Simply Supported Beam Calculator — Reactions, SFD & BMD

Enter the span, add point loads and UDLs, then view reactions plus interactive SFD/BMD. Private by design—everything runs locally in your browser.

Quick Answer

This simply supported beam calculator finds the left and right support reactions, then plots the shear force diagram (SFD) and bending moment diagram (BMD) for point loads and uniformly distributed loads.

  • Supports: pin at x = 0 and roller at x = L
  • Loads: point loads and full or partial UDLs
  • Outputs: RA, RB, V(x), M(x), max shear, max bending moment
  • Limits: statics only; no deflection, stress, buckling, or code checks

Calculation Checks and Assumptions

  • Transparent formulas: reactions use static equilibrium, with steps shown for the entered beam.
  • Input warnings: the tool flags loads outside the span and negative support reactions.
  • Local calculation: beam inputs and results stay in the browser.
  • Design boundary: educational statics only; no deflection, stress, buckling, or code checks.

Inputs

Point Loads (downward +)

Load P (kN)Position x (m)Actions

Uniformly Distributed Loads (UDL)

Intensity w (kN/m)Start a (m)End b (m)Actions

Results

Support Reactions
RA (left): –  |  RB (right): –
Upward reactions in kN (pin at x=0, roller at x=L).
Extrema
Max |V|: – at –
Max M (sagging +): – at –
Notes
Statics only (no deflection). Sign convention: downward loads positive; V positive upward on left face; M positive for sagging.

Key Shear and Moment Values

LocationEventV beforeV afterM

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Solution Steps

    Simply supported beam with point load and UDL Diagram showing a simply supported beam with left reaction RA, right reaction RB, a point load P, a uniformly distributed load w, and corresponding shear and moment diagrams. Load model RA RB P w kN/m SFD x BMD
    Simply supported beam with point load, UDL, support reactions, shear force diagram and bending moment diagram.

    What This Calculator Does

    Release Updates

    v1.1 (June 7, 2026)

    • Added live solution steps for the entered beam, including total point load, UDL load, moment about A, RA, RB, and maximum moment location.
    • Added a key shear and moment values table plus a static SVG beam diagram for support reactions, point loads, UDLs, SFD, and BMD concepts.

    We assume a simply supported beam (pin at the left end, roller at the right). You can add point loads and uniformly distributed loads (UDLs). Reactions are found by static equilibrium: ∑F = 0 and ∑MA = 0. The shear function \(V(x)\) is evaluated piecewise across all load breakpoints; the bending moment is obtained by integrating shear numerically along the span.

    Shear Force & Bending Moment — Concepts, Examples, and Tips

    This tool helps you sketch the shear force diagram (SFD) and bending moment diagram (BMD) for a simply supported prismatic beam (pin at the left, roller at the right). It supports point loads and uniformly distributed loads (UDLs), computes support reactions, and draws SFD/BMD live. All calculations run privately in your browser.

    When to Use This

    • Quick checks during early sizing or coursework.
    • Understanding how loads change shear/moment shape before a detailed design package.
    • Exploring “what if” scenarios (moving a point load, adjusting a UDL length, etc.).

    Inputs & Units

    • Beam length \(L\): metres (m)
    • Point load \(P\): kilonewtons (kN), downward taken as positive here
    • UDL \(w\): kilonewtons per metre (kN/m)
    • Shear \(V\): kN   |   Moment \(M\): kN·m

    How the Calculations Work

    Reactions from equilibrium with origin at the left support:

    \( R_B = \dfrac{\sum_i P_i x_i + \sum_j w_j(b_j-a_j)\,\dfrac{a_j+b_j}{2}}{L}, \quad R_A = \sum_i P_i + \sum_j w_j(b_j-a_j) - R_B. \)

    Shear at position \(x\):

    \( V(x) = R_A - \sum_{x_i \le x} P_i - \sum_j w_j\,\max\!\bigl(0,\,\min(x,b_j)-a_j\bigr). \)

    Moment from \( dM/dx = V \): \( M(x) = \int_0^x V(s)\,ds \) with \(M(0)=0\).

    Worked Example (Step-by-Step)

    Beam: \(L=6\,\text{m}\), \(P_1=10\,\text{kN}\) at \(x=2\,\text{m}\), UDL \(w=3\,\text{kN/m}\) from \(3\) to \(6\,\text{m}\).

    1. Total load \(=10+3(6-3)=19\,\text{kN}\).
    2. Moment about A \(=10\cdot2 + 3\cdot3 \cdot \tfrac{3+6}{2} = 60.5\,\text{kN·m}\).
    3. \(R_B=60.5/6=10.083\,\text{kN}\), \(R_A=19-10.083=8.917\,\text{kN}\).
    4. Shear: jump at point load; linear drop under UDL with slope \(-w\).

    Interpreting the Diagrams

    • Point load → jump in SFD; slope change in BMD.
    • UDL → linear SFD; parabolic BMD.
    • Where \(V=0\) → local extremum of \(M\).

    Common Mistakes (and Fixes)

    • Keep all loads within \(0\le x\le L\).
    • Downward loads positive; reactions positive upward.
    • If a UDL start/end is flipped, the tool swaps the two positions automatically.
    • Negative reactions → uplift: re-check inputs/supports.

    Assumptions & Limits

    • Statically determinate, simply supported beam; prismatic section.
    • Linear statics only; no deflection/stress checks.
    • No axial or torsion; vertical loads only.

    Disclaimer: Educational tool. Not a substitute for professional structural analysis or code-compliant design checks.

    How Loads Affect Shear and Moment Diagrams

    Load or event Effect on SFD Effect on BMD
    Support reaction Vertical jump upward Changes the slope after the support
    Point load Vertical jump downward by P Moment stays continuous; slope changes
    UDL Linear change with slope -w Parabolic curve
    Zero shear V(x) = 0 Possible maximum or minimum moment

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