## Ll

which gives P = P2 as

7. Since a sufficient number of plastic hinges have formed in the beams (Figure 2.96g) to arrive at a collapse mechanism, the computed load, P2, is the exact plastic limit load.

### EXAMPLE 2.12 Portal Frame

A pin-based rectangular frame is subjected to a vertical load V and a horizontal load H as shown in Figure 2.97a. All the members of the frame are made of the same section with moment capacity Mp. The objective is to determine the limit value of H if the frame's width-to-height ratio L/h is 1.0.

### Procedure

The frame has one degree of redundancy. The redundancy for this structure can be chosen as the horizontal reaction at E. Figure 2.97b and c show the resulting determinate frame loaded by the applied loads and redundant force. The moment diagrams corresponding to these two loading conditions are shown in Figure 2.97d and e.

The horizontal reaction S should be chosen in such a manner that all three conditions, equilibrium, plastic moment, and mechanism, are satisfied. Formation of two plastic hinges is necessary to form a mechanism. The plastic hinges may be formed at B, C, and D. Assuming that a plastic hinge is formed at D as shown in Figure 2.97e, we have

Corresponding to this value of S, the moments at B and C can be expressed as

The condition for the second plastic hinge to form at B is |MB| > |MC|. From Equations 2.216 and 2.217 we have

Hh VL

The condition for the second plastic hinge to form at C is |MC| > |MB|. From Equations 2.216 and 2.217 we have

Hh VL

For a particular combination of V, H, L, and h, the collapse load for H can be calculated.

Since |Mb| > |MC|, the second plastic hinge will form at B and the corresponding value for H is

Since | MC | > | MB | , the second plastic hinge will form at C and the corresponding value for H is

1.6Mp

2.11.8 Mechanism Method

This method, which is based on the upper bound theorem, states that the load computed on the basis of an assumed failure mechanism is never less than the exact plastic limit load of a structure. Thus, it always predicts the upper bound solution of the collapse limit load. It can also be shown that the minimum upper bound is the limit load itself. The procedure of using the mechanism method has the following two steps:

1. Assume a failure mechanism and form the corresponding work equation from which an upper bound value of the plastic limit load can be estimated.

2. Write the equilibrium equations for the assumed mechanism and check the moments to see whether the plastic moment condition is met everywhere in the structure.

To obtain the true limit load using the mechanism method, it is necessary to determine every possible collapse mechanism of which some are the combinations of a certain number of independent mechanisms. Once the independent mechanisms have been identified, a work equation may be established for each combination and the corresponding collapse load is determined. The lowest load among those obtained by considering all the possible combinations of independent mechanisms is the correct plastic limit load.

2.11.8.1 Independent Mechanisms

The number of possible independent mechanisms, n, for a structure can be determined from the following equation:

where N is the number of critical sections at which plastic hinges might form and R indicates the degrees of redundancy of the structure.

Critical sections generally occur at the points of concentrated loads, at joints where two or more members meet at different angles, and at sections where there is an abrupt change in section geometries or properties. To determine the number of redundancies (R) of a structure, it is necessary to free sufficient supports or restraining forces in structural members so that the structure becomes an assembly of several determinate substructures.

Figure 2.98 shows two examples. The cuts that are made in each structure reduce the structural members to either cantilevers or simply supported beams. The fixed-end beam requires a shear force and a moment to restore continuity at the cut section, and thus R = 2. For the two-store frame, an axial force, shear, and moment are required at each cut section for full continuity and thus R = 12.

### 2.11.8.2 Types of Mechanism

Figure 2.99a shows a frame structure subjected to a set of loading. The frame may fail by different types of collapse mechanisms dependent on the magnitude of loading and the frame's configurations. The collapse mechanisms are

1. Beam mechanism. Possible mechanisms of this type are shown in Figure 2.99b.

2. Panel mechanism. The collapse mode is associated with side sway, as shown in Figure 2.99c.

3. Gable mechanism. The collapse mode is associated with the spreading of column tops with respect to the column bases, as shown in Figure 2.99d.

4. Joint mechanism. The collapse mode is associated with the rotation of joints of which the adjoining members developed plastic hinges and deformed under an applied moment, as shown in Figure 2.99e.

5. Combined mechanism. It can be a partial collapse mechanism as shown in Figure 2.99f or it may be a complete collapse mechanism as shown in Figure 2.99g.

Cut section

FIGURE 2.98 Number of redundants in a (a) beam and (b) frame.

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