# Bending Stress Calculator Calculate the maximum bending (flexural) stress σ_max = M·c/I = M/Z at the outermost fiber of a beam section. Supports solid circle, rectangle, hollow tube, and I-beam; computes I, section modulus Z, and distance c automatically. Stress in MPa. > Canonical page: https://elysiatools.com/en/tools/bending-stress-calculator - **Category:** Math & Numbers - **Keywords:** bending stress, flexural stress, section modulus, sigma = M·y/I, M/Z, I-beam, beam section, strength of materials, mechanics of materials, structural, civil engineering ## Overview The Bending Stress Calculator computes the maximum flexural stress at the outermost fiber of a beam section using the formula σ_max = M·c/I = M/Z. By selecting a cross-section shape—such as a rectangle, solid circle, hollow tube, or I-beam—and entering the bending moment and section dimensions, the tool automatically calculates the moment of inertia, section modulus, and distance to the extreme fiber to output the final stress in MPa. ## Inputs - **Cross-Section Shape** (select) - **Bending Moment M (N·mm)** (number): e.g. 5000000 - **Diameter D (mm)** (number): Required for Solid Circle. - **Width b (mm)** (number): Section width (horizontal). Required for Rectangle. - **Height h (mm)** (number): Section height (vertical). Required for Rectangle. - **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 structural beams to ensure the maximum bending stress does not exceed the material's allowable yield strength. - When analyzing mechanical shafts, axles, or structural profiles subjected to bending moments. - When comparing the structural efficiency of different cross-sectional shapes under the same bending load. ## How it works - Select the cross-section shape of the beam: Rectangle, Solid Circle, Hollow Circle (Tube), or I-Beam. - Input the bending moment (M) in N·mm and the required dimensional parameters of the chosen shape in mm. - The calculator computes the area moment of inertia (I), the distance to the outermost fiber (c), and the section modulus (Z = I/c). - The tool applies the flexure formula to determine the maximum bending stress (σ_max) in MPa. ## Use cases - Verifying the structural integrity of a rectangular timber joist under a known bending moment. - Calculating the maximum bending stress on a hollow steel tube axle to prevent mechanical failure. - Determining the required section modulus for an I-beam structural support during building design. ## Frequently asked questions ### What units should I use for the inputs? The bending moment (M) must be in Newton-millimeters (N·mm), and all dimensional inputs must be in millimeters (mm). ### What shapes are supported by this calculator? It supports four standard cross-sections: Rectangle, Solid Circle, Hollow Circle (Tube), and I-Beam. ### How is the section modulus (Z) calculated? The section modulus is calculated as Z = I / c, where I is the area moment of inertia and c is the distance from the neutral axis to the outermost fiber. ### Why is the output stress in MPa? Using Newtons (N) for force and millimeters (mm) for dimensions yields stress in N/mm², which is equivalent to Megapascals (MPa). ### Does this tool account for shear stress or axial loads? 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