Continuous Struts

Abstract

In this chapter, we consider the dynamics of a thin elastic strut including axial effects, arising primarily from one of two situations:• the axial load is applied externally (including postbuckling), or • deformation is sufficient to cause coupling between axial and bending behavior (the membrane effect).In the first section, attention is focused on a traditional approach to setting up the equations of motion (by means of D'Alembert's principle) for the simple case based on engineering beam theory (Euler-Bernoulli) with the addition of axial loads. The resulting partial differential equation of motion is then separated into temporal and spatial ordinary differential equations and the response analyzed for various magnitudes of the axial load [1,2]. Then an energy approach is used together with Rayleigh's method [3]. In this case, additional terms are retained in the potential energy to allow postbuckled effects to be analyzed [4] and the effect of initial geometric imperfections are included. An alternative approach is developed based on a simple application of Hamilton's principle, and in this instance stretching effects are also included and a solution developed by use of Galerkin's method. This approach will be similar to that used in the previous chapter on strings but now bending strain energy enters into the analysis (as well as compressive axial loading). In the final part of the chapter we consider the dynamics of struts that are loaded by gravity through self-weight. The next chapter will then continue the study of axially loaded members but with the scope opened to include a wider class of problem. Basic FormulationIn this section, we develop the governing equation of motion for a thin, elastic, prismatic beam subject to a constant axial force. In Fig. 7.1(a) a schematic of the beam is shown. It has mass per unit length m, constant flexural rigidity EI, and is subject to an axial load P. The length is L, the coordinate along the beam is x, and the lateral (transverse) deflection is w(x, t). In part (b) is shown an element of the beam between locations x and x + x, which is subject to the D'Alembert forces R(x, t) = m∂ 2 w/∂t 2 . (7.1) 111Cambridge Books Online