Report essay how to increase of disciplinary problems in the school free essay sample

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How to Write a Proposal

How to Write a Proposal How to Write a Proposal How to Write a Proposal By Mark Nichol Writing a proposal is similar to but not exactly the same as crafting a persuasive essay or producing a report. Here are suggestions for developing a proposal, including some pertinent to its specific purpose. 1. A proposal should define a problem and describe a solution that will persuade busy, thrifty, skeptical readers to support it. 2. Employ facts, not opinions, to bolster the argument for approval. Research similar plans or projects and cite them, emphasizing their successes and/or how your proposal resolves the weaknesses, omissions, or mistaken priorities apparent in them. 3. Analyze your plan or project, demonstrating possible outcomes. If possible, model a small-scale version of the plan or project, report on the results, and extrapolate how the full-scale plan or project will turn out based on the test. 4. Any discussion of financial or other resources should be conducted carefully and should present a realistic picture of the expense required. 5. Be meticulous in writing, editing, and design of the proposal. Revise as necessary to make it clear and concise, ask others to critique and edit it, and make sure the presentation is attractive and engaging as well as well organized and helpful. A proposal should include the following elements: Executive Summary: State the rationale for putting the proposal into effect, and summarize the proposal. (This allows a decision maker to quickly get the gist of the proposal, hence the name.) Statement of Need: Detail why the plan or project the proposal recommends is necessary. Project Description: Explain specifics of the plan or project, and how it will go into effect and how it will be evaluated. Budget Analysis: Provide and explain how the plan or project will be financed and categorize and annotate operating expenses. Organization Details: If the proposal is being submitted to an outside party, provide information about the beneficiary organization, including its mission, its stakeholders and who its serves, and the scope of its programs and services. Conclusion: Summarize the proposal’s main points. Want to improve your English in five minutes a day? Get a subscription and start receiving our writing tips and exercises daily! Keep learning! Browse the Business Writing category, check our popular posts, or choose a related post below:Inquire vs EnquireUsing the Active Voice to Strengthen Your WritingHonorary vs. Honourary

The Black and the Blues Research Paper Example | Topics and Well Written Essays - 1000 words

The Black and the Blues - Research Paper Example † which depicts the distinct difference between their personality and insights can be considered as the focal points of the story. Truly in the trajectory of life, people are pushed and weathered with numerous battles either economically, personally, socially, or conflicts within the family. Through these most who prefer to stay on the safe side, surrender and be submissive to what is dictated by the society as they deem that this is a more practical and rational path to take. For the other free spirits, they opt to follow their dreams no matter how surreal the idea may seem yet for them it is something worth believing and following. Baldwin’s story though written in 1957 depicted a crucial message on bitter realities within the society which strongly affected him and the black community of his time and still somewhat remained unchanged in some parts of the world today. The story focused on the narration of an algebra teacher in Harlem who struggled to earn and maintain a middle class status and keep his wife and kids on a safe part of the society; and his antonym relationship with his brother Sonny, a man who elected to navigate the road less traveled and pursued to become a jazz pianist though ended up behind bars from possession and addiction to heroin. Trimmer and Urbana (p.165) discussed that the differences between the brothers represent the division within the black community where the narrator represents the middle class men who strives to live the American dream and aspires for an improved standard of living in the United States; while the other group, as represented by Sonny, are those who just accept the fact that they will forever be locked up from the privileges the white people savor. Take note that the setting happened on a post war as the narrator cited his military service to the US government. It is an era where people, particularly the blacks, are rallying for equality and their war against racism. The narrator tried to understan d his brother and struggled to reach out to find answers to his questions though remained blindly to accept the truthful answers. The story reminds us on the importance of listening and true understanding in every form of relationship and in using these rather than coercion or rigid authority. There was a scene in the story, after their mother’s funeral, when they discussed the future of the teenage Sonny. There was tension between the brothers as Sonny declared that he wants to be a jazz musician while the older brother sees that dream to be irrational. This scene together with the one when he offers his wife Isabel’s place to be Sonny’s home manifests the narrator’s sense of responsibility to his younger brother’s welfare. Throughout the story, issues on racism and inequality were strongly reflected. During his time there was strong discrimination for the black people in the racist America. On strong reflection of Baldwin’s personal strugg le is the part of the story where the narrator recollects the death of their uncle who was driven over by a group of drunken white Americans and recklessly left the carcass on the street. Depression from racism was not confined to the character’s family but it extends to the entire black community during that time (Reilly 56). Baldwin succeeded in

Managing Case Study on Colgate Co Example | Topics and Well Written Essays - 2500 words

Managing on Colgate Co - Case Study Example The report will also outline some of the competitors in this market that pose a challenge to the operations of Colgate in the oral and dental health care industry. 1.1 Background information about Colgate According to its official website, Colgate was established in 1806 and it was founded by William Colgate. To date, Colgate is regarded as the world’s leader in oral and dental health care and it offers products such as toothbrushes as well as toothpastes. Colgate operates in more than 200 countries though it is based in the US. According to Berger (2010), Colgate has a market share of 44,4 % in the world’s toothpaste industry and it operates in more than 200 countries. The company posted a profit of $3, 615 in the financial year of 2010. The brand is very popular in different countries. For instance, Colgate enjoys a lot of positive consumer response in large emerging economies such as India and Brazil. The company also offers different sizes of toothpaste to different customers. 2.0 Situational analysis of the marketing environment (SWOT) Basically, â€Å"SWOT analysis is a useful instrument that is used to help managers to identify internal strengths and weaknesses of a business and external opportunities and threats facing it,† (Strydom, 2004, p. 31). SWOT stands for strengths (S), weaknesses (W) while on the other hand the external environmental factors are regarded as either opportunities (O) or threats (T) and these factors help the company determine its current position in the market in order to map an effective strategy to realize its intended goals. The table below shows the SWOT analysis for Colgate. Table 1: SWOT analysis for Colgate Strengths The company has good human resources management practices which entails that it has a dedicated workforce The company also uses effective marketing strategies in selling its products Innovation is prioritized in terms of product development and it incorporates dental professionals in its o perations Colgate is a leader in dental hygiene The company operates in many countries across the globe Weaknesses The major weakness of this company is that it spends less on media advertising compared to its rival competitors Opportunities The major opportunity of the company is based on the aspect related to growing dental hygiene market across the whole world. The emerging markets also provide a good opportunity for Colgate Threats The major threat faced by Colgate is related to competition from other rival competitors. There are chances of substitute products being offered by other companies and this affects price of products offered. The other threat is related to currency fluctuations in different countries where Colgate operates Basically, the managers in a company should take advantage of the company’s strengths and opportunities while they put measures in place that are designed to minimise the weaknesses and threats. This section explains in detail the contents of the

Family Study Essay Example | Topics and Well Written Essays - 500 words

Family Study - Essay Example Immediately after a wedding, couples would show substantial commitment towards these vows. However, circumstances may change along the journey of a marriage. One partner may develop negative habits and even abusive behavior towards the other partner. In addition, one partner may experience deteriorating health; hence dimming the hope of married couples. Today, such worsening situations often call for divorce, which in this case constitutes breaking of the wedding vows. Personally, I believe that divorce is not necessary. Wedding vows are synonymous to oaths of duty. Soldiers swear to protect their nation even if protection calls for sacrifice of one’s life. Similarly, marriage vows, especially the one saying, ‘for better or for worse’ should possess its traditional meaning throughout the relationship. The willingness to commit oneself into a marriage should not depend on prevailing circumstances. I believe each of the committed partners is cognizant of the fact that the world is not perfect. Individuals have strengths and weaknesses. Also; love, companionship and forgiveness are the essential pillars of marriage, which are explicitly expressed in form of wedding vows (Keller, 32). In this regard, I am of the opinion that marriage relationships should be permanent. Inasmuch as I discredit divorce in marriages, I believe that crossing certain boundaries necessitates separation between couples. Morally, most societies and religions regard marriage as a sacred covenant. Married couple must remain loyal and faithful to one another. However, these moral platforms assert that lack of faithfulness cannot be tolerated. Nowadays, couples dissolve their marriages for slights reasons like financial insecurity, psychological dissatisfaction and for selfish pursuit of self-centered pleasures (Keller, 34). Such slight reasons are not

Earthquake Simulation for Buildings

Earthquake Simulation for Buildings Abstract Earthquake is an independent natural phenomenon of vibration of the ground which can become dangerous mainly when it is considered in relation with structures. Earthquakes can be very weak, without even realizing them but (they) can also be strong enough to result serious damages to buildings which can lead to injures or even loss of human lives. In order to avoid any structural damage the legislation sets conditions on the building design. For that purpose, Eurocode 8 is established in European countries and sets up all the appropriate criteria and measures for the design of buildings for earthquake resistance (Eurocode 8 is established in Europe and suggests 4 different methods of analysis.) In this project the response of eight buildings is examined (investigated) under seismic excitation. Firstly, is examined the case of four buildings (1 storey, 2 storey, 3 storey and 4 storey) where all the storeys are facsimile (replica). Afterwards, is examined the case of four buildings (aga in 1-4 storeys) where while the storeys of each building are increased, the mass, the stiffness and the height of each floor are decreased. Both the lateral method of analysis and the modal response spectrum analysis are used as recommended by EC8 to calculate the inter-storey drifts, the total shear forces and the overturning moments at the base of each building. The results are plotted and compared so that useful outcomes can be obtained. 1. Introduction One of the most frightening and destructive phenomena of nature is a severe earthquake and its terrible aftereffects especially when they are associated with structures. An earthquake is a sudden movement of the Earth, caused by the abrupt release of strain that has accumulated over a long time. Earthquake intensity and magnitude are the most common used parameters in order to understand and compare different earthquake events.( Ά° are the most common parameters used to appreciate and compare.) In recent years have been giving increasing attention to the design of buildings for earthquake resistance. Specific (particular) legislation is (have been) established to make structures able to resist at any seismic excitation. In Europe, Eurocode 8 explains how to make buildings able to resist to earthquakes, and recommends the use of linear and non-linear methods for the seismic design of the buildings Simple structures can be modelled either as equivalent single degree of freedom systems (SDOF) or as a combination of SDOF systems. In this project 8 different buildings with a variation either on the number of storeys or on their characteristics are simulated as a combination of SDOF systems for which the mode shapes and their corresponding eigenfrequencies and periods are calculated. Afterwards the fundamental frequency is obtained for each case and the elastic design is used in order to obtain the base shear forces and the overturning moments. (INELASTIC DESIGN AND LATERAL FORCE METHOD) 2. Literature review 2.1 Introduction to earthquake engineering Definition and earthquake derivation or generation or creation or production or formation or genesis The lithosphere is the solid part of Earth which includes or consists of the crust and the uppermost mantle. The sudden movement of the earths lithosphere is called earthquake (technical name seism). Fractures in Earths crust where sections of rock have slipped past each other are called Faults. Most earthquakes occur along Faults. Generally, earthquakes are caused by the sudden release of built-up stress within rocks along geologic faults or by the movement of magma in volcanic areas. The theory of plate tectonics provides geology with a comprehensive theory that explains how the Earth works. The theory states that Earths outermost layer, the lithosphere, is broken into 7 large, rigid pieces called plates: the African, North American, South American, Australian- Indian, Eurasian, Antarctic, and Pacific plates. Several subcontinental plates also exist, including the Caribbean, Arabian, Nazca, Philippines and Cocos plates. Boundaries of tectonic plates are found at the edge of the lithospheric plates and can be of various forms, depending on the nature of relative movements. By their distinct motions, three main types can be characterized. The three types are: subduction zones (or trenches), spreading ridges (or spreading rifts) and transform faults.. convergent, divergent and conservative. At subduction zone boundaries, plates move towards each other and the one plate subducts underneath the other (ÃŽÂ ® ÃŽ ¼Ãâ‚¬ÃŽ ¿Ã Ãƒ Ã… ½ ÃŽ ½ÃŽ ± πω: one plate is overriding another, thereby forcing the other into the mantle beneath it.) The opposite form of movement takes place at spreading ridge boundaries. At these boundaries, two plates move away from one another. As the two move apart, molten rock is allowed to rise from the mantle to the surface and cool down to form part of the plates. This, in turn, causes the growth of oceanic crust on either side of the vents. As the plates continue to move, and more crust is formed, the ocean basin expands and a ridge system is created. Divergent boundaries are responsible in part for driving the motion of the plates. At transform fault boundaries, plate material is neither created nor destroyed at these boundaries, but rather plates slide past each other. Transform faults are mainly associated with spreading ridges, as they are usually formed by surface movement due to perpendicular spreading ridges on either side. Earthquake Location When an earthquake occurs, one of the first questions is where was it?. An earthquakes location may tell us what fault it was on and where the possible damage most likely occurred. The hypocentre of an earthquake is its location in three dimensions: latitude, longitude, and depth. The hypocentre (literally meaning: below the center from the Greek υπà Ã…’Î ºÃŽ µÃŽ ½Ãâ€žÃ ÃŽ ¿ÃŽ ½), or focus of the earthquake, refers to the point at which the rupture initiates and the first seismic wave is released. As an earthquake is triggered, the fault is associated with a large area of fault plane. The point directly above the focus, on the earths surface where the origin of an earthquake above ground. The epicentre is the place on the surface of the earth under which an earthquake rupture originates, often given in degrees of latitude (north-south) and longitude (east-west). The epicentre is vertically above the hypocentre. The distance between the two points is the focal depth. The location of any station or observation can be described relative to the origin of the earthquake in terms of the epicentral or hypocentral distances. Propagation of seismic waves Seismic waves are the energy generated by a sudden breaking of rock within the earth or an artificial explosion that travels through the earth and is recorded on seismographs. There are several different kinds of seismic waves, and they all move in different ways. The two most important types of seismic waves are body waves and surface waves. Body waves travel deep within the earth and surface waves travel near the surface of the earth. Body waves: There are two types of body waves: P-waves (also pressure waves) and S-waves (also shear waves). P-waves travel through the Earth as longitudinal waves whose compressions and rarefactions resemble those of a sound wave. The name P-wave comes from the fact that this is the fastest kind of seismic wave and, consequently, it is the first or ‘Primary wave to be detected at a seismograph. Speed depends on the kind of rock and its depth; usually they travel at speeds between 1.5 and 8 kilometers per second in the Earths crust. P waves are also known as compressional waves, because of the pushing and pulling they do. P waves shake the ground in the direction they are propagating, while S waves shake perpendicularly or transverse to the direction of propagation. The P-wave can move through solids, liquids or gases. Sometimes animals can hear the P-waves of an earthquake S-waves travel more slowly, usually at 60% to 70% of the speed of P waves. The name S-wave comes from the fact that these slower waves arrive Secondary after the P wave at any observation point. S-waves are transverse waves or shear waves, so that particles move in a direction perpendicular to that of wave propagation. Depending in whether this direction is along a vertical or horizontal plane, S-waves are subcategorized into SV and SH-waves, respectively. Because liquids and gases have no resistance to shear and cannot sustain a shear wave, S-waves travel only through solids materials. The Earths outer core is believed to be liquid because S-waves disappear at the mantle-core boundary, while P-waves do not. (3: http://www.globalchange.umich.edu/globalchange1/current/lectures/nat_hazards/nat_hazards.html) Surface waves: The surface waves expand, as the name indicates, near the earths surface. The amplitudes of surface waves approximately decrease exponentially with depth. Motion in surface waves is usually larger than in body waves therefore surface waves tend to cause more damage. They are the slowest and by far the most destructive of seismic waves, especially at distances far from the epicenter. Surface waves are divided into Rayleigh waves and Love waves. Rayleigh waves, also known as ground roll, are the result of an incident P and SV plane waves interacting at the free surface and traveling parallel to that surface. Rayleigh waves (or R-waves) took their name from (named for) John Strutt, Lord Rayleigh who first described them in 1885 (ÃŽÂ ® who mathematically predicted the existence of this kind of wave in 1885) and they are an important kind of surface wave. Most of the shaking felt from an earthquake is due to the R-wave, which can be much larger than the other waves. In Rayleigh waves the particles of soil move vertically in circular or elliptical paths, just like a wave rolls across a lake or an ocean. As Rayleigh wave particle motion is only found in the vertical plane, this means that they most commonly found on the vertical component of seismograms. The Rayleigh equation is: Love waves (also named Q waves) are surface seismic waves that cause horizontal shifting of the earth during an earthquake. They move the ground from side to side in a horizontal plane but at right angles to the direction of propagation. Love waves took their name from A.E.H. Love, a British mathematician who worked out the mathematical model for this kind of wave in 1911. Love waves are the result from the interaction with SH-waves. They travel with a slower velocity than P- or S- waves, but faster than Rayleigh waves, their speed relate to the frequency of oscillation. Earthquake size: Earthquake measurement is not a simple problem and it is hampered by many factors. The size of an earthquake can be quantified in various ways. The intensity and the magnitude of an earthquake are terms that were developed in an attempt to evaluate the earthquake phenomenon and they are the most commonly used terms to express the severity of an earthquake. Earthquake intensity: Intensity is based on the observed effects of ground shaking on people, buildings, and natural features. It varies from place to place within the disturbed region depending on the location of the observer with respect to the earthquake epicenter. Earthquake magnitude: The magnitude is the most often cited measure of an earthquakes size. The most common method of describing the size of an earthquake is the Richter magnitude scale, ML. This scale is based on the observation that, if the logarithm of the maximum displacement amplitudes which were recorded by seismographs located at various distances from the epicenter are put on the same diagram and this is repeated for several earthquakes with the same epicentre, the resulting curves are parallel to each other. This means that if one of these earthquakes is taken as the basis, the coordinate difference between that earthquake and every other earthquake, measures the magnitude of the earthquake at the epicentre. Richter defined as zero magnitude earthquake one which is recorded with 1ÃŽ ¼m amplitude at a distance of 100 km. Therefore, the local magnitude ML of an earthquake is based on the maximum trace amplitude A and can be estimated from the relation: ML= log A log A (3) Where A is the amplitude of the zero magnitude earthquake (ML=0). The Richter magnitude scale can only be used when seismographs are within 600 km of the earthquake. For greater distances, other magnitude scales have been defined. The most current scale is the moment magnitude scale MW, which can be used for a wide range of magnitudes and distances. Two main categories of instruments are used for the quantitative evaluation (estimation, assessment) of the earthquake phenomenon: the seismographs which record the displacement of the ground as a function of time, and the accelerographs (or accelerometers) which record the acceleration of the ground as a function of time, producing accelerograms. X the accelerogram of the 1940 El Centro earthquake. For every earthquake accelerogram, elastic or linear acceleration response spectrum diagrams can be calculated. (obtained, estimated) The response spectrum of an earthquake is a diagram of the peak values of any of the response parameters (displacement, acceleration or velocity) as a function of the natural vibration period T of the SDOF system, subjected to the same seismic input. All these parameters can be plotted together in one diagram which is called the tripartite plot (also known as four coordinate paper). 2.2 Earthquake and Structures simulation 2.2.1 Equation of motion of SDOF system Introduction Vibration is the periodic motion or the oscillation of an elastic body or a medium, whose state of equilibrium has been disturbed. ÃŽâ€" ÃŽ ¼Ãâ‚¬ÃŽ ¿Ã Ãâ€° ÃŽ ½ÃŽ ± πω: whose position of equilibrium has been displaced. There are two types of vibrations, free vibration and forced vibration. Vibration can be classified as either free or forced. A structure is said to be in a state of free vibration when it is disturbed from its static equilibrium by given a small displacement or deformation and then released and allowed to vibrate without any external dynamic excitation. Number of Degrees of Freedom (DOF) is the number of the displacements that are needed to define the displaced position of the masses relative to their original position. Simple structures can be idealised as a system with a lumped mass m supported by a massless structure with stiffness k. It is assumed that the energy is dissipated through a viscous damper with damping coefficient c. Only one displacement variable is required in order to specify the position of the mass in this system, so it is called Singe Degree of Freedom (SDOF) system. Undamped Free Vibration of SDOF systems Furthermore, if there is no damping or resistance in the system, there will be no reduction to the amplitude of the oscillation and theoretically the system will vibrate forever. Such a system is called undamped and is represented in the below: By taking into consideration the inertia force fin and the elastic spring force fs the equation of the motion is given by: fin + fs = 0 → m+ ku = 0 Considering the initial conditions u(0) and (0), where u(0) is the displacement and (0) is the velocity at the time zero, the equation (4) has the general solution: u(t) = u(0) cosωnt + sinωnt where ωn is the natural frequency of the system and is given by, ωn = (6) The natural period and the natural frequency can be defined by the above equations: Tn = (7) fn = (8) Viscously damped Free Vibration of SDOF systems The equation of motion of such a system can be developed from its free body diagram below: Considering the inertia force fin, the elastic spring force fs and the damping force fD, the equation of the motion is given by: m+ c+ ku = 0 (9) Dividing by m the above equation gives: + 2ÃŽ ¾Ãâ€°n+ ω2u = 0 (10) where ÃŽ ¾ is the critical damping and is given by: ÃŽ ¾ = (11) and Cc is the critical damping ratio given by: Cc = 2mωn * If ÃŽ ¾ > 1 or c > Cc the system is overdamped. It returns to its equilibrium position without oscillating. * If ÃŽ ¾ = 1 or c = Cc the system is critically damped. It returns to its equilibrium position without oscillating, but at a slower rate. * If ÃŽ ¾ Taking into account that all the structures can be considered as underdamped systems, as typically their damping ratio ÃŽ ¾ is less than 0.10 the equation (9) for the initial conditions u (0) and (0) gives the solution below: U (t) = e†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦[u(0)cosωn+[†¦.+sinωDt] (13) where ωD is the natural frequency of damped vibration and is given by: ωD = ωn (14) Hence the natural period is: TD = (15) Undamped Forced Vibration of SDOF system The equation of motion of such a system can be developed from its free body diagram below: Considering the inertia force fin, the elastic spring force fs and the external dynamic load f(t), the equation of the motion is given by: m+ ku = f(t) (16) where f(t) = f0 sinωt is the maximum value of the force with frequency ω By imposing the initial conditions u(0) and (0) the equation (16) has a general solution: u(t) = u(0)cosωnt + sinωnt + sinωt (17) Damped Forced Vibration of SDOF system The equation of motion of such a system can be developed from its free body diagram below: Considering the inertia force fin, the elastic spring force fs, the damping force fD and the external dynamic load f(t), the equation of the motion is given by: m+ c+ ku = f(t) (18) where f(t) = f0 sinωt The particular solution of equation (18) is: up = Csinωt + Dcosωt (19) And the complementary solution of equation (18) is: (20) uc = e(AcosωDt + Bsinωnt) (20) 2.2.2 Equation of motion of MDOF system The equation of motion of a MDOF elastic system is expressed by: M+ C+ Ku = -MAI(t) (21) where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, u is the acceleration vector, u is the velocity vector and u is the displacement vector. Finally, AI is a vector with all the elements equal to unity and ug(t) is the ground acceleration. 2.2 Earthquake and Structures simulation 2.2.1 Equation of motion of SDOF system Introduction Vibration is the periodic motion or the oscillation of an elastic body or a medium, whose state of equilibrium has been disturbed. ÃŽâ€" ÃŽ ¼Ãâ‚¬ÃŽ ¿Ã Ãâ€° ÃŽ ½ÃŽ ± πω: whose position of equilibrium has been displaced. There are two types of vibrations, free vibration and forced vibration. Vibration can be classified as either free or forced. A structure is said to be in a state of free vibration when it is disturbed from its static equilibrium by given a small displacement or deformation and then released and allowed to vibrate without any external dynamic excitation. Number of Degrees of Freedom (DOF) is the number of the displacements that are needed to define the displaced position of the masses relative to their original position. Simple structures can be idealised as a system with a lumped mass m supported by a massless structure with stiffness k. It is assumed that the energy is dissipated through a viscous damper with damping coefficient c. Only one displacement variable is required in order to specify the position of the mass in this system, so it is called Singe Degree of Freedom (SDOF) system. Undamped Free Vibration of SDOF systems Furthermore, if there is no damping or resistance in the system, there will be no reduction to the amplitude of the oscillation and theoretically the system will vibrate forever. Such a system is called undamped and is represented in the below: By taking into consideration the inertia force fin and the elastic spring force fs the equation of the motion is given by: fin + fs = 0 → m+ ku = 0 Considering the initial conditions u(0) and (0), where u(0) is the displacement and (0) is the velocity at the time zero, the equation (4) has the general solution: u(t) = u(0) cosωnt + sinωnt where ωn is the natural frequency of the system and is given by, ωn = (6) The natural period and the natural frequency can be defined by the above equations: Tn = (7) fn = (8) Viscously damped Free Vibration of SDOF systems The equation of motion of such a system can be developed from its free body diagram below: Considering the inertia force fin, the elastic spring force fs and the damping force fD, the equation of the motion is given by: m+ c+ ku = 0 (9) Dividing by m the above equation gives: + 2ÃŽ ¾Ãâ€°n+ ω2u = 0 (10) where ÃŽ ¾ is the critical damping and is given by: ÃŽ ¾ = (11) and Cc is the critical damping ratio given by: Cc = 2mωn * If ÃŽ ¾ > 1 or c > Cc the system is overdamped. It returns to its equilibrium position without oscillating. * If ÃŽ ¾ = 1 or c = Cc the system is critically damped. It returns to its equilibrium position without oscillating, but at a slower rate. * If ÃŽ ¾ Taking into account that all the structures can be considered as underdamped systems, as typically their damping ratio ÃŽ ¾ is less than 0.10 the equation (9) for the initial conditions u (0) and (0) gives the solution below: U (t) = e†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦[u(0)cosωn+[†¦.+sinωDt] (13) where ωD is the natural frequency of damped vibration and is given by: ωD = ωn (14) Hence the natural period is: TD = (15) Undamped Forced Vibration of SDOF system The equation of motion of such a system can be developed from its free body diagram below: Considering the inertia force fin, the elastic spring force fs and the external dynamic load f(t), the equation of the motion is given by: m+ ku = f(t) (16) where f(t) = f0 sinωt is the maximum value of the force with frequency ω By imposing the initial conditions u(0) and (0) the equation (16) has a general solution: u(t) = u(0)cosωnt + sinωnt + sinωt (17) Damped Forced Vibration of SDOF system The equation of motion of such a system can be developed from its free body diagram below: Considering the inertia force fin, the elastic spring force fs, the damping force fD and the external dynamic load f(t), the equation of the motion is given by: m+ c+ ku = f(t) (18) where f(t) = f0 sinωt The particular solution of equation (18) is: up = Csinωt + Dcosωt (19) And the complementary solution of equation (18) is: uc = (AcosωDt + Bsinωnt) (20) 2.2.2 Equation of motion of MDOF system The equation of motion of a MDOF elastic system is expressed by: M+ C+ Ku = -MAI(t) (21) where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, u is the acceleration vector, u is the velocity vector and u is the displacement vector. Finally, AI is a vector with all the elements equal to unity and g(t) is the ground acceleration. 3. Description of the Method 3.1 Simplified Multi-Storey Shear Building Model It is almost impossible to predict precisely which seismic action a structure will undergo during its life time. Each structure must be designed to resist at any seismic excitation without failing. For this reason each structure is designed to meet the requirements of the design spectrum analysis based in EC8. Also some assumptions are necessary in order to achieve the best and the simplest idealization for each multi store building. Initially it is assumed that the mass of each floor is lumped at the centre of the floor and the columns are massless. The floor beams are completely rigid and incompressible; hence the floor displacement is being transferred equally to all the columns. The columns are flexible in horizontal displacement and rigid in vertical displacement, while they are provided with a fully fixed support from the floors and the ground. The building is assumed to be symmetric about both x and y directions with symmetric column arrangement. The consequence of this is tha t the centre of the mass of each floor to coincide with the centre of the stiffness of each floor. The position of this centre remains stable up the entire height of the building. Finally, it is assumed that there are no torsional effects for each of the floors. If all the above assumptions are used the building structure is idealised as a model where the displacement at each floor is described by one degree of freedom. Thus, for a jth storey building, j degrees of freedom required to express the total displacement of the building. The roof of the building has always to be considered as a floor. The mass matrix M is a symmetric diagonal nxn matrix for a n-storey building and is given below. Each diagonal value in the matrix represents the total mass of one beam and its two corresponding columns which are assumed to be lumped at each level. M = Stiffness method is used to formulate the stiffness matrix. K is the lateral stiffness of each column and is given by the relationship: K = (22) where EI is the flexural stiffness of a column. The lateral stiffness of each column is clamped at the ends and is imposed in a unit sway. The stiffness of each floor is the sum of the lateral force of all columns in the floor. The stiffness matrix is for a n-storey building is: K = In order to calculate the natural modes of the vibration, the system is assumed that vibrates freely. Thus, g(t)=0, which for systems without damping (c=0) the equation (21) specializes to: M+ Ku = 0 (23) The displacement is assumed to be harmonic in time, this is: = -ω2Ueiωt (24) Hence equation (23) becomes: (K ω2M)U = 0 (25) The above equation has the trivial solution u=0. For non trivial solutions, u≠ 0 the determinant for the left hand size must be zero. That is: |K ω2 M| = 0 (26) This condition leads to a polynomial in terms of ω2 with n roots, where n is the size of matrices and vectors as cited above. These roots are called eigenvalues. By applying the equation (6) (7), the natural frequency and the natural period of vibration for each mode shape can be determined. Each eigenvalue has a relative eigenvector which represent the natural ith mode shape. After the estimation of the eigenvector in order to compare the mode shapes, scale factors are applied to natural modes to standarise their elements associated with various degrees of freedom (X). This process is called normalization. Hence, after the estimation of the eigenvectors each mode is normalised so that the biggest value is X: eigenvector notation. unity. The eigenvectors of a symmetric matrix corresponding to distinct eigenvalues are orthogonal. This aspect is expressed by the following expression: UiTKUij = UiTMUij (27) The classical eigenvalue problem has the following form: (M-1K ÃŽ » I) u = 0 (28) where ÃŽ » =ω2 and I is the identity matrix. EC8 suggests that the response in two modes i and j can be assumed independent of each other when Tj ≠¤ 0.9 Ti where Ti and Tj are the periods of the modes i and j respectively (always Ti ≠¥ Tj). The calculated fundamental period can be checked by the equation that EC8 suggests: T = Ct*H3/4 where T is the fundamental period of the building, Ct is a coefficient and H is the total height of the building; this expression is valid buildings that their total height is not more than forty metres 3.2 Elastic Analysis The response method is used to estimate the maximum displacement (uj), pseudo- velocity (j) and acceleration (j) for each calculated natural frequency. It is assumed that the MDOF system oscillates in each of its modes independently and displacements, velocities and accelerations can be obtained for each mode separately considering modal responses as SDOF responses. Each maximum, displacement velocity and acceleration read from the design spectrum is multiplying by the participation factor ÃŽ ±i to re-evaluate the maximum values expressed ujmax, jmax, jmax respectively. The participation factor ÃŽ ±i is defined by the following equation: (28) where UijT is the transpose vector of each of the mode vectors, M is the mass matrix, AI is the unit vector and Uij is the mode shape vector. The actual maximum displacements of the jth mode are given by: u = ujmaxΆ¡Uj Afterwards, the root-mean-square (RMS) approximation is used in order to calculate the maximum displacement for each floor. In this approach, all the maximum values for each mode, are squared and summed and their square root is derived. If we let Dmax be the maximum displacement then: Dmax = (29) A very variable parameter to characterise the seismic behaviour of a building is the Inter-Storey Drift which can be obtained by the following equation: ÃŽ ´i = Di Di-1/hi (30) where Di, Di-1, are the horizontal displacements for two contiguous floors and hi is the corresponding height of the floor. The calculated values must be lower than 4% in order to agree with the Eurocode. Afterwards the horizontal inertia forces Fjs applied at each floor are obtained by applying the following equation: Fj = MΆ¡UjΆ¡jmax (31) where M is the mass matrix, Uj is the eigenvector for each mode and jmax is the maximum acceleration. As it is suggested from the EC8, the root-mean-approximation is used again in order to obtain the total lateral forces. EC8 suggests that the combined lateral force at each floor is given by the square root of the sum of the squares of each lateral force at each floor of all the modes. If we let Ftotal,i the maximum base shear force then: Ftotal,j = [1] (32) where Fij is the lateral force at floor i of the mode j. Once the total lateral forces and the shear forces have been obtained, the maximum overturning moment is calculated. 3.3 Inelastic Analysis The inelastic response spectra are generally obtained by the scaling of the elastic design spectra via the use of response modification factors. No effect of the energy absorption was assumed in the structure for the calculated values by using the elastic design spectrum. By introducing the ductility factor this parameter is taking into consideration. Newmark has described the ductility parameter ÃŽ ¼ as the ratio of maximum displacement to the displacement at yield. Apparently when yielding does not take place the concept of ductility is not relevant and ÃŽ ¼ is taken equal to unity. ÃŽ ¤he system is described by the damping ratio Ï‚, the natural frequency ωn, and the ductility factor ÃŽ ¼. In order to calculate the new set of values of acceleration, displacement and velocity the design response spectrum has to be constructed. Newmarks procedure leads to the construction of two modified spectra. 1. For maximum acceleration: In this case the elastic design spectrum is reduced by the appropriate coefficients. The acceleration region of the graph is multiplie

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Free Essays on First Year in College

My First College Year Everyday in my life since I started college has been the same. It seems like I do the same thing everyday without remembering that I did that the day before or just yesterday. It's like a somewhat automatic repeat of my day and I don’t even realize it. I usually always wake up at around 8:00am every Monday, Wednesday, and Friday to go to school. Only on those three days I wake up early. On Tuesday and Thursday I wake up just a little bite later, around 11am or 12pm. I usually study all day long when I don’t go to school on those two days. There has never been weekdays were I haven't followed this incidental procedure, well maybe only on long weekends and holidays. My Saturdays and Sundays are also usually similar to any other regular weekend. I always get loads of homework on Fridays, so both days of my free weekend is gone by studying. I never studied on the weekends when I was in high school. Know that I am in college, I have become more mature, and focused then ever, mentally. College has totally changed my sleeping habits also. I usually sleep at either 2 or 3 am everyday regardless of having school the next day or not. I think the reason why I will not go to sleep is because know I am more worried about getting my studying done ahead of time, rather then waiting to the last minute. I am glad that college has made me become more positive about studying. This procedure has really made me a better and a positive person, because I have actually opened myself up to ask more questions, and paying more attention during classes. My thoughts and feeling are opening up more often. The people who already had there first year in college might actually understand my viewpoint. First year in college is the best year, in which you’ve become mature, focused, metally alert, and also trying to find out what career opportunities are available.... Free Essays on First Year in College Free Essays on First Year in College My First College Year Everyday in my life since I started college has been the same. It seems like I do the same thing everyday without remembering that I did that the day before or just yesterday. It's like a somewhat automatic repeat of my day and I don’t even realize it. I usually always wake up at around 8:00am every Monday, Wednesday, and Friday to go to school. Only on those three days I wake up early. On Tuesday and Thursday I wake up just a little bite later, around 11am or 12pm. I usually study all day long when I don’t go to school on those two days. There has never been weekdays were I haven't followed this incidental procedure, well maybe only on long weekends and holidays. My Saturdays and Sundays are also usually similar to any other regular weekend. I always get loads of homework on Fridays, so both days of my free weekend is gone by studying. I never studied on the weekends when I was in high school. Know that I am in college, I have become more mature, and focused then ever, mentally. College has totally changed my sleeping habits also. I usually sleep at either 2 or 3 am everyday regardless of having school the next day or not. I think the reason why I will not go to sleep is because know I am more worried about getting my studying done ahead of time, rather then waiting to the last minute. I am glad that college has made me become more positive about studying. This procedure has really made me a better and a positive person, because I have actually opened myself up to ask more questions, and paying more attention during classes. My thoughts and feeling are opening up more often. The people who already had there first year in college might actually understand my viewpoint. First year in college is the best year, in which you’ve become mature, focused, metally alert, and also trying to find out what career opportunities are available....

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