Truss

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Template:Otheruses3 Image:RRTrussBridgeSideView.jpg

In architecture and structural engineering, a truss is a static structure consisting of straight slender members inter-connected at joints into triangular units.

1 History
2 Statics of trusses
- 2.1 Vierendeel truss
3 Analysis of trusses
4 See also
5 External links

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History

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

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Statics of trusses

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

<math> m \ge 2j - 3 \qquad \qquad \mathrm{(a)} </math>

where m is the total number of truss members and j is the total number of joints in a 2-dimensional structure.

When <math> m=2j - 3</math>, 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.

Image:Shanghai Shimao Plaza Construction.jpg

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Vierendeel truss

A special truss is the Vierendeel truss, named after the Belgian engineer A. Vierendeel[1]. Also described as a Vierendeel frame, this truss has rigid upper and lower beams, connected by vertical beams. The joints are also rigid. In this statically indeterminate truss, all members are subject to bending moments. Trusses of this type are used in some bridges, and were also used in the frame of the 'Twin Towers' World Trade Center. 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.

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Analysis of trusses

Image:Cremonadiagram.jpg

The analysis assumes that loads are applied to joints only, not to the members. The estimated weights of bars are either omitted or, if required, they are applied to the joints (a half of the weight to each of the bar joints). As long as loads are applied only at the joints of a truss, and the joints act like "hinges", every member of the truss is in pure compression or pure tension -- shear, bending moments, 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 stress. Structural analysis of trusses of any type can readily be carried out using a matrix method such as the matrix stiffness method or the flexibility method.

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Forces in members

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).

In the Cremona method, first the external forces and reactions are drawn (to scale) forming a vertical line in the lower right side of the picture. This is the sum of all the force vectors and is equal to zero as there is mechanical equilibrium.

Since the equilibrium holds for the external forces on the entire truss construction, it also holds for the internal forces acting on each joint. For a joint to be at rest the sum of the forces on a joint must also be equal to zero. Starting at joint A, the internal forces can be found by drawing lines in the Cremona diagram representing the forces in the members 1 and 4, going clockwise; V_A</sup> (going up) load at A (going down), force in member 1 (going down/left), member 4 (going up/right) and closing with V_A</sup>. As the force in member 1 is towards the joint, the member is under compression, the force in member 4 is away from the joint so the member 4 is under tension. The length of the lines for members 1 and 4 in the diagram, multiplied with the chosen scale factor is the magnitude of the force in members 1 and 4.

Now, in the same way the forces in members 2 and 6 can be found for joint C; force in member 1 (going up/right), force in C going down, force in 2 (going down/left), force in 6 (going up/left) and closing with the force in member 1.

The same steps can be taken for joints D, H and E resulting in the complete Cremona diagram where the internal forces in all members are known.

In a next phase the forces caused by wind must be considered. Wind will cause pressure on the upwind side of a roof (and truss) and suction on the downwind side. This will translate to asymmetrical loads but the Cremona method is the same. Wind force may introduce larger forces in the individual truss members than the static vertical loads.

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Design of members

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 (typically 240 MPa).
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.

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Design of joints

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.

Image:The Little Belt Bridge (1935).jpeg

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