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Theology/Seminary Notes/OT01

Key Terms: -Stoic Ethics: very rational and ethical 1. Believed in brotherhood of man (because of the logos, we are all related as one family 2. Believed that God was our common father 3. The only appropriate behavior is apathy. You need to rise above the world and your feelings and pay attention to the logos (reason) Plotinus - greatest of the pagan mystics. He said we become one with god. We are united with God and it is proper aspiration to become one with God -He was a Platonist but he took Plato in a mystical direction -had notion of great chain of being. Top is God and then you move down toward nothingness and Satan. If you exalt your animal nature you move towards nothingness. Gnostics grasped this great chain of being and believed that Jesus is the one who comes down and helps us get turned in the right direction. He does this by secret knowledge and not faith.
 * Socrates - The doctrine and greatest discovery that was taught primarily by Socrates was the beautiful philosophical theory called the immortality of the soul. That was Socrates' so according to Dr. Partee "all you soul-type people are Socratic". Forget the Socratic method (answers through questions), that we all have aspiration for high things, that we especially have aspiration for the beautiful soul and beauty itself and that people should think about the care for the Socratic "soul" It is us that insist that it is the person that survives. We are one entity. We believe in resurection. -Socrates believed death was a friend a freeing of the soul. Jesus called death an enemy. What up with that?
 * Plato - continued Socrates idea of aspiration (that we strive toward God) -wrote many different works but there are a few that are more important to Christianity. -Euthypho (dialogue concerned with what is holiness) Ion- argues that the poet (Homer) is inspired -He also had different accounts of what the soul was in his different works that often contradict. What is really important is the Platonic synthesis:
 * Platonic dualism -forms (ideals) and things Forms (heaven) things (earth) and God is above all of this The world is a world of things and ideas the ideas (ideals) never changing. They don’t change because you can’t change their substance but things are ever changing. Ideas are thing to aspire to. (good and God are in the realm of ideals) The ideas are the cause of the physical world. -This whole scheme helps form the chain of being. This chain of being influence New Testament writers
 * Aristotle -took the idea of ideas and things and put them together -the form was immanent in matter. Form and matter constitute concrete individual realities. -in everything, the form is already in the material Ex. Acorn becoming a tree. From the moment of becoming an acorn, it has a cause of moving in a certain direction. Everything in the universe has its end. -the substance tends to be the form and the attributes are changeable. (This idea of substance develops in the Eucharist (transubstaciation, consubstantiation, symbolic, Calvin's view
 * Epicureans -believed the world was causeless -everything was a matter of chance and fortune -Developed view of Metaphysics: atoms have a property called the swerve. These atoms fly around and this swerve introduces chance. -Life Theme "Eat drink and be merry for tomorrow you die." -Ethics:- sought pleasure; hedonists -Tried to be free from evil. Believed that death was the end.
 * Theology of Epicureans: 1.Believe that gods exist because they dream about them but the gods can’t know about the human condition 2.Since gods are without concern for the world, there can be no possible doctrine of providence. 3.There can be no creation by the gods because then they would be responsible for the evil that crept into the world.
 * Stoics - Stoic View of Fate The concept of providence was influced by the stoic view -they believed that nature, God, fate, and providence were all essentially the same idea. Nature is determining who you are. We get in on that because in each of us there is a logos (a divine spark). This means something that is human and divine. We participate in a secondary and subordinate reality

The next question is how w/out confusion against Eutichism w/out transmuting one nature into the other- against Apoll W/out dividing them into 2 separate categories- against Nestorius Thomistic synthesis: 1274 (Angelic doctor) Theological synthesis: Grace perfects nature. Aquainas view: Cooperating grace is a gift of God infused by God into the soul (justification) and becomes habitual grace (sanctification), which is the principle of meritorious works (Christian perfection). (this is Aquainas’ position.) this depends on human ability to do good. -If you are just, you are creating meritorious works. He said that habits are the intrinsic principles of human actions of mind or will. A habit is always potential. Aristotle- the major philosophical influence for Aquanans Operating grace-grace given by God…then we have to cooperate with it Relation of philosophy and theology- you can’t be a theologian until you first learn philosophy Philosophy is the truth which can be discovered and demonstrated by reason. -we are all philosophers -philosophical theological (natural theology) appeals only to reason. One of the facets of this is the ability to prove the existence of God. Cosmological argument 1. Motion to first mover 2. Effect to cause. First cause is uncaused 3. We are all contingent. We are not necessary. Therefore, there must be one who is necessary 4. Degrees of perfection: must be a best 5. Order of universe: must be an ordered Philosophical reflection on God’s nature 1. God is pure actuality. God sees everything as one event 2. God is perfectly simple (can’t add to God). God is immutable 3. God is perfect and perfectly good How we know: 3 kinds of knowledge 1. Univocal- what we say applies in the same way to dif realities (whatever we mean by wise applies equally to 2 people, events, etc 2. Equivocal- similar but mostly deferent (she’s an angel but u don’t mean that) 3. Analogical- neither univocal or equivocal. There is something in the world that we know that we apply to ‘God (like the term father) His cosmology 1. Adopts chain of being. There is a hierchy. (angels higher than bishops, bishops higher than other people etc 2. In chain of being it is obvious that there are angels 3. Humans are composed of matter and form. We differ only in appearance but we are all human (angels, on the other hand, are different species Ethics: humans seek happiness. Perfect happiness is perfect contemplation of God (Beatific vision) Law Eternal law- way God thinks Natural law- reflection of God’s eternal reason. (children being produced form nature, etc) Divine law- God’s revelation which reaches those people who have been reached by revelation Human law- rules we make for ourselves. We must act in terms of our conscience Believes in condign merit which requires co operating grace -when God’s sovereign grace is bestowed, an adequate human response is enabled and the result is salvation. In other words: elect r predestined to grace. On that basis, they earn glory…
 * Ebionites- said that Jesus was only human *Docetists- Jesus only appeared human. He was only divine
 * Arius - He wanted to preserve the humanity of Jesus and monotheism. He thought that since God was immutable and the son changes it must mean that the son is a creature and not THE God. -He believed that the son had to learn obedience (but God does not have to learn obedience) -He believed that Jesus was not co eternal. If he were then God would have a twin brother and not a son -God was not always the father but he became the father when the son was begotten. So he is not the eternal father. -For Arius, to same substance meant two principles -He made Jesus a bridge to God. An intermediatery at the top of the list. He rejected the distinction between incarnate logos and divine logos. -The son is equal in glory but not in substance. He was the highest human being. Be only becomes God by participation.
 * Athansius Arius was opposed at Nicea by Athansius who is the father of orthodoxy Arius was opposed at Nicea by Athansius who is the father of orthodoxy -He insisted that only God can save. Jesus is our savior so he is not a creature 2. the Logos is fully divine. The Logos is now personal. All of the persons of the trinity are co eternal and co equal 3. the HS is not a creature. 4. Jesus is fully god and fully man.
 * Nicea was an attack against Arius. It answered the who question. 1. There was an emphasis on the Eternal son. 2. Said Jesus was one substance 3. Jesus was made man 4. Had filioque clause (and the son) The Spirit proceeds from the father and the son.
 * Apollanarius said Jesus had 3 different forms (mind, body and soul) He said the divine logos took on the human nature but not the human personality. He had a dual nature. He had the mind of God. -The divine nature took the place of the human soul of God.
 * Nestorius tended to go more toward emphasizing Jesus’ humanity -He opposed the phrase theotocus (mother of God). He though this was to speak of Mary as the mother of God. He said that Jesus 2 natures are conjoined but not united. He wants to keep the two natures apart (turning Jesus into two people) -He destroys the one personality -Jesus is an appearance, behind which one finds the man Jesus and the divine logos. He loses the unity of the person.
 * Eutichian Christ was of two natures before incarnation but not in 2 natures. In incarnation, he was one person IN two natures. -emphasized the divinity -one nature became subsumed by the other. The humanity of Christ is o only a memory of Christ -denied the reality of the human nature Divine nature always overcomes the human attributes.
 * Chalcedon: answered the how -it was a decision and definition and not a compromise -set flexible but definite boundaries -talked about hypostatic union Jesus is fully divine, fully human, 2 natures, united in one person

•	Ignatius •	Justin Martin •	The Gnostics •	Marcion •	Montanus •	Irenaeus •	Tertullian •	Origen •	Arius •	Athanasius •	The Cappadocians •	Apollinaris •	Nestorius •	Eutyches •	Gregory the Great •	Anslem •	Aquinas •	Occam The Historical Impact of classical philosophy on Christian Theology. Socrates, Plato, Aristotle, Epicureans, Stoics and Plotinus all helped build the foundation of Christianity. The Historical Impact of classical philosophy on Christian Theology. Socrates provided a basis and method of thought and reflection without this and the Socratic Method there would be no creed, doctrine or dogma. He aspired for better and encouraged others to think about the care for the soul. Plato, expanded the idea of aspiration to that holiness and introduced to idea that the poet (Homer) is inspired. From Aristotle we get concepts of form, matter and being. Form and matter constitute concrete individual realities. Looking at an acorn Aristotle saw it as the substance (being) of an Oak Tree. From this seed come the complex views that develop into the Eucharist, transubstaciation and consubstantiation. The Epicureans influenced the view that God is to Holy to be in contact with fallen humanity. This restriction was taken up by Christians and melded into the debate over the substance of Jesus. We must thank the Stoics for the concept of providence, fate and predestination because they taught that nature, God, fate, and providence were all essentially the same idea. Nature is determining who you are. It gets deeper when they mix in that in each of us there is a logos (a divine spark). This means something that is human and divine. We participate in a secondary and subordinate reality. Stop the bus because I want OFF! Before I get off I need to talk about the greatest of the pagan mystics. Plotinus said we must seek to become one with god. We are united with God and it is the only proper aspiration to become one with God. In 2002 Mulholland wrote a book for modern Christian with the central theme of "Finding Christ-likeness. Without the milk of Philosophy, the fledgling Christian Church of the Patristic Period would have had nothing to suckle. Question is was it milk or hemlock.

Christianity developed with the aid of Ebionites, Docetists, Arius, Athanasius, Nestorius, Apollinaris, Nicea and Chalcedon. Christianity developed with the aid of Ebionites, Docetists, Arius, Athanasius, Nestorius, Apollinaris, Nicea and Chalcedon. Ebionites and Muslims said that Jesus was only human. Agreeing with them, Arius wanted to preserve the humanity of Jesus and monotheism. He thought that since God was immutable and the son changes it must mean that the son is a creature and not THE God. Opposing them Docetists said that Jesus only appeared human that He was 100% divine. Athanasius condemned them all as Heretic at Nicea. Vehemently stating that despite all reason four things are mysteriously true: That in order for Jesus to be our Savior he can not be a creature; that the Logos is fully divine; The Logos is now personal; All of the persons of the trinity are co eternal and co equal; That the Holy Spirit is not a creature; And Jesus is 100% fully God and 100% fully man. For his trouble Athanasius was told by the bishops of Nicea the he was less wrong than Arius who was really wrong. Therefore, Council of Nicea in an attempt to suppress the Arius Heresy and answer the question of "Who (What) Jesus is (Is) (was)." question. 1. There was an emphasis on the Eternal son 2. Said Jesus was one substance 3. Jesus was made man 4. The Spirit proceeds from the Father and The Son Unfortunately, left three questions unanswered. What does this mean to mean to me I just want to love Jesus as my Savior and Lord? Who put the hemlock in my milk? How in the world does this make any sense?

The Council at Chalcedon answered the "how" 1 It was a decision and definition and not a compromise 2 Set flexible but definite boundaries 3 Talked about hypostatic union 4 Jesus is fully divine, fully human, 2 natures, united in one person 5 There is No con-fusion against Eutichism 6 Transmuting one nature into the other against Apoll 7 There is no dividing them into 2 separate categories against Nestorius The Bishops of the Medieval Period had there hand full. The chaos and persecution of the Patrictic Period had melded into a period of intellectual and spiritual torture. The Lions and Gladiators had made way for double edged dogma, razor sharp vocabulary and an unquenchable fire of Christians shouting unspeakable heresies. They spent centuries in a proverbial Ferrari speeding down mountain roads of ultra-thin truth and faith in an unseen and inscrutable God.

The Thomastic Synthesis involved Aristotle, Dante, operating grace and condign and congruent merit. The essential difference between condign merit and congruent merit is based on the fact that, besides those works which claim a remuneration under pain of violating strict justice (as in contracts between employer and employee, in buying and selling, etc.), there are also other meritorious works which at most are entitled to reward or honour for reasons of equity or mere distributive justice, as in the case of gratuities and military decorations. St Thomas Aquainas synthesized the various problems that developed over time in the Church. He appointed himself the one to finally and for all blend the sacred words of Paul and the sacrilegious philosophical and historical foundation of the society where Paul wrote his letters. Hoping to challenge and simplify the issues that perplexed so many good men through the centuries. So many before him attempting to meet this challenge found themselves on the wrong end of the heretic's pyre. There are few ideas in political theory, in philosophy or Christianity as a whole which cannot be traced back in some form to Plato, Aristotle, and their amazingly fertile period of Greek speculative thought. Indeed, the fact that he sat down to interpret works done centuries before Christ demonstrates that the philosophical efforts of the ancient Greeks rank far above even this genius work on the subject. At any rate, when one considers the enduring quality of ancient political thought, lasting in spite of the entirely different theology and doctrine that developed in it's wake, one must realize how little man has changed in his quest for God even awash in the Holy Spirit. History demonstrates that though Christian creeds and political institutions rise and fall, the frailties of man limit the mind and soul. We are a social worm in the garden of the Divine. Theological synthesis: Grace perfects nature. Aquainas view: Cooperating grace is a gift of God infused by God into the soul (justification) and becomes habitual grace (sanctification), which is the principle of meritorious works (Christian perfection). (this is Aquainas’ position.) this depends on human ability to do good. The Augustinian Synthesis involved Plato, Pelagius and the Donatists. The Augustinian synthesis had at its roots the teachings of Plato, Pelagius and resisance to the views of the Donatists. Being one of the earliest attempts to marry the secular reason with prayful revelation this work has had implications that reverberate throughout history. Augustine had studied Plato through the writings of Plotinus. He accepted Plato's view that idea, God and spirit were combined, and he accepted Plotinus' view that the power of God touched everything, molding and giving meaning to passive matter. From Plotinus, Augustine believed he had gained an understanding of a permanence that was God. Augusine now saw God as utterly transcendent, as the creator of all, all-knowing and the source of human knowledge. He had come to believe that materiality was not evil, that the universe was a continuous active whole and that evil was merely the turning away from God. Yet Augustine believed that Plato was right about God but wrong about gods. Augustine used the creation story to illustrate how he and Ambrose viewed God as an all powerful and intelligent designer. That creation had not just happened and that God had not created the universe with a compass and a level but instead had commanded it. But Augustine's legacy was not limited to his writings. Sadly, much of his legacy was from conflict. Augustine railed against the remnant paganism among his parishioners, including astrology. He attacked the notion that humanity's course of action could be determined by the stars while animals remained free to chose between doing something and not doing it. He condemned the idea that people born in the same month - even the same hour - had some type of unified destiny in their life or a lifetime. Augustine believed that the Church needed to exclude ideas that were contrary to fundamental Christianity. So Augustine was a word warrior. Augustine also came into conflict with Donatists Christians who believed that the Church should be restricted to those who maintained the purity they had acquired at baptism, who believed that the Church was a source of holiness and that no sinner should have a part in it, that the Church should expel those who were guilty of mortal sins. For Augustine, sin was not just a matter of choice, sin was inherited and ingrained, that the Church should embrace all of humanity, saints and sinners alike, that the good and bad would be together until Armageddon, when they would be separated. Augustine claimed that the good Christian must try to become holy but must also coexist with sinners in the same community and be prepared to rebuke and correct them. His neo-Platonic education led him to see Christians as part of a world of development, as imperfection struggling toward the ideal as manifested in God. Augustine saw the Church not as a body of purists defying society but a body that should master society, a body capable of bringing truth to the masses. Augustine led the drive against Donatism. He wanted the Donatists to come under the discipline of both church and state. Like others of his time, Augustine believed that people lacked the will and wisdom to govern themselves. The battle between the Donatists and Augustinians was akin to a civil war. With one side gaining the upper hand and then the other. At one point Augustine's life was in danger. When peace was achieved by suppression of the Donatist a new villain lay on the horizon. Pelagius became disturbed by the moral laxity among Christians and began advocating a stricter morality for all Christians. Pelagius and those known as Pelagians came to believe that people could be good. The stated that rather than being born sinful, people had no excuse for sinful behavior and that every sin was a deliberate act of contempt for God. What began as an influence on those who wished to reform the Church became a threat to the church in Augustine’s mind. The ideas of Pelagius were the ideas of Dotanists. This disturbed Augustine, and once again he led the attack. This time against the Pelagians. Once again his argument involved inner feelings and patience, a belief that people should merely try to do right while convalescing within the Church. Augustine reiterated his belief in humanity's power to choose, and he added that freedom of choice was limited and, in having only a limited power to choose, people could not live flawlessly. Augustine supported his belief in the limits of will by holding up his own experinces and understanding of the scriptures. Although Augustine saw the world that God had created as overwhelmingly good, he believed that humanity was destined to envy and to lust for power. Though he had been extraordinarily active sexually in his younger days, now in his old age he saw humanity as gluttonous. Augustine described infants at the breast as filled with lust, jealousy and other vices. Adam and Eve could have had sex without lust, he wrote, but they chose instead to have it with lust. A carpenter moved his hands without lust, he added, and so too could people in sexual intercourse. Virtue, claimed Augustine, demanded complete control over one's body, but absolute control was impossible, he claimed, because of Adam' fall. The Ecumenical Councils: There are seven councils that provide an ecumenical or mutual doctrine of the Christian Church. These seven councils were all called by the emperor. Their decisions were widely received as authoritative by "Christians" in both East and West. The Arians, however, continued to exist as a separate church long after their condemnation at Nicaea I. The main dogmatic pronouncements of the seven councils are: (1) nicaea I (325): that the Son is ‘of the substance of the Father’; that is, the Son is consubstantial, homoousios, with the Father. This council condemned Arianism. (2) constantinople I (381): that the Holy Spirit is fully God. The restatement of the Nicene faith attributed to this council declares the Spirit to be ‘Lord and Life-giver, proceeding from the Father, worshipped and glorified together with the Father and the Son’. (3) ephesus (431): that Mary is theotokos, the one who gave birth to God. This council condemned Nestorius, bishop of Constantinople. (4) chalcedon (451): Defined the Divine and Human nature of Christ and condemned the view of Eutyches.

SEVEN ECUMENICAL COUNCILS Council of Nicaea (325) Bishops from all over the Christian world came to answer key issues and to set a Christian Creed. Pope Sylvester. The Emperor Constantine. Of primary importance to those in attendance was the differing view of the Divinity of Jesus. "Was Jesus Human?" (as viewed by Arius) or Divine as ultimately presented in the The Creed Of Nicaea. First Council of Constantinople (381) Defined the Divinity of the Holy Ghost. Opposed to Macedonius. Added wording to state divinity of Holy Ghost. Council of Ephesus (431) Defined the unity of Christ, defined Mary "Theotokos",and condemned the view of Pelagius. Council of Chalcedon (451) Defined the Divine and Human nature of Christ and condemned the view of Eutyches. Second Council of Constantinople (553) of 165 bishops under Pope Vigilius and Emperor Justinian I, condemned the errors of Origen and certain writings (The Three Chapters) of Theodoret, of Theodore, Bishop of Mopsuestia and of Ibas, Bishop of Edessa; it further confirmed the first four general councils, especially that of Chalcedon whose authority was contested by some heretics. http://www.piar.hu/councils/ number 5 Third Council of Constantinople (680-681) Addressed the Heresy of Monothelism. Which maintained that even though Jesus had two natures there was only One Will. This Council sought to Define the Two Wills of Jesus Christ. It anathematized (To bring about the worst imaginable curse upon, The Curse of Judas) Sergius, Pyrrhus, Paul, Macarius, and all their followers. Second Council of Nicaea (787) Later Councils: Fourth Council of Constantinople (869) --Eastern Orthodox Church Breaks With The Roman Catholic Church (1054) First Lateran Council (1123) Second Lateran Council (1139) Heresy of Arnold of Brescia Third Lateran Council (1179) Heresy of Albigenses and Waldenses Fourth Lateran Council (1215), Heresy of Albigenses again and the arrant if not heretical views of Abbot Joachim (Trinitarian). "This is the most important council of the Middle Ages, it marks the culminating point of ecclesiastical life and papal power". First Council of Lyons (1245). Excommunicated and deposed Emperor Frederick II and directed a crusade, under the command of St. Louis, against the Saracens and Mongols. Council of Lyons (1274) Council of Vienne in France (1311-1313) Council of Constance (1414-1418) Council of Basle (1431). The Crusades. The Papacy. Discuss Origen and his influence. Origen believed our body keeps us on earth. It is not evil, but it is a hindrance. Origen also: •	had a theology of scripture. He described three levels of scripture (literal/historical, moral, and spiritual. •	He believed in a transcendental fall. Adam and Eve were not historical, but spiritual. Means alienation from God. The fall occurs before Adam and Eve. We are born blind, and our sight is restored. •	Believe in a distinction between essence and existence. Salvation is restoration to our essential being. We remove ourselves from existence. In God's eternity all are saved •	thought logos was eternally generated but subordinate. Was never a time when the son was not

Discuss Anselem and the Ontological Argument and Atonement. An Ontological Arguement is a method of proving the existence of God. For Anslem the method was logical and similar to the Socratic Method of using questions until a proof is made. Anslem is famous for two proofs of God. The first that God exists in reality the second that God is necessary. God exists in reality because: If God is that thing that is greater than everything then God is the greatest power. If this concept exists in human understanding then God exists in one's mind. If God existed in reality then God would be greater than what exists in a man's. Since, existence both in reality and in imagination is greater than just one or the other. Therefore, God in reality must exist. God is necessary because: God is that entity than which nothing greater can be conceived. If it is greater to be necessary than not then God must be necessary. Therefore, God is the greatest necessary entity to exist. Anslem's view of Attonement is seated in what was his modern view of chivalry, justice and logic. This is different than our modern view but in the time of knights and Nobels is was a clear model. The first part of the model is chivalry. For Anslem, sin was an attack on the Honor of God. Each act of sin, disobedience or lowliness is a slap to the face of God. The honor of God must be re-cooped or satisfied. Yet, we are lowly creatures more lowly than a peasant to a King and only an equal can propose to repair the honor of another. This leaves only one person in a position to be able the restore God's Honor and that is Jesus. The Second part of the model is justice. Our dishonor requires an action. Justice in that period emphasizes the importance of character. In this case, our action to dishonor God demands response and justice. Otherwise, God's Honor is lost or somehow reduced. This Justice demands that the sin be punished and action be taken. The third part is logically, logic. We are all sinners. Lowly reprobates and because even the best of us would fail to live a life good enough to honor God. So, all of us fall short. Yet, it is illogical to believe that God created an entity or species for the purpose of insulting himself. Therefore, by logic there must be a means of atonement. This again leads the realization that Jesus is the only possible source for reconciliation to God. Patristc verses Medival History

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User:Eml4500.f08.RAMROD.F/HW7

Mtg. 36 November 19 2008

$$\tilde{K}_{6x6}^{(e)}\tilde{d}_{6x1}^{(e)}=\tilde{F}_{6x1}^{(e)} $$

$$\tilde{d}_{6x1}^{(e)}=\begin{Bmatrix} \tilde{d}_{1}^{(e)}\\ \tilde{d}_{2}^{(e)}\\ \tilde{d}_{3}^{(e)}\\ \tilde{d}_{4}^{(e)}\\ \tilde{d}_{5}^{(e)}\\ \tilde{d}_{6}^{(e)} \end{Bmatrix} $$

$$\tilde{F}_{6x1}^{(e)}=\begin{Bmatrix} \tilde{F}_{1}^{(e)}\\ \tilde{F}_{2}^{(e)}\\ \tilde{F}_{3}^{(e)}\\ \tilde{F}_{4}^{(e)}\\ \tilde{F}_{5}^{(e)}\\ \tilde{F}_{6}^{(e)} \end{Bmatrix}$$

Note: $$\tilde{F}_{3}^{(e)}=F_{3}^{(e)}$$ and $$\tilde{F}_{6}^{(e)}=F_{6}^{(e)}$$ are moments around the z-axis.

$$\tilde{K}_{6x6}^{(e)}=\begin{bmatrix} \frac{EA}{L} & 0 & 0 & \frac{-EA}{L} & 0 & 0\\ & \frac{12EI}{L^3} & \frac{6EI}{L^2} & 0 & \frac{-12EI}{L^3} & \frac{-6EI}{L^2}\\ & & \frac{4EI}{L} & 0 & \frac{-6EI}{L^2} & \frac{2EI}{L}\\ & &  & \frac{EA}{L} & 0 & 0\\ & SYM. & &  & \frac{12EI}{L^3} & \frac{-6EI}{L^2}\\ & &  &  &  & \frac{4EI}{L} \end{bmatrix}$$

Dimensional Analysis
$$\left[\tilde{d}_1 \right]=L=\left[\tilde{d}_i \right] \; \; i=1,2,4,5$$

$$ \left[\tilde{d}_3 \right]=L=\left[\tilde{d}_6 \right]$$

Let's take an arc of a circle of radius r

The arc length $$\bar{AB}=R\theta $$ where $$\bar{AB}$$ is the arc length and $$\theta$$ is the angle measure.

$$ \theta =\frac{\bar{AB}}{R}$$

$$\left[\theta \right]=\frac{\left[\bar{AB} \right]}{\left[R \right]}= \frac{L}{L}=1$$ no dimension

$$ \sigma =E\varepsilon \Rightarrow \left[\sigma \right]=\left[E \right]\left[\varepsilon  \right]$$

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User:Eml4500.f08.team.dwyer.jd/HW7 Nov 17 (Meeting 35)

Frame Elements

 * Frame element = truss(bar) element + beam element

 * A truss has axial deformation

 * A beam has transverse deformation (bends)

Model Frame Problem with 2 Elements:

FBD's:

 * Because the connection between the two elements is rigid, the angle between them will stay constant after deformation.

The free body diagrams of each element is shown below:

$$ rotational dofs and $$\left.\begin{matrix}f_{3}^{(e)}\\ f_{6}^{(e)} \end{matrix}\right\} $$ bending moments
 * Where $$\left.\begin{matrix}d_{3}^{(e)}\\ d_{6}^{(e)}\end{matrix}\right\}

2-D Frame Global DOF's:

There are now 9 total degrees of freedom, so:

 * 2 element stiffness matrix: $$\underline{k}_{(6x6)}^{(e)}$$, e=1,2
 * Global stiffness matrix: $$\underline{k}_{(9x9)}=A\underline{k}_{(6x6)}^{(e)}$$

The 9x9 global stiffness matrix will now look like this:


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User:Eml4500.f08.team.dwyer.jd/HW7 Nov 19 (Meeting 36)

Frame Elements & Dimensional Analysis

 $$ \tilde {\mathbf{k}} _ {6 x 6} ^ {(e)}\ \tilde {\mathbf{d}} _ {6 x 1} ^ {(e)}\ =\ \tilde {\mathbf{f}} _ {6 x 1} ^ {(e)}$$

$$ \tilde {\mathbf{d}} ^ {(e)} = \begin{Bmatrix} d_1^{(e)} \\ \vdots \\ d_6^{(e)} \end{Bmatrix} = \tilde {\mathbf{f}} ^ {(e)} = \begin{Bmatrix} f_1^{(e)} \\ \vdots \\ f_6^{(e)} \end{Bmatrix} $$

Note: $$ \tilde {f} _{3}^ {(e)}\ =\ f_3^{(e)}\, \ \tilde {f} _{6}^ {(e)}\ =\ f_6^{(e)} $$

moments (about $$  \tilde {z}\ =\ z) $$

$$\tilde{\mathbf{k}}=\begin{bmatrix} \frac{EA}{L} & 0 & 0 & \frac{-EA}{L} & 0 & 0\\ & \frac{12EI}{L^{3}} & \frac{6EI}{L^2} & 0 & \frac{-12EI}{L^{3}} & \frac{6EI}{L^2}\\ & & \frac{4EI}{L} & 0 & \frac{-6EI}{L^2} & \frac{2EI}{L}\\ & &  & \frac{EA}{L} & 0 & 0\\ & sym & &  & \frac{12EI}{L^{3}} & \frac{-6EI}{L^2}\\ & &  &  &  & \frac{4EI}{L} \end{bmatrix}$$

$$\tilde{\mathbf{k}}^{(e)}_{6x6}\tilde{\mathbf{d}}^{(e)}_{6x1}=\tilde{\mathbf{f}}^{(e)}_{6x1}$$

$$\tilde{\mathbf{d}}^{(e)}=\begin{Bmatrix} \tilde{\mathbf{d}}^{(e)}_1\\ \tilde{\mathbf{d}}^{(e)}_2\\ \tilde{\mathbf{d}}^{(e)}_3\\ \tilde{\mathbf{d}}^{(e)}_4\\ \tilde{\mathbf{d}}^{(e)}_5\\ \tilde{\mathbf{d}}^{(e)}_6 \end{Bmatrix},\tilde{\mathbf{f}}^{(e)}=\begin{Bmatrix} \tilde{\mathbf{f}}^{(e)}_1\\ \tilde{\mathbf{f}}^{(e)}_2\\ \tilde{\mathbf{f}}^{(e)}_3\\ \tilde{\mathbf{f}}^{(e)}_4\\ \tilde{\mathbf{f}}^{(e)}_5\\ \tilde{\mathbf{f}}^{(e)}_6 \end{Bmatrix}$$

Dimensional Analysis
 $$\begin{bmatrix} \tilde{d}_1 \end{bmatrix}=L=\begin{bmatrix} \tilde{d}_i \end{bmatrix}\rightarrow i=1,2,4,5$$ $$\begin{bmatrix} \tilde{d}_1 \end{bmatrix}=1$$ $$\sigma =E\varepsilon\rightarrow[\sigma] =[E][\varepsilon] $$


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User:EML4500.f08.delta 6.ramirez/Matlab

2-Element Frame and Truss System
Here are the properties of the system we will be solving using MATLAB.

$$

E^{(1)}=3$$

$$

A^{(1)}=1$$

$$

L^{(1)}=4$$

$$

\theta^{(1)}=30$$

$$

I^{(1)}=.0833$$

$$

E^{(2)}=5$$

$$

A^{(2)}=2$$

$$

L^{(2)}=2$$

$$

\theta^{(2)}=-45$$

$$

P=7$$

title: User:EML4500.f08.delta 6.ramirez/Matlab


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User:EML4500.F08.RAMROD.B/Homework 7

=Motivation=

The motivation for this section is to find the deformed shape of the truss element through interpolation of the transverse displacement $$\displaystyle v(s) = v(\tilde{x})$$. We will first look at the Principal of Virtual Work for beams;

$$\displaystyle \int_{0}^{L}{w(\tilde{x})\left[-\frac{\partial ^2}{\partial x^2}\left((EI)\frac{\partial^2v}{\partial x^2} \right)+f_t -m\ddot{u} \right]dx} = 0$$ for all possible $$\displaystyle w(x)$$

For the first term we must integrate by parts. This is done below.

$$\displaystyle \alpha := \int_{0}^{L}{w(\tilde{x}) \left[-\frac{\partial ^2}{\partial x^2}\left((EI)\frac{\partial^2v}{\partial x^2} \right)\right]}$$

$$\displaystyle \alpha := \frac{\partial }{\partial x}\left[ \frac{\partial }{\partial x}\left((EI)\frac{\partial^2v}{\partial x^2} \right)\right]$$

What lies inside the square brackets will be designated as $$\displaystyle r(x)$$ and the whole expression is designated as $$\displaystyle r'(x)$$. Performing the integration by parts yields;

$$\displaystyle \alpha := \left[w\frac{\partial }{\partial x} \left( (EI)\frac{\partial^2 v}{\partial x^2} \right)\right]- \int_{0}^{L}{\frac{dw}{dx}\frac{\partial }{\partial x}\left((EI)\frac{\partial ^2v}{\partial x^2} \right)dx}$$

The first term (non integral term) will be denoted by $$\displaystyle \beta_1$$, the first derivative in the interal will be represented by $$\displaystyle s'(x)$$, and as your can see $$\displaystyle r(x)$$ is imbedded in the integral. Performing another integration by parts yields the following;

$$\displaystyle \alpha := \beta_1 - \alpha := \left[ \frac{dw}{dx} (EI)\frac{\partial^2 v}{\partial x^2} \right] + \int_{0}^{L}{\frac{d^2w}{dx^2}\left((EI)\frac{\partial ^2v}{\partial x^2} \right)dx}$$

The term after $$\displaystyle \beta_1 $$ will be denoted $$\displaystyle \beta_2$$ and the whole integral term will be denoted as $$\displaystyle \gamma$$. It is important to note that this new equation is symmetric. Inputting the new variables the new equation of the Principal of Virtual Work for beams is as follows;

$$\displaystyle -\beta_1 + \beta_2 - \gamma +\int_{0}^{L}{w(\tilde{x})f_t dx} - \int_{0}^{L}{Wm\ddot{u}dx} = 0$$ for all possible $$\displaystyle w(x)$$.

=Beam Stiffness Matrix=

Let us now focus on the stiffness term $$\displaystyle \gamma$$ so we can derive the beam stiffness matrix and identify the beam shape.

We need to plot the following eqaution;

$$\displaystyle v(\tilde{x}) = N_2(\tilde{x})\tilde{d_2} + N_3(\tilde{x})\tilde{d_3} + N_5(\tilde{x})\tilde{d_5} + N_6(\tilde{x})\tilde{d_6}$$

We recall that

$$\displaystyle u(\tilde{x}) = N_1(\tilde{x})\tilde{d_1} +N_4(\tilde{x})\tilde{d_4}$$

The equations for $$\displaystyle N_2(\tilde{x}),N_3(\tilde{x}),N_5(\tilde{x}),N_6(\tilde{x})$$ can now be found from these drawings and are shown below.

$$\displaystyle N_2(\tilde{x}) = 1 - \frac{3\tilde{x}^2}{L^2} + \frac{2\tilde{x}^3}{L^3}$$, $$\displaystyle \tilde{d_2}$$

$$\displaystyle N_3(\tilde{x}) = \tilde{x} - \frac{2\tilde{x}^2}{L} + \frac{\tilde{x}^2}{L^2}$$, $$\displaystyle \tilde{d_3}$$

$$\displaystyle N_5(\tilde{x}) = \frac{3\tilde{x}^2}{L^2} - \frac{2\tilde{x}^3}{L^3}$$, $$\displaystyle \tilde{d_5}$$

$$\displaystyle N_6(\tilde{x}) = -\frac{\tilde{x}^2}{L} + \frac{\tilde{x}^3}{L^2}$$, $$\displaystyle \tilde{d_6}$$

title: User:EML4500.F08.RAMROD.B/Homework 7


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User:Eml4500.f08.gravy.bje/Notes7

$$\ \underline{\tilde{k}}^{(e)} \underline{\tilde{d}}^{(e)} = \underline{\tilde{f}}^{(e)} \,$$

$$\underline{k}^{(e)} = \underline{\tilde{T}}^{(e)T} \underline{\tilde{k}}^{(e)} \underline{\tilde{T}}^{(e)} $$

$$ \underline{\tilde{T}}^{(e)} = \begin{bmatrix} l^{(e)} & -m^{(e)} & 0 & 0 & 0 & 0\\ m^{(e)} & l^{(e)} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & l^{(e)} & -m^{(e)} & 0 \\ 0 & 0 & 0 & m^{(e)} & l^{(e)} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}_{6x6}\,$$

$$ \underline{\tilde{T}}^{(e)T} = \begin{bmatrix} l^{(e)} & m^{(e)} & 0 & 0 & 0 & 0\\ -m^{(e)} & l^{(e)} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & l^{(e)} & m^{(e)} & 0 \\ 0 & 0 & 0 & -m^{(e)} & l^{(e)} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}_{6x6}\,$$

$$\ l^{(e)}= \vec \tilde{i} \cdot \vec i= cos\theta^{(e)}\,$$

$$\ m^{(e)}= \vec i \cdot \vec j= cos\left (\frac{\pi}{2}- \theta^{(e)} \right )= sin\left (\theta^{(e)} \right )\,$$

$$ \underline{\tilde{T}}^{(e)T} * \underline{\tilde{T}}^{(e)} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}_{6x6} $$

Similarly

$$\underline{d}^{(e)} = \underline{\tilde{T}}^{(e)T} \underline{\tilde{d}}^{(e)} \underline{\tilde{T}}^{(e)} $$

And

$$\underline{f}^{(e)} = \underline{\tilde{T}}^{(e)T} \underline{\tilde{f}}^{(e)} \underline{\tilde{T}}^{(e)} $$

Therefore:

$$\underline{f}^{(e)} = \underline{\tilde{k}}^{(e)} \underline{\tilde{d}}^{(e)} \underline{\tilde{T}}^{(e)} \underline{\tilde{T}}^{(e)T} $$\

Which is equal to:

$$\ \underline{k}^{(e)} \underline{d}^{(e)} = \underline{f}^{(e)} \,$$

title: User:Eml4500.f08.gravy.bje/Notes7


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User:Eml4500.f08.gravy.sms/HW7

Two Bar Frame Element in Matlab
The two bar frame system is the same one analyzed in HW2 and HW5, with the same lengths, areas and elasticities for both elements, but for this case, the elements are considered to be frame elements with square cross sections. This requires adding the moment of inertia to the previous code and using PlaneFrameElement.m, taken from here, to build the global stiffness matrix.

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User:EAS4200c.f08.Blue.E/dec1st

resulting for the consequences of the first method,

$$\tilde q_{j6}=\tilde q_{2j}-\tilde q_{j5}- \tilde q_{j8}+q(j)$$

then if the $$\tilde q_{6j}$$ is reversed

then

$$-\tilde q_{j6}=\tilde q_{2j}-\tilde q_{j5}- \tilde q_{j8}+q(j)$$

The question was raised that if one stringer was isolated if the system could still be solved.

this is not true because when one stringer is isolated it would yield that the q of the actual stringer is zero which is known not to be true.

$$\tilde q_{j6}=\tilde q_{2j}-\tilde q_{j5}- \tilde q_{j8}+q(j)$$ which would yeild $$0=0+0+0+q(j)$$ and q(j) is a number that is non zero $$q(j) \neq 0$$ so this yields that the system is unsolvable whenever a stringer is isolated.

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User:Eas4200c.f08.blue.a/Lecture 38

resulting for the consequences of the first method,

$$\tilde q_{j6}=\tilde q_{2j}-\tilde q_{j5}- \tilde q_{j8}+q(j)$$

then if the $$\tilde q_{6j}$$ is reversed

then

$$-\tilde q_{j6}=\tilde q_{2j}-\tilde q_{j5}- \tilde q_{j8}+q(j)$$

The question was raised that if one stringer was isolated if the system could still be solved.

this is not true because when one stringer is isolated it would yield that the q of the actual stringer is zero which is known not to be true.

$$\tilde q_{j6}=\tilde q_{2j}-\tilde q_{j5}- \tilde q_{j8}+q(j)$$ which would yeild $$0=0+0+0+q(j)$$ and q(j) is a number that is non zero $$q(j) \neq 0$$ so this yields that the system is unsolvable whenever a stringer is isolated.

title: User:Eas4200c.f08.blue.a/Lecture 38


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User:Eml4500.f08.RAMROD.A/MyContributionHW7

Dimensional Analysis
Dimensions of common parameters

$$ [\varepsilon ]=\frac{[du]}{[dx]}=\frac{L}{L}=1 $$

$$ [\sigma ]=[E]=\frac{F}{L^2} $$

$$ [A]=L^2 $$

$$ [I]=L^4 $$

$$ [\frac{EA}{L}]=[K_{11}^{N}]=\frac{(F/L^2)(L^2)}{L}=\frac{F}{L} $$

$$ [\tilde{K}_{11}\tilde{d}_1]=[\tilde{K}_{11}][\tilde{d}_1]=F $$

$$[\tilde{K}_{23}\tilde{d}_3]=[\tilde{K}_{23}][\tilde{d}_3]=F $$

FD relationship and the Transpose
Converting the element FD relationship in global coordinates from element FD in local coordianates

The relationship is presented as follows:

$$ K^{(e)}=\tilde{T}^{(e)T}\tilde{k}^{(e)}\tilde{T}^{(e)} $$

$$ k^{(e)}_{6x6}d^{(e)}_{6x1}=f^{(e)}_{6x1} $$

$$ k^{(e)}_{6x6}=\tilde{T}^{(e)T}_{6x6}\tilde{k}^{(e)}_{6x6}\tilde{T}^{(e)}_{6x6} $$

$$ \tilde{k}^{(e)}_{6x6}\tilde{d}^{(e)}_{6x1}=\tilde{f}^{(e)}_{6x1} $$

where

$$ \tilde{d} = \begin{Bmatrix}\tilde{d}_1\\\tilde{d}_2\\\tilde{\underline{d}}_3\\\tilde{d}_4\\\tilde{d}_5 \\\tilde{\underline{d}}_6\end{Bmatrix}$$

And

$$ \tilde{T} = \begin{bmatrix}\underline{R}_{2x2} & & 0 & 0 & 0 & 0\\ &  & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & \underline{R}_{2x2} & & 0\\0 & 0 & 0 &  &  & 0\\0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} $$

Friday Dec 5
Equation One: $$ \begin{Bmatrix}U_x(\tilde{x})\\\mu_y(y)\end{Bmatrix} = R^T \begin{Bmatrix}U(\tilde{x})\\V(\tilde{x}) \end{Bmatrix}$$

Equation Two: $$ \begin{Bmatrix}U(\tilde{x}\\V(\tilde{x})\end{Bmatrix} = \begin{bmatrix}N_1 & 0 & 0 & N_4 & 0 & 0\\ 0 & N_2 & N_3 & 0 & N_5 & N_6\end{bmatrix} \begin{Bmatrix}\tilde{d}_1\\\tilde{d}_2\\\tilde{d}_3\\\tilde{d}_4\\\tilde{d}_5 \\\tilde{d}_6\end{Bmatrix}^{(e)} $$

Equation Three: $$ \begin{Bmatrix}U_x(\tilde{x})\\U_y(\tilde{x})\end{Bmatrix} = R_{2x2}^TN_{2x6}(\tilde{x})\tilde{T}^{(e)}d^{(e)} $$

Once again, we check the dimensional analysis:

$$ [U]=[V]=L $$

$$ [N_1]=[N_4]=1 $$

$$ [N_1][d_1]+[N_4][d_4]=L+L $$

d = displacement or rotation

displacement:

$$ [d_1]=[d_2]=[d_4]=[d_5]=L $$

rotation:

$$ [d_3]=[d_6]=\frac{L}{[\theta] } =\frac{L}{rad}=L $$

Equations of V and V' of Ni
Looking at N5:

$$V(0)=0,V'(L)=-1$$

$$V'(0)=0,V(L)=0 $$

Looking at N6:

$$V(0)=0,V(L)=0 $$

$$V'(0)=0,V'(L)=1 $$

Derived Coefficient in $$ \tilde{k}$$:

$$ \tilde{k}_{22}=\frac{12EI}{L^3}=\int_{0}^{L}{\frac{d^2N_2}{dx^2}(EI)\frac{d^2N_2}{dx^2}dx} $$

$$ \tilde{k}_{23}=\frac{6EI}{L^2}=\int_{0}^{L}{\frac{d^2N_2}{dx^2}(EI)\frac{d^2N_2}{dx^2}dx} $$

Elastodynamics
Studying frames, beams and trusses in 3-D

discrete PVW:

$$w\cdot{(m\ddot{d}+kd-F)}=0$$, for all w

$$d(0)$$ is denoted d0

and $$\dot{d}(0)$$ is denoted V0

Method:
1) Complete ordinary diff. equations (ODE's)

(2nd order in time) and initial conditions are used

Next: Governing the elastodynamics of discrete and continuous problems

For Unforced Vibration:

general equation:

$$ M\ddot{V}+kV=0$$

Assume:

$$ v(t)=\sin(wt)\phi$$

$$-w^2\sin(wt)M\phi+\sin(t)k\phi=0$$

$$ -w^2k\phi =w^2M\phi $$ general Eigenvalue

General Form Eigenvalue:

$$ A_x=\lambda B_x$$

$$I=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$$

And:

$$ \lambda=w^2$$

2) Model Superposition

$$ M\phi=\lambda^{-1}k\phi $$

finally,

$$ M(\varepsilon,\ddot{\xi},\phi)+k(\varepsilon,\dot{\xi},\phi)=F $$


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User:Eas4200c.f08.WIKI.E/extra3

Wikimedia vs. E-learning
During this course, we used wikimedia to turn in our homework. Our homework consisted of writing up the class notes and doing some matlab assignments. We were asked to form teams to work on this Wikimedia homework. The format of these homework assignments forced us to attend class once every two weeks in order to get the notes for our assigned day. When doing this type of work, Wikimedia is a much more convenient and usable program to upload and format than E-learning. E-learning was not designed to do what was require for this course. Wikimedia's ability to save all archived versions of pages as well as the ability to easily interface with equation editors made it user friendly for students. It also allowed for easier grade formatting and uploading. I see E-learning as an older, less usable and less versatile version of Wikimedia.

I do believe that Wikimedia is a more versatile program than E-learning and was a much better decision for us to accomplish our homework as assigned. However, I do not believe that Wikimedia allowed me to learn Aerospace structures effectively. We would prefer to have traditional homework problems that would allow us to practice the method rather than regurgitating theory that we may or may not understand. Also, assigning Matlab to try to patch this significant problem only compounded it because most of us have never used Matlab, adding another program we needed to learn to succeed in this class, without learning any actual Aerospace Structures. I believe assigning homework in this way has hindered my learning. An argument could be made that I can always look back at these pages to find information about structures. I would respond to this argument by saying that none of our homework was graded carefully enough to catch errors or that it was written with any understanding or authority of the subject it was discussing. I would much rather turn to a validated source like a textbook than look on Wikiveristy. I strongly disagree with the style in which this class was taught. Because I have learned near nothing in the class, I would venture to say that the homework's effectiveness on my learning experience was a utter failure.

title: User:Eas4200c.f08.WIKI.E/extra3


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User:Eml4500.f08.lulz.abcd/HW7

Solve 2-element frame system and plot undeformed shape, deformed shape of a 2-bar truss, and deformed shape of a 2-element frame
To accomplish this, the Matlab script our group has been using was modified to include calculation and plotting of the 2-element frame system. This requires use of the TwoBarTrussSystemDebugged.m function to calculate the displacements for the truss condition, which in turn requires the NodalSoln.m and PlaneFrameElement.m functions that are provided from our textbook's publisher's website and are herein. Our code and the publisher-provided code are reproduced for extra rigor. The plot is provided below showing the dotted line as the undeformed shape, the red line as the deformed frame shape, and the blue line as the deformed truss shape.

TwoBarTrussDisplayMod.m
This code plots the image required in the assignment.

TwoBarTrussSystemDebugged.m
This code calculates the displacement values for the standard two-bar truss system and outputs them as a matrix, d.

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User:Eml4500.f08.echo.sell.cm/HW7

HW7
In order to run this coding, the following three codes used in previous homeworks were used, i.e. PlaneTrussElements, PlaneTrussResults, and NodalSln, and they were all taken from the website for the text Fundamental Finite Element Analysis and Applications by M. Asghar Bhatti.

You will also need the following code circle2.m for formatting purposes

Matlab code for plot:

title: User:Eml4500.f08.echo.sell.cm/HW7


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Comparative Mythology/Angelology

Welcome to the Angelology resource.

Angelology is the study of Angels it comes from Old English enġel, influenced in Middle English by Old French angele, both from Late Latin angelus, from Ancient Greek ἄγγελος (angelos), “‘messenger’”)

In this section I will try to present (remember) information concerning those Beings of Light known to us as Angels. You will find information that I have found and gathered together on various topics listed on the right and much more inside those pages. The portrayal of the synthesis for the sections: The Angelic Orders - The 7 Heavens, and The Archangels & Angels; was made by me using as much as possible the information in hand.

I came to believe that assisting others, brings joy to life, so I plan to share this information with everyone because... "freely you received... freely you shall give".

"We are, each of us angels with only one wing; and we can only fly by embracing one another".

title: Comparative Mythology/Angelology


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User:Eml4500.f08.bottle.loschak/HW7

HW problems
from 11/21 notes

Revisitng the matrix below we must verify the dimensions for all terms multiplied by $$\tilde{d}$$. $$\tilde{k} =\! $$ $$\begin{bmatrix} \frac{EA}{L} & 0 & 0 & \frac{-EA}{L} &0 &0 \\ \\ & \frac{12EI}{L^{3}} & \frac{6EI}{L^{2}} & 0 & \frac{-12EI}{L^{3}} &\frac{6EI}{L^{2}} \\ \\ & &  \frac{4EI}{L}&0  & \frac{-6EI}{L^{2}} & \frac{2EI}{L}\\ \\ & &  & \frac{EA}{L} &0  & 0\\ \\ & &  &  &\frac{12EI}{L^{3}}  &\frac{-6EI}{L^{2}} \\ \\ symmetric & &  &  &  & \frac{4EI}{L} \end{bmatrix} \! $$ Verify all $$\tilde{k}_{ij}\tilde{d}_{j}$$ for i = 1, ..., 6 and j = 1, ..., 6 We know from the lecture notes that the dimensions of the first term are

$$[\tilde{k}_{11}\tilde{d}_1] = \frac{[E][A]}{[L]}[\tilde{d}_{1}] = \frac{(\frac{F}{L^2})L^2}{L}(L) = F$$

Calculating the dimensions of the four other terms results in the dimensions

$$[\tilde{k}_{22}\tilde{d}_2] = \frac{[12][E][I]}{[L]^3}[\tilde{d}_{2}] = \frac{(\frac{F}{L^2})L^4}{L^3}(L) = F$$

$$[\tilde{k}_{23}\tilde{d}_2] = \frac{[6][E][I]}{[L]^2}[\tilde{d}_{2}] = \frac{(\frac{F}{L^2})L^4}{L^2}(L) = FL$$

$$[\tilde{k}_{33}\tilde{d}_3] = \frac{[4][E][I]}{[L]}[\tilde{d}_{3}] = \frac{(\frac{F}{L^2})L^4}{L}(1) = FL$$

$$[\tilde{k}_{36}\tilde{d}_6] = \frac{[2][E][I]}{[L]}[\tilde{d}_{6}] = \frac{(\frac{F}{L^2})L^4}{L}(1) = FL$$

After the dimensions of all terms have been determined, the results can be displayed in the dimension matrix below.

$$ \begin{bmatrix} F & 0 & 0 & -F & 0 & 0\\ 0 & F & FL & 0 & -F & FL\\ 0 & FL & FL & 0 & -FL & FL\\ -F & 0 & 0 & F & 0 & 0\\ 0 & -F & -FL & 0 & F & -FL\\ 0 & FL & FL & 0 & -FL & FL \end{bmatrix} $$ To solve for the global stiffness matrix k(e) in $$\mathbf{k}^{(e)}\mathbf{d}^{(e)}=\mathbf{f}^{(e)}$$ from $$\tilde{\mathbf{k}}^{(e)}\tilde{\mathbf{d}}^{(e)}=\tilde{\mathbf{f}}^{(e)}$$ we use the equation $$\mathbf{k}^{(e)}=\tilde{\mathbf{T}}^{(e)^{T}}\tilde{\mathbf{k}}^{(e)}\tilde{\mathbf{T}}^{(e)}$$ The stiffness matrix is first given as $$\tilde{k} = \begin{bmatrix} \frac{EA}{L} & 0 & 0 & \frac{-EA}{L} &0 &0 \\ \\ 0 & \frac{12EI}{L^{3}} & \frac{6EI}{L^{2}} & 0 & \frac{-12EI}{L^{3}} &\frac{6EI}{L^{2}} \\ \\ 0 & \frac{6EI}{L^{2}} & \frac{4EI}{L}     & 0 & \frac{-6EI}{L^{2}}  & \frac{2EI}{L}\\ \\ \frac{-EA}{L} & 0     & 0                 & \frac{EA}{L}  &0        & 0\\ \\ 0 & \frac{-12EI}{L^{3}} & \frac{-6EI}{L^{2}} & 0 & \frac{12EI}{L^{3}} & \frac{-6EI}{L^{2}} \\ \\ 0 & \frac{6EI}{L^{2}} & \frac{2EI}{L} & 0 & \frac{-6EI}{L^{2}} & \frac{4EI}{L} \end{bmatrix} \! $$ The Transfer matrix and its transpose are below. The R matrix used to assemble the Transfer matrix was discussed in Lecture 19. $$\tilde{T}^{(e)} = \begin{bmatrix} l^{(e)} & m^{(e)} & 0 & 0 & 0 & 0\\ -m^{(e)} & l^{e)} & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & l^{(e)} & m^{(e)} & 0\\ 0 & 0 & 0 & -m^{(e)} & l^{e)} & 0\\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}$$ $$\tilde{T}^{(e)^{T}} = \begin{bmatrix} l^{(e)} & -m^{(e)} & 0 & 0 & 0 & 0\\ m^{(e)} & l^{e)} & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & l^{(e)} & -m^{(e)} & 0\\ 0 & 0 & 0 & m^{(e)} & l^{e)} & 0\\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}$$ Multiply the three matrices according to the equation listed above for the global stiffness matrix

We know from previously in the lecture notes that the units of displacement are length (L) and the units of rotation are simply 1. For the transverse displacement term N5 and the rotation of N6, dimensional analysis gives us the following results. $$ \left[N_{5} \right]\left[\tilde{d}_{5} \right] = (1)(L) = L $$ $$ \left[N_{6} \right]\left[\tilde{d}_{6} \right] = (L)(1) = L $$

title: User:Eml4500.f08.bottle.loschak/HW7


title: File:Nex1010.png


text:
User:Eml4500.f08.bottle.ranto/HW7

Electric Pylon Analysis
Our team was asked to modify the pylon analysis code to use frame analysis. The properties of the pylon are as follows:

Material = 300 M Steel Height = 60 m Moment of Intertia = 1.3333e-8 Young's Modulus = 200 GPa Area = 4 cm2 Density = 7.8 g/cm3

MATLAB was implemented in solving for the maximum bending moment, shear stress, eigenvalues, and vibrational periods.

Code
The code implemented is as follows:

%********************************************************************* % Filename: pylon2.m % % PURPOSE: % FEA Frame Analysis % %  %  Team Bottle % EML 4500 %*********************************************************************  clear; close; dof = 2; % Node coordinates n_node = 46;            % number of nodes n_elem = 91;            % number of elements total_dof = 2 * n_node; % total dof of system position(:, 1) = [ 1.5; 0]; position(:, 2) = [ 4.53; 0]; position(:, 3) = [ 1.88; 1.18]; position(:, 4) = [ 3.03; 1.18]; position(:, 5) = [ 4.17; 1.18]; position(:, 6) = [ 2.2; 2.181]; position(:, 7) = [ 3.03; 2.181]; position(:, 8) = [ 3.85; 2.181]; position(:, 9) = [ 2.53; 3.181]; position(:, 10) = [ 3.53; 3.181]; position(:, 11) = [ 2.34; 3.681]; position(:, 12) = [ 3.03; 3.681]; position(:, 13) = [ 3.71; 3.681]; position(:, 14) = [ 2.16; 4.181]; position(:, 15) = [ 2.62; 4.181]; position(:, 16) = [ 3.44; 4.181]; position(:, 17) = [ 3.89; 4.181]; position(:, 18) = [ 2.36; 4.5]; position(:, 19) = [ 3.7; 4.5]; position(:, 20) = [ 1.8; 5.172]; position(:, 21) = [ 4.25; 5.172]; position(:, 22) = [ 0; 6]; position(:, 23) = [ 1.06; 6]; position(:, 24) = [ 1.5; 6]; position(:, 25) = [ 1.82; 6]; position(:, 26) = [ 2.22; 6]; position(:, 27) = [ 2.71; 6]; position(:, 28) = [ 3.34; 6]; position(:, 29) = [ 3.84; 6]; position(:, 30) = [ 4.23; 6]; position(:, 31) = [ 4.55; 6]; position(:, 32) = [ 5; 6]; position(:, 33) = [ 6.05; 6]; position(:, 34) = [ .8; 6.23]; position(:, 35) = [ 5.25; 6.23]; position(:, 36) = [ 1.24; 6.355]; position(:, 37) = [ 4.82; 6.355]; position(:, 38) = [ 1.5; 6.43]; position(:, 39) = [ 2; 6.43]; position(:, 40) = [ 2.48; 6.43]; position(:, 41) = [ 3.03; 6.43]; position(:, 42) = [ 3.58; 6.43]; position(:, 43) = [ 4.05; 6.43]; position(:, 44) = [ 4.55; 6.43]; position(:, 45) = [ 1.5; 7.43]; position(:, 46) = [ 4.55; 7.43]; scale = 8.0753701211306; for i = 1 : n_node; x(i) = scale * position(1,i); y(i) = scale * position(2,i); end node_connect(1, 1) = 1;  % element 1 node_connect(2, 1) = 3; node_connect(1, 2) = 1;  % element 2 node_connect(2, 2) = 4; node_connect(1, 3) = 3;  % element 5 node_connect(2, 3) = 4; node_connect(1, 4) = 4;  % element 3 node_connect(2, 4) = 2; node_connect(1, 5) = 4;  % element 6 node_connect(2, 5) = 5; node_connect(1, 6) = 2;  % element 4 node_connect(2, 6) = 5; node_connect(1, 7) = 3;  % element 8 node_connect(2, 7) = 7; node_connect(1, 8) = 5;  % element 9 node_connect(2, 8) = 7; node_connect(1, 9) = 3;  % element 7 node_connect(2, 9) = 6; node_connect(1, 10) = 5;  % element 10 node_connect(2, 10) = 8; node_connect(1, 11) = 6;  % element 11 node_connect(2, 11) = 7; node_connect(1, 12) = 7;  % element 12 node_connect(2, 12) = 8; node_connect(1, 13) = 6;  % element 13 node_connect(2, 13) = 9; node_connect(1, 14) = 7;  % element 14 node_connect(2, 14) = 9; node_connect(1, 15) = 7;  % element 15 node_connect(2, 15) = 10; node_connect(1, 16) = 8;  % element 16 node_connect(2, 16) = 10; node_connect(1, 17) = 9;  % element 17 node_connect(2, 17) = 10; node_connect(1, 18) = 9;  % element 18 node_connect(2, 18) = 11; node_connect(1, 19) = 9;  % element 19 node_connect(2, 19) = 12; node_connect(1, 20) = 10;  % element 20 node_connect(2, 20) = 12; node_connect(1, 21) = 10;  % element 21 node_connect(2, 21) = 13; node_connect(1, 22) = 11;  % element 22 node_connect(2, 22) = 12; node_connect(1, 23) = 12;  % element 23 node_connect(2, 23) = 13; node_connect(1, 24) = 11;  % element 24 node_connect(2, 24) = 14; node_connect(1, 25) = 11;  % element 25 node_connect(2, 25) = 15; node_connect(1, 26) = 12;  % element 26 node_connect(2, 26) = 15; node_connect(1, 27) = 12;  % element 27 node_connect(2, 27) = 16; node_connect(1, 28) = 13;  % element 28 node_connect(2, 28) = 16; node_connect(1, 29) = 13;  % element 29 node_connect(2, 29) = 17; node_connect(1, 30) = 14;  % element 30 node_connect(2, 30) = 15; node_connect(1, 31) = 15;  % element 31 node_connect(2, 31) = 16; node_connect(1, 32) = 16;  % element 32 node_connect(2, 32) = 17; node_connect(1, 33) = 14;  % element 33 node_connect(2, 33) = 20; node_connect(1, 34) = 14;  % element 34 node_connect(2, 34) = 18; node_connect(1, 35) = 15;  % element 35 node_connect(2, 35) = 18; node_connect(1, 36) = 16;  % element 36 node_connect(2, 36) = 19; node_connect(1, 37) = 17;  % element 37 node_connect(2, 37) = 19; node_connect(1, 38) = 17;  % element 38 node_connect(2, 38) = 21; node_connect(1, 39) = 18;  % element 39 node_connect(2, 39) = 20; node_connect(1, 40) = 19;  % element 40 node_connect(2, 40) = 21; node_connect(1, 41) = 20;  % element 41 node_connect(2, 41) = 24; node_connect(1, 42) = 20;  % element 42 node_connect(2, 42) = 26; node_connect(1, 43) = 21;  % element 43 node_connect(2, 43) = 29; node_connect(1, 44) = 21;  % element 44 node_connect(2, 44) = 31; node_connect(1, 45) = 22;  % element 45 node_connect(2, 45) = 23; node_connect(1, 46) = 23;  % element 46 node_connect(2, 46) = 24; node_connect(1, 47) = 24;  % element 47 node_connect(2, 47) = 25; node_connect(1, 48) = 25;  % element 48 node_connect(2, 48) = 26; node_connect(1, 49) = 26;  % element 49 node_connect(2, 49) = 27; node_connect(1, 50) = 27;  % element 50 node_connect(2, 50) = 28; node_connect(1, 51) = 28;  % element 51 node_connect(2, 51) = 29; node_connect(1, 52) = 29;  % element 52 node_connect(2, 52) = 30; node_connect(1, 53) = 30;  % element 53 node_connect(2, 53) = 31; node_connect(1, 54) = 31;  % element 54 node_connect(2, 54) = 32; node_connect(1, 55) = 32;  % element 55 node_connect(2, 55) = 33; node_connect(1, 56) = 22;  % element 56 node_connect(2, 56) = 34; node_connect(1, 57) = 23;  % element 57 node_connect(2, 57) = 34; node_connect(1, 58) = 23;  % element 58 node_connect(2, 58) = 36; node_connect(1, 59) = 24;  % element 59 node_connect(2, 59) = 36; node_connect(1, 60) = 24;  % element 60 node_connect(2, 60) = 38; node_connect(1, 61) = 25;  % element 61 node_connect(2, 61) = 38; node_connect(1, 62) = 25;  % element 62 node_connect(2, 62) = 39; node_connect(1, 63) = 26;  % element 63 node_connect(2, 63) = 39; node_connect(1, 64) = 26;  % element 64 node_connect(2, 64) = 40; node_connect(1, 65) = 27;  % element 65 node_connect(2, 65) = 40; node_connect(1, 66) = 27;  % element 66 node_connect(2, 66) = 41; node_connect(1, 67) = 28;  % element 67 node_connect(2, 67) = 41; node_connect(1, 68) = 28;  % element 68 node_connect(2, 68) = 42; node_connect(1, 69) = 29;  % element 69 node_connect(2, 69) = 42; node_connect(1, 70) = 29;  % element 70 node_connect(2, 70) = 43; node_connect(1, 71) = 30;  % element 71 node_connect(2, 71) = 43; node_connect(1, 72) = 30;  % element 72 node_connect(2, 72) = 44; node_connect(1, 73) = 31;  % element 73 node_connect(2, 73) = 44; node_connect(1, 74) = 31;  % element 74 node_connect(2, 74) = 37; node_connect(1, 75) = 32;  % element 75 node_connect(2, 75) = 37; node_connect(1, 76) = 32;  % element 76 node_connect(2, 76) = 35; node_connect(1, 77) = 33;  % element 77 node_connect(2, 77) = 35; node_connect(1, 78) = 34;  % element 78 node_connect(2, 78) = 36; node_connect(1, 79) = 35;  % element 79 node_connect(2, 79) = 37; node_connect(1, 80) = 36;  % element 80 node_connect(2, 80) = 38; node_connect(1, 81) = 37;  % element 81 node_connect(2, 81) = 44; node_connect(1, 82) = 38;  % element 82 node_connect(2, 82) = 39; node_connect(1, 83) = 39;  % element 83 node_connect(2, 83) = 40; node_connect(1, 84) = 40;  % element 84 node_connect(2, 84) = 41; node_connect(1, 85) = 41;  % element 85 node_connect(2, 85) = 42; node_connect(1, 86) = 42;  % element 86 node_connect(2, 86) = 43; node_connect(1, 87) = 43;  % element 87 node_connect(2, 87) = 44; node_connect(1, 88) = 38;  % element 88 node_connect(2, 88) = 45; node_connect(1, 89) = 39;  % element 89 node_connect(2, 89) = 45; node_connect(1, 90) = 43;  % element 90 node_connect(2, 90) = 46; node_connect(1, 91) = 44;  % element 91 node_connect(2, 91) = 46;

figure(1) for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x(node_1),x(node_2)]; yy = [y(node_1),y(node_2)]; plot(xx,yy,'--') hold on  end E = 200e9; A = 4e-4; P = -1000; dof = 2*n_node; K=zeros(dof); R = zeros(dof,1); R(66) = P;  debc = [1,2,3,4]; ebcVals = zeros(length(debc),1);

for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); location = [x(node_1),y(node_1); x(node_2),y(node_2)]; lm = [ node_1*2-1, node_1*2, node_2*2-1, node_2*2 ]; k = PlaneTrussElement(E,A,location); K(lm,lm) = K(lm,lm) + k;  end [d, reactions] = NodalSoln(K, R, debc, ebcVals) results=[]; for i=1:n_elem node_1 = node_connect(1,i); node_2 = node_connect(2,i); location = [x(node_1),y(node_1); x(node_2),y(node_2)]; lm = [ node_1*2-1, node_1*2, node_2*2-1, node_2*2 ]; results = [results; PlaneTrussResults(E, A, ...      location, d(lm))]; end format short g  multiplier = 20; d = d*multiplier; for i = 1:n_node position_d(:,i) = [ position(1,i)+d(2*i-1), position(2,i)+d(2*i) ]; end for i = 1 : n_node; x(i) = scale * position_d(1,i); y(i) = scale * position_d(2,i); end

for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x(node_1),x(node_2)]; yy = [y(node_1),y(node_2)]; plot(xx,yy,'-black') hold on  end dof = 3; % Node coordinates n_node = 46;              % number of nodes n_elem = 91;              % number of elements total_dof = dof * n_node; % total dof of system position(:, 1) = [ 1.5; 0]; position(:, 2) = [ 4.53; 0]; position(:, 3) = [ 1.88; 1.18]; position(:, 4) = [ 3.03; 1.18]; position(:, 5) = [ 4.17; 1.18]; position(:, 6) = [ 2.2; 2.181]; position(:, 7) = [ 3.03; 2.181]; position(:, 8) = [ 3.85; 2.181]; position(:, 9) = [ 2.53; 3.181]; position(:, 10) = [ 3.53; 3.181]; position(:, 11) = [ 2.34; 3.681]; position(:, 12) = [ 3.03; 3.681]; position(:, 13) = [ 3.71; 3.681]; position(:, 14) = [ 2.16; 4.181]; position(:, 15) = [ 2.62; 4.181]; position(:, 16) = [ 3.44; 4.181]; position(:, 17) = [ 3.89; 4.181]; position(:, 18) = [ 2.36; 4.5]; position(:, 19) = [ 3.7; 4.5]; position(:, 20) = [ 1.8; 5.172]; position(:, 21) = [ 4.25; 5.172]; position(:, 22) = [ 0; 6]; position(:, 23) = [ 1.06; 6]; position(:, 24) = [ 1.5; 6]; position(:, 25) = [ 1.82; 6]; position(:, 26) = [ 2.22; 6]; position(:, 27) = [ 2.71; 6]; position(:, 28) = [ 3.34; 6]; position(:, 29) = [ 3.84; 6]; position(:, 30) = [ 4.23; 6]; position(:, 31) = [ 4.55; 6]; position(:, 32) = [ 5; 6]; position(:, 33) = [ 6.05; 6]; position(:, 34) = [ .8; 6.23]; position(:, 35) = [ 5.25; 6.23]; position(:, 36) = [ 1.24; 6.355]; position(:, 37) = [ 4.82; 6.355]; position(:, 38) = [ 1.5; 6.43]; position(:, 39) = [ 2; 6.43]; position(:, 40) = [ 2.48; 6.43]; position(:, 41) = [ 3.03; 6.43]; position(:, 42) = [ 3.58; 6.43]; position(:, 43) = [ 4.05; 6.43]; position(:, 44) = [ 4.55; 6.43]; position(:, 45) = [ 1.5; 7.43]; position(:, 46) = [ 4.55; 7.43]; scale = 8.0753701211306; for i = 1 : n_node; x(i) = scale * position(1,i); y(i) = scale * position(2,i); end node_connect(1, 1) = 1;  % element 1 node_connect(2, 1) = 3; node_connect(1, 2) = 1;  % element 2 node_connect(2, 2) = 4; node_connect(1, 3) = 3;  % element 5 node_connect(2, 3) = 4; node_connect(1, 4) = 4;  % element 3 node_connect(2, 4) = 2; node_connect(1, 5) = 4;  % element 6 node_connect(2, 5) = 5; node_connect(1, 6) = 2;  % element 4 node_connect(2, 6) = 5; node_connect(1, 7) = 3;  % element 8 node_connect(2, 7) = 7; node_connect(1, 8) = 5;  % element 9 node_connect(2, 8) = 7; node_connect(1, 9) = 3;  % element 7 node_connect(2, 9) = 6; node_connect(1, 10) = 5;  % element 10 node_connect(2, 10) = 8; node_connect(1, 11) = 6;  % element 11 node_connect(2, 11) = 7; node_connect(1, 12) = 7;  % element 12 node_connect(2, 12) = 8; node_connect(1, 13) = 6;  % element 13 node_connect(2, 13) = 9; node_connect(1, 14) = 7;  % element 14 node_connect(2, 14) = 9; node_connect(1, 15) = 7;  % element 15 node_connect(2, 15) = 10; node_connect(1, 16) = 8;  % element 16 node_connect(2, 16) = 10; node_connect(1, 17) = 9;  % element 17 node_connect(2, 17) = 10; node_connect(1, 18) = 9;  % element 18 node_connect(2, 18) = 11; node_connect(1, 19) = 9;  % element 19 node_connect(2, 19) = 12; node_connect(1, 20) = 10;  % element 20 node_connect(2, 20) = 12; node_connect(1, 21) = 10;  % element 21 node_connect(2, 21) = 13; node_connect(1, 22) = 11;  % element 22 node_connect(2, 22) = 12; node_connect(1, 23) = 12;  % element 23 node_connect(2, 23) = 13; node_connect(1, 24) = 11;  % element 24 node_connect(2, 24) = 14; node_connect(1, 25) = 11;  % element 25 node_connect(2, 25) = 15; node_connect(1, 26) = 12;  % element 26 node_connect(2, 26) = 15; node_connect(1, 27) = 12;  % element 27 node_connect(2, 27) = 16; node_connect(1, 28) = 13;  % element 28 node_connect(2, 28) = 16; node_connect(1, 29) = 13;  % element 29 node_connect(2, 29) = 17; node_connect(1, 30) = 14;  % element 30 node_connect(2, 30) = 15; node_connect(1, 31) = 15;  % element 31 node_connect(2, 31) = 16; node_connect(1, 32) = 16;  % element 32 node_connect(2, 32) = 17; node_connect(1, 33) = 14;  % element 33 node_connect(2, 33) = 20; node_connect(1, 34) = 14;  % element 34 node_connect(2, 34) = 18; node_connect(1, 35) = 15;  % element 35 node_connect(2, 35) = 18; node_connect(1, 36) = 16;  % element 36 node_connect(2, 36) = 19; node_connect(1, 37) = 17;  % element 37 node_connect(2, 37) = 19; node_connect(1, 38) = 17;  % element 38 node_connect(2, 38) = 21; node_connect(1, 39) = 18;  % element 39 node_connect(2, 39) = 20; node_connect(1, 40) = 19;  % element 40 node_connect(2, 40) = 21; node_connect(1, 41) = 20;  % element 41 node_connect(2, 41) = 24; node_connect(1, 42) = 20;  % element 42 node_connect(2, 42) = 26; node_connect(1, 43) = 21;  % element 43 node_connect(2, 43) = 29; node_connect(1, 44) = 21;  % element 44 node_connect(2, 44) = 31; node_connect(1, 45) = 22;  % element 45 node_connect(2, 45) = 23; node_connect(1, 46) = 23;  % element 46 node_connect(2, 46) = 24; node_connect(1, 47) = 24;  % element 47 node_connect(2, 47) = 25; node_connect(1, 48) = 25;  % element 48 node_connect(2, 48) = 26; node_connect(1, 49) = 26;  % element 49 node_connect(2, 49) = 27; node_connect(1, 50) = 27;  % element 50 node_connect(2, 50) = 28; node_connect(1, 51) = 28;  % element 51 node_connect(2, 51) = 29; node_connect(1, 52) = 29;  % element 52 node_connect(2, 52) = 30; node_connect(1, 53) = 30;  % element 53 node_connect(2, 53) = 31; node_connect(1, 54) = 31;  % element 54 node_connect(2, 54) = 32; node_connect(1, 55) = 32;  % element 55 node_connect(2, 55) = 33; node_connect(1, 56) = 22;  % element 56 node_connect(2, 56) = 34; node_connect(1, 57) = 23;  % element 57 node_connect(2, 57) = 34; node_connect(1, 58) = 23;  % element 58 node_connect(2, 58) = 36; node_connect(1, 59) = 24;  % element 59 node_connect(2, 59) = 36; node_connect(1, 60) = 24;  % element 60 node_connect(2, 60) = 38; node_connect(1, 61) = 25;  % element 61 node_connect(2, 61) = 38; node_connect(1, 62) = 25;  % element 62 node_connect(2, 62) = 39; node_connect(1, 63) = 26;  % element 63 node_connect(2, 63) = 39; node_connect(1, 64) = 26;  % element 64 node_connect(2, 64) = 40; node_connect(1, 65) = 27;  % element 65 node_connect(2, 65) = 40; node_connect(1, 66) = 27;  % element 66 node_connect(2, 66) = 41; node_connect(1, 67) = 28;  % element 67 node_connect(2, 67) = 41; node_connect(1, 68) = 28;  % element 68 node_connect(2, 68) = 42; node_connect(1, 69) = 29;  % element 69 node_connect(2, 69) = 42; node_connect(1, 70) = 29;  % element 70 node_connect(2, 70) = 43; node_connect(1, 71) = 30;  % element 71 node_connect(2, 71) = 43; node_connect(1, 72) = 30;  % element 72 node_connect(2, 72) = 44; node_connect(1, 73) = 31;  % element 73 node_connect(2, 73) = 44; node_connect(1, 74) = 31;  % element 74 node_connect(2, 74) = 37; node_connect(1, 75) = 32;  % element 75 node_connect(2, 75) = 37; node_connect(1, 76) = 32;  % element 76 node_connect(2, 76) = 35; node_connect(1, 77) = 33;  % element 77 node_connect(2, 77) = 35; node_connect(1, 78) = 34;  % element 78 node_connect(2, 78) = 36; node_connect(1, 79) = 35;  % element 79 node_connect(2, 79) = 37; node_connect(1, 80) = 36;  % element 80 node_connect(2, 80) = 38; node_connect(1, 81) = 37;  % element 81 node_connect(2, 81) = 44; node_connect(1, 82) = 38;  % element 82 node_connect(2, 82) = 39; node_connect(1, 83) = 39;  % element 83 node_connect(2, 83) = 40; node_connect(1, 84) = 40;  % element 84 node_connect(2, 84) = 41; node_connect(1, 85) = 41;  % element 85 node_connect(2, 85) = 42; node_connect(1, 86) = 42;  % element 86 node_connect(2, 86) = 43; node_connect(1, 87) = 43;  % element 87 node_connect(2, 87) = 44; node_connect(1, 88) = 38;  % element 88 node_connect(2, 88) = 45; node_connect(1, 89) = 39;  % element 89 node_connect(2, 89) = 45; node_connect(1, 90) = 43;  % element 90 node_connect(2, 90) = 46; node_connect(1, 91) = 44;  % element 91 node_connect(2, 91) = 46; % Undeformed plot for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x(node_1),x(node_2)]; yy = [y(node_1),y(node_2)]; plot(xx,yy,':') hold on  end % Frame analysis E = 200e9; A = 4e-4; P = -1000; I = 1.3333e-8; K = zeros(dof*n_node); R = zeros(dof*n_node,1); R(66) = P;  debc = [1,2,3,4,5,6]; ebcVals = zeros(length(debc),1); % Assemble global matrix for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); coord = [x(node_1),y(node_1); x(node_2),y(node_2)]; lm = [ node_1*3-2, node_1*3-1, node_1*3, node_2*3-2, node_2*3-1, node_2*3 ]; [ke rq] = PlaneFrameElement(E,I,A,0,0,coord); K(lm, lm) = K(lm, lm) + ke; R(lm) = R(lm) + rq; end % Compute reactions [d, reactions] = NodalSoln(K, R, debc, ebcVals) fa = []; bma = []; Va = []; for i = 1:n_elem node_1 = node_connect(1,i); node_2 = node_connect(2,i); coord = [x(node_1),y(node_1); x(node_2),y(node_2)]; lm = [ node_1*3-2, node_1*3-1 node_1*3, node_2*3-2, node_2*3-1, node_2*3 ]; [f, bm, V] = PlaneFrameResults(E,I,A,0,0,location,d(lm)); fa = [fa; f]; bma = [bma; bm]; Va = [Va; V]; end % Highest bending moment moment = 0; for i = 1:n_elem*2 if abs(bma(i,3)) > moment moment = abs(bma(i,3)); momentnode = i;     end end moment if mod(momentnode,2) == 0 momentelement = momentnode/2; else momentelement = (momentnode+1)/2; end momentelement % Shear stress element shearstress = 0; for i = 1:n_elem*2 if abs(Va(i,3)) > shearstress shearstress = abs(Va(i,3)); shearstressnode = i;     end end shearstress if mod(momentnode,2) == 0 shearstresselement = shearstressnode/2; else shearstresselement = (shearstressnode+1)/2; end shearstresselement

multiplier = 20; d = d*multiplier; % Deformed positions for i = 1:n_node position_d(:,i) = [ position(1,i)+d(3*i-2), position(2,i)+d(3*i-1) ]; end for i = 1 : n_node; x(i) = scale * position_d(1,i); y(i) = scale * position_d(2,i); end for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x(node_1),x(node_2)]; yy = [y(node_1),y(node_2)]; axis([-5 53 -5 70]) plot(xx,yy,'--') hold on  end title('Electric Pylon in Deformed and Undeformed State') xlabel('x') ylabel('y') rho = 7.8e3; M = zeros(dof); for i = 1 : n_node; x(i) = scale * position(1,i); y(i) = scale * position(2,i); end for i = 1 : n_elem node_1 = node_connect(1,i); node_2 = node_connect(2,i); location = [x(node_1),y(node_1); x(node_2),y(node_2)]; L(i) = PlaneTrussLength(location); m(i) = rho*L(i)*A; end i = 1; for j = 1 : n_node mass(j) = 0; end for j = 1 : n_node for k = 1 : n_elem if node_connect(1,k) == j              mass(j) = mass(j) + m(k)/2; elseif node_connect(2,k) == j                  local_elems(j,i) = k;                   mass(j) = mass(j) + m(k)/2; end end end for i = 1 : 46 M(3*i-2,3*i-2) = mass(i); M(3*i-1,3*i-1) = mass(i); M(3*i,3*i) = 0; end M;  Mf = ReduceMassMatrix(M, R, debc, ebcVals); Kf = ReduceStiffnessMatrix(K, R, debc, ebcVals); A = pinv(Mf)*Kf; [V2,D2] = eig(A); eig = [999999999999,9999999999999,9999999999999]; for i = 1 : 132 if (eig(1) > D2(i,i) && D2(i,i) ~= 0) eig(1) = D2(i,i); column(1) = i;          end end for i = 1 : 132 if (eig(2) > D2(i,i) && D2(i,i) > eig(1) && D2(i,i) ~= 0) eig(2) = D2(i,i); column(2) = i;      end end for i = 1 : 132 if (eig(3) > D2(i,i) && D2(i,i) > eig(2) && D2(i,i) ~= 0) eig(3) = D2(i,i); column(3) = i;      end end eig % Period Calculation for i = 1 : 3 omega(i) = sqrt(eig(i)); T(i) = 2*pi/omega(i); end T % Eigenvalue plots % First eigenvalue plot figure(2) for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x(node_1),x(node_2)]; yy = [y(node_1),y(node_2)]; plot(xx,yy,'--') hold on  end

Z = zeros(6,132); V2 = [Z;V2]; for i = 1:n_node position_d(:,i) = [ scale*position(1,i) + multiplier*V2(3*i-2,column(1)), scale*position(2,i) + multiplier*V2(3*i-1,column(1))  ]; end for i = 1 : n_node dx(i) = position_d(1,i); dy(i) = position_d(2,i); end for i = 1 : n_elem node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [dx(node_1),dx(node_2)]; yy = [dy(node_1),dy(node_2)]; plot(xx,yy,'-black') hold on  end title('Effects of the Smallest Eigenvalue') xlabel('x') ylabel('y') % Second eigenvalue plot

figure(3)

for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x(node_1),x(node_2)]; yy = [y(node_1),y(node_2)]; axis([-5 53 -5 70]) plot(xx,yy,'--') hold on  end for i = 1:n_node position_d(:,i) = [ scale*position(1,i) + multiplier*V2(3*i-2,column(2)), scale*position(2,i) + multiplier*V2(3*i-1,column(2)) ]; end for i = 1 : n_node dx(i) = position_d(1,i); dy(i) = position_d(2,i); end for i = 1 : n_elem node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [dx(node_1),dx(node_2)]; yy = [dy(node_1),dy(node_2)]; axis([-5 53 -5 70]) plot(xx,yy,'-black') hold on  end title('Effects of the Second Smallest Eigenvalue') xlabel('x') ylabel('y') % plot eigenvectors for 3rd smallest eigenvalue multiplier = 10; figure(4) for i = 1 : n_elem; node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [x(node_1),x(node_2)]; yy = [y(node_1),y(node_2)]; plot(xx,yy,'--') hold on  end for i = 1:n_node position_d(:,i) = [ scale*position(1,i) + multiplier*V2(3*i-2,column(3)), scale*position(2,i) + multiplier*V2(3*i-1,column(3)) ]; end for i = 1 : n_node dx(i) = position_d(1,i); dy(i) = position_d(2,i); end for i = 1 : n_elem node_1 = node_connect(1,i); node_2 = node_connect(2,i); xx = [dx(node_1),dx(node_2)]; yy = [dy(node_1),dy(node_2)]; plot(xx,yy,'-black') hold on  end title('Effects of the Third Smallest Eigenvalue') xlabel('x') ylabel('y')

Results
The displacement results were:

d = 0           0            0            0  -0.00019023   0.00017465   -0.0001277 -1.4074e-006 -0.00016006  -0.0002967  -0.00012671   0.00032602  -0.00012798    -0.000128  -0.00012792  -0.00054327   0.00044932   0.00030901   0.00046381  -0.00061057     0.001017   0.00064583    0.0010325  -0.00019427    0.0010502  -0.00097027    0.0017167    0.0010338    0.0017162   0.00023259    0.0016833  -0.00064959     0.001683   -0.0013293    0.0023452   0.00064079    0.0021863   -0.0010297    0.0047122    0.0023902    0.0039576   -0.0024165    0.0063605    0.0057328    0.0063605    0.0037877    0.0063605    0.0029803    0.0063614     0.002384    0.0063657    0.0016463    0.0063711   0.00086194    0.0063182  -0.00024268    0.0062289   -0.0013182    0.0061363   -0.0023672    0.0060413   -0.0033231    0.0058839   -0.0051278    0.0055152    -0.012587    0.0067826    0.0042648     0.006996   -0.0063359     0.007012    0.0034574    0.0074006   -0.0043591    0.0071505    0.0029772    0.0071478    0.0020508    0.0071421    0.0012278     0.007182   0.00032026    0.0072816  -0.00073249    0.0074073   -0.0018633    0.0075643   -0.0033872    0.0090005    0.0029772     0.010455   -0.0033872

The maximum bending moment is:

moment = 472.49

This moment occurs at:

momentelement = 45

The maximum transverse shear stress is:

shearstress = 85.852

This stress occurs at:

shearstresselement =  45

Below is a plot of the undeformed (blue) and deformed (black) pylon structure:

Effects of Eigenvectors
The lowest eigenvalues and their corresponding oscillation periods are:

eig =  132.86       2471.2       2942.6

T =  0.54511      0.12639      0.11583

Plots of the effects of the three lowest eigenvalues are shown below. The blue dashed line represents the undeformed structure while the solid black line represents the deformed structure.

Statically Determinant
The final step in this problem was to determine whether the structure can be solved using statics. Since there are only two fixed points and four bars for which reactions must be computed, it can be determined that this problem is statically indeterminate.

title: User:Eml4500.f08.bottle.ranto/HW7


title: File:Lukas3.jpg


text:
User:Herrific

I'm as yet completely unfamiliar with all this, so I'm just going to experiment a little and make this a decent page as soon as I'm able. Though I'm a native Dutch speaker, I'll put this all in English - much easier.

title: User:Herrific


title: File:Lukas4.jpg


text:
User:Eas4200c.f08.WIKI.E/hw7 other days

Mtg 39, December 1 2008
Solving P2: For each cell,

-Follow path $$S_i$$

-Find the equilibrium of each stringer on this path: There are two ways of solving

1) Complete method FBD as seen before

2) Consequence of the first method:

$$\tilde{q}_{j6}=\tilde{q}_{2j}-\tilde{q}_{j5}-\tilde{q}_{j8}+\tilde{q}^{j}$$

In this example, $$\tilde{q}_{6j}$$ is now going in the opposite direction as previous but the answer will remain the same as long as the conventions stay consistent.

$$-\tilde{q}_{6j}=\tilde{q}_{2j}-\tilde{q}_{j5}-\tilde{q}_{j8}+\tilde{q}^{j}$$

Note: What if we cut cell walls such that one stringer was isolated?

$$\tilde{q}_{31}=\tilde{q}_{23}-\tilde{q}_{34}+\tilde{q}^{3}$$

This equation says that $$\tilde{q}^{3}= 0$$ which is not true. Therefore you cannot isolate the stinger in this way because it results in an answer that is not valid.

Trying to make different cuts to isolate stringer 2 will result in the same false answer.

$$\tilde{q}_{31}=\tilde{q}_{12}-\tilde{q}_{24}+\tilde{q}^{2}$$

$$\tilde{q}^{2}= 0$$

title: User:Eas4200c.f08.WIKI.E/hw7 other days


text:
User:Eas4200c.f08.aeris.guan/webctvswiki

E-learning versus MediaWiki
From previous experience using E-learning, once the server gets busy, the entire system is slow. Using Wikiversity and MediaWiki, the traffic of the server never seems to overwhelm the system and cause slow loading and sluggish performance. This increases the accessibility and reliability of MediaWiki. E-learning's information is only available for a semester and then whatever is posted is taken off. On Wikiversity, the information available on it is essentially ever-lasting so long as the user and Wikiversity continues to exist. Using E-learning, the user interface of the system was more cluttered and unorganized than Wikiversity. I believe that Wikiversity has a more user-friendly system. Learning to use MediaWiki was not difficult although typing in the mathematical formulas were very tedious. As for listing MediaWiki experience on a resume, this is a very minor thing. The MediaWiki software is not difficult to use. MATLAB experience, C++ programming experience or web page design experience are far better things to have and list on a resume. Also, a positive aspect of the wiki system is that anyone can edit the information provided by the wiki pages. This allows for universal editing, and accessing. MediaWiki allows for text, images, equations, video, and audio to be displayed on one page all at once, available for immediate use. E-Learning presents the media in file or link form. Text, equations and some images are commonly presented as Word documents on E-Learning. Video, audio and some images are presented as separate files, or as links to those files. Both E-Learning and MediaWiki can present any media type; they just differ on how it is presented. E-Learning has its advantages by presenting the material as files. Printing documents is simpler when they are already in document form, as opposed to trying to print a webpage. Also, if any of the video or audio files were wanted to be on an mp3 player or some other portable device, they can be moved quickly and simply with E-Learning’s file format.


text: File:Arc length.png

title: File:Arc length.png


text:
User:Eml4500.f08.team.Bengtson/teamHWStaticallydeterminate

The stability of the Electric Pylon
How does one find if something is statically indeterminate? When the amount of equations is not sufficient for finding all the unknowns of the system. $$   * \sum \vec F = 0$$ : the vectorial sum of the forces acting on the body equals zero. This translates to

Σ H = 0: the sum of the horizontal components of the forces equals zero; Σ V = 0: the sum of the vertical components of forces equals zero;

$$\sum \vec M = 0$$ : the sum of the moments (about an arbitrary point) of all forces equals zero. Reference used

If we look at the electric pylon we find that it has two nodes but four beams coming out of the nodes. Thus the system is not made of two force beams at the reaction (constrained) sites and the amount of equations is less than the unknowns.

Also, as the system is a frame system, the rotational displacements cause bending moments within the beams.

In summary, each of the four beams at the constrained sites has an axial force and a bending moment associated with it. This makes a total of 8 unknowns, and there are only 3 equations to solve for these unknowns using the statics method.

Thus the electric pylon is statically indeterminate.


text: File:Twobarzoom.jpg

title: File:Twobarzoom.jpg


title: File:Twobarhw7.jpg


text:
User:Eml4500.f08.delta 6.guzman/Evaluation of Wikiversity vs E-learning

Wikiversity: Pros

 * Reports can be easily edited by any group member.
 * Dr. VuQuoc insists in using it because he strongly believes Wikiversity would be greatly accepted in the future.

Wikiversity: Cons

 * Professionals often discourage the use of Wikipedia as a valid source of information.
 * The quality of the work done does not equal the amount of time spent on Wikiversity due to the unfamiliarity with the system.
 * The grade for each team member is given in proportion to their Wiki contributions instead of the equal distribution that would be given to any other group effort.
 * Wikiversity content can be changed by any person with a valid IP address.
 * The quality of work decreases since any other member from another group can observe the Wiki report and just copy its contents as its own.

E-learning: Pros

 * System provided just to the University of Florida students.
 * Secured system that can only be accessed with a valid Gator username and password.
 * Members from other groups can not observe/modify the reports. Only the group members, professors and TA are able to.
 * The system provides a feature that compares reports/essays to any available information online making cheating and copy right infringements almost impossible.
 * The system provides the chat feature which allows students to contribute to other group members without meetings in person facilitating and improving the quality of the reports.
 * Students are familiar with E-learning making the time spent in projects truly reflect the quality of the work.
 * Program usage of Word, Excel and Power Point are encouraged by E-learning since these programs are greatly accepted in the industry.
 * The grade is equally distributed since there was just one report presented by the group instead of having a grade count by the number of collaborations made in the report.
 * E-learning motivates confidentiality since all the other groups’ work is kept private just like in real life project.
 * Students actually learn the material since they do not have to spent time to figure out unfamiliar commands in E-learning.

E-learning: Cons

 * Editing one report is usually done by one group member instead of all the collaborating members.

Conclusion

I honestly believe that Wikiversity and E-learning can not be compared to each other. The two systems have different purposes. While Wikiversity is intended for easy learning the E-learning system is intended for students who are part of an institution and this system reflects the principles of confidentiality and integrity that the University of Florida stands for. The online editing feature of Wikiversity is the only disadvantage that I believe is holding back the E-learning system. Once this feature is available there won’t be any reason to use Wikiversity. I do enjoy E-learning more because it reflects many of the aspects that I will experience when I get out of college. The use of programs widely accepted is encourage same as the privacy and ethics principles I have learnt while completing my degree. Also the amount of time spent on the reports is truly reflected when using the E-learning system. For these reasons I believe that the E-learning system is a better approach for the college classroom.


text:
User:Eas4200c.f08.gator.edwards/Wk7 Matlab

The objective of this addition to the MATLAB code is to determine the shear flow in the airfoil. Part 1 deals with a single-cell airfoil while Part 2 addresses the three-cell airfoil.

This segment was added to perform buckling analysis on the airfoil.


text:
User:Eml4500.f08/HW report table/The best of HW7

 Under construction; not final (but close to being stable after a couple of weeks). The intention here is to document the best features in any HW report for the readers (including you); if you see excellent features in any HW report (including your team's) that I may have missed noticing, don't hesitate to let me know. I don't have time to read all HW reports in detail. In fact, due to time constraint, I only selectively looked at a few important features. On the other hand, I also added some annotations (e.g., related to errors) in the HW reports that I looked through for the benefit of everyone. Eml4500.f08 21:26, 9 December 2008 (UTC)

 * classnotes graded over 100%
 * 2-D frame elements (truss + beam)
 * 2-D frame model problem
 * FBDs
 * element dofs: displacements, rotations
 * assembly of element stiffness matrices into global stiffness matrix
 * element stiffness matrix $$\displaystyle \tilde \mathbf k ^{(e)}$$ in local coordinates
 * dimensional analysis of element stiffness coefficients $$\displaystyle \tilde k ^{(e)}_{ij}$$
 * transformation of FD relation from local coordinates to global coordinates
 * PVW for elastodynamics of beams
 * stiffness term
 * shape functions for truss and for beam
 * true deformed shape of a frame element (truss + beam)
 * disp components in global coordinates for a point on a frame element in terms of element dofs
 * dimensional analysis of shape functions and overall relation
 * derivation of beam shape functions
 * elastodynamics of trusses, frames, 2-D, 3-D elasticity
 * discrete equations of motion (MDOF; coupled ODEs) and initial conditions
 * unforced vibration problem
 * eigenvalue problem: derivation
 * generalized eigenvalue problem
 * standard eigenvalue problem
 * mass orthogonality property
 * stiffness orthogonality property
 * decouple system of equations of motion into uncoupled ODEs
 * recommended software to improve productivity. Most useful: Team Bike
 * writing wiki articles
 * writing latex equations
 * drawing figures
 * Mediawiki (on Wikiversity) vs. WebCT (on E-learning at UF): To set the proper context for this section, I included below my original e-mail requesting a team consensus on a comparison between Mediawiki vs. WebCT. Best team-consensus comparison: Team Bike. NOTE: Team Delta_6 (with annotations), parallel non-cooperative individual work and comparison.  General comment: Non-cooperative work. Some teams even admit that they never met together as instructed per cooperative learning techniques, with each team member taking turn to work on a lecture.  Team Aero (with annotations): "the scope of collaboration did not extend as far as was perhaps intended. In general, the assignment was divided based on lecture days and each person made their own posts with little editing." There was not much of a cooperation. Here is a most-telling example.  The result is often a mosaic of incoherent pieces put together, without a flowing story.  Absenteism and workload. If each student (in a typical team of six) did 1/6 of the work, took care of one lecture once every two weeks, but received 34% credit for the course grade, and yet did not attend all lectures, it is not clear why some of these students said that there was too much work, and what they would use the rest of their freed-up time for !?  At least, Mediawiki allowed some students to free up their lecture attendance.  Young Einstein was famous for skipping his classes (for thinking about physics), and learned from the math notes of his friend, but he was one in a billion... It would be a wild stretch of imagination to think that we had a Young Einstein in the class... Students who regularly attended lectures would do better in exams than they potentially could otherwise, regardless of their own level.  Lecture notes taking seems to be a lost art with in the digital age of powerpoint presentations.  While many students continued to take good notes (and even scanned them to e-mail out to their teammates who did not attend lectures), many others just relied either on their teammates' notes, the textbook (some lectures cannot be found in the textbook), or the digital photos they took of my transparencies (hand-written in real time in class) using their cell phones, even though I mentioned in class that it was not possible to do math or mechanics without writing, and did referred to the effect of writing on remembering that I read in Carl Sagan's The Dragons of Eden.
 * collaboration
 * parallel team work across the internet
 * math equations
 * incorporation other media (video, audio)
 * accessibility

EAS 4200C, The best of HW7
 * matlab problem graded over 100%
 * 2-bar truss system
 * uniform bar elements
 * tapered bar elements
 * comparison of deformed shapes
 * 2-element frame system
 * boundary conditions: clamped, hinged
 * plot of complete (true) deformed shape
 * comparison with 2-bar truss system
 * electric pylon problem
 * frame model (instead of truss model)
 * finite element analysis
 * plot undeformed and deformed shape
 * compare to deformed shape of truss model
 * frame element with highest nodal bending moment
 * frame element with highest transverse shear force
 * frame model statically determinate?
 * compute reactions from frame model
 * compare to reactions from truss model
 * vibration analysis
 * lumped mass matrix
 * generalized eigenvalue problem
 * lowest three eigenpairs
 * compare with 3 lowest eigenvalues of truss system
 * Results: eigenvalue (period in seconds)
 * Team Bike 132.86 (0.54511 s), 2471.2 (0.12639 s), 2942.6 (0.11583 s); very close to those of the truss model; see The best of HW6; at least 3 teams got the same results.
 * Team ATeam(Sean) (0.54511 s), (0.12639 s), (0.11583 s); too close to those of the truss model (0.54654 s), (0.12647 s), (0.11587 s); at least 3 teams got the same results.
 * Team Delta_6 (with annotations) (0.54511 s), (0.12639 s), (0.11583 s); same comments.
 * plot lowest three eigenvectors: animate these 3 modes. Best animation: Team Bike (gif file) Team ATeam(Sean) (video file)


text:
User:Eml4500.f08.team.foskey.ckf/hw7 2barframe

 Note: A detailed explanation of the code was added. Also, extra functions used to solve the problem were also added.

Jin Yu Guan Eml4500.f08.team.guan 18:19, 10 December 2008 (UTC)

Two bar Frame system

Problem
This problem solves a two bar frame system.

MATLAB Code
The code starts out by computing the stiffness matrices of each of the frame elements using the function PlaneFrameElement. Then, it computes the results of the loading and returns the displacement and reactions of the loading by utilizing the PlaneFrameResults function. Upon obtaining the displacements, the function customized function segmentedcory.m was called to calculating the transverse and axial displacements of 50 discrete portions of each frame element. In doing so, the x and y global coordinates of each small displaced discrete element were then returned. Using two for loops (one for an element, and one for plotting the frame), the frames were then plotted according to the displacements obtained from segmentedcory.m.     It is obvious to see that the deformed frames are not linear. This is due to the fact that frames are subject to transverse and axial loading and thus suffer from transverse and axial displacement. Thus, frames are designed to carry transverse and axial loading as opposed to trusses in which only carry axial loading.

This code uses the follwing function, in addition to the functions used in the truss problem

The above code works basically like every other truss code, with the exception that it now takes into account rotational displacements and bending moments.

The function segments an element into pieces and helps plot the bent frame bars.

The PlaneFrameElement.m function is below.

The PlaneFrameResults.m function is below.

Output results
Data for Frame System

Figure
The following figure is generated by the above code


text: File:Twobarframeteam.png

title: File:Twobarframeteam.png