Guide

How beam analysis works

Shear force diagram (SFD)

The shear force V(x) at a section is the algebraic sum of vertical forces to one side of the cut. Downward point loads cause downward jumps; support reactions cause upward jumps; distributed loads create sloped lines (the slope equals −w). Crossings through zero mark potential extrema of the bending moment.

Bending moment diagram (BMD)

The bending moment M(x) is the area under the shear curve: M(x) = M(0) + ∫V dx. Curved sagging segments appear under UDLs (parabolas), straight lines under point loads, and applied couples cause instantaneous jumps. Interior hinges force M = 0 at the hinge position.

Sign conventions

This calculator defaults to the sagging-positive convention: a moment that sags the beam (smile shape) is positive; downward deflection is plotted downward (negative). You can flip the moment/shear sign convention with the toggle under the results. Applied couples are clockwise-positive in inputs.

Supports

  • Pin: restrains vertical translation; free to rotate.
  • Roller: same as pin in this 2D vertical model; free horizontal.
  • Fixed: restrains translation and rotation (adds a fixing moment reaction).
  • Spring: vertical spring with stiffness k; the reaction equals k × settlement-adjusted deflection.
  • Internal hinge: releases the moment, so rotation is discontinuous across the hinge.

Method

The engine is a classical Euler–Bernoulli finite-element solver: two degrees of freedom per node (v, θ), exact Hermitian beam elements, consistent nodal loads for distributed loads, and piecewise-exact post-processing. Solve time is under a millisecond for typical models and everything runs locally in your browser: nothing is sent to a server.

Try it on a real case

The fastest way to internalize these diagrams is to build a case and watch them change. Start with the cantilever under uniform load to see the classic linear shear and parabolic moment, or the cantilever beam calculator for tip loads and moments. For full background on the tool's assumptions, see the about page.

Stresses and serviceability

Bending stress is computed as σ = M·c/I separately for the top and bottom fibres (cTop, cBot may differ for asymmetric sections). Average shear stress is V/A: a coarse check; webs of I-sections carry significantly higher peak shear. Deflection badges compare against the common serviceability limits L/240, L/360 and L/480.

Calculator capabilities (API & LLM Reference)

This tool is a client-side web application. It does not provide a REST API, but the following plain-text description is provided for AI crawlers, LLMs, and technically-minded users to understand its exact capabilities without executing the JavaScript interface.

Supported inputs

  • Beam length: any positive value.
  • Supports: Pin, Roller, Fixed, Spring (with adjustable stiffness), Internal Hinge (moment release). Can be placed anywhere, including overhangs.
  • Loads: Point load, Applied moment, Uniform distributed load (UDL), Triangular load, Trapezoidal load (linearly varying). Loads can start and end anywhere on the beam.
  • Materials: Custom Young's Modulus (E), Density, and Allowable Yield Stress. Presets included for Steel, Wood, Aluminum, and Concrete.
  • Sections: Custom Area (A) and Moment of Inertia (I). Presets included for Rectangular, Circular, Hollow sections, I-beams, and AISC W-shapes.
  • Other: Toggleable self-weight (calculated from area and density), support settlement.

Computed outputs

  • Reactions: Vertical force and bending moment at every support.
  • Diagrams: Piecewise-exact Shear Force (SFD), Bending Moment (BMD), Slope, and Deflection diagrams.
  • Extrema: Exact magnitude and location (x) of the maximum shear force, maximum sagging/hogging moment, and maximum deflection.
  • Stress: Maximum bending stress (calculated as M·c/I) and average shear stress (V/A).
  • Serviceability: Pass/fail checks for L/240, L/360, and L/480 deflection limits.