## Hinge

FIGURE 3.71 Beams and corresponding conjugate beams for various types of supports.

FIGURE 3.72 Deflection calculations for a cantilever by the conjugate beam method.

(a) Cantilever beam with a load on the end.

(b) Bending-moment diagram. (c) Conjugate beam loaded with M/EI distribution. (d) Deflection at B equals the bending moment at B due to the M/EI loading.

FIGURE 3.72 Deflection calculations for a cantilever by the conjugate beam method.

(a) Cantilever beam with a load on the end.

(b) Bending-moment diagram. (c) Conjugate beam loaded with M/EI distribution. (d) Deflection at B equals the bending moment at B due to the M/EI loading.

### 3.33.2 Moment-Area Method

Similar to the conjugate-beam method, the moment-area method is based on Eqs. (3.130a) to (3.130d). It expresses the deviation in the slope and tangential deflection between points A and B on a deflected beam:

Equation (3.131a) indicates that the change in slope of the elastic curve of a beam between any two points equals the area under the M/EI diagram between these points. Similarly, Eq. (3.131b) indicates that the tangential deviation of any point on the elastic curve with respect to the tangent to the elastic curve at a second point equals the moment of the area under the M/EI diagram between the two points taken about the first point.

For example, deflection SB and rotation dB at point B in the cantilever shown in Fig. 3.72a are f3L/4 M(x) ,

## 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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