Beam Calculator: Shear Force, Bending Moment and Deflection

Beam calculator with solutions: reactions, SFD, BMD and deflection

This beam calculator solves a loaded beam the way a structures course teaches it: reactions from equilibrium (plus compatibility for indeterminate beams), shear force and bending moment along the span, then slope and deflection by double integration. Choose a simply supported, cantilever, fixed-ended, propped or overhanging beam, add point loads, uniform, triangular or trapezoidal loads and applied moments anywhere on the span, and read the maximum moment, shear, reactions and deflection with their positions.

The shear force diagram, bending moment diagram and deflected shape are drawn as you type, the Macaulay equations for V(x) and M(x) appear in the solution steps, and a table lists values at 21 stations. It works in metric (m, kN, GPa, cm⁴) and imperial (ft, kip, ksi, in⁴), with E presets and a section helper for rectangles, circles and hollow sections.

Everything runs in your browser. These are estimates for planning and study; a qualified engineer must check any design.

How to use the Beam Calculator

1. Pick the unit system and the beam type. For an overhanging beam, enter where supports A and B sit.
2. Enter the beam length, choose a material preset or a custom E, and give I directly or from the section dimensions.
3. Add loads: a point load at x, a UDL or varying load from x₁ to x₂, or a clockwise moment. Downward is positive; positions are measured from the left end.
4. Read the reactions, peak shear, moments and deflection with their positions.
5. Open Solution steps for the equations and copy the station table as CSV.

Beam Calculator features

How the beam is solved

Every reaction and load is written as a singularity function: a force F at a contributes F⟨x − a⟩⁰ to shear and F⟨x − a⟩¹ to moment; a uniform load w from a adds w⟨x − a⟩¹ and w⟨x − a⟩²/2 and is cancelled past its end; a couple adds a step to the moment. The unknown reactions and integration constants come from one small linear system: forces and moments sum to zero, deflection is zero at every support and slope is zero at every fixed end. That handles the indeterminate fixed and propped cases exactly. Deflection follows from EI·v″ = M with constant EI.

The method reproduces the standard textbook results (AISC Manual Table 3-23 “Beam diagrams and formulas”, Roark’s Formulas):

Worked example: 6 m steel beam with 10 kN/m

A simply supported ISMB 300 (I = 8,603.6 cm⁴, E = 200 GPa) spanning 6 m under 10 kN/m: each reaction is 30 kN, V(x) = 30 − 10x, M(x) = 30x − 5x², so the moment peaks at x = 3 m where V = 0: M = 45 kN·m. The deflection is 5 × 10 × 6⁴ / (384 × 1.72 × 10⁷) = 9.8 mm, or L/612 — inside L/360. Switch to “Fixed at both ends” and the end moments become −30 kN·m, the mid-span moment drops to 15 kN·m and the deflection to 2.0 mm. Count the bars for a concrete version with the rebar calculator.

Sign conventions, limits and tips

• Loads act downward when positive; enter a negative value for an upward load. Applied moments are positive clockwise, which makes the bending moment diagram jump upward. Sagging moments are positive and drawn above the axis; hogging moments at fixed ends and over supports are negative.
• Deflections use Euler–Bernoulli theory: no shear deformation, small displacements, constant EI, linear elastic material. Deep beams, cracked concrete and composite members need a more refined model.
• E for concrete follows IS 456 (5000√fck MPa) or ACI 318 (57,000√f′c psi); use a cracked or long-term stiffness for deflection checks of reinforced concrete.
• Serviceability limits vary: IS 456 and Eurocode use L/250 for total deflection, AISC and IBC commonly L/360 for live load. For shafts, see the torque calculator; the integral calculator helps when repeating the double integration by hand.