Beam Load Calculator — Shear Force & Bending Moment (Simply Supported)

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

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.

What This Calculator Does

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.
  • Ensure UDL \(a\le b\); the tool warns if flipped.
  • 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.

5 Fun Facts about Shear & Moment

Zero shear = peak moment

Where the shear diagram crosses zero is where the bending moment hits an extremum—often the critical bending spot to check.

Hotspot finder

Area tells the story

The area under the load diagram gives shear; the area under the shear diagram gives moment. Stack diagrams and you can sanity-check slopes by eye.

Integral ladder

Triangles in disguise

A UDL makes a triangular shear and a parabolic moment. A triangular load makes a parabolic shear and a cubic moment—shapes keep “integrating up.”

Shape cascade

Moving loads reshuffle reactions

Sliding a point load just 10% along the span can swing support reactions by that same 10%—handy for checking temporary shoring or cranes.

Leverage lesson

Sign conventions matter

Flip the shear sign and your moment curvature flips too. Being consistent is why your BMD shows sagging (negative) or hogging (positive) in the right spots.

Stay consistent

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