Plastic analysis

 In plastic analysis and design of a structure, the ultimate load of the structure as a whole is regarded as the design criterion. The term plastic has occurred due to the fact that the ultimate load is found from the strength of steel in the plastic range. This method is rapid and provides a rational approach for the analysis of the structure. It also provides striking economy as regards the weight of steel since the sections required by this method are smaller in size than those required by the method of elastic analysis. Plastic analysis and design has its main application in the analysis and design of statically indeterminate framed structures.

 Basics of plastic analysis

Plastic analysis is based on the idealization of the stress-strain curve as elastic-perfectly-plastic. It is further assumed that the width-thickness ratio of plate elements is small so that local buckling does not occur- in other words the sections will classify as plastic. With these assumptions, it can be said that the section will reach its plastic moment capacity and then undergo considerable rotation at this moment. With these assumptions, we will now look at the behaviour of a beam up to collapse.

Consider a simply supported beam subjected to a point load W at midspan. as shown in Fig. 2.14(a). The elastic bending moment at the ends is   and at mid-span is where L is the span. The stress distribution across any cross section is linear [Fig. 2.15(a)]. As W is increased gradually, the bending moment at every section increases and the stresses also increase. At a section Design of Steel Structures Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar Indian Institute of Technology Madras close to the support where the bending moment is maximum, the stresses in the extreme fibers reach the yield stress. The moment corresponding to this state is called the first yield moment My, of the cross section. But this does not imply failure as the beam can continue to take additional load. As the load continues to increase, more and more fibers reach the yield stress and the stress distribution is as shown in Fig 2.15(b). Eventually the whole of the cross section reaches the yield stress and the corresponding stress distribution is as shown in Fig. 2.15(c). The moment corresponding to this state is known as the plastic moment of the cross section and is denoted by Mp. In order to find out the fully plastic moment of a yielded section of a beam, we employ the force equilibrium equation, namely the total force in compression and the total force in tension over that section are equal.

Formation of a collapse mechanism in a fixed beam

Plastification of cross-section under

The ratio of the plastic moment to the yield moment is known as the shape factor since it depends on the shape of the cross section. The cross section is not capable of resisting any additional moment but may maintain this moment for some amount of rotation in which case it acts like a plastic hinge. If this is so, then for further loading, the beam, acts as if it is simply supported with two additional moments Mp on either side, and continues to carry additional loads until a third plastic hinge forms at mid-span when the bending moment at that section reaches Mp. The beam is then said to have developed a collapse mechanism and will collapse as shown in Fig 2.14(b). If the section is thinwalled, due to local buckling, it may not be able to sustain the moment for additional rotations and may collapse either before or soon after attaining the plastic moment. It may be noted that formation of a single plastic hinge gives a collapse mechanism for a simply supported beam. The ratio of the ultimate rotation to the yield rotation is called the rotation capacity of the section. The yield and the plastic moments together with the rotation capacity of the crosssection are used to classify the sections.

Shape factor

As described previously there will be two stress blocks, one in tension, the other in compression, both of which will be at yield stress. For equilibrium of the cross section, the areas in compression and tension must be equal. For a rectangular cross section, the elastic moment is given by,

The plastic moment is obtained from,

Thus, for a rectangular section the plastic moment Mp is about 1.5 times greater than the elastic moment capacity. For an I-section the value of shape factor is about 1.12. Theoretically, the plastic hinges are assumed to form at points at which plastic rotations occur. Thus the length of a plastic hinge is considered as zero. However, the values of moment, at the adjacent section of the yield zone are more than the yield moment upto a certain length ∆L, of the structural member. This length ∆L, is known as the hinged length. The hinged length depends upon the type of loading and the geometry of the cross-section of the structural member. The region of hinged length is known as region of yield or plasticity.

In a simply supported beam (Fig. 2.16) with central concentrated load, the maximum bending moment occurs at the centre of the beam. As the load is    increased gradually, this moment reaches the fully plastic moment of the section Mp and a plastic hinge is formed at the centre.

Let x (= ∆L) be the length of plasticity zone.

From the bending moment diagram shown in Fig. 2.16

Therefore the hinged length of the plasticity zone is equal to one-third of the span in this case.