Shear (V)

Shear (V) is the tendency for one part of a beam to slide past another part. The magnitude of the shear at any section is equal to the algebraic sum of loads and reactions acting perpendicular to that section.



SIGN CONVENTION
If the tendency of the section to the left of the cut is to move upward, the shear is positive; if it has a tendency to move down, it is negative.




For simplicity in obtaining the correct sign, one may say that it is equal to the algebraic sum of loads and reactions to the left of the cut. Note that the internal shear on the end of the FBD acts in the opposite direction of the algebraic sum of the loads and reactions in order to balance these forces (SV = 0).

A shear diagram is a graphic representation of the shear at every point along the length of a member.

To plot a shear diagram on a beam, the shear must be calculated at each point along the length of the beam. One way is to simply plot the shear as the algebraic sum of the loads and reactions acting perpendicular to the beam at the left side of each increment along the length of the beam. Positive values are shown above and negative values below a reference axis.

How to draw a SHEAR DIAGRAM
Start at the left end and plot the external shear values with regard to the following:

• The shear diagram is the graphic representation of the shear force at successive points along the beam. Forces acting upward are assumed positive and downward forces negative. 
• The shear force (V) at any point is equal to the algebraic sum of the external loads and reactions, perpendicular to the beam, to the left of that point. 
• Since the entire beam must be in equilibrium (sum of V = 0), the shear diagram must close to zero at the right end. 
• Consider the loading for increments along the length of the beam in order to determine the shape of the curve.
  
  

  if there is no change in the load along the incremental length under consideration, the shear curve is a straight horizontal line (or a curve of zero slope). The slope at any point is defined as the tangent to the curve at that point.
  
  if a load exists, but does not change in magnitude over successive increments (uniformly distributed), the slope of the shear curve is constant and non-horizontal.
  
  if a load exists, and increases in magnitude over successive increments, the slope of the shear curve is positive (approaches the vertical); if the magnitude decreases, the slope of the shear curve is negative (approaches the horizontal).

• Abrupt changes in loading cause abrupt changes in the slope of the shear curve. Concentrated loads produce vertical lines (a jump) in the shear curve.

Supports

Structural elements carry their loading to other elements or the ground through connections or supports. In order to be able to analyze a structure it is necessary to be clear about the forces that can be resisted at each support. The actual behaviour of a support or connection can be quite complicated. So much so, that if all of the various conditions were considered, the design of each support would be a terribly lengthy process. And yet, the conditions of the supports is very important to the behaviour of the elements which are being supported.

In order to facilitate the analysis of a structure, it is often necessary to idealize the behaviour of a support. This is similar to the massless, frictionless pulley in physics homework problems. Even though these pulleys do not exist, they are useful to enable learning about certain issues. It is important to realize that all of the grpahical representations of supports are idealizations of a real connection. Effort should be made to search out and compare the reality with the grpahical and/or numerical model. It is often very easy to forget that the reality can be strikingly different than what is assumed in the idealization!

The four types of supports that can be found in structures are; roller, frictionless surface, pinned, and fixed. The type of support affects the forces and moments that are used to represent these supports. It is expected that these representative forces and moments, if properly calculated, will bring about equilibrium in the structural element.

There is not a single accepted graphical method to represent each of these support types. However, no matter what the representation looks like, the forces that the type can resist is indeed standardized. Almost all supports can be assigned to one of the four types. The usual methods of graphical designation are:

These supports can be at the ends or at any intermediate points along the structural member. They are used to draw free body diagrams (FBD's) which aid in the analysis of all structural members. Each of the support types is a representation of an actual support.

ROLLER SUPPORTS
Roller supports are free to rotate and translate along the surface upon which the roller rests. The surface can be horizontal, vertical, or sloped at any angle. The resulting reaction force is always a single force that is perpendicular to, and away from, the surface. Roller supports are always found at at least one end of long bridges so that forces due to thermal expansion and contraction are minimalized. These supports can also take the form of rubber bearings which are designed to allow a limited amount of lateral movement. A roller support cannot provide any resistance to lateral forces. The representation of a roller support includes one force perpendicular to the surface.

FRICTIONLESS SUPPORTS
Frictionless surface supports are similar to roller supports. The resulting reaction force is always a single force that is perpendicular to, and away from, the surface. They too are often found as supports for long bridges or roof spans. These are often found supporting large structures in zones of frequent seismic activity. The representation of a frictionless support includes one force perpendicular to the surface.

PINNED SUPPORTS
A pinned support can resist both vertical and horizontal forces but not a moment. They will allow the structural member to rotate, but not to translate in any direction. Many connections are assumed to be pinned connections even though they might resist a small amount of moment in reality. It is also true that a pinned connection could allow rotation in only one direction; providing resistance to rotation in any other direction. The knee can be idealized as a connection which allows rotation in only one direction and provides resistance to lateral movement. The design of a pinned connection is a good example of the idealization of the reality. A single pinned connection is usually not sufficient to make a structure stable. Another support must be provided at some point to prevent any rotation of the structure. The representation of a pinned support includes both horizontal and vertical forces.

FIXED SUPPORTS
Fixed supports can resist vertical and horizontal forces as well as a moment. Since they restrain both rotation and translation, they are also known as rigid supports. This means that a structure only needs one fixed support in order to be stable. All three equations of equilibrium can be satisfied. A flagpole set into a concrete base is a good example of this kind of support. The representation of fixed supports always includes two forces (horizontal and vertical) and a moment.

Moments

The Moment of a force is a measure of its tendency to cause a body to rotate about a specific point or axis. This is different from the tendency for a body to move, or translate, in the direction of the force. In order for a moment to develop, the force must act upon the body in such a manner that the body would begin to twist. This occurs every time a force is applied so that it does not pass through the centroid of the body. A moment is due to a force not having an equal and opposite force directly along it's line of action.

Imagine two people pushing on a door at the doorknob from opposite sides. If both of them are pushing with an equal force then there is a state of equilibrium. If one of them would suddenly jump back from the door, the push of the other person would no longer have any opposition and the door would swing away. The person who was still pushing on the door created a moment. 

The magnitude of the moment of a force acting about a point or axis is directly proportinoal to the distance of the force from the point or axis. It is defined as the product of the force (F) and the moment arm (d). The moment arm or lever arm is the perpendicular distance between the line of action of the force and the center of moments. The Center of Moments may be the actual point about which the force causes rotation. It may also be a reference point or axis about which the force may be considered as causing rotation. It does not matter as long as a specific point is always taken as the reference point. The latter case is much more common situation in structural design problems. A moment is expressed in units of foot-pounds, kip-feet, newton-meters, or kilonewton-meters. A moment also has a sense; it is either clockwise or counter-clockwise. The most common way to express a moment is

Moment = Force x DistanceM = F x d

The example shown is a wrench on a nut (at C) that has a force applied to it. The force is applied at a distance of 12 inches from the nut. The center of moments could be point C or points A or B. The moment arm for calculating the moment around point C is 12 inches. The magnitude of the moment of the 100 pound force about point C is 12 inches multiplied by the force of 100 pounds to give a moment of 1200 inch-pounds (or 100 foot-pounds). Similarly, the moment about point A can be found to be 800 inch-pounds.

The direction of the rotation is important to understand in order to describe its effect on the body (structure). A moment will cause either a clockwise or counter-clockwise rotation about the center of moments. It is essential that the direction of rotation about the center of moments be understood. Since an international convention does not exist, in this course, a clockwise rotation about the center of moments will be considered as positive; a counter-clockwise rotation about the center of moments will be considered as negative.

A moment causes a rotation about a point or axis. Thus, the moment of a force taken about any point that lies on its own line of action is zero. Such a force cannot cause a rotation because the moment arm is non-existant.

In this second example, a 200 pound force is applied to the wrench. The moment of the 200 pound force applied at C is zero because 200 pounds x 0 inches = 0 inch-pounds (Fxd = M). In other words, there is no tendency for the 200 pound force to cause the wrench to rotate the nut. One could increase the magnitude of the force until the bolt finally broke off.

A moment can also be considered to be the result of forces detouring from a direct line drawn between the point of loading of a system and its supports. In this case, the blue force is an eccentric force. In order for it to reach the base of the column, it must make a detour through the beam. The greater the detour, the greater the moment. The most efficient structural systems have the least amount of detours possible. 

Equilibrium

Architectural structures are normally stationary. Most clients, building officials and designers prefer that their structures remain static rather than move dynamically. There are specific loading conditions which are dynamic loads, but in each and every case a return to a stable and static state is desireable. Such a condition is known as equilibrium.

Various states of static equilibrium are experinced throughout one's life. Think of the "teeter-totter" at a playground or of a game of "tug-of-war." In the first case, two or more individuals sit upon a board which has been fixed to a fulcrum which allows rotation. If each of the individuals on the teeter totter weight exaclty the same amount and sit at exactly the same distance from the fulcrum the teeter-totter will not move. A state of equilibrium has been achieved. The two will remain at rest until an action takes them out of equilibrium.

Such an action could be the addition of another person to the system or it could be that one of the original two would change their position slightly. In either case, the teeter-totter would most likely swing to one side and rest upon the ground. A new state of equilibrium would have been found. 

In this case, another boy climbed on behind the one already sitting on the right. In order to put the system back into equilibrium the girl on the left had to move backwards along the board until she was far away. The moment two boys created around the fulcrum doubled when the second boy climbed on. The girl knew that the only way that she could increase the magnitude of the moment she created would be to increase the moment arm. Thus she moved back until she was twice as far from the fulcrum. Now the system would be back in equilibrium.

Another example of a state of equilibrium is the game of "tug-of-war." A rope is pulled taught between two teams; each pulls with a force that equals the force of the oppostie team. Assume in the figure that each team is pulling with a force of 220 pounds. As long as each team maintained a pull of 220 pounds the system is in equilibrium. If during this time a device would be inserted between the two teams to measure the magnitude of the tension force that the rope has anywhere along its length, it would read 220 pounds at each and every point. This would be true at ANY point along the rope.

A structure is in equilibrium when all forces or moments acting upon it are balanced. This means that each and every force acting upon a body, or part of the body, is resisted by either another equal and opposite force or set of forces whose net result is zero. Issac Newton addressed this issue when he noted that a body is at rest will remain at rest until acted upon by an external force. Every structure that can be seen to remain standing on a daily basis is in equilibrium; it is at rest and each of its members, combination of its members or any part of a member that is supporting a load are also at rest. There is a net result of zero in all directions for all of the applied loads and reactions.

In both of the the illustrations above, the split bamboo beam is held in a state of equilibrium. The beam of the top figure illustrates the sytsm at rest under it's own weight. The lower figure shown another state in which a pair of scissors sits at one end and a thermos of coffee on the opposite side near the fulcrum. All of the forces and moments are balanced so that the system is in a stable equilibrium.

There are two types of equilibrium; External and Internal. External equilibrium encompasses the loads upon, and reactions of, a structural system as a whole. Internal equilibrium describes the various forces the are acting within every member of the system. There are conditions of equilibrium that must be satisfied for each case. These are:

Sum of All Vertical Forces (Fy) = 0

Sum of All Horizontal Forces (Fx) = 0

Sum of All Moments (Mz) = 0

(Sum of All Forces (Fz) = 0)

(Sum of All Moments (My) = 0)

(Sum of All Moments (Mx) = 0)

These six equations are all that can be used to determine every one of the forces that are acting with a structure. They are few, but very powerful. The first three are the most common equations and will be utilized in all of the problems asociated with thid course. The other three are only necessary when considering three-dimensional force systems.

What is a Force?

The one constant around the world is the action of gravity upon each and every structure that is erected. The primary function of all structural design is to make a building stand-up. Understanding architectonics will enable a designer to include these issues as part of a design language that will create a significantly clearer architectural expression. Primary to this study is the concept of a force. A force is actually a very abstract conception. It can be defined, but it cannot become physically apparent until it meets resistance. Imagine a six foot tall block of ice sliding along a frictionless surface laid inside of a hockey rink. If a person props herself against the wall and tries to stop the ice she will then perceive the force imparted by the block of ice. She transfers the force of the block of ice into a force that moves into the ground. Thus, if the ground could actually experience a push, it would as well as the slide is arrested.

A "force" is an action that changes, or tends to change, the state of motion of the body upon which it acts. It is a vector quantity that can can be represented either mathematically or graphically.

A complete description of a force MUST include its:

1. MAGNITUDE
2. DIRECTION and SENSE
3. POINT OF APPLICATION

The Magnitude is most often expressed in the units of pounds (lbs), newtons(N), kilo-newtons(KN) or kilo-pounds(KIPS; 1 Kip = 1000 lbs). The magnitude is represented graphically by the scaled length of the arrow which represents the force. Graphic statics depends upon the accurate representation of the magnitude of each force acting upon a body.

The Direction of a force is discerned by oberserving the line of action of the vector of the force. The Sense refers to the direction of the movement of the vector along that line of action. The sense is always represented on the vector by an arrowhead. These two attributes can be either a written verbal description or more conveniently expressed in terms of 360 degrees. In the later case, one begins with zero and increases clock-wise with the direction of the arrowhead until 360 is reached. The sense of the force can also be expressed as a positive or negative sign. This is especially useful when combining forces algebraically. It is very important not to confuse the direction and sense. The direction always relates to the line of action of the vector, and the sense is the way in which the vector would move along that line. In the example, the sense of the 100 lb force would be "down and to the left" or "210 degrees."

The Point of Application is often overlooked in the description of a force. However, it is fully as important as the magnitude of the force! It is the exact location of the application of a force on a body. It can either be a relative measurement or a set of coordinates.

A 100 kip force is applied to the stone column in the diagram. The colum will experience this force at every point along the line of action of the force. As a matter of fact, the force will also be transferred to the ground which is supporting the column. Thus, the earth below the column along the line of action of the force will also experience the 100 kip load. This illustrates the Principle of Transmissibility. The point of application of an external force acting on a body (structure) may be transmitted anywhere along the force's line of action without affecting the other external forces (reactions and loads) acting on that body. This means that there is NO NET CHANGE in the static effect upon any body if the body is in equilibrium. This can be illustrated with the following diagram. 

Assume that a beam is supported at its ends. A load is applied to the top of the beam that is acting downward. This load could be a person standing on the beam. The load creates reactions that push up at the two points of support. The line of action of the load (person) on the beam also passes through a hook that is attached to the underside of the beam. Now, if the person standing on top of the beam would climb down and hold on to the hook exactly below the point where they were previously located so that the lines of action were exactly the same, the reactions at the ends of the beams would not change. This is because the load of the person is still acting along the same line of action. As long as a load is applied at any point along the line of action the external reactions will not change.

The truss above is loaded with a force that is applied at point C. This load creates reactions at the two supports A and B. The load on the truss could move anywhere along the line of action and the external reactions at A and B would remain the same. That means that if the load was applied at points D, E, F or G the reactions at A and B would not change. Note that the only point of discussion at this moment is the fact that the external reactions will not change. It is clear that the internal forces will vary greatly within the truss as the force is moved along the line of action.

This strangly shaped beam is another example of a structure that is loaded at a specific point, namely E. In order for the structure to remain at rest there must be reactions of some kind at points A and B. The reaction at B is a tension reaction since it is a cable. Again, if the load was applied at points C, D, E, F or G on this rigid body (structure) the reactions at A and at B would remain exactly the same. ONLY the EXTERNAL forces (reactions) remain unchanged. Some of the internal resisting forces within the elements of the structure change as the load is applied at different points along its line of action. This illustrates one important difference between INTERNAL and EXTERNAL forces. The Principle of Transmissibility applies to any body (blobs, balloons, simple beams, crooked beams, trusses, shells, etc.). It is independent of the body's size or shape.

What is STRUCTURE?

Structure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society. From a child's verbal description of a snowflake, to the detailed scientific analysis of the properties of magnetic fields, the concept of structure is now often an essential foundation of nearly every mode of inquiry and discovery in science, philosophy, and art.^[1] In early 20th-century and earlier thought, form^{[disambiguation needed ]} often plays a role comparable to that of structure in contemporary thought. The neo-Kantianism of Ernst Cassirer (cf. his Philosophy of Symbolic Forms, completed in 1929 and published in English translation in the 1950s) is sometimes regarded as a precursor of the later shift to structuralism and poststructuralism.^[2]

The description of structure implicitly offers an account of what a system is made of: a configuration of items, a collection of inter-related components or services. A structure may be ahierarchy (a cascade of one-to-many relationships), a network featuring many-to-many links, or a lattice featuring connections between components that are neighbors in space. -Wikipedia-

One of the greatest problems of designing today is the fact that engineers can solve ANY problem. Anything can be built. Structural "realities" are perceived as no longer imposing limitations upon the design architect. Form does not have to be dictated by structure or even follow a function. Many of the seemingly undeniable "truths" of architectural design have been rendered meaningless. Yet, gravity persists despite this incredible freedom of choice. Buildings must stand up at the end of a real or virtual working day.

Architectural design cannot be based soley upon one of the many aspects that make up the profession. It surely should never be based on architectonics alone. Yet, structure is the very raw material of building. To use structure without understanding its implications is irresponsible and results in meaningless formalism. An architect is supposed to be a specialist in building, not just a creator of arbitrary form. The word structure can be used alone or in conjunction with many other descriptive words. Dictionaries can be consulted to find the following definitions:

manner of construction

the arrangement of particles or parts in a substance or body

arrangement or interrelation of parts as dominated by the general character of the whole

the aggregate of elements of an entity in their relationships to each other

the composition of conscious experience with its elements and their combinations

something that is constructed

something that is arranged in a definate pattern of organization

the action of building

There are multitudes of different scales at which one should perceive structures. Each scale reveals beauty and provides an amazing amount of information at the same time. Seeing the information at each level of perception is critical. Learning to see the structure of the world around us is an important part of life and of this course. It is critical to the success of an architect that she/he be able to see beyond the skin of a building; beyond the surfaces of a space and into the load-bearing structure. This is the fabric from which space is molded. Understanding the nature of the fabric enables one to create the seams between spaces. Understanding the load-bearing structure of a building is to understand the space that is being created.

There is a fundamental rightness in a structurally correct concept. It leads to an economy of means that can be understood by all. Designs which are inherently structurally correct are often perceived as objects of great beauty, even if only truely comprehended by few. One can find structure in everything. Look at landscapes, cities, roofs, walls, and at the veins in a leaf from both afar and as close as you can. Record what you see. What are the similarities? What is unique about each? Look at the:

• external expression of internal structure
• relationship between natural and built forms
• relationship between size and internal forces
• articulation and supporting structure of vertical surfaces
• articulation and supporting structure of horizontal surfaces
• nature of scale in relation to the elements of a system
• nature of scale in relation to a system
• openings in a wall
• relationship between loading and structural form

Tutorial 7 : Trusses

Tutorial 6 : Shear And Moment Diagram

Tutorial 5 : Support

Tutorial 4 : Moment

Toturial 3 : Moment

Toturial 2 : Equilibrium system for cOncurrent Force

Toturial 1 : Reaction Force

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Bob the Builder!
Can we fix it?
Bob the Builder!
Yes we can!

Scoop, Muck and Dizzy,
And Roley too.
Lofty and Wendy
Join the crew.
Bob and the gang
Have so much fun.
Working together
They get the job done.

Bob the Builder!
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Bob the Builder!
Yes we can! (I think so!)