Note: For more detailed stress & strain
notes refer to webpage Stress & Strain

Strain = Change in length (dL)over original length (L)

e = dL / L

Stress = Force (F) divided by Area withstanding Force (A)

σ = F / A

Young's Modulus E = Stress ( s ) / Strain(e). This is a property of a material

E = s / e

General Formula for Bending

A beam with a moment of inertia I and with Young's modulus E will have a
bending stress f at a distance from the Neutral Axis (NA) y and the NA will bend
to a radius R ...in accordance with the following formula.

M / I = s / y = E / R

W and w as used below for beam concentrated load, total load and uniform distributed load are assumed to be in units of force i.e. Newtons If they are provided in units of weight i.e kg then they should be converted into units of force by mutliplying by the gravity constant g (9.81)

__ Simply Supported Beam . Concentrated Load__

__Simply Supported Beam . Uniformly Distributed Load__

__Cantilever . Concentrated Load__

__Cantilever . Uniformly Distributed Load __

__ Fixed Beam . Concentrated Load __

__Fixed Beam . Uniformly Distributed Load__

Poisson's Ratio = ν = (lateral strain / primary strain )

Shear Modulus G = Shear Stress /Shear Strain

G = τ / ε = E / (2 .( 1 + ν ))

General Formula for Torsion

A shaft subject to a torque T having a polar moment of inertia J and a shear Modulus G will have a shear stress q at a radius r and an angular deflection θ over a length L as calculated from the following formula.

T / J = G . θ / L = t / r

More detailed notes on torsion calculations are found at webpage Torsion

For a thin walled cylinder subject to internal pressure P the circumferential
stress = σ_{c}

This stress tends to stretch the cylinder along its length. This is also called the longitudinal stress.

σ_{c} = P . d / ( 4 . t )

For a thin walled cylinder subject to internal pressure P the tangential
stress = σ_{c}

This stress tends to increase the diameter). This is also called the hoop stress.

σ_{t} = P . d / ( 2 . t )

The above two formulae are only valid if the ratio of thickness to dia is less than 1:20

The equations for the stresses in thick walled cylinders are derived on web page Cylinders

r_{1} = internal radius

r_{2} =outer radius

p_{1} = internal pressure

p_{2} = external pressure

σ _{r} = radial stress

σ _{t} =tangential stress

Consider a cylinder with and internal diameter d _{1}, subject to an
internal pressure p _{1}. The external diameter is d _{2} which
is subject to an external pressure p _{2}.
The radial pressures at the surfaces are the same as the applied pressures therefore

σ _{r} = A + B / r ^{2}

σ _{t} = A - B / r ^{2}

The radial pressures at the surfaces are the same as the applied pressures therefore

- p_{1} = A + B / r _{1}^{2}

-p_{2} = A + B / r _{2}^{2}

__ The resulting general equations are known as Lame's Euqations and are shown as follows __

If the external pressure is zero this reduces to

If the internal pressure is zero this reduces to