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Shakedown analysis of truss structures with nonlinear kinematic hardening
International Journal of Mechanical Sciences 103 (2015) 172–180
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Shakedown analysis of truss structures with nonlinear kinematic hardening S.-Y. Leu n, J.-S. Li Department of Aviation Mechanical Engineering China University of Science and Technology No.200, Jhonghua Street, Hengshan Township, Hsinchu County 312, Taiwan, ROC
art ic l e i nf o
a b s t r a c t
Article history: Received 20 September 2014 Received in revised form 7 September 2015 Accepted 10 September 2015 Available online 24 September 2015
The paper aims to perform shakedown analysis of truss structures with nonlinear kinematic hardening. Both the static and kinematic shakedown analyses of truss structures were conducted analytically and numerically to bound the shakedown limit. First, the problem statement leads to the lower bound (primal) formulation accounting for nonlinear kinematic hardening extended from the Melan's static shakedown theorem. In particular, the Hölder inequality is then utilized to establish the corresponding upper bound (dual) formulation from the lower bound formulation as well as to confirm the duality relationship between them. The derived upper bound formulation is an equivalent form of the Koiter’s kinematic shakedown formulation for trusses without involving time integrals. Further, both the primal and the dual analyses of truss structures were conducted using the optimization algorithms provided by MATLAB. Accordingly, the primal analysis and the dual analysis are validated by converging to the shakedown limit efficiently. Finally, the step-by-step finite-element analysis by using ABAQUS is also performed to verify the analytical formulation and numerical implementation. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Shakedown analysis Truss structure Hölder inequality Duality
1. Introduction Engineering structures are often subjected to cyclic loads. Structures of elastic–plastic materials under cyclic loads may behave in some different types, namely pure elastic, elastic shakedown, ratcheting (incremental collapse), plastic shakedown (reversed plasticity) and plastic collapse (unconstrained plastic flow) [1,2]. Among them, elastic shakedown, ratcheting and plastic shakedown are three types of elastic–plastic responses induced by the cyclic loads ranging between the elastic limit and plastic collapse loads. An elastic–plastic structure is said to shake down to an elastic state if it deforms plastically in the initial loading cycles and then reacts purely elastically in the sequential cycles e.g. [1–5]. As well known, shakedown analysis is a well-established direct method to determine the shakedown limit based on the lower bound (Melan's static) or upper bound (Koiter's kinematic) theorem e.g. [1–5]. By the shakedown theorems, we can formulate shakedown analysis into constrained optimization problems by mathematical programming techniques e.g. [6] to seek the shakedown limit. On the one hand, we seek the greatest lower bound by the static theorem e.g. [7,8]. On the other hand, we search for the least upper bound by the kinematic theorem e.g. [9,10]. n
Corresponding author. Tel.: þ 886 3 5935707x200; fax: þ886 3 5936297. E-mail address: [email protected] (S.-Y. Leu).
http://dx.doi.org/10.1016/j.ijmecsci.2015.09.003 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
Moreover, the duality holding between the lower and the upper bound formulations has been well revealed in shakedown analysis e.g. [11–22]. Accordingly, we can apply the lower and upper bound theorems to bound the exact shakedown limit from below and above, respectively. The Melan's static shakedown theorem was originally stated for structures made of elastic-perfectly plastic materials. However, reallife materials generally demonstrate kinematic hardening behavior [23]. Accordingly, it is more realistic to take kinematical hardening into account while dealing with shakedown analysis problems. In literature, much effort has been made to extend the classical shakedown theorems to consider the effects of hardening [24–29]. Compared to other hardening models, the Armstrong–Frederick nonlinear kinematic hardening model [30,31] is more realistic one for shakedown analysis of metals. However, it seems that no effort has been made to shakedown analyses of truss structures with nonlinear kinematic hardening. The paper aims to perform shakedown analysis of truss structures with nonlinear kinematic hardening. It may be formidable to derive the corresponding shakedown limits directly. However, it is possible to acquire the exact solutions by the duality between static and kinematic formulations. Namely, we can approach from below and above the shakedown limit by conducting static and kinematic shakedown analyses of truss structures, respectively. In the paper, we first state the problem statement of shakedown analysis in the form of the lower bound (primal) formulation. The lower bound
S.-Y. Leu, J.-S. Li / International Journal of Mechanical Sciences 103 (2015) 172–180
(primal) formulation is then transformed to the upper bound (dual) formulation. On the other hand, limit analysis is a special case of shakedown analysis. Accordingly, there exist many similarities between limit and shakedown analysis. In limit analysis, Yang [32] applied the Hölder inequality [33] to establish the kinematic (dual) formulation transformed from the corresponding static (primal) formulation. In particular, Yang [32] stated the primal (lower bound) formulation with the yield criterion denoted in the form of l1 -norm while dealing with limit analysis of truss structures. Yang [32] applied the Hölder inequality [33] to establish the corresponding dual (upper bound) formulation with the l1 -norm on plastic deformation rate of truss members. Following the successful experience in limit analysis of truss structures [32], the paper aims to perform shakedown analysis of truss structures with nonlinear kinematic hardening. Both the static and kinematic shakedown analyses of truss structures were conducted to bound the exact shakedown limit. First, the problem statement of shakedown analysis is to be stated in the form of the lower bound (primal) formulation involving with l1 -norm of axial forces [32]. By the Hölder inequality [33], the lower bound (primal) formulation is then transformed to the upper bound (dual) formulation with the l1 -norm [32]. The equality relationship between the greatest lower bound and the least upper bound is to be analytically confirmed to illustrate the strong duality between the lower and upper bound formulations. To illustrate numerically the strong duality between the lower and upper bound formulations, the primal and the dual analysis are to be performed by the computing tool MATLAB [34], respectively. Finally, the step-bystep finite-element analysis by using ABAQUS is also performed to verify the analytical formulation and numerical implementation.
2. Analytical background We consider truss members made of materials with nonlinear kinematic hardening. The Armstrong–Frederick kinematic hardening model [30,31] is adopted. Corresponding to the Armstrong– Frederick nonlinear kinematic hardening for a von Mises material, the yield function is denoted as [35] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ðS X dev Þ : ðS X dev Þ σ 0 f ðσ XÞ ¼ 2
ð1Þ
where S is the deviatoric stress tensor, X dev is the deviatoric part of the backstress tensor X acting to translate the center of the yield surface, σ 0 is the yield strength. It is noted that the backstress X denotes the movement of the yield surface center. Accordingly, the convexity of the yield surface preserves for a von Mises material with nonlinear kinematic hardening. By the Armstrong–Frederick kinematic hardening model [30,31], the backstress rate X_ is described as 2 p X_ ¼ C ε_ p γ X ε_ 3
ð2Þ
p where C and γ are material parameters, ε_ p is the plastic strain rate, ε_ denotes the equivalent plastic strain rate. As well known, truss members carry only axial forces. For uniaxial loading in the 1-direction, we have stress tensors σ ij ¼ 0 except σ 11 a 0. By the incompressibility condition and symmetry, we have the strain rate tensors ε_ p22 ¼ ε_ p33 ¼ ε_ p11 =2. On the other hand, the backstress is also a deviatoric tensor with X 22 ¼ X 33 ¼ X 11 =2[35].
173
Furthermore, we have the equivalent stress σ associated with the von Mises yield criterion expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ½ðσ 11 X 11 Þ ðσ 22 X 22 Þ2 þ ½ðσ 22 X 22 Þ ðσ 33 X 33 Þ2 σ¼ 2 2 1 2 þ ½ðσ 33 X 33 Þ ðσ 11 X 11 Þ 2 ¼ ðσ 11 X 11 Þ ðσ 22 X 22 Þ 3 ¼ σ 11 X 11 ð3Þ 2 Thus, the yield function can be simplified as [36] 3 f ¼ σ 11 X 11 σ 0 2
ð4Þ
Due to the uniaxial loading condition, it is convenient to describe the yield behavior of truss members in terms of axial forces. Thus, the yield condition for the i th truss member with initial cross sectional area A0 and yield strength σ 0 can be generally described as 3 ðiÞ ðiÞ 3 ðiÞ A0 σ ðiÞ ð5Þ 11 X 11 ¼ t x r A0 σ 0 2 2 ðiÞ ðiÞ where t ðiÞ ¼ A0 σ ðiÞ 11 , x ¼ A0 X 11 are the axial force and the (forceðiÞ like) axial backstress of the i th truss member, σ ðiÞ 11 and X 11 are the corresponding axial stress and axial backstress, respectively. Note that, if we normalize the constitutive model as follows ðiÞ t 3xðiÞ ðiÞ 3 ðiÞ t x ¼ r1 ð6Þ A σ 2 2A0 σ 0 0 0
with ðiÞ
¼
σ t ðiÞ ¼ 11 A0 σ 0 σ 0
xðiÞ ¼
X xðiÞ ¼ 11 A0 σ 0 σ0
t
ðiÞ
ð7Þ
ðiÞ
ð8Þ
Then we can state the constitutive model in the form of l1 -norm as 3 t x1 ¼ max t ðiÞ 3xðiÞ r 1 ð9Þ 2 2 i ðiÞ
where t, x are vectors with components t , xðiÞ , respectively. p On the other hand, the equivalent plastic strain rate ε_ associated with the von Mises yield criterion is expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2h p p ðε_ 11 ε_ p22 Þ2 þðε_ p22 ε_ p33 Þ2 þ ðε_ p33 ε_ p11 Þ2 ε_ ¼ 9 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 3 p 2 ε_ 11 ¼ 9 2 p ð10Þ ¼ ε_ 11 For uniaxial loading with the initial condition Xð0Þ ¼ 0, the values of the backstress X 11 can be obtained by Eq. (2) as follows For uniaxial tension, we have [37]
pðiÞ 2 C ð11Þ X ðiÞ 1 e γε11 11 ¼ 3γ where εpðiÞ 11 is the axial plastic strain of the i th truss member. For uniaxial compression, we have [37]
pðiÞ 2 C ð12Þ X ðiÞ 1 þ eγε11 11 ¼ 3γ Thus, the backstress of the i th truss member, X ðiÞ 11 , will be expressed as Eqs. (11) or (12) depending on the truss member subjected to tensile or compressive loading.
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According to Eqs. (11) and (12), we have 2 C 2 C r xðiÞ r 3 γσ 0 3 γσ 0
ð13Þ
where u is a kinematically admissible displacement field. The superscript T stands for the transpose of the matrix. From Eq. (17), we obtain E
quT ðRBt Þ ¼ uT ðRBtÞ
Since u appears homogeneously and linearly in Eq. (18), we can normalize the relationship by setting the following normalization
3. Static and kinematic shakedown analysis In the following formulation, we consider shakedown analysis of truss structures. The yield behavior is modeled by the von Mises yield criterion. 3.1. Static shakedown analysis For a n bar truss structure, the equilibrium condition can be expressed in the following matrix equation Rt ¼ qf
ð14Þ
where R is the equilibrium matrix, t is the vector of axial forces t ðiÞ of truss members, qf is the applied force with f the unit vector and q the load factor. For a n bar truss structure, the equilibrium condition can be stated in term of the normalized axial force vector t as Rt ¼ RBB−1 t ¼ RBt ¼ qf
ð15Þ
with the diagonal matrix B ¼ diagðA0 σ 0 ; A0 σ 0 ; …; A0 σ 0 Þ ¼ diag ðt 0 ; t 0 ; …; t 0 Þ, the inverse matrix B 1 ¼ diagf1=½A0 σ 0 ; 1=½A0 σ 0 ; …; 1=½A0 σ 0 g ¼ diagð1=t 0 ; 1=t 0 ; …; 1=t 0 Þ and B 1 t ¼ t. By static shakedown analysis, we aim to seek the greatest lower bound under constraints of static and constitutive admissibility. Obviously, the problem statement naturally leads to the lower bound formulation to maximize the load factor q such that maximize q subject to ‖t−32x‖1 r 1 E
t ¼ qt þ ρ
ð18Þ
ð16Þ
RBρ ¼ 0 −23 γσC0 r xðiÞ r 23 γσC0
E
uT ðRBt Þ ¼ 1 Thus, we can rewrite Eq. (18) as 3 3 q ¼ uT ðRBtÞ ¼ ðBT R T uÞT t ¼ ðBT R T uÞT t− x þ x 2 2 3 3 ¼ ðBT R T uÞT t− x þðBT R T uÞT x 2 2
E
as described in Eqs. (8) and (9), qt is the normalized elastic axial force, ρ ¼ B 1 ρ is the normalized time-independent residual axial force, ρ is the time-independent residual axial force. In the lower bound formulation (16), the static admissibility is stated by the equilibrium equation and the static boundary condition. And the constitutive admissibility is about the yield criterion and the backstress in an inequality form. Note that the equilibrium equation is linear and the constitutive inequalities are convex and bounded [32]. Accordingly, the intersection of statically-admissible set and constitutively-admissible set is convex and bounded. Thus, there exists a unique maximum to the convex programming problem.
ð20Þ
Note that, the lower bound q denotes the plastic dissipation of the structure as shown in Eq. (20). By the Hölder inequality [33], it results in T T T ðB R uÞ t−3x r‖BT RT u‖ ‖t−3x‖1 ð21Þ 1 2 2 with the l1 -norm of BT RT u defined as n
ðiÞ X BT R T u ‖BT R T u‖1 ¼
ð22Þ
i¼1
where ðBT R T uÞðiÞ is the i th component of the vector BT RT u. Actually, inequality (21) is sharp, namely equality holds because it can be shown the vector BT R T u is derived according to the normality condition [38] as shown below. ðiÞ As shown in Eq. (6), the yield function ϕ for the i th truss member can be stated as 3 ϕðiÞ ¼ t ðiÞ xðiÞ r1 ð23Þ 2 The plastic axial deformation ΔLðiÞ of the i th truss member is derived by the normality condition [38] as ΔLðiÞ ¼
where ‖t 3x=2‖1 r 1 denotes the von Mises yield criterion in the form of l1 -norm as defined in Eq. (9) and the unknowns t, x are the normalized axial force and the normalized (force-like) ðiÞ axial backstress with components t ¼ t ðiÞ =ðA0 σ 0 Þ ¼ σ ðiÞ 11 =σ 0 and xðiÞ ¼ xðiÞ =ðA0 σ 0 Þ ¼ X ðiÞ 11 =σ 0 , respectively, of the i th truss member
ð19Þ
∂ϕðiÞ ðiÞ λ ∂t ðiÞ
ð24Þ
with ðiÞ
ðiÞ
signðσ 11 Þ ∂ϕ ¼ ∂t ðiÞ t ðiÞ 0
ð25Þ
ðiÞ
and λ the plastic multiplier corresponding to the i th truss member. Namely, t 0 ðiÞ ΔLðiÞ ¼ t ðiÞ 0
ðiÞ
∂ϕ ðiÞ ðiÞ λ ¼ signðσ ðiÞ 11 Þλ ∂t ðiÞ
ð26Þ
Thus, Eq. (26) can be restated as the following matrix-vector form BT ΔL ¼ BT
∂ϕ λ ¼ signðσ11 Þλ ∂t
ð27Þ
With Eq. (15), we then state the principal of virtual work as t ΔL ¼ qf u ¼ ðRtÞT u ¼ tT R T u T
T
ð28Þ
Accordingly, we acquire the compatible condition 3.2. Kinematic shakedown analysis Now we establish the upper bound (dual) formulation from the corresponding lower bound (primal) formulation. The equilibrium equation of the self-equilibrium time-independent residual force can be restated weakly in the form as h i E uT ðRBρÞ ¼ uT RBðtqt Þ ¼ 0 ð17Þ
ΔL ¼ R T u
ð29Þ
Further,
λ ¼ signðσ11 ÞBT ΔL ¼ signðσ11 ÞBT R T u T
ð30Þ
Namely, R u indicates the plastic axial deformation of the truss members and signðσ11 ÞBT R T u denotes the plastic dissipation of the truss.
S.-Y. Leu, J.-S. Li / International Journal of Mechanical Sciences 103 (2015) 172–180
By the Hölder inequality [33], the lower bound q is bounded above by the upper bound q as 3 3 q ¼ ðBT R T uÞT t x þ ðBT R T uÞT x 2 2 3 3 T T T T r ‖B R u‖1 t x1 þ ðB R uÞT x 2 2 3 r ‖BT RT u‖1 þ ðBT RT uÞT x ¼ q ð31Þ 2 Recall Eqs. (11) and (12), we can express the normalized (forcelike) axial backstress x in terms of the plastic axial deformation and then the displacement field u as shown below. For uniaxial tension, we have
εpðiÞ 11 ¼ ln
LðiÞ LðiÞ 0
¼ ln
ðiÞ LðiÞ 0 þ ΔL
ð32Þ
LðiÞ 0
ðiÞ are the initial and current lengths of the i th truss where LðiÞ 0 , L member. Based on Eqs. (11) and (32), we have
pðiÞ 2 C ¼ 1 e γε11 3 γσ 0 " !γ # " !γ # LðiÞ LðiÞ 2 C 2 C ¼ 1 ðiÞ 0 ðiÞ 1 ðiÞ 0 T ðiÞ ¼ 3 γσ 0 3 γσ 0 L0 þ ΔL L0 þ ðR uÞ
xðiÞ ¼
X ðiÞ 11
σ0
ð33Þ
For uniaxial compression, we have
εpðiÞ 11 ¼ ln
LðiÞ LðiÞ 0
¼ ln
ðiÞ LðiÞ 0 þ ΔL
ð34Þ
LðiÞ 0
According to Eqs. (12) and (34), we obtain
pðiÞ 2 C ¼ 1 þ eγε11 3 γσ 0 " !γ # " !γ # LðiÞ þ ΔLðiÞ LðiÞ þ ðRT uÞðiÞ 2 C 2 C 1 þ 0 ðiÞ 1þ 0 ¼ ¼ 3 γσ 0 3 γσ 0 L0 LðiÞ 0
xðiÞ ¼
X ðiÞ 11
σ0
ð35Þ Therefore, the upper bound formulation can be stated as follows minimize subject to
T
T
q
q ¼ ‖B R u‖1 þ BT R T u
E uT RBt ¼ 1
T
3 2 x ð uÞ
ð36Þ
ΔL ¼ R T u Note that, the problem statement (36) is an equivalent form of the Koiter’s kinematic shakedown formulation for trusses without involving time integrals [39]. Also, the unique minimum to the convex programming problem is the least upper bound. In addition, the equality relation between the greatest lower bound and the least upper bound is confirmed due to the sharpness condition as shown in the Hölder inequality (21). Namely, the unique exact solution q is obtained by equating the extreme values of the lower bound q to its corresponding upper bound q. Namely maximize q ¼ q ¼ minimize q
175
for the Armstrong–Frederick nonlinear kinematic hardening [30,31]. Also, the duality between the primal and dual formulations has been demonstrated analytically. To perform numerically static and kinematic shakedown analysis, we adopt the optimization algorithms linprog and fmincon of MATLAB [34] to solve the primal problem (36) and the dual problem (36), respectively. Namely, we conduct the primal and the dual analyses to get the greatest lower bound and the least upper bound, respectively. Without loss of generality, we assume all truss members have the same cross sectional area A and the yield strength σ 0 . Truss members made of perfectly plastic and nonlinear kinematic hardening materials, respectively, are considered. The kinematic hardening parameters C ¼6.92 GPa, γ ¼ 153:4 and the initial yield strength σ 0 ¼ 122:24 MPa, Young’s modulus E ¼210 GPa, Poisson's ratio υ ¼ 0:3 are adopted for the following case studies. As shown in Fig. 1, we consider a three-bar truss structure with the initial configuration θ ¼ 60o . The three-bar truss is subjected to a cyclic force V in the vertical direction as shown in Fig.1. As shown in Fig. 2, we also consider a bridge truss subjected to a cyclic load V in the vertical direction. The bridge truss concerned is hinged at both ends. We consider the loading ranges ½0; P and ½P; 3P for case studies of the three-bar truss and the bridge truss, respectively, as detailed below. First, we consider the three-bar truss with the loading range ½0; P. Table 1 shows the iterative results corresponding to the primal problem (16) and the dual problem (36), respectively. By the optimization algorithms of MATLAB [34], both the lower and upper bounds converge to the shakedown limit, q =ðAσ 0 Þ, at 5 decimal places within 10 iterations. Accordingly, the shakedown limit are P=ðAσ 0 Þ ¼ 2:00000 and P=ðAσ 0 Þ ¼ 2:50000, respectively, for the three-bar truss made of perfectly plastic and nonlinear kinematic hardening materials. On the other hand, the step-by-step finite-element analysis by using ABAQUS [40] is also conducted to calculate the shakedown limit load for comparisons and validation. The shakedown condition is recognized if the plastic dissipation of the structure is bounded e.g. in Ref. [41]. Fig. 3 shows that the accumulated plastic dissipation energy for the truss with perfectly plastic members has been in stable status within 50 loading cycles for the loading case with P=ðAσ 0 Þ ¼ 2:000 but keeps abrupt increasing for a higher cyclic load with P=ðAσ 0 Þ ¼ 2:001 as shown in Fig. 4. Accordingly, it is sure that the truss structure collapses accompanied by unconstrained flow if subjected to the loading case P=ðAσ 0 Þ ¼ 2:001. Thus, the shakedown limit acquired by using ABAQUS [40] is P=ðAσ 0 Þ ¼ 2:000. Very good agreement is obtained to verify the result by shakedown analysis. Fig. 5 shows that the accumulated plastic dissipation energy for the truss with nonlinear kinematic hardening members with P=ðA σ 0 Þ ¼ 2:500 has been in stable status within 50 loading cycles. However, the accumulated plastic dissipation energy keeps moderate increasing as seen in Fig. 6a and then ratcheting is observed as shown in Fig. 6b if we impose the cyclic load with P=ðAσ 0 Þ ¼ 2:501.
ð37Þ
As shown above, we have analytically illustrated the strong duality holding between the primal problem (16) and the dual problem (36).
4. Numerical computations In the paper, we have established the upper bound (dual) formulation from the lower bound (primal) formulation accounting
Fig. 1. A three-bar truss subjected to a cyclic force in the vertical direction.
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S.-Y. Leu, J.-S. Li / International Journal of Mechanical Sciences 103 (2015) 172–180
Fig. 2. A bridge truss subjected to a cyclic force in the vertical direction. 120
Table 1 Convergence of the lower and upper bounds of shakedown limit corresponding to a three-bar truss with loading range ½0; P by MATLAB.
100
Inter. No.
Perfectly plastic
1 2 3 4 5 6 7 8 9 10
Nonlinear kinematic hardening
Lower bound
Upper bound
Lower bound
Upper bound
0.68107 0.68751 1.21090 1.55236 1.76897 1.96970 1.99997 2.00000 2.00000 2.00000
6.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000
0.79166 1.31127 1.78772 2.26453 2.49729 2.49928 2.50000 2.50000 2.50000 2.50000
6.00000 2.50000 2.50000 2.50000 2.50000 2.50000 2.50000 2.50000 2.50000 2.50000
Plastic Dissipation (kJ)
Material type 80
60 Load Factor=2.001
40
20
0 0
10
20
30
40
50
Loading Cycles
Fig. 4. Evolution of plastic dissipation (perfectly plastic three-bar truss, ½0; P) by ABAQUS.
100
2500
80
2000
60 Load Factor=2.000 40
20
Plastic Dissipation (kJ)
Plastic Dissipation (kJ)
120
1500 Load Factor=2.500 1000
500
0 0
10
20
30
40
50
Loading Cycles
Fig. 3. Evolution of plastic dissipation (perfectly plastic three-bar truss, ½0; P) by ABAQUS.
Thus, the shakedown limit acquired by using ABAQUS in Ref. [40] is P=ðAσ 0 Þ ¼ 2:500. Again, very good agreement is obtained to verify the developed approach in the paper. Second, we consider the three-bar truss subjected to the loading range ½P; 3P. Table 2 shows the iterative results corresponding to the primal problem (16) and the dual problem (36), respectively. As shown in Table 2, both the lower and upper bounds converge to the shakedown limit, q =ðAσ 0 Þ, at 5 decimal
0 0
10
20
30
40
50
Loading Cycles
Fig. 5. Evolution of plastic dissipation (nonlinear kinematic hardening three-bar truss, ½0; P) by ABAQUS.
places within 15 iterations. For the three-bar truss made of perfectly plastic materials, the shakedown limit is P=ðAσ 0 Þ ¼ 0:66667 as shown in Table 2. On the other hand, the shakedown limit is P=ðAσ 0 Þ ¼ 0:91269 for the three-bar truss made of nonlinear kinematic hardening materials.
S.-Y. Leu, J.-S. Li / International Journal of Mechanical Sciences 103 (2015) 172–180
177
120
2500
100
Plastic Dissipation (kJ)
Plastic Dissipation (kJ)
2000
1500 Load Factor=2.501 1000
500
80
60 Load Factor=0.666
40
20
0
0 0
10
20
30
40
0
50
10
20
30
40
50
Loading Cycles
Loading Cycles
Fig. 7. Evolution of plastic dissipation (perfectly plastic three-bar truss, ½P; 3P) by ABAQUS.
180 160
120
140
100
100
Plastic Dissipation (kJ)
Stress (MPa)
120
Load Factor=2.501 80
60 40
80
60 Load Factor=0.667
40
20
20
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Strain %
0
Fig. 6. a. Evolution of plastic dissipation (nonlinear kinematic hardening three-bar truss, ½0; P) by ABAQUS. b. Evolution of stress–strain (nonlinear kinematic hardening three-bar truss, ½0; P) by ABAQUS. Table 2 Convergence of the lower and upper bounds of shakedown limit corresponding to a three-bar truss with loading range ½P; 3P by MATLAB.
Inter. no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Material type Perfectly plastic
Nonlinear kinematic hardening
Lower bound
Upper bound
Lower bound
Upper bound
0.41463 0.50565 0.53841 0.56829 0.59790 0.62750 0.64000 0.66514 0.66665 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667
6.00000 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667
0.34855 0.59039 0.68279 0.75276 0.80949 0.83377 0.86714 0.88418 0.90681 0.91035 0.91269 0.91269 0.91269 0.91269 0.91269
4.10710 4.10691 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269
In addition, the step-by-step finite-element results are acquired by using ABAQUS [40]. Fig. 7 shows that the accumulated plastic dissipation energy corresponding to the perfectly plastic truss with
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Loading Cycles
Fig. 8. Evolution of plastic dissipation (perfectly plastic three-bar truss, ½P; 3P) by ABAQUS.
P=ðAσ 0 Þ ¼ 0:666 has been in stable status within 50 loading cycles. However, the accumulated plastic dissipation energy keeps abrupt increasing if we impose the cyclic load with P=ðAσ 0 Þ ¼ 0:667. Thus, the shakedown limit acquired by using ABAQUS [40] is P=ðAσ 0 Þ ¼ 0:666. Moreover, it can be recognized that the truss structure will collapse accompanied by unconstrained flow if subjected to the cyclic load with P=ðAσ 0 Þ ¼ 0:667 as shown in Fig. 8. Fig. 9 shows that the accumulated plastic dissipation energy corresponding to the nonlinear kinematic hardening truss with P =ðAσ 0 Þ ¼ 0:912 has been in stable status within 50 loading cycles. However, the accumulated plastic dissipation energy keeps abrupt increasing if we impose the cyclic load with P=ðAσ 0 Þ ¼ 0:913. Thus, the shakedown limit acquired by using ABAQUS [40] is P=ðAσ 0 Þ ¼ 0:912. Moreover, it can be recognized that the truss structure collapses accompanied by unconstrained flow if subjected to the cyclic load with P=ðAσ 0 Þ ¼ 0:913 as shown in Fig. 10. Note that, the ratio of the kinematic hardening shakedown limit 0.91269 to the perfectly plastic shakedown limit 0.66667 is 1.36903. It is approximately equal to the value of the ratio of kinematic hardening plastic limit to perfectly plastic one C=ðγσ 0 Þ þ 1. Obviously, the axial stress σ 11 of each perfectly plastic member is equal to the yield strength σ 0 if the three-bar truss structure collapses. Also, the axial stress σ 11 of each nonlinear
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25000
Plastic Dissipation (kJ)
20000
15000 Load Factor=0.912 10000
5000
0 0
10
20
30
40
50
Loading Cycles
Fig. 9. Evolution of plastic dissipation (nonlinear kinematic hardening three-bar truss, ½P; 3P) by ABAQUS. 30000
Plastic Dissipation (kJ)
25000
load larger than its shakedown limit load. Therefore, the ratio of the kinematic hardening shakedown limit to the perfectly plastic one is approximately equal to the ratio of kinematic hardening plastic limit to perfectly plastic one C=ðγσ 0 Þ þ1. Fourth, we consider the bridge truss with the loading range ½P; 3P. Table 4 lists the iterative results corresponding to the primal problem (16) and the dual problem (36), respectively. As shown in Table 4, both the lower and upper bounds also converge to the shakedown limit, q =ðAσ 0 Þ, at 5 decimal places within 15 iterations. For the bridge truss structure made of perfectly plastic members, the shakedown limit is P=ðAσ 0 Þ ¼ 0:43301 as shown in Table 4. On the other hand, the shakedown limit is P=ðAσ 0 Þ ¼ 0:59281 for the bridge truss structure made of nonlinear kinematic hardening members. Once again, the ratio of the kinematic hardening shakedown limit 0.59281 to the perfectly plastic shakedown limit 0.43301 is 1.36904. It is close to the value of the ratio of kinematic hardening plastic limit to perfectly plastic one C=ðγσ 0 Þ þ 1. Similarly, the bridge truss structure concerned will collapse if subjected to cyclic load larger than its shakedown limit load. Thus, the ratio of the kinematic hardening shakedown limit to the perfectly plastic one is approximately equal to the ratio of kinematic hardening plastic limit to perfectly plastic one C=ðγσ 0 Þ þ 1. Table 3 Convergence of the lower and upper bounds of shakedown limit corresponding to a bridge truss with loading range ½0; P by MATLAB.
20000
15000 Load Factor=0.913
Inter. no.
10000
5000
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Loading Cycles
Fig. 10. Evolution of plastic dissipation (nonlinear kinematic hardening three-bar truss, ½P; 3P) by ABAQUS.
kinematic hardening member is equal to the yield strength C=γ þ σ 0 if the three-bar truss structure collapses according to Eqs. (3) and (13). Further, it is illustrated in Figs. 8 and 10 that the three-bar truss structure will collapse accompanied by unconstrained flow if subjected to cyclic load larger than its shakedown limit load. Accordingly, the ratio of the kinematic hardening shakedown limit to the perfectly plastic one is approximately equal to the ratio of kinematic hardening plastic limit to perfectly plastic one C=ðγσ 0 Þ þ1. Third, we consider the bridge truss with the loading range ½0; P. Table 3 lists the iterative results corresponding to the primal problem (16) and the dual problem (36), respectively. As shown in Table 3, both the lower and upper bounds also converge to the shakedown limit, q =ðAσ 0 Þ, at 5 decimal places within 10 iterations. For the bridge truss made of perfectly plastic members, the shakedown limit is P=ðAσ 0 Þ ¼ 1:29904 as shown in Table 3. On the other hand, the shakedown limit is P=ðAσ 0 Þ ¼ 1:77843 for the bridge truss with nonlinear kinematic hardening. Again, the ratio of the kinematic hardening shakedown limit 1.77843 to the perfectly plastic shakedown limit 1.29904 is 1.36903. It is close to the value of the ratio of kinematic hardening plastic limit to perfectly plastic one C=ðγσ 0 Þ þ 1. Obviously, the bridge truss structure concerned will collapse subjected to cyclic
1 2 3 4 5 6 7 8 9 10
Material type Perfectly plastic
Nonlinear kinematic hardening
Lower bound
Upper bound
Lower bound
Upper bound
1.28566 1.29880 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904
10.00000 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904
1.17865 1.18035 1.49825 1.69251 1.76062 1.76449 1.77754 1.77842 1.77843 1.77843
2.73807 2.02966 1.77843 1.77843 1.77843 1.77843 1.77843 1.77843 1.77843 1.77843
Table 4 Convergence of the lower and upper bounds of shakedown limit corresponding to a bridge truss with loading range ½P; 3P by MATLAB.
Inter. no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Material type Perfectly plastic
Nonlinear kinematic hardening
Lower bound
Upper bound
Lower bound
Upper bound
0.29966 0.33541 0.36683 0.39784 0.42880 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301
4.00000 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301
0.17296 0.25430 0.34458 0.42793 0.45525 0.49373 0.55207 0.58595 0.59244 0.59250 0.59281 0.59281 0.59281 0.59281 0.59281
6.84517 5.15985 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281
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Finally, as shown in Tables 1–4, the shakedown limits are bounded from below and above by the greatest lower and least upper bounds, respectively. The greatest lower and least upper bounds are approached monotonically by the lower and upper bounds corresponding to the primal problem (16) and the dual problem (36), respectively. Certainly, the equality relation between the greatest lower bound and the least upper bound also illustrates numerically the strong duality holding between the primal problem (16) and the dual problem (36). Moreover, the analytical formulation and numerical implementation have been verified by the results of the finite-element analysis based on ABAQUS. By the results of ABAQUS mentioned above, an interesting find has been further obtained that in some case studies the trusses will collapse accompanied by unconstrained flow if subjected to higher cyclic loads than shakedown limit loads. Taking the case study with a three-bar truss structure for example, the axial stress σ 11 of each perfectly plastic member is equal to the yield strength σ 0 if the three-bar truss structure collapses. On the other hand, the axial stress σ 11 of each kinematic hardening member is equal to the yield strength C=γ þ σ 0 if the three-bar truss structure collapses according to Eqs. (3) and (13). Accordingly, the kinematic hardening shakedown factors are C=ðγ σ 0 Þ þ 1 ffi 1:369 times of the perfectly plastic ones for such case studies.
5. Conclusions In the paper, we have conducted shakedown analysis of truss structures made of Armstrong–Frederic nonlinear kinematic hardening materials. Both static and kinematic shakedown analyses were performed to acquire the corresponding shakedown limit by equating the greatest lower bound to the least upper bound. By the Hölder inequality [33], the upper bound (dual) formulation was derived from the lower bound (primal) formulation. In particular, the derived upper bound formulation is an equivalent form of the Koiter’s kinematic shakedown formulation for trusses without involving time integrals. The sharpness condition of the Hölder inequality [33] was also shown. Moreover, the optimization algorithms linprog and fmincon of MATLAB [34] was adopted to perform the primal analysis and the dual analysis of a three-bar truss and a bridge truss, respectively. Especially, the primal analysis and the dual analysis have been validated by converging to the shakedown limit efficiently. Moreover, the analytical formulation and numerical implementation have been also verified by the step-by-step finite-element results provided by ABAQUS. In the paper, we adopt the existing Armstrong–Frederick kinematic hardening model [30,31] in the formulation of static and kinematic shakedown analyses. The case studies illustrated in the paper focus on the loading ranges ½0; P and ½P; 3P such that each truss member is subjected to tensile or compression force. However, the approach developed in the paper is also applicable to other loading ranges which may induce the incremental collapse phenomenon and other kinematic hardening models including the straightforward extension to the modified version of the Armstrong–Frederick kinematic hardening model e.g. [42].
Acknowledgments The authors are deeply grateful to the referees’ comments for leading to substantial enhancement of the paper’s quality. The authors also gratefully acknowledge the financial support by Ministry of Science and Technology, ROC, through the grant MOST 102-2221-E-157-003.
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A primal-dual algorithm for shakedown analysis of structure. Comput Methods Appl Mech Eng 2004;193:4663–74. [16] Vu DK, Yan AM, Hung ND. A dual form for discretized kinematic formulation in shakedown analysis. Int J Solids Struct 2004;41:267–77. [17] Vu DK, Staat M, Tran IT. Analysis of pressure equipment by application of the primal–dual theory of shakedown. Commun Numer Methods Eng 2007;23:213–25. [18] Garcea G, Leonetti L. Numerical methods for the evaluation of the shakedown and limit loads. In: Ambrosio J et al., editors. Proceedings of the 7th EUROMECH solid mechanics conference, Lisbon; 2009. [19] Chen H. Lower and upper bound shakedown analysis of structures with temperature-dependent yield stress. J Press Vessel Technol 2010;132:011202– 1-011202-8. [20] Tran TN, Liu GR, Nguyen-Xuan H, Nguyen-Thoi T. An edge-based smoothed finite element method for primal–dual shakedown analysis of structures. Int J Num Methods Eng 2010;82:917–38. [21] Tran TN. A dual algorithm for shakedown analysis of plate bending. Int J Num Methods Eng 2011;86:862–75. [22] Leu SY, Li JS. Shakedown analysis of framed structures: Strong duality and primal-dual analysis. Procedia Eng 2014;79:204–11. [23] de Souza Neto EA, Perić D, Owen DRJ. Computational methods for plasticity: theory and applications. West Sussex: John Wiley & Sons Ltd; 2008. [24] Weichert D, Groβ-Weege J. The numerical assessment of elastic-plastic sheets under variable mechanical and thermal loads using a simplified two surface yield condition. Int J Mech Sci 1988;30:757–67. [25] Groβ-Weege J, Weichert D. Elastic-plastic shells under variable mechanical and thermal loads. Int J Mech Sci 1992;34:863–80. [26] Pham D, Weichert D. Shakedown analysis for elastic-plastic bodies with limited kinematical hardening. Proc R Soc Lond A 2001;457:1097–110. [27] Nguyen QS. On shakedown analysis in hardening plasticity. J Mech Phys Solid 2003;51:101–25. [28] Pham D. Shakedown theory for elastic plastic kinematic hardening bodies. Int J Plast 2007;23:1240–59. [29] Simon JW. Direct evaluation of the limit states of engineering structures exhibiting limited, nonlinear kinematical hardening. Int J Plast 2013;42:141–67. [30] Armstrong PJ, Frederick CO. A mathematical representation of the multiaxial bauschinger effect (CEGB Report RD/B/N/731). Berkeley: Berkeley Nuclear Laboratories; 1966. [31] Frederick CO, Armstrong PJ. A mathematical representation of the multiaxial Bauschinger effect. Mater High Temp 2007;24:1–26. [32] Yang WH. Large deformation of structures by sequential limit analysis. Int J Solids Struct 1993;30:1001–13. [33] Goffman C, Pedrick G. First course in functional analysis. New Jersey: PrenticeHall; 1965. [34] MATLAB, 〈http://www.mathworks.com/help/optim/〉 (2015). [35] Dunne F, Petrinic N. Introduction to computational plasticity. 1st ed. New York: Oxford University Press Inc; 2005. [36] Hu W, Wang CH, Barter S. 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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Shakedown analysis of truss structures with nonlinear kinematic hardening S.-Y. Leu n, J.-S. Li Department of Aviation Mechanical Engineering China University of Science and Technology No.200, Jhonghua Street, Hengshan Township, Hsinchu County 312, Taiwan, ROC
art ic l e i nf o
a b s t r a c t
Article history: Received 20 September 2014 Received in revised form 7 September 2015 Accepted 10 September 2015 Available online 24 September 2015
The paper aims to perform shakedown analysis of truss structures with nonlinear kinematic hardening. Both the static and kinematic shakedown analyses of truss structures were conducted analytically and numerically to bound the shakedown limit. First, the problem statement leads to the lower bound (primal) formulation accounting for nonlinear kinematic hardening extended from the Melan's static shakedown theorem. In particular, the Hölder inequality is then utilized to establish the corresponding upper bound (dual) formulation from the lower bound formulation as well as to confirm the duality relationship between them. The derived upper bound formulation is an equivalent form of the Koiter’s kinematic shakedown formulation for trusses without involving time integrals. Further, both the primal and the dual analyses of truss structures were conducted using the optimization algorithms provided by MATLAB. Accordingly, the primal analysis and the dual analysis are validated by converging to the shakedown limit efficiently. Finally, the step-by-step finite-element analysis by using ABAQUS is also performed to verify the analytical formulation and numerical implementation. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Shakedown analysis Truss structure Hölder inequality Duality
1. Introduction Engineering structures are often subjected to cyclic loads. Structures of elastic–plastic materials under cyclic loads may behave in some different types, namely pure elastic, elastic shakedown, ratcheting (incremental collapse), plastic shakedown (reversed plasticity) and plastic collapse (unconstrained plastic flow) [1,2]. Among them, elastic shakedown, ratcheting and plastic shakedown are three types of elastic–plastic responses induced by the cyclic loads ranging between the elastic limit and plastic collapse loads. An elastic–plastic structure is said to shake down to an elastic state if it deforms plastically in the initial loading cycles and then reacts purely elastically in the sequential cycles e.g. [1–5]. As well known, shakedown analysis is a well-established direct method to determine the shakedown limit based on the lower bound (Melan's static) or upper bound (Koiter's kinematic) theorem e.g. [1–5]. By the shakedown theorems, we can formulate shakedown analysis into constrained optimization problems by mathematical programming techniques e.g. [6] to seek the shakedown limit. On the one hand, we seek the greatest lower bound by the static theorem e.g. [7,8]. On the other hand, we search for the least upper bound by the kinematic theorem e.g. [9,10]. n
Corresponding author. Tel.: þ 886 3 5935707x200; fax: þ886 3 5936297. E-mail address: [email protected] (S.-Y. Leu).
http://dx.doi.org/10.1016/j.ijmecsci.2015.09.003 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
Moreover, the duality holding between the lower and the upper bound formulations has been well revealed in shakedown analysis e.g. [11–22]. Accordingly, we can apply the lower and upper bound theorems to bound the exact shakedown limit from below and above, respectively. The Melan's static shakedown theorem was originally stated for structures made of elastic-perfectly plastic materials. However, reallife materials generally demonstrate kinematic hardening behavior [23]. Accordingly, it is more realistic to take kinematical hardening into account while dealing with shakedown analysis problems. In literature, much effort has been made to extend the classical shakedown theorems to consider the effects of hardening [24–29]. Compared to other hardening models, the Armstrong–Frederick nonlinear kinematic hardening model [30,31] is more realistic one for shakedown analysis of metals. However, it seems that no effort has been made to shakedown analyses of truss structures with nonlinear kinematic hardening. The paper aims to perform shakedown analysis of truss structures with nonlinear kinematic hardening. It may be formidable to derive the corresponding shakedown limits directly. However, it is possible to acquire the exact solutions by the duality between static and kinematic formulations. Namely, we can approach from below and above the shakedown limit by conducting static and kinematic shakedown analyses of truss structures, respectively. In the paper, we first state the problem statement of shakedown analysis in the form of the lower bound (primal) formulation. The lower bound
S.-Y. Leu, J.-S. Li / International Journal of Mechanical Sciences 103 (2015) 172–180
(primal) formulation is then transformed to the upper bound (dual) formulation. On the other hand, limit analysis is a special case of shakedown analysis. Accordingly, there exist many similarities between limit and shakedown analysis. In limit analysis, Yang [32] applied the Hölder inequality [33] to establish the kinematic (dual) formulation transformed from the corresponding static (primal) formulation. In particular, Yang [32] stated the primal (lower bound) formulation with the yield criterion denoted in the form of l1 -norm while dealing with limit analysis of truss structures. Yang [32] applied the Hölder inequality [33] to establish the corresponding dual (upper bound) formulation with the l1 -norm on plastic deformation rate of truss members. Following the successful experience in limit analysis of truss structures [32], the paper aims to perform shakedown analysis of truss structures with nonlinear kinematic hardening. Both the static and kinematic shakedown analyses of truss structures were conducted to bound the exact shakedown limit. First, the problem statement of shakedown analysis is to be stated in the form of the lower bound (primal) formulation involving with l1 -norm of axial forces [32]. By the Hölder inequality [33], the lower bound (primal) formulation is then transformed to the upper bound (dual) formulation with the l1 -norm [32]. The equality relationship between the greatest lower bound and the least upper bound is to be analytically confirmed to illustrate the strong duality between the lower and upper bound formulations. To illustrate numerically the strong duality between the lower and upper bound formulations, the primal and the dual analysis are to be performed by the computing tool MATLAB [34], respectively. Finally, the step-bystep finite-element analysis by using ABAQUS is also performed to verify the analytical formulation and numerical implementation.
2. Analytical background We consider truss members made of materials with nonlinear kinematic hardening. The Armstrong–Frederick kinematic hardening model [30,31] is adopted. Corresponding to the Armstrong– Frederick nonlinear kinematic hardening for a von Mises material, the yield function is denoted as [35] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ðS X dev Þ : ðS X dev Þ σ 0 f ðσ XÞ ¼ 2
ð1Þ
where S is the deviatoric stress tensor, X dev is the deviatoric part of the backstress tensor X acting to translate the center of the yield surface, σ 0 is the yield strength. It is noted that the backstress X denotes the movement of the yield surface center. Accordingly, the convexity of the yield surface preserves for a von Mises material with nonlinear kinematic hardening. By the Armstrong–Frederick kinematic hardening model [30,31], the backstress rate X_ is described as 2 p X_ ¼ C ε_ p γ X ε_ 3
ð2Þ
p where C and γ are material parameters, ε_ p is the plastic strain rate, ε_ denotes the equivalent plastic strain rate. As well known, truss members carry only axial forces. For uniaxial loading in the 1-direction, we have stress tensors σ ij ¼ 0 except σ 11 a 0. By the incompressibility condition and symmetry, we have the strain rate tensors ε_ p22 ¼ ε_ p33 ¼ ε_ p11 =2. On the other hand, the backstress is also a deviatoric tensor with X 22 ¼ X 33 ¼ X 11 =2[35].
173
Furthermore, we have the equivalent stress σ associated with the von Mises yield criterion expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ½ðσ 11 X 11 Þ ðσ 22 X 22 Þ2 þ ½ðσ 22 X 22 Þ ðσ 33 X 33 Þ2 σ¼ 2 2 1 2 þ ½ðσ 33 X 33 Þ ðσ 11 X 11 Þ 2 ¼ ðσ 11 X 11 Þ ðσ 22 X 22 Þ 3 ¼ σ 11 X 11 ð3Þ 2 Thus, the yield function can be simplified as [36] 3 f ¼ σ 11 X 11 σ 0 2
ð4Þ
Due to the uniaxial loading condition, it is convenient to describe the yield behavior of truss members in terms of axial forces. Thus, the yield condition for the i th truss member with initial cross sectional area A0 and yield strength σ 0 can be generally described as 3 ðiÞ ðiÞ 3 ðiÞ A0 σ ðiÞ ð5Þ 11 X 11 ¼ t x r A0 σ 0 2 2 ðiÞ ðiÞ where t ðiÞ ¼ A0 σ ðiÞ 11 , x ¼ A0 X 11 are the axial force and the (forceðiÞ like) axial backstress of the i th truss member, σ ðiÞ 11 and X 11 are the corresponding axial stress and axial backstress, respectively. Note that, if we normalize the constitutive model as follows ðiÞ t 3xðiÞ ðiÞ 3 ðiÞ t x ¼ r1 ð6Þ A σ 2 2A0 σ 0 0 0
with ðiÞ
¼
σ t ðiÞ ¼ 11 A0 σ 0 σ 0
xðiÞ ¼
X xðiÞ ¼ 11 A0 σ 0 σ0
t
ðiÞ
ð7Þ
ðiÞ
ð8Þ
Then we can state the constitutive model in the form of l1 -norm as 3 t x1 ¼ max t ðiÞ 3xðiÞ r 1 ð9Þ 2 2 i ðiÞ
where t, x are vectors with components t , xðiÞ , respectively. p On the other hand, the equivalent plastic strain rate ε_ associated with the von Mises yield criterion is expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2h p p ðε_ 11 ε_ p22 Þ2 þðε_ p22 ε_ p33 Þ2 þ ðε_ p33 ε_ p11 Þ2 ε_ ¼ 9 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 3 p 2 ε_ 11 ¼ 9 2 p ð10Þ ¼ ε_ 11 For uniaxial loading with the initial condition Xð0Þ ¼ 0, the values of the backstress X 11 can be obtained by Eq. (2) as follows For uniaxial tension, we have [37]
pðiÞ 2 C ð11Þ X ðiÞ 1 e γε11 11 ¼ 3γ where εpðiÞ 11 is the axial plastic strain of the i th truss member. For uniaxial compression, we have [37]
pðiÞ 2 C ð12Þ X ðiÞ 1 þ eγε11 11 ¼ 3γ Thus, the backstress of the i th truss member, X ðiÞ 11 , will be expressed as Eqs. (11) or (12) depending on the truss member subjected to tensile or compressive loading.
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According to Eqs. (11) and (12), we have 2 C 2 C r xðiÞ r 3 γσ 0 3 γσ 0
ð13Þ
where u is a kinematically admissible displacement field. The superscript T stands for the transpose of the matrix. From Eq. (17), we obtain E
quT ðRBt Þ ¼ uT ðRBtÞ
Since u appears homogeneously and linearly in Eq. (18), we can normalize the relationship by setting the following normalization
3. Static and kinematic shakedown analysis In the following formulation, we consider shakedown analysis of truss structures. The yield behavior is modeled by the von Mises yield criterion. 3.1. Static shakedown analysis For a n bar truss structure, the equilibrium condition can be expressed in the following matrix equation Rt ¼ qf
ð14Þ
where R is the equilibrium matrix, t is the vector of axial forces t ðiÞ of truss members, qf is the applied force with f the unit vector and q the load factor. For a n bar truss structure, the equilibrium condition can be stated in term of the normalized axial force vector t as Rt ¼ RBB−1 t ¼ RBt ¼ qf
ð15Þ
with the diagonal matrix B ¼ diagðA0 σ 0 ; A0 σ 0 ; …; A0 σ 0 Þ ¼ diag ðt 0 ; t 0 ; …; t 0 Þ, the inverse matrix B 1 ¼ diagf1=½A0 σ 0 ; 1=½A0 σ 0 ; …; 1=½A0 σ 0 g ¼ diagð1=t 0 ; 1=t 0 ; …; 1=t 0 Þ and B 1 t ¼ t. By static shakedown analysis, we aim to seek the greatest lower bound under constraints of static and constitutive admissibility. Obviously, the problem statement naturally leads to the lower bound formulation to maximize the load factor q such that maximize q subject to ‖t−32x‖1 r 1 E
t ¼ qt þ ρ
ð18Þ
ð16Þ
RBρ ¼ 0 −23 γσC0 r xðiÞ r 23 γσC0
E
uT ðRBt Þ ¼ 1 Thus, we can rewrite Eq. (18) as 3 3 q ¼ uT ðRBtÞ ¼ ðBT R T uÞT t ¼ ðBT R T uÞT t− x þ x 2 2 3 3 ¼ ðBT R T uÞT t− x þðBT R T uÞT x 2 2
E
as described in Eqs. (8) and (9), qt is the normalized elastic axial force, ρ ¼ B 1 ρ is the normalized time-independent residual axial force, ρ is the time-independent residual axial force. In the lower bound formulation (16), the static admissibility is stated by the equilibrium equation and the static boundary condition. And the constitutive admissibility is about the yield criterion and the backstress in an inequality form. Note that the equilibrium equation is linear and the constitutive inequalities are convex and bounded [32]. Accordingly, the intersection of statically-admissible set and constitutively-admissible set is convex and bounded. Thus, there exists a unique maximum to the convex programming problem.
ð20Þ
Note that, the lower bound q denotes the plastic dissipation of the structure as shown in Eq. (20). By the Hölder inequality [33], it results in T T T ðB R uÞ t−3x r‖BT RT u‖ ‖t−3x‖1 ð21Þ 1 2 2 with the l1 -norm of BT RT u defined as n
ðiÞ X BT R T u ‖BT R T u‖1 ¼
ð22Þ
i¼1
where ðBT R T uÞðiÞ is the i th component of the vector BT RT u. Actually, inequality (21) is sharp, namely equality holds because it can be shown the vector BT R T u is derived according to the normality condition [38] as shown below. ðiÞ As shown in Eq. (6), the yield function ϕ for the i th truss member can be stated as 3 ϕðiÞ ¼ t ðiÞ xðiÞ r1 ð23Þ 2 The plastic axial deformation ΔLðiÞ of the i th truss member is derived by the normality condition [38] as ΔLðiÞ ¼
where ‖t 3x=2‖1 r 1 denotes the von Mises yield criterion in the form of l1 -norm as defined in Eq. (9) and the unknowns t, x are the normalized axial force and the normalized (force-like) ðiÞ axial backstress with components t ¼ t ðiÞ =ðA0 σ 0 Þ ¼ σ ðiÞ 11 =σ 0 and xðiÞ ¼ xðiÞ =ðA0 σ 0 Þ ¼ X ðiÞ 11 =σ 0 , respectively, of the i th truss member
ð19Þ
∂ϕðiÞ ðiÞ λ ∂t ðiÞ
ð24Þ
with ðiÞ
ðiÞ
signðσ 11 Þ ∂ϕ ¼ ∂t ðiÞ t ðiÞ 0
ð25Þ
ðiÞ
and λ the plastic multiplier corresponding to the i th truss member. Namely, t 0 ðiÞ ΔLðiÞ ¼ t ðiÞ 0
ðiÞ
∂ϕ ðiÞ ðiÞ λ ¼ signðσ ðiÞ 11 Þλ ∂t ðiÞ
ð26Þ
Thus, Eq. (26) can be restated as the following matrix-vector form BT ΔL ¼ BT
∂ϕ λ ¼ signðσ11 Þλ ∂t
ð27Þ
With Eq. (15), we then state the principal of virtual work as t ΔL ¼ qf u ¼ ðRtÞT u ¼ tT R T u T
T
ð28Þ
Accordingly, we acquire the compatible condition 3.2. Kinematic shakedown analysis Now we establish the upper bound (dual) formulation from the corresponding lower bound (primal) formulation. The equilibrium equation of the self-equilibrium time-independent residual force can be restated weakly in the form as h i E uT ðRBρÞ ¼ uT RBðtqt Þ ¼ 0 ð17Þ
ΔL ¼ R T u
ð29Þ
Further,
λ ¼ signðσ11 ÞBT ΔL ¼ signðσ11 ÞBT R T u T
ð30Þ
Namely, R u indicates the plastic axial deformation of the truss members and signðσ11 ÞBT R T u denotes the plastic dissipation of the truss.
S.-Y. Leu, J.-S. Li / International Journal of Mechanical Sciences 103 (2015) 172–180
By the Hölder inequality [33], the lower bound q is bounded above by the upper bound q as 3 3 q ¼ ðBT R T uÞT t x þ ðBT R T uÞT x 2 2 3 3 T T T T r ‖B R u‖1 t x1 þ ðB R uÞT x 2 2 3 r ‖BT RT u‖1 þ ðBT RT uÞT x ¼ q ð31Þ 2 Recall Eqs. (11) and (12), we can express the normalized (forcelike) axial backstress x in terms of the plastic axial deformation and then the displacement field u as shown below. For uniaxial tension, we have
εpðiÞ 11 ¼ ln
LðiÞ LðiÞ 0
¼ ln
ðiÞ LðiÞ 0 þ ΔL
ð32Þ
LðiÞ 0
ðiÞ are the initial and current lengths of the i th truss where LðiÞ 0 , L member. Based on Eqs. (11) and (32), we have
pðiÞ 2 C ¼ 1 e γε11 3 γσ 0 " !γ # " !γ # LðiÞ LðiÞ 2 C 2 C ¼ 1 ðiÞ 0 ðiÞ 1 ðiÞ 0 T ðiÞ ¼ 3 γσ 0 3 γσ 0 L0 þ ΔL L0 þ ðR uÞ
xðiÞ ¼
X ðiÞ 11
σ0
ð33Þ
For uniaxial compression, we have
εpðiÞ 11 ¼ ln
LðiÞ LðiÞ 0
¼ ln
ðiÞ LðiÞ 0 þ ΔL
ð34Þ
LðiÞ 0
According to Eqs. (12) and (34), we obtain
pðiÞ 2 C ¼ 1 þ eγε11 3 γσ 0 " !γ # " !γ # LðiÞ þ ΔLðiÞ LðiÞ þ ðRT uÞðiÞ 2 C 2 C 1 þ 0 ðiÞ 1þ 0 ¼ ¼ 3 γσ 0 3 γσ 0 L0 LðiÞ 0
xðiÞ ¼
X ðiÞ 11
σ0
ð35Þ Therefore, the upper bound formulation can be stated as follows minimize subject to
T
T
q
q ¼ ‖B R u‖1 þ BT R T u
E uT RBt ¼ 1
T
3 2 x ð uÞ
ð36Þ
ΔL ¼ R T u Note that, the problem statement (36) is an equivalent form of the Koiter’s kinematic shakedown formulation for trusses without involving time integrals [39]. Also, the unique minimum to the convex programming problem is the least upper bound. In addition, the equality relation between the greatest lower bound and the least upper bound is confirmed due to the sharpness condition as shown in the Hölder inequality (21). Namely, the unique exact solution q is obtained by equating the extreme values of the lower bound q to its corresponding upper bound q. Namely maximize q ¼ q ¼ minimize q
175
for the Armstrong–Frederick nonlinear kinematic hardening [30,31]. Also, the duality between the primal and dual formulations has been demonstrated analytically. To perform numerically static and kinematic shakedown analysis, we adopt the optimization algorithms linprog and fmincon of MATLAB [34] to solve the primal problem (36) and the dual problem (36), respectively. Namely, we conduct the primal and the dual analyses to get the greatest lower bound and the least upper bound, respectively. Without loss of generality, we assume all truss members have the same cross sectional area A and the yield strength σ 0 . Truss members made of perfectly plastic and nonlinear kinematic hardening materials, respectively, are considered. The kinematic hardening parameters C ¼6.92 GPa, γ ¼ 153:4 and the initial yield strength σ 0 ¼ 122:24 MPa, Young’s modulus E ¼210 GPa, Poisson's ratio υ ¼ 0:3 are adopted for the following case studies. As shown in Fig. 1, we consider a three-bar truss structure with the initial configuration θ ¼ 60o . The three-bar truss is subjected to a cyclic force V in the vertical direction as shown in Fig.1. As shown in Fig. 2, we also consider a bridge truss subjected to a cyclic load V in the vertical direction. The bridge truss concerned is hinged at both ends. We consider the loading ranges ½0; P and ½P; 3P for case studies of the three-bar truss and the bridge truss, respectively, as detailed below. First, we consider the three-bar truss with the loading range ½0; P. Table 1 shows the iterative results corresponding to the primal problem (16) and the dual problem (36), respectively. By the optimization algorithms of MATLAB [34], both the lower and upper bounds converge to the shakedown limit, q =ðAσ 0 Þ, at 5 decimal places within 10 iterations. Accordingly, the shakedown limit are P=ðAσ 0 Þ ¼ 2:00000 and P=ðAσ 0 Þ ¼ 2:50000, respectively, for the three-bar truss made of perfectly plastic and nonlinear kinematic hardening materials. On the other hand, the step-by-step finite-element analysis by using ABAQUS [40] is also conducted to calculate the shakedown limit load for comparisons and validation. The shakedown condition is recognized if the plastic dissipation of the structure is bounded e.g. in Ref. [41]. Fig. 3 shows that the accumulated plastic dissipation energy for the truss with perfectly plastic members has been in stable status within 50 loading cycles for the loading case with P=ðAσ 0 Þ ¼ 2:000 but keeps abrupt increasing for a higher cyclic load with P=ðAσ 0 Þ ¼ 2:001 as shown in Fig. 4. Accordingly, it is sure that the truss structure collapses accompanied by unconstrained flow if subjected to the loading case P=ðAσ 0 Þ ¼ 2:001. Thus, the shakedown limit acquired by using ABAQUS [40] is P=ðAσ 0 Þ ¼ 2:000. Very good agreement is obtained to verify the result by shakedown analysis. Fig. 5 shows that the accumulated plastic dissipation energy for the truss with nonlinear kinematic hardening members with P=ðA σ 0 Þ ¼ 2:500 has been in stable status within 50 loading cycles. However, the accumulated plastic dissipation energy keeps moderate increasing as seen in Fig. 6a and then ratcheting is observed as shown in Fig. 6b if we impose the cyclic load with P=ðAσ 0 Þ ¼ 2:501.
ð37Þ
As shown above, we have analytically illustrated the strong duality holding between the primal problem (16) and the dual problem (36).
4. Numerical computations In the paper, we have established the upper bound (dual) formulation from the lower bound (primal) formulation accounting
Fig. 1. A three-bar truss subjected to a cyclic force in the vertical direction.
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Fig. 2. A bridge truss subjected to a cyclic force in the vertical direction. 120
Table 1 Convergence of the lower and upper bounds of shakedown limit corresponding to a three-bar truss with loading range ½0; P by MATLAB.
100
Inter. No.
Perfectly plastic
1 2 3 4 5 6 7 8 9 10
Nonlinear kinematic hardening
Lower bound
Upper bound
Lower bound
Upper bound
0.68107 0.68751 1.21090 1.55236 1.76897 1.96970 1.99997 2.00000 2.00000 2.00000
6.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000
0.79166 1.31127 1.78772 2.26453 2.49729 2.49928 2.50000 2.50000 2.50000 2.50000
6.00000 2.50000 2.50000 2.50000 2.50000 2.50000 2.50000 2.50000 2.50000 2.50000
Plastic Dissipation (kJ)
Material type 80
60 Load Factor=2.001
40
20
0 0
10
20
30
40
50
Loading Cycles
Fig. 4. Evolution of plastic dissipation (perfectly plastic three-bar truss, ½0; P) by ABAQUS.
100
2500
80
2000
60 Load Factor=2.000 40
20
Plastic Dissipation (kJ)
Plastic Dissipation (kJ)
120
1500 Load Factor=2.500 1000
500
0 0
10
20
30
40
50
Loading Cycles
Fig. 3. Evolution of plastic dissipation (perfectly plastic three-bar truss, ½0; P) by ABAQUS.
Thus, the shakedown limit acquired by using ABAQUS in Ref. [40] is P=ðAσ 0 Þ ¼ 2:500. Again, very good agreement is obtained to verify the developed approach in the paper. Second, we consider the three-bar truss subjected to the loading range ½P; 3P. Table 2 shows the iterative results corresponding to the primal problem (16) and the dual problem (36), respectively. As shown in Table 2, both the lower and upper bounds converge to the shakedown limit, q =ðAσ 0 Þ, at 5 decimal
0 0
10
20
30
40
50
Loading Cycles
Fig. 5. Evolution of plastic dissipation (nonlinear kinematic hardening three-bar truss, ½0; P) by ABAQUS.
places within 15 iterations. For the three-bar truss made of perfectly plastic materials, the shakedown limit is P=ðAσ 0 Þ ¼ 0:66667 as shown in Table 2. On the other hand, the shakedown limit is P=ðAσ 0 Þ ¼ 0:91269 for the three-bar truss made of nonlinear kinematic hardening materials.
S.-Y. Leu, J.-S. Li / International Journal of Mechanical Sciences 103 (2015) 172–180
177
120
2500
100
Plastic Dissipation (kJ)
Plastic Dissipation (kJ)
2000
1500 Load Factor=2.501 1000
500
80
60 Load Factor=0.666
40
20
0
0 0
10
20
30
40
0
50
10
20
30
40
50
Loading Cycles
Loading Cycles
Fig. 7. Evolution of plastic dissipation (perfectly plastic three-bar truss, ½P; 3P) by ABAQUS.
180 160
120
140
100
100
Plastic Dissipation (kJ)
Stress (MPa)
120
Load Factor=2.501 80
60 40
80
60 Load Factor=0.667
40
20
20
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Strain %
0
Fig. 6. a. Evolution of plastic dissipation (nonlinear kinematic hardening three-bar truss, ½0; P) by ABAQUS. b. Evolution of stress–strain (nonlinear kinematic hardening three-bar truss, ½0; P) by ABAQUS. Table 2 Convergence of the lower and upper bounds of shakedown limit corresponding to a three-bar truss with loading range ½P; 3P by MATLAB.
Inter. no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Material type Perfectly plastic
Nonlinear kinematic hardening
Lower bound
Upper bound
Lower bound
Upper bound
0.41463 0.50565 0.53841 0.56829 0.59790 0.62750 0.64000 0.66514 0.66665 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667
6.00000 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667 0.66667
0.34855 0.59039 0.68279 0.75276 0.80949 0.83377 0.86714 0.88418 0.90681 0.91035 0.91269 0.91269 0.91269 0.91269 0.91269
4.10710 4.10691 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269 0.91269
In addition, the step-by-step finite-element results are acquired by using ABAQUS [40]. Fig. 7 shows that the accumulated plastic dissipation energy corresponding to the perfectly plastic truss with
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Loading Cycles
Fig. 8. Evolution of plastic dissipation (perfectly plastic three-bar truss, ½P; 3P) by ABAQUS.
P=ðAσ 0 Þ ¼ 0:666 has been in stable status within 50 loading cycles. However, the accumulated plastic dissipation energy keeps abrupt increasing if we impose the cyclic load with P=ðAσ 0 Þ ¼ 0:667. Thus, the shakedown limit acquired by using ABAQUS [40] is P=ðAσ 0 Þ ¼ 0:666. Moreover, it can be recognized that the truss structure will collapse accompanied by unconstrained flow if subjected to the cyclic load with P=ðAσ 0 Þ ¼ 0:667 as shown in Fig. 8. Fig. 9 shows that the accumulated plastic dissipation energy corresponding to the nonlinear kinematic hardening truss with P =ðAσ 0 Þ ¼ 0:912 has been in stable status within 50 loading cycles. However, the accumulated plastic dissipation energy keeps abrupt increasing if we impose the cyclic load with P=ðAσ 0 Þ ¼ 0:913. Thus, the shakedown limit acquired by using ABAQUS [40] is P=ðAσ 0 Þ ¼ 0:912. Moreover, it can be recognized that the truss structure collapses accompanied by unconstrained flow if subjected to the cyclic load with P=ðAσ 0 Þ ¼ 0:913 as shown in Fig. 10. Note that, the ratio of the kinematic hardening shakedown limit 0.91269 to the perfectly plastic shakedown limit 0.66667 is 1.36903. It is approximately equal to the value of the ratio of kinematic hardening plastic limit to perfectly plastic one C=ðγσ 0 Þ þ 1. Obviously, the axial stress σ 11 of each perfectly plastic member is equal to the yield strength σ 0 if the three-bar truss structure collapses. Also, the axial stress σ 11 of each nonlinear
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S.-Y. Leu, J.-S. Li / International Journal of Mechanical Sciences 103 (2015) 172–180
25000
Plastic Dissipation (kJ)
20000
15000 Load Factor=0.912 10000
5000
0 0
10
20
30
40
50
Loading Cycles
Fig. 9. Evolution of plastic dissipation (nonlinear kinematic hardening three-bar truss, ½P; 3P) by ABAQUS. 30000
Plastic Dissipation (kJ)
25000
load larger than its shakedown limit load. Therefore, the ratio of the kinematic hardening shakedown limit to the perfectly plastic one is approximately equal to the ratio of kinematic hardening plastic limit to perfectly plastic one C=ðγσ 0 Þ þ1. Fourth, we consider the bridge truss with the loading range ½P; 3P. Table 4 lists the iterative results corresponding to the primal problem (16) and the dual problem (36), respectively. As shown in Table 4, both the lower and upper bounds also converge to the shakedown limit, q =ðAσ 0 Þ, at 5 decimal places within 15 iterations. For the bridge truss structure made of perfectly plastic members, the shakedown limit is P=ðAσ 0 Þ ¼ 0:43301 as shown in Table 4. On the other hand, the shakedown limit is P=ðAσ 0 Þ ¼ 0:59281 for the bridge truss structure made of nonlinear kinematic hardening members. Once again, the ratio of the kinematic hardening shakedown limit 0.59281 to the perfectly plastic shakedown limit 0.43301 is 1.36904. It is close to the value of the ratio of kinematic hardening plastic limit to perfectly plastic one C=ðγσ 0 Þ þ 1. Similarly, the bridge truss structure concerned will collapse if subjected to cyclic load larger than its shakedown limit load. Thus, the ratio of the kinematic hardening shakedown limit to the perfectly plastic one is approximately equal to the ratio of kinematic hardening plastic limit to perfectly plastic one C=ðγσ 0 Þ þ 1. Table 3 Convergence of the lower and upper bounds of shakedown limit corresponding to a bridge truss with loading range ½0; P by MATLAB.
20000
15000 Load Factor=0.913
Inter. no.
10000
5000
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Loading Cycles
Fig. 10. Evolution of plastic dissipation (nonlinear kinematic hardening three-bar truss, ½P; 3P) by ABAQUS.
kinematic hardening member is equal to the yield strength C=γ þ σ 0 if the three-bar truss structure collapses according to Eqs. (3) and (13). Further, it is illustrated in Figs. 8 and 10 that the three-bar truss structure will collapse accompanied by unconstrained flow if subjected to cyclic load larger than its shakedown limit load. Accordingly, the ratio of the kinematic hardening shakedown limit to the perfectly plastic one is approximately equal to the ratio of kinematic hardening plastic limit to perfectly plastic one C=ðγσ 0 Þ þ1. Third, we consider the bridge truss with the loading range ½0; P. Table 3 lists the iterative results corresponding to the primal problem (16) and the dual problem (36), respectively. As shown in Table 3, both the lower and upper bounds also converge to the shakedown limit, q =ðAσ 0 Þ, at 5 decimal places within 10 iterations. For the bridge truss made of perfectly plastic members, the shakedown limit is P=ðAσ 0 Þ ¼ 1:29904 as shown in Table 3. On the other hand, the shakedown limit is P=ðAσ 0 Þ ¼ 1:77843 for the bridge truss with nonlinear kinematic hardening. Again, the ratio of the kinematic hardening shakedown limit 1.77843 to the perfectly plastic shakedown limit 1.29904 is 1.36903. It is close to the value of the ratio of kinematic hardening plastic limit to perfectly plastic one C=ðγσ 0 Þ þ 1. Obviously, the bridge truss structure concerned will collapse subjected to cyclic
1 2 3 4 5 6 7 8 9 10
Material type Perfectly plastic
Nonlinear kinematic hardening
Lower bound
Upper bound
Lower bound
Upper bound
1.28566 1.29880 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904
10.00000 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904 1.29904
1.17865 1.18035 1.49825 1.69251 1.76062 1.76449 1.77754 1.77842 1.77843 1.77843
2.73807 2.02966 1.77843 1.77843 1.77843 1.77843 1.77843 1.77843 1.77843 1.77843
Table 4 Convergence of the lower and upper bounds of shakedown limit corresponding to a bridge truss with loading range ½P; 3P by MATLAB.
Inter. no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Material type Perfectly plastic
Nonlinear kinematic hardening
Lower bound
Upper bound
Lower bound
Upper bound
0.29966 0.33541 0.36683 0.39784 0.42880 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301
4.00000 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301 0.43301
0.17296 0.25430 0.34458 0.42793 0.45525 0.49373 0.55207 0.58595 0.59244 0.59250 0.59281 0.59281 0.59281 0.59281 0.59281
6.84517 5.15985 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281 0.59281
S.-Y. Leu, J.-S. Li / International Journal of Mechanical Sciences 103 (2015) 172–180
Finally, as shown in Tables 1–4, the shakedown limits are bounded from below and above by the greatest lower and least upper bounds, respectively. The greatest lower and least upper bounds are approached monotonically by the lower and upper bounds corresponding to the primal problem (16) and the dual problem (36), respectively. Certainly, the equality relation between the greatest lower bound and the least upper bound also illustrates numerically the strong duality holding between the primal problem (16) and the dual problem (36). Moreover, the analytical formulation and numerical implementation have been verified by the results of the finite-element analysis based on ABAQUS. By the results of ABAQUS mentioned above, an interesting find has been further obtained that in some case studies the trusses will collapse accompanied by unconstrained flow if subjected to higher cyclic loads than shakedown limit loads. Taking the case study with a three-bar truss structure for example, the axial stress σ 11 of each perfectly plastic member is equal to the yield strength σ 0 if the three-bar truss structure collapses. On the other hand, the axial stress σ 11 of each kinematic hardening member is equal to the yield strength C=γ þ σ 0 if the three-bar truss structure collapses according to Eqs. (3) and (13). Accordingly, the kinematic hardening shakedown factors are C=ðγ σ 0 Þ þ 1 ffi 1:369 times of the perfectly plastic ones for such case studies.
5. Conclusions In the paper, we have conducted shakedown analysis of truss structures made of Armstrong–Frederic nonlinear kinematic hardening materials. Both static and kinematic shakedown analyses were performed to acquire the corresponding shakedown limit by equating the greatest lower bound to the least upper bound. By the Hölder inequality [33], the upper bound (dual) formulation was derived from the lower bound (primal) formulation. In particular, the derived upper bound formulation is an equivalent form of the Koiter’s kinematic shakedown formulation for trusses without involving time integrals. The sharpness condition of the Hölder inequality [33] was also shown. Moreover, the optimization algorithms linprog and fmincon of MATLAB [34] was adopted to perform the primal analysis and the dual analysis of a three-bar truss and a bridge truss, respectively. Especially, the primal analysis and the dual analysis have been validated by converging to the shakedown limit efficiently. Moreover, the analytical formulation and numerical implementation have been also verified by the step-by-step finite-element results provided by ABAQUS. In the paper, we adopt the existing Armstrong–Frederick kinematic hardening model [30,31] in the formulation of static and kinematic shakedown analyses. The case studies illustrated in the paper focus on the loading ranges ½0; P and ½P; 3P such that each truss member is subjected to tensile or compression force. However, the approach developed in the paper is also applicable to other loading ranges which may induce the incremental collapse phenomenon and other kinematic hardening models including the straightforward extension to the modified version of the Armstrong–Frederick kinematic hardening model e.g. [42].
Acknowledgments The authors are deeply grateful to the referees’ comments for leading to substantial enhancement of the paper’s quality. The authors also gratefully acknowledge the financial support by Ministry of Science and Technology, ROC, through the grant MOST 102-2221-E-157-003.
179
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