Natural frequencies of beams under tensile axial loads

Abstract

The effect of a constant axial tensile force on natural frequencies and mode shapes of a uniform single-span beam, with different combinations of end conditions, is presented. Numerical observations indicate that the variation of normalized natural frequency parameter, Ω − , with normalized tension parameter, Ω − , is almost the same for clamped-pinned and pinned-free, and similarly for clamped-clamped and clamped-sliding beams; the variation of the sliding-free beam is only slightly different from that of the latter pair and the free-free beam. For pinned-pinned, pinned-sliding and sliding-sliding beams, this variation may exactly be expressed as Ω − = √1 + U − . This formula may be used for beams with other types of end constraints when the beam vibrates in a third mode or higher. It also gives the upper bound approximation to the fundamental natural frequency of a pinned-free beam. For beams with other types of boundary conditions, this approximation may be expressed as Ω − = √1 + γ U − (γ < 1) , where the coefficient γ depends only on the type of the end constraints. It is found that when the dimensionless tension parameter U is greater than about 12, then U can be expressed as an analytical function of Ω U , where Ω is the dimensionless natural frequency parameter. For such a beam in the first few modes, the natural frequency is independent of the flexural rigidity and the beam behaves like a string. The string solution gives a lower bound approximation to the natural frequency.