Abstract
The Euler–Bernoulli theory of beams is usually presented in two forms: (i) in the linear case of a small slope using Cartesian coordinates along and normal to the straight undeflected position; and (ii) in the non-linear case of a large slope using curvilinear coordinates along the deflected position, namely, the arc length and angle of inclination. The present paper starts with the exact equation in a third form, that is, (iii) using Cartesian coordinates along and normal to the undeflected position like (i), but allowing exactly the non-linear effects of a large slope like (ii). This third form of the equation of the elastica shows that the exact non-linear shape is a superposition of linear harmonics; thus, the non-linear effects of a large slope are equivalent to the generation of harmonics of a linear solution for a small slope. In conclusion, it is shown that: (i) the critical buckling load is the same in the linear and non-linear cases because it is determined by the fundamental mode; (ii) the buckled shape of the elastica is different in the linear and non-linear cases because non-linearity adds harmonics to the fundamental mode. The non-linear shape of the elastica, for cases when powers of the slope cannot be neglected, is illustrated for the first four buckling modes of cantilever, pinned, and clamped beams with different lengths and amplitudes.
Highlights
The Bernoulli [1] and Euler [2] theory of beams is a standard introductory subject in textbooks on elasticity [3,4,5,6,7,8,9,10,11] and leads to the phenomenon of buckling, which has been considered in several conditions: (i) geometric and material non-linearities [12]; (ii) in combination with shear [13,14] that is more significant for thin-walled beams [15,16,17,18,19]; (iii) constraints [20,21,22], such as hyper or non-local elasticity [23,24]; (iv) vibrations [25,26], that can be excited by unsteady applied forces [27,28,29,30,31,32,33], leading to control problems [34]; (v) steady mechanical [35] or thermal [36,37,38] effects; and (vi) vibrations of tapered beams [39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55], with multiple applications like airplane wings and flexible aircraft and helicopters [56,57,58,59,60,61,62,63,64]. Among this wide range of topics related to the buckling of elastic beams, the present paper focuses on geometric non-linearities associated with a large slope of the elastica
The equation of the elastica of a beam is usually written in one of the two forms: (i) in Cartesian coordinates, y = ζ(x), (1a) with the x-axis along the undeformed beam; or (ii) in curvilinear coordinates, s = ξ(θ), (1b) with the arc length s as a function of the angle of inclination
A linear deflection is defined by a small slope and implies that the maximum deflection is small compared with the distance between the supports
Summary
The Bernoulli [1] and Euler [2] theory of beams is a standard introductory subject in textbooks on elasticity [3,4,5,6,7,8,9,10,11] and leads to the phenomenon of buckling, which has been considered in several conditions: (i) geometric and material non-linearities [12]; (ii) in combination with shear [13,14] that is more significant for thin-walled beams [15,16,17,18,19]; (iii) constraints [20,21,22], such as hyper or non-local elasticity [23,24]; (iv) vibrations [25,26], that can be excited by unsteady applied forces [27,28,29,30,31,32,33], leading to control problems [34]; (v) steady mechanical [35] or thermal [36,37,38] effects; and (vi) vibrations of tapered beams [39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55], with multiple applications like airplane wings and flexible aircraft and helicopters [56,57,58,59,60,61,62,63,64]. Both methods provide almost the same stable and unstable equilibrium solutions and, for some cases of the unstable equilibrium configuration, the elastica can form a single loop or snap-back bending Continuing in this line of investigation, the paper [92] considers the large deflection problem of variable deformed arc-length beams, with a uniform flexural rigidity, but under a point load. A beam of variable cross-sections can taper in two directions [104], for example, in the case of a pyramidal beam representing an airplane wing with chords much larger than the thickness affecting the natural frequencies of bending modes Following this introduction (Section 1) to the Euler–Bernoulli theory of beams, the core of the paper focuses on geometric non-linearities associated with a large slope of the elastica. The conclusion (Section 5) highlights the use of linear buckling harmonics to specify the shape of the elastica for non-linear buckling with a large slope
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