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FIGURE 3.24 For a rectangular beam (a) in the plastic range, strain distribution (b) is linear, while stress distribution (c) is rectangular.

FIGURE 3.24 For a rectangular beam (a) in the plastic range, strain distribution (b) is linear, while stress distribution (c) is rectangular.

To determine these stresses, consider the portion of a beam with length dx between vertical sections 1-1 and 2-2 (Fig. 3.25). At a horizontal section a distance y from the neutral axis, the horizontal shear force AH(y) equals the difference between the normal forces acting above the section on the two faces:

where f2( y) and f1( y) are the bending-stress distributions at sections 2-2 and 1-1, respectively.

If the bending stresses vary linearly with depth, then, according to Eq. (3.61),

M2 y

where M2 and M1 are the internal bending moments at sections 2-2 and 1-1, respectively, and I is the moment of inertia about the neutral axis of the beam cross section. Substitution in Eq. (3.67) gives

where Q(y) = fyyb(y) dy = static moment about neutral axis of the area above the plane at a distance y from the neutral axis b(y) = width of beam dM = M2 - Mj

Division of AH(y) by the area b(y) dx yields the shear stress at y:

Integration of v(y) over the cross section provides the total internal vertical shear force V on the section:

J cb

To satisfy equilibrium requirements, V must be equal in magnitude but opposite in direction to the shear at the section due to the loading. Substitution of Eq. 3.70 in Eq. 3.71 gives

inasmuch as I = SC£Q(y) dy. Equation (3.72) indicates that shear is the rate of change of bending moment alobng the span of the beam.

Substitution of Eq. (3.72) into Eq. (3.70) yields an expression for calculating the shear stress at any section depth:

According to Eq. (3.73), the maximum shear stress occurs at a depth y when the ratio Q(y)/b(y) is maximum.

For rectangular cross sections, the maximum shear stress occurs at middepth and equals

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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