Advanced Methods of Structural Analysis

12.3 Fundamental Methods of Plastic Analysis 431

P=∞

2.

P=6Ny

1.

Ny Ny

Ny Ny

Ny Ny Ny

Ny Ny Ny

Ny

P=4Ny

3.

Ny

P= 3 Ny

4.

D

a

P

1.

Ny

Ny Ny Ny

Ny Ny Ny

Ny Ny Ny

Ny Ny

N 4 =0

P

3.

N 2 =∝

P

4.

N 1 =3Ny

P

2.

N 3 =−Ny

b

Fig. 12.4 (a) Kinematical method. Different forms of failure of the structure; (b) Static method.

Different stress conditions of the structure

Let us consider variousdistributions of internal forcesin the structure. For all

these schemes, the following two conditions should be met:

1.Since the structure contains four hangers, then forces in any three members have
to be equal to the limit loadNy, which means that these three members corre-
spond to horizontal part of Prandtl diagram
2.The force in the remaining forth member should be less thanNy.

There is possible the following cases.

1.N 1 DN 2 DN 3 DNy
2.N 1 DN 2 DN 4 DNy
3.N 1 DN 3 DN 4 DNy
4.N 2 DN 3 DN 4 DNy

The force in the remaining forth member should be defined from equilibrium con-

dition. The sum of the moments with respect to point of application of the forceP

leads to the following results:

1: N 4 D0; 2: N 3 DNy;3:N 4 D1;4:N 4 D3Ny

These internal forces are shown in Fig.12.4b.