APOLLONIUS, HOOKE, AND BUCKLING
Dylan Buren
APOLLONIUS OF PERGA
oGreek geometer and astronomer
oKnown for writings on conic sections
oWritings influenced Ptolemy, Francesco, Maurolico, Kepler, Newton, and Descartes
oHypothesis of eccentric orbits and deferent epicycles was credited to him
APOLLONIUS’S WORK ON CONICS
oNamed the Parabola, Hyperbola, and Ellipse
oHis first four books on conics were based on the essential
principles
oWorks with generation of the curves and fundamental properties
oThe others were specialized for particular directions
oDefines conic properties as the equivalent of the Cartesian
equation applied to the oblique axes
oObtained by cutting an oblique circular cone
oConics 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
Hooke discovered the Law of Elasticity
Applied to springs
When compressed, springs push out
When extended, springs pull in
In the most basic form 𝐹 = −𝑘𝑋
Force is proportional to the displacement
1
Also goes to energy 𝑈𝑒𝑙 𝑥 = 2 𝑘𝑥 2
Force is always towards equilibrium
Applied to rigid bodies
𝜏 = 𝐺 ∗ ∆𝜖 » stress = modulus of rigidity * strain
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
P = pressure exerted on column
Y = height
u = displacement, and a function of Y
𝑑2𝑢
𝐶 2
𝑑𝑦
+ 𝑃 ∗ 𝑢 = 0, 𝑢 0 = 0 𝑢 𝑙 = 0
l = highest point, C is a constant
MODES OF BUCKLING
EXAMPLES OF FAILURES
APPLICATIONS TO AIRPLANES
WORK CITED
"Apollonius of Perga." Wikipedia. Wikimedia Foundation, n.d. Web. 29 July 2015.
"Basics of Space Flight: Orbital Mechanics." Basics of Space Flight: Orbital Mechanics. N.p., n.d. Web.
29 July 2015.
"Buckling." Wikipedia. Wikimedia Foundation, n.d. Web. 29 July 2015.
"Hooke's Law." Wikipedia. Wikimedia Foundation, n.d. Web. 29 July 2015.
Johnston, Burce G. "COLUMN BUCKLING THEORY: HISTORIC HIGHLIGHTS." Ascelibrary.org. N.p., n.d.
Web. 29 July 2015.
"Robert_Hooke." Wikipedia. Wikimedia Foundation, n.d. Web. 29 July 2015.
VIDEOS AND IMAGE CREDIT
http://alert.air-worldwide.com/Alert/alertdata/Protect/Data/Chile__Earthquake/L3/Fig2_Shear.jpg
http://www.nptel.ac.in/courses/Webcourse-contents/IITROORKEE/strength%20of%20materials/lects%20&%20picts/image/lect36/lecture36.htm
http://www.aafo.com/hangartalk/attachment.php?attachmentid=15020&d=1272517145
http://www.myparkingsign.com/MPS/Buckle_Up_Signs.aspx
http://www.activatingart.com/portfolio/buckleUp/buckleUp.html
http://www.wgnflag.com/xcart/images/P/G-51_BudkleUpStockSign.jpg
http://www.superservicellc.com/news/buckle-up-3/
https://en.wikipedia.org/wiki/File:ColumnEffectiveLength.png
http://www.ratomodeling.com/articles/stressed_skin/
https://www.youtube.com/watch?v=Mm7gYl18Muk
https://www.youtube.com/watch?v=DFeHYFPElvE
https://www.youtube.com/watch?v=yqAFSKlALwk