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Beam reaction calculator • shear force calculator • bending moment and deflection calculator

Simply Supported Beam Calculator

Calculate support reactions, shear force, bending moment, slope, deflection, and bending stress for a simply supported beam. You can combine point loads, applied moments, uniform distributed loads, and linearly varying distributed loads on the same beam.

The calculator updates automatically as you type. It also shows the formulas, sign convention diagrams, applied load totals, and a worked example so the result is easier to check and explain.

On this page

Simply supported beam calculator inputs

Enter the beam span, material stiffness, section properties, and any loads you want to apply. This page recalculates automatically, so there is no need to press Calculate after every change. The results are solved in the browser and shown together with the formulas, sign diagrams, and substituted values.

Current position unit used for L and load positions.
Advanced numerical settings

This setting mainly affects slope and deflection accuracy. Most users can leave it at 1200.

Unit tip: keep the section units compatible. If I is entered in in⁴, enter c in in. The beam span can still be in ft or m. That avoids unit mistakes in the bending-stress formula σ = Mc / I.

Point loads

Applied moments

Uniform distributed loads (UDL)

Linearly varying loads (VDL / trapezoidal)

How to use this calculator

1. Enter beam data

Set the beam length L, the report position x, and the material and section properties E, I, and c.

2. Add loads

Add any mix of point loads, applied moments, uniform distributed loads, and varying loads. Positions are measured from the left support.

3. Check units

Use consistent units. The beam span and the fiber distance can use different units, but c must match the unit system of I.

4. Read the results

The page updates automatically and reports reactions, shear, moment, slope, deflection, stress, and applied-load totals.

What the inputs mean

InputMeaning
LTotal beam span between the two supports.
xPosition where the calculator reports a local set of results.
EYoung’s modulus. This controls material stiffness.
ISecond moment of area of the cross-section. Larger values reduce bending and deflection.
cDistance from the neutral axis to the outer fiber used in the stress formula σ = Mc / I.
Point loadA concentrated vertical force applied at one position.
Applied momentA concentrated external moment applied at one position.
UDLA constant distributed load over part or all of the beam.
Varying loadA linearly changing distributed load from one intensity to another.
Numerical resolutionThe number of internal stations used for slope and deflection. Higher values can improve accuracy but are usually not needed for routine use.

Beam calculation results

Values are shown at the selected position x = 10 and as absolute extrema over the full beam span.

Shear force and bending moment charts

These live diagrams update with your current inputs and unit selections. The dashed line marks the selected report position x.

Shear force diagram

Bending moment diagram

Applied loads summary

This section shows the totals of all loads and applied moments currently on the beam.

Simple beam sign convention drawings

These simple SVG drawings show each case separately so the sign convention is easier to understand.

Positive point load

+P

Positive uniform distributed load

+w

Positive varying distributed load

+w₁ +w₂

Positive applied moment

+M

Positive support reactions

+R₁ +R₂

Positive shear at a cut

+V

Positive bending moment at a cut

+M at cut positive sagging moment

Beam model and x direction

x from left support

Beam calculation details and formulas

The formulas below show how the simply supported beam reactions, applied loads, internal forces, slope, deflection, and bending stress are calculated using your entered values.

Sign convention used

  • x is measured from the left support.
  • Downward applied loads are entered as positive.
  • Positive applied moments follow the moment drawing shown above.
  • Positive shear acts upward on the left face of the cut.
  • Positive bending moment is sagging.
  • Positive bending stress is reported from $\sigma = Mc/I$ using the chosen sign of $M$.

Simply supported beam formulas

This simply supported beam calculator uses static equilibrium for support reactions, exact load resultants for finite load segments, and numerical integration of the beam curvature equation to solve slope and deflection: $$EI\,y''(x)=M(x).$$

  • Support equilibrium: $R_1 + R_2 = \sum P + \sum W$
  • Moment equilibrium about the left support: $R_2L = \sum(Pa) + \sum(W\bar{x}) - \sum M$
  • Bending stress at fiber distance $c$: $\sigma = Mc/I$
  • UDL resultant on $[s,e]$: $W = w(e-s)$ at $\bar{x}=(s+e)/2$
  • Trapezoidal load resultant on $[s,e]$: $W = \frac{w_1+w_2}{2}(e-s)$
  • Trapezoidal centroid from the left support: $\bar{x}=s+(e-s)\frac{w_1+2w_2}{3(w_1+w_2)}$ when $w_1+w_2\neq0$

Important correctness note: finite UDL and VDL contributions are handled as true finite beam load segments. This avoids the common error where bending moment, slope, and deflection become incorrect to the right of the loaded segment.

Worked example

Use this sample problem to see how the calculator works in practice. It loads a 20 ft beam with a point load, an applied moment, a full-span UDL, and a triangular load segment.

  • Beam length: 20 ft
  • Report position: 10 ft
  • Point load: 1000 lbf at 6 ft
  • Applied moment: 1500 lbf·ft at 10 ft
  • UDL: 50 lbf/ft from 0 to 20 ft
  • Triangular load: 0 to 120 lbf/ft from 5 ft to 15 ft
  • E: 29,000 ksi
  • I: 200 in⁴
  • c: 6 in

Click the button below, then review the reactions, diagrams, and values at x = 10 ft. This is a good way to verify the load definitions and learn the input format.

FAQ

Can I mix point loads, applied moments, UDLs, and varying loads?

Yes. Any combination of supported load types can be included in the same simply supported beam model.

Are the reactions exact?

Yes. The support reactions are obtained from static equilibrium using exact resultants and centroids for each distributed-load segment.

Does the page update automatically?

Yes. The page recalculates whenever you change a beam input, unit selection, or load value.

Does numerical resolution affect the calculation?

Yes, but mainly for slope and deflection because those are obtained by numerical integration. Support reactions and load resultants come from equilibrium and exact segment resultants, so they are not meaningfully affected by the numerical resolution setting.

Which beam theory is assumed?

Euler–Bernoulli beam theory, small deflections, constant material modulus, and constant second moment of area along the span.