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SW = Fj cos a2 Ss - F2 cos a2 Ss + F3 cos a3 Ss (3.96)

Factoring Ss from the right side of Eq. (3.96) and substituting the equilibrium relationship provided in Eq. (3.95) gives

SW = (Fj cos aj - F2 cos a2 + F3 cos a3) Ss = 0 (3.97)

Similarly, the virtual work is zero for the components along the y and z axes. In general, Eq. (3.97) requires

That is, virtual work must be equal to zero for a single particle in equilibrium under a set of forces.

In a rigid body, distances between particles remain constant, since no elongation or compression takes place under the action of forces. The virtual work done on each particle of the body when it is in equilibrium is zero. Hence the virtual work done by the entire rigid body is zero.

In general, then, for a rigid body in equilibrium, SW = 0.

Virtual Work on a Rigid Body. This principle of virtual work can be applied to idealized systems consisting of rigid elements. As an example, Fig. 3.59 shows a horizontal lever, which can be idealized as a rigid body. If a virtual rotation of SO is applied, the virtual displacement for force Wj is a SO, and for force W2, b SO. Hence the virtual work during this rotation is

If the lever is in equilibrium, SW = 0. Hence Wj a = W2 b, which is the equilibrium condition that the sum of the moments of all forces about a support should be zero.

When the body is not rigid but can be distorted, the principle of virtual work as developed above cannot be applied directly. However, the principle can be modified to apply to bodies that undergo linear and nonlinear elastic deformations.

Strain Energy in a Bar. The internal work U done on elastic members is called elastic potential energy, or strain energy. Suppose, for example, that a bar (Fig. 3.60a) made of an elastic material, such as steel, is gradually elongated an amount Af by a force Pf. As the

FIGURE 3.59 Virtual rotation of a lever.

bar stretches with increases in force from 0 to Pf, each increment of internal work dU may be expressed by Eq. (3.91) with a = 0:

where dA = the current increment in displacement in the direction of P P = the current applied force, 0 < P < Pf

Equation (3.100) also may be written as dA = P (3.101)

which indicates that the derivative of the internal work with respect to a displacement (or

FIGURE 3.60 (a) Bar in tension elongates. (b) Energy stored in the bar is represented by the area under the load-displacement curve.

rotation) gives the corresponding force (or moment) at that location in the direction of the displacement (or rotation).

After the system comes to rest, a condition of equilibrium, the total internal work is

The current displacement A is related to the applied force P by Eq. (3.51); that is, P = EAA/ L. Substitution into Eq. (3.102) yields

When the force is plotted against displacement (Fig. 3.60b), the internal work is the shaded area under the line with slope k = EA/L.

When the bar in Fig. 3.60a is loaded and in equilibrium, the internal virtual work done by Pf during an additional virtual displacement SA equals the change in the strain energy of the bar:

where Af is the original displacement produced by Pf.

Principle of Virtual Work. This example illustrates the principal of virtual work. If an elastic body in equilibrium under the action of external loads is given a virtual deformation from its equilibrium condition, the work done by the external loads during this deformation equals the change in the internal work or strain energy, that is,

Hence, for the loaded bar in equilibrium (Fig. 3.60a), the external virtual work equals the internal virtual strain energy:

[For rigid bodies, no internal strain energy is generated, that is, SU = kAf SA = 0, and Eq. (3.106) reduces the Eq. (3.98).] The example may be generalized to any constrained (supported) elastic body acted on by forces P1, P2, P3,. . . for which the corresponding displacements are A1, A2, A3, . . . . Equation (3.100) may then be expanded to dU = 2PtdAt (3.107)

Similarly, Eq. (3.101) may be generalized to dU

The increase in strain energy due to the increments of the deformations is given by substitution of Eq. (3.108) into Eq. (3.107):

dU dU dU dU

dU = 2 dv^ = dAT"dA1 + ^ dA2 + dA3 + • • • (3.109)

If specific deformations in Eq. (3.109) are represented by virtual displacements, load and deformation relationships for several structural systems may be obtained from the principle of virtual work.

Strain energy also can be generated when a member is subjected to other types of loads or deformations. The strain-energy equation can be written as a function of either load or deformation.

Strain Energy in Shear. For a member subjected to pure shear, strain energy is given by

V 2L

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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