Structural Analysis II Course Code: CIVL322 Dr. Aeid A. Abdulrazeg Dr. Aeid A. Abdulrazeg

Outline Plastic Analysis of Beams Number of Plastic hinges Partial Collapse Simply Supported beam Propped cantilever Continuous Beams

Plastic Analysis of Beams (Number of Plastic hinges) The value of Plastic Moment (Mp) is the maximum value of moment which can be applied to a cross-section before a plastic hinge develops. Consider structural collapse in which either individual members may fail or the entire structure may fail as a whole due to the development of plastic hinges. According to the theory of plasticity, a structure is deemed to have reached the limit of its load carrying capacity when it forms sufficient hinges to convert it into a mechanism with consequent collapse. This is normally one hinge more than the number of degrees of-indeterminacy (DoI) in the structure as indicated in Figure 1.

Ignoring horizontal forces: Number of degrees of indeterminacy is Minimum number of hinges required Ignoring horizontal forces: Number of degrees of indeterminacy is Minimum number of hinges required Ignoring horizontal forces: Number of degrees of indeterminacy is Minimum number of hinges required

Plastic Analysis of Beams (Partial Collapse) It is possible for part of a structure to collapse whilst the rest remains stable. In this instance full collapse does not occur and the number of hinges required to cause partial collapse is less than the (DoI+1.0).

Example 1 (Simply Supported beam) Kinematic method P L/2 L/2 Ɵ Ɵ δ Ɵ Ɵ Mp Mp

Ɵ Ɵ δ Ɵ Ɵ Mp Mp Internal work External work

Example 2 (Simply Supported beam) Kinematic method W L dx Wdx x Ɵ Ɵ Y δ Ɵ Ɵ Mp Mp L/2 L/2

Internal work External work

Example 3 ( propped cantilever) Plastic Hinge

Plastic hinge on the cantilever which do work during the virtual displacement are comprised solely of WU since the vertical reactions at A and C are not displaced. The internal forces which do work consist of the plastic moments , MP, at A and Band which resist rotation. Hence

Possible collapse mechanisms in a propped cantilever supporting two concentrated loads

Example 4 ( propped cantilever)

The total load on AC is wx and its centroid (at x/2 from A) will be displaced a vertical distance δ/2. The total load on CB is w(L−x) and its centroid will suffer the same vertical displacement δ/2. Then, from the principle of virtual work Note that the beam at B is free to rotate so that there is no plastic hinge at B. Substituting for δ from Eq. (i) and φ from Eq. (ii) we obtain

Group Work (Fixed beam) L/2 L/2 W L

Example 5 (Continuous Beams) A non-uniform, three-span beam is fixed at support A, simply supported on rollers at D, F and G and carries unfactored loads as shown in Figure. Determine the minimum Mp value required to ensure a minimum load factor equal to 1.7 for any span.

There are a number of possible elementary beam mechanisms and it is necessary to ensure all possibilities have been considered. It is convenient in multi-span beams to consider each span separately and identify the collapse mechanism involving the greatest plastic moment Mp; this is the critical one and results in partial collapse. The number of elementary independent mechanisms can be determined from evaluating (the number of possible hinge positions—the degree-of indeterminacy) The degree-of indeterminacy = 3 The number of possible hinge positions = 7 (A, B, C, D, E, F and between F and G) Number of independent mechanisms =(7−3)=4

Consider span AD:

Consider span DF:

Consider span FG:

Group Work (Continuous Beams)