APOLLONIUS, HOOKE, AND BUCKLING Dylan Buren

APOLLONIUS OF PERGA o Greek geometer and astronomer o Known for writings on conic sections o Writings influenced Ptolemy, Francesco, Maurolico, Kepler, Newton, and Descartes o Hypothesis of eccentric orbits and deferent epicycles was credited to him

APOLLONIUS’S WORK ON CONICS o Named the Parabola, Hyperbola, and Ellipse o His first four books on conics were based on the essential principles o Works with generation of the curves and fundamental properties o The others were specialized for particular directions o Defines conic properties as the equivalent of the Cartesian equation applied to the oblique axes o Obtained by cutting an oblique circular cone o Conics are used for the orbits of satellites

ROBERT HOOKE  English natural philosopher, architect and polymath  Studied Mechanics, Gravitation, Horology, Microscopy, Paleontology, Astronomy, and the Human Memory  He studied at Oxford  Widely reported to have corresponded with Thomas Newcomen with the invention of the steam engine

HOOKE’S LAW

BUCKLING

DEFINITION OF BUCKLING  Buckling is a mathematical instability that leads to a failure mode  It is characterized by a sideways failure subjected to high compressive stress  The compressive stress is considered the axial load  Force is applied from the ends  The point at which the load makes the object buckle is called the critical load

THE BEGINNINGS OF THE THEORY In 1678 Robert Hooke provided a necessary preliminary to the development of the elastic buckling theory when he stated that the displacement of any springy body was in proportion to the load causing the displacement. Hooke affirmed that this relationship, now known as Hooke's Law, could be applied to all "springy bodies,... metal, wood, stone, baked earth, hair, horns, silk, bones, sinews, glass and the like." Jacob Bernoulli studied the deflection and curvature in a cantilever beam. He asserted, in 1705, on the basis of Hooke's Law, that the curvature at any point in a bent rod was in proportion to the resisting moment developed in the rod at that point.

EULER'S CONTRIBUTION Leonard Euler studied under Jacob Bernoulli's brother John. Euler adopted Jacob's assumption regarding the moment/curvature relationship and in the Appendix to his 1744 book on variational calculus he presented the column formula that still bears his name. The "Euler load" is the critical load at which a slender elastic column can be held in a bent configuration under axial load alone. There were two limitations with Euler’s discoveries: (1) Before loading, the column is perfectly straight and carries a load which, prior to buckling, is coincident with the longitudinal centroidal axis of the member; and (2) although perfect with respect to dimensional requirements, our hypothetical column is made of "real" material; structural steel, structural aluminum alloy, stainless steel, or other nonferrous structural metal. In the case of structural steel, the stress-strain relationship is assumed to be linear up to the yield stress level, after which the material (on the average) deforms plastically without change in stress until a strain is reached that is several times that of the elastic range.

BASIC CALCULATIONS FOR CRITICAL LOAD  F = the critical force  E = modulus of elasticity  I = area moment of inertia of the cross section of the rod  L = unsupported length of the column  K = column effective length factor  KL = the effective length of the column

K? K = the effective length factor => constant Relates to the conditions of the end supports 4 options for each end: fixed, pinned, slide, free Values and shapes shown on right

BASIC CALCULATIONS FOR CURVE

MODES OF BUCKLING

EXAMPLES OF FAILURES

APPLICATIONS TO AIRPLANES

WORK CITED "Apollonius of Perga." Wikipedia. Wikimedia Foundation, n.d. Web. 29 July "Basics of Space Flight: Orbital Mechanics." Basics of Space Flight: Orbital Mechanics. N.p., n.d. Web. 29 July "Buckling." Wikipedia. Wikimedia Foundation, n.d. Web. 29 July "Hooke's Law." Wikipedia. Wikimedia Foundation, n.d. Web. 29 July Johnston, Burce G. "COLUMN BUCKLING THEORY: HISTORIC HIGHLIGHTS." Ascelibrary.org. N.p., n.d. Web. 29 July "Robert_Hooke." Wikipedia. Wikimedia Foundation, n.d. Web. 29 July 2015.

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