Truss

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In architecture and structural engineering, a truss is a structure comprising one or more triangular units constructed with straight slender members whose ends are connected at joints.

A plane truss is one where all the members and joints lie within a 2-dimensional plane, while a space truss has members and joints extending into 3 dimensions.

The Auckland Harbour Bridge from Watchman Island, west of it.

Truss bridge for a single track railway, converted to pedestrian use and pipeline support

1 History
2 Truss types
3 Statics of trusses
4 Analysis of trusses
5 See also
6 References
7 External links

The earliest trusses were made out of wood. The ancient Greeks used truss construction for their dwellings. In 1570 Andrea Palladio published I Quattro Libri dell'Architettura, which contained instructions for wooden trussed bridges.

Support structure under the Auckland Harbour Bridge.

Pre fabricated steel bow string roof trusses built 1942 for war department properties in Northern Australia.

A metal plate-connected wood truss is a roof or floor truss whose wood members are connected with metal connector plates.

There are two basic types of truss. The pitched truss or common truss is characterized by its triangular shape. It is most often used for roof construction. Some common trusses are named according to their web configuration. The chord size and web configuration are determined by span, load and spacing. The parallel chord truss or flat truss gets its name from its parallel top and bottom chords. It is often used for floor construction. A combination of the two is a truncated truss, used in hip roof construction.

Named for its distinctive shape, thousands of bow strings were used during World War II for aircraft hangars and other military buildings.

A Vierendeel bridge; note the lack of diagonal elements in the primary structure and the way bending loads are carried between elements

A special truss is the Vierendeel truss, named after the Belgian engineer Arthur Vierendeel [1], who developed the design in 1896. The Vierendeel truss is a truss 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. In this statically indeterminate structure the individual horizontal and vertical members are designed as beams. Diagonal bracing is omitted as the joints are designed to withstand the moments that occur at the ends of the members. Trusses of this type are used in some bridges (see Vierendeel bridge), and were also used in the frame of the Twin Towers of the World Trade Center [2]. By eliminating diagonal members, the creation of rectangular openings for windows and doors is simplified, since this truss can reduce or eliminate the need for compensating shear walls.

King Post Truss

Main article: King post

One of the simplest truss styles to implement, the king post consists of two angled supports leaning into a common vertical support.

Queen Post Truss

Main article: Queen post

The queen post truss, sometimes queenpost or queenspost, is similar to a king post truss in that the outer supports are angled towards the center 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. ^[1]

See Ithiel Town's lattice truss

A truss that is assumed to comprise of members that are connected by means of pin joints and which is supported at both ends by means of a hinged joints or 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 which may be subjected to an external load or force to remain static in space the following conditions are required to be true: the sum of all horizontal forces, and the sum of all vertical forces as well as the sum of all moments acting about the node need to equate to zero. Analysis of these conditions at each node yields the magnitude of the forces in each member of the truss. These may be compression or tension forces.

Trusses that are supported at more than two positions are said to be statically indeterminate and the application of Newton's Laws alone is not sufficient to determine the member forces.

In order for a truss with pin-connected members to be stable, it must be composed entirely of triangles. In mathematical terms, we have the following necessary condition for stability:

$m \ge 2j - r \qquad \qquad \mathrm{(a)}$

where m is the total number of truss members, j is the total number of joints and r is the number of reactions (equal to 3 generally) in a 2-dimensional structure.

When $m = 2 j - 3$ , the truss is said to be statically determinate because the (m+3) internal member forces and support reactions can then be completely determined by 2j equilibrium equations, once we know the external loads and the geometry of the truss. Given a certain number of joints, this is the minimum number of members, in the sense that if any member is taken out (or fails), then the truss as a whole fails. While the relation (a) is necessary, it is not sufficient for stability, which also depends on the truss geometry, support conditions and the load carrying capacity of the members.

Some structures are built with more than this minimum number of truss members. Those structures may survive even when some of the members fail. They are called statically indeterminate structures, because their member forces also depend on the relative stiffness of the members, in addition to the equilibrium condition.

Cremona diagram for a plane truss

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

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. If required, half of the weight of each member may be applied to the adjacent joints. Provided the members are long and slender, the moments transmitted through the joints are negligible and they can be treated as "hinges" or 'pin-joints'. Every member of the truss is then in pure compression or pure tension – shear, bending moment, and other more complex stresses are all practically zero. This makes trusses easier to analyze. This also makes trusses physically stronger than other ways of arranging material – because nearly every material can hold a much larger load in tension and compression than in shear, bending, torsion, or other kinds of force.

Structural analysis of trusses of any type can readily be carried out using a matrix method such as the matrix stiffness method, the flexibility method or the finite element method.

On the right is a simple, statically determinate flat truss with 9 joints and (2 x 9 − 3 =) 15 members. External loads are concentrated in the outer joints. Since this is a symmetrical truss with symmetrical vertical loads, it is clear to see that the reactions at A and B are equal, vertical and half the total load.

The internal forces in the members of the truss can be calculated in a variety of ways including the graphical methods:

Cremona diagram
Culmann diagram

Or the analytical Ritter method (method of sections).

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.

The diagonal and vertical members form the truss web, and carry the shear force. Individually, they are also in tension and compression, the exact arrangement of forces depending on the type of truss and again on the direction of bending. In the truss shown above right, the vertical members are in tension, and the diagonals are in compression.

In addition to carrying the static forces, the members serve additional functions of stabilizing each other, preventing buckling. In the picture, the top chord is prevented from buckling by the presence of bracing and by the stiffness of the web members.

The inclusion of the elements shown is largely an engineering decision based upon economics, being a balance between the costs of raw materials, off-site fabrication, component transportation, on-site erection, the availability of machinery and the cost of labor. In other cases the appearance of the structure may take on greater importance and so influence the design decisions beyond mere matters of economics. Modern materials such as prestressed concrete and fabrication methods, such as automated welding, have significantly influenced the design of modern bridges.

A building under construction in Shanghai. The truss sections stabilize the building and will house mechanical floors.

Once the force on each member is known, the next step is to determine the cross section of the individual truss members. For members under tension the cross-sectional area A can be found using A = F × γ / σ_y, where F is the force in the member, γ is a safety factor (typically 1.5 but depending on building codes) and σ_y is the yield tensile strength of the steel used.
The members under compression also have to be designed to be safe against buckling.

The weight of a truss member depends directly on its cross section -- that weight partially determines how strong the other members of the truss need to be. Giving one member a larger cross section than on a previous iteration requires giving other members a larger cross section as well, to hold the greater weight of the first member -- one needs to go through another iteration to find exactly how much greater the other members need to be. Sometimes the designer goes through several iterations of the design process to converge on the "right" cross section for each member. On the other hand, reducing the size of one member from the previous iteration merely makes the other members have a larger (and more expensive) safety factor than is technically necessary, but doesn't require another iteration to find a buildable truss.

The effect of the weight of the individual truss members in a large truss, such as a bridge, is usually insignificant compared to the force of the external loads.

After determining the minimum cross section of the members, the last step in the design of a truss would be detailing of the bolted joints, e.g., involving shear of the bolt connections used in the joints, see also shear stress.