M dx J 2eix Lx dx 2EIJ

where C3 is a constant. When x = L/2, O = 0. Hence C3 = -wL3/24EI, and the equation for slope is w

FIGURE 3.30 Diagrams for moments applied at both ends of a simple beam.

FIGURE 3.32 Simple beam with concentrated load at the third points.

FIGURE 3.31 Shears, moments, and deformations for uniformly loaded simple beam.

FIGURE 3.30 Diagrams for moments applied at both ends of a simple beam.

(e) ELASTIC CURVE

FIGURE 3.31 Shears, moments, and deformations for uniformly loaded simple beam.

FIGURE 3.33 Diagrams for simple beam loaded at quarter points.

FIGURE 3.32 Simple beam with concentrated load at the third points.

FIGURE 3.33 Diagrams for simple beam loaded at quarter points.

FIGURE 3.34 Diagrams for concentrated load on a simple beam.

FIGURE 3.36 Concentrated load on a beam overhang.

FIGURE 3.35 Symmetrical triangular load on a simple beam.

FIGURE 3.34 Diagrams for concentrated load on a simple beam.

FIGURE 3.35 Symmetrical triangular load on a simple beam.

FIGURE 3.37 Uniformly loaded beam with overhang.

FIGURE 3.36 Concentrated load on a beam overhang.

FIGURE 3.37 Uniformly loaded beam with overhang.

FIGURE 3.38 Shears, moments, and deformations for moment at one end of a cantilever.

FIGURE 3.39 Diagrams for concentrated load on a cantilever.

FIGURE 3.38 Shears, moments, and deformations for moment at one end of a cantilever.

FIGURE 3.39 Diagrams for concentrated load on a cantilever.

Cantilever Beam With Triangular Load

FIGURE 3.40 Shears, moments, and deformations FIGURE 3.41 Triangular load on cantilever with for uniformly loaded cantilever.

maximum at support

Triangular Load

FIGURE 3.42 Uniform load on beam with one end FIGURE 3.43 Triangular load on beam with one fixed, one end on rollers. end fixed, one end on rollers.

The deflected-shape curve at any point is, by Eq. (3.80b), w

where C4 is a constant. In this case, when x = 0, S = 0. Hence C4 = 0, and the equation for deflected shape is w

as shown in Fig. 3.31e. The maximum deflection occurs at midspan, where x = L/2, and equals -5wL4/384El.

For concentrated loads, the equations for shear and bending moment are derived in the region between the concentrated loads, where continuity of these diagrams exists. Consider the simply supported beam subjected to a concentrated load at midspan (Fig. 3.28a). From equilibrium equations, the reactions R1 and R2 equal P/2. With the origin taken at the left end of the span, w(x) = 0 when x + L/2. Integration of Eq. (3.80e) gives V(x) = C3, a constant, for x = 0 to L/2, and V(x) = C4, another constant, for x = L/2 to L. Since V = R1 = P/2 at x = 0, C3 = P/2. Since V = -R2 = -P/2 at x = L, C4 = -P/2. Thus, for 0 < x < L/2, V = P/2, and for L/2 < x < L, V = -P/2 (Fig. 3.28b). Similarly, equations

FIGURE 3.44 Moment applied at one end of a beam with a fixed end.

FIGURE 3.45 Load at midspan of beam with one fixed end, one end on rollers.

FIGURE 3.44 Moment applied at one end of a beam with a fixed end.

FIGURE 3.45 Load at midspan of beam with one fixed end, one end on rollers.

for bending moment, slope, and deflection can be expressed from x = 0 to L/2 and again for x = L/2 to L, as shown in Figs. 3.28c, 3.28d, and 3.28e, respectively.

In practice, it is usually not convenient to derive equations for shear and bending-moment diagrams for a particular loading. It is generally more convenient to use equations of equilibrium to plot the shears, moments, and deflections at critical points along the span. For example, the internal forces at the quarter span of the uniformly loaded beam in Fig. 3.31 may be determined from the free-body diagram in Fig. 3.50. From equilibrium conditions for moments about the right end,

Also, the sum of the vertical forces must equal zero:

Several important concepts are demonstrated in the preceding examples:

FIGURE 3.46 Shears, moments, and deformations for uniformly loaded fixed-end beam.

FIGURE 3.48 Shears, moments, and deformations for load at midspan of a fixed-end beam.

Triangular Loaded Beams

FIGURE 3.47 Diagrams for triangular load on a fixed-end beam.

FIGURE 3.46 Shears, moments, and deformations for uniformly loaded fixed-end beam.

FIGURE 3.47 Diagrams for triangular load on a fixed-end beam.

FIGURE 3.49 Diagrams for concentrated load on a fixed-end beam.

Renewable Energy 101

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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