Statically Indeterminate Structures And the Principle of Least Work

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Assume D E F to represent the line of resistance of this load. Then a certain thrust T, say, acts on the arch along the line D E, o A F B F B and another thrust along the line E F. Let G he the point at which the line D E cuts the vertical through 0. Now the action of the left-hand part of the arch on the right-hand part can be replaced by a force ARCHES. 41 equal to T, acting in the direction indicated in Fig. 14, whilst the action of the right-hand part of the arch on the left-hand part can b...e replaced by a numeri- cally equal force T acting as indicated in Fig. 13.
These forces T may be resolved into their horizontal and vertical components P and S, as shown. Now consider a section H J taken at the right-hand side of the arch. Then the bending moment at this section is evidently P(«+!/) + Sj!-W«i or if Pa= N, the above expression will reduce to N + Py + Si-ffj.
Turning now to the left-hand portion of the arch Fig. 12, it will be seen that the bending moment at H'J'is N + Py — S x where P and S now tend to bend the beam in opposite directions.


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