# Statics : Trusses, Cables and Frames

The analysis of trusses, cables and frames is based on the static equilibrium of these structures. So, the equations of equilibrium are extensively used. There are certain differences as far as the definitions are concerned, but the analysis is similar.

• # Trusses

• ## Structure of Howrah Bridge

The Howrah bridge is a cantilever truss bridge, built in 1943. Typical structure of a cantilever bridge is shown below, with specific details mentioned.

The bridge contains an engineering structure, known as truss.

• ## What is a Truss?

A truss is a structure which contains only two-force members. A two-force member is the one, where force is applied at only two points. The members of a truss are pinned to each other. The members are generally straight and their ends are connected at joints. These joints are referred to as joints. In general, 5 (or more) triangular units are constructed and their ends are pinned at nodes.

The forces are considered to act only on the nodes. No torque acts on any member (An assumption, generally holds well). The forces can be tensile or compressive.

Trusses are used in construction of bridges, towers, roofs etc.

• ## Terms Associated with Trusses

I) Top Chords – The top beams in a truss, generally are in compression

II) Bottom Chords – The bottom beams in a truss, generally are in tension

III) Webs– The interior beams, the areas inside the webs are called panels

IV) Plane Truss– All members are in a plane

V) Space Truss – Members form a 3D structure

VI) Perfect Truss– Let $n$ be the number of members, $j$ be the number of joints and $r$ be the number of total reactions. Then a truss is perfect, when

$n = 2j-r$

VII) Redundant Truss$n > 2j-r$

VIII) Deficient Truss$n < 2j - r$

IX) Cantilever Truss– It is fixed on one side and free on the other. Howrah bridge contains 2 cantilever arms, as shown.

• ## Analysis of Truss – Method of Joints

We will be dealing with perfect planar trusses.

The deformation due to loads is neglected and the friction between the members is assumed to be negligible, so that no torque acts on the member. So, the external forces are in complete balance with the reactions. This proves that although a truss contains multiple members, as a whole, it behaves as a single unit. The general problem is to find the members with highest tension or compression. (Recall that the forces can be tensile or compressive, but they cannot be shear).

In the method of joints, we assume that all forces are tensile. We draw the free body diagram at each joint. The starting point is generally the load application point/ the supports. The unknowns are obtained by

$\sum F_x = 0, \sum F_y = 0, \sum M = 0$

Sometimes, Lami’s theorem is useful.

• ## Analysis of Truss – Method of Sections

The method is useful, when we are asked to find the forces in a limited number of members (generally three). We choose an imaginary plane, cutting those members and apply the conditions of equilibrium on either of the 2 sections. The unknown forces are obtained by solving a system of simultaneous equations.

• # Cables

The famous Vandre-Worli sea link is a bridge supported by cables. Such bridges are known as cable stayed bridges. (See figure below). Another application of cables is in the transmission lines.

• ## Analysis of Cables

The analysis of cables is done by considering the usual equilibrium equations, i.e.

$\sum F_x = 0, \sum F_y = 0, \sum M = 0$

The loads are generally point loads, thus the flexible and inextensible cable will sag or hog only at the load points. Weight of cable is neglected. The forces are assumed to act along the cable.

The analysis is similar to the analysis of trusses by the method of joints.

• ## Frames

A frame is a structure, made up of members pinned (hinged) together. The members may be straight or bent into a shape. The difference between a frame and a truss is that, there exists at least one multi-force member in a frame. Rest of the analysis is similar, because frame is in static equilibrium. The equations

$\sum F_x = 0, \sum F_y = 0, \sum M = 0$

hold true for frames as well.

The load analysis of machine frames plays an important role in the design and theory of machines.