In continuous structural systems with many members, there are several ways that mechanisms can develop. The limit load, or load creating a mechanism, lies between the loads computed from upper-bound and lower-bound theorems. The upper-bound theorem states that a load computed on the basis of an assumed mechanism will be greater than, or at best equal to, the true limit load. The lower-bound theorem states that a load computed on the basis of an assumed bending-moment distribution satisfying equilibrium conditions, with bending moments nowhere exceeding the plastic moment Mp , is less than, or at best equal to, the true limit load. The plastic moment is Mp = ZFy , where Z = plastic section modulus and Fy = yield stress. If both theorems yield the same load, it is the true ultimate load.

In the application of either theorem, the following conditions must be satisfied at the limit load: External forces must be in equilibrium with internal forces; there must be enough plastic hinges to form a mechanism; and the plastic moment must not be exceeded anywhere in the structure.

The process of investigating mechanism failure loads to determine the maximum load a continuous structure can sustain is called plastic analysis.

## Equilibrium Method

The statical or equilibrium method is based on the lower-bound theorem. It is convenient for continuous structures with few members. The steps are

Select and remove redundants to make the structure statically determinate.

Draw the moment diagram for the given loads on the statically determinate structure.

Sketch the moment diagram that results when an arbitrary value of each redundant is applied to the statically determinate structure.

Superimpose the moment diagrams in such a way that the structure becomes a mechanism because there are a sufficient number of the peak moments that can be set equal to the plastic moment Mp .

Compute the ultimate load from equilibrium equations.

Check to see that Mp is not exceeded anywhere.

To demonstrate the method, a plastic analysis will be made for the two-span continuous beam shown in Fig. 3.97a. The moment at C is chosen as the redundant. Figure 3.97b shows the bending-moment diagram for a simple support at C and the moment diagram for an assumed redundant moment at C. Figure 3.97c shows the combined moment diagram. Since the moment at D appears to exceed the moment at B, the combined moment diagram may be adjusted so that the right span becomes a mechanism when the peak moments at C and D equal the plastic moment Mp (Fig. 3.97d).

If MC = MD = Mp , then equilibrium of span CE requires that at D,

## Mechanism Method

As an alternative, the mechanism method is based on the upper-bound theorem. It includes the following steps:

Determine the locations of possible plastic hinges.

Select plastic-hinge configurations that represent all possible mechanism modes of failure.

Using the principle of virtual work, which equates internal work to external work, calculate the ultimate load for each mechanism.

Assume that the mechanism with the lowest critical load is the most probable and hence represents the ultimate load.

Check to see that Mp is not exceeded anywhere.

To illustrate the method, the ultimate load will be found for the continuous beam in Fig. 3.97a. Basically, the beam will become unstable when plastic hinges form at B and C (Fig. 3.98a) or C and D (Fig. 3.98b). The resulting constructions are called either independent or fundamental mechanisms. The beam is also unstable when hinges form at B, C, and D (Fig. 3.98c). This configuration is called a composite or combination mechanism and also

will be discussed.

Applying the principle of virtual work (Art. 3.23) to the beam mechanism in span AC (Fig. 3.98d), external work equated to internal work for a virtual end rotation Q gives

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.

## Extension of Classical Plastic Analysis

The methods of plastic analysis presented in Secs. 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).

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