7.2 Methods of plastic analysis

The criteria to be followed in plastic analysis to identify a correct load factor:

• 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 $\left\vert M\right\vert \le \left\vert M_{pl}\right\vert$;
• mechanism - at collapse, the structure, or a part of it, can deform as a mechanism;
• dissipation energy $D \geq 0$ 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 $W_{b} = - D \leq 0$ must be negative (Eq. (7.19).

Three fundamental theorems are based on the criteria named (see [Cpr11]):

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

Theorem 7.2.1 (the lower bound (safe) theorem or static theorem)  
A load computed on the basis of a bending moment distribution in which the moment nowhere exceeds $M_{pl}$ 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 $\lambda_{i}$ in Eq. (7.1) is either less than or equal to the true load factor at collapse (see Fig. 7.6).

Figure 7.6: Upper and lower bound values

Theorem 7.2.2 (the upper bound (unsafe) theorem or kinematic theorem)   A load computed on the basis of an assumed mechanism is either equal to or greater than the true collapse load. When several mechanisms are tried, the true collapse load will be the smallest of them.

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

Theorem 7.2.3 (the uniqueness theorem)  
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 $\lambda_{i}$ of Eq. (7.1) is the true load factor at collapse.         

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

• the kinematic (mechanism) method,
• the equilibrium (statical) method,
• the load incremental method.

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 $F_{i}$ are given,
• the cross-sectional stiffness parameters and the full plastic moment $M_{pl}$ are known.

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

1. Increasing the load $\lambda_{o}$ times ( $\lambda_{o} F_{i}$) to the appearance of a full plastic moment $M_{pl}^{\left( 0\right)}$. For that
   • compose a structural system for an n-1 times statically indeterminate structure with a full plastic moment $M_{pl}^{\left( 0\right)}$ in the plastic hinge,
   • find the load factor $\lambda_{o}$,
   • check if the boundary work at the plastic hinge satisfies the condition of Eq. (7.19).
2. In an n-1 times statically indeterminate structure, increasing the load $\lambda_{1}$ times to the appearance of a second full plastic moment $M_{pl}^{\left( 1\right)}$. For that
   • compose a structural system for an n-2 times statically indeterminate structure with the full plastic moments $M_{pl}^{\left( 0\right)}$ and $M_{pl}^{\left( 1\right)}$ in the plastic hinges,
   • find the load factor $\lambda_{o} + \lambda_{1}$,
   • check if the boundary work at the plastic hinges satisfies the condition of Eq. (7.19) $W_{b}=M^{r}_{pl}\varphi_{r}+M^{l}_{pl}\varphi_{l} = -{D} \le 0$.
3. Repeating the step-by-step procedure until the structure is statically determinate (n=0).
4. Increasing the load in the statically determinate structure (n=0) to the appearance of a full plastic moment $M_{pl}^{\left( n\right)}$. A plastic hinge forms and the structure becomes a mechanism. The structure cannot sustain any more load and thus will collapse at $F_{lim}$. The load computed in which the moment exceeds $M_{pl}^{\left( n\right)}$ is equal to the true collapse load. For that
   • compose a structural system for a statically determinate structure with full plastic moments $M_{pl}^{\left( 0\right)}$, $M_{pl}^{\left( 1\right)}$, ... , $M_{pl}^{\left( n \right)}$ in the plastic hinges,
   • find the load factor $\lambda_{o} + \lambda_{1} + ... + \lambda_{n}$,
   • check if the boundary work at the plastic hinge satisfies the condition of Eq. (7.19) $W_{b}=M^{r}_{pl}\varphi_{r}+M^{l}_{pl}\varphi_{l} = -{D} \le 0$.