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Lesson 7 · Margin and heat
Margin, and the stress nobody applied
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Margin, and the stress nobody applied

FS=σuσallowFS = \dfrac{\sigma_u}{\sigma_{allow}} — F S equals sigma-u over sigma-allowable. σu\sigma_u is the strength of the material, the stress at which it yields or breaks; σallow\sigma_{allow} is the working stress the design actually puts into it; FSFS is the factor of safety, a bare ratio with no units at all, because it is a stress divided by a stress.

Read it as a declaration of ignorance rather than of strength. A factor of safety of 3 does not say the part is three times better than it needs to be; it says the engineer is keeping a threefold reserve against the load being heavier than assumed, the steel weaker than certified and the weld worse than drawn — all on the same bad morning. FSFS below 1 means the working stress has already passed the strength: the part is failing on paper. FSFS at 1.4 means it survives the load you wrote down, and nothing else.

Then a stress nobody applies. Heat a bar and it wants to grow by α ΔT\alpha \, \Delta T per unit of length; hold both ends and it cannot, so the stress arrives anyway: σ=Eα ΔT\sigma = E \alpha \, \Delta T — sigma equals E alpha delta-T. α\alpha (alpha) is the coefficient of thermal expansion, about 12×10−612 \times 10^{-6} per °C for carbon steel; ΔT\Delta T is the temperature change from the condition it was installed at; EE is the same Young's modulus as ever. Notice what is NOT in that formula: length. A restrained run builds the same stress at two metres as at two hundred — which is why expansion is designed around anchors, never around lengths.

σuσallowFS

FS=σuσallowFS = \frac{\sigma_{u}}{\sigma_{allow}}

  • FSFS= Factor of safety
  • σu\sigma_{u}= Ultimate or yield strength (pressure)
  • σallow\sigma_{allow}= Allowable working stress (pressure)
Factor of Safety solver →
σαΔTE

σ=EαΔT\sigma = E \alpha \Delta T

  • σ\sigma= Thermal stress (pressure)
  • EE= Young's modulus (pressure)
  • α\alpha= Coefficient of thermal expansion (thermal expansion coefficient)
  • ΔT\Delta T= Temperature change (temperature difference)
Thermal Stress in a Restrained Member solver →