# Section Modulus Calculator Calculate the elastic section modulus W = I/y_max and associated geometric properties. Supports rectangle, solid circle, hollow tube, and I-beam; computes I, distance c, area A, and radius of gyration r. Units in mm. > Canonical page: https://elysiatools.com/en/tools/section-modulus-calculator - **Category:** Math & Numbers - **Keywords:** section modulus, moment of inertia, W = I/y, radius of gyration, cross-section, beam section, I-beam, strength of materials, structural, civil engineering ## Overview The Section Modulus Calculator computes the elastic section modulus (W = I/y_max), moment of inertia (I), cross-sectional area (A), distance to the extreme fiber (c), and radius of gyration (r) for standard structural shapes. Supporting rectangles, solid circles, hollow tubes, and I-beams with dimensions in millimeters, this tool helps engineers and designers quickly evaluate beam strength and bending resistance. ## Inputs - **Cross-Section Shape** (select) - **Width b (mm)** (number): Section width (horizontal). Required for Rectangle. - **Height h (mm)** (number): Section height (vertical). Required for Rectangle. - **Diameter D (mm)** (number): Required for Solid Circle. - **Outer Diameter Do (mm)** (number): Required for Hollow Circle. - **Inner Diameter Di (mm)** (number): Required for Hollow Circle. Must be less than Do. - **Flange Width B (mm)** (number): Required for I-Beam. - **Total Height H (mm)** (number): Required for I-Beam. - **Web Thickness tw (mm)** (number): Required for I-Beam. - **Flange Thickness tf (mm)** (number): Required for I-Beam. - **Decimal Places** (number) ## When to use - When designing or analyzing structural beams to determine their bending stress resistance under load. - When calculating the moment of inertia and radius of gyration for standard shapes like rectangles, tubes, solid shafts, or I-beams. - When verifying cross-sectional properties during mechanical design or civil engineering calculations using millimeter dimensions. ## How it works - Select the cross-section shape from the options: Rectangle, Solid Circle, Hollow Circle (Tube), or I-Beam. - Input the required dimensions in millimeters, such as width, height, diameters, or flange and web thicknesses. - Set the desired decimal precision for the output calculations. - Click calculate to instantly generate the area, moment of inertia, distance to the extreme fiber, elastic section modulus, and radius of gyration. ## Use cases - Sizing a rectangular wooden beam to ensure it meets the minimum section modulus required for a specific bending moment. - Calculating the moment of inertia and radius of gyration of a hollow steel tube to evaluate its buckling resistance as a column. - Determining the elastic section modulus of an I-beam to calculate maximum bending stress under a concentrated load. ## Frequently asked questions ### What shapes does this calculator support? It supports four standard cross-sections: Rectangles, Solid Circles, Hollow Circles (Tubes), and I-Beams. ### What units are used for the inputs and outputs? All dimensional inputs are in millimeters (mm). The outputs are in mm² for area, mm⁴ for moment of inertia, mm³ for section modulus, and mm for radius of gyration and distance c. ### How is the elastic section modulus (W) calculated? It is calculated using the formula W = I / c, where I is the area moment of inertia and c is the distance from the neutral axis to the outermost fiber. ### What is the radius of gyration (r) and how is it computed? The radius of gyration describes the distribution of the cross-sectional area around its centroidal axis, calculated as the square root of the moment of inertia divided by the area (r = √(I/A)). ### Can I calculate properties for custom asymmetric shapes? 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