Bending Moment Diagram (BMD) Calculator
Plot exact bending moment diagrams, pinpoint maximum flexural stress points, and locate inflection points for any determinate or indeterminate beam.
dM(x)/dx = V(x) ⟹ M(x) = M₀ + ∫ V(x) dx
Beam Master handles any combination of point loads, uniform distributed loads (UDL), varying loads (UVL), and applied moments with instant SFD, BMD, and deflection curves.
Understanding Bending Moments in Beams
The internal bending moment $M(x)$ represents the internal couple developed inside a structural cross-section to resist external transverse forces. Sagging bending causes tension in bottom fibers and compression in top fibers.
Engineering Formulas & Governing Equations
Fundamental Slope Relation
Equation 01The derivative (slope) of the bending moment diagram at any point is equal to the value of the shear force.
Area Integration Law
Equation 02The change in bending moment between two points equals the area under the shear force diagram.
Point of Inflection (Contraflexure)
Equation 03The coordinate along the beam where the moment changes sign from sagging (+) to hogging (-).
Point Moment Discontinuity
Equation 04An applied external concentrated couple causes a vertical step jump in the bending moment diagram.
Calculation Details & Clarifications
Solve in Beam Master
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