2

FIGURE 3.27 (a) Portion of an unloaded beam. (b) Deformed portion after beam is loaded.

When the stress-strain diagram for the material is linear, et = ft/E and eb = fb /E, where ft and fb are the unit stresses at top and bottom surfaces and E is the modulus of elasticity. By Eq. (3.60), ft = M(x)ct/I(x) and fb = M(x)cb/I(x), where x is the distance along the beam span where the section dx is located and M(x) is the moment at the section. Substitution for et and ft or eb and fb in Eq. (3.78) gives

dx d6 dx

Equation (3.79) is of fundamental importance, for it relates the internal bending moment along the beam to the curvature or second derivative of the elastic curve S(x), which represents the deflected shape. Equations (3.72) and (3.76) further relate the bending moment M(x) and shear V(x) to an applied distributed load w(x). From these three equations, the following relationships between load on the beam, the resulting internal forces and moments, and the corresponding deformations can be shown:

S(x) = elastic curve representing the deflected shape

S = —— = curvature of the deflected shape and also the El(x)

moment-curvature relationship

= shear-deflection relationship

d 3S

d

M(x)

V(x)

dx3

dx

_El(x)_

El(x)

d 4S

d

r v(x) i

w(x)

dx 4

dx

El(x)

El(x)

= shear-deflection relationship

= load-deflection relationship

= load-deflection relationship

These relationships suggest that the shear force, bending moment, and beam slope and deflection may be obtained by integrating the load distribution. For some simple cases this approach can be used conveniently. However, it may be cumbersome when a large number of concentrated loads act on a structure. Other methods are suggested in Arts. 3.32 to 3.39.

Shear, Moment, and Deflection Diagrams. Figures 3.28 to 3.49 show some special cases in which shear, moment, and deformation distributions can be expressed in analytic form. The figures also include diagrams indicating the variation of shear, moment, and deformations along the span. A diagram in which shear is plotted along the span is called a shear diagram. Similarly, a diagram in which bending moment is plotted along the span is called a bending-moment diagram.

Consider the simply supported beam subjected to a downward-acting, uniformly distributed load w (units of load per unit length) in Fig. 3.31a. The support reactions R1 and R2 may be determined from equilibrium equations. Summing moments about the left end yields

R1 may then be found from equilibrium of vertical forces:

FIGURE 3.28 Shears moments, and deformations FIGURE 3.29 Diagrams for moment applied at one for midspan load on a simple beam. end of a simple beam.

With the origin taken at the left end of the span, the shear at any point can be obtained from Eq. (3.80e) by integration: V = f -w dx = - wx + C1, where C1 is a constant. When x = 0, V = R1 = wL/2, and when x = L, V = -R2 = -wL/2. For these conditions to be satisfied, C1 = wL/2. Hence the equation for shear is V(x) = - wx + wL/2 (Fig. 3.31b).

The bending moment at any point is, by Eq. (3.80d), M(x) = fV dx = f(-wx + wL/2) dx = - wx2/2 + wLx/2 + C2, where C2 is a constant. In this case, when x = 0, M = 0. Hence C2 = 0, and the equation for bending moment is M(x) = z^w (-x2 + Lx), as shown in Fig. 3.31c. The maximum bending moment occurs at midspan, where x = L/2, and equals wL2/8.

From Eq. (3.80c), the slope of the deflected member at any point along the span is M(x) , f w / , r N , w I x2 Lx2

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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