Build bending moment diagrams with confidence quickly. Choose loads, set positions, and review reactions clearly. Download results, share designs, and validate beam safety easily.
This tool assumes a simply supported beam with supports at x = 0 and x = L. Loads are superposed.
Shear and bending moment are computed along the beam:
Here H(·) is the Heaviside step function. The partial UDL moment term Mw(x) is evaluated piecewise for x inside and beyond the loaded segment.
Example inputs and key outputs for a typical case.
| L (m) | P1 (kN) @ a1 (m) | w (kN/m) over x1–x2 (m) | RA (kN) | RB (kN) |
|---|---|---|---|---|
| 6 | 10 @ 2 | 3 over 1–5 | 10.6667 | 11.3333 |
A bending moment diagram summarizes how internal flexural demand varies along a beam. Peak positive or negative moments control section sizing, reinforcement layout, and allowable stress checks. Engineers also use moment locations to place stiffeners, connections, and inspection points where fatigue or cracking risk is highest.
The solver treats the beam as simply supported at x = 0 and x = L, with linear elastic behavior and small deflections. Loads are idealized as concentrated forces, a partial uniform load segment, and an optional concentrated couple moment. Results are reported in kN and kN·m.
Support reactions are found using global force balance and a moment balance about a support. For example, a 10 kN load at 2 m on a 6 m span contributes 6.667 kN to the left reaction and 3.333 kN to the right. A partial UDL is converted to an equivalent resultant W = w(x2 − x1) acting at its centroid.
Shear starts at RA and changes wherever a load is crossed. A point load creates an instantaneous drop equal to its magnitude. A UDL produces a linear shear slope of −w within its interval. These rules make it easy to spot zero-shear locations, which often coincide with extreme bending moments.
The moment curve is the integral of shear: dM/dx = V. Between concentrated loads, M(x) varies linearly; under a UDL, M(x) becomes quadratic. In this calculator, moments are accumulated directly from reaction and load contributions, with optional moment jumps from a concentrated couple.
Expect step changes in V(x) at point loads, but continuity in M(x) unless a couple is applied. Over a UDL, the moment diagram is curved; higher intensity or longer loaded length increases curvature. Checking both the sign and magnitude helps distinguish sagging from hogging regions for design detailing.
A couple moment does not change shear directly; it produces a discrete jump in M(x) at its application point. The solver models this as +M0 added after x ≥ am. Reactions shift by ±M0/L to preserve equilibrium, which can meaningfully move peak values.
Use higher sampling points when you need smooth plots or accurate peak searches for mixed loading. After computing, export the full-resolution CSV for further processing in spreadsheets or scripts, and export the PDF for reporting. Always validate sign conventions against your project standard before final sizing.
Point loads are treated as positive when acting downward, creating negative steps in shear. A positive applied moment input adds an upward jump in the moment diagram at its location.
This page supports two point loads plus one partial UDL and one applied moment. To model more loads, combine equivalent resultants or export the CSV and superpose additional contributions externally.
Set x1 = 0 and x2 = L, then enter w. The reactions and diagrams update automatically using the equivalent resultant at the span centroid.
A concentrated couple imposes a discontinuity in bending moment without adding vertical force. The diagram must step by the couple magnitude to satisfy internal moment equilibrium at that point.
For quick checks, 201 points is usually enough. For sharper peak localization under mixed loading, increase to 501–1001 points. Very high values mainly improve plot smoothness and CSV resolution.
The CSV contains x, V(x), and M(x) at every sample point. The PDF captures the results panel, including reactions, extrema, the diagrams, and the sample table for documentation.
The analysis is statically determinate and assumes a simply supported beam. It does not include fixed-end moments, continuous spans, variable EI, or nonlinear effects. For complex structures, use a dedicated structural solver.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.