8.1 Illustrative problems

*Let us assume that the flexural rigidity of a column
and that of the beam
; the axial rigidity of a column
and that of the beam
; the shear rigidity of a column
and that of the beam
.
*

*We wish to compute the collapse load factor at which the frame will actually fail.
*

* Problem Solving.
We use the EST method to solve the problem of an n times statically indeterminate frame. The steps of the load incremental method are listed below.
*

- The load of an n=3 times statically indeterminate frame is increased times
(
, see Fig. 8.1)
to the appearance of the full plastic moment . For that
- compose a structural system for an n=3 times statically indeterminate frame (Fig. 8.2(a)) with a load factor ;
- find the node where a full plastic moment or will appear. We find that this is node 4, where (see Fig. 8.2(a));
- find the load factor for a moment to appear at node 4. We find that for a full plastic moment to develop at node 4, the load factor is to be as shown in Fig. 8.2(b);
- find the next node where a full plastic moment or will appear. We find that this is node 5 (see Fig. 8.2(b)). We also find that at this node to achieve , is needed to add, and that as can be seen in Figs. 8.2(b) and 8.3(a).

- In an n=2 times statically indeterminate frame, load is increased (
)
times to the appearance of the second full plastic moment . For that
- compose a structural system for an n=2 times statically indeterminate frame (Fig. 8.3(a)) with a load factor ;
- find the load factor for a moment to appear at node 5. We find that for the appearance of a plastic moment at node 5, the load factor is to be as shown in Fig. 8.3(a);
- add the boundary moments that are equal to the full plastic moment
(related to )
to the ends of elements 3 and 4 at
hinge 4 (see Fig. 8.3(b));

- find the displacements and internal forces of the frame shown in Fig. 8.3(b). Here, the load factor , the corresponding loads and ;
- check if the dissipation D at plastic hinge 4 satisfies the condition (7.19). We find that ;
- find the next node at which a full plastic moment or will form. We find that this is node 3 (see Fig. 8.3(b)). We also find that at this node, is needed to add to achieve , and that as shown in Figs. 8.3(b) and 8.4(a).

- In an n=1 times statically indeterminate frame, load is increased (
times to the appearance of the third full plastic moment . For that
- compose a structural system for an n=1 times statically indeterminate frame (Fig. 8.4(a)) with a load factor ;
- find the load factor for a moment to appear at node 3. We find that for the appearance of a plastic moment at node 3, the load factor is to be as shown in Fig. 8.4(a);
- add the boundary moments that are equal to the full plastic moment
(related to )
to the ends of elements 3 and 4 at
hinge 4 (see Fig. 8.3(b));

- add the boundary moments that are equal to the full plastic moment
(related to )
to the end of element 4 at plastic hinge 5 (see Fig. 8.4(b));

- find the displacements and internal forces of the frame shown in Fig. 8.4(b). Here, the load factor is , the corresponding loads are and ;
- check if the dissipation D at plastic hinges 4 and 5 satisfies the condition (7.19). We find that and ;
- find the next node at which a full plastic moment or will appear. We find that this is node 1 (see Fig. 8.4(b)). We also find that at this node, is needed to add to achieve , and that as shown in Figs. 8.4(b) and 8.5(a).

- As the last step of the procedure, load is increased in a statically determined frame (n=0)
to the appearance of the full plastic moment
. A plastic hinge forms and the frame
becomes a mechanism. The load computed in which the moment exceeds
is equal to the true collapse load.
In the n=0 times statically indeterminate (statically determinate) frame, load is increased (
times to the appearance of the fourth full plastic moment . For that
- compose a structural system for an n=0 times statically indeterminate frame with a load factor (see Fig. 8.5(a));
- find the load factor for a moment to develop at node 1. We find that for the appearance of the plastic moment at node 1, the load factor is to be as shown in Fig. 8.5(a);
- add the boundary moments that are equal to the full plastic moment (related to ) to the ends of elements 3 and 4 at hinge 4 (see Fig. 8.5(b));
- add the boundary moments that are equal to the full plastic moment (related to ) to the end of element 4 at plastic hinge 5 (see Fig. 8.5(b));
- add the boundary moments that are equal to the full plastic moment (related to ) to the end of elements 2 and 3 at plastic hinge 3 (see Fig. 8.5(b));
- find the displacements and internal forces of the frame shown in Fig. 8.5(b). Here, the load factor is , the corresponding loads are and ;
- check if the dissipation D at plastic hinges 4 and 5 satisfies the condition
(7.19).
We find that

,

and

.

*The four
plastic hinges (nodes 1, 3, 4, and 5) produce a mechanism
leading to collapse at the loads
and
(related to the load factor
).
*

*An idealized relation between the load and deflection at node 3 is displayed in Fig. 8.6. The first yield moment occurs at node 4 shown in Fig. 8.5(b) and is marked with the letter A in Fig. 8.6.
The following yield moments occurring at nodes 5, 3, and 1 shown in Fig. 8.5(b) are marked with the letters B, C, D in Fig. 8.6.
*

- 8.1.1 The n=3 times statically indeterminate frame
- 8.1.2 The n=2 times statically indeterminate frame
- 8.1.3 The n=1 times statically indeterminate frame
- 8.1.4 The n=0 times statically indeterminate frame

2014-09-09