# Truss

Truss bridge for a single track railway, converted to pedestrian use and pipeline support. Outer vertical members are in tension, lower horizontal members in tension, shear, and bending, diagonal and top members are in compression. The central vertical member stabilizes the upper compression member, preventing it from buckling. If the top member is sufficiently stiff then this vertical element may be eliminated. If the lower chord is sufficiently resistant to bending and shear, the outer vertical elements may be eliminated. 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 post-stressed concrete and fabrication methods, such as automated welding, have significantly influenced the design of modern bridges.

In architecture and structural engineering, a truss is a structure consisting of straight slender members (struts) connected at joints.

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## Statics of trusses

In order for a truss to be rigid, it must be composed entirely of triangles.

In mathematical terms, we have the following condition for its proper stability (the truss will not collapse):

m = 2 · j - 3

where m is the total number of truss members and j is the total number of joints.

Such trusses are statically determinate, because the stress in each and every member can be completely determined, 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.

Some structures are built with more than this minimum number of truss members. Those structures can survive even when some of the members fail. They are called statically indeterminate structures, because they can store "internal tension" and "internal compression" that is impossible to determine from the external loads and the geometry of the truss.

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

### Vierendeel truss

A special truss is the Vierendeel truss, named after the Belgian engineer A. Vierendeel[1] (http://users.pandora.be/karel.roose/vierendeel/). 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.

## Analysis of trusses

Missing image
Cremona diagram for a plane truss

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 or bending or torsion or other kinds of stress.

### 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 (devised by Carl Culmann)
• or the analytical Ritter method.

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; VA</sup> (going up) load at A (going down), force in member 1 (going down/left), member 4 (going up/right) and closing with VA</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.

### 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 / σy, where F is the force in the member, γ is a safety factor (typically 1.5 but depending on building codes) and σy 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.

### 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.

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The_Little_Belt_Bridge_(1935).jpeg

### History of Metal Truss Bridges

Truss bridges were a common type of bridge to see built from the 1870s through the 1930s. Their ornate decorations and builder plaques, coupled with their complex design makes these bridges unparalleled in beauty. Examples of these bridges still remain across the United States, but their numbers are dropping rapidly, as they are demolished and replaced with new structures. Wrought iron bridges in the US started being built on a large scale in the 1870s. Bowstring truss bridges were a common truss design seen during this time, with their arched top chords. Companies like the Wrought Iron Bridge Company of Canton, Ohio and the King Bridge Company of Cleveland, Ohio became well-known companies, as they marketed their designs to different cities and townships. The bowstring truss design fell out of favor due to a lack of durability, and gave way to the Pratt truss design, which was stronger. Again, the bridge companies marketed their designs, with the Wrought Iron Bridge Company in the lead. As the 1880s and 1890s progressed, steel began to replace wrought iron as the preferred material. Other truss designed were used during this time, including the Warren truss, Whipple truss (double-intersection Pratt), and the Thatcher Truss. The railroad companies also designed some configurations which also were used on highways, including the Pensylvania truss and the Baltimore truss. In the eastern US, the Lenticular truss was patented and built by the Berlin Bridge and Iron Company. As the 1900s progressed, states began to evolve their road system management, and developed standard plans for bridges. The large bridge firms that held a sort of monopoly over the states started to fade away. By the 1910s, many states developed standard plan truss bridges, including steel warren pony truss bridges. As the 1920s and 1930s progressed, some states, like Pennsylvania continued to build steel truss bridges, including massive steel through truss bridges for long spans. Other states, like Michigan, utilized standard plan concrete girder and beam bridges, and only a limited number of truss bridges were built.

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