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The timbers were not selected on the basis of modern engineering analysis They certainly do not bear inspection marks attesting to their grade and specie It should be obvious to the engineer that a safe oor structure should not fail in bending due to the actual loads imposed But it is important to recognize that excessive de ection, excessive vibration, or lack of stiffness, should not automatically categorize a oor structure as unsafe Strict de ection limitations should be set, however, for oors that support plaster ceilings in lieu of wood or tin ceilings, or no ceilings at all But for comfort, the live load de ection limitation set in most building codes for oors, no matter what the ceiling, is 1/360th of the span It must be said that overstressed structural members may also be perfectly safe It is, however, important to evaluate the basis for the conclusion.



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Leonardo of Pisa lived from 1170 to 1250, and he is regarded as perhaps the greatest mathematician of the Middle Ages. His father William had the nickname Bonaccio, which means good-natured, so Leonardo was subsequently known by the nickname of Fibonacci (short for lius Bonacci, or the son of Bonaccio). William worked in North Africa at a trading post, and even though his home country of Italy had used the Roman numeral system for centuries, he came into contact with traders who instead used the Arabic method of numbers (which is what we use today 1, 2, 3, 4, and so forth, as opposed to numbers such as XI and MMC). Leonardo helped his father with his work, and in the course of doing so, he became acquainted with and enamored by the Arabic style of numbering, which seemed far more sensible and ef cient than the Roman style. The world owes Leonardo a debt of gratitude for help347





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Figure 6.3.5 Coherent wave attenuation rate 2K i in cm- 1 as a function of frequency. The permittivity of the particles is E. = (3.2 + iO.OO1)Eo. Rayleigh size distribution is considered with (a) = 0.035 cm and fractional volume f = 0.3. The cases of independent scattering and QCA are compared.

QCA (single size a=0.035 em. vs Rayleigh size <a>=0.035 em.)

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The loads assumed for design should be reconciled with the actual loads that will occur in service The design values that we assume are very critical to the computed capacity of some oor systems For structures such as mill buildings, average design values yield results that fall well above the minimum code requirements for an adaptive reuse occupancy such as of ce or retail..

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ing to popularize this system of numbers, since it s doubtful anyone today would appreciate having to use Roman numerals in everyday life. Another great contribution Leonardo made was with respect to his writings related to what we today call Fibonacci numbers. This is a sequence of numbers which begins like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1,597. Each number is the sum of the two preceding numbers. These numbers have a variety of interesting properties, including the fact that, as the sequence progresses, each number divided by its preceding number approaches the mathematical gure phi (approximately 1.618033989), known as the golden ratio, which has a wide variety of expressions in nature. In the world of stock charts, the concepts put forth by Fibonacci are manifested in drawn objects retracements, time zones, arcs, and fans. We will explore each of these. There are entire volumes written on the golden ratio and Fibonacci numbers, including their relationship to nature. We will focus on how these mathematical insights can help make us better, more pro table traders.

Figure 6.3.6 Coherent wave attenuation rate 2Ki in cm- 1 under QCA as a function of frequency. The permittivity of the particles is t s = (3.2 + iO.OOl)Eo. The fractional volume is f = 0.3. The cases of identical sizes of a = 0.035 cm and Rayleigh size distribution with (a) = 0.035 cm are compared.

<a>=0.035 em.)

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