2d And 3d Truss

[Llanque Mazurek]

In engineering, a truss is a structure that "consists of two-force members only, where the members are organized so that the assemblage as a whole behaves as a single object".[2] A "two-force member" is a structural component where force is applied to only two points. Although this rigorous definition allows the members to have any shape connected in any stable configuration, trusses typically comprise five or more triangular units constructed with straight members whose ends are connected at joints referred to as nodes.

In this typical context, external forces and reactions to those forces are considered to act only at the nodes and result in forces in the members that are either tensile or compressive. For straight members, moments (torques) are explicitly excluded because, and only because, all the joints in a truss are treated as revolutes, as is necessary for the links to be two-force members.

A planar truss is one where all members and nodes lie within a two-dimensional plane, while a space frame has members and nodes that extend into three dimensions. The top beams in a truss are called 'top chords' and are typically in compression, the bottom beams are called 'bottom chords', and are typically in tension. The interior beams are called webs, and the areas inside the webs are called panels,[3] or from graphic statics (see Cremona diagram) 'polygons'.[4]

Truss derives from the Old French word trousse, from around 1200 AD, which means "collection of things bound together".[5][6] The term truss has often been used to describe any assembly of members such as a cruck frame[7][8] or a couple of rafters.[9][10] One engineering definition is: "A truss is a single plane framework of individual structural member [sic] connected at their ends of forms a series of triangle [sic] to span a large distance".[11]

A truss consists of typically (but not necessarily) straight members connected at joints, traditionally termed panel points. Trusses are typically (but not necessarily[12]) 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.[13] In comparison, both the angles and the lengths of a four-sided figure must be fixed for it to retain its shape.

The simplest form of a truss is one single triangle. This type of truss is seen in a framed roof consisting of rafters and a ceiling joist,[14] and in other mechanical structures such as bicycles and aircraft. Because of the stability of this shape and the methods of analysis used to calculate the forces within it, a truss composed entirely of triangles is known as a simple truss.[15] However, a simple truss is often defined more restrictively by demanding that it can be constructed through successive addition of pairs of members, each connected to two existing joints and to each other to form a new joint, and this definition does not require a simple truss to comprise only triangles.[12] The traditional diamond-shape bicycle frame, which utilizes two conjoined triangles, is an example of a simple truss.[16]

The depth of a truss, or the height between the upper and lower chords, is what makes it an efficient structural form. A solid girder or beam of equal strength would have substantial weight and material cost as compared to a truss. For a given span, a deeper truss will require less material in the chords and greater material in the verticals and diagonals. An optimum depth of the truss will maximize the efficiency.[18]

A space frame truss is a three-dimensional framework of members pinned at their ends. A tetrahedron shape is the simplest space truss, consisting of six members that meet at four joints.[15] Large planar structures may be composed from tetrahedrons with common edges, and they are also employed in the base structures of large free-standing power line pylons.

Truss members are made up of all equivalent equilateral triangles. The minimum composition is two regular tetrahedrons along with an octahedron. They fill up three dimensional space in a variety of configurations.

The Pratt truss was patented in 1844 by two Boston railway engineers,[19] Caleb Pratt and his son Thomas Willis Pratt.[20] The design uses vertical members for compression and diagonal members to respond to tension. The Pratt truss design remained popular as bridge designers switched from wood to iron, and from iron to steel.[21] This continued popularity of the Pratt truss is probably due to the fact that the configuration of the members means that longer diagonal members are only in tension for gravity load effects. This allows these members to be used more efficiently, as slenderness effects related to buckling under compression loads (which are compounded by the length of the member) will typically not control the design. Therefore, for given planar truss with a fixed depth, the Pratt configuration is usually the most efficient under static, vertical loading.

The Southern Pacific Railroad bridge in Tempe, Arizona is a 393 meter (1,291 foot) long truss bridge built in 1912.[22][23] The structure is composed of nine Pratt truss spans of varying lengths. The bridge is still in use today.

Thousands of bowstring trusses were used during World War II for holding up the curved roofs of aircraft hangars and other military buildings. Many variations exist in the arrangements of the members connecting the nodes of the upper arc with those of the lower, straight sequence of members, from nearly isosceles triangles to a variant of the Pratt truss.

The queen post truss, sometimes queenpost or queenspost, is similar to a king post truss in that the outer supports are angled towards the centre of the structure. The primary difference is the horizontal extension at the centre which relies on beam action to provide mechanical stability. This truss style is only suitable for relatively short spans.[25]

Lenticular trusses, patented in 1878 by William Douglas (although the Gaunless Bridge of 1823 was the first of the type), have the top and bottom chords of the truss arched, forming a lens shape. A lenticular pony truss bridge is a bridge design that involves a lenticular truss extending above and below the roadbed.

American architect Ithiel Town designed Town's Lattice Truss as an alternative to heavy-timber bridges. His design, patented in 1820 and 1835, uses easy-to-handle planks arranged diagonally with short spaces in between them, to form a lattice.

The Vierendeel truss is a structure where the members are not triangulated but form rectangular openings, and is a frame with fixed joints that are capable of transferring and resisting bending moments. As such, it does not fit the strict definition of a truss (since it contains non-two-force members): regular trusses comprise members that are commonly assumed to have pinned joints, with the implication that no moments exist at the jointed ends. This style of structure was named after the Belgian engineer Arthur Vierendeel,[26] who developed the design in 1896. Its use for bridges is rare due to higher costs compared to a triangulated truss.

The utility of this type of structure in buildings is that a large amount of the exterior envelope remains unobstructed and can be used for windows and door openings. In some applications this is preferable to a braced-frame system, which would leave some areas obstructed by the diagonal braces.

A truss that is assumed to comprise members that are connected by means of pin joints, and which is supported at both ends by means of hinged joints and rollers, is described as being statically determinate. Newton's Laws apply to the structure as a whole, as well as to each node or joint. In order for any node that may be subject to an external load or force to remain static in space, the following conditions must hold: the sums of all (horizontal and vertical) forces, as well as all moments acting about the node equal zero. Analysis of these conditions at each node yields the magnitude of the compression or tension forces.

In order for a truss with pin-connected members to be stable, it does not need to be entirely composed of triangles.[12] In mathematical terms, the following necessary condition for stability of a simple truss exists:

Some structures are built with more than this minimum number of truss members. Those structures may survive even when some of the members fail. Their member forces depend on the relative stiffness of the members, in addition to the equilibrium condition described.

Because the forces in each of its two main girders are essentially planar, a truss is usually modeled as a two-dimensional plane frame. However if there are significant out-of-plane forces, the structure must be modeled as a three-dimensional space.

The analysis of trusses often assumes that loads are applied to joints only and not at intermediate points along the members. The weight of the members is often insignificant compared to the applied loads and so is often omitted; alternatively, half of the weight of each member may be applied to its two end joints. Provided that the members are long and slender, the moments transmitted through the joints are negligible, and the junctions can be treated as "hinges" or "pin-joints".

These simplifications make trusses easier to analyze. Structural analysis of trusses of any type can readily be carried out using a matrix method such as the direct stiffness method, the flexibility method, or the finite element method.

A truss can be thought of as a beam where the web consists of a series of separate members instead of a continuous plate. In the truss, the lower horizontal member (the bottom chord) and the upper horizontal member (the top chord) carry tension and compression, fulfilling the same function as the flanges of an I-beam. Which chord carries tension and which carries compression depends on the overall direction of bending. In the truss pictured above right, the bottom chord is in tension, and the top chord in compression.

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