Simply supported beam calculator

What it computes

You give the span, the uniform load, the section depth, the material, a deflection limit, and whether to include the beam's own weight. The sheet derives the section properties from the depth, adds the own weight to the load, and reports the bending moment, shear force, stress, deflection and two utilisations (stress over allowable stress, deflection over its limit), with a plain sentence under each saying whether it passes. A moment diagram is plotted along the span.

The formulas

• Section from the depth h: flange width b = h / 2, flange thickness t_f = h / 20, web thickness t_w = h / 40.
• Area A = 2 b t_f + (h - 2 t_f) t_w. Second moment of area I = (b h^3 - (b - t_w)(h - 2 t_f)^3) / 12. Section modulus Z = 2 I / h.
• Own weight w_sw = density x A x g, total load w_tot = w + w_sw.
• Maximum moment M = w_tot L^2 / 8 at mid-span. Maximum shear V = w_tot L / 2 at the supports.
• Bending stress sigma = M / Z. Deflection delta = 5 w_tot L^4 / (384 E I). Limit = L / 250, 360 or 500.
• Utilisations: sigma / allowable stress, and delta / limit.

The materials table holds three entries (structural steel, aluminium and timber) with a modulus, an allowable bending stress and a density. A new material is one more entry in each list.

Worked example

Defaults: span 6 m, load 12 kN/m, a 300 mm I-beam in structural steel (E = 200 GPa, allowable stress 165 MPa), a limit of L/250, own weight included. These are the values you get when you open the template. Results from the engine:

Both utilisations are below 1, so the sheet prints "Bending: OK" and "Deflection: OK". Stress uses about 0.493 of the allowable stress and deflection about 0.424 of its limit.

Change one input and watch what governs

With the 200 mm section instead, the total load is 12.22 kN/m, the stress becomes 268.36 MPa and the deflection 50.32 mm, giving utilisations of 1.63 for bending and 2.1 for deflection. Both fail, and the sheet says to choose a deeper section.

Back at 300 mm, switching the material to aluminium gives a stress of 79.19 MPa (utilisation 0.792, which passes) but a deflection of 28.28 mm against a limit of 24 mm (utilisation 1.18, which fails). Stiffness, not strength, decides it. That is the kind of thing a form makes quick to see.

What this is, and is not

The sheet source

This is the template exactly as it ships. The lines with a question mark are the inputs, and the span and load show as sliders in Input mode; everything else is calculation and text. You can copy it into the app, or open the template and edit it.

# Simply supported beam check

A template with inputs. Choose **Input** in the results bar to fill in the form; every change
recalculates the page. The beam is a doubly symmetric I-beam, so one depth $h$ describes it:
flange width $b = h/2$, flange thickness $t_f = h/20$ and web thickness $t_w = h/40$. The check
compares the bending stress $\sigma = M/Z$ with an allowable stress and the mid-span deflection
$\delta = 5 w L^4 / (384 E I)$ with a limit.

## Inputs
L:     ? 6 m [2 .. 12 step 0.5] | span
w:     ? 12 kN/m [2 .. 40 step 1] | applied uniform load
h:     ? [200 mm deep = 200 mm, *300 mm deep = 300 mm, 400 mm deep = 400 mm] | I-beam section
mat:   ? [Structural steel = 1, Aluminium = 2, Timber = 3] | material
limit: ? (L/250 = 250, L/360 = 360, L/500 = 500) | deflection limit
sw:    ? on | include the own weight of the beam

## Material properties
The input above picks one entry of each table, so a new material is one more entry in each.

E_tab:   [200 GPa, 70 GPa, 11 GPa] | modulus of elasticity
fa_tab:  [165 MPa, 100 MPa, 12 MPa] | allowable bending stress
rho_tab: [7850 kg/m^3, 2700 kg/m^3, 500 kg/m^3] | density
E:   E_tab.mat -> GPa
f_a: fa_tab.mat -> MPa
rho: rho_tab.mat -> kg/m^3

if mat == 1: "The material is structural steel."
if mat == 2: "The material is aluminium."
if mat == 3: "The material is timber."

## Section
b:   h / 2 | flange width
t_f: h / 20 | flange thickness
t_w: h / 40 | web thickness
A:   2 * b * t_f + (h - 2 * t_f) * t_w -> cm^2 | area
I:   (b * h^3 - (b - t_w) * (h - 2 * t_f)^3) / 12 -> cm^4 | second moment of area
Z:   2 * I / h -> cm^3 | elastic section modulus

## Load
g_acc: 9.81 m/s^2 | gravitational acceleration
w_sw:  sw * rho * A * g_acc -> kN/m | own weight, when included
w_tot: w + w_sw | total uniform load

## Results
M_max: w_tot * L^2 / 8 -> kN*m | maximum bending moment, at mid-span
V_max: w_tot * L / 2 -> kN | maximum shear force, at the supports
sigma: M_max / Z -> MPa | maximum bending stress
delta: 5 * w_tot * L^4 / (384 * E * I) -> mm | mid-span deflection
delta_lim: L / limit -> mm | deflection limit
util_b: sigma / f_a | bending utilisation
util_d: delta / delta_lim | deflection utilisation

if util_b <= 1: "Bending: OK, the stress is within the allowable stress." else: "Bending: the stress exceeds the allowable stress, choose a deeper section or a stronger material."
if util_d <= 1: "Deflection: OK, within the limit." else: "Deflection: over the limit, choose a deeper section or a stiffer material."

## Moment diagram
M(x): w_tot * x * (L - x) / 2 -> kN*m | bending moment along the span

plot M(x * 1 m) / (1 kN*m) for x in 0..L / (1 m)

How to use it

1. Open the template. Choose Open this template on this page. The sheet opens in a new tab of CivilKit Calcs, with the inputs set to the values used on this page.
2. Switch to Input mode. In the results bar choose Input. The input lines become boxes, lists, buttons and ticks.
3. Enter your values. Change the inputs to your own. The span, load, section, material, limit and own weight are the six inputs. Every change recalculates the whole page.
4. Read and export the result. Read the results and the OK or not OK lines under them. Use Report, then Print PDF, Word or Excel, to keep the calculation with its workings.