
        },
        {
            "key": "effective_length",
            "label": "Effective length KL",
            "format": "number"
        },
        {
            "key": "radius_gyration",
            "label": "Radius of gyration r",
            "format": "number"
        },
        {
            "key": "slenderness",
            "label": "Slenderness ratio KL/r",
            "format": "number"
        },
        {
            "key": "sigma_cr",
            "label": "Critical stress",
            "format": "number"
        },
        {
            "key": "validity",
            "label": "Euler validity check",
            "format": "text"
        }
    ],
    "formula": {
        "text": "P_cr = π² E I ÷ (K L)²;  r = √(I ÷ A);  slenderness = K L ÷ r;  σ_cr = P_cr ÷ A",
        "variables": {
            "E": "elastic modulus",
            "I": "least second moment of area",
            "K": "effective-length factor",
            "L": "unbraced length",
            "A": "cross-sectional area",
            "r": "radius of gyration"
        }
    },
    "worked_example": {
        "inputs": {
            "e": "200",
            "i": "500",
            "length": "5",
            "k": "1",
            "area": "20",
            "fy": "250"
        },
        "steps": [
            "EI = 200 × 10⁹ × 500 × 10⁻⁸ = 1 × 10⁶ N·m².",
            "P_cr = π² × 10⁶ ÷ (1 × 5)² = 9.8696 × 10⁶ ÷ 25 = 394,784 N = 394.78 kN.",
            "r = √(500 ÷ 20) = 5 cm, so KL/r = 500 ÷ 5 = 100.",
            "σ_cr = 394,784 N ÷ 2,000 mm² = 197.4 MPa, below fy = 250 MPa, so elastic buckling governs."
        ],
        "result": "A 5 m pinned-end steel column with I = 500 cm⁴ buckles elastically at about 395 kN."
    },
    "interpretation": "The Euler load is an upper bound for a perfect column. If σ_cr approaches or exceeds the yield strength (stocky columns), failure is inelastic and design codes use column curves that give lower capacities. Apply a safety factor before using the load.",
    "assumptions": [
        "Perfectly straight, centrally loaded, linear-elastic prismatic column.",
        "The effective-length factor K represents the end restraint you enter."
    ],
    "limitations": [
        "Real columns have initial crookedness and residual stresses, so the true capacity is lower than P_cr, especially at intermediate slenderness.",
        "Not a code design check."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Which axis should I use?",
            "answer": "The one with the smaller I (and its matching length and K), because the column buckles about its weakest axis."
        },
        {
            "question": "Why does K = 0.5 give four times the load?",
            "answer": "Load varies with 1 ÷ (KL)², so halving the effective length quadruples P_cr."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Beam Deflection Calculator",
            "url": "https://calculators.geekmidst.com/calculators/beam-bending-deflection-calculator"
        },
        {
            "name": "Factor of Safety Calculator",
            "url": "https://calculators.geekmidst.com/calculators/safety-factor-calculator"
        },
        {
            "name": "Young’s Modulus, Stress and Strain Calculator",
            "url": "https://calculators.geekmidst.com/calculators/young-modulus-strain-calculator"
        }
    ],
    "disclaimer": "Quantities are estimates. Confirm requirements with your engineer or supplier before ordering materials.",
    "publisher": "GM¹X CalcVerse"
}