See how a beam bends.

All reported deflections, slopes and internal moments are magnitudes. The load and displacement act downward; reactions act upward. The cantilever’s external fixed-support moment is counterclockwise. The simply supported beam has an ideal pin at the left and roller at the right. The cantilever is fully fixed at the left.

The example uses L = 1000 mm, P = 1000 N, E = 200 GPa, I = 1,000,000 mm⁴ and Z = 20,000 mm³. For the simply supported case this gives δ = 0.10416667 mm, maximum moment = 250,000 N·mm and nominal bending stress = 12.5 MPa.

• Euler–Bernoulli theory: straight, slender, prismatic beam; constant EI; homogeneous linear-elastic material; static transverse loading; small deflections and rotations.
• Use the centroidal principal bending axis. Shear deformation, self-weight, support flexibility, local contact effects, stress concentrations and torsion are excluded.
• No buckling, yielding, fatigue, allowable stress, serviceability limit or safety factor is assessed. Section slenderness cannot be checked from I alone.
• A warning appears when δ/L exceeds 1% or maximum slope exceeds 0.1 rad. These are calculator screening thresholds, not limits prescribed by the references or a design code. Values below these thresholds still require independent model validation.
• The optional stress is nominal elastic bending stress. Its validity depends on your geometry and material staying within the model assumptions.

Supported numerical bounds: span up to 10⁹ mm, load up to 10¹² N, E up to 10⁶ GPa, and I or Z up to 10¹⁸ in their displayed units. Nonzero inputs must be at least 10⁻¹². Numerical acceptance does not imply physical validity. Display precision is not measurement uncertainty.

• MIT OpenCourseWare — Deflections due to Bending, chapter 10 (PDF): small-rotation moment–curvature relation, end-loaded cantilever and simply supported point-load solution. The central-load case follows by setting b = L/2.
• MIT OpenCourseWare — Stresses: Beams in Bending, chapter 9 (PDF): elastic bending stress and centroidal second moment of area. With Z = I/c, |σ|max = |M|max/Z.
• Iowa State University — Beam Deflection Formulas (PDF): cases 1 and 6 independently tabulate the two deflection and slope formulas.

Support reactions and maximum moments follow static force and moment equilibrium. Alkoc Labs beam calculator v0.5.0. Independently verify inputs and results before consequential engineering use.