## 41

1. The sum of the horizontal forces acting on the arch must be zero. This relates the horizontal components of the reactions:

2. The sum of the moments about the left support must be zero. For the arch in Fig. 4.1, this determines the vertical component of the reaction at the right support:

where P = load at distance kL from left support L = span

3. The sum of the moments about the right support must be zero. This gives the vertical component of the reaction at the left support:

4. The bending moment at the crown hinge must be zero. (The sum of the moments about the crown hinge also is zero but does not provide an independent equation for determination of the reactions.) For the right half of the arch in Fig. 4.1, Hh - VRb = 0, from which

The influence line for H for this portion of the arch thus is a straight line, varying from zero for a unit load over the support to a maximum of ab/Lh for a unit load at C.

Reactions of three-hinge arches also can be determined graphically by taking advantage of the fact that the bending moment at the crown hinge is zero. This requires that the line of action of reaction RR at the right support pass through C. This line intersects the line of action of load P at X (Fig. 4.1). Because P and the two reactions are in equilibrium, the line of action of reaction RL at the left support also must pass through X. As indicated in Fig. 4.1b, the magnitudes of the reactions can be found from a force triangle comprising P and the lines of action of the reactions.

For additional concentrated loads, the results may be superimposed to obtain the final horizontal and vertical reactions. Since the three hinged arch is determinate, the same four equations of equilibrium can be applied and the corresponding reactions determined for any other loading condition. It should also be noted that what is important is not the shape of the arch, but the location of the internal hinge in relation to the support hinges.

After the reactions have been determined, the stresses at any section of the arch can be found by application of the equilibrium laws (Art. 4.4).

(T. Y. Lin and S.D. Stotesbury, Structural Concepts and Systems for Architects and Engineers, 2d Ed., Van Nostrand Reinhold Company, New York.)

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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