## Continuous Beam Plastic Hinge

FIGURE 3.98 Plastic analysis of two-span continuous beam by the mechanism method. Beam mechanisms form when plastic hinges occur at (a) B and C, (b) C and D, and (c) B, C, and D. (d), (e) (/) show virtual displacements assumed for the mechanisms in (a), (b), and (c), respectively.

Similarly, by assuming a virtual end rotation a at E, a beam mechanism in span CE (Fig. 3.98e) yields

Of the two independent mechanisms, the latter has the lower critical load. This suggests that the ultimate load is Pu = l5Mp/4L.

For the combination mechanism (Fig. 3.98f), application of virtual work yields

OpL + 2P 3 2a = 20Mp + 0Mp + 2aMp + 3aMp from which

In this case, the ultimate load is a function of the value assumed for the ratio 6/a. If 6/ a equals zero, the ultimate load is P = l5Mp/4L (the ultimate load for span CE as an independent mechanism). The limit load as 6/a approaches infinity is P = 6Mp/L (the ultimate load for span AC as an independent mechanism). For all positive values of 6/a, this equation predicts an ultimate load P such that l5Mp/4L < P < 6MpL. This indicates that for a mechanism to form span AC, a mechanism in span CE must have formed previously. Hence the ultimate load for the continuous beam is controlled by the load required to form a mechanism in span CE.

In general, it is useful to determine all possible independent mechanisms from which composite mechanisms may be generated. The number of possible independent mechanisms m may be determined from m = p - r (3.188)

where p = the number of possible plastic hinges and r = the number of redundancies. Composite mechanisms are selected in such a way as to maximize the total external work or minimize the total internal work to obtain the lowest critical load. Composite mechanisms that include the displacement of several loads and elimination of plastic hinges usually provide the lowest critical loads.

### 3.50.3 Extension of Classical Plastic Analysis

The methods of plastic analysis presented in Sees. 3.50.1 and 3.50.2 can be extended to analysis of frames and trusses. However, such analyses can become complex, especially if they incorporate second-order effects (Art. 3.46) or reduction in plastic-moment capacity for members subjected to axial force and bending (Art. 3.49).

(E. H. Gaylord, Jr., et al., Design of Steel Structures, McGraw-Hill, Inc., New York; W. Prager, An Introduction to Plasticity, Addison-Wesley Publishing Company, Inc., Reading, Mass., L. S. Beedle, Plastic Design of Steel Frames, John Wiley & Sons; Inc., New York: Plastic Design in Steelâ€”A Guide and Commentary, Manual and Report No. 41, American Society of Civil Engineers; R. O. Disque, Applied Plastic Design in Steel, Van Nostrand Reinhold Company, New York.)

## 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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