Fixedfixed

Fixedfixed, often written fixed-fixed, designates a two-end boundary condition where both ends of a structural member are restrained from translation and rotation. It is typically applied to beams in structural analysis and to analogous conditions in plates or shells. Under this condition, the transverse displacement y is zero at the ends and the transverse slope y' is zero, eliminating both vertical and angular movement at the boundaries.

In Euler-Bernoulli beam theory, fixed-fixed boundary conditions lead to mode shapes and natural frequencies that differ from other end conditions. For a uniform beam of length L, the governing equation is EI y'''' + ρ A ω^2 y = 0 with y(0)=y'(0)=y(L)=y'(L)=0. The resulting eigenvalues require solving cosh(kL) cos(kL) = 1, yielding discrete mode shapes that begin with a lower deflection pattern than simply supported end conditions for the same frequency. The fundamental frequency is higher than that of a simply supported beam of equal length and mass.

In engineering practice, fixed-fixed conditions arise when ends are clamped or welded to supports, or when joints constrain rotation. They are common in flywheels, shafts, and long continuous beams, and are supported in finite element analyses by appropriate boundary constraints.

Alternative terms include clamped-clamped or built-in. Fixed-fixed analysis contrasts with fixed-free (cantilever) and simply supported conditions, each producing different stiffness, deflection, and vibration responses.