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EasyBeam

Beam Calculator

Compute support reactions, shear force, bending moment, deflection and bending stress for a simply supported beam. Add point loads, distributed loads, linearly varying loads and applied moments. Switch between SI and imperial units at any time.

Beam
Supports
Loads
Cross-section & material
Project (for reports)
Beam layout
Beam layout diagramA simple beam, total length 6.00 m, with 2 supports.

Support reactions

SupportLocation (m)Force (N)Moment (N·m)
R10.005.00k—
R26.005.00k—
Max shear
5.00k N
Max moment
15.00k N·m
Max deflection
0.50 mm
Max stress
5.00 MPa
Shear force
Shear force along the beam length in m. Values in N. Use left and right arrow keys to read values at a point.-4.00k-2.00k02.00k4.00k0.001.002.003.004.005.006.00
Bending moment
Bending moment along the beam length in m. Values in N·m. Use left and right arrow keys to read values at a point.02.00k4.00k6.00k8.00k10.00k12.00k14.00k0.001.002.003.004.005.006.00
Deflection
Deflection along the beam length in m. Values in mm. Use left and right arrow keys to read values at a point.-4.00e-1-3.00e-1-2.00e-1-1.00e-10.00e+00.001.002.003.004.005.006.00
Bending stress
Bending stress along the beam length in m. Values in MPa. Use left and right arrow keys to read values at a point.01.002.003.004.005.000.001.002.003.004.005.006.00
Data table (accessibility fallback)
x (m)Shear (N)Moment (N·m)Deflection (mm)Stress (MPa)
0.005.00k00.00e+00
0.555.00k2.73k-1.35e-10.91
1.095.00k5.46k-2.61e-11.82
1.645.00k8.19k-3.69e-12.73
2.185.00k10.92k-4.50e-13.64
2.735.00k13.65k-4.94e-14.55
3.28-5.00k13.62k-4.94e-14.54
3.82-5.00k10.89k-4.49e-13.63
4.37-5.00k8.16k-3.68e-12.72
4.91-5.00k5.43k-2.60e-11.81
5.46-5.00k2.70k-1.34e-10.90
6.00000.00e+00

Multi-span beam? Try thecontinuous beam calculatoror compute section properties with thecross-section calculator.

How the simply supported beam calculator works

A simply supported beam rests on a pin at one end and a roller at the other, so it is statically determinate: the two support reactions follow from equilibrium alone. The calculator applies the two equations of static equilibrium, ΣM = 0 and ΣF = 0, to every load you add, then walks the beam to build the shear and bending-moment diagrams.

Support reactions

For a single point load P at distance a from the left support on a beam of span L:

RA = P(L − a) / L     RB = P·a / L

For a uniformly distributed load w over the full span, each reaction is wL / 2.

Maximum bending moment

Central point load: M = PL / 4. Full-span uniform load: M = wL² / 8. The bending-moment diagram shows the full variation, including partial and varying loads where the peak is not at midspan.

Deflection

Deflection follows from the double integration of the moment diagram divided by the flexural rigidity EI:

EI · d²y/dx² = M(x)

Central point load: δ = PL³ / 48EI. Uniform load: δ = 5wL⁴ / 384EI. Deflection is usually the governing serviceability check — a common limit is span/360 for live load.

Bending stress

The extreme-fibre bending stress isσ = M·y / I, wherey is the distance from the neutral axis and I the second moment of area. Compare it with the allowable stress of the material.

Worked example

A steel beam, span 6 m, rectangular section 200 mm × 300 mm, carries a 10 kN point load at midspan. Second moment of areaI = bh³/12 = 4.5 × 10⁻⁴ m⁴.

  • Reactions: R = P/2 = 5 kN at each support.
  • Maximum moment: M = PL/4 = 15 kN·m at midspan.
  • Midspan deflection: δ = PL³/48EI = 0.5 mm (E = 200 GPa).
  • Maximum bending stress: σ = My/I = 5 MPa.

Load the same inputs into the calculator above and the diagrams match these hand calculations exactly.

Frequently asked questions

How do I calculate the reactions of a simply supported beam?+

For a beam of length L with a single point load P at distance a from the left support, the reactions are R_left = P(L−a)/L and R_right = Pa/L. For a uniformly distributed load w over the full span, each reaction is wL/2. The calculator solves these automatically for any combination of point, distributed, varying and moment loads.

What is the maximum bending moment of a simply supported beam?+

For a central point load P the maximum moment is PL/4 at midspan. For a full-span uniform load w it is wL²/8, also at midspan. The bending moment diagram shows how the moment varies along the whole beam, including the effect of partial and varying loads.

How is beam deflection calculated?+

Deflection comes from integrating the bending moment diagram twice and dividing by the flexural rigidity EI: EI·d²y/dx² = M(x). The boundary conditions (zero deflection at the supports) fix the constants of integration. For a central point load the midspan deflection is PL³/48EI; for a uniform load it is 5wL⁴/384EI.

What is bending stress and how is it found?+

Bending stress is σ = My/I, where M is the bending moment, y the distance from the neutral axis, and I the second moment of area. The maximum bending stress occurs where the moment is largest, at the extreme fibre y. Compare it against the allowable stress of the material to check the beam.

Does this calculator use SI or imperial units?+

Both. You can switch between SI (metres, newtons, MPa) and imperial (feet, pounds-force, ksi) at any time and the physical beam is preserved — only the units change.