Vibration and strength of struts and continuous beams under end thrusts

Abstract

On January 3, 1918, the authors presented a paper on “The Critical Loading of Struts and Structures” Since then the method of analysis has been extended to include the more general problem of the vibration of a continuous beam simply supported at any number of non-collinear points, the external loading being a constant longitudinal end thrust, varying from bay to bay and a periodic lateral loading. In addition, the supports are assumed in a state of periodic vibration. The flexural rigidity constant along each bay varies, however, from bay to bay. These conditions will then correspond closely with those originated in a wing spar of an aeroplane when in actual flight and influenced by engine throbbing. A very generalised form of the equation of three moments is derived, and the conditions for resonance and crippling expressed in a convenient determinantal form. The general equation where the restrictions are removed as regards constancy of end thrust and of flexural rigidity is then treated and illustrated in the particular case of the crippling of a strut of variable flexural rigidity. Attention has at intervals been drawn to the effect of vibrations set up in aeroplane spars by external forces during flight, such as engine throbbing and wind forces. For very simple cases a few investigations exist to find under what conditions the results of this become serious on account of resonance, but no proper treatment of the actual problem and of the stresses brought into being appears to exist, especially in the case where the spar is under end thrust. In a previous communication* the static case of a beam supported at any number of non-collinear points and under the influence of lateral loading and end thrust varying from bay to bay was investigated, and the conditions for instability derived in a determinantal form. It will be seen that the previous discussion falls out as a mere particular case of the more general question developed in the present paper.