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