1 Introduction

Shakedown should be conducted for a structure with elastic-plastic material under repeated loads to prevent low-cycle fatigue and progressive plastic deformation failure. The determination of structural shakedown behavior under variable loading is important in structural evaluation and design. Complete analytical shakedown solutions are solved for some relatively simple structure. For complex structures and boundary conditions, the solutions of the shakedown limit load are calculated employing finite element numerical methods. This leads to develop numerical method by applying shakedown theorem. The classic shakedown theorems are the static (Melan) shakedown theorem^[1] and kinematic shakedown theorem^[2]. Melan's shakedown theorem is particularly popular as it leads to conservative lower-bound solution which is desirable in engineering field.

Melan's theorem uses the statically admissible residual stress field to check structural shakedown behavior. For simplicity, it neglects material hardening and adopts an elastic-perfectly plastic material model. The geometric effect, inertia and viscous forces are all neglected. The static shakedown theorem in its original formulation is only valid for proportional loading and not combined loading. To address situations that are more realistic, the classic theorems have been extensively extended; e.g., the extensions for considering a combined loading with a constant load and cyclic load^[3] and extensions for considering the geometry effect^[4] and strain hardening^[5]. The methods that apply Melan's theorem solve the shakedown limit load by finding the optimal residual stress field. It is thus always the case that a fictitious statically admissible residual stress field is constructed, and the optimal stress field is then searched. Most of the methods are reformulated to mathematical programming problems. To improve the computational efficiency and accuracy and to overcome potential dimension obstacles, great efforts have been made to construct statically admissible residual stress fields and to develop mathematical programming algorithms^[6-8]. Among these programming algorithms, the interior-point algorithm has been well developed to solve problems of large-scale engineering structures, multidimensional loading spaces, and kinematical hardening material. There are also methods that avoid mathematical programming. Nonlinear Superposition Method^[9] and a simplified technique^[10] construct the statically admissible stress field by performing elastic analysis and elastic-plastic analysis, respectively. The statically admissible stress field is constructed by superposing the results of the two analyses. The shakedown limit load is then determined by the superposition of stress fields^[10]. These methods have been applied to a number of practical components^[11-12]. Another method is to apply iterative elastic techniques; e.g., the GLOSS R-Node method^[13], the Elastic Compensation Method^[14], and the Linear Matching Method^[15-16]. Among these, the Linear Matching Method has been developed to be suitable for practical engineering applications involving a complicated thermo-mechanical load history^[15], and developed into a software tool of an Abaqus computer aided engineering plug-in with graphical user interfaces^[16]. Another new numerical procedure based on residual stress decomposition with a Fourier series has also been proposed and applied to thermomechanical loading^[17-18].

In many engineering applications, structures are subjected to combined loading with constant and cyclic loads. It needs to clarify the influence of combined loading in structural shakedown for structural design and evaluation. In general, specific shakedown limit loads are obtained employing the numerical method above or the analytical method. As an example, the shakedown analysis of a hollow tensile specimen subjected to combined constant tension and cyclic torsion has been conducted employing the interior-point method^[7], the shakedown limit load of a square plate with a central small hole subjected to a combined constant horizontal tensile stress and cyclic vertical tensile stress has been calculated employing the Nonlinear Superposition Method^[9]. The shakedown limit loads of a 90° mitred pipe bends subjected to steady internal pressure and cyclic bending moments^[19-20], a functionally graded rotating disk subjected to cyclic temperature gradient loading and a constant rotational speed are also been developed by numerical methods^[21].

However the general effect rules of the combined loading in structural shakedown have not yet been given. The general analytical relationship between combined loads and the shakedown limit load is not clear. The present paper investigates the general effect of the combination loading on structural shakedown behavior employing an analytic method. Polizzotto's extended static shakedown theory is applied. The stress field in equilibrium with the external constant load required in Polizzotto's extended theorem for the lower-bound shakedown analysis is constructed. The shakedown condition of the structure is then derived. The elastic shakedown of the structure under combined loading is discussed through analytical analysis, and the general analytical solution of the shakedown limit load is derived. The general effect of the combined loading on the shakedown limit load is presented. The general result is applied to a hollow tension specimen under constant tension and alternating torsion and a plate with a central hole under constant and cyclic tension. The results are compared with solutions from the literature.

2 Static Shakedown Theorem 2.1 Classic Static Shakedown Theorem

In many situations, it is useful to employ the fictitious perfectly elastic structure of the actual structure with only its elastic properties. In static shakedown theory, the fictitious perfectly elastic structure is used to develop the reference elastic stress field. Melan's classic static shakedown theorem states the following^[1].

For a given cyclic load set, a structure made of elastic-perfectly plastic material will shake down if a time-independent self-equilibrium residual stress field can be found such that the combination of the reference elastic stress and residual stress satisfies the condition^{[1, 9]}

$ {f_Y}\left[ {\mathit{\boldsymbol{\sigma }}_{ij}^{\rm{e}}\left( {x,t} \right) + \mathit{\boldsymbol{\sigma }}_{ij}^{\rm{r}}\left( x \right),{\sigma _Y}} \right] \le 0 $

for any loads in the loading domain and anywhere in the structure, where f_Y(σ_ij, σ_Y)=0 is the yield condition and σ_ij^e(x, t) is the reference elastic stress field. σ_ij^r(x) is the self-equilibrium time-independent residual stress field and σ_Y is the yield stress of the material. It is obvious that Melan's theorem is not suitable for combined loading with constant and cyclic loads.

2.2 Extended Static Shakedown Theorem

In reality, many engineering structures are subjected to a combined loading with constant and cyclic loads. The combined loads can be denoted by P(t) and expressed as

$ P\left( t \right) = {P_0} + {P_{\rm{c}}}\left( t \right) $

where P₀ is the constant load and P_c(t) is the cyclic load with peak cyclic load P_cmax and valley cyclic load P_cmin; i.e., P_c(t)∈[P_cmin, P_cmax].

Polizzotto's extended static theorem makes use of time-independent stress fields, which are in static equilibrium with the external constant load, instead of the self-equilibrium residual stress field used in Melan's theorem. Polizzotto's extended static shakedown theorem for the combined loading P(t) states the following^{[3, 9]}.

For elastic-perfectly plastic structures, the necessary and sufficient condition for elastic shakedown to occur under the combined cyclic and constant loads is that a time-independent stress field σ_ij^a(x), which is in equilibrium with the external constant load P₀, can be found such that the combination of the reference elastic stress of the fictitious perfectly elastic structure for the cyclic load and the stress σ_ij^a(x) does not violate yield condition^{[3, 9]}:

$ {f_Y}\left[ {\mathit{\boldsymbol{\sigma }}_{ij}^{\rm{e}}\left( {x,t} \right) + \mathit{\boldsymbol{\sigma }}_{ij}^{\rm{a}}\left( x \right),{\sigma _Y}} \right] \le 0 $

The shakedown limit load can then be searched for by searching the stress field σ_ij^a(x).

3 Shakedown Analysis of the Structure Under Combined Loading

Because the stress distribution of the structure is not uniform, only the critical points with high stress should be checked to determine the shakedown behavior of the structure. For simplicity, only one structural critical point is discussed in the following to illustrate the analysis process.

3.1 Shakedown Condition and Its Analytical Analysis

For shakedown analysis under a combined load P(t), it is reasonable to assume that, when the cyclic load P_c(t) reaches its peak value (i.e., P(t)=P₀+P_cmax), the structure is in an elastic-plastic state. Then, according to Polizzotto's theorem, a stress field σ_ij^a(x) in equilibrium with the external constant load P₀ is constructed as

$ \mathit{\boldsymbol{\sigma }}_{ij}^{\rm{a}}\left( x \right) = \mathit{\boldsymbol{\sigma }}_{ij}^{\rm{0}}\left( x \right) - \mathit{\boldsymbol{\sigma }}_{ij\max }^{\rm{e}}\left( x \right) $

(1)

where σ_ij⁰(x) is the elastic-plastic stress field for the combined load at which the cyclic load reaches its peak (i.e., P₀+P_cmax) and σ_ijmax^e(x) is the reference elastic stress field of the peak cyclic load P_cmax. The combination of the reference elastic stress of the cyclic load and the stress σ_ij^a(x) should then not violate the yield condition anywhere or at any time, and the shakedown conditions of the structure based on the stress constructed with Eq. (1) can thus be concluded as

$ {f_Y}\left[ {\mathit{\boldsymbol{\sigma }}_{ij\max }^{\rm{e}}\left( x \right) + \mathit{\boldsymbol{\sigma }}_{ij}^{\rm{a}}\left( x \right),{\sigma _Y}} \right] \le 0 $

(2)

$ {f_Y}\left[ {\mathit{\boldsymbol{\sigma }}_{ij\min }^{\rm{e}}\left( x \right) + \mathit{\boldsymbol{\sigma }}_{ij}^{\rm{a}}\left( x \right),{\sigma _Y}} \right] \le 0 $

(3)

where σ_ijmin^e(x) is the reference elastic stress field of the valley cyclic load P_cmin.

Substituting Eq. (1) into Eq. (2) yields

$ {f_Y}\left( {\mathit{\boldsymbol{\sigma }}_{ij}^0\left( x \right),{\sigma _Y}} \right) \le 0 $

(4)

Because the structure is assumed to be in an elastic-plastic state, the elastic-plastic stress field σ_ij⁰(x) naturally satisfies Eq. (4).

Substituting Eq. (1) into Eq. (3) yields

$ {f_Y}\left[ {\boldsymbol{\sigma }_{ij\min }^{\rm{e}}\left( x \right) + \boldsymbol{\sigma }_{ij}^0\left( x \right) - \boldsymbol{\sigma }_{ij\max }^{\rm{e}}\left( x \right),{\sigma _Y}} \right] \le 0 $

(5)

It follows that

$ {f_Y}\left[ {\boldsymbol{\sigma }_{ij}^0\left( x \right) - \boldsymbol{\sigma }_{ij}^{\rm{E}}\left( x \right),{\sigma _Y}} \right] \le 0 $

(6)

where σ_ij^E(x)= σ_ijmax^e(x)-σ_ijmin^e(x) is the reference elastic stress field of the scope of the cyclic load P_cmax-P_cmin. The shakedown condition of the structure under the combined constant and cyclic load is thus expressed as Eq. (6). The shakedown limit load can be obtained by searching for the maximum load that satisfies Eq. (6).

Materials are assumed to follow the von Mises criterion. The yield condition f_Y(σ_ij, σ_Y)=0 can be written as

$ {f_Y}\left( {{\boldsymbol{\sigma }_{ij}},{\sigma _Y}} \right) = 1/2\left( {{\boldsymbol{\sigma }_{ij}}{\boldsymbol{\sigma }_{ij}} - 3\sigma _m^2} \right) - \sigma _Y^2/3 = 0 $

where σ_ij is elastic-plastic stress of the structure, σ_m is the average value of the three normal stress components of σ_ij; i.e., the hydrostatic stress, σ_m=(σ₁+σ₂+σ₃)/3. The left term of Eq. (6) can then be expressed as

$ \begin{array}{l} {f_Y}\left[ {\boldsymbol{\sigma }_{ij}^0 - \boldsymbol{\sigma }_{ij}^{\rm{E}}\left( x \right),{\sigma _Y}} \right] = \\ \;\;\;\;1/2\left[ {\left( {\boldsymbol{\sigma }_{ij}^0 - \boldsymbol{\sigma }_{ij}^{\rm{E}}} \right)\left( {\boldsymbol{\sigma }_{ij}^0 - \boldsymbol{\sigma }_{ij}^{\rm{E}}} \right) - 3\left( {\sigma _m^0 - } \right.} \right.\\ \;\;\;\;\left. {{{\left. {\sigma _m^{\rm{E}}} \right)}^2}} \right] - \sigma _Y^2/3 = 1/2\left[ {\boldsymbol{\sigma }_{ij}^0\boldsymbol{\sigma }_{ij}^0 - 3{{\left( {\sigma _m^0} \right)}^2} - \left( {2\boldsymbol{\sigma }_{ij}^0 - } \right.} \right.\\ \;\;\;\;\left. {\left. {\boldsymbol{\sigma }_{ij}^{\rm{E}}} \right)\boldsymbol{\sigma }_{ij}^{\rm{E}} + 3\left( {2\sigma _m^0 - \sigma _m^{\rm{E}}} \right)\sigma _m^{\rm{E}}} \right] - \sigma _Y^2/3 \end{array} $

(7)

where σ_m⁰ and σ_m^E are the hydrostatic stresses of σ_ij⁰ and σ_ij^E, respectively. As it is assumed that the structure is in an elastic-plastic state under P₀+P_cmax, the critical point of the structure reaches a plastic state. The elastic-plastic stress σ_ij⁰ thus satisfies the yield condition

$ {f_Y}\left( {\boldsymbol{\sigma }_{ij}^0,{\sigma _Y}} \right) = 1/2\left[ {\boldsymbol{\sigma }_{ij}^0\boldsymbol{\sigma }_{ij}^0 - 3{{\left( {\sigma _m^0} \right)}^2}} \right] - {\left( {{\sigma _Y}} \right)^2}/3 = 0 $

(8)

where σ_m⁰ is the hydrostatic stress of σ_ij⁰.

Substituting Eq. (8) into Eq. (7) yields

$ \begin{array}{l} {f_Y}\left( {\boldsymbol{\sigma }_{ij}^0 - \boldsymbol{\sigma }_{ij}^{\rm{E}},{\sigma _Y}} \right) = {f_Y}\left( {\boldsymbol{\sigma }_{ij}^0,{\sigma _Y}} \right) - 1/2\left( 2{\boldsymbol{\sigma }_{ij}^0 - \boldsymbol{\sigma }_{ij}^{\rm{E}}} \right)\boldsymbol{\sigma }_{ij}^{\rm{E}} + \\ \;\;\;\;\;3/2\left( {2\sigma _m^0 - \sigma _m^{\rm{E}}} \right)\sigma _m^{\rm{E}} = - 1/2\left( 2{\boldsymbol{\sigma }_{ij}^0 - \boldsymbol{\sigma }_{ij}^{\rm{E}}} \right)\boldsymbol{\sigma }_{ij}^{\rm{E}} + \\ \;\;\;\;\;3/2\left( {2\sigma _m^0 - \sigma _m^{\rm{E}}} \right)\sigma _m^{\rm{E}} \end{array} $

(9)

According to the generalized Hooke law,

$ \boldsymbol{\sigma }_{ij}^{\rm{E}} = 2G\boldsymbol{\varepsilon }_{ij}^{\rm{E}} + \lambda \varepsilon _v^{\rm{E}}{\delta _{ij}} $

(10)

$ \sigma _m^{\rm{E}} = {\rm{K}}\varepsilon _{\rm{v}}^{\rm{E}} $

(11)

where

$ \lambda = \mu E/\left[ {\left( {1 + \mu } \right)\left( {1 - 2\mu } \right)} \right] $

(12)

$ K = E/3\left( {1 - 2\mu } \right) $

(13)

$ G = E/2\left( {1 + \mu } \right) $

(14)

Here ε_ij^E is the elastic strain corresponding to σ_ij^E, ε_v^E is the corresponding volumetric strain and the sum of the three principle strains of ε_ij^E; i.e., ε₁^E+ε₂^E+ε₃^E.δ_ij is the unit tensor. If i = j, then δ_ij=1. If i ≠ j, then δ_ij=0. E is the elastic modulus, and μ is the Poisson ratio, and G is the shear modulus. Substituting Eqs. (10)-(14) into Eq. (9), we obtain

$ \begin{array}{l} {f_Y}\left( {\boldsymbol{\sigma }_{ij}^0 - \boldsymbol{\sigma }_{ij}^{\rm{E}},{\sigma _Y}} \right) = - G\left( 2{\boldsymbol{\sigma }_{ij}^0 - \boldsymbol{\sigma }_{ij}^{\rm{E}}} \right)\boldsymbol{\varepsilon }_{ij}^{\rm{E}} + G\left( {2\sigma _m^0 - } \right.\\ \;\;\;\;\;\;\;\;\;\;\left. {\sigma _m^{\rm{E}}} \right)\varepsilon _v^{\rm{E}} \end{array} $

(15)

According to elastic-plastic theory,

$ {\boldsymbol{\sigma }_{ij}} = {{\boldsymbol{\sigma '}}_{ij}} + {\sigma _m}{\delta _{ij}} $

(16)

$ \boldsymbol{\varepsilon }_{ij}^{\rm{E}} = \boldsymbol{\varepsilon }_{ij}^{{\rm{E'}}} + \varepsilon _m^{\rm{E}}{\delta _{ij}} $

(17)

where ε_ij^E′ is the deviatoric stress of ε_ij^E. ε_m^E is the average strain of the three principal strain components of ε_ij^E; i.e., ε_m^E=(ε₁^E+ε₂^E+ε₁^E)/3. Substituting the expressions in Eqs. (16) and (17) into Eq. (15) yields

$ {f_Y}\left( {\boldsymbol{\sigma }_{ij}^0 - \boldsymbol{\sigma }_{ij}^{\rm{E}},{\sigma _Y}} \right) = - G\left( {2\boldsymbol{\sigma }_{ij}^{0'} - \boldsymbol{\sigma }_{ij}^{{\rm{E'}}}} \right)\boldsymbol{\varepsilon }_{ij}^{{\rm{E'}}} $

(18)

where σ_ij^0′ is the deviatoric stress of σ_ij⁰ and σ_ij^E' is the deviatoric stress of σ_ij^E. Eq. (18) is the expression of the left term of Eq. (6) for the critical point. Substituting the expression in Eq. (18) into Eq. (6), the shakedown condition of the structure is obtained as

$ - G\left( {2\boldsymbol{\sigma }_{ij}^{0'} - \boldsymbol{\sigma }_{ij}^{{\rm{E'}}}} \right)\boldsymbol{\varepsilon }_{ij}^{{\rm{E'}}} \le 0 $

(19)

3.2 Shakedown Limit Load

For the structure under combined loading in the shakedown state, the shakedown limit load is the maxium range of the cyclic load for a given constant load. The shakedown limit load can then be expressed as [P_cmin^S, P_cmax^S], where P_cmax^S and P_cmin^S are the peak and valley of the shakedown limit load respectively, P_cmax^S-P_cmin^S is the scope of the shakedown limit load.

In a coordinate system on the π plane, vectors are used to represent the deviatoric stress field σ _ij^′, σ_ij^E′ and the deviatoric strain field ε_ij^E′, as illustrated in Fig. 1. The solid concentric circle is the yield curve. The radius of the dotted concentric circle is twice that of the solid circle.

We set $ \overrightarrow {\mathit{\boldsymbol{OA}}} = \mathit{\boldsymbol{\sigma }}_{ij}^{0'}$, $ \overrightarrow {\mathit{\boldsymbol{OB}}} = 2\mathit{\boldsymbol{\sigma }}_{ij}^{0'} = 2\overrightarrow {\mathit{\boldsymbol{OA}}} $, $ \overrightarrow {\mathit{\boldsymbol{OC}}}= \mathit{\boldsymbol{\sigma }}_{ij}^{{\rm{E}}'} $ and ∠AOC=θ.θ is then the angle between the deviatoric stress σ_ij^0′ and deviatoric stress σ_ij^E′. It is clear that the angle between the two stresses and the angle between their corresponding deviatoric stresses are the same according to elastic-plastic theory. Therefore, the angle θ is equal to the angle between the stress σ_ij⁰ and elastic stress σ_ij^E. The stress σ_ij⁰ is thus generated in the structure under combined loading with constant and peak cyclic loads. The elastic stress σ_ij^E is the reference elastic stress generated in the structure by the scope of the cyclic load P_cmax-P_cmin. The angle θ thus reflects the effect of the interaction between the constant load and cyclic load. According to the geometry, $ \overrightarrow {\mathit{\boldsymbol{CB}}} = \overrightarrow {\mathit{\boldsymbol{OB}}}-\overrightarrow {\mathit{\boldsymbol{OC}}} = 2\mathit{\boldsymbol{\sigma }}_{ij}^{0'}-\mathit{\boldsymbol{\sigma }}_{ij}^{{\rm{E}}'} $. According to the generalized Hooke law $ \mathit{\boldsymbol{\varepsilon }}_{ij}^{{\rm{e}}'} = \frac{1}{{2G}}\mathit{\boldsymbol{\sigma }}_{ij}^{{\rm{e}}'} $, the vectors of ε_ij^E′ and σ_ij^E′ are in the same direction. $ \overrightarrow {\mathit{\boldsymbol{BD}}} = \mathit{\boldsymbol{\varepsilon }}_{ij}^{{\rm{E}}'} $, which is in the same direction of the vector $ \overrightarrow {\mathit{\boldsymbol{OC}}} $, is then set. Eq. (18) can be expressed as

$ {f_Y}\left( {\boldsymbol{\sigma }_{ij}^0 - \boldsymbol{\sigma }_{ij}^{\rm{E}},{\sigma _Y}} \right) = - G \cdot \overrightarrow {\boldsymbol{CB}} \cdot \overrightarrow {\boldsymbol{BD}} $

(20)

Given that G>0, the sign of f_Y(σ_ij⁰-σ_ij^E, σ_Y) can be determined by the angle ψ between the vectors $ \overrightarrow {\mathit{\boldsymbol{CB}}} $ and $ \overrightarrow {\mathit{\boldsymbol{BD}}} $. If ψ∈[0, 90°), $ \overrightarrow {\mathit{\boldsymbol{CB}}} \cdot \overrightarrow {\mathit{\boldsymbol{BD}}} > 0 $ and $ {f_Y}\left( {\mathit{\boldsymbol{\sigma }}_{ij}^0-\mathit{\boldsymbol{\sigma }}_{ij}^E, {\sigma _Y}} \right) =-G \cdot \overrightarrow {\mathit{\boldsymbol{CB}}} \cdot \overrightarrow {\mathit{\boldsymbol{BD}}} < 0 $, and Eq. (19) is satisfied and the structural critical point satisfies the shakedown condition. If ψ=(90°, 180°] then $\overrightarrow {\mathit{\boldsymbol{CB}}} \cdot \overrightarrow {\mathit{\boldsymbol{BD}}} < 0 $ and $ {f_Y}\left( {\mathit{\boldsymbol{\sigma }}_{ij}^0-\mathit{\boldsymbol{\sigma }}_{ij}^E, {\sigma _Y}} \right) =-G \cdot \overrightarrow {\mathit{\boldsymbol{CB}}} \cdot \overrightarrow {\mathit{\boldsymbol{BD}}} > 0 $, and Eq. (19) is violated and the structural critical point does not satisfy the shakedown condition. If ψ=90°, $ \overrightarrow {\mathit{\boldsymbol{CB}}} \cdot \overrightarrow {\mathit{\boldsymbol{BD}}} = 0 $ and $ {f_Y}\left( {\mathit{\boldsymbol{\sigma }}_{ij}^0-\mathit{\boldsymbol{\sigma }}_{ij}^E, {\sigma _Y}} \right) =-G \cdot \overrightarrow {\mathit{\boldsymbol{CB}}} \cdot \overrightarrow {\mathit{\boldsymbol{BD}}} = 0 $, and Eq. (19) is again satisfied and the structural critical point satisfies the shakedown condition.

By analyzing the geometric relationship in Fig. 1, it is clear that the value of ψ depends on θ and |$ \overrightarrow {\mathit{\boldsymbol{OC}}} $|, where |$ \overrightarrow {\mathit{\boldsymbol{OB}}} $| is fixed for a structure with given material.

The situation that θ∈(0°, 90°) is illustrated in Fig. 1. By point B, line s running perpendicular to vectors $ \overrightarrow {\mathit{\boldsymbol{BD}}} $ and $ \overrightarrow {\mathit{\boldsymbol{OC}}} $ is created. The whole π plane is then divided into two parts, denoted M and N. If point C is located within the half-plane M, or on the line s, then ψ≤90° and f_Y(σ_ij⁰-σ_ij^E, σ_Y)≤0; i.e., the shakedown condition Eq.(19) is satisfied. If point C is located within the half-plane N, then ψ>90° and f_Y(σ_ij⁰-σ_ij^E, σ_Y)>0, and the shakedown condition Eq. (19) is not satisfied. It is assumed that line OC and line s intersect at point C_s. The shakedown condition with the vectors of the stresses is then

$ \left| {\overrightarrow {\boldsymbol{OC}} } \right| \le \left| {\overrightarrow {\boldsymbol{O}{\boldsymbol{C}_s}} } \right| $

(21)

From the geometry,

$ \left| {\overrightarrow {\boldsymbol{O}{\boldsymbol{C}_s}} } \right| = \left| {\overrightarrow {\boldsymbol{OB}} } \right|\cos \theta $

(22)

Substituting Eq. (22) into Eq. (21), we obtain

$ \left| {\overrightarrow {\boldsymbol{OC}} } \right| \le \left| {\overrightarrow {\boldsymbol{OB}} } \right|\cos \theta $

(23)

According to $ \overrightarrow {\mathit{\boldsymbol{OC}}} = \mathit{\boldsymbol{\sigma }}_{ij}^{{\rm{E}}'}$, $ |\overrightarrow {\mathit{\boldsymbol{OB}}}| = 2\overrightarrow {\mathit{\boldsymbol{OA}}} = 2\mathit{\boldsymbol{\sigma }}_{ij}^{0'} $, and $ |\overrightarrow {\mathit{\boldsymbol{OB}}}| = 2{\delta _Y} $, δ_Y is the radius of the yield surface in the principal stress space, and Eq. (23) can be rewritten with stress vectors in the form

$ \overline {\boldsymbol{\sigma }_{ij}^{{\rm{E'}}}} \le 2{\delta _Y}\cos \theta $

(24)

where $ \overline {\mathit{\boldsymbol{\sigma }}_{ij}^{{\rm{E}}'}} $ is the equivalent stress of the reference elastic stress σ_ij^E′. Eq. (24) is the shakedown condition of the stresses. Then, according to Eq. (24), the shakedown condition of the cyclic load is

$ {P_{{\rm{cmax}}}} - {P_{{\rm{cmin}}}} \le 2P\left( {{\delta _Y}} \right)\cos \theta $

(25)

where P(δ_Y) is the load under which the equivalent stress of the reference elastic stress is equal to the radius of the yield surface in the principal stress space. For a perfectly elastic-plastic material structure,

$ P\left( {{\delta _Y}} \right) = P_{\rm{c}}^{\rm{e}} $

(26)

where P_c^e is the elastic limit load of the actual structure subjected to only the cyclic load P_c, which is referred to as the elastic limit load of only the cyclic load. Substituting Eq. (26) into Eq. (25), we obtain

$ {P_{{\rm{cmax}}}} - {P_{{\rm{cmin}}}} \le 2P_{\rm{c}}^{\rm{e}}\cos \theta $

(27)

The shakedown limit load thus satisfies

$ P_{{\rm{cmax}}}^{\rm{s}} - P_{{\rm{cmin}}}^{\rm{s}} = 2P_{\rm{c}}^{\rm{e}}\cos \theta $

(28)

It is concluded that the structural shakedown behavior is affected by the scope of the cyclic load P_cmax-P_cmin and the angle θ in this case. The value and direction of the constant load affect the structural shakedown behavior through the angle θ. Once θ is specified, the shakedown behavior depends on the scope of the cyclic load. If the cyclic load satisfies Eq. (27), the structure is shakedown. Because θ∈(0°, 90°), it follows that 0 < cos θ < 1, and the scope of the shakedown limit load P_cmax^S－P_cmin^S is thus smaller than twice the elastic limit load of only the cyclic load. If P_cmin=0, then the peak of the shakedown limit load satisfies P_cmax^S=2P_c^ecos θ.

The situation that θ=0° is illustrated in Fig. 2. By point B, line s that runs perpendicular to vectors $ \overrightarrow {\mathit{\boldsymbol{BD}}} $ and $ \overrightarrow {\mathit{\boldsymbol{OC}}} $ is created. The whole π plane is then divided into two parts, denoted M and N. If point C is located within the half-plane M, then ψ=0° and f_Y(σ_ij⁰-σ_ij^E, σ_Y) < 0, and therefore, the shakedown condition Eq. (19) is satisfied. If point C is located within the half-plane N, then ψ=180° and f_Y(σ_ij⁰-σ_ij^E, σ_Y)>0, and therefore, the shakedown condition Eq. (19) is not satisfied. The shakedown condition of the stress vectors is then $ |\overrightarrow {\mathit{\boldsymbol{OC}}} | \le |\overrightarrow {\mathit{\boldsymbol{OB}}} | $. Because $ \overrightarrow {\mathit{\boldsymbol{OC}}} = \mathit{\boldsymbol{\sigma }}_{ij}^{{\rm{E}}'} $, $ \overrightarrow {\mathit{\boldsymbol{OB}}} = 2\overrightarrow {\mathit{\boldsymbol{OA}}} = 2\mathit{\boldsymbol{\sigma }}_{ij}^{0'} $ and $ |\overrightarrow {\mathit{\boldsymbol{OB}}} | = 2{\delta _Y} $, the shakedown condition of stress is then expressed as

$ \overline {\boldsymbol{\sigma }_{ij}^{{\rm{E'}}}} \le 2{\delta _Y} $

(29)

The shakedown condition of the cyclic load in this case is

$ {P_{{\rm{cmax}}}} - {P_{{\rm{cmin}}}} \le 2P_{\rm{c}}^{\rm{e}} $

(30)

The scope of the shakedown limit load therefore satisfies

$ P_{{\rm{cmax}}}^S - P_{{\rm{cmin}}}^S = 2P_{\rm{c}}^{\rm{e}} $

(31)

In particular, if P_cmin=0, the peak of the shakedown limit load P_cmax is twice the elastic limit load of only the cyclic load; i.e., P_cmax^S=2P_c^e.

The situation that θ=90° is illustrated in Fig. 3. By point B, line s that runs perpendicular to vectors $ \overrightarrow {\mathit{\boldsymbol{BD}}} $ and $ \overrightarrow {\mathit{\boldsymbol{OC}}} $ is created. The whole π plane is then divided into two parts, denoted M and N. It is clear that point C can be only located within the half-plane N, and therefore ψ>90 and f_Y(σ_ij⁰-σ_ij^E, σ_Y)>0 always holds. The structure thus fails by plastic collapse rather than by exceeding the shakedown limit state. The plastic limit load should substitute the safety load for the shakedown limit load in this case.

The situation that θ∈(90°, 180°) is illustrated in Fig. 4. Because ψ>90°, it follows that f_Y(σ_ij⁰-σ_ij^E, σ_Y)>0 always holds. The situation that θ=180° is illustrated in Fig. 5. It is clear that point C can be only located within the half-space N, and ψ=180° and f_Y(σ_ij⁰-σ_ij^E, σ_Y)>0 always holds.

Therefore, if θ∈[90, 180°], the structure will not reach shakedown under the combined loading. θ∈[90, 180°] means that the constant load controls the stress state in the structure. The shakedown limit is no longer the major factor affecting structural safety. The major factor affecting structural safety is the plastic limit state. The safety load of the structure should be evaluated using the plastic limit load rather than the shakedown limit load.

4 Discussion

As examples, two structures under combined loading are presented and analyzed.

4.1 Hollow Tension Specimen

A hollow tension specimen is subjected to alternating torsion with zero mean shear stress, such that -τ_min=τ_max, and a constant tensile stress, σ>0. The geometry of the specimen is given in Fig. 6. For this problem, the shakedown limit load of the shear stress given in Ref. [8] is

$ \sqrt 3 \tau _{\max }^S = \sqrt {\sigma _Y^2 - {\sigma ^2}} $

(32)

For the specimen with an elastic-perfectly plastic material model under only cyclic torsion, it is clear that the shear stresses reach the maximum value at the outer radius, and the elastic limit of the shear stress load τ_max^e satisfies

$ \sqrt 3 \tau _{\max }^{\rm{e}} = {\sigma _Y} $

(33)

Because the angle between the shear stress τ and tensile stress σ is a right angle, as shown in Fig. 7, the angle θ between the stress of the combined loading with the peak cyclic torsion load and the constant tensile load and the reference elastic stress of the cyclic torsion load in this problem is θ∈[0, 90°]. If merely cyclic torsion is applied to the specimen, θ=0. If merely constant tension is applied to the specimen, θ=90°.

If θ∈(0°, 90°), according to Eq.(28), the shakedown limit load should satisfy

$ \tau _{\max }^S - \tau _{\min }^S = 2\tau _{\max }^{\rm{e}}\cos \theta $

(34)

Substituting -τ_min=τ_max into Eq. (34) yields

$ \tau _{\max }^S = \tau _{\max }^{\rm{e}}\cos \theta $

(35)

Substituting Eq. (33) into Eq. (35) yields

$ \sqrt 3 \tau _{\max }^S = \sqrt 3 \tau _{\max }^{\rm{e}}\cos \theta = {\sigma _Y}\cos \theta $

(36)

If θ=0°, according to Eqs. (31) and (33) and -τ_min=τ_max, the shakedown limit load of the shear stress satisfies

$ \sqrt 3 \tau _{\max }^S = \sqrt 3 \tau _{\max }^e = {\sigma _Y} $

(37)

If θ=90°, according to the analysis results presented in Section 4, the safety load of the tensile stress is the plastic limit load in this case.

Because the angle between the shear stress and tensile stress is a right angle, θ is thus shown in Fig. 7, and it is clear that Eq. (32) is equivalent to Eqs. (36) and (37). Therefore, the result in Ref.[8] is consistent with the analytical analysis results of the present paper.

4.2 Square Plate with a Central Hole

The second example is a square plate with a central hole under combined loading with cyclic and constant loads. Fig. 8 is a schematic diagram of the plate and applied loads. The horizontal stress P₁ is assumed to be constant, and the vertical stress P₂ is assumed to vary cyclically between zero and its corresponding peak value; i.e., P₂∈[0, P_2max].

In Ref. [9], shakedown was analyzed employing the Nonlinear Superposition Method. The elastic limit load, shakedown limit load and plastic limit load of the plate are given for different ratios λ of the constant load P₁ and cyclic load P₂; i.e., λ=P₁/P₂. Analysis results reported in Ref. [9] are illustrated in Fig. 9.

For discussion, the stress states under different load ratios for this problem are first clarified. The problem is analyzed in this paper employing elastic-plastic finite element analysis and the same basic conditions as used in Ref. [9]. In this case, the diameter of the hole D and the length of the plate L are 20 and 100 mm, respectively. The material of the plate adopts an elastic-perfectly plastic material model and follows the von Mises criterion. The elastic modulus is 207 GPa, Poisson's ratio is 0.3 and the yield stress is 300 MPa. All finite element analyses are carried out in MSC Marc with plane stress four-noded quad elements. The finite element meshes and boundary conditions are shown in Fig. 10. A quarter model is used owing to the symmetry of the geometry and loading. Approximately 17 000 elements are meshed.

For the convenience of illustration, we take the example of P₁/P₂=0.25, P₁=25 MPa, and P₂=100 MPa for λ=P₁/P₂∈[0, 1]. The equivalent Mises stress contours under the combined loading with constant load P₁ and peak cyclic load P₂ are shown in Fig. 11. It is seen that the maximum stress is located at point E. The shakedown of the square plate thus depends on the shakedown behavior of the critical point E. Figs. 12(a) and 12(b) show the stress components along the directions of the constant load P₁ and cyclic load P₂, respectively. In this situation, the component of the stress along the direction of constant load P₁ at point E is 5.2 MPa. The component of the stress along the direction of the cyclic load P₂ at point E is 305.2 MPa, which is much larger than 5.2 MPa. Therefore, the stress under the combined load (P₁+P_2max) is almost exactly along the direction of the cyclic load P₂. Because the direction of the reference elastic stress generated at point E by the cyclic load P₂ is the same as the direction of the cyclic load P₂. The angle θ between the stress generated by the combined loading with the constant load and the peak cyclic load and the reference elastic stress generated by the cyclic load is thus almost zero.

According to Eq. (31), the peak of the shakedown limit load P_2max^S is equal to twice the elastic limit load of the structure subjected to only the cyclic load P₂^e; i.e., P_2max^S=2P₂^e.Fig. 9 shows that the shakedown limit load of the cyclic load is twice the elastic limit load of only the cyclic load within allowable error. The results in Ref. [9] are therefore consistent with the analytical results of the present paper.

5 Conclusions

According to Polizzotto's extended static shakedown theorem for combined loading with a constant load and cyclic load, the shakedown condition of a structure was derived. A general expression for the shakedown limit load was then derived through the analytic analysis of the shakedown condition. The general effect of the combined loading on the structural shakedown limit load was proposed. The following conclusions are drawn from the results of the study.

(1) For combined loading, shakedown is affected by the scope of the cyclic load and the angle θ between the stress of the combined loading with the constant load and the peak of the cyclic load and the reference elastic stress of the cyclic load. The constant load affects the structural shakedown behavior through the angle θ.

(2) If the angle θ is zero, the scope of the shakedown limit load is equal to twice the elastic limit load of only the cyclic load.

(3) If θ∈(0°, 90°), the scope of the cyclic load is not larger than twice the product of the elastic limit load of only the cyclic load and the cosine of θ for the structure to shake down. The scope of the shakedown limit load is less than twice the elastic limit load of only the cyclic load.

(4) If the angle θ is greater than or equal to 90°, the structure cannot reach shakedown under the combined loading. The structural safety load should be evaluated using the plastic limit load.

(5) The analytical solutions are applied to two examples, which are the hollow tension specimen subjected to constant tension and cyclic torsion, and the square plate with a central hole subjected to constant and cyclic tension. The obtained results are consistent with the ones in the literature.