68 Mechanics of Materials 2 §3.4 about some convenient axis as described in Chapter 4.7A typical position of the elastic N.A.is shown in the figure.Application of the simple blending theory about the N.A.will then yield the value of ME as described in the previous paragraph. Whatever the state of the section,be it elastic,partially plastic or fully plastic,equilibrium of forces must always be maintained,i.e.at any section the tensile forces on one side of the N.A.must equal the compressive forces on the other side. ∑stress×area above N.A.=∑stress×area below N.A. In the fully plastic condition,therefore,when the stress is equal throughout the section, the above equation reduces to ∑areas above N.A.=∑areas below N.A. (3.6) 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 (3.7) 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 F1+F2=F3+F4 (3.8) Yielded area Fig.3.7.Partially plastic bending of unsymmetrical section beam. and this is an equation in terms of a single unknown yp,which can then be determined,as can the independent values of F1,F2,F3 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. EJ.Hearn,Mechanics of Materials 1,Butterworth-Heinemann,1997.68 Mechanics of Materials 2 g3.4 about some convenient axis as described in Chapter 4.t A typical position of the elastic N.A. is shown in the figure. Application of the simple blending theory about the N.A. will then yield the value of ME as described in the previous paragraph. Whatever the state of the section, be it elastic, partially plastic or fully plastic, equilibrium of forces must always be maintained, i.e. at any section the tensile forces on one side of the N.A. must equal the compressive forces on the other side. 1 stress x area above N.A. = 1 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 areas above N.A. = areas below N.A. (3.6) 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 (3.7) 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 ----- =* Fig. 3.7. Partially plastic bending of unsymmetrical section beam. and this is an equation in terms of a single unknown 7,. which can then be determined, as can the independent values of F 1, F2, F3 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. E.J. Hearn, Mechanics of Materials 1, Butterworth-Heinemann, 1991