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.

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.