Plastic Bending of Beams | Steel Structure

In this article we will discuss about the plastic bending of beams.

Let us consider a beam of homogeneous material and symmetrical section subjected to a bending moment M. The distribution of bending stress follows a linear law with zero stress at the neutral axis and a maximum stress at the outermost fibres, when the deformations are within the elastic limit. In case the magnitude of M increases, the stress distribution also changes.

These are shown in the following stages:

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Stage 1:

The deformation is within the elastic limit. The maximum bending stress is ƒ. If the section modulus is Z, we have M = ƒZ [Fig. 8.14(a)].

Stage 2:

If the bending moment is gradually increased so that the extreme fibre reaches the yield stress ƒ_y, the corresponding bending moment is given by, M_y = ƒ_y Z [Fig. 8.14(b)].

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Stage 3:

The bending moment, if further increased, will not increase the maximum fibre stress which remains at the yield stress value ƒ_y, but the yield will spread into the fibres for a depth e called the depth of penetration [Fig. 8.14(c)].

Stage 4:

If the bending moment is further increased, a stage will be reached when the yield will spread into all the fibres resulting in a stress diagram as shown in Fig. 8.14(d).

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The beam section in this stage has reached its maximum resisting capacity. Any further increase in the bending moment cannot be resisted by the section and an instability is reached as would happen if a hinge was provided at the section. A plastic hinge has formed at the section. (A plastic hinge is sometimes also called a rusty hinge or a friction hinge).

At this stage area of the compression zone or tension zone of the section equals A/2 where A is the cross sectional area of the beam.

Total compression on the section = Total tension on the section = ƒ_y (A/2)

This means, the neutral axis at this stage, called the plastic neutral axis divides the beam section into two equal areas.

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The moment of resistance of the section at this stage is called the plastic moment of resistance (or plastic moment) denoted by M_p and is given by-

M_p = Moment of total compression about the plastic neutral axis + Moment of total tension about the plastic neutral axis.

= ƒ_y (A/2) × Distance of the centroid of the compression zone from the plastic neutral axis + ƒ_y (A/2) × Distance of the centroid of the tension zone from the plastic neutral axis

= ƒ_y [Sum of the moments of the compression and tension zones about the plastic neutral axis]

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= ƒ_y Z_p

where, Z_p = Sum of the moments of the compression and tension zones about the plastic neutral axis and is called the plastic modulus.

Thus when a beam section develops a plastic hinge-

Plastic moment of resistance = M_p = ƒ_y Z_p

The ratio of the moment of inertia of the beam section about the elastic i.e., the centroidal neutral axis to the distance of the most distant edge of the section is the section modulus Z of the beam section.

The ratio of the plastic modulus Z_p to the section modulus Z is called the shape factor or form factor, denoted by K_s of the section. This is a measure of the reserve strength the section possesses after the initial yielding.

Shape Factor = K_s = Z_p/Z

If M_y = Moment of resistance of the section when the most extreme fibre of the section reaches the yield stress ƒ_y (this is also called the yield moment)-

M_y = ƒ_y Z

∴ M_p/M_y = (ƒ_y Z_p)/(ƒ_y Z) = Z_p/Z = Shape factor

1. Fixed Beams:

(i) Fixed Beam Carrying a Point Load of the Centre:

In this case corresponding to the collapse condition plastic hinges are developed at the ends and at the centre (Fig. 8.76).

By the geometry of the collapse B.M. diagram, we have-

(ii) Fixed Beam Carrying a Point Load Eccentrically on the Span:

Consider the fixed beam shown in Fig. 8.77.

When the beam deforms within the elastic limit we know that the hogging moments at the ends A and B of the fixed beam carrying a point load at distances a and b from the ends A and B are given by-

The maximum sagging moment occurs under the load itself. For the position of the load shown in the figure, as the load goes on increasing in its magnitude, a plastic hinge is first developed at the end B, then at the end A. As the load is increased further the sagging moment under the load goes on increasing till a plastic hinge is developed under the load. With three plastic hinges, at A, B and under the load, the beam develops a mechanism leading to collapse.

Thus at collapse state the hogging moments at the ends of the beam and the sagging moment under the load are all equal to the plastic moment M_p. Fig. 8.77 shows the collapse B.M. diagram for the beam.

If C is the collapse load, then by the geometry of the collapse B.M. diagram-

Condition for Minimum Value of Collapse Load:

(iii) Fixed Beam Carrying a Uniformly Distributed Load over the Whole Span:

We know when the fixed beam is loaded within the elastic limit the hogging bending moment at each end of the beam due to a uniformly distributed load of w per unit run = (wl²)/12 and the sagging moment at the centre = (wl²)/24.

As the magnitude of the load intensity is increased plastic hinges will be developed at the two ends and as the load intensity is further increased a stage is reached when another plastic hinge is developed at the centre. With three plastic hinges the beam develops a mechanism leading to the collapse of the beam. Thus at collapse state hogging moment at each end and the sagging moment at the centre are each equal to plastic moment Fig. 8.78 shows the collapse B.M. diagram.

From this diagram, it is clear-

(iv) Fixed Beam Carrying a General Load System:

We know for a fixed beam subjected to a loading, at the stage of collapse plastic hinges exist at each end and also at the section of maximum sagging moment. Hence corresponding to any collapse loading, M_p = Half the maximum Free B.M (Fig. 8.79).

2. Continuous Beams:

Two span beam with equal spans carrying uniformly distributed load.

Fig. 8.98 shows a continuous beam ABC consisting of two equal spans AB and BC each of length I carrying a uniformly distributed collapse load of intensity c. In order one of the spans say AB may reach a collapse condition, it is necessary that the maximum sagging bending moment for AB and the hogging bending moment at B should reach the plastic moment M_p. This condition is shown in Fig. 8.98. Analysis of the span AB is similar to that of a beam fixed at B and propped at A.

Thus, the plastic moments at B and between A and B will be reached at collapse condition-

The positions of the plastic hinges are, one at the support B and one on each side of the support B at a distance of 0.586 l.

Three Span Beam, With Equal Spans Carrying Uniformly Distributed Load:

(i) When the section of the beam is uniform throughout:

In this case the beam is designed for the collapse condition of the end span. For the continuous beam shown in Fig. 8.110, the B.M. diagram for the end span at collapse state is identical with that of a propped cantilever carrying the collapse uniformly distributed load.

For the end span AB, plastic hinges are formed at B and at 0.586 I form B. Note at collapse state only the end spans collapse due to plastic instability and the middle span does not collapse since plastic hinges exist only at B and C. Since the bending moment at the middle of the middle span is less than the plastic moment, a plastic hinge has not formed.

The beam is therefore designed for a plastic moment-

(ii) Beam with Cover Plates for End Spans:

At the collapse condition the outer spans collapse while the central span does not collapse since for the central span, plastic hinges are developed at its ends only and the bending moment at the centre has not yet reached the plastic moment. Hence to obtain a more economical design it is desirable to make the central span of a lighter section and this section with cover plates is provided for the end spans.

A beam section is first designed for the collapse condition of the central span. The section so selected is strengthened with cover plates for the end spans. Thus the beam is so designed that at collapse state all the three spans collapse simultaneously.

Consider the three span beam ABCD shown in Fig. 8.111. Corresponding to the collapse condition of the central span, the plastic moment requirement of the beam section is M_p = (cl²)/16. Fig. 8.111 shows the collapse bending moment diagram. Let us determine the maximum bending moment for the end span corresponding to the collapse condition of the central span.

At any section of the end span distant x from the end, the bending moment is given by-

Hence for the condition that the end spans should collapse corresponding collapse condition of the central span, the plastic moment requirement for the end span-

The increase in the plastic moment of resistance required for the end span is obtained by providing cover plates for the end span.

Curtailment of Cover Plates:

The cover plates for the end span are needed only in the region where the bending moment is greater than (cl²)/16.

Let the bending moment for the end span reach the value (cl²)/16 at a distance x from the end.

For this condition-

Hence, for the end span the cover plates are required between x = 0.18 l and x = 0.695 l from the end.

Plastic Bending of Beams:

These are shown in the following stages: