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## Q. 6.3

Bridge Design

The loads a bridge structure must support and pin supports where the structure is to be attached are shown in Fig. 6.14(1). Assigned to design the structure, a civil engineering student proposes the structure shown in Fig. 6.14(2). What are the axial forces in the members?

## Verified Solution

The vertical members AG, BH, CI, DJ, and EK are subjected to compressive forces of magnitude F. From the free-body diagram of joint C, we obtain $T_{BC} = T_{CD} = -1.93F$. We draw the free-body diagram of joint B in Fig. a.

From the equilibrium equations

$\sum{F_{x} } = – T_{AB} \cos α + T_{BC} \cos 15º = 0$,

$\sum{F_{y} } = – T_{AB} \sin α + T_{BC} \sin 15º -F = 0$,

we obtain $T_{AB} = —2.39F and α = 38.8°$ . By symmetry,$T_{DE} = T_{AB}$. The axial forces in the members are shown in Table 6. 1

Table 6.1  Axial forces in the members of the bridge structure.

 Members Axial Force AG, BH, CI, DJ, EK F (C) AB, DE 2.39 F (C) BC, CD 1.93 F (C)

Design Issues
The bridge was an early application of engineering. Although initially the solution was as primitive as laying a log between the banks, engineers constructed surprisingly elaborate bridges in the remote past. For example, archaeologists have identified foundations of the seven piers of a 120-m (400-ft) highway bridge over the Euphrates that existed in Babylon at the time of Nebuchadnezzar II (reigned 605-562 B.C.).
The basic difficulty in bridge design is that a single beam extended between the banks will fail if the distance between banks, or span, is too large.
To meet the need for bridges of increasing strength and span, civil engineers created ingenious and aesthetic designs in antiquity and continue to do so today.
The bridge structure proposed by the student in Example 6.3. called an arch, is an ancient design. Notice in Table 6.1 that all the members of the structure are in compression. Because masonry (stone, brick, or concrete) is weak in tension but very strong in compression, many bridges made of these materials were designed with arched spans in the past. For the same reason, modem concrete bridges are often built with arched spans (Fig. 6.15).

Unlike masonry, wood and steel can support substantial forces in both compression and tension. Beginning with the wooden truss bridges designed by the architect Andrea Palladio (1518-1580), both of these materials have been used to construct a large variety of trusses to support bridges. For example, the forces in Fig. 6.14(1) can be supported by the Pratt truss shown in Fig. 6.16. Its members are subjected to both tension and compression (Table 6.2). The Forth Bridge (Fig. 6.17) has a truss structure.

 Table 6.2 Axial forces in the members of the Pratt truss. Members Axial Force AB, BC, CD, DE 1.5F (T) AG. EI 2.12F (C) CG.CI 0.71 F (T) GH.HI 2F (C) BG.DI F (T) CH 0

Truss structures are too heavy for the largest bridges. (The Forth Bridge contains 58.000 tons of steel.) By taking advantage of the ability of relatively light cables to support large tensile forces, civil engineers use suspension structures to bridge very large spans. The system of five forces we are using as an example can be supported by the simple suspension structure in Fig. 6.18. In effect, the compression arch used since antiquity is inverted. (Compare Figs. 6.14(2) and 6.18.) The loads in Fig. 6.18 are “suspended” from members AB, BC, CD. and DE. Every member of this structure except the towers AG and EK is in tension (Table 6.3).

 Table 6.3 Axial forces in the members of the suspension structure. Members Axial Force BH. CI, DJ F (T) AB,DE 2.39 F (T) BC,CD 1.93F (T)

The largest existing bridges, such as the Golden Gate
Bridge (Fig. 6.19), consist of cable-suspended spans supported by towers.