The Bree problem with the primary load cycling out-of-phase with the secondary load
Introduction
In this, the 50th anniversary year of Bree's classic analysis of ratcheting [1], it is particularly appropriate to return to the algebraic method of solving ratcheting problems. Analytical solutions of ratcheting problems are relatively rare, being confined to cases of sufficient geometrical and loading simplicity. In real engineering applications, numerical methods, such as the linear matching method [2], [3], are far more versatile and are now becoming mature techniques. Nevertheless, complete analytical solutions remain of considerable value, partly because they provide validation cases for numerical methods and partly because, in the few cases where they are tractable, they provide greater physical insight into the problem. This is exemplified by the Bree analysis, Refs. [1], [4], which is still the chief landmark in this field even after half a century. A further advantage of a complete analytical solution is that all load magnitudes are addressed, identifying the behaviour at all points on the X,Y ratchet-shakedown diagram.
The Bree problem addresses uniaxial ratcheting under a constant primary membrane load plus a secondary wall-bending load which cycles between zero and some maximum value. Whilst the application that Bree had in mind in his original analysis was fast reactor fuel clad, and hence a cylindrical geometry, he analysed the problem as if for uniaxial stressing. Consequently the problem may be considered as relating to a beam of rectangular section.
A modified Bree problem consists of considering the primary membrane load to also cycle. If the primary and secondary loads cycle precisely in-phase (Fig. 1a), or if they cycle precisely in anti-phase (Fig. 1b), the complete solution to the resulting ratcheting and shakedown problem has been published, [5], [6], [7]. The in-phase and anti-phase cases have an identical ratchet boundary, but this lies at substantially greater loads than for the original Bree problem, i.e., in-phase or anti-phase cycling of the primary load is a significantly more benign loading as regards ratcheting than if the primary load were held constant.
However, whilst realistic plant operation does frequently involve both the primary and secondary loads cycling, the occurrence of precisely in-phase, or precisely anti-phase, cycling is improbable. This raises the question as to where the ratchet boundary lies when the primary and secondary loads cycle with some intermediate phase. There are published analyses in which the primary load cycles as well as the secondary load, with various phase relations, for example [8], [9], [10]. However, none have presented the complete solution for the out-of-phase case addressed here.
Section snippets
Definition of the problem addressed
The problem may be considered as relating to a beam of rectangular section subject to uniaxial stressing: a primary membrane load and a secondary bending load. Both loads vary between zero and some maximum, their temporal variation being 'square waves' with the same period but not in phase, as depicted by Fig. 2. Note that because a square-wave profile is assumed, each load is either acting fully or not acting at all. This simplifying assumption makes the problem analytically tractable.
The
Formulation of the problem and solution method
All dimensions are normalised by the section thickness, . (This may differ from the normalisation convention used elsewhere, e.g. [7], normalised using ). All stresses are normalised by the yield stress, , and all strains by the yield strain, . The applied (elastic) primary membrane stress, after normalisation, is (hence is plastic collapse). Similarly the peak (elastic) thermal bending stress after normalisation is . The through-thickness dimensionless coordinate , is zero
Ratchet solutions
Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10a are schematic diagrams of possible stress distributions. Each Figure shows the stress distributions for the first five quarter-cycles. The stress distribution for the fifth quarter cycle differs from that of the first quarter cycle despite corresponding to the same loading. Thereafter the distributions repeat, i.e., the sixth cycle has the same stress distribution as the second cycle, the seventh cycle has the same stress
Distributions of Fig. 11
For Fig. 11 to be valid it is required that which, using the parameter solutions of Table 9, implies . This is merely the condition that yielding does occur when both loads act, and establishes that the region in question lies above the 'elastic line', UE in Fig. 15.
For Fig. 11 to be valid it is also required that which implies that , i.e., that the region in question lies below line BE on Fig. 15.
Finally, for Fig. 11 to be valid it is also required that which
Ratchet-shakedown diagram
The resulting ratchet-shakedown diagram is shown, with all its sub-regions, in Fig. 15. Coordinates of salient points on Fig. 15 are given in Table 11a. Table 11b gives the algebraic equations of the various boundary curves on Fig. 15, except in two cases, curves GAB and JAD, which are given numerically in Table 11c. Fig. 16 compares the ratcheting and shakedown regions with the equivalents for the original Bree loading [1], [4], and also with the in-phase or anti-phase case, [5], [6], [7].
Ratchet strains
The ratchet strain (normalised by the yield strain ) is given by for all ratchet regions except R1f for which the ratchet strain is . The parameters and may be found using Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 for each of the eight ratchet regions separately. The results are plotted against for a range of values in Fig. 17.
Cyclic plastic strains
In the cyclic plasticity regions, P1 and P2, the cyclic plastic strain ranges are given by on the left-hand surface, and on the right-hand surface. Note that these are ranges, not amplitudes. (Recall that the two surfaces differ because the right-hand side is the positive side of the secondary bending load). The parameters and have been found numerically by applying the rules of §3.
It may sometimes be overlooked that a structure undergoing ratcheting will also be
Conclusion
The complete analytic solution for the positive out-of-phase cycling variant of the Bree problem has been derived. Fig. 15 shows the resulting ratcheting, shakedown and cyclic plasticity regions on the diagram. Fig. 16 compares the ratcheting and shakedown regions obtained for positive out-of-phase cycling with the original Bree loading [1], [4], and also with the in-phase or anti-phase case, [5], [6], [7].
In the region where design code limits on primary loading are likely to be respected,
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