Bug Rivet Paradox Calculator
Calculate thermal expansion force when a heated rivet cannot expand freely in a hole. This free browser tool solves the bug rivet paradox fast, with no signup.
The Bug and Rivet Paradox
A bug sits on a metal rivet in a plate. When both are heated equally, the bug expands freely while the rivet cannot because the hole constrains it. Even though both experience the same temperature rise, the rivet develops internal stress and pushes on the plate.
Free Thermal Expansion
$$\Delta L = \alpha L \Delta T$$\(\alpha\) is the coefficient of linear thermal expansion, \(L\) is the original length, and \(\Delta T\) is the temperature change. An unconstrained body lengthens by \(\Delta L\).
Stress When Expansion Is Prevented
$$\sigma = E \alpha \Delta T \qquad F = \sigma A$$If expansion is fully prevented, the material behaves as if strained by \(\alpha \Delta T\). Hooke's law gives compressive stress \(\sigma\), and multiplying by cross-sectional area \(A\) gives axial force \(F\) on the constraint.
Frequently Asked Questions
Why is it called a paradox?
Bug and rivet heat equally and would expand the same amount in free space, yet only the constrained rivet generates large force. The difference is boundary conditions, not temperature.
Does the bug exert force on the rivet?
The bug's weight is tiny compared to thermal stress in a constrained steel rivet. The paradox highlights that constraint, not temperature difference, creates force.
What is a typical steel expansion coefficient?
Carbon steel is about \(12 \times 10^{-6}\) per °C. Aluminum is roughly \(23 \times 10^{-6}\) per °C.
What if expansion is only partially constrained?
Partial constraint produces stress between zero and the fully constrained value \(E \alpha \Delta T\). Real joints also allow some compliance from the surrounding plate.
Can thermal stress exceed yield strength?
Yes. Large \(\Delta T\) in a fully constrained member can generate stress above yield, causing permanent deformation or failure.
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