Equation Pad

Beam deflection calculator: write it once, change any input

Deflection of a simply supported beam under a uniform load and a central point load, its end slope, the deflection of a cantilever, and the maximum bending moment and shear force. Every line is typed exactly as shown, and every result updates when an input changes.

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Beam deflection calculation in Equation Pad with typeset equations and results in mm.

The calculation

w := 12 kN/m
L := 6 m
E := 210 GPa
I := 8360 cm^4
P := 25 kN
delta_w := 5*w*L^4/(384*E*I) = mm
delta_P := P*L^3/(48*E*I) = mm
delta := delta_w + delta_P = mm
L/delta =
theta := w*L^3/(24*E*I) + P*L^2/(16*E*I) =
theta*180/pi =
L_c := 2.5 m
delta_c := w*L_c^4/(8*E*I) + P*L_c^3/(3*E*I) = mm
M_max := w*L^2/8 + P*L/4 = kN*m
V_max := w*L/2 + P/2 = kN

Results for these inputs

  • Midspan deflection: 11.53 mm from the uniform load plus 6.41 mm from the point load, 17.94 mm in total, which is span / 334.
  • End slope 0.0094 rad (0.54°).
  • A 2.5 m cantilever with the same loads deflects 10.75 mm at its tip.
  • Maximum bending moment 91.5 kN·m and shear force 48.5 kN.

How it works

The classical elastic formulas: 5wL⁴/384EI for a uniform load and PL³/48EI for a central point load on a simply supported beam, wL⁴/8EI and PL³/3EI for a cantilever. Superposition adds the cases. Units are checked: kN, m, GPa and cm⁴ combine to give millimetres without any conversion factor.

These calculations use classical textbook mechanics, not a design code. For design, use the method, factors and checks that apply to your work and have the result checked by a competent person.

Try other cases

Change w, P, L or I and every line follows. Plot the deflected shape with plot(y(x), x, 0 m, L), or compare sections in a spreadsheet with units.

Questions

How do I calculate the deflection of a simply supported beam?

For a uniform load w on a span L: delta = 5wL⁴/(384EI). For a central point load P: delta = PL³/(48EI). Equation Pad evaluates both with units and adds them.

Can I use cm⁴ and GPa directly?

Yes. Write I := 8360 cm^4 and E := 210 GPa; the result is converted to the unit you ask for, for example = mm.

Also used for

Equation Pad runs in your browser, with nothing to install, and we never see or store your calculations. Results must be checked by a competent person before they are relied on; see the disclaimer.

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