7.2 Methods of plastic analysis

- equilibrium - the internal bending moments must be in equilibrium with the external loads;
- yield - no point in the structure can have a moment greater than the plastic moment ;
- mechanism - at collapse, the structure, or a part of it, can deform as a mechanism;
- dissipation energy in Eq. (7.13) must be positive, i.e. the sum of the amounts of work done by boundary forces of members at a plastic hinge must be negative (Eq. (7.19).

- the lower bound (safe) theorem or static theorem,
- the upper bound (unsafe) theorem or kinematic theorem,
- the uniqueness theorem.

A load computed on the basis of a bending moment distribution in which the moment nowhere exceeds is either equal to or less than the true collapse load.

A bending moment diagram is found which satisfies the conditions of equilibrium. The load factor in Eq. (7.1) is either less than or equal to the true load factor at collapse (see Fig. 7.6).

A bending moment diagram is found which satisfies the conditions of equilibrium. The load factor in Eq. (7.1) is either greater than or equal to the true load factor at collapse (see Fig. 7.6).

A load computed on the basis of bending moment distribution which satisfies both the plastic moment and mechanism conditions is ae true plastic collapse load.

A bending moment diagram is found which satisfies the conditions of equilibrium.
The load factor of Eq. (7.1) is the true load factor at collapse.

Basically there are three methods of analysis (see [Cpr11] p. 28):

The load incremental method

This method is most readily suited for computer implementation. We will concentrate only on this one, using the EST method (see Example 8.1). Let us assume that

- for an n times statically indeterminate structure, the forces are given,
- the cross-sectional stiffness parameters and the full plastic moment are known.

The method is based on an incremental process producing a sequence of the load factor of Eq. (7.1). The following steps are made:

- Increasing the load times (
) to the appearance of a full plastic moment
. For that
- compose a structural system for an n-1 times statically indeterminate structure with a full plastic moment in the plastic hinge,
- find the load factor ,
- check if the boundary work at the plastic hinge satisfies the condition of Eq. (7.19).

- In an n-1 times statically indeterminate structure, increasing the load
times to the appearance of a second full plastic moment
. For that
- compose a structural system for an n-2 times statically indeterminate structure with the full plastic moments and in the plastic hinges,
- find the load factor ,
- check if the boundary work at the plastic hinges satisfies the condition of Eq. (7.19) .

- Repeating the step-by-step procedure until the structure is statically determinate (n=0).
- Increasing the load in the statically determinate structure (n=0)
to the appearance of a full plastic moment
. A plastic hinge forms and the structure
becomes a mechanism. The structure cannot sustain any more load and thus will collapse at .
The load computed in which the moment exceeds
is equal to the true collapse load. For that
- compose a structural system for a statically determinate structure with full plastic moments , , ... , in the plastic hinges,
- find the load factor ,
- check if the boundary work at the plastic hinge satisfies the condition of Eq. (7.19) .

2014-09-09