What beam types and loads does this support?

A simply supported prismatic beam (pin at the left end, roller at the right end) with downward point loads and uniformly distributed loads (UDLs).

How are support reactions calculated?

From static equilibrium: RB equals the total moment about the left support divided by the span; RA equals the total applied load minus RB.

What do the shear and moment diagrams show?

The shear diagram shows internal transverse force along the beam; the bending moment diagram is the integral of shear and highlights where peak sagging typically occurs (near where shear crosses zero).

Which units are used?

Length in metres (m), forces in kilonewtons (kN), distributed loads in kN/m, and bending moment in kN·m.

Is this suitable for final design?

No. This is an educational statics tool. It does not check deflection, stresses, stability, or code compliance.

Beam Load Calculator (Shear & Moment)

Beam Load Calculator (Simply Supported)

Enter the beam length, add point loads and UDLs, and we’ll compute support reactions plus draw the shear force and bending moment diagrams. Private, in-browser.

Inputs

Beam length L (m)

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:

R_A (left): – | R_B (right): –

Upward reactions in kN (pin at x=0, roller at x=L).

Extrema:

Max |V|: – at –

Max M (sagging +): – at –

Notes:

Static analysis only (no deflection). Sign convention: downward loads positive; shear V is positive upward on the left face; bending moment 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 ∑M_A = 0. The shear function V(x) is evaluated piecewise across all load breakpoints; the bending moment is computed via numerical integration of V(x).

FAQs

Can I add moments or overhangs?

Not yet. This v1 focuses on the most common case: simply supported with point loads and UDLs. If you need concentrated moments or cantilevers, we may add those in a future version.

Why is a reaction negative?

That indicates uplift would be required at that support under the given loading, which a simple pin/roller cannot provide. Re-check inputs or support conditions.

How accurate are the diagrams?

They’re high-fidelity for statics. We sample densely between every breakpoint so the SFD/BMD capture the correct linear/parabolic behavior for UDLs and point loads.

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 are found from static equilibrium with the origin at the left support:

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

The shear function at a position \(x\) (0 at the left support) is:

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

Moment follows from \( \dfrac{dM}{dx} = V \). We integrate shear numerically from 0 to \(x\):

\( M(x) = \int_0^x V(s)\,ds \) (we choose \(M(0)=0\)).

Worked Example (Step-by-Step)

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

Total load: \( \sum P = 10 + 3(6-3) = 19\,\text{kN}. \)
Moment about left support: \( \sum M_A = 10\cdot2 + 3(3)\cdot\dfrac{3+6}{2} = 20 + 9\cdot4.5 = 60.5\,\text{kN·m}. \)
Reactions: \( R_B = 60.5/6 = 10.083\,\text{kN}, \quad R_A = 19 - 10.083 = 8.917\,\text{kN}. \)
Shear just right of loads:
- For \(0\le x<2\): \(V = 8.917\) (horizontal line)
- At \(x=2\): jump down by \(10\) → \(V = -1.083\)
- For \(3\le x\le 6\): UDL reduces shear linearly with slope \(-w=-3\) kN/m
Moment: area under the shear curve. The BMD rises to a maximum near where \(V(x)=0\).

Tip: In a simply supported beam without applied end moments, the global maximum sagging moment occurs where \(V(x)=0\).

Interpreting the Diagrams

Point load → shear diagram has a jump; moment diagram has a change in slope.
UDL → shear diagram is linear; moment diagram is quadratic (parabolic).
Zero-shear location (\(V=0\)) marks a local extremum of the bending moment.

Common Mistakes (and Fixes)

Loads outside the span: Keep all point-load positions and UDL intervals within \(0 \le x \le L\).
Sign confusion: Here, downward loads are positive; support reactions are positive upwards.
UDL endpoints reversed: Enter start \(a\) ≤ end \(b\); the tool will warn if they’re flipped.
Unexpected negative reactions: Indicates uplift at a support — re-check load positions or the support model.

Assumptions & Limits

Statically determinate, simply supported beam; prismatic section.
Linear statics only (no deflection, slope, stresses, or shear deformation effects).
Loads are vertical; no axial forces or torsion.
Use with judgement; for real projects, defer to a qualified structural engineer and the applicable design standards.

Glossary

Shear force \(V\): internal transverse force at a section.
Bending moment \(M\): internal couple at a section; \(dM/dx = V\).
UDL: uniformly distributed load, constant \(w\) over an interval.
Reaction: support force that balances applied loads.

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