beams formulae
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Home Formulae_Index Nomenclature
Stress/Strain Formulae
Basic Definitions
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 ﴾ ﴿ / Strain﴾e﴿. This is a property of a material
E = / 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 NeutralAxis ﴾NA﴿ y and the NA will bend to a radius R ...in accordance with the following formula.
M / I = / y = E / R
Important noteW 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 gravityconstant g ﴾9.81﴿
Simply Supported Beam . Concentrated Load
Simply Supported Beam . Uniformly Distributed Load
Strength of MaterialRs.578 New Deals Every Day on Amazon IndiaAmazon India
Cantilever . Concentrated Load
Cantilever . Uniformly Distributed Load
Fixed Beam . Concentrated Load
Fixed Beam . Uniformly Distributed Load
Torsion /Shear
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 aradius r and an angular deflection θ over a length L as calculated from the following formula.
T / J = G . θ / L = / r
More detailed notes on torsion calculations are found at webpage Torsion
Pressure Vessels Thin Walled Cylinders
For a thin walled cylinder subject to internal pressure P the circumferential stress = σc
This stress tends to stretch the cylinder along its length. This is also called the longitudinal stress.
σc = P . d / ﴾ 4 . t ﴿
For a thin walled cylinder subject to internal pressure P the tangential stress = σc This stress tends to increase the diameter﴿. This is also called the hoop stress.
σt = 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
The equations for the stresses in thick walled cylinders are derived on web page Cylinders
r1 = internal radiusr2 =outer radiusp1 = internal pressurep2 = external pressure
σ r = radial stressσ 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 whichis subject to an external pressure p 2. The radial pressures at the surfaces are the same as the applied pressurestherefore
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
p1 = A + B / r 12
p2 = A + B / r 22
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
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Last Updated 06/02/2011
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