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.

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