Beam Calculator — Shear, Moment & Deflection

A beam calculator finds the internal forces and the deformation of a beam under load: the shear force diagram (SFD), the bending moment diagram (BMD) and the deflected shape. Those three curves are the starting point of every steel-beam design — the peak moment sizes the section for strength, and the peak deflection checks serviceability.

Most free tools on the web evaluate closed-form textbook formulas, which means they only work for the handful of cases the formulas cover (one load, symmetric, uniform). This calculator is different: it runs a real finite-element solver — the same direct-stiffness method used by commercial structural software. The beam is meshed into 60 Euler-Bernoulli elements with 2 degrees of freedom per node (deflection v and rotation θ), boundary conditions are applied by direct elimination, and the system K·u = F is solved with Gaussian elimination with partial pivoting. That is why you can stack any number of point loads and uniform or trapezoidal distributed loads, mix support types, and still get exact linear-elastic results — including statically indeterminate cases such as fixed–fixed and propped cantilevers, which simple formula tools cannot handle at all.

Because the engine is real, the section matters too. Pick any of the 974+ flexural catalog profiles — Brazilian NBR/Gerdau W sections, European IPE/HEA/HEB/HEM, AISC W shapes, channels (U/C) and hollow sections (RHS/SHS); non-flexural families such as angles and round bars are deliberately kept out of the bending screening — and the calculator computes the flexural stiffness EI from the actual cross-section, the elastic bending stress σ = M/Sx, and the utilization against your yield strength — or type a custom EI if your section is not steel.

1. Set the span and pick the support type — simply supported, cantilever, fixed–fixed or propped cantilever. The support symbols in the sketch follow standard drafting conventions (pin triangle, roller circles, fixed hatching).
2. Add loads. Click Add in either list — point loads (magnitude + position) or distributed loads (start/end intensity for trapezoids and start/end position for partial coverage). Positive values point down; enter a negative value for uplift. There is no limit on the number of loads. Tick Include self-weight to add the selected profile's kg/m as an extra distributed load in one click.
3. Choose the section. Select a catalog profile (grouped by standard) or switch to Manual EI and type the stiffness directly. The yield strength fy and the deflection limit (L/180 … L/500) drive the checks.
4. Read the results — they update instantly, no “calculate” button. The KPI strip shows Mmax, Vmax, δ, σ and the utilization; the maxima are annotated directly on each diagram, the SFD carries the shear ordinate at both supports, and the sketch draws the solved reaction arrows (R_A, R_B) at the supports. NBR 8800 and AISC 360 are checked side by side — click either card to make that code govern the ranking.
5. Check the lightest profiles that pass table: the whole flexural catalog is screened against the bending and deflection checks and the five lightest sections are ranked. Click Use to adopt one.
6. Export or share for free — PNG/SVG of the sketch and diagrams, CSV with every V/M/δ point plus the ranking table (no watermark, no paywall), and Copy link to this beam produces a permalink that reproduces your exact model for a colleague.
7. Press Open in 3D editor to convert this exact model — nodes, bar, supports, loads, profile — into a CalcSteel project and continue with the full NBR 8800 / AISC 360 design verification, load combinations and connection design.

Tip: the SI ⇄ imperial toggle above converts every input and output (kN ↔ kip, m ↔ ft, mm ↔ in, MPa ↔ ksi) — the math always runs in SI internally, so nothing is lost in translation.

The solver does not evaluate these formulas — it solves the stiffness system numerically — but for the classic textbook cases its output matches them to machine precision, which is a good way to validate any beam tool you use:

After the analysis, the stress and the checks are:

• Elastic bending stress: σ = Mmax / Sx (Sx = elastic section modulus about the strong axis).
• Bending check, both codes side by side (screening): NBR 8800 style σ ≤ fy / γa1 with γa1 = 1.10, and AISC 360 LRFD style σ ≤ φb · fy with φb = 0.90 (use fy in MPa). For fy = 250 MPa that is 227.3 MPa vs 225.0 MPa — the calculator shows both utilizations in parallel so you see exactly how much each code governs.
• Deflection check: δmax ≤ L / n with n selectable (250 is the common floor-beam limit; 360 for beams supporting brittle finishes; 180 for roofs). Serviceability is code-independent.
• Plastic-moment screening (compact I sections): when the selected W/I shape meets the compact limits (flange b/2tf ≤ 0.38√(E/fy), web hw/tw ≤ 3.76√(E/fy) — same λp in NBR 8800 Tabela G.1 and AISC 360 B4), the calculator also shows Mp = Zx·fy with the two design capacities side by side: Mp/γa1 (NBR) and φb·Mp (AISC F2.1). Valid for continuous lateral restraint (Lb ≤ Lp) — the full FLT check runs in the 3D editor.

Section properties are computed from the nominal plate dimensions of each catalog profile (fillets neglected — slightly conservative: for an IPE 300 the calculator uses Ix = 7,999 cm⁴ vs the 8,356 cm⁴ handbook value that includes root fillets).

• Loads: positive input = downward (gravity). Enter negative values for uplift (wind suction, prestress ballast…). Internally the solver works y-up, so your +10 kN becomes fy = −10 kN — the calculator handles the flip.
• Shear V: positive shear is plotted above the axis. At a downward point load the SFD steps down by P; at supports it jumps by the reaction. The ordinate is annotated at both supports, drafting-board style.
• Bending moment M: sagging (tension in the bottom fiber) is positive and is plotted below the axis — on the tension side — the Brazilian/European drafting convention. If you are used to the US convention (positive up), the curve is simply mirrored; the values and positions are identical.
• Deflection δ: plotted as the beam actually moves — downward deflection draws downward. Values are reported as magnitudes.
• Reactions: listed positive upward and drawn as arrows at the supports of the sketch (R_A, R_B…); fixed supports also report the clamping moment (counter-clockwise positive).
• Positions x: always measured from the left support/end.

The engine is a direct-stiffness (matrix) finite-element solver for Euler-Bernoulli bending — the exact same code that powers CalcSteel's profile pages, not a lookup table:

• 60 beam elements, 122 degrees of freedom, cubic Hermitian shape functions;
• distributed loads converted to consistent nodal forces (works for trapezoids);
• boundary conditions imposed by direct elimination (no penalty-number ill-conditioning);
• K·u = F solved by Gaussian elimination with partial pivoting;
• V(x) and M(x) recovered by section equilibrium using the same (snapped) load positions the FEM uses, so reactions and diagrams are always mutually consistent and the BMD closes to zero at pinned/roller supports.

Against closed-form solutions the results agree to better than 0.001% (e.g. simple beam + UDL: δ engine 10.5482 mm vs theory 10.5482 mm). Element size L/60 means point-load positions snap to the nearest 1.7% of the span — irrelevant for design, but if you type x = 2.99 m on a 6 m beam the load acts on the node at 3.00 m, in the FEM solve and in the diagrams alike.

Assumptions: linear elastic material, small deflections, shear deformation neglected (fine for span/depth > 10), prismatic member (constant EI), loads in the plane of bending, lateral-torsional buckling prevented. Self-weight is one click away — the Include self-weight toggle adds the selected profile's kg/m as an extra distributed load (1 kg/m ≈ 0.00981 kN/m; e.g. IPE 300 at 42.2 kg/m ≈ 0.41 kN/m).