List of FIFA World Cup finals

The FIFA World Cup is an international association football competition established in 1930. It is contested by the men's national teams of the members of Fédération Internationale de Football Association (FIFA), the sport's global governing body. The tournament has taken place every four years, except in 1942 and 1946, when the competition was cancelled due toWorld War II. The most recent World Cup, hosted by South Africa in 2010, was won by Spain, who beat the Netherlands 1–0 after extra time. The next World Cup is currently being held in Brazil until 13 July 2014.

The World Cup final matches are the last of the competition, and the results determine which country's team is declared world champions. If after 90 minutes of regular play the score is a draw, an additional 30-minute period of play, called extra time, is added. If such a game is still tied after extra time it is decided by kicks from the penalty shoot-out. The winning penalty shoot-out team are then declared champions.The tournament has been decided by a one-off match on every occasion except 1950, when the tournament winner was decided by a final round-robin group contested by four teams (Uruguay, Brazil, Sweden, and Spain). Uruguay's 2–1 victory over Brazil was the decisive match (and one of the last two matches of the tournament) which put them ahead on points and ensured that they finished top of the group as world champions. Therefore, this match is regarded by FIFA as the de facto final of the 1950 World Cup.

In the 19 tournaments held, 76 nations have appeared at least once. Of these, 12 have made it to the final match, and eight have won. With five titles, Brazil is the most successful World Cup team and also the only nation to have participated in every World Cup finals tournament.Italy have four titles and Germany have three. The other former champions areUruguay and Argentina with two titles each, and England, France, and Spain with one each. The current champions, Spain, took their first title in 2010. The team that wins the finals receive the FIFA World Cup Trophy, and their name is engraved in the bottom side of the trophy.

List of finals matches, their venues and locations, 

Year	Winners	Final score[3]	Runners-up	Venue	Location

1930	Uruguay	4–2	Argentina	Estadio Centenario	Montevideo, Uruguay

1934	Italy	2–1 [n 2]	Czechoslovakia	Stadio Nazionale PNF	Rome, Italy

1938	Italy	4–2	Hungary	Stade Olympique de Colombes	Paris, France

1950	Uruguay	2–1 [n 3]	Brazil	Estádio do Maracanã	Rio de Janeiro, Brazil

1954	West Germany	3–2	Hungary	Wankdorf Stadium	Bern, Switzerland

1958	Brazil	5–2	Sweden	Råsunda Stadium	Solna, Sweden

1962	Brazil	3–1	Czechoslovakia	Estadio Nacional	Santiago, Chile

1966	England	4–2 [n 4]	West Germany	Wembley Stadium	London, England

1970	Brazil	4–1	Italy	Estadio Azteca	Mexico City, Mexico		[
1974	West Germany	2–1	Netherlands	Olympiastadion	Munich, West Germany		]
1978	Argentina	3–1 [n 5]	Netherlands	Estadio Monumental	Buenos Aires,Argentina		[28][29]
1982	Italy	3–1	West Germany	Santiago Bernabéu	Madrid, Spain
1986	Argentina	3–2	West Germany	Estadio Azteca	Mexico City, Mexico
1990	West Germany	1–0	Argentina	Stadio Olimpico	Rome, Italy
1994	Brazil	0–0 [n 6]	Italy	Rose Bowl	Pasadena, California,United States
1998	France	3–0	Brazil	Stade de France	Saint-Denis, France
2002	Brazil	2–0	Germany	International Stadium Yokohama	Yokohama, Japan
2006	Italy	1–1 [n 7]	France	Olympiastadion	Berlin, Germany
2010	Spain	1–0 [n 8]	Netherlands	Soccer City	Johannesburg, South Africa
2014				Estádio do Maracanã	Rio de Janeiro, Brazil

RAMANUJAN @ 125

The story of mathematician genius Srinivasa Ramanujan is a story of human triumph and an example of what genius can accomplish against the odds, Robert Kanigel, Ramanjuan's biographer said on Monday.

Delivering a lecture hosted by the Organising Committee of Ramanujan 125, TNQ Books and Journals and the Institute of Mathematical Sciences, Mr. Kanigel said that while illuminating the genius of Ramanujan, “we should also momentarily hold the spotlight briefly” on the scores of people who did not make it due to adverse circumstances.

After giving a bare-bone account of the Ramanujan story, Mr. Kanigel who for his research visited India, which remains a jumble of mostly happy experiences and memories, and the beautiful but slightly forbidding Cambridge, took several questions from the audience.

Responding to a question on the everyday applicability of Ramanujan's theorems, Mr. Kanigel pointed out that Ramanujan's work was in pure mathematics and he took delight in exploring numbers without attaching any larger purpose.

Saying that he felt good when his book was translated into German, Italian or Greek, Mr. Kanigel also admitted to a certain sadness that it was not available yet in Tamil. His delight would know no end if “The Man Who Knew Infinity” were to be available in Ramanujan's own language, he said.

Offering his reasons for the title of the book, Mr. Kanigel said it defined though in a narrow sense the intimacy with which Ramanujan worked with numbers and theorems; it was as if “he knew infinity as his homeland.”

Mr. Kanigel was amused by a suggestion to get Italian filmmaker Bernardo Bertolucci to make a film based on his biography and to a questioner, who wanted to know whether Ramanujan would have felt more comfortable and enjoyed greater creative freedom in the U.S. than in Cambridge, he said he would like to think that Ramanujan in the U.S. might have felt more at home and a little “less stiff.”

When someone wanted to know whether working on the biography on Ramanujan had in any way changed him as a person, he remarked in a lighter vein that the impact would have been more on his wife when he got immersed in work.

N. Ram, Editor-in-Chief of The Hindu, said Mr. Kanigel's biography on Ramanujan, which was first published in 1991 and has undergone several printings, was perhaps his best and most influential works.

Among the many features of this book was its perfectly legitimate practice of the art of narrative journalism, or the use of a novelistic imagination in bringing a story alive without ever crossing the line between fact and fiction, he said.

The biography counters the flatness of the picture of Ramanujan's origins and life in 19 century south India, locates him in everyday life, brings alive the emotional geography of family relationships and the imprint on a young man of the cultural ethos, religious values and rituals of the times.

Mariam Ram (TNQ Books and Journals) and Balasubramanian of the Institute of Mathematical Sciences also participated.

collector's edition of Ramanujan notebook

The familiar face of Srinivasa Ramanujan looks up in relief from two classy hardback volumes in black. Inside is proof of the mathematician's genius. For both form and content, there can be no doubt that the second edition of Notebooks of Srinivasa Ramanujan is a collector's edition.

The notebooks of the ‘man who knew infinity,' originally believed to have been written down on loose sheets of paper, have kept mathematicians busy for over a century, scrambling to provide derivations for the results. An effort by the Tata Institute of Fundamental Research, Mumbai, the two volumes contain over 700 pages in Ramanujan's neat hand, some of his finest work. The books also have a foreword from Bruce Berndt, American mathematician known for his work in explaining the results Ramanujan postulated in his notebooks.

It is believed that Ramanujan actually worked out the problems on a slate in an attempt to save paper, using the sheets only to note down results. Since the discovery of the notebooks, mathematicians have wrestled with the results trying to arrive at plausible derivations. Most of the work has now been solved, thanks to Professor Brendt, according to M.S. Raghunathan, Vice-Chairman of the National Committee supervising the Ramanujan 125 year celebrations.

The books were launched at a function in which Prime Minister Manmohan Singh participated here on Monday. The release was possible with help from the archiving and digitising team at the Roja Muthiah Research Library (RMRL) in Chennai.

The production quality is of excellent standard, a huge improvement over the first edition published in 1957, again by the TIFR.

A brief overview of combinatorics in ancient and medieval India

The beginnings of combinatorics in India date back to Bharata's Natyasastra and the last chapter of the work Chandahsastra (Sanskrit Prosody) by Pingala (c. 300 BCE). Pingala deals in a few cryptic sutras with the combinatorics underlying the metres of Vedic Hymns and classical Sanskrit poetry. Metre is the basic rhythmic structure of a verse, here characterised by a finite sequence of syllables, some short and some long. We recall that syllables are irreducible units of speech, some being short like ka and some being long like kaa. We define the length of a metre as the number of its constituent syllables. We shall abbreviate a short syllable by l and a long syllable by g. For example, the length of a metre gll is 3.

Pingala gives a method of enumeration, which is called prastara, of metres of a given length n (which indeed are 2n in number), using a recursive procedure.

For example, the prastara of metres of length 1 is g, l (which are displayed as an array with two rows). According to Pingala, the prastara of metres of length 2 is obtained from the above by adding a g to the right of the prastara above, and then an l to this prastara, so that we get gg; lg; gl; ll (which are displayed as an array with four rows). The prastara of metres of length 3 is the array comprising the eight rows ggg; lgg; glg; llg; ggl; lgl; gll; lll which is obtained from the prastara of metres of length 2 above by adding first a g and then an l to the right. The above recursive procedure is then continued.

In this remarkable manner, Pingala's method leads one to a construction of the binary expansion of integers, by setting g = 0 and l = 1 in the successive rows of the prastara, the fifth row of the prastara of length n being a mnemonic for the binary expansion of i — 1.

For example, the 5th row ggl in the prastara of the metres of length 3 stands for the binary expansion 0:1 + 0:2 + 1:22 of 4. The last row, that is, the (2n — 1)-th row, of the prastara of metres of length n is ll; … l which yields the formula 1 + 2 + + 2n - 1 = 2n — 1:

With the advent of Prakrit and Apabhramsa poetry, came the idea of extending the above theory to matra metres where the value of a long syllable g is assumed to be 2 and that of a short syllable l is assumed to be 1, the value of the metre being defined to be the sum of the values of its constituent syllables. The construction of the prastara of metres of value n is achieved as above by a recursive procedure which is more subtle. The prastara of metres of value 1 is l; that with value 2 is g; ll; the prastara of metres of value 3 is obtained from these two by adding a g to the right of the prastara of metres of value 1 to get lg, and an l to the right of the prastara of metres of value 2 to get gl; lll; thus the prastara of metres of value 3 is lg; gl; lll. Similarly, we can write down the prastaras of metres of value n, using the prastaras of metres of value n _ 1 and n _ 2. If sn is the number of elements of in the prastara of metres of length n, we have s1 = 1; s2 = 2, and for n _ 3, sn = sn_1 +sn_2. This relation was noticed by Virahanka (c.600 CE).

The study of matra metres thus led the ancient Indian mathematicians to the sequence sn = 1, 2, 3, 5, 8, …, (what is generally known as the Fibonacci sequence, though discovered centuries before Fibonacci). As in the case of binary expansions, we obtain now unique expansions for natural numbers in terms of the Fibonacci numbers.

Combinatorics evolved in time not merely to apply to Sanskrit prosody but to many other problems of enumeration: in medicine by Sushruta, perfumery by Varahamihira, music by Sarngadeva (which shall be briefly discussed below), and so on.

Sarngadeva (c.1225 CE), who lived in Devagiri in Maharashtra, under the patronage of King Singhana, wrote his magnum opus Sangitaratnakara, a comprehensive treatise on music which gives in its first chapter a prastara enumerating all the 7! = 7_6_5_4_3_2_1 = 5040 permutations of the swaras S;R; G;M; P;D;N. The prastara of a single swara is S, that of two swaras S;R, is SR;RS; the prastara of three swaras, S;R;G is SRG;RSG; SGR;GSR;RGS;GRS:

More generally, the prastara of all the seven swaras, starts with the swaras in the natural order SRGMPDN and ends in the last or the 5040th row with the swaras in the reverse order, NDPMGRS, and the intermediate rows are constructed by a rule formulated by Sarngadeva.

It is indeed a remarkable fact that if we start more generally with n elements a1; a2; _ _ _ an and arrange their permutations in a prastara following Sarangadeva's rule, the ith row of the prastara is a mnemonic for a unique expansion of i; 1 _ i _ n!, as a sum of factorials, i = 1 _ 0! + c1 _ 1! + c2 _ 2! + _ _ _ + cn_1 _ (n _ 1)!; (with the convention 0! = 1), where the coefficient cj of j! lies between 0 and j _ 1. In particular, we have the beautiful formula n! = 1 _ 0! + 1 _ 1! + 2 _ 2! + _ _ _ + (n _ 1) _ (n _ 1)!; implicit in the work of Sarngadeva.

Indian combinatorics continued to flourish till the 14th century, when the celebrated mathematician Narayana Pandita wrote his comprehensive Ganitakaumudi (“Moonlight of Mathematics") in 1356 CE, placing the earlier work on combinatorics in a general mathematical context. He has in this great work a chapter on magic squares entitled Bhadraganita, where among other things, he constructs a class of 384 pan-diagonal 4 _ 4 magic squares with entries 1; 2; 3; _ _ _ ; 16.

We recall that in a magic square, the numbers in the rows, columns and diagonals sum to the same magic total. In a pan-diagonal magic square, the broken diagonals also yield the same magic total.

Succinctly put, pan-diagonal magic squares have the remarkable property that they can be considered as a magic squares “on the torus". It is of interest to note that Rosser and Walker proved in 1936 (the proof was simplified by Vijayaraghavan in 1941) that there are only 384 pan-diagonal 4 _ 4 magic squares with entries 1; 2; _ _ _ ; 16.

Curiously enough, Ramanujan, in his Notebooks of probably his earliest school days, has the magic square

1 14 11 8

12 7 2 13

6 9 16 3

15 4 5 10

This turns out to be one of the 384 magic squares considered by Narayana Pandita. (Notice that rows, columns, diagonals and broken diagonals add to 34. For example, 5+3+12+14 = 12 + 9 + 5 + 8 = 34).

Xenophanes, the founder of the Eleatic School of Philosophy, had the well known dictum - Ex nihilo nihil fit, “Out of nothing, nothing comes." One wonders whether after all Ramanujan was indeed influenced somewhat by the mathematical tradition of his ancestors.

Ramanujan mathematics centre to be set up in Chennai

The Ramanujan Mathematical Society (RMS) will hold a series of activities in 2012 (National Mathematical Year) to mark the 125th birth anniversary of mathematician Srinivasa Ramanujan.

RMS president and chair of the organising committee M.S. Raghunathan said on Monday a mathematics centre named after Ramanujan would be set up in Chennai. It would have a host of facilities, including a museum.

Efforts were on to bring out the biography of Ramanujan by Robert Kanigel in regional languages. A documentary, tracing the history of mathematics in India, would also be made.

The activities were planned to involve different sections, including school-goers, college students, students pursuing their research in mathematics and others interested in developments in the discipline.

In the programmes involving school students, for instance, talks and activities would focus on helping them get rid of their fear of the subject, Professor Raghunathan said.

The year-long celebrations of Ramanujan birth anniversary would culminate in an international conference of mathematicians in New Delhi in December next, he said.

LITHUANIAN

Lithuanian
Lithuanian is the most ancient living Indo-European language. It has many similarities with Slavic languages, but, unlike Russian, it uses the Latin alphabet. In 1864, the ruling Russian government banned the Lithuanian language, but it has since been restored as independent Lithuania's official language. Lithuanian is spoken by approximately three million people there and by an additional half million around the world. How did Lithuanians obtain books written in Lithuanian during Russian rule?
 

 

CONVERTING SOUND ENERGY INTO ELECTRICITY USING PIEZOELECTRIC MATERIAL

“There is definitely energy contained in that sound,” says David Cohen-Tanugi, vice 

president of the MIT Energy Club and a John S. Hennessy Fellow in MIT’s Materials Science and 

Engineering department. “But the density of the energy is very low, and there is no way to capture it 

all. You’d have to have obscenely loud, continuous noise for harvesting to be worthwhile.” Sound 

energy is the energy produced by sound vibrations as they travel through a specific medium. 

Speakers use electricity to generate sound waves and now by using zinc oxide, the main ingredient of 

calamine lotion, to do the reverse - convert sound waves into electricity. Piezoelectrics are materials 

capable of turning mechanical energy into electricity, and can be substances as simple as cane sugar, 

bones, or quartz. Much research in this field has been focused on transforming the movement of a 

person running, or even the impact of a bullet, into a small electrical current, but although these 

advanced applications are not yet available in consumer products, scientists have been using 

piezoelectric materials in environmental sensors and speakers for years. Piezoelectrics create an 

electrical charge under stress, and thus zinc oxide, the main ingredient of calamine lotion, was bent 

into a field of nanowires sandwiched between two electrodes. The researchers subjected the 

sandwich to sound waves of 100 decibels which produced an electrical current of about 50 millivolts. 

Passing trains and subways aren’t only loud, but their surroundings rattle and vibrate as they pass, 

and part of the thrill of a rock concert is feeling the whole auditorium shake. Piezo material 

converts mechanical strain into electric energy this property of piezo material could be used to 

make a device which would be able to sustainably convert the sound energy to electric energy as 

piezo materials convert sound energy to electric energy. Transducer is also used to convert 

Mechanical energy to electric energy i.e.it can convert sound energy to electric energy the simple 

e.g. of use of transducer to convert sound to electric and vice versa is in speakers, headset also it 

could be converted into electric energy. 

CONCEPT 

 

Suppose we create a very thin curtain like diphagram which will get fluctuated by the 

oscillation and pressure created by the sound wave and a conductor will be attached to it which will 

be placed between magnetic bars these fluctuation in the curtain will create a movement in conductor 

which will affect the magnetic field of the magnet this will generate motional emf and will generate 

voltage across it. As per faradays law generated emf is given by Generated voltage = Emf =velocity 

of conductor X magnetic field X length of conductor thus the oscillation created by the sound wave 

could be converted into electricity and as the frequency is high the movement will be fast due to it 

we will get appreciable amount of electric energy. 

Piezo electric materials are transducers its crystals could convert mechanical strain to 

electricity, the crystals are formed naturally e.g. quartz, bone, DNA whereas artificially ZnO, lithium 

niobatet Lead Metaniobate the sound energy could be converted into electricity using piezo electric 

material. Let us see the properties of piezo electric material. Certain single crystal materials exhibit 

the following phenomenon: when the crystal is mechanically strained, (here sound energy) or when 

the crystal is deformed by the application of an external stress, electric charges appear on the crystal 

surfaces; and when the direction of the strain reverses, the polarity of the electric charge is reversed. 

This is called the direct piezo electric effect, and the crystals that exhibit it are classed as 

piezoelectric crystal. 

First let’s understand concept to produce current. When coil of aluminum comes in between two 

magnets opposite polarity say P N pole, and some force is applied on coil to rotate on its axis it’ll 

produce magnetic field and due to electromagnet flux charge/current flows. Its shows that to produce current, force or pressure are required or can say force is a main key 

to produce current. This paper suggesting to utilizing sound vibration as an applied force to produce 

current. Piezo electric material has ability to convert mechanical stress into electricity. 

 

History of Piezoelectricity 

The first scientific publication describing the phenomenon, later termed as piezoelectricity, 

appeared in 1880. It was co-authored by Pierre and Jacques Curie, who were conducting a variety of 

experiments on a range of crystals at the time. In those experiments, they cataloged a number of 

crystals, such as tourmaline, quartz, topaz, cane sugar and Rochelle salt that displayed surface 

charges when they were mechanically stressed. 

Without any external stress Centers of charges coincide, charges are reciprocally cancelled 

and formed electrical neutral unit cell.  

Applied external stress Internal structure is deformed, separation of charge centers and dipoles are generated Poles inside material are mutually Cancelled and charge occurs on surface creates 

polarization on the surface of material. 

Mathematical modeling 

Piezoelectricity is the combination of the materials electrical behavior: 

D =ε E, And Hook’s laws = s T where D: electric displacement, ε: permittivity, 

E: electric field strength, S: strain, s: compliance, T: stress 

 

The coupled strain-voltage equation 

S = sET + dtE converse piezoelectric effect 

D= ε 

T

E + dT direct piezoelectric effect 

d ij , k = ∂Sij /∂Ek piezoelectric coefficient 

 

When a poled piezoelectric material mechanically strained it became electrically polarized, 

producing an electric charge on the surface of the material. 

 

Example to explain concept 

 A human walking, for example is a low frequency event that can be captured in the form of 

stress on a piezoelectric platform. A person walking across a room may complete 1-2 steps per 

second. Each step introduces a stress in the floor of the room, and the frequency of that alternating 

stress would be about 1-2 vibrations per second, and this waste vibration energy can be harvested. 

Vibrations per second are a measure of frequency, often stated in Hertz (Hz). One vibration per 

second is equal to 1 Hz. Two vibrations per second are equal to 2 Hz. The common United States 

household’s electrical circuit carries electricity oscillating at 60 cycles per second, or 60 Hz, which is 

evidenced by the low frequency buzz of an electric shaver. 

 To determine how much energy piezoelectric can produce, a few metrics need to be defined 

that will be useful for the discussion. The first metric is power. Power is defined in Watts (W), which 

is defined as units of energy per second. Power is an indication of how quickly energy can be 

delivered. A powerful air conditioner can cool a room quickly, whereas a weakly powered heater may 

require a long time to heat a room. Other examples include a solar panel which may be rated at 200 W 

in peak sunlight at noon in the middle of a summer. The second metric is energy. Energy is defined in 

many units. In standard units, energy is stated in Joules (J), but for electricity it is often most useful to 

define energy in terms of watt-hours (W-h), for example, how many watts are produced in an hour. In 

the examples above, the solar panel would produce 200 W-h from noon to 1 PM. The natural gas 

power plant would produce 200 million watt-hours (200 megawatt-hours, or MWh) in the same hour. 

Again, the two examples are different by a factor of one million. 

 One study used lead zirconate titanate (PZT) wafers and flexible, multilayer polyvinylidene 

fluoride (PVDF) films inside shoes to convert mechanical walking energy into usable electrical 

energy [1], [2]. This system has been proposed for mobile computing and was ultimately able to provide 

continuously 1.3 mW at 3 V when walking at a rate of 0.8 Hz. 

 

METHOD 1 

Suppose we create a very thin curtain like diphagram which will get fluctuated by the 

oscillation and pressure created by the sound wave and a conductor will be attached to it which will 

be placed between magnetic bars these fluctuation in the curtain will create a movement in conductor 

which will affect the magnetic field of the magnet this will generate motional emf and will generate 

voltage across it. As per faradays law generated emf is given by Generated voltage = Emf =velocity 

of conductor X magnetic field X length of conductor Thus the oscillation created by the sound wave 

could be converted into electricity and as the frequency is high the movement will be fast due to it 

we will get appreciable amount of electric energy. It would work similar as the working of turbine 

this type of device could be made but its limitation will be that it will be efficient only in the place where 

high decibel of sound is available, for example nuclear power plant, industries using huge and noisy 

machines.

 

METHOD 2 

In this method we could convert sound energy to heat energy as sound wave travel by 

oscillating the particles of the medium so when sound energy travel through the medium it will 

disturbs the particle of the medium these disturbance created by sound will be used to convert it into 

heat energy as when the particles of the medium will be pushed by the sound wave it will collides 

with adjacent particle of the medium this collision will result in production of heat energy the 

production of heat energy will be more in the denser medium so for more heat production we will 

need a material with very high density. This heat energy will be converted into electricity. 

 

METHOD 3 

Converting sound energy to electricity by piezo electric material (piezo electric materials are 

the crystal which convert mechanical strain to electric energy) device could be made using piezo 

electric material which will collect the sound wave which are travelling near it and that sound wave 

will be used to cause a strain due to pressure created by its oscillation in the piezo crystal and that 

will create the disturbance in its atoms resulting in the flow of electric charge on the surface of the 

crystal thus sound energy could be converted into electricity as the piezo electric material convert 

mechanical strain to electric energy. And thus this sound energy could be used to perform various 

tasks by converting it into useful electric energy

conclusion

 

• As sound has enormous amount energy with it, it could be used by converting it into electric 

energy for various purposes. Sound energy is a mechanical energy so according to law of 

thermodynamics mechanical energy could be converted into electric energy. 

• Sound energy could be converted by different methods: 

 

Method 1- By creating apparatus using curtain (diphagram) magnet and conductor. 

 

Methods 2- By converting Sound energy to heat energy and then heat energy to electric energy. 

 

Method 3- By using transducers such as piezo electric material which converts mechanical strain 

to electric energy and vice4 versa. 

FINITE ELEMENT MODELING OF REINFORCED CONCRETE BEAM COLUMN JOINTS RETROFITTED WITH GFRP WRAPPING

Recent earthquakes have demonstrated that most of the reinforced concrete 

structures were severely damaged during earthquakes and they need major repair works. 

Beam column joints, being the lateral and vertical load resisting members in reinforced 

concrete structures are particularly vulnerable to failures during earthquakes. The existing 

reinforced concrete beam-column joints which are not designed as per code IS 

13920:1993 must be strengthened since they do not meet the ductility requirements. The 

finite element method (FEM) has become a staple for predicting and simulating the 

physical behavior of complex engineering systems. The commercial finite element 

analysis (FEA) programs have gained common acceptance among engineers in industry 

and researchers. The details of the finite element analysis of beam column joints 

retrofitted with glass fiber reinforced polymer sheets (GFRP) carried out using the 

package ANSYS are presented in this paper. Three exterior reinforced concrete beam 

column joint specimens were modeled using ANSYS package. The first specimen is the 

control specimen. This had reinforcement as per code IS 456:2000. The second specimen 

which is also the control specimen. This had reinforcement as per code IS 13920:1993. 

The third specimen had reinforcement as per code IS 456:2000 and was retrofitted with 

glass fiber reinforced polymer (GFRP) sheets. During the analysis both the ends of 

column were hinged. Static load was applied at the free end of the cantilever beam up to a 

controlled load. The performance of the retrofitted beam-column joint was compared 

with the control specimens and the results are presented in this paper. 

Key words: Beam column joint, Retrofitting, FRP sheets. The techniques of using fiber sheets for strengthen the beam column joints have a number of favorable characteristics such as ease to install, immunity to corrosion and high strength. The simplest way to strengthen the joints is to wrap fiber sheets in the joint 

region in two orthogonal directions. Many fiber reinforced polymer sheets are available the market for strengthening reinforced concrete members .Glass fiber reinforced 

polymer sheets are commonly used for retrofitting the structural elements.

The beam column joint considered for analysis consists of a cantilever portion and 

column portion as shown in Figure 1.a and Figure 1.b. The column had a cross section of 

200 mm x 200 mm with an overall length of 1500 mm and the beam had a cross section 

of 200 mm x 200 mm and the length of the cantilevered portion was 600 mm. The control 

specimens were designated as C1 and C2. C1 had reinforcement as per code IS 456-2000 

and C2 had reinforcement as per code IS 13920-1993. The specimen retrofitted with glass 

reinforced polymer sheet was designated as C3 which had reinforcement as per code IS 

456-2000. The column portion was reinforced with 4 numbers of 12mm diameter Fe 415 

rods and the beam portion was reinforced with 2 numbers of 16 mm diameter Fe 415 rods 

each in the tension and compression zones. The lateral ties in the columns of the 

specimens C1 and C3 were 6 mm diameter Fe 250 bars with the spacing of 180 mm c/c 

as per code IS 456:2000, clause 26.5.3.2(c). Beam had vertical stirrups of 6 mm diameter 

Fe 250 bar at 120 mm c/c as per code IS 456:2000, clause 26.5.1.6. The development 

length of the tension and compression rods in beam were also provided as per clause 

26.2.1 of code IS 456:2000. For the specimen C2, the lateral ties in the columns consisted 

of 8 mm diameter Fe 415 bar at 75 mm c/c for the central distance of 1100 mm as per 

code IS 13920:1993, clause 7.4.6 and 6mm diameter Fe250 bars at 100 mm c/c for the 

remaining length of the column. Beams had vertical stirrups of 6 mm diameter Fe 250 bar 

at 40 mm c/c. up to a distance of 340 mm from the face of the column as per code IS 

13920:1993, clause 6.3.5 and 6 mm diameter Fe250 bar at 80 mm c/c for remaining 

length of the beam. The development length of the beam rods were also provided as per 

code IS 13920:1993, clause 6.2.5. M25 grade concrete was adopted. 13920:1993 

Meshing was done for both control and retrofitted specimens using ANSYS. Both ends of 

the column were hinged. The concrete was modeled using Solid 65 element. The 

reinforcement was modeled using Link 8 element. The wrapping was modeled using 

Solid 45 element. The static load was applied at the free end of the cantilever beam at a 

regular load interval of 5 kN for the control and retrofitted reinforced concrete beam 

column joint models. The performance of the retrofitted beam column joint specimen was 

compared with the control beam-column joint specimens

NON LINEAR MODELING OF THE BEAM-COLUMN JOINTS 

Non linear analysis was done for three beam column specimens using the 

software ANSYS. A transverse static was applied at the free end of the beam to develop a 

bending moment at the joint. The load was increased in steps till a controlled load of 22 

kN. The deflection at the free end of the cantilever beam was noted. The deflections of 

the specimen C1 were found to be 12.5 mm for the load of 15 kN, 35 mm for the load of 

20 kN, and 56 mm for the load of 22 kN. The same procedure was repeated for the 

specimen detailed as per code IS 13920-1993 and for the retrofitted specimen. Figure 3a, 

Figure 3b, and Figure 3c show the typical views of the deflected control and retrofitted 

specimens. Figure 4 shows the load deflection curve for the control and retrofitted 

specimens. Table 1 shows comparison between the deflections and energy absorption 

capacity of the control and retrofitted specimens. 

DISCUSSION OF THE RESULTS 

 It can be found from the Table 1 that the deflection of the beam column joint 

specimen detailed as per code IS 13920-1993 is 19.64 % less than that of the specimen 

detailed as per code IS 456-2000 and deflection of the beam column joint specimen 

retrofitted with glass reinforced polymer sheet was 42.85 % less than that of the 

specimen detailed as per code IS 456-2000.The energy absorption capacity of the 

specimen beam column joint specimen detailed as per code IS 13920-1993 is 15.93 % 

more than that of the specimen detailed as per code IS 456-2000 and energy absorption 

capacity of the beam column joint specimen retrofitted with glass reinforced polymer 

sheet was 34.22 % more than that of the specimen detailed as per code IS 456-2000. 

CONCLUSIONS 

 Based on the ANSYS modeling and analysis carried out on the control and retrofitted beam 

column joint specimens using GFRP sheets, the following conclusions were drawn: 

• The deflection of the beam column joint specimen detailed as per code IS 13920-1993 

was found to be 19.64 % lower than that of the specimen detailed per code IS 456-

2000. 

• The deflection of the beam column joint specimen retrofitted with GFRP sheet reduced 

the deflection about 42.85 %.when compared with the deflection of specimen detailed 

as per code IS 456-2000. 

• The energy absorption capacity of the beam column joint specimen detailed as per code 

IS 13920-1993 was found to be 15.93 % higher than that of the specimen detailed per 

code IS 456-2000. 

• The energy absorption capacity of the beam column joint specimen retrofitted with 

GFRP sheet increased about 34.22 %.when compared with the energy absorption 

capacity of specimen detailed as per code IS 456-2000.