Shakedown analysis of engineering structures under multiple variable mechanical and thermal loads using the stress compensation method
Graphical abstract
Introduction
In many practical engineering fields, such as electric power, nuclear energy, aerospace, petrochemical and civil industries, structural components are usually subjected to variable repeated mechanical and thermal loads. On the one hand, for making full use of the load-carrying capability of materials, these structural components are allowed to operate in plasticity state. On the other hand, in order to ensure structures to be safe and serviceable, the applied variable loads cannot be beyond the safety margin, i.e. shakedown domain, so that the structural components cannot fail due to alternating plasticity (low-cycle fatigue) or ratcheting (incremental collapse). Therefore, the shakedown analysis has a wide application prospect because of its important theoretical significance and practical engineering value for strengthening the security of structures and reducing costs. Moreover, the determination of shakedown load or shakedown domain of structures becomes the important task in structural design and integrity assessment.
Many designers hope to determine the shakedown limit by the step-by-step incremental elastic-plastic analysis [1], [2], but for complicated loading history the computation is cumbersome and time-consuming. In addition, the exact loading history is often uncertain in practical situations. The shakedown analysis [3], [4], [5], [6] based on the lower bound theorem by Melan [3] and the upper bound theorem by Koiter [4] provides an effective approach to calculating the shakedown limit of structures, where the exact loading history is not concerned but only the bounding box of these loads. Since the two classical shakedown theorems [3], [4] were established, the studies on shakedown analysis have attracted broad attention in structural engineering and academic circles (see Refs. [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44]), mainly involving the theoretical extensions [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] and development of numerical methods [7], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44] for shakedown analysis.
The two classical shakedown theorems rest on the assumptions [5] of perfectly plastic material, associated temperature-independent constitutive laws, small displacement, negligible inertia and creeping effects. In some engineering situations, these assumptions may be unrealistic. To extend the theory to make it applied in more practical applications, some researchers [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] got rid of some coercive assumptions. The shakedown problems of non-associated flow rules [6], [7], geometrical nonlinearities [8], dynamic effects [9], [10], damaging inelastic material [11], [13] and nonlinear kinematic hardening material [12], [13], [14], [15], [16] have been investigated.
However, although the shakedown theories are proposed and extended, a bigger difficulty in practical engineering applications lies on the numerical method for solving the shakedown problem. Shakedown analysis based on the upper and lower bound theorem is mostly transformed as a mathematical programming problem [7], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], which aims to minimize or maximize a goal function with plenty of independent variables and constraint conditions [17]. As one of pioneers in limit and shakedown analysis field, Maier [6] adapted shakedown theory to the linear programming method using piecewise linearization of yield surfaces. If the von Mises yield criterion is used, the mathematical programming formulation for shakedown analysis leads to a complicated nonlinear optimization problem. Over the last four decades, with the rapid development of numerical methods, some powerful algorithms such as the nonlinear Newton-type iteration algorithm [7], [20], [21], [22], the second order cone programming (SQCP) [23], [24] and the interior point method (IPM) [25], [26], [27], [28], [29] have been developed to solve the nonlinear optimization problem. Besides, some other computational methods [30], [31], [32], [33], [34], [35] of structural analysis instead of traditional finite element method have been combined with shakedown theory to solve the shakedown problem.
Going around the difficulties of optimization, Ponter and Chen [36–[39], [45] developed the elastic compensation method (ECM) or the linear matching method (LMM) to solve the shakedown problem. Using more physical arguments, the LMM matches the linear behavior to the nonlinear plastic behavior by performing a sequence of linear solutions with spatially varying moduli [38], and the incompressible and kinematically admissible strain rate history is also constructed at the same time. Then a series of monotonically reducing upper bounds are generated by an iterative scheme making full use of the upper shakedown theorem. More recently, the residual stress decomposition method for shakedown (RSDM-S) [41], [46] was proposed for the shakedown analysis of some simple two-dimensional structures under mechanical and thermal loads.
Using these proposed numerical methods, the shakedown limits or shakedown domains of some structures such as tubes, holed plates, continuous beams, pressure vessels and piping, are calculated. However, it should be mentioned that most of these applications are restricted to some specific cases (plane problem and axisymmetric shells under two loads) and the computational models are relatively simple. In practical industrial applications, engineering structures are often complex and subjected to multiple variable loads. After mesh discretization, the large number of optimization variables and constraints generally result in a tremendous mathematical programming problem, which implies these methods are of low computational efficiency. Moreover, the computing scale of the mathematical programming problem is multiplied with the increase of the vertices of the loading domain.
The purpose of this paper is to develop a novel and effective numerical procedure based on the stress compensation method (SCM) to solve the practical shakedown problems of large-scale engineering structures under multiple variable mechanical and thermal loads. Differing from the LMM that modifies elastic moduli of the material to match the stress to the yield surface, the SCM directly adjusts the stress to the yield surface by applying the compensation stress on the yield regions. The residual stresses for static shakedown analysis are calculated iteratively at the end of a load cycle instead of at every load vertex, by which the proposed method achieves the good performance that the computational time has little relationship with the number of dimensions of loading domain. Moreover, an iterative procedure rather than mathematical programming formulation is established to generate a sequence of descending load multipliers approaching to the shakedown limit. Over the whole procedure, the global stiffness matrix is decomposed only once, which ensures the high computational efficiency of shakedown analysis regardless of the number of the vertices of the loading domain. Different types of Bree problem with two-dimensional loading domain are tested for the verification purpose of the proposed method. A square plate with a central circular hole considering different load combinations in three-dimensional loading space is calculated and analyzed. Finally, the method is effectively applied for solving the practical shakedown problems of a thick vessel with nozzles from nuclear reactor plant.
Section snippets
Basic theory of shakedown analysis
If a structure made up of elastic-perfectly plastic material is subjected to some complex cyclic history of mechanical and thermal loads, the following situations are possible with the increase of the applied loads [5]:
- (1)
Elastic behavior: If the loads remain sufficiently low, the structural response is perfectly elastic throughout the cycle.
- (2)
Shakedown: The plastic deformation occurs in some local parts of the structure during the initial several load cycles. Afterwards, the development of plastic
Novel SCM for mechanical and thermal loads
We suppose that the structure is made up of elastic-perfectly plastic material obeying the Drucker's postulate. The strain rate is decomposed into three parts:where is the elastic strain rate corresponding to the fictitious elastic stress rate ; is the thermal strain rate; and is the residual strain rate. It is worth noting that the residual strain rate consists of the plastic part and the elastic part , and the
Numerical procedure of the SCM for shakedown analysis
In Section 3, the SCM presents an approach to calculating constant residual stress field for shakedown analysis and provides a symbol to estimate whether the structure made up of the elastic-perfectly plastic material shakes down. In this section, an iterative procedure well suitable for shakedown analysis is proposed.
Numerical applications
In this section, three different numerical examples of shakedown analysis for structures under mechanical and thermal loads that vary within multi-dimensional loading domain are considered. The presented algorithm is implemented into the commercial finite element software ABAQUS [47] via user subroutine UMAT and is used to calculate the shakedown limits of these structures.
All the structures are made up of homogeneous, isotropic and elastic-perfectly plastic material with von Mises yield
Conclusions
A novel numerical procedure based on the stress compensation method (SCM) for shakedown analysis of engineering structures under multiple variable mechanical and thermal loads is proposed. The presented methodology has been implemented into ABAQUS platform to investigate the Bree problem and the shakedown domains of a square plate with a central circular hole under various three-dimensional loading domains, and to solve the practical shakedown problems of a thick vessel with nozzles. The
Acknowledgments
The authors would like to acknowledge the support of the National Science Foundation for Distinguished Young Scholars of China (Grant no. 11325211), the National Natural Science Foundation of China (Grant no. 11672147) and the Project of International Cooperation and Exchange NSFC (Grant no. 11511130057) during this work.
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2022, Ocean EngineeringCitation Excerpt :Later, the residual stress decomposition method (RSDM) (Spiliopoulos and Panagiotou, 2014) which solves shakedown problem by decomposing the residual stresses into Fourier series and RSDM-S (Spiliopoulos and Panagiotou, 2017) that adopts a new convergence criterion and measures the shakedown load for 2D loading domain were developed. Recently, the stress compensation method (SCM) (Peng, 2018) was developed which directly adjusts the stress to the yield surface by applying a compensation stress and is effective to solve shakedown problem of the large-scale structure. Among them, the LMM is capable of calculating limit loads (Chen et al., 2014) and strict shakedown boundaries (Cho et al., 2018; Giugliano et al., 2019), as well as ratchet boundaries (Cho et al., 2020).