## M x ry r A cos sin sin 14

By application of virtual work, the downward vertical displacement dV of the crown produced by the unit vertical load is obtained by dividing the rib into elements of length As and computing dV = | ^ (4.31)

where E = modulus of elasticity of steel

I = moment of inertia of cross section about horizontal axis

The summation extends over the length of the rib.

FIGURE 4.7 Reactions for a three-hinged rib (a) for a vertical downward load and (b) for a horizontal load at the crown.

FIGURE 4.7 Reactions for a three-hinged rib (a) for a vertical downward load and (b) for a horizontal load at the crown.

Next, apply at the crown a unit horizontal load acting to the right (Fig. 4.7b). This produces vertical reactions V1 = - Vv = -h/2r = -(1 - sin fa)/2 cos fa and H1 =

For fa < fa < it/2, the bending moment at any point (x, y) due to the unit horizontal load at the crown is

= _hx + y = h / cos± _ ! + sin * - sin *i , (4.32)

By application of virtual work, the displacement dH of the crown to the right induced by the unit horizontal load is obtained from the summation over the arch rib dH = £ ^ (4.34)

Now, apply an upward vertical load PV on rib 1C1' at (xp, yp), with the rib still disconnected from the other ribs. This produces the following reactions:

V 1 = -PV —1 = —;V 1 + ^^r I (4.35) 1 V 2r 2 \ cos fa/

1 1 1 h 2 1 - sin fa v y where fa is the angle that the radius vector to the load point (xp, yp) makes with the horizontal <^/2. By application of virtual work, the horizontal and vertical components of the crown displacement induced by PV may be computed from

## 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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