{"id":3128,"date":"2026-03-06T14:33:37","date_gmt":"2026-03-06T13:33:37","guid":{"rendered":"https:\/\/solveredu.com\/?p=3128"},"modified":"2026-03-06T14:33:42","modified_gmt":"2026-03-06T13:33:42","slug":"beam-deflection-calculation","status":"publish","type":"post","link":"https:\/\/solveredu.com\/en\/post\/obliczanie-ugiecia-belki\/","title":{"rendered":"Beam Deflection Calculation: From Bending Moment Integration to an Online Calculator"},"content":{"rendered":"

Determining deflection lines is one of the key steps in structural design. Whether you are a student preparing for a strength of materials colloquium or an engineer verifying the stiffness of a component, you need to know how a beam \u201eworks\u201d under load.<\/p>\n\n\n\n

In this article, we'll walk the full path: from classical theory Integration of the differential equation of the deflection line<\/strong>, through practical drawing of rotation angle diagrams, to examples for simply supported beam<\/strong> and bracket<\/strong><\/p>\n\n\n\n

And if you value your time and want to avoid tedious hand calculations, at the end of the article you will find my proprietary beam calculator<\/strong>, which will perform these operations for you in a few seconds. Let's get started!<\/p>\n\n\n\n

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  1. Theoretical basis: Differential equation of the deflection line<\/a><\/strong><\/li>\n\n\n\n
  2. Analytical method: Step-by-step integration<\/a><\/strong><\/li>\n\n\n\n
  3. Example 1 : Freely supported beam<\/a><\/strong><\/li>\n\n\n\n
  4. Example 2: cantilever beam<\/a><\/strong><\/li>\n\n\n\n
  5. Quick verification: Use the online beam calculator<\/a><\/strong><\/li>\n<\/ol>\n\n\n\n

    1 Theory and Methodology: Where does deflection come from?<\/h2>\n\n\n\n

    Calculation of beam deflection<\/strong> is one of the most important elements of checking the Serviceability Limit State (SGU). In order to understand this process, we need to go to the foundation of the differential equation of the beam deflection line<\/strong>.<\/p>\n\n\n\n

    The basic relationship that binds bending moment vs. deflection<\/strong>, is described by the formula:<\/p>\n\n\n\n

    E<\/mi>I<\/mi>\u22c5<\/mo>y<\/mi>\u2032<\/mo>\u2032<\/mo><\/mrow><\/msup>(<\/mo>x<\/mi>)<\/mo>=<\/mo>\u2212<\/mo>M<\/mi>(<\/mo>x<\/mi>)<\/mo><\/mrow>EI \\dot y\u201d(x) = -M(x).<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    Where:<\/p>\n\n\n\n