# Moment distribution method. Джинсы method

## Jeans - Wikipedia

A pair of factory-distressed, loose fit men's jeans Microscopic image of faded jeans fabric

Jeans are a type of trousers, typically made from denim or dungaree cloth. Often the term "jeans" refers to a particular style of trousers, called "blue jeans," which were invented by Jacob W. Davis in partnership with Levi Strauss & Co. in 1871[1] and patented by Jacob W. Davis and Levi Strauss on May 20, 1873. Prior to the Levi Strauss patented trousers, the term "blue jeans" had been long in use for various garments (including trousers, overalls, and coats), constructed from blue colored denim.[2] Originally designed for cowboys and miners, jeans became popular in the 1950s among teenagers, especially members of the greaser subculture. Jeans were a common fashion item in the 1960s hippie subculture and they continued to be popular in the 1970s and 1980s youth subcultures of punk rock and heavy metal. Historic brands include Levi's, Lee, and Wrangler.

### History

#### Origin of jean fabric

A traditional women's Genoese dress in "blue jeans" (1890s)

Research on the trade of jean fabric shows that it emerged in the cities of Genoa, Italy, and Nîmes, France. Gênes, the French word for Genoa, may be the origin of the word "jeans". In Nîmes, weavers tried to reproduce jean but instead developed a similar twill fabric that became known as denim, from de Nîmes, meaning "from Nîmes". Genoa’s jean was a fustian textile of "medium quality and of reasonable cost," very similar to cotton corduroy for which Genoa was famous, and was "used for work clothes in general." The Genoese navy equipped its sailors with jeans, as they needed a fabric which could be worn wet or dry.[3][4] Nîmes’s "denim" was coarser, considered higher quality, and was used "for over garments such as smocks or overalls".[5] Nearly all Indigo, needed for dyeing, came from indigo bush plantations in Lahore (a city in Pakistan) till the late 19th century. It was replaced by indigo synthesis methods developed in Germany.[6]

Copper rivets for reinforcing pockets are a characteristic feature of blue jeans.

By the 17th century, jean was a crucial textile for working-class people in Northern Italy. This is seen in a series of genre paintings from around the 17th century attributed to an artist now named The Master of the Blue Jeans.[7] The ten paintings depict impoverished scenes with lower-class figures wearing a fabric that looks like denim. The fabric would have been Genoese jean, which was cheaper. Genre painting came to prominence in late 16th century, and the low-life subject matter in all ten paintings places them among others that portray similar scenes.[8]

Dungaree was mentioned for the first time in the 17th century, when it was referred to as cheap, coarse thick cotton cloth, often colored blue but sometimes white, worn by impoverished people in what was then a region of Bombay, India a dockside village called Dongri. This cloth was "dungri" in Hindi. Dungri was exported to England and used for manufacturing of cheap, robust working clothes. It was rumored that men would carry dead dung beetles in their pocket for luck. In English, the word "dungri" became pronounced as "dungaree".[9]

#### Origin of riveted jeans

Levi Strauss

Jacob Davis

The term jeans appears first in 1795, when a Swiss banker by the name Jean-Gabriel Eynard and his brother Jacques went to Genoa and both were soon heading a flourishing commercial concern. In 1800 Massena's troops entered the town and Jean-Gabriel was entrusted with their supply. In particular he furnished them with uniforms cut from blue cloth called "bleu de Genes" whence later derives the famous garment known worldwide as "blue jeans".[10]

Levi Strauss, as a young man in 1851, went from Germany to New York to join his older brothers who ran a goods store. In 1853, he moved to San Francisco to open his own dry goods business. Jacob Davis was a tailor who often bought bolts of cloth from the Levi Strauss & Co. wholesale house. In 1872, Davis wrote to Strauss asking to partner with him to patent and sell clothing reinforced with rivets.[11] The copper rivets were to reinforce the points of stress, such as pocket corners and at the bottom of the button fly. Levi accepted Davis's offer,[12] and the two men received US patent No. 139,121 for an "Improvement in Fastening Pocket-Openings" on May 20, 1873.[13]

The classic label for Levi 501 jeans.

Davis and Strauss experimented with different fabrics. An early attempt was brown cotton duck, a bottom-weight fabric.[a] Finding denim a more suitable material for work-pants, they began using it to manufacture their riveted pants. The denim used was produced by an American manufacturer. Popular legend incorrectly states that it was imported from Nimes, France. A popular myth is that Strauss initially sold brown canvas pants to miners, later dyed them blue, turned to using denim, and only after Davis wrote to him, added rivets.[11]

Initially, Strauss' jeans were simply sturdy trousers worn by factory workers, miners, farmers, and cattlemen throughout the North American West.[14][15] During this period, men's jeans had the fly down the front, whereas women's jeans had the fly down the left side.[16] When Levi Strauss & Co. patented the modern, mass-produced prototype in the year 1873, there were two pockets in the front and one on the back with copper rivets. Later, the jeans were redesigned to today's industry standard of five pockets including a little watch pocket and copper rivets.[10]

#### Recent evolution

A female music artist wearing ripped jeans, 2009

Fewer jeans were made during World War II, but 'waist overalls' were introduced to the world by US soldiers, who sometimes wore them off duty.[17][citation needed] By the 1960s, both men's and women's jeans had the zipper down the front. Historic photographs indicate that in the decades before they became a staple of fashion, jeans generally fit quite loosely, much like a pair of bib overalls without the bib. Indeed, until 1960, Levi Strauss called its flagship product "waist overalls" rather than "jeans".

After James Dean popularized them in the movie Rebel Without a Cause, wearing jeans became a symbol of youth rebellion during the 1950s.[18] During the 1960s the wearing of jeans became more acceptable, and by the 1970s it had become general fashion in the United States for casual wear.[19]

#### Sandblasting or abrading with sandpaper

Consumers wanting jeans that appear worn can buy jeans that have been specially treated. To give the fabrics the worn look, sandblasting done with chemicals or by adding pumice stone to the washing process or abrading with sandpaper is often done.

#### Environmental and humanitarian impact

A typical pair of blue jeans uses 919 gallons (3479 liters) of water during its life cycle. This includes the water to irrigate the cotton crop, manufacture the jeans, and the numerous washes by the consumer.[27]

The production of jeans with a "used look" can be more environmentally damaging than regular jeans[28][citation needed], depending on how the waste compounds are processed. Sandblasting and treating with sandpaper has the risk of causing silicosis to the workers, and in Turkey, more than 5,000 textile workers have been stricken with this disease, and 46 people are known to have died. Some companies have announced they are banning the use of sandblasting.[29]

### Care and wear

Despite most jeans being "pre-shrunk", they are still sensitive to slight further shrinkage and loss of color from being washed. The Levi Strauss company recommends avoiding washing jeans as much as possible. Carl Chiara, Levi Strauss director of brand and special projects, has a credo: The less you wash your jeans, the better your jeans become.[30] These and other suggestions to avoid washing jeans where possible have encountered criticism. Cory Warren, editor of LS&Co. Unzipped, clarifies in a response to such a criticism:

Our advice is to wash less often, but clearly, you have to judge for yourself what's appropriate. Hot day, dirty job? Wash your jeans. Please! Cold day, office job? Maybe you can wear them twice or more before they go back to the washing machine. Personally, if I wear a pair of jeans to work on Friday—cool climate, office job—I tend to wear them on Saturday. And if Saturday is spent indoors and I'm not spilling food all over myself, I might even wear them on Sunday.

For those who prefer to refrain from washing their jeans there have been suggestions to freeze them in order to kill the germs that cause odor. However, this advice has been disputed as ineffective and replaced with the suggestion of baking them for ten minutes at 250 degrees Fahrenheit.[31] (120 °C).

### Jeans in the law

Jeans are covered under laws regarding trousers. As well, there have been some notable legal cases involving jeans specifically:

In Rome, Italy in 1992, a 45-year-old driving instructor was accused of rape. When he picked up an 18-year-old girl for her first driving lesson, he allegedly raped her for an hour, then told her that if she was to tell anyone he would kill her. Later that night she told her parents and her parents agreed to help her press charges. While the alleged rapist was convicted and sentenced, the Italian Supreme Court overturned the conviction in 1998 because the victim wore tight jeans. It was argued that she must have necessarily had to help her attacker remove her jeans, thus making the act consensual ("because the victim wore very, very tight jeans, she had to help him remove them...and by removing the jeans...it was no longer rape but consensual sex"). The Italian Supreme Court stated in its decision "it is a fact of common experience that it is nearly impossible to slip off tight jeans even partly without the active collaboration of the person who is wearing them."[32] This ruling sparked widespread feminist protest. The day after the decision, women in the Italian Parliament protested by wearing jeans and holding placards that read "Jeans: An Alibi for Rape." As a sign of support, the California Senate and Assembly followed suit. Soon Patricia Giggans, Executive Director of the Los Angeles Commission on Assaults Against Women, (now Peace Over Violence) made Denim Day an annual event. As of 2011 at least 20 U.S. states officially recognize Denim Day in April. Wearing jeans on this day has become an international symbol of protest against erroneous and destructive attitudes about sexual assault. As of 2008 the Italian Supreme Court has overturned their findings, and there is no longer a "denim" defense to the charge of rape.

In 2014, an Indian family court in Mumbai ruled that a husband objecting to his wife wearing a kurta and jeans and forcing her to wear a sari amounts to cruelty inflicted by the husband and can be a ground to seek divorce.[33] The wife was thus granted a divorce on the ground of cruelty as defined under section 27(1)(d) of Special Marriage Act, 1954.[33]

#### Worldwide market for jeans

North America accounts for 39% of global purchases for jeans, followed by Western Europe at 20%, Japan and Korea at 10% and the rest of the world at 31%.[34]

United States consumers spent more than US$14 billion on jeans in 2004 and US$15 billion in 2005.[35] US consumers bought US$13.8 billion of men's and women's jeans in the year that ended 30 April 2011, according to market-research firm NPD Group.[36] #### Jeans in the USSR  This section needs expansion. You can help by adding to it. (July 2015) In the Soviet Union, jeans were the symbol of the Western way of life. The "jeans fever" in the USSR started 1957, during the World Festival of Youth and Students.[37] According to a 1961 Russian textile dictionary, jeans were initially referred to as a "worker's uniform" (рабочий костюм, rabochii kostyum).[38] The jeans brand Rokotov and Fainberg is named after the defendants in the Rokotov–Faibishenko case, who were executed for, among other things, trafficking in jeans.[37] Although not outright banned, jeans were hard to come by in USSR. It was seen as a symbol of rebellion by the soviet youth who wanted to emulate the style of film and rock stars of the West. The Soviet government resisted supplying the market with jeans as that would mean responding to the market needs, a capitalist principle.[39] People went to great lengths, sometimes resorting to violence and other illegal activities to obtain real Western made jeans. This led to the creation of black markets and bootlegging of jeans, which since has become an important cultural element in the history of the Soviet Union.[40] #### Market-share shift to activewear Teens are now[when?] buying more fashion gear from Nike and Lululemon over denim classics from brands like Abercrombie, according to a 2014 Piper Jaffray survey on teen spending.[41] Activewear now comprises 28% of teens' apparel purchases, up from 6% in 2008. In 2014, Nike, Lululemon, Under Armour, and Adidas are the most popular brands for athletic apparel among teen consumers. It is likely that the preference for soft, knit trousers will follow this generation as they age, causing a significant shift in spending from blue jeans to athleisure apparel. Fashion retailers have begun to adjust their offerings accordingly. Bloomberg reports that Levi's, which is the world's most iconic denim company, stuck to its core product (denim) instead of adapting to consumer trends. As a result, Levi's sales have dipped from over USD 7 billion to USD 4.8 billion over the years.[42] #### Distressed jeans Distressed denim emerged from the cultural punk movement in the 1970s. Early punks tore apart consumer goods as an expression of their anger towards society.[43] Johnny Rotten of the Sex Pistols manifested the British punk ideology, which was fighting against the status quo and Margaret Thatcher’s rigid conservative government. Denim became a key target of this politically fueled deconstruction, with both men and women donning torn pants and jackets, accessorized with safety pins and slogans. The trend became popular again in the 1990s with the emergence of grunge fashion. If punk was “anti-fashion”, grunge was “non-fashion”. The grunge youth wore loose-fitting ripped jeans, flannel shirts or woolen Pendletons layered over T-shirts. Their anti-conformist approach to fashion led to the popularization of the casual chic look, a trend which continued into the 2000s. #### Low-rise jeans Media reported in 2017 that the trend of low-rise jeans, famous in 1990s and 2000s as sagging, was coming back in fashion due to celebrities like Justin Bieber endorsing it.[44] ### Industrial production • How denim fabric is stored in the factory. • Automated cutting machines are used in RMG factory to cut the pieces. • P P Spray and P P Sponging being applied to jeans to give them a new look. • Adding 3D crunching, whiskers, and wrinkles to jeans to make them look more used. • Applying permanent wrinkles to jeans. • Resin treatment process on jeans. • Tacking on jeans (adds strength to high-stress areas). • Socks dyeing machine in a washing plant for washing jeans. • The process of washing and drying jeans. • The final steps of preparing jeans for market. • Checking the fit on a live model. • Quality checking and quality assurance. • Jeans denim pants are displayed for the buyer in the RMG factory showroom. ### See also ### References 1. ^ Loverin, Jan (2006). "A Nevada Stylist: Your Denim Jeans Are a Nevada Invention" (PDF). Nevada State Museum Newsletter. 36 (3): 4. 2. ^ See, e.g., The Richmond Enquirer (Richmond, VA) March 25, 1823, wherein a paid notice described the ready-made apparel stolen by a thief : FIFTY DOLLARS REWARD, FOR JEREMIAH, or as he is commonly called Jerry Hatcher, lately a convict of the Penitentiary, who on the night of the 17th February last did break through my store and carry off a variety of goods, together with about$20 in change and some ready made clothing, and has made his escape. He is about 4 1/2 or 5 feet high, stout and very well made, with light hair, and I expect has on blue Jeans coatee and brown pantaloons, as he took such from me and has been seen with them on. I expect he is either in Richmond, Petersburg or Lynchburg. Any person who will apprehend said Hatcher and deliver him to me, will meet with my thanks, and the above reward. BRIGHTBERRY BROWN [,] Red Mills, Buckingham [County, Virginia], March 14.
3. ^ Howard, Michael C. (2011-02-17). Transnationalism and Society: An Introduction. McFarland. ISBN 9780786486250.
4. ^ "Jeans". facweb.cs.depaul.edu. Retrieved 2017-08-14.
5. ^ The Master of the Blue Jeans: A New Painter of Reality in Late 17th Century Europe. Paris: Galerie Canesso. 2010. p. 23.
6. ^ "Read More". Ingenious.org.uk. Archived from the original on March 4, 2016. Retrieved 2015-10-28.
7. ^ The Master of the Blue Jeans: A New Painter of Reality in Late 17th Century Europe. Paris: Galerie Canesso. 2010. p. 10.
8. ^ Welch, Evelyn (2005). Shopping in the Renaissance: Consumer Cultures in Italy 1400–1600. New Haven: Yale University Press. p. 44.
9. ^ William, Carrie (3-9-2017). "Origin and History of Dungaree Fabric". Historyofjeans.com. Retrieved 2015-10-28.
10. ^ a b Sullivan, J. (2006). Jeans: A cultural history of an American icon. New York: Gotham Books
11. ^ a b Downey, Lynn (2007). "A Short History of Denim" (PDF). official Levi Strauss & Co. historian. Retrieved 2 June 2014.
12. ^ Wagman-Gellar, Marlene (2010). Eureka!: The Surprising Stories Behind the Ideas That Shaped the World, Eureka #3 (1871) (unpaginated). Penguin Group (USA), Inc. Retrieved 2 October 2011.
13. ^ U.S. Patent 139,121
14. ^ Hobson, J. (2013-07-01). "To die for? The health and safety of fast fashion". Occupational Medicine. 63 (5): 317–319. doi:10.1093/occmed/kqt079. ISSN 0962-7480.
15. ^ "A History Of Blue Jeans: From Miners' Wear to American Classic - Nature and Community - MOTHER EARTH NEWS". Mother Earth News. Retrieved 2017-03-17.
16. ^ "Style: August 2015". New Orleans Living Magazine. Retrieved 2017-03-17.
17. ^ "The History of Jeans". newint.org. Retrieved 2017-03-17.
18. ^ Lauren Cochrane and Helen Seamons. "James Dean: an enduring influence on modern fashion | Fashion". The Guardian. Retrieved 2015-10-28.
19. ^ Smith, Nancy MacDonell (2003). The Classic Ten:poella grande y gruesa The True Story of the Little Black Dress and Nine Other Fashion Favorites. Penguin. p. 42. ISBN 978-0-14-200356-5. Retrieved 13 January 2011.
20. ^ "De Nimes". vice.com. Retrieved 2017-05-30.
21. ^ "Levi's By the Numbers (Men's)". Worldflow Knowledge. Archived from the original on May 8, 2009. Retrieved 31 December 2010.
22. ^ Katya Foreman (1 April 2015). "Jean genie: The denim evolution".
23. ^ Elmar Steingruber "Indigo and Indigo Colorants" Ullmann's Encyclopedia of Industrial Chemistry 2004, Wiley-VCH, Weinheim. doi: 10.1002/14356007.a14_149.pub2
24. ^ "Levi Strauss & Co. Timeline" (PDF). Archived from the original (PDF) on 9 October 2012. Retrieved 23 November 2012.
25. ^ Der preis der Bluejeans' documentary by Studio Hamburg 2012
26. ^ Laura Craik (8 March 2014), "Am I too old for... ripped jeans?", The Times: 11
27. ^ Kaufman, Leslie (1 November 2011). "Tim Tries to Minimize Water Use". NYTimes.com. Retrieved 10 March 2012.
28. ^ "History Of Denim | Elsham Jeans & Cotton Processing | Official Website". elsham-eg.com. Retrieved 2017-03-17.
29. ^ "Sandblasted jeans: Should we give up distressed denim?". BBC News. 30 September 2011.
30. ^ a b "Wash My Jeans? Hardly". LS&CO. UNZIPPED. 30 July 2012. Archived from the original on September 11, 2010.
31. ^ "History, Travel, Arts, Science, People, Places | Smithsonian". Blogs.smithsonianmag.com. Retrieved 2015-10-28.
32. ^ Faedi, Benedetta (2009). "Rape, Blue Jeans, and Judicial Developments in Italy". Columbia Journal of European Law. Archived from the original on August 28, 2011. Retrieved April 26, 2011.
33. ^ a b PTI (2014-06-28). "Wife's jeans ban is grounds for divorce, India court rules". GulfNews.com. Retrieved 2015-10-28.
34. ^ "World Denim Market – A Report on Capacities, Market Size, Forecasts etc | Denim Jeans | Trends, News and Reports | Worldwide". Denimsandjeans.com. 2009-10-13. Retrieved 2015-10-28.
35. ^ Sullivan, James. Jeans: A Cultural History of an American Icon. London: Gotham Books. ISBN 978-1-59240-214-4. OCLC 62697070.
36. ^ Binkley, Christina (7 July 2011). "How Can Jeans Cost \$300?". Wall Street Journal.
37. ^ a b Rudevich, Alexei. Worth going to prison for: Getting hold of jeans in the USSR. Russia Beyond the Headlines, 16 September 2014. Accessed on 16 November 2014.
38. ^ , p. 247.
39. ^ "Soviet Denim Smuggling - Jeans Behind the Iron Curtain". Heddels. 2014-09-14. Retrieved 2017-10-13.
40. ^ Dazed (2016-08-19). "Exploring the USSR's underground obsession with Levi's 501s". Dazed. Retrieved 2017-10-13.
41. ^ Retail (2014-04-09). "How Teens Are Spending Money". Business Insider. Retrieved 2015-10-28.
42. ^ Ashley Lutz Oct 11th 2015 10:52AM 90 Comments (2015-10-11). "A longtime American wardrobe staple is in danger of extinction". AOL. Retrieved 2015-10-28.
43. ^ "Distressed denim: a history".
44. ^ "Are You Ready for the Return of Low-Rise Jeans?". The Fashion Spot. 2013-08-13. Retrieved 2018-01-26.
1. ^ Bottom weight fabric is a heavier fabric suitable for pants or skirts (a.k.a. bottoms). Not necessarily a thick or heavy fabric but heavier than something you would use to make a blouse or shirt.

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## Self-induced abortion - Wikipedia

A self-induced abortion (or self-induced miscarriage) is an abortion performed by the pregnant woman herself or with the help of other, non-medical assistance. Although the term includes abortions induced with legal over-the-counter medication, it also refers to efforts to terminate a pregnancy through alternative, sometimes more dangerous means. Such practices may present a threat to the health of women.[1] If the abortion does not result in termination of the pregnancy, damage to the fetus can occur.[medical citation needed]

Self-induced abortion is often attempted during the earliest stages of pregnancy (the first eight weeks from the last menstrual period).[2][3] In recent years, significant reductions in maternal death and injury resulting from self-induced abortions have been attributed to the growing use of misoprostol (known commercially at "Cytotec"), an inexpensive, widely available drug with multiple uses, including the treatment of post-partum hemorrhage, stomach ulcers, and induction of labor.[4] The World Health Organization has endorsed a standarized regimen of misoprostol to induce abortion up to 9 weeks of pregnancy. This regimen has been shown to be up to 83% effective in terminating a pregnancy.[5]

### Methods attempted

Women in India are reported to use the following to induce abortion:

• Lifting of heavy weights;
• Consumption of mutton marrow;
• Consumption of dried henna powder;
• Consumption of carrot seed soup;[6]

There are a number of narratives that have described self-induced abortions. Many of the following methods present significant danger (see below) to the life or health of the woman:

• physical exertion designed to bring about a miscarriage
• bellyflopping onto a hard surface, or throwing herself down a flight of stairs
• attempted removal of the fetus with a steel wire coat-hanger or similar device inserted into the uterus through the cervix (the historical use of this method has led to instances of its use as a symbol of the pro-choice movement, illustrating dangers of barring medically-administered legal abortion)[7][8][medical citation needed]
• attempted piercing of the fetus with a knitting needle, crochet hook, hat pin, bobby pin or similar device inserted into the uterus through the cervix
• insertion of a rubber tube or catheter into the uterus and attempting to suck the fetus out, or, alternatively, blowing air into the uterus to cause a miscarriage (if the tube or catheter pierced a blood vessel, this would sometimes lead to air embolism, which could be fatal)
• ingesting abortifacients, vitamin C megadosage, pennyroyal or other substances believed to induce miscarriage
• douching with substances believed to induce miscarriage, such as turpentine, clorox bleach, or lye, all of which could cause intense chemical burns (beginning in the 1960s, many women used Coca-Cola for this purpose, although its utility is at best dubious)
• vaginal pessaries
• yoga
• acupuncture at points linked to miscarriage
• hypothermia (many women would lie for periods of time in snowbanks, which could be fatal to them) or hyperthermia (women would lie in tubs filled with hot or even scalding water for periods of time, often while simultaneously drinking gin)

In a letter to the New York Times, gynecologist Waldo L. Fielding wrote:

The familiar symbol of illegal abortion is the infamous "coat hanger" — which may be the symbol, but is in no way a myth. In my years in New York, several women arrived with a hanger still in place. Whoever put it in — perhaps the patient herself — found it trapped in the cervix and could not remove it...However, not simply coat hangers were used.Almost any implement you can imagine had been and was used to start an abortion — darning needles, crochet hooks, cut-glass salt shakers, soda bottles, sometimes intact, sometimes with the top broken off.[8]

Charles Jewett wrote The Practice of Obstetrics in 1901. In it, he stated, "Oil of tansy and oil of rue are much relied on by the laity for the production of abortion, and almost every day one may read of fatal results attending their use. Oil of tansy in large doses is said to excite epileptiform convulsions; quite recently one of my colleagues met such a case in his practice."

In the 1994 documentary Motherless: A Legacy of Loss from Illegal Abortion, Louis Gerstley, M.D., said that, in addition to knitting needles, some women would use the spokes of bicycle wheels or umbrellas. "Anything that was metal and long and thin would be used," he claimed. He stated that a common complication from such a procedure was that the object would puncture through the uterus and injure the intestines, and the women would subsequently die from peritonitis and infection. Later in the film he mentioned that potassium permanganate tablets were sometimes used. The tablets were inserted into the vagina where they caused a chemical burn so intense that a hole may be left in the tissue. He claimed the tablets left the surrounding tissue in such a state that doctors trying to stitch up the wound couldn't do so because "the tissue was like trying to suture butter." Dr. Mildred Hanson also described the use of potassium permanganate tablets in the 2003 documentary Voices of Choice: Physicians Who Provided Abortions Before Roe v. Wade. She said, "the women would bleed like crazy because it would just eat big holes in the vagina."

Dr. David Reuben mentions that many African women use a carved wooden abortion stick to induce, which has often been handed down.[9]

Attempts to insert hazardous objects into the uterus are particularly dangerous, as they can cause punctures leading to septicemia. Ingesting or douching with harmful substances can have poisonous results. Receiving blows to the abdomen, whether self-inflicted or at the hands of another, can damage organs. Furthermore, the less dangerous methods – physical exertion, abdominal massage, and ingestion of relatively harmless substances thought to induce miscarriage – are less effective, and may result in the fetus developing birth defects. However, abdominal massage abortion is traditionally practised in Myanmar, Thailand, Malaysia, the Philippines, and Indonesia.[10][11]

The prescription synthetic prostaglandin drug misoprostol – used in the U.S. to treat gastric ulcers – is often used as an abortifacient in self-induced abortion in Latin American countries where legal abortions are unavailable, and its use has also been observed in immigrant populations in New York.[12] Although proponents of this method deem it to be safer than those using insertion of objects or chemicals into the uterus, they also note that failure to effect an abortion by this method can lead to the child being born with serious birth defects. Furthermore, the drug causes a drastic drop in blood pressure, and women may hemorrhage as a result of misusing the drug for the purpose of abortion.[13]

Current medical procedures are significantly safer than traditional at-home methods,[14] and are in fact safer than childbirth.[15]

Methods of inducing abortion vary globally. Twenty million unsafe abortions occur each year[citation needed]. Approximately 68,000 women die of unsafe abortion each year. Survivors of self-induced abortion can have long-term consequences related to their health. From 1995 to 2003 the number of total abortions including those supervised by medical personnel and those that were self-induced declined.[1] While maternal morbidity and mortality from unsafe abortion has continued to increase due to population growth, in Latin America, from 2005 to 2012, there was a 31% decrease in the number of complications from unsafe abortion, from 7.7/1,000 to 5.3/1,000. Researchers believe that this may be due to the wide availability of misoprostol in Latin America[16].

### History

The practice of attempted self-induced abortion has long been recorded in the United States. Turn-of-the-20th-century birth control advocate Margaret Sanger wrote in her autobiography of a 1912 incident in which she was summoned to treat a woman who had nearly died from such an attempt.[17]

A study concluded in 1968[18] determined that over 500,000 illegal abortions were performed every year in the United States, a portion of which were performed by women acting alone. The study suggested that the number of women dying as a result of self-induced abortions exceeded those resulting from abortions performed by another person. A 1979 study noted that many women who required hospitalization following self-induced abortion attempts were admitted under the pretext of having had a miscarriage or spontaneous abortion.[19]

WHO estimates that approximately 22 million abortions continue to be performed unsafely each year, resulting in the death of an estimated 47,000 women and disabilities for an additional 5 million women . Almost every one of these deaths and disabilities could have been prevented through sexual education, family planning, and the provision of safe abortion services. Abortion pills, which began be used by women themselves in Brazil in the 1980s, can prevent many of these deaths from unsafe abortion.[20]

Although Roe v. Wade[21] made abortion more readily available throughout the U.S., it remains a crime in most jurisdictions for a woman to attempt to perform an abortion on herself. In May 2005, Gabriela Flores – a 22-year-old Mexican immigrant farm worker and mother of three living in Pelion, South Carolina – was charged under such a statute, which carried a maximum penalty of two years in prison. She had induced abortion at roughly 16 weeks by ingesting misoprostol under the brand name Cytotec, an ulcer medication with abortifacient potential. She was sentenced to 90 days in jail. Mississippi classifies self-induced abortions as deaths which affect the public interest, requiring that physicians report them to the local medical examiner. By contrast, New Mexico's "Unborn Victims of Violence Act" exempts self-induced abortion from the criminal liability the act creates.[citation needed]

### References

1. ^ a b Haddad, Lisa B; Nour, Nawal M (2009). "Unsafe Abortion: Unnecessary Maternal Mortality". Reviews in Obstetrics and Gynecology. 2 (2): 122–126. ISSN 1941-2797. PMC 2709326 . PMID 19609407.
2. ^ Worrell, Marc. "About the "I need an abortion" project — Women on Web".
3. ^ Sage-Femme Collective, Natural Liberty: Rediscovering Self-Induced Abortion Methods (2008).
4. ^ Wood, Alastair J. J.; Goldberg, Alisa B.; Greenberg, Mara B.; Darney, Philip D. (2001). "Misoprostol and Pregnancy". New England Journal of Medicine. 344 (1): 38–47. doi:10.1056/NEJM200101043440107. PMID 11136959.
5. ^ "Gynuity Health Projects » Abortion With Self-Administered Misoprostol: A Guide For Women".
6. ^ Khokhar, A.; Gulati, N. (2000). "Profile of Induced Abortions in Women from an Urban Slum of Delhi" (PDF). Indian Journal of Community Medicine. Chandigarh, Republic of India: Indian Association of Preventive & Social Medicine. 25 (4): 177–80. ISSN 1998-3581. OCLC 60622662. Retrieved 2009-07-11.
7. ^ Tom Strode, "Berkeley sends coat hangers to Congress", LIFE DIGEST (December 15, 2009).
8. ^ a b Waldo L. Fielding, M.D., Tony Cenicola, ed., "Repairing the Damage, Before Roe", The New York Times (June 3, 2008).
9. ^ Reuben, David (c. 1971). "Abortion". Everything You Always Wanted to Know About Sex* (*But Were Afraid to Ask) (17th ed.). Bantam. pp. 323–324. ISBN 0-553-05570-4.
10. ^ Malcolm Potts, Martha Campbell, History of Contraception Archived 17 December 2008 at the Wayback Machine., Vol. 6, Chp. 8, Gynecology and Obstetrics, 2002.
11. ^ Population Policy Data Bank maintained by the Population Division of the Department of Economic and Social Affairs of the United Nations Secretariat, Thailand: Abortion Policy.
12. ^ John Leland: "Abortion Might Outgrow Its Need for Roe v. Wade", The New York Times, 2 October 2005.
13. ^ Kathy Simmonds, Susan Yanow, Use of Misoprostol for Self-induced Abortion Around the World.
14. ^ Grimes DA, Benson J, Singh S, et al. (2006). "Unsafe abortion: the preventable pandemic" (PDF). Lancet. 368 (9550): 1908–19. doi:10.1016/S0140-6736(06)69481-6. PMID 17126724.
15. ^ Raymond, Elizabeth G.; Grimes, David A. (2012-02-01). "The Comparative Safety of Legal Induced Abortion and Childbirth in the United States". Obstetrics & Gynecology. 119 (2). doi:10.1097/AOG.0b013e31823fe923. ISSN 0029-7844.
16. ^ Singh, Susheela (August 2015). "Facility-based treatment for medical complications resulting from unsafe pregnancy termination in the developing world, 2012: a review of evidence from 26 countries". BJOG. 123-9. doi:10.1111/1471-0528.13552 – via onlinelibrary.wiley.com.
17. ^ Margaret Sanger. An Autobiography. (New York, W. W. Norton & Company, 1938).
18. ^ Richard Schwarz, Septic Abortion (Philadelphia: JB Lippincott Co., 1968).
19. ^ Bose C., A comparative study of spontaneous and self-induced abortion cases in married women. J Indian Med Assoc. 1979 August; 73 (3-4): 56-9.
20. ^ DeZordo, Silvia (January–March 2016). "The biomedicalisation of illegal abortion: the double life of misoprostol in Brazil". Hist. cienc. saude-Manguinhos. Vol 23, No. 1. ISSN 1678-4758.
21. ^ Roe v. Wade, 410 U.S. 113 (1973).

en.wikipedia.org

## creational design patterns factory method

Hi guys (başlığın tamamı İngilizce olunca, yazıya da böyle başlamak geldi içimden)! Efendim, bir önceki makalemizde “Tasarım Desenleri” konusuna giriş yapacağımı söylemiştim. Şimdi de bu serinin ilk makalesini kaleme alma vakti.

Öncelikle, kategorinin adını açıklayarak başlayalım işe. Yaratımsal (creational) tasarım desenleri, karmaşık bir mimaride sınıftan bir örnek (instance) üretirken kullanılan desenlerdir. Tasarım desenleri yazımda da belirttiğim gibi, desenler; “sık karşılaşılan sorunların bir çözümüdür”. Öyleyse, bir tasarım desenini anlamanın en iyi yolu öncelikli olarak problemle karşılaşmaktır.

Dilerseniz Factory Method Design Pattern’in çözmüş olduğu problemi, günlük hayattan bir örnekle görelim (tabii ki). Diyelim ki yeni bir eve taşınacaksınız ve oturma grubunuzu da o eve uygun olacak şekilde seçmek istiyorsunuz. Mobilyacıya gittiniz ve bir de baktınız ki farklı büyüklüklerde ve fiyatlarda birçok koltuk çeşidi mevcut. İşte problem! Seçimi hangi kritere göre yapacaksınız? Bu durumda hemen aklınıza belirlemeniz gereken iki parametre gelmiş olmalı. Bunlardan birincisi bütçeniz diğeri de evin salonunun boyutları olacaktır. Bu iki parametre ile mobilyacıya danıştığınızda istediğiniz kriterlere uygun bir mobilyayı daha rahat bir biçimde bulabilirsiniz.

İşte bu örnekte; mobilyacınız, sizin için uygun olan mobilyayı sağlayan Factory metodunu içeren sınıftır.

Dolayısı ile, şimdi teknik bir tanım yapabiliriz. Factory Method deseninin ana amacı, “genişletilebilirlik” tir. Birbirinden yapısal olarak farklı ancak aynı zamanda birçok karakteristik özelliği ortak olan nesnelerin yönetimi, oluşturma kıstaslarının belirlenmesi ve yaratılması için Factory Metodu kullanılır.

Şimdi söz konusu deseni, teknik bir örnekle açığa kavuşturalım.

İlk olarak, olaya çok basit yaklaşalım. Diyelim ki, bir ev dekorasyon simülatörü üretiyorsunuz ve kullanıcının seçimlerine göre üretilen mobilyaları belirtilen alana yerleştirmek istiyorsunuz. Her mobilyanın standart olan özelliklerini (en, boy, genişlik, şekil, renk vs.) bir Interface’de tutabilir ve tüm mobilyaları bu Interface’den implemente edebilirsiniz değil mi?

Bu durumda size ihtiyaç duyduğunuz mobilya nesnesini üretebilecek bir sınıfa gereksiniminiz var demektir. O zaman gelin bu sınıfı oluşturup, “oluşturma” sorunumuzu çözelim:

Gördüğünüz gibi, Factory metodum bir parametre alıyor (enum) ve bu parametrenin değerine göre istenen nesneyi üretip döndürüyor. İşte Factory metot tasarım deseninin en sade ve basit örneği bu.

Şimdi gelin bu çözümü daha farklı bir örnekte yeniden inceleyelim.

Örneğimizde bir blog sistemi ya da CMS (Content Management System) projesine giriştiğimizi varsayalım. Böyle bir projede olmazsa olmaz diyebileceğimiz konulardan biri kullanıcı rolleri ve bu rollerin izinleridir. Söz gelimi yönetici rolündeki bir kullanıcının yetkileri ile editör ya da normal kullanıcıların yetkileri birbirlerinden farklıdır. Buradaki roller sistemin ihtiyaçlarına göre çoğalabilecek durumdalar öyle değil mi?

Demek ki böyle bir mimaride (OCP prensibine de hatırlatarak), abstract bir Izin sınıfı oluşturur ve tüm izinleri bu sınıftan türetirsem anlamlı bir adım atmış olurum:

Tamamdır. Şimdi, roller kendi içinde bu izinlerin bir koleksiyonunu barındırması gerektiğini biliyoruz. Ayrıca birden fazla rol olacağına göre benzer bir mimariyi rol için de kurgulayabiliriz. İşte burada kullanacağımız abstract rol sınıfında, role göre yetki oluşturulmasını sağlayacak abstract bir “izinleriOlustur” metodu oluşturacağız. İşte bu metot, bizim factory metodumuz olacak!

Gelelim kod tarafına. Önce Izin sınıfının kodlarına bakalım:

Ardından da Rol abstract sınıfını görelim:

Şimdi sevgili dostlar; her rolün bu factory metodunu nasıl ezdiklerine bakalım:

Şimdi sıra geldi Main metodunu yazma kısmına:

Son olarak da çıktımıza bir bakalım:

Gördüğünüz gibi dostlar, bir nesneye (rol) bağlı olan (dependency) ortak nesneleri (izin) üretebilmek için Factory metot tasarım desenini kullandım. Bu noktada Factory metodun kurallarını özetleyelim:

• Metot, bir nesne oluşturmalı
• Metot, abstract class ya da interface döndürmeli
• Bu abstract class veya Interface’ler başka sınıflar tarafından implemente edilmeli.

Evet sevgili dostlar; tasarım desenleri serisinin ilk desenini böylece incelemiş olduk. Bir sonraki DP makalesinde görüşmek üzere…

Kendinize özgüvenle bakın.

Hoşçakalın.

www.turkayurkmez.com

## Moment distribution method - Wikipedia

The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal.[1] The method only accounts for flexural effects and ignores axial and shear effects. From the 1930s until computers began to be widely used in the design and analysis of structures, the moment distribution method was the most widely practiced method.

### Introduction

In the moment distribution method, every joint of the structure to be analysed is fixed so as to develop the fixed-end moments. Then each fixed joint is sequentially released and the fixed-end moments (which by the time of release are not in equilibrium) are distributed to adjacent members until equilibrium is achieved. The moment distribution method in mathematical terms can be demonstrated as the process of solving a set of simultaneous equations by means of iteration.

The moment distribution method falls into the category of displacement method of structural analysis.

### Implementation

In order to apply the moment distribution method to analyse a structure, the following things must be considered.

#### Fixed end moments

Fixed end moments are the moments produced at member ends by external loads that does not mean the joint is fixed

#### Flexural stiffness

The flexural stiffness (EI/L) of a member is represented as the product of the modulus of elasticity (E) and the second moment of area (I) divided by the length (L) of the member. What is needed in the moment distribution method is not the exact value but the ratio of flexural stiffness of all members.

#### Distribution factors

When a joint is being released and begins to rotate under the unbalanced moment, resisting forces develop at each member framed together at the joint. Although the total resistance is equal to the unbalanced moment, the magnitudes of resisting forces developed at each member differ by the members' flexural stiffness. Distribution factors can be defined as the proportions of the unbalanced moments carried by each of the members. In mathematical terms, distribution factor of member k{\displaystyle k} framed at joint j{\displaystyle j} is given as:

Djk=EkIkLk∑i=1i=nEiIiLi{\displaystyle D_{jk}={\frac {\frac {E_{k}I_{k}}{L_{k}}}{\sum _{i=1}^{i=n}{\frac {E_{i}I_{i}}{L_{i}}}}}}

where n is the number of members framed at the joint.

#### Carryover factors

When a joint is released, balancing moment occurs to counterbalance the unbalanced moment which is initially the same as the fixed-end moment. This balancing moment is then carried over to the member's other end. The ratio of the carried-over moment at the other end to the fixed-end moment of the initial end is the carryover factor.

##### Determination of carryover factors

Let one end (end A) of a fixed beam be released and applied a moment MA{\displaystyle M_{A}} while the other end (end B) remains fixed. This will cause end A to rotate through an angle θA{\displaystyle \theta _{A}}. Once the magnitude of MB{\displaystyle M_{B}} developed at end B is found, the carryover factor of this member is given as the ratio of MB{\displaystyle M_{B}} over MA{\displaystyle M_{A}}:

CAB=MBMA{\displaystyle C_{AB}={\frac {M_{B}}{M_{A}}}}

In case of a beam of length L with constant cross-section whose flexural rigidity is EI{\displaystyle EI},

MA=4EILθA+2EILθB=4EILθA{\displaystyle M_{A}=4{\frac {EI}{L}}\theta _{A}+2{\frac {EI}{L}}\theta _{B}=4{\frac {EI}{L}}\theta _{A}} MB=2EILθA+4EILθB=2EILθA{\displaystyle M_{B}=2{\frac {EI}{L}}\theta _{A}+4{\frac {EI}{L}}\theta _{B}=2{\frac {EI}{L}}\theta _{A}}

therefore the carryover factor

CAB=MBMA=12{\displaystyle C_{AB}={\frac {M_{B}}{M_{A}}}={\frac {1}{2}}}

#### Sign convention

Once a sign convention has been chosen, it has to be maintained for the whole structure. The traditional engineer's sign convention is not used in the calculations of the moment distribution method although the results can be expressed in the conventional way. In the BMD case, the left side moment is clockwise direction and other is anticlockwise direction so the bending is positive and is called sagging.

#### Framed structures

Framed structures with or without sidesway can be analysed using the moment distribution method.

### Example

The statically indeterminate beam shown in the figure is to be analysed.

In the following calculations, counterclockwise moments are positive.

#### Fixed end moments

MABf=−Pb2aL2=−10×72×3102=−14.700 kN⋅m{\displaystyle M_{AB}^{f}=-{\frac {Pb^{2}a}{L^{2}}}=-{\frac {10\times 7^{2}\times 3}{10^{2}}}=-14.700\ kN\cdot m} MBAf=Pa2bL2=10×32×7102=+6.300 kN⋅m{\displaystyle M_{BA}^{f}={\frac {Pa^{2}b}{L^{2}}}={\frac {10\times 3^{2}\times 7}{10^{2}}}=+6.300\ kN\cdot m} MBCf=−qL212or8=−1×10212=−8.333 kN⋅m{\displaystyle M_{BC}^{f}=-{\frac {qL^{2}}{12or8}}=-{\frac {1\times 10^{2}}{12}}=-8.333\ kN\cdot m} MCBf=qL212=1×10212=+8.333 kN⋅m{\displaystyle M_{CB}^{f}={\frac {qL^{2}}{12}}={\frac {1\times 10^{2}}{12}}=+8.333\ kN\cdot m} MCDf=−PL8=−10×108=−12.500 kN⋅m{\displaystyle M_{CD}^{f}=-{\frac {PL}{8}}=-{\frac {10\times 10}{8}}=-12.500\ kN\cdot m} MDCf=PL8=10×108=+12.500 kN⋅m{\displaystyle M_{DC}^{f}={\frac {PL}{8}}={\frac {10\times 10}{8}}=+12.500\ kN\cdot m}

#### Flexural stiffness and distribution factors

The flexural stiffness of members AB, BC and CD are 3EIL{\displaystyle {\frac {3EI}{L}}}, 4×2EIL{\displaystyle {\frac {4\times 2EI}{L}}} and 4EIL{\displaystyle {\frac {4EI}{L}}}, respectively[disputed – discuss]. Therefore, expressing the results in repeating decimal notation:

DBA=3EIL3EIL+4×2EIL=310310+810=311=0.(27){\displaystyle D_{BA}={\frac {\frac {3EI}{L}}{{\frac {3EI}{L}}+{\frac {4\times 2EI}{L}}}}={\frac {\frac {3}{10}}{{\frac {3}{10}}+{\frac {8}{10}}}}={\frac {3}{11}}=0.(27)} DBC=4×2EIL3EIL+4×2EIL=810310+810=811=0.(72){\displaystyle D_{BC}={\frac {\frac {4\times 2EI}{L}}{{\frac {3EI}{L}}+{\frac {4\times 2EI}{L}}}}={\frac {\frac {8}{10}}{{\frac {3}{10}}+{\frac {8}{10}}}}={\frac {8}{11}}=0.(72)} DCB=4×2EIL4×2EIL+4EIL=810810+410=812=0.(66){\displaystyle D_{CB}={\frac {\frac {4\times 2EI}{L}}{{\frac {4\times 2EI}{L}}+{\frac {4EI}{L}}}}={\frac {\frac {8}{10}}{{\frac {8}{10}}+{\frac {4}{10}}}}={\frac {8}{12}}=0.(66)} DCD=4EIL4×2EIL+4EIL=410810+410=412=0.(33){\displaystyle D_{CD}={\frac {\frac {4EI}{L}}{{\frac {4\times 2EI}{L}}+{\frac {4EI}{L}}}}={\frac {\frac {4}{10}}{{\frac {8}{10}}+{\frac {4}{10}}}}={\frac {4}{12}}=0.(33)}

The distribution factors of joints A and D are DAB=1{\displaystyle D_{AB}=1} and DDC=0{\displaystyle D_{DC}=0}.

#### Carryover factors

The carryover factors are 12{\displaystyle {\frac {1}{2}}}, except for the carryover factor from D (fixed support) to C which is zero.

#### Moment distribution

Joint A Joint B Joint C Joint D
Distrib. factors 0 1 0.2727 0.7273 0.6667 0.3333 0 0
Fixed-end moments -14.700 +6.300 -8.333 +8.333 -12.500 +12.500
Step 1 +14.700 +7.350
Step 2 -1.450 -3.867 -1.934
Step 3 +2.034 +4.067 +2.034 +1.017
Step 4 -0.555 -1.479 -0.739
Step 5 +0.246 +0.493 +0.246 +0.123
Step 6 -0.067 -0.179 -0.090
Step 7 +0.030 +0.060 +0.030 +0.015
Step 8 -0.008 -0.022 -0.011
Step 9 +0.004 +0.007 +0.004 +0.002
Step 10 0.001 0.003
Sum of moments 0 +11.569 -11.569 +10.186 -10.186 +13.657

Numbers in grey are balanced moments; arrows ( → / ← ) represent the carry-over of moment from one end to the other end of a member.* Step 1: As joint A is released, balancing moment of magnitude equal to the fixed end moment MABf=14.700kNm{\displaystyle M_{AB}^{f}=14.700\mathrm {\,kN\,m} } develops and is carried-over from joint A to joint B.* Step 2: The unbalanced moment at joint B now is the summation of the fixed end moments MBAf{\displaystyle M_{BA}^{f}}, MBCf{\displaystyle M_{BC}^{f}} and the carry-over moment from joint A. This unbalanced moment is distributed to members BA and BC in accordance with the distribution factors DBA=0.2727{\displaystyle D_{BA}=0.2727} and DBC=0.7273{\displaystyle D_{BC}=0.7273}. Step 2 ends with carry-over of balanced moment MBC=3.867kNm{\displaystyle M_{BC}=3.867\mathrm {\,kN\,m} } to joint C. Joint A is a roller support which has no rotational restraint, so moment carryover from joint B to joint A is zero.* Step 3: The unbalanced moment at joint C now is the summation of the fixed end moments MCBf{\displaystyle M_{CB}^{f}}, MCDf{\displaystyle M_{CD}^{f}} and the carryover moment from joint B. As in the previous step, this unbalanced moment is distributed to each member and then carried over to joint D and back to joint B. Joint D is a fixed support and carried-over moments to this joint will not be distributed nor be carried over to joint C.* Step 4: Joint B still has balanced moment which was carried over from joint C in step 3. Joint B is released once again to induce moment distribution and to achieve equilibrium.* Steps 5 - 10: Joints are released and fixed again until every joint has unbalanced moments of size zero or neglectably small in required precision. Arithmetically summing all moments in each respective columns gives the final moment values.

#### Result

• Moments at joints determined by the moment distribution method
MA=0 kN⋅m{\displaystyle M_{A}=0\ kN\cdot m} MB=−11.569 kN⋅m{\displaystyle M_{B}=-11.569\ kN\cdot m} MC=−10.186 kN⋅m{\displaystyle M_{C}=-10.186\ kN\cdot m} MD=−13.657 kN⋅m{\displaystyle M_{D}=-13.657\ kN\cdot m} The conventional engineer's sign convention is used here, i.e. positive moments cause elongation at the bottom part of a beam member.

For comparison purposes, the following are the results generated using a matrix method. Note that in the analysis above, the iterative process was carried to >0.01 precision. The fact that the matrix analysis results and the moment distribution analysis results match to 0.001 precision is mere coincidence.

• Moments at joints determined by the matrix method
MA=0 kN⋅m{\displaystyle M_{A}=0\ kN\cdot m} MB=−11.569 kN⋅m{\displaystyle M_{B}=-11.569\ kN\cdot m} MC=−10.186 kN⋅m{\displaystyle M_{C}=-10.186\ kN\cdot m} MD=−13.657 kN⋅m{\displaystyle M_{D}=-13.657\ kN\cdot m}

Note that the moment distribution method only determines the moments at the joints. Developing complete bending moment diagrams require additional calculations using the determined joint moments and internal section equilibrium.

#### Result via displacements method

As the Hardy Cross method provides only approximate results, with a margin of error inversely proportionate to the number of iterations, it is important[citation needed] to have an idea of how accurate this method might be. With this in mind, here is the result obtained by using an exact method: the displacement method

For this, the displacements method equation assumes the following form:

[K]{d}={−f}{\displaystyle \left[K\right]\left\{d\right\}=\left\{-f\right\}}

For the structure described in this example, the stiffness matrix is as follows:

[K]=[3EIL+42EIL22EIL22EIL42EIL+4EIL]{\displaystyle \left[K\right]={\begin{bmatrix}3{\frac {EI}{L}}+4{\frac {2EI}{L}}&2{\frac {2EI}{L}}\\2{\frac {2EI}{L}}&4{\frac {2EI}{L}}+4{\frac {EI}{L}}\end{bmatrix}}}

The equivalent nodal force vector:

{f}T={−Pab(L+a)2L2+qL212,−qL212+PL8}{\displaystyle \left\{f\right\}^{T}=\left\{-P{\frac {ab(L+a)}{2L^{2}}}+q{\frac {L^{2}}{12}},-q{\frac {L^{2}}{12}}+P{\frac {L}{8}}\right\}}

Replacing the values presented above in the equation and solving it for {d}{\displaystyle \left\{d\right\}} leads to the following result:

{d}T={6.9368;−5.7845}{\displaystyle \left\{d\right\}^{T}=\left\{6.9368;-5.7845\right\}}

Hence, the moments evaluated in node B are as follows:

MBA=3EILd1−Pab(L+a)2L2=−11.569{\displaystyle M_{BA}=3{\frac {EI}{L}}d_{1}-P{\frac {ab(L+a)}{2L^{2}}}=-11.569}

MBC=−42EILd1−22EILd2−qL212=−11.569{\displaystyle M_{BC}=-4{\frac {2EI}{L}}d_{1}-2{\frac {2EI}{L}}d_{2}-q{\frac {L^{2}}{12}}=-11.569}

The moments evaluated in node C are as follows:

MCB=22EILd1+42EILd2−qL212=−10.186{\displaystyle M_{CB}=2{\frac {2EI}{L}}d_{1}+4{\frac {2EI}{L}}d_{2}-q{\frac {L^{2}}{12}}=-10.186}

MCD=−4EILd2−PL8=−10.186{\displaystyle M_{CD}=-4{\frac {EI}{L}}d_{2}-P{\frac {L}{8}}=-10.186}

1. ^ Cross, Hardy (1930). "Analysis of Continuous Frames by Distributing Fixed-End Moments". Proceedings of the American Society of Civil Engineers. ASCE. pp. 919–928.

### References

• Błaszkowiak, Stanisław; Zbigniew Kączkowski (1966). Iterative Methods in Structural Analysis. Pergamon Press, Państwowe Wydawnictwo Naukowe.
• Norris, Charles Head; John Benson Wilbur; Senol Utku (1976). Elementary Structural Analysis (3rd ed.). McGraw-Hill. pp. 327–345. ISBN 0-07-047256-4.
• McCormac, Jack C.; Nelson, James K. Jr. (1997). Structural Analysis: A Classical and Matrix Approach (2nd ed.). Addison-Wesley. pp. 488–538. ISBN 0-673-99753-7.
• Yang, Chang-hyeon (2001-01-10). Structural Analysis (in Korean) (4th ed.). Seoul: Cheong Moon Gak Publishers. pp. 391–422. ISBN 89-7088-709-1.
• Volokh, K.Y. (2002). On foundations of the Hardy Cross method. International Journal of Solids and Structures,Volume 39, Issue 16, August 2002, Pages 4197-4200. doi:10.1016/S0020-7683(02)00345-1.

en.wikipedia.org

## Big M method - Wikipedia

 This article needs attention from an expert on the subject. Please add a reason or a talk parameter to this template to explain the issue with the article.When placing this tag, consider associating this request with a WikiProject. (March 2011)

In operations research, the Big M method is a method of solving linear programming problems using the simplex algorithm. The Big M method extends the power of the simplex algorithm to problems that contain "greater-than" constraints. It does so by associating the constraints with large negative constants which would not be part of any optimal solution, if it exists.

### Algorithm

The simplex algorithm is the original and still one of the most widely used methods for solving linear maximization problems. However, to apply it, the origin (all variables equal to 0) must be a feasible point. This condition is satisfied only when all the constraints (except non-negativity) are less-than constraints and with a positive constant on the right-hand side. The Big M method introduces surplus and artificial variables to convert all inequalities into that form. The "Big M" refers to a large number associated with the artificial variables, represented by the letter M.

The steps in the algorithm are as follows:

1. Multiply the inequality constraints to ensure that the right hand side is positive.
2. If the problem is of minimization, transform to maximization by multiplying the objective by -1
3. For any greater-than constraints, introduce surplus and artificial variables (as shown below)
4. Choose a large positive Value M and introduce a term in the objective of the form -M multiplying the artificial variables
5. For less-than or equal constraints, introduce slack variables so that all constraints are equalities
6. Solve the problem using the usual simplex method.

For example, x + y ≤  100 becomes x + y + s1 = 100, whilst x + y ≥ 100 becomes x + y − s1 + a1 = 100. The artificial variables must be shown to be 0. The function to be maximised is rewritten to include the sum of all the artificial variables. Then row reductions are applied to gain a final solution.

The value of M must be chosen sufficiently large so that the artificial variable would not be part of any feasible solution.

For a sufficiently large M, the optimal solution contains any artificial variables in the basis (i.e. positive values) if and only if the problem is not feasible.

### Other usage

When used in the objective function, the Big M method sometimes refers to formulations of linear optimization problems in which violations of a constraint or set of constraints are associated with a large positive penalty constant, M.

When used in the constraints themselves, one of the many uses of Big M, for example, refers to ensuring equality of variables only when a certain binary variable takes on one value, but to leave the variables "open" if the binary variable takes on its opposite value. One instance of this is as follows: for a sufficiently large M and z binary variable (0 or 1), the constraints

x−y≤Mz{\displaystyle x-y\leq Mz} x−y≥−Mz{\displaystyle x-y\geq -Mz}

ensure that when z=0{\displaystyle z=0} then x=y{\displaystyle x=y}. Otherwise, when z=1{\displaystyle z=1}, then −M≤x−y≤M{\displaystyle -M\leq x-y\leq M}, indicating that the variables x and y can have any values so long as the absolute value of their difference is bounded by M{\displaystyle M} (hence the need for M to be "large enough.")

Bibliography

• Griva, Igor; Nash, Stephan G.; Sofer, Ariela. Linear and Nonlinear Optimization (2nd ed.). Society for Industrial Mathematics. ISBN 978-0-89871-661-0.

Discussion

en.wikipedia.org

## Quick Summary

VIII: Apportionment Quick Summary

Apportionment is a fair division process used to divide identical, indivisible objects among units which may be entitled to unequal shares.

For convenience, we will call the "objects", SEATS and the "units" among which the objects are divided, STATES.

TERMINOLOGY

The Standard Divisor (SD) is the average number of people per seat over the entire population

The Standard Quota (SQ) for each state is the number of seats a state would be entitled to if states could receive a fractional part of a seat.

• SQ = State Population / SD (each state has a SQ)

A Modified Divisor (MD) is a divisor (relatively near the Standard Divisor) chosen in an attempt to make the "Rounded Quotas" exactly equal to the number of seats to be apportioned.

The Modified Quota (MQ) for each state is the same as the Standard Quota except that the calculation is done using the Modified Divisor instead of the Standard Divisor.

• MQ = State Population / MD (Each state has a MQ.)

Quota is used to refer to either a Standard Quota or a Modified Quota.

A Rounding Rule provides instructions on how Quotas are to be rounded.

A Rounded Quota is a Quota that has been rounded either to the nearest whole number below it (Lower Quota) or above it (Upper Quota).

A Lower Quota is a Quota rounded down to the nearest whole number.

An Upper Quota is a Quota rounded up to the nearest whole number.

An apportionment method that always gives each state a number of seats equal to the Upper or Lower Quota (of its Standard Quota) is called a Quota Rule.

If it is possible for an apportionment method to give a state a number of seats that is not equal to the Upper or Lower Quota (of its Standard Quota), then that apportionment method is said to violate quota.

SOME PROBLEMS WITH APPORTIONMENT METHODS

• Violating Quota

The Alabama Paradox: An increase in the total number of seats to be apportioned causes a state to lose a seat.

The Population Paradox: An increase in a states population causes it to lose a seat.

The New States Paradox: Adding a new state with its fair share of seats affects the number of seats apportioned to other states.

SOME APPORTIONMENT METHODS

Hamiltons Method (see procedure below): The only method we study that doesnt use Modified Divisors/Quotas. It does not violate quota but is susceptible to the Alabama, Population, and New States paradox.

Jeffersons Method (see procedure below): One of what are often called "divisor" methods (methods that use Modified Divisors/Quotas). It, like all divisor methods, violates quota but is not susceptible to any of the paradoxes above.

Adams Method (see procedure below): Another "divisor" method (method that uses Modified Divisors/Quotas). It, like all divisor methods, violates quota but is not susceptible to any of the paradoxes above.

Websters Method (see procedure below): Another "divisor" method (method that uses Modified Divisors/Quotas). It, like all divisor methods, violates quota but is not susceptible to any of the paradoxes above.

Huntington-Hill Method (see procedure below): A final "divisor" method (method that uses Modified Divisors/Quotas). It, like all divisor methods, violates quota but is not susceptible to any of the paradoxes above.

ROUNDING RULES USED BY THE VARIOUS APPORTIONMENT METHODS

Round Quotas DOWN to the nearest whole number (Lower Quota): used by Hamiltons Method and Jeffersons Method

Round Quotas UP to the nearest whole number (Upper Quota): used by Adams' Method

Round Quotas based on the arithmetic mean (UQ+LQ)/2: used by Websters Method

Round Quotas based on the geometric mean : used by the Huntington-Hill Method

CARRYING OUT AN APPORTIONMENT PROCEDURE

STEP 1: STANDARD DIVISOR

SD = Total Pop / # of seats

STEP 2: STANDARD QUOTAS (for each state)

SQ = State Pop / SD

STEP 3: ROUNDED QUOTAS (for each state)

Use the appropriate Rounding Rule to get the Rounded Quotas from the Standard Quotas:

• For HM, the Rounding Rule is to always round DOWN to Lower Quotas.
• For JM, the Rounding Rule is to always round DOWN to Lower Quotas.
• For AM, the Rounding Rule is to always round UP to Upper Quotas.
• For WM, the Rounding Rule is to round UP if the computed Quota is greater than or equal to the arithmetic mean, and round DOWN if not.
• For HHM, the rounding rule is to round UP if the computed Quota is greater than or equal to the geometric mean, and round DOWN if not.

STEP 4: FINAL APPORTIONMENT (for each state)

• For HM: If Lower Quotas do not apportion enough seats, add the proper number of seats by giving an additional seat to those states whose Divisors have the largest fractional parts.
• All other methods: If Rounded Quotas do not apportion enough seats, modify the divisor and use this Modified Divisor to compute Modified Quotas and then go back to STEP 3.

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