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The lever arm is defined as the perpendicular distance from the axis of rotation to the line of action of the force.

Two equal forces acting in opposite directions, i.e., clockwise and counterclockwise, and applied to a uniform lever at equal distances from the fulcrum counteract each other and establish a state of equilibrium, or balance

Product of a force by its effort arm is called a moment of the force

Effort

● Force applied on the machine

Load

● Force exerted by the machine

NB: The SI unit of effort and load is Newton(N)

Mechanical Advantage (M.A)

● The ratio of load to effort

Mathematically,

M.A is a ratio of two units hence it has no unit

M.A depends on:

1) Friction between the moving parts of the machine

2) The weight of the parts of the machine that have to lifted when operating it.

Therefore, M.A cannot be 100%

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Velocity Ratio (V.R)

● The ratio of effort distance to load distance

Mathematically,

V.R is a ratio of two distances hence it has no unit

NB: If two machines have velocity ratios VR1 and VR2 then, the resultant VR is given by;

VR = VR1 x VR2

E=( LoadEffort )∗( LoaddistanceEffortdistance )∗100 00

E=M . AV .R

∗100 00

● E of a machine is always less than 100% because some energy is lost in overcoming the friction force and the weight of the parts that have to be lifted

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• Steel is very ductile material and from the stress strain curve it is observed that higher loads than in the elastic method can be applied over the structure. This is due to the fact that a major portion of the curve lies beyond the elastic limit. This extra strength is termed reserve strength and forms the basis of plastic design method.

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• This is an aspect of limit design, which confines the structural usefulness up to the plastic strength or ultimate load carrying capacity. this method is based on failure condition. In this method of design failure implies collapse or extremely large deformations, thus the structure fails at a much higher load, called the collapse load, than working load.

Shape factor-

• For a ductile material like structural steel a member reaching yield at the extreme fibers retains a reserve of strength that varies with the shape factor.

• Shape factor=plastic moment/yield moment

• It is a function of the cross section form or shape.

Load factor-

Load factor is defined as the ratio of collapse load to the working load. It is represented by F.

F = Pu/ Pw = Mp/ Mw = fy Zp/f Ze = fy S /f = (F.O.S)S

What is difference between structure and mechanism?when a load act in a member, the structure will retain the load or deformation takes place whereas in mechanism it will not take the load. It will fail without taking the load

MECHANISM-

When a structure is subjected to a system of loads ,it is stable and hence functional until a sufficient number of plastics things have been formed to render the structure unstable. As soon as the structure reaches an unstable condition it is considered to have been failed. The segments of the beam between the plastic hinges are able to move without an increase of load. This condition in a member is called mechanism.

The concept of mechanism formation in a structure due to loading beyond the elastic limit of a virtual work are used in plastic analysis has redundancy r,the collapse of number of plastic hinges required (r+1).

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• The plastic collapse of a structure depends upon its redundancy .When a sufficient number of plastic hinges are formed to convert a structure into mechanism the structure collapse as such a stage, the deflection increases very fast at a constant load. The collapse of a structure can be partial, complete and over complete.

• This terms can be explained indeterminacy (r)

• Number of Plastic hinges(N)

• The number of plastic hinges in collapse mechanism are less than (r+1).The collapse is called partial collapse.

• The number of plastic hinges in collapse mechanism are equal to (r+1).The collapse is called complete collapse.

• The number of plastic hinges in collapse mechanism are greater than (r+1).The collapse is called over complete collapse.

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CONDITION IN PLASTIC ANALYSIS

1) Equilibrium condition

2) Mechanism condition

3) Yield condition

PRINCIPLE OF VIRTUAL WORK

If a system of forces in equilibrium is subjected to a virtual displacement the work done by the external forces equals the work done by internal forces.

We=Wi

It is to a object to express an equilibrium condition.

Engineers and research workers have been stimulated to study the plastic strength of steel structures and its application to design for three principle reasons:

a)it has a more logical design basis.

b)it is more economical in the use of steel.

c)it represents a substantial saving of time in the design

the calculation of load carrying capacity by use of limit theorems is much easier than the calculation of stress

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The simplicity of limit analysis opens the way to limit design, to direct design as contrasted with the trial and error procedure normally followed in conventional design

The ratio of the plastic moment to the yield moment is known as the shape factor since it depends on the shape of the cross section. The cross section is not capable of resisting any additional moment but may maintain this moment for some amount of rotation in which case it acts like a plastic hinge. If this is so, then for further loading, the beam, acts as if it is simply supported with two additional moments Mp on either side, and continues to carry additional loads until a third plastic hinge forms at mid-span when the bending moment at that section reaches Mp. The beam is then said to have developed a collapse mechanism and will collapse as shown in Fig 2.14

If the section is thinwalled, due to local buckling, it may not be able to sustain the moment for additional rotations and may collapse either before or soon after attaining the plastic moment. It may be noted that formation of a single plastic hinge gives a collapse mechanism for a simply supported beam. The ratio of the ultimate rotation to the yield rotation is called the rotation capacity of the section. The yield and the plastic moments together with the rotation capacity of the cross section are used to classify the sections.

Theoretically, the plastic hinges are assumed to form at points at which plastic rotations occur. Thus the length of a plastic hinge is considered as zero. However, the values of moment, at the adjacent section of the yield zone are more than the yield moment upto a certain length ΔL, of the structural member. This length ΔL, is known as the hinged length. The hinged length depends upon the type of loading and the geometry of the cross-section of the structural member. The region of hinged length is known as region of yield or plasticity.

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Equilibrium: the internal bending moments must be in equilibrium with the external loading. Mechanism: at collapse the structure, or a part of, can deform as a mechanism Yield: no point in the structure can have a moment greater than the plastic moment capacity of the section it is applied to.

STRUCTURE VS MECHANISM

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Thus far the cross sections considered are only carrying moment. In the presence of axial force, clearly some material must be given over to carry the axial force and so is not available to carry moment, reducing the capacity of the section. Further, it should be apparent that the moment capacity of the section therefore depends on the amount of axial load being carried. Considering a compression load as positive, more of the section will be in compression and so the neutral axis will drop.

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• M – Moment corresponding to working load• My – Moment at which external fibre of the section yields• MP – Moment at which entire section is under yield stress

Elastic Analysis - Factor of SafetyPlastic Analysis - Load Factor

A statically determinate beam will collapse if one plastic hinge is developedFor a statically indeterminate beam to collapse, more than one plastic hinge should be developed

As the load is increased, there is a redistribution of moment, as the plastic hinge cannot carry any additional moment.

Plastic hinges develop at the ends first Beam becomes a simple beam Plastic hinge develops at the centre Beam collapses

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• Collapse load (Wc): Minimum load at which collapse will occur – Least value• Fully plastic moment (MP): Maximum moment capacity for design – Highest value

The applied moment to the cross section is such that all fibres in the cross section are at yield stress. This is termed the Plastic Moment Capacity of the section since there are no fibres at an elastic stress

At the plastic hinge stresses remain constant, but strains and hence rotations can increase.

Once the plastic moment capacity is reached, the section can rotate freely – that is, it behaves like a hinge, except with moment of Mp at the hinge.

For a rectangular cross-section shape factor, f=1.5 means rectangular section can sustain 50% more moment than the yield moment before a plastic hinge is formed.Therefore the shape factor is a good measure of the efficiency of a cross section in bending.

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Methods of Plastic Analysis There are three main approaches for performing a plastic analysis:

The Incremental Method This is probably the most obvious approach: the loads on the structure are incremented until the first plastic hinge forms. This continues until sufficient hinges have formed to collapse the structure. This is a labour-intensive, ‘brute-force’, approach, but one that is most readily suited for computer implementation.

The Equilibrium (or Statical) Method In this method, free and reactant bending moment diagrams are drawn. These diagrams are overlaid to identify the likely locations of plastic hinges. This method therefore satisfies the equilibrium criterion first leaving the two remaining criterion to derived therefrom.

The Kinematic (or Mechanism) Method In this method, a collapse mechanism is first postulated. Virtual work equations are then written for this collapse state, allowing the calculations of the collapse bending moment diagram. This method satisfies the mechanism condition first, leaving the remaining two criteria to be derived therefrom.

The Upperbound (Unsafe) Theorem ( Plastic analysis)If a bending moment diagram is found which satisfies the conditions of equilibrium and mechanism (but not necessarily yield), then the corresponding load factor is either greater than or equal to the true load factor at collapse.This is called the unsafe theorem because for an arbitrarily assumed mechanism the load factor is either exactly right (when the yield criterion is met) or is wrong and is too large, leading a designer to think that the frame can carry more load than is actually possible.Think of it like this: unless it’s exactly right, it’s dangerousThis is why plastic analyses are not used as often in practice as one might suppose.

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The Lowerbound (Safe) Theorem (elastic analysis)If a bending moment diagram is found which satisfies the conditions of equilibrium and yield (but not necessarily that of mechanism), then the corresponding load factor is either less than or equal to the true load factor at collapse.This is a safe theorem because the load factor will be less than (or at best equal to) the collapse load factor once equilibrium and yield criteria are met leading the designer to think that the structure can carry less than or equal to its actual capacity.Think of it like this: it’s either wrong and safe or right and safe.Since an elastic analysis will always meet equilibrium and yield conditions, an elastic analysis will always be safe. This is the main reason that it is elastic analysis that is used, in spite of the significant extra capacity that plastic analysis offers.

The Uniqueness TheoremIf a bending moment distribution can be found which satisfies the three conditions of equilibrium, mechanism, and yield, then the corresponding load factor is the true load factor at collapse.The Uniqueness Theorem does not claim that any particular collapse mechanism is unique – only that the collapse load factor is unique. Although rare, it is possible for more than one collapse mechanism to satisfy the Uniqueness Theorem, but they will have the same load factor.

If the collapse loads are determined for all possible mechanisms, then the actual collapse load will be the lowest of these (Upperbound Theorem);

The collapse load of a structure cannot be decreased by increasing the strength of any part of it (Lowerbound Theorem);

The collapse load of a structure cannot be increased by decreasing the strength of any part of it (Upperbound Theorem);

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What's the difference between the centre of gravity and the centre of mass?

Center of mass and center of gravity are two terms that are often used interchangeably, but they're really not the same.Let's take an object, like, for example, a 5 kilogram bowling ball. If you drop a bowling ball, it will fall to the ground because of the force of gravity. But did you know that the bowling ball will fall to the ground in the same way that a 5 kilogram point mass would if the point mass was placed at the very center of the bowling ball?

The bowling ball is a uniform object with a center of mass at the very center of the bowling ball. The center of mass is the mean position of the mass in an object. If you have the same amount of mass to your right as you have to your left and the same amount above as you have below and the same amount in front as you have behind, then you must be at the center of mass.

The bowling ball also has a center of gravity, which is the point where gravity appears to act. Or in other words, it's the sum total of all the forces of gravity on all the particles in the object. It doesn't take much understanding of physics to realize that for the bowling ball, this is also at the very center of the object. For the bowling ball, the center of mass and center of gravity are pretty much in the same place.

But they're NOT the same thing. It turns out that they're only the same when the gravitational field is uniform across the object, or at least close enough to be uniform that it isn't worth discussing. With small objects near the surface of the Earth, that's always the case. But once you start putting spaceships in space, suddenly things get weird.

Centre of mass is the point at which the distribution of mass is equal in all directions, and does not depend on gravitational field. Centre of gravity is the point at which the distribution of weight is equal in all directions, and does depend on gravitational field.

The centre of mass and the centre of gravity of an object are in the same position if the gravitational field in which the object exists is uniform.But if the gravitational field strength were greater towards your feet and weaker towards your head, then your centre of gravity would be below your centre of mass, perhaps somewhere around your knees. If the gravitational field strength were greater towards your head, and weaker towards your feet, then your centre of gravity would be above your centre of mass, perhaps somewhere around your shoulders.

The object on the left, in a uniform gravitational field, has overlapping centres of gravity and mass. For the object on the right, in which the gravitational field is stronger towards its base, the centre of gravity is below the centre of mass. 

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Tetracalcium Aluminoferrite (C4AF) contributes very slightly to strength gain. However, acts as a flux during manufacturing. Contributes to the color effects that makes cement gray. tetracalcium aluminoferrite (C4AF) is primarily a result of materials used in the cement manufacturing process to lower the temperatures required in the kilns. C4AF hydrates rapidly, but contributes little actual strength. Perhaps its most significant effect on concrete is its influence on color.

Higher concentrations of C4AF will result in darker color concretes. In white cement the percentage of C4AF is kept low, often about 1% to 2%.

This is a fluxing agent which reduces the melting temperature of the raw materials in the kiln (from 3,000o F to 2,600o F). It hydrates rapidly, but does not contribute much to strength of the cement paste.

Gives resistance to Sulphate attack

Build–operate–transfer (BOT) or build–own–operate–transfer (BOOT) is a form ofproject financing, wherein a private entity receives a concession from the private or public sector to finance, design, construct, and operate a facility stated in the concession contract.A type of arrangement in which the private sector builds an infrastructure project, operates it and eventually transfers ownership of the project to the government. In many instances, the government becomes the firm's only customer and promises to purchase at least a predetermined amount of the project's output. This ensures that the firm recoups its initial investment in a reasonable time span.This type of arrangement is used typically in complicated long-term projects as seen in power plants and water treatment facilities. In some arrangements, the government does not assume ownership of the project. In those cases, the company continues running the facility and the government acts as both the consumer and regulator.The main reasons for this trend are a shortage of public funds and a hands-off approach of government agencies. The Build Operate Transfer (BOT) approach is an option for the government to outsource public projects to the private sector

The most common examples are roads, bridges, water and sewer systems, airports, ports and public buildings

Build-Own-Operate-Transfer BOOT is a founding model and a form of concession in which a public authority makes an agreement with a private company (concessionaire) to Design Build, Own and Operate a specific piece of an infrastructure such as power, transport, water, and telecom industries, within receiving the right to achieve income from the facility under a period of time (concession period approximately 15-25 years), and later transferring it back into public ownership

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Public Private Partnership

The P3 procurement model is unique in that the private sector assumes a major share of the responsibility for the delivery and the performance of the infrastructure from designing the concept, architectural and structural planning to its long-term maintenance. When applied effectively, the P3 model can provide additional value to taxpayers by leveraging the right capabilities to complete the job on time and on budget, allowing for greater integration of project planning and design and eliminating major shortfalls around building, construction and maintenanceP3 is a service contract between a public authority and a private sector concessionaire, where the public authority pays the concessionaire to deliver infrastructure and related services, Typically, the concessionaire, who builds the infrastructure asset, is financially responsible for its condition and performance throughout the asset lifetime, or the duration of the agreement [13], or it describes a government service or private business venture which is funded and operated through a partnership of government and one or more private sector companies.P3 main features and benefits are, Delivers value for money, Engages in a competitive process to achieve the best project for the best cost, Transfers appropriate risks, Establishes performance standards and payment mechanisms, Maintains government involvement to oversee public interest, improve project delivery, better project discipline, reduce scope creep, faster procurementIn PPP, private sector has a role as engineer or constructor. Ownership, operation and financing are the public role. On the other hand a pure private is responsible for all matter. In BOOT final owner is public, but concession for a long period of time (25-30 year) is regarded to private. The ownership shifts from public to private as we move from PPP to BOOT. Also private sector accepts more risk and preparing capital investment in BOOT/BOT.