On the Interaction of Thermal Membrane and Thermal Bending Stress in Shakedown Analysis

Abstract

When the primary plus secondary stress range exceeds 3 Sm, the current ASME Code rules on simplified elastic-plastic analysis impose two separate requirements to evaluate the potential for ratcheting. The range of primary plus secondary stress excluding thermal bending must be less than 3 Sm, and the thermal stress must satisfy the Bree criterion for thermal stress ratchet. It has been shown previously that this method can be unconservative, i.e. predict shakedown when elastic-plastic analysis shows ratcheting. This paper clarifies the interaction between thermal membrane and bending stress in the presence of a primary membrane stress. An analytical model is used to derive the closed-form ratchet boundary for combined uniform loading of this type. The impact of having stress gradients along the wall that are typical for discontinuities is studied numerically. Simple modifications of the current Code methods are suggested that would achieve a clearer and better-justified set of rules.

A new four-dimensional ratcheting boundary: Derivation and numerical validation

“…Thus, the detail expressions of Eq. (40) for the five cases in Figure 3 The results for the maximum allowable mechanical membrane stresses are consistent with Wolf Reinhardt's work [8], which however ignored the influence of the mechanical bending stress pb  and the Eqs. (41)~(45) of which were not established on the equilibrium equation.…”

“…Several analytical derivations of the ratchet boundary based on the noncyclic method can be found in previous literatures. For example, the Bree problem and inverse Bree problem have been demonstrated in Ref [13], and the expression of ratchet boundary for the rectangular beam subjected to steady mechanical membrane, cyclic thermal membrane and cyclic thermal bending stress simultaneously is derived in Ref [8], and the ratchet limit solution of a beam with arbitrary cross section is obtained analytically in Ref [9] for the case that the cyclic thermal bending and the combination of a steady mechanical membrane and mechanical bending load act simultaneously. However, only three types of stresses or less were considered at the same time in the analytic solutions mentioned above.…”

“…For the case where the thermal membrane and thermal bending stresses act simultaneously, according to whether the thermal stress amplitude exceeds the cyclic yield stress or not, there are five characteristic distributions of the thermal cyclic stress amplitude [8], as shown in Figure 3. ) [8] It is clear from Figure 3 that the five characteristic distributions of the thermal cyclic stress amplitude can be separated into three typical cases: the first case denoted '1' (i.e. case 1a and 1b) corresponds to the elastic cycling situation when the cyclic thermal stress is low enough not to cause yielding through all the cross section.…”

“…Until now, many investigations [6][7][8][9] about ratcheting boundary for cylindrical shell have only considered a few stress parameters of these four stresses: constant mechanical membrane stress, mechanical bending stress, cyclic thermal membrane stress, and thermal bending stress. W.Bradford et al [6,7] derived shakedown and ratcheting behavior for uniaxial primary membrane and thermal bending stresses which was compared with the results of the linear matching method (LMM) [10,11] and excellent agreement was found.…”

“…W.Bradford et al [6,7] derived shakedown and ratcheting behavior for uniaxial primary membrane and thermal bending stresses which was compared with the results of the linear matching method (LMM) [10,11] and excellent agreement was found. The proposed ratcheting evaluation method [8,9] considering three stresses of the four was already adopted by ASME VIII-2 Code 2013 [12] and later versions. Nevertheless, the unified ratcheting analytical solution for these four types of stresses is still unavailable due to complex effects of interaction on each other, as well as the difficulty of building the numerical model for validation.…”