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)
s = F / A
Young's Modulus E = Stress ( s ) / Strain(e). This is a property of a material
E = s / e
Bending
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
Torsion /Shear
Poisson's Ratio = n = (lateral strain / primary strain )
Shear Modulus G = Shear Stress /Shear Strain
G = t / g = E / (2 .( 1 + n ))
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 q over a length L as calculated from the following formula.
T / J = G . q / L = t / r
Pressure Vessels - Thin Walled Cylinders
For a thin walled cylinder subject to internal pressure P the circumferential
stress = p_c.
This stress tends to stretch the cylinder along its length. This is also called the longitudinal stress.
p_c = P . d / ( 4 . t )
For a thin walled cylinder subject to internal pressure P the tangential stress = p_t
This stress tends to increase the diameter). This is also called the hoop stress.
p_t = P . d / ( 2 . t )
The above two formulae are only valid if the ratio of thickness to dia is less than 1:20
Pressure Vessels - Thick Walled Cylinders
