Wednesday, 2 May 2012
Tuesday, 1 May 2012
Trusses
In architecture and structural engineering, a truss is a structure comprising one or more triangular units constructed with straight members whose ends are connected at joints referred to as nodes. External forces and reactions to those forces are considered to act only at the nodes and result in forces in the members which are either tensile or compressive forces. Moments (torques) are explicitly excluded because, and only because, all the joints in a truss are treated as revolutes.
A planar truss is one where all the members and nodes lie within a two dimensional plane, while a space truss has members and nodes extending into three dimensions.
A truss consists of straight members connected at joints, traditionally termed panel points. Trusses are composed of triangles because of the structural stability of that shape and design. A triangle is the simplest geometric figure that will not change shape when the lengths of the sides are fixed.[1] In comparison, both the angles and the lengths of a four-sided figure must be fixed for it to retain its shape. -wikipedia-
Monday, 30 April 2012
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:
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.
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.
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.
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.
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
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.
Subscribe to:
Posts (Atom)