## Ll

Beam mechanisms

X"

X"

FIGURE 2.99 Typical plastic mechanisms.

Gable mechanism

Ar Ar

. Independent

f mechanisms

Gable mechanism

Joint mechanism

Partial collapse

Partial collapse

Complete collapse

Complete collapse

Combined mechanisms

The principal rule for combining independent mechanisms is to obtain a lower value of collapse load. The combinations are selected in such a way that the external work becomes a maximum and the internal work becomes a minimum. Thus, the work equation would require that the mechanism involve as many applied loads as possible and at the same time eliminate as many plastic hinges as possible. This procedure will be illustrated in the following example.

### EXAMPLE 2.13 Rectangular frame

A fixed-end rectangular frame has a uniform section with Mp = 20 and carries the load shown in Figure 2.100. Determine the value of load ratio 1 at collapse.

Solution

Number of possible plastic hinges N = 5

Number of redundancies R = 3 Number of independent mechanisms N — R = 2

The two independent mechanisms are shown in Figure 2.100b and c and the corresponding work equations are

Panel mechanism 201 = 4(20) = 80 ) 1 = 4 Beam mechanism 301 = 4(20) = 80 ) 1 = 2.67

The combined mechanisms are now examined to see whether they will produce a lower 1 value. It is observed that only one combined mechanism is possible. The mechanism is shown in Figure 2.100c involving cancellation of plastic hinge at B. The calculation of the limit load is described below:

Panel mechanism 201 = 4(20)

Beam mechanism 301 = 4(20)

Addition 501 = 8(20)

Cancellation of plastic hinge _— 2(20)

Combined mechanism 501 = 6(20)

The combined mechanism results in a smaller value for l and no other possible mechanism can produce a lower load. Thus, l = 2.4 is the collapse load.

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