328 Determinacy And Geometric Stability
In a statically determinate system, all reactions and internal member forces can be calculated solely from equations of equilibrium. However, if equations of equilibrium alone do not provide enough information to calculate these forces, the system is statically indeterminate. In this case, adequate information for analyzing the system will only be gained by also considering the resulting structural deformations. Static determinacy is never a function of loading. In a statically determinate system, the distribution of internal forces is not a function of member cross section or material properties.
In general, the degree of static determinacy n for a truss may be determined by n = m  aj + R (3.125)
where m = number of members j = number of joints including supportjs a = dimension of truss (a = 2 for a planar truss and a = 3 for a space truss) R = number of reaction components
Similarly, the degree of static determinacy for a frame is given by n = 3(a  1)(m  j) + R (3.126)
where a = 2 for a planar frame and a = 3 for a space frame.
If n is greater than zero, the system is geometrically stable and statically indeterminate; if n is equal to zero, it is statically determinate and may or may not be stable; if n is less than zero, it is always geometrically unstable. Geometric instability of a statically determinate truss (n = 0) may be determined by observing that multiple solutions to the internal forces exist when applying equations of equilibrium.
Figure 3.65 provides several examples of statically determinate and indeterminate systems. In some cases, such as the planar frame shown in Fig. 3.65e, the frame is statically indeterminate for computation of internal forces, but the reactions can be determined from equilibrium equations.
FIGURE 3.65 Examples of statically determinate and indeterminate systems: (a) Statically determinate truss (n = 0); (b) statically indeterminate truss (n = 1); (c) statically determinate frame (n = 0); (d) statically indeterminate frame (n = 15); (e) statically indeterminate frame
FIGURE 3.65 Examples of statically determinate and indeterminate systems: (a) Statically determinate truss (n = 0); (b) statically indeterminate truss (n = 1); (c) statically determinate frame (n = 0); (d) statically indeterminate frame (n = 15); (e) statically indeterminate frame
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 nonrenewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a nonrenewable supply, the nonrenewable energy sources release emissions into the air, which has an adverse effect on the environment.
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