# Cantilever Beam Calculator Calculate the maximum bending moment M_max, maximum shear force V_max and free-end deflection δ_max of a cantilever beam under a free-end point load P or a uniformly distributed load q. Point load: δ = PL³/(3EI); UDL: δ = qL⁴/(8EI). > Canonical page: https://elysiatools.com/en/tools/cantilever-beam-calculator - **Category:** Math & Numbers - **Keywords:** cantilever beam, bending moment, shear force, deflection, PL³/3EI, qL⁴/8EI, strength of materials, mechanics of materials, structural, civil engineering, fixed end ## Overview The Cantilever Beam Calculator computes the structural response of a cantilever beam under a point load at the free end or a uniformly distributed load (UDL). By inputting the span length, Young's modulus, second moment of area, and load magnitude, you can instantly determine the maximum bending moment, maximum shear force, and free-end deflection. ## Inputs - **Load Type** (select) - **Point Load P at Free End (N)** (number): Used in 'Point Load at Free End' mode. - **Uniformly Distributed Load q (N/mm)** (number): Used in 'Uniformly Distributed Load' mode. - **Span Length L (mm)** (number): e.g. 2000 - **Young's Modulus E (MPa)** (number): Modulus of elasticity in MPa (= N/mm²). Steel ≈ 210000, aluminum ≈ 70000. - **Second Moment of Area I (mm⁴)** (number): Area moment of inertia of the cross-section about the bending axis, in mm⁴. - **Decimal Places** (number) ## When to use - When designing cantilevered structures like balconies, overhangs, or crane arms to ensure they meet deflection limits. - When verifying structural calculations for bending moments and shear forces against material yield strengths. - When comparing different beam cross-sections or materials to optimize structural stiffness. ## How it works - Select the load type: either a point load at the free end (P) or a uniformly distributed load (UDL) across the span (q). - Enter the physical properties of the beam, including the span length (L), Young's modulus (E), and the second moment of area (I). - Input the load magnitude and specify the desired decimal precision for the output. - Click calculate to obtain the maximum bending moment, maximum shear force, and free-end deflection based on standard Euler-Bernoulli beam theory formulas. ## Use cases - Sizing a steel cantilever support beam for an outdoor awning to prevent excessive sagging. - Checking the maximum bending stress at the fixed support of a machine lever arm under a point load. - Academic verification of shear force and bending moment calculations for civil engineering coursework. ## Frequently asked questions ### What formulas are used to calculate deflection? For a point load at the free end, deflection is calculated as PL³/(3EI). For a uniformly distributed load, it is qL⁴/(8EI). ### What units should I use for the inputs? Use Newtons (N) for point load, N/mm for UDL, millimeters (mm) for span length, MPa (N/mm²) for Young's Modulus, and mm⁴ for the second moment of area. ### Where do the maximum bending moment and shear force occur? For both load types, the maximum bending moment and maximum shear force occur at the fixed end of the cantilever beam. ### How do I find the second moment of area (I) for my beam? You can calculate it based on the beam's cross-sectional shape (such as bh³/12 for a solid rectangle) or look it up in standard structural steel section tables. ### Can this calculator handle multiple combined loads? 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