E

stress x area above N.A.

stress x area below N.A.

In the fully plastic condition, therefore, when the stress is equal throughout the section, the above equation reduces to y) areas above N.A. = ^ areas below N.A.

and in the special case shown in Fig. 3.5 the N.A. will have moved to a position coincident with the lower edge of the flange. Whilst this position is peculiar to the particular geometry chosen for this section it is true to say that for all unsymmetrical sections the N.A. will move from its normal position when the section is completely elastic as plastic penetration proceeds. In the ultimate stage when a plastic hinge has been formed the N.A. will be positioned such that eqn. (3.6) applies, or, often more conveniently, area above or below N.A. = \ total area

In the partially plastic state, as shown in Fig. 3.7, the N.A. position is again determined by applying equilibrium conditions to the forces above and below the N.A. The section is divided into convenient parts, each subjected to a force = average stress x area, as indicated, then

Yielded oreo

N.A. partially plastic

N.A. partially plastic

Fig. 3.7. Partially plastic bending of unsymmetrical section beam.

and this is an equation in terms of a single unknown y , which can then be determined, as can the independent values of F\, Fi, Ft, and F4.

The sum of the moments of these forces about the N.A. then yields the value of the partially plastic moment Mpp. Example 3.2 describes the procedure in detail.

3.5. Shape factor - unsymmetrical sections

Whereas with symmetrical sections the position of the N.A. remains constant as the axis of symmetry through the centroid, in the case of unsymmetrical sections additional work is required to take account of the movement of the N.A. position. However, having determined the position of the N.A. in the fully plastic condition using eqn. (3.6) or (3.7), the procedure outlined in ยง3.2 can then be followed to evaluate shape factors of unsymmetrical sections - see Example 3.2.

3.6. Deflections of partially plastic beams

Deflections of partially plastic beams are normally calculated on the assumption that the yielded areas, having yielded, offer no resistance to bending. Deflections are calculated therefore on the basis of the elastic core only, i.e. by application of simple bending theory and/or the standard deflection equations of Chapter 5^ to the elastic material only. Because the second moment of area I of the central core is proportional to the fourth power of d, and I appears in the denominator of deflection formulae, deflections increase rapidly as d approaches zero, i.e. as full plasticity is approached.

If an experiment is carried out to measure the deflection of beams as loading, and hence B.M., is increased, the deflection graph for simply supported end conditions will appear as shown in Fig. 3.8. Whilst the beam is elastic the graph remains linear. The initiation of yielding in the outer fibres of the beam is indicated by a slight change in slope, and when plastic penetration approaches the centre of the section deflections increase rapidly for very small increases in load. For rectangular sections the ratio MFP/ME will be 1.5 as determined theoretically above.

Fig. 3.8. Typical load-deflection curve for plastic bending.

3.7. Length of yielded area in beams

Consider a simply supported beam of rectangular section carrying a central concentrated load W. The B.M. diagram will be as shown in Fig. 3.9 with a maximum value of WL/4 at w

the centre. If loading is increased, yielding will commence therefore at the central section when (WL/4) = (BD2/6)ay and will gradually penetrate from the outside fibres towards the N.A. As this proceeds with further increase in loads, the B.M. at points away from the centre will also increase, and in some other positions near the centre it will also reach the value required to produce the initial yielding, namely BD2ay/6. Thus, when full plasticity is achieved at the central section with a load Wp, there will be some other positions on either side of the centre, distance x from the supports, where yielding has just commenced at the outer fibres; between these two positions the beam will be in some elastic-plastic state. Now at distance x from the supports:

The central third of the beam span will be affected therefore by plastic yielding to some depth. At any general section within this part of the beam distance x' from the supports the B.M. will be given by

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