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2.11.6.1 Principle of Virtual Work

The virtual work principle may be applied to relate a system of forces in equilibrium to a system of compatible displacements. For example, if a structure in equilibrium is given a set of small compatible displacements, then the work done by the external loads on these external displacements is equal to the work done by the internal forces on the internal deformation. In plastic analysis, internal deformations are assumed to be concentrated at plastic hinges. The virtual work equation for hinged structures can be written in explicit form as

where Pi is an external load and Mi is the internal moment at a hinge location. Both Pi and Mi constitute an equilibrium set and they must be in equilibrium. dj are the displacements under the point loads Pi and in the direction of the loads. y are the plastic hinge rotations under the moment Mi. Both dj and y constitute a displacement set and they must be compatible with each other.

### 2.11.6.2 Lower Bound Theorem

For a given structure, if there exists any distribution of bending moments in the structure that satisfies both the equilibrium and the plastic moment conditions, then the load factor, 1L, computed from this moment diagram must be equal to or less than the collapse load factor, lc, of the structure. The lower bound theorem provides a safe estimate of the collapse limit load, that is, 1L < lc.

### 2.11.6.3 Upper Bound Theorem

For a given structure subjected to a set of applied loads, a load factor, lu, computed based on an assumed collapse mechanism must be greater than or equal to the true collapse load factor, lc. The upper bound theorem, which uses only the mechanism condition, estimates correctly or overestimates the collapse limit load, that is, 1u > lc.

### 2.11.6.4 Uniqueness Theorem

A structure at collapse has to satisfy three conditions. First, a sufficient number of plastic hinges must be formed to turn the structure, or part of it, into a mechanism; this is called the mechanism condition. Second, the structure must be in equilibrium, that is, the bending moment distribution must satisfy equilibrium with the applied loads. Finally, the bending moment at any cross-section must not exceed the full plastic value of that cross-section; this is called the plastic moment condition. The theorem simply implies that the collapse load factor, Ac, obtained from the three basic conditions (mechanism, equilibrium, and plastic moment) has a unique value.

The proof of the three theorems can be found in Chen and Sohal (1994). A useful corollary of the lower bound theorem is that if at a load factor, l, it is possible to find a bending moment diagram that satisfies both the equilibrium and the moment conditions but not necessarily the mechanism condition, then the structure will not collapse at that load factor, unless the load happens to be the collapse load. A corollary of the upper bound theorem is that the true load factor at collapse is the smallest possible one that can be determined from a consideration of all possible mechanisms of collapse. This concept is very useful in finding the collapse load ofthe system from various combinations ofmechanisms. From these theorems, it can be seen that the lower bound theorem is based on equilibrium approach while the upper bound technique is based on mechanism approach. These two alternative approaches to an exact solution, called the equilibrium method and mechanism method, will be discussed in the sections that follow.

### 2.11.7 Equilibrium Method

The equilibrium method, which employs the lower bound theorem, is suitable for the analysis of continuous beams and frames in which the structural redundancies do not exceed two. The procedures for obtaining the equilibrium equations of a statically indeterminate structure and evaluating its plastic limit load are as follows:

To obtain the equilibrium equations of a statically indeterminate structure

1. Select the redundant(s).

2. Free the redundants and draw a moment diagram for the determinate structure under the applied loads.

3. Draw a moment diagram for the structure due to the redundant forces.

4. Superimpose the moment diagrams in steps 2 and 3.

5. Obtain maximum moment at critical sections of the structure utilizing the moment diagram in step 4.

To evaluate the plastic limit load of the structure

6. Select the value(s) of redundant(s) such that the plastic moment condition is not violated at any section in the structure.

7. Determine the load corresponding to the selected redundant(s).

8. Check for the formation of a mechanism. If a collapse mechanism condition is met, then the computed load is the exact plastic limit load. Otherwise, it is a lower bound solution.

9. Adjust the redundant(s) and repeat steps 6 to 9 until the exact plastic limit load is obtained.

### EXAMPLE 2.11 Continuous Beam

Figure 2.96a shows a two-span continuous beam analyzed using the equilibrium method. The plastic limit load of the beam is calculated based on the step-by-step procedure described in the previous section as follows:

1. Select the redundant force as M1 which is the bending moment at the intermediate support, as shown in Figure 2.96b.

2. Free the redundants and draw a moment diagram for the determinate structure under the applied loads, as shown in Figure 2.96c.

3. Draw a moment diagram for the structure due to the redundant moment M1, as shown in Figure 2.96d.

4. Superimpose the moment diagrams in Figure 2.96c and d and the results are shown in Figure 2.96e.

5. The moment diagram in Figure 2.96e is redrawn on a single straight base line. The critical moment in the beam is

The maximum moment at critical sections of the structure is obtained by using the moment diagram in Figure 2.96e. By letting Mcr = Mp, the resulting moment distribution is shown in Figure 2.96f.

6. A lower bound solution may be obtained by selecting a value of redundant moment M1.

For example, if M1 = 0 is selected, the moment diagram is reduced to that shown in Figure 2.96c. By equating the maximum moment in the diagram to the plastic moment, Mp, we have

which gives P = P1 as

The moment diagram in Figure 2.96c shows a plastic hinge formed at each span. Since two plastic hinges in each span are required to form a plastic mechanism, the load P1 is a lower bound solution. However, if the redundant moment M1 is set equal to the plastic moment Mp, and by letting the maximum moment in Figure 2.96f equal the plastic moment, we have

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