**Related Resources: calculators**

### Bolt Preload Temperature Effects Formulae and Calculator

Fastener, Bolt and Screw Design Torque and Force Calculations

Bolt Preload Effects Due to Temperature Load Calculator and EquationsJoints with bolts, or other threaded fasteners, which are exposed to a temperature change after installation will have a change in preload. The change can either be an increase or decrease.

Preview Bolt Preload Effects Due to Temperature Calculator

The following equations assume no washer deformation and no gap exists.

Eq. 1

${e}_{B}={\alpha}_{B}{L}_{B}(T-{T}_{0})+\frac{P}{{E}_{B}}(\frac{{L}_{s}}{{A}_{s}}+\frac{{L}_{r}}{{A}_{r}}+\frac{{L}_{g}}{2{A}_{r}})$Eq. 2

${e}_{m}=(T-{T}_{0})({\alpha}_{a}{t}_{a}+{\alpha}_{b}{t}_{b}+{\alpha}_{c}{t}_{c})-P(\frac{{t}_{a}}{{A}_{a}{E}_{a}}+\frac{{t}_{b}}{{A}_{b}{E}_{b}}+\frac{{t}_{c}}{{A}_{c}{E}_{c}})$Eq. 3

$e}_{B}={e}_{M$Eq. 4

$P}_{T}=\frac{(T-{T}_{0})({\alpha}_{a}{t}_{a}+{\alpha}_{b}{t}_{b}+{\alpha}_{c}{t}_{c}-{\alpha}_{B}{L}_{B})}{\frac{1}{{E}_{B}}(\frac{{L}_{s}}{{A}_{s}}+\frac{{L}_{r}}{{A}_{r}}+\frac{{L}_{g}}{2{A}_{r}})+(\frac{{t}_{a}}{{A}_{a}{E}_{a}}+\frac{{t}_{b}}{{A}_{b}{E}_{b}}+\frac{{t}_{c}}{{A}_{c}{E}_{c}})$P_{T} will tension if T > T_{0} and α_{M} > α_{B}

(+) is tension in bolt, (-) would unload a preload bolt

If all three materials are the same, equation 4 becomes

Eq. 5

$P}_{T}=\frac{(T-{T}_{0})({\alpha}_{M}{t}_{M}-{\alpha}_{B}{L}_{B})}{\frac{1}{{E}_{b}}(\frac{{t}_{s}}{{A}_{s}}+\frac{{t}_{r}}{{A}_{r}}+\frac{{t}_{g}}{2{A}_{r}})+\frac{{t}_{m}}{{A}_{M}{E}_{M}}$If gap exists, no preload in bolt, equations 4 and 5 become

Eq. 6

$P}_{T}=\frac{-g+(T-{T}_{0})({\alpha}_{a}{t}_{a}+{\alpha}_{b}{t}_{b}+{\alpha}_{c}{t}_{c}-{\alpha}_{B}{L}_{B})}{\frac{1}{{E}_{B}}(\frac{{t}_{s}}{{A}_{s}}+\frac{{t}_{r}}{{A}_{r}}+\frac{{t}_{g}}{2{A}_{r}})+(\frac{{t}_{a}}{{A}_{a}}+\frac{{t}_{b}}{{A}_{b}{E}_{b}}+\frac{{t}_{c}}{{A}_{c}{E}_{c}})$(+) is tension in bolt, (-) would unload a reloaded bolt

Eq. 7

$P}_{T}=\frac{-g(T-{T}_{0})({\alpha}_{M}{t}_{M}-{\alpha}_{B}{L}_{B})}{\frac{1}{{E}_{B}}(\frac{{t}_{s}}{{A}_{s}}+\frac{{t}_{r}}{{A}_{r}}+\frac{{t}_{g}}{2{A}_{r}})+\frac{{t}_{M}}{{A}_{M}{E}_{M}}$(+) is tension in bolt, (-) would unload a reloaded bolt

Eq. 8

${A}_{M}=3\pi {D}_{B}^{2}/4$Eq. 9

L_{B} = L_{s} + L_{r} + L_{g} / 2

Eq. 10

t_{M} = t_{a} + t_{b} + t_{c}

Figure 1 - Partially Restrained Rectangle Plate

Figure 2 - Bolt Nomenclature

Where

A_{r} = Section area of root of bolt

A_{s} = Section area of shank of bolt

A_{a} = effective area of material under thermal load
= A_{b} = A_{c}

D_{B} = bolt diameter

D_{M} = Effective diameter of material thermal load on bolt; assumed to be 2 D_{B}

e_{B} = Deformation of bolt over length, L_{B}

L_{B} = effective bolt length

e_{M} = Deformation of material over thickness, L_{M}

L_{M} = Materials thickness

g = gap

t_{M} = Total material thickness

E = modulus of elasticity

T_{0} = Initial temperature

T = final temperature

α = Coefficient of thermal expansion

P = bolt preload

P_{T} = Preload tension with temperature effects

E_{a} = modulus of elasticity
material a

E_{b} = modulus of elasticity material b

E_{c} = modulus of elasticity material c

- Bell Helicopter Structural Design Manual, 1977

Related

- Fastener Relation Between Torque, Thread Pitch and Clamping Force Formula and Calculator
- Fastener Relation Between Bolt Torque and Clamping Force Formula and Calculator per. JIS B 1083
- Bolt Elongation Equation and Calculator while under Axial Stress
- Fastener / Thread Tensile Area of External Thread Formula
- Fastener / Threaded Pitch Circle Diameter Formula and Calculation
- Fastener / Threaded Shear Area Formula and Calculation
- Bolt Elongation Stress Formulas and Calculator per. MIL-HDBH-60 Micrometer Bolt Elongation Method: Below the proportional limit. bolt elongation is directly proportional to axial stress, i.e., elongation increases in the same proportion as stress increases.
- Bolt Fastener Preload Torsion Load Stress Formulas and Calculator per. MIL-HDBH-60
- Bolt Elongation by Turn Formulas and Calculator per. MIL-HDBH-60
- Bolt Torque on Assembly for Gasket Contact Stress per. ASME PCC-1 Equations and Calculator
- Force Required to Strip the Bolt Threads Any Length Formula and Calculator
- Force Required to Strip the Bolt Threads Formula and Calculator
- Torque Clamp Force Calculator
- Stud Preload Calculator
- Bolt Preload Tension Equation and Calculator
- Bolt Preload Tension Force Calculator
- Torque vs Tension Bolts Table Chart SAE J429 Bolts
- Bolt Multiple Diameters Change of Length Under Axial Force Formula and Calculator