Cantilever beam calculator with a tip load

What it computes

You give the length, the tip load, the width and depth of the section, the material, a deflection limit (the length over 90, 180 or 360) and whether to include the own weight. The sheet works out the section properties, adds the own weight as a uniform load, and reports the moment and shear at the fixed end, the bending stress, the tip deflection and rotation, and two utilisations, with a plain sentence under each saying whether it passes. A moment diagram is plotted along the length.

The formulas

• Section: A = b d, I = b d^3 / 12, Z = b d^2 / 6.
• Own weight w = density x A x g, when the box is ticked.
• Moment at the fixed end M = P L + w L^2 / 2. Shear at the fixed end V = P + w L.
• Bending stress sigma = M / Z.
• Tip deflection delta = P L^3 / (3 E I) + w L^4 / (8 E I). Tip rotation theta = P L^2 / (2 E I) + w L^3 / (6 E I), in radians.
• Limit = L / 90, 180 or 360. Utilisations: sigma / allowable stress, and delta / limit.

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

Worked example

Defaults: a 2.5 m cantilever carrying 8 kN at the tip, a 100 mm by 200 mm section in structural steel (E = 200 GPa, allowable stress 165 MPa), a limit of L/90, own weight included. These are the values you get when you open the template. Results from the engine:

Both utilisations are well below 1, so the sheet prints "Bending: OK" and "Deflection: OK".

Change the length or the material

With the length raised to 4 m the moment grows to 44.32 kN·m and the tip deflection to 16.5 mm. Deflection grows with the cube of the length for the point load and with the fourth power for the own weight, so 60 percent more length gives more than four times the deflection. Both checks still pass, at 0.403 for bending and 0.371 for deflection.

Back at 2.5 m, switching the material to timber gives a stress of 30.46 MPa, a little lower than for steel because timber is lighter, against an allowable stress of only 12 MPa. The tip deflection is 57.47 mm against a limit of 27.78 mm. The utilisations of 2.54 and 2.07 both fail, and the sheet says to enlarge the section or change the material.

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; the length and the load show as sliders in Input mode.

# Cantilever with a load at the tip

A template with inputs. Choose **Input** in the results bar to fill in the form. A cantilever of
length $L$ with a solid rectangular section $b \times d$ carries a point load $P$ at the free end
and, if wanted, its own weight $w$. At the fixed end $M = P L + w L^2 / 2$ and
$V = P + w L$, and the tip deflection is $\delta = P L^3 / (3 E I) + w L^4 / (8 E I)$.

## Inputs
L:     ? 2.5 m [0.5 .. 6 step 0.1] | length
P:     ? 8 kN [1 .. 100 step 1] | point load at the free end
b:     ? 100 mm | section width
d:     ? 200 mm | section depth
mat:   ? [Structural steel = 1, Aluminium = 2, Timber = 3] | material
limit: ? (L/90 = 90, L/180 = 180, L/360 = 360) | tip deflection limit
sw:    ? on | include the own weight of the beam

## Material properties
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
A: b * d -> cm^2 | area
I: b * d^3 / 12 -> cm^4 | second moment of area
Z: b * d^2 / 6 -> cm^3 | elastic section modulus

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

## Results at the fixed end and at the tip
M_max: P * L + w * L^2 / 2 -> kN*m | maximum bending moment, at the fixed end
V_max: P + w * L -> kN | maximum shear force, at the fixed end
sigma: M_max / Z -> MPa | maximum bending stress
delta: P * L^3 / (3 * E * I) + w * L^4 / (8 * E * I) -> mm | tip deflection
theta: P * L^2 / (2 * E * I) + w * L^3 / (6 * E * I) | tip rotation, in radians
delta_lim: L / limit -> mm | tip 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, enlarge the section or change the material."
if util_d <= 1: "Deflection: OK, within the limit." else: "Deflection: over the limit, enlarge the section or change the material."

## Bending moment diagram
M(x): P * (L - x) + w * (L - x)^2 / 2 -> kN*m | bending moment at a distance x from the fixed end

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. Length, load, section width and depth, material, deflection limit and own weight are the 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.