Shakedown reduced kinematic formulation, separated collapse modes, and numerical implementation

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Shakedown reduced kinematic formulation, separated collapse modes,

and numerical implementation

T.D. Tran

a

_{, C.V. Le}

b

_{, D.C. Pham}

c,⇑

_{, H. Nguyen-Xuan}

d,e a

Faculty of Construction & Electricity, Ho Chi Minh City Open University, 97 Vo Van Tan Street, Ho Chi Minh City, Viet Nam b

Department of Civil Engineering, International University – VNU HCMC, Viet Nam c

Institute of Mechanics – VAST, 264 Doi Can, Hanoi, Viet Nam d

Duy Tan University, Da Nang, Viet Nam

e_{Department of Computational Engineering, Vietnamese-German University, Binh Duong New City, Viet Nam}

a r t i c l e

i n f o

Article history:

Received 16 December 2013 Received in revised form 14 April 2014 Available online 28 April 2014 Keywords:

Plastic collapse Shakedown

Elastic–plastic material Cyclic loading

Reduced kinematic formulation Incremental plasticity Alternating plasticity

Second-order cone programming

a b s t r a c t

Our shakedown reduced kinematic formulation is developed to solve some typical plane stress problems, using ﬁnite element method. Whenever the comparisons are available, our results agree with the available ones in the literature. The advantage of our approach is its simplicity, computational effective-ness, and the separation of collapse modes for possible different treatments. Second-order cone program-ming developed for kinematic plastic limit analysis is effectively implemented to study the incremental plasticity collapse mode. The approach is ready to be used to solve general shakedown problems, includ-ing those for elastic–plastic kinematic hardeninclud-ing materials and under dynamic loadinclud-ing.

1. Introduction

Shakedown analysis determines the load limits on the loading cycles, under which a structure should be safe (i.e. it would behave elastically after some possible initial limited plastic dissipation), while over the limits the structure would collapse incrementally or fail by alternating plasticity (Melan, 1938; Koiter, 1963; Gokhfeld, 1966; Debordes, 1977; König, 1987; Bree, 1989; Pham and Stumpf, 1994; Pham, 2000b, 2003b, 2008, 2013). Melan’s static and Koiter’s kinematic shakedown theorems are generalizations of the respective plastic limit ones (Gvozdev, 1960; Drucker et al., 1951; Hill, 1951), from the instantaneous collapse considerations to those over loading processes. Original shakedown theorems are restricted to quasi-static loading processes, however latter have been extended for the larger class of dynamic problems (Gavarini, 1969; Ho, 1972; Corradi and Maier, 1974; Ceradini, 1980; Pham, 1992, 2000a, 2003a, 2010; Corigliano et al., 1995). Preliminarily restricted to elastic-perfectly plastic bodies, the shakedown theorems have been developed for more general kinematic hardening materials (Maier, 1972; Ponter, 1975; König,

1987; Weichert and Gross-Weege, 1988; Polizzotto et al., 1991; Stein and Huang, 1994; Fuschi, 1999; Pham and Weichert, 2001; Weichert and Maier, 2002; Bousshine et al., 2003; Pham, 2007, 2008, 2013; Simon, 2013; . . .).

Implementing the shakedown theorems in applications, one faces the difﬁculty of solving nonlinear optimization problems for structures with complex geometries and under complicated loading programs (Belytschko, 1972; Corradi and Zavelani, 1974; Zouain et al., 2002; Magoariec et al., 2004; Vu et al., 2004a; Liu et al., 2005; Garcea et al., 2005; Chen et al., 2008; Tran et al., 2010). Various iterative optimization algorithms have been devel-oped to provide solution of such the non-linear programming (Zouain et al., 2002; Vu et al., 2004b; Garcea et al., 2005; Li and

Yu, 2006). However, these methods can tackle problems with a

moderate number of variables, and hence it is still desirable to develop an alternative solution procedure that can solve large-scale shakedown analysis problems in engineering practices. Shakedown analysis is a generation of limit analysis, and hence optimization algorithms initially developed for the latter can usu-ally be extended to former in a relatively straightforward manner. In the context of limit analysis, a primal–dual interior-point method proposed inAndersen et al. (2001, 2003)) has been proved to be one of the most robust and efﬁcient algorithms in treating

http://dx.doi.org/10.1016/j.ijsolstr.2014.04.016

⇑ Corresponding author. Tel.: +84 4 37 628 125; fax: +84 437622039. E-mail address:pdchinh@imech.ac.vn(D.C. Pham).

Contents lists available atScienceDirect

International Journal of Solids and Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j s o l s t r

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such the large-scale non-linear optimization problem (Ciria et al.,

2008; Munoz et al., 2009; Le et al., 2009, 2010, 2012). The

algorithm has been extended to both static and kinematic shake-down analysis problems (Le et al., 2010; Bisbos et al., 2005; Makrodimopoulos, 2006; Weichert and Simon, 2012).

From the original Koiter’s shakedown kinematic theorem, reduced shakedown kinematic formulations have been deduced with separated collapse modes and applied to various simple typ-ical structures (of plastic limit analysis), yielding certain interest-ing semi-analytical results (Pham, 1992, 2000a,b, 2003a,b, 2008, 2010, 2013; Pham and Stumpf, 1994;. . .). The purpose of this research is to develop this approach with numerical ﬁnite element implementation for more complex engineering structures. The pri-mal–dual interior-point algorithm will be developed with the reduced shakedown kinematic formulation, making the use of the optimization method in direct manner. It is worth noting that with the use of the reduced kinematic formulation the number of kinematic variables in the resulting optimization is kept to a minimum. This is because there is no need to approximate displacement ﬁelds at every vertex of a polyhedral load domain as required in the original kinematic formulation. Furthermore, the reduced kinematic formulation is able to produce indepen-dently the incremental and alternative modes, leading to possible different treatments of the modes in analysis of structures.

The layout of the paper is as follows. The next section recalls kinematic formulations including both uniﬁed and reduced kine-matic shakedown ones. In Section3, the kinematic formulation is discretized using ﬁnite element method, and an optimization strat-egy based on second-order cone programming is described. Numerical examples are provided in Section4to illustrate the per-formance of the proposed solution strategy, and non-shakedown modes for various loading domains are also shown. The conclusion completes the paper.

2. Shakedown kinematic formulations

Let

r

e_{ðx; tÞ denote the ﬁctitious elastic stress response of the body}

V to external agencies over a period of time ðx 2 V; t 2 ½0; TÞ under the assumption of perfectly elastic behavior, called a loading process (history). The actions of all kinds of external agencies upon V can be expressed explicitly through

r

e_{. At every point x 2 V, the elastic}

stress response

r

e_{ðx; tÞ is conﬁned to a bounded time-independent}

domain with prescribed limits in the stress space, called a local load-ing domain Lx. As a ﬁeld over V;

r

eðx; tÞ belongs to the

time-indepen-dent global loading domain L:

L ¼ f

r

e_j

_r

e_{ðx; tÞ 2 L}

x; x 2 V; t 2 ½0; Tg: ð1Þ

In the spirit of classical shakedown analysis, the bounded load-ing domain L, instead of a particular loadload-ing history

r

e_{ðx; tÞ, is}

given a priori. Shakedown of a body in L means it shakes down for all possible loading histories

r

e_{ðx; tÞ 2 L .}

A signiﬁes the set of compatible-end-cycle (deviatoric) plastic strain rate ﬁelds ep_{over time cycles 0 6 t 6 T:}

A ¼ ep_j

_e

p_¼Z T 0 ep_{dt 2 C} ; ð2Þ

where C is the set of strain ﬁelds that are both deviatoric and com-patible on V. Let ks be the shakedown safety factor: at ks>1 the

structure will shake down, while it will not at ks<1, and ks¼ 1

deﬁnes the boundary of the shakedown domain. Koiter’s shake-down kinematic theorem can be stated as (Koiter, 1963; Pham, 2003a): k1 s ¼ sup ep_2A;_re_2L RT 0dt R V

r

e_:_ep_dV RT 0dt R VDðepÞdV ; ð3Þ

where Dðep_{Þ ¼}

_r

_:_ep _{is the dissipation function determined by the}

yield stress

r

Y and the respective yield criterion; e.g. for a Mises

material we have

Dðep_{Þ ¼}pffiffiffiffiffiffiffiffi₂₌₃

_r

Yðep:epÞ

1=2

: ð4Þ

Alternatively,(3)can also be presented as

ui;jþ

D

uj;iÞ on V;

D

ui¼R T 0 _uidt on V;

D

ui¼ 0 on @Vu; 8 > > > > > < > > > > > : ð6Þ

where kþ_s is the upper bound on the actual shakedown load multi-plier,D

e

p

ijis the admissible cycle of plastic strain ﬁelds

correspond-ing to a cycle of displacement ﬁelds Dui, and the constrained

boundary @Vuis ﬁxed. Note that at each instant during the time

cycle t, the plastic strain rates ep_ijmay be not compatible, but the plastic strain accumulated over the cycleD

e

ijmust be compatible.

In order to perform numerical shakedown analysis of struc-tures, the time integration in problem (6) must be removed because the evaluation of plastic strains over a loading cycle would be difﬁcult. Based on the two convex-cycle theorems presented in König (1987), the problem(6)can be expressed as

kþs ¼ min XM k¼1 Z V Dðep kÞdV s:t PM k¼1 R V

r

e k:e p kdV ¼ 1

D

e

p_¼X M k¼1 ep k on V;

D

u ¼ 0 on @Vu; 8 > > > > < > > > > : ð7Þ

where M ¼ 2N _{is the number of vertices of the convex polyhedral}

load domain L; N is the number of variable loads.

The so-called uniﬁed optimization problem(7)can provide a shakedown load multiplier that is the smaller one of incremental plasticity limit (ratchetting or progressive deformation limit) and alternating plasticity limit (low-cycle fatigue or plastic shakedown limit), and it has been solved numerically using various discretiza-tion method and optimizadiscretiza-tion algorithms.

From Koiter’s shakedown kinematic theorem(3),Pham (1992)

and Pham and Stumpf (1994)have deduced the much simpler

reduced shakedown kinematic formulation

k1s Pk 1

sr ¼ max fI; Ag; ð8Þ

ks6ksr¼ min fI; Ag; ð11Þ

e

p_{ðxÞ compatible}

over V, is global; while the latter one, which can be checked at every point x 2 V separately, is local. The alternating plasticity collapse criterion can be solved explicitly for a particular criterion and par-ticular loading conditions. The incremental plasticity collapse crite-rion appears similar to the respective plastic limit one (Pham, 2003b), except for the appearance of the under-integral maximum operation in the expressions of Eqs.(9)or(12), which makes the incremental collapse more dangerous. Still, the similarity indicates that the available kinematic methods of plastic limit analysis could be developed to be used there, as shall be done in the remaining part of this work. Pham and Stumpf (1994) have proved that ks¼ ksr for a large class of practical problems. We suggest that ksr

may equal to ks generally but can not yet prove it. We have not

found any particular case where ks<ksr. Thus, our simple reduced

kinematic formulation could provide, if not the exact values, very good upper bound estimates for the shakedown safety factor and the shakedown load limits.

3. Solution of discrete kinematic formulations

In the kinematic formulation, plastic strains must be approxi-mated using discretization methods, i.e. ﬁnite element method (FEM). The problem domain is discretized into elements such as V X1[X2[ . . . [Xnel and Xi\Xj¼ ø; i – j, where nel is the number of elements. Then, the ﬁnite element approximation, elas-tic strains and stresses can be expressed as

uh_{¼ Nd;}

_e

h_{¼ Bd;}

_r

e_{¼ DBd;} _ð14Þ

where N are the shape functions, B is the strain matrix, D is the constitutive matrix and d are corresponding nodal displacement values, for elastic problems d will be obtained using the well-known Galerkin procedure.

The incremental collapse mode can be determined by the normalized form of the problem(12)given as

I ¼ inf re_2L;_ep_2C Z V Dð

e

p_ÞdV _ð15Þ s:t Z V max 06t6T

r

e_{ðx; tÞ :}

_e

p_{ðxÞdV ¼ 1}

Note that the expression of problem(15)is closely similar to that of the respective plastic limit kinematic theorem, the only difference is that the under-integral maximum operation over time parameter t is taken at every point x 2 V. Hence, available optimization algo-rithms of limit analysis can be used to solve problem(15). However, one should pay attention on searching maximum external power at every point in the problem domain.

By means of ﬁnite element discretization and Gaussian integra-tion technique, the problem(15)can be rewritten as

I ¼ infX NG i¼1

r

Yni ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðBidiÞ T HBidi q ð16Þ s: t XNG i¼1 nimaxk¼1;...;Mf

r

eikBidig ¼ 1 di¼ 0 on @Vu; 8 > < > :

where niis the integral weight at the i Gaussian integral point and

NG is the total number of integration points over V. It should be noted that difﬁculty arises when solving the above optimization problem is to identify the maximum external power at every point over loading domain L in the ﬁrst constraint while diis unknown.

In the following we will describe a solution strategy based on a second-order cone programming, ensuring that the optimization CPU time is kept to a minimum.

Instead of solving directly (16), following the upper bound approach of (Pham, 1992, 2000a; Pham and Stumpf, 1994;. . .), we shall take the trial incremental collapse ﬁelds dikas the plastic

limit solutions of the problems (k ¼ 1; . . . ; M)

v

j ¼ ð

v

T

_v

_Þ1=2

and

q

ðdiÞ ¼

q

1

q

2

q

3 2 6 4 3 7 5 ¼ CTBidik ð20Þ

in which C is the so-called Cholesky factor of H. The problem(19)is a standard second order cone programming form involving equality, inequality and quadratic cone constraints, which can be solved by a highly efficient primal–dual interior point algorithm (Andersen et al., 2001, 2003). It is also worth noting that the number of degrees of freedom used in the problem(19)is very much smaller (M times) than that employed in the unified optimization problem (7)for each finite element mesh.

4. Numerical examples

This section will investigate the performance of the proposed approach for kinematic shakedown analysis via a number of benchmark plane stress problems in some of which other numeri-cal solutions are available for comparison. Mosek optimization sol-ver sol-version 6.0 was used to obtain solutions (using a 2.8 GHz Intel Core i5 PC running Window 7).

eik 22

r

eij 22 þ 3½

r

eik 12

r

eij 12 2o1=2 : ð22Þ

It can be seen from the problem(22)that only elastic stresses are involved in the computation of the alternating plasticity limit. Therefore, the accuracy of the solution relies on the use of discret-ization methods or adaptive mesh refinement. However, for the sake of simplicity the finite element method with uniform mesh refinement will be used in this paper.

4.1. Square plate with a circular hole

The ﬁrst example deals with a square plate with central circular hole, which is subjected to quasi-static biaxial uniform loads (Fig. 1)

0 6 p0

16p1; 0 6 p026p2: ð23Þ

Due to symmetry, only the upper-right quarter of the plate is mod-eled, and symmetry conditions are enforced on the left and bottom edges. The input data was assumed as follows: E ¼ 2:1 105MPa,

m

¼ 0:3; L ¼ 10 m, D=L ¼ 0:2, and

r

Y¼ 200 MPa. The limit and

shakedown load factors of the problem were obtained numerically by many authors through different numerical of the methods (Belytschko, 1972; Gross-Weege, 1997; Zhang et al., 2002; Zhang and Raad, 2002; Zouain et al., 2002; Magoariec et al., 2004; Vu et al., 2004a; Garcea et al., 2005;. . .). Three loading cases of (23) have been considered in the mentioned references

ðaÞ : p1¼ p2¼ p0; ðbÞ : 1

2p1¼ p2¼ p0; ðcÞ : p1¼ p0; p2¼ 0: ð24Þ

The solutions of Eqs. (11), (18) and (22) yield the estimates ks¼ 0:44 (case a), ks¼ 0:51 (case b), ks¼ 0:60 (case c) – in

agree-ment with the results of the agree-mentioned references. However our results indicate additionally that in all the cases the collapse mode is the simple alternating plasticity one (in all the cases I ¼ 0:81, which coincides with kpof plastic limit). Graphics of the

incremen-tal plasticity collapse curve I ¼ 1, alternating plasticity collapse curve A ¼ 1, proportional plastic limit curve kprop ¼ 1 (in

presump-tion that p0

1; p02increases proportionally up to p1; p2), plastic limit

curve kp¼ 1 (for all possible loads in the range(23)), for all the

range (23) are plotted in the plane of load coordinates

p₁=

r

Y;p2=

r

Y ofFig. 2. On a curve, the plate fails by the respective

collapse mode, while under the curve it is safe with respect to the mode. Nonshakedown limit curve ks¼ 1 is the lower envelope of

the incremental collapse I ¼ 1 and alternating plasticity collapse

A ¼ 1 curves. It is clear that, not only in the cases(24), but for all the range(23)the nonshakedown collapse mode is the alternating plasticity one ks¼ A ¼ 1, while the incremental collapse and plastic

limit curves coincide I ¼ kp¼ 1. The nonshakedown curve

ks¼ A ¼ 1 also agrees with that of direct application of Koiter’s

theorem(7), using FEM-DUAL method ofVu et al. (2004b). Now instead of(23), we consider the loads’ ranges

0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Fig. 1. The upper-right quarter of the square plate with a circular hole subjected to quasi-static biaxial uniform loads, and a ﬁnite element mesh.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p₁/σY p2 / σY

Proportional plastic Limit Plastic Limit Incremental plasticity Alternating plasticity Koiter FEM−DUAL

Fig. 2. The incremental plasticity collapse curve I ¼ 1 (coincides with the plastic limit curve), alternating plasticity collapse curve A ¼ 1, proportional plastic limit curve, and the nonshakedown curve using FEM-DUAL method, for the square plate with a circular hole subjected to biaxial uniform loads 0 6 p0

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0:4p16p016p1; 0:4p26p026p2: ð25Þ

Graphics of the collapse curves are presented in Fig. 3, which appears much more interesting than theFig. 2. One sees that, as p1=

r

Y increases from 0.4, the nonshakedown mode switches from

the incremental plasticity collapse mode to the alternating plastic-ity one and vice versa, at certain values of the external load limits; the incremental plasticity collapse curve I ¼ 1 does not coincide with the plastic limit ones. The lower envelope of the incremental plasticity collapse I ¼ 1 and alternating plasticity collapse A ¼ 1 curves agrees with the result ks¼ 1 of direct application of Koiter’s

theorem(7), using FEM-DUAL method ofVu et al. (2004b). 4.2. Grooved rectangular plate

A grooved rectangular plate subjected to in-plane tension pN

and bending pM, as shown inFig. 4, was also considered. The load

domain is given by

0 6 p0

N6pN; 0 6 p0M6pM: ð26Þ

The problem was investigated by Prager and Hodge (1951), Vu (2001), Tran (2008) and Nguyen-Xuan et al. (2012). With particular values E ¼ 2:1 105_MPa,

_m

¼ 0:3; R ¼ 250 mm, L ¼ 4R, and

r

Y¼

116:2 MPa, pN¼ pM¼

r

Y, our calculations yield ks¼ 0:23, which

agrees with those obtained inVu (2001), Tran (2008) and Nguyen-Xuan et al. (2012), however again, the nonshakedown mode is the simple alternating plasticity one (I ¼ 0:28 in the case).

Now instead of(26)we consider the loads’ ranges

0:035pN6p0N6pN; 0:035pM6p0M6pM: ð27Þ

Graphics of the collapse curves are projected inFig. 5. One observes that, as pM=

r

Y increases from 0.035, the nonshakedown mode

switches from the alternating plasticity collapse mode to the mental plasticity one, at about the middle of the range; the incre-mental plasticity collapse curve I ¼ 1 lies strictly below the plastic limit curve, the latter coincides with the proportional plastic limit one. The lower envelope of the incremental plasticity collapse I ¼ 1 and alternating plasticity collapse A ¼ 1 curves also agrees

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 p₁/σY p2 / σY

Proportional plastic Limit Plastic Limit Incremental plasticity Alternating plasticity Koiter FEM−DUAL

Fig. 3. The incremental plasticity collapse curve I ¼ 1, alternating plasticity collapse curve A ¼ 1, proportional plastic limit curve, plastic limit curve, and the nonshakedown curve using FEM-DUAL method, for the square plate with a circular hole subjected to biaxial uniform loads 0:4p16p016p1;0:4p26p026p2.

0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4

Fig. 4. A grooved rectangular plate subjected to varying tension and bending, and a ﬁnite element mesh.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 p_M/σ_Y pN / σY

Proportional plastic Limit Plastic Limit Incremental plasticity Alternatiimg plasticity Koiter FEM−DUAL

Fig. 5. The incremental plasticity collapse curve I ¼ 1, alternating plasticity collapse curve A ¼ 1, proportional plastic limit curve (coincides with the plastic limit one), and the nonshakedown curve using FEM-DUAL method, for the grooved rectangular plate subjected to varying tension and bending.

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with the result ks¼ 1 of direct application of Koiter’s theorem(7),

using FEM-DUAL method ofVu et al. (2004b).

4.3. Square plate with a circular hole under dynamic loads

We come back to the square plate with a circular hole ofFig. 1, which is subjected to dynamic uniform load

p1¼ p0ð1 þ 0:1 sin

x

tÞ; p2¼ 0; ð28Þ

where p0increases quasi-statically from 0 to the nonshakedown

limit. The alternating plasticity value A from(21)is found directly from dynamic stress solution

r

e_{ðx; tÞ ¼}

_r

0_{ðxÞ sin}

_x

_{t of the}

respec-tive dynamic elastic problem, and we get the curve A ¼ 1 as shown in the plane of load’s amplitude p0=

r

Y and dimensionless load’s

t; ð29Þ

the particular results of which are projected inFig. 7.

Separation of collapse modes is not only to make the shakedown analysis simpler. We have known that the incremental

plasticity collapse mode is global, while the alternating plasticity collapse mode is local and may or may not lead to global collapse of structures (Pham, 2000b).

Shakedown theorems have been extended to elastic plastic kinematic hardening materials satisfying positive hysteresis postu-late and restricted by initial and ultimate yield stresses (Pham, 2007, 2008, 2013). Our reduced formulation applied for them has almost the same forms: one only has to substitute the yield stress

r

Yin the alternating plasticity mode(21)or(22)by the initial yield

stress

r

i

Y, and the yield stress

r

Yin the incremental plasticity mode (16)–(18)by the ultimate yield stress

r

u

Y. The initial

r

iY and

ulti-mate

r

u

Yyield stresses are not ﬁxed, but should be taken according

to the particular loading processes considered; e.g. for high-cycle loading

r

i

Ymay be as low as the fatigue limit.

5. Conclusion

A solution strategy for a kinematic shakedown analysis formu-lation based on ﬁnite element method has been described. The underlying optimization problem is cast in the form of a standard second-order cone programming, so that it can be solved using highly efﬁcient primal–dual interior point algorithm. It is has been shown in numerical examples that the incremental collapse mode always leads to the loss of load bearing capacity of a structure, the local alternating plasticity collapse one sometimes may not. Hence, the separation of collapse modes should have not only the practical value for an easy application, but also for possible different treatments of the modes in analysis of structures.

Though the simple elastic perfectly plastic model has been used in this paper, the method can also be applied to more general elas-tic plaselas-tic kinemaelas-tic hardening materials satisfying the positive hysteresis postulate and being bounded by initial and ultimate yield stresses. The only additional step to be taken is to substitute the yield stress

r

Yin the expressions of the incremental and

alter-nating plasticity collapse modes by the ultimate (

r

u

Y) and initial

(

r

i

Y) yield stresses, respectively.

Finally, although only plane stress problems are considered here, the numerical procedure can be extended to tackle more complex structural conﬁgurations. It would also be interesting to combine the proposed method with an adaptive reﬁnement based on dissipation error indicator to speed-up the computational process. 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4

ϖ

p0 / σY Incremental plasticity−Static Incremental plasticity−Dynamic Alternating plasticity

Fig. 6. The incremental plasticity-static curve, incremental plasticity-dynamic curve, alternating plasticity collapse curve, for the square plate with a circular hole subjected to uniform dynamic load p1¼ p0ð1 þ 0:1 sinxtÞ,-¼xDLh

ffiffiffi q E q . 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4

ϖ

p0 / σY Incremental plasticity−Static Incremental plasticity−Dynamic Alternating plasticity

Fig. 7. The incremental plasticity-static curve, incremental plasticity-dynamic curve, alternating plasticity collapse curve, for the square plate with a circular hole subjected to uniform dynamic loads p1¼ p0;p2¼ 0:1 sinxt.

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Acknowledgment

This research has been supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant reference 107.02-2013.11.

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