fully plastic torque

where xy is the shear stress at the elastic limit, or shear yield stress. Angles of twist of partially plastic shafts are calculated on the basis of the elastic core only. For hollow shafts, inside radius R\, outside radius R yielded to radius R2,



For eccentric loading of rectangular sections the fully plastic moment is given by

45(7 v where P is the axial load, N the load factor and B the width of the cross-section. The maximum allowable moment is then given by


AN y 4Boy

For a solid rotating disc, radius R, the collapse speed cop is given by

2 3cry where p is the density of the disc material.

For rotating hollow discs the collapse speed is found from


When the design of components is based upon the elastic theory, e.g. the simple bending or torsion theory, the dimensions of the components are arranged so that the maximum stresses which are likely to occur under service loading conditions do not exceed the allowable working stress for the material in either tension or compression. The allowable working stress is taken to be the yield stress of the material divided by a convenient safety factor (usually based on design codes or past experience) to account for unexpected increase in the level of service loads. If the maximum stress in the component is likely to exceed the allowable working stress, the component is considered unsafe, yet it is evident that complete failure of the component is unlikely to occur even if the yield stress is reached at the outer fibres provided that some portion of the component remains elastic and capable of carrying load, i.e. the strength of a component will normally be much greater than that assumed on the basis of initial yielding at any position. To take advantage of the inherent additional strength, therefore, a different design procedure is used which is often referred to as plastic limit design. The revised design procedures are based upon a number of basic assumptions about the material behaviour.

Figure 3.1 shows a typical stress-strain curve for annealed low carbon steel indicating the presence of both upper and lower yield points and strain-hardening characteristics.

Fig. 3.1. Stress-strain curve for annealed low-carbon steel indicating upper and lower yield points and strain-

hardening characteristics.

Fig. 3.2. Assumed stress-curve for plastic theory - no strain-hardening, equal yield points, <tv, = crVl = f.v

Figure 3.2 shows the assumed material behaviour which:

(a) ignores the presence of upper and lower yields and suggests only a single yield point;

(b) takes the yield stress in tension and compression to be equal;

Stress a


Upper yield point * hordening

Fig. 3.1. Stress-strain curve for annealed low-carbon steel indicating upper and lower yield points and strain-

hardening characteristics.


Upper yield point * hordening

Stress c

(c) assumes that yielding takes place at constant strain thereby ignoring any strain-hardening characteristics. Thus, once the material has yielded, stress is assumed to remain constant throughout any further deformation.

It is further assumed, despite assumption (c), that transverse sections of beams in bending remain plane throughout the loading process, i.e. strain is proportional to distance from the neutral axis.

It is now possible on the basis of the above assumptions to determine the moment which must be applied to produce:

(a) the maximum or limiting elastic conditions in the beam material with yielding just initiated at the outer fibres;

(b) yielding to a specified depth;

(c) yielding across the complete section.

The latter situation is then termed a fully plastic state, or "plastic hinge". Depending on the support and loading conditions, one or more plastic hinges may be required before complete collapse of the beam or structure occurs, the load required to produce this situation then being termed the collapse load. This will be considered in detail in ยง3.6.

3.1. Plastic bending of rectangular-sectioned beams

Figure 3.3(a) shows a rectangular beam loaded until the yield stress has just been reached in the outer fibres. The beam is still completely elastic and the bending theory applies, i.e.

BD3 2

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