Structural Engineering

Static Beam Calculator

Compute support reactions, shear force diagram, bending moment diagram, and deflection curve for simply supported beams. Add any combination of point loads and uniform distributed loads.

Simply Supported Beam Shear Force Diagram Bending Moment Deflection Bernoulli-Euler

About This Simulator

This calculator analyzes simply supported beams — one of the most fundamental structural elements in civil and mechanical engineering. The beam rests on a pin support at end A and a roller support at end B, making it statically determinate: reactions, internal forces, and deflections can all be computed from equilibrium equations alone.

The simulator applies the principle of superposition: each load (point or distributed) is processed independently, its contribution to shear V(x) and bending moment M(x) calculated analytically, and the results summed across the full beam length. This produces exact diagrams — not approximations — for any combination of loads within the isostatic model.

Deflections are computed by double-integrating the curvature ? = M(x)/EI using the Bernoulli-Euler beam theory. This assumes the beam is linearly elastic, cross-sections remain plane after bending, and deformations are small relative to beam length.

How to Use

  1. Set beam length L. Enter the span between support A (x = 0) and support B (x = L). Valid range: 0.1 m to 100 m.
  2. Set material and section properties. Enter E (GPa) and I (cm4). These affect deflection but not the shear or moment diagrams.
  3. Add loads. Click Point Load or Distributed to add a load card. Enter the magnitude and position(s) within [0, L].
  4. Click Calculate. The simulator computes reactions, critical values, and renders three diagrams: Q(x), M(x), and y(x).
  5. Read the results. Hover over any chart to see exact values at any position. Critical maximums and their locations are shown in the summary cards.

Understanding the Results

Support Reactions RA and RB

Vertical forces exerted by the supports. Found from SF = 0 and SM = 0. Check: RA + RB must equal the total applied load.

Shear Force Diagram Q(x)

Algebraic sum of vertical forces to the left of any cross-section. Constant between point loads, linear under distributed loads. Maximum shear typically occurs at supports.

Bending Moment Diagram M(x)

Sum of moments of all forces to the left of a section. Linear between point loads, parabolic under UDLs. Maximum moment occurs where Q = 0 — the critical section for flexural design.

Deflection Curve y(x)

Obtained by double-integrating M(x)/EI with y(0) = y(L) = 0. Results in mm. Compare against code limits: Eurocode 3 limits L/300 for floors, L/500 for elements supporting brittle finishes.

Frequently Asked Questions

What sign convention is used?
Downward loads are positive. Shear Q is positive when the left portion tends to slide upward. Moment M is positive (sagging) when the beam bends concave upward — tension on the bottom fiber.
Can I add multiple loads at the same position?
Yes. Superposition handles any combination correctly. Multiple point loads at the same x, overlapping distributed loads — all are summed algebraically.
Why do I enter I directly instead of the cross-section dimensions?
The second moment of area I is the only cross-section property that affects deflection under Bernoulli-Euler theory. Entering I directly makes the calculator applicable to any section shape: I-beams, rectangular, circular pipe, composite sections.
How do I find I for a standard steel profile?
Section tables are published by steel manufacturers and standards bodies. Reference values: IPE 200 ? 1 943 cm4; IPE 300 ? 8 356 cm4; HEB 200 ? 5 696 cm4; HEB 300 ? 25 170 cm4. For custom sections, compute I from the parallel-axis theorem.
What are the limitations?
The model assumes: linear elastic, isotropic material; plane sections remain plane; small deformations; transverse loads only (no axial force or torsion); simply supported ends. It does not cover cantilevers, fixed-end beams, continuous beams, or non-uniform sections.